Rothe H.J.; Lattice Gauge Theories an Introduction

World Scientific Lecture Notes in Physics -Vol . 74

LATTICE GAUGE THEORIESAn Introduction

Third Edition

World Scientific Lecture Notes in Physics

Published

Vol. 53: Introduction to Nonlinear Dynamics for PhysicistsH DI Abarbanel, et al.

Vol. 54: Introduction to the Theory of Spin Glasses and Neural NetworksV Dotsenko

Vol. 55: Lectures in Particle PhysicsD Green

Vol. 56: Chaos and Gauge Field TheoryT S Biro, et al.

Vol. 57: Foundations of Quantum Chromodynamics (2nd ed.): An Introductionto Perturbative Methods in Gauge Theories7 Mute

Vol. 58: Concepts in Solids, Lectures on the Theory of SolidsP. W. Anderson and H. Bunke

Vol. 59: Lattice Gauge Theories: An Introduction (2nd ed.)H J Rothe

Vol. 60: Massive Neutrinos in Physics and Astrophysics (2nd ed.)R N Mohapatra and P B Pal

Vol. 61: Modern Differential Geometry for Physicists (2nd ed.)C J Isham

Vol. 62: ITEP Lectures on Particle Physics and Field Theory (In 2 Volumes)MA Shifman

Vol. 64: Fluctuations and Localization in Mesoscopic Electron SystemsMJanssen

Vol. 65: Universal Fluctuations: The Phenomenology of Hadronic MatterR Botet and M Ploszajczak

Vol. 66: Microcanonical Thermodynamics: Phase Transitions in "Small" SystemsDHE Gross

Vol. 67: Quantum Scaling in Many-Body SystemsM A Continentino

Vol. 69: Deparametrization and Path Integral Quantization of Cosmological ModelsC Simeone

Vol. 70: Noise Sustained Patterns: Fluctuations & NonlinearitiesMarkus Loecher

Vol. 71: The QCD Vacuum, Hadrons and Superdense Matter (2nd Edition)Edward VShuryak

Vol. 72: Massive Neutrinos in Physics and Astrophysics (3rd ed.)R. Mohapatra and P. B. Pal

Vol. 73: The Elementary Process of BremsstrahlungW. Nakel and E. Haug

World Scientific Lecture Notes in Physics - Vol. 74

LATTICE GAUGE THEORIESAn Introduction

Third Edition

Heinz J. RotheUniversitdt Heidelberg, Germany

Y£> World ScientificNEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Cover: 'CUBIC SPACE DIVISION (1952)1 by M. C. Escher© 1952 M. C. Escher/Cordon Art - Baam - Holland

World Scientific Lecture Notes in Physics — Vol. 74LATTICE GAUGE THEORIES (Third Edition)An Introduction

Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright ClearanceCenter, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopyis not required from the publisher.

ISBN 981-256-062-9ISBN 981-256-168-4 (pbk)

Printed in Singapore by World Scientific Printers (S) Pte Ltd

To the memory of my father

PREFACE TO THE THIRD EDITION

Apart from minor modifications, this new edition includes a number of topics,some of which are of great current interest. These concern in particular a discussionin chapter 17 of instantons and calorons, and of the role played by vortices forthe confinement problem. Furthermore we have included in chapter 4 a sectionon Ginsparg-Wilson fermions. In chapter 10 we have added a section on theperturbative verification of the energy sum rule obtained in section 10.3. Somedetails of the calculations have been delegated to an appendix. New sectionshave also been added in chapters 14 and 15. In chapter 14 we come back tothe Ginsparg-Wilson discretization of the action and discuss the ABJ anomalywithin this framework. In the same chapter we also have included a detailedanalysis of the renormalization of the axial vector current in one-loop order, sinceit provides an instructive example of how lattice regulated Ward identities can beused to determine the renormalization constants for currents. In chapter 15 wehave included a very general treatment of the ABJ anomaly in QCD and showthat in the continuum limit one recovers the well known result, irrespective of theprecise way in which the action has been discretized.

Following our general principle which we have always tried to implement, wehave done our best to convey the main ideas in a transparent way as possible, andhave presented most of the non-trivial calculations in sufficient detail, so that thereader can verify them without too much effort. As always we have only includedresults of numerical calculations of pioneering work, be it in the early days of thelattice formulation of gauge field theories, or in more recent days.

Finally, we want to thank W. Wetzel and I. 0. Stamatescu for a numberof very fruitful discussions and constructive comments, and in particular Prof.Stamatescu for providing me with some unpublished plots relevant to instantonson the lattice.

Note from the author

Corrections to this book which come to the author's attention, will be posted onthe World-Wide Web at

http://www.thphys.uni-heidelberg.de/~rotheJi/LGT.htmlWe would be grateful if the reader would inform us about any errors he may find.The e-mail address is: [email protected]

vii

CONTENTS

Preface to the Third Edition viiPreface to the Second Edition ixPreface to the First Edition xi

1. INTRODUCTION 1

2. THE PATH INTEGRAL APPROACH TO QUANTIZATION 72.1 The Path Integral Method in Quantum Mechanics 82.2 Path Integral Representation of Bosonic Green Functions in

Field Theory 152.3 The Transfer Matrix 222.4 Path Integral Representation of Fermionic Green Functions . . 232.5 Discretizing Space-Time. The Lattice as a Regulator of a

Quantum Field Theory 33

3. T H E F R E E S C A L A R FIELD O N T H E LATTICE 36

4. F E R M I O N S O N T H E LATTICE 434.1 The Doubling Problem 434.2 A Closer Look at Fermion Doubling 484.3 Wilson Fermions 564.4 Staggered Fermions 574.5 Technical Details of the Staggered Fermion Formulation . . . . 614.6 Staggered Fermions in Momentum Space 694.7 Ginsparg-Wilson Fermions 73

5. ABELIAN GAUGE FIELDS ON THE LATTICE ANDCOMPACT QED 775.1 Preliminaries 775.2 Lattice Formulation of QED 80

6. NON-ABELIAN GAUGE FIELDS ON THE LATTICECOMPACT QCD 87

7. THE WILSON LOOP AND THE STATICQUARK-ANTIQUARK POTENTIAL 957.1 A Look at Non-Relativistic Quantum Mechanics 96

xiii

xiv Lattice Gauge Theories

7.2 The Wilson Loop and the Static gg-Potential in QED 977.3 The Wilson Loop in QCD 105

8. THE QQ POTENTIAL IN SOME SIMPLE MODELS 1098.1 The Potential in Quenched QED 1098.2 The Potential in Quenched Compact QEDi 114

9. THE CONTINUUM LIMIT OF LATTICE QCD 1199.1 Critical Behaviour of Lattice QCD and the Continuum Limit . . 1199.2 Dependece of the Coupling Constant on the Lattice Spacing

and the Renormalization Group /^-Function 122

10. LATTICE SUM RULES 13010.1 Energy Sum Rule for the Harmonic Oscillator 13010.2 The SU(N) Gauge Action on an Anisotropic Lattice 13610.3 Sum Rules for the Static gg-Potential 13810.4 Determination of the Electric, Magnetic and Anomalous

Contribution to the gg-Potential 14610.5 Sum Rules for the Glueball Mass 148

11. THE STRONG COUPLING EXPANSION 15111.1 The gg-Potential to Leading Order in Strong Coupling 15111.2 Beyond the Leading Approximation 15411.3 The Lattice Hamiltonian in the Strong Coupling Limit and

the String Picture of Confinement 158

12. THE HOPPING PARAMETER EXPANSION 17012.1 Path Integral Representation of Correlation Functions

in Terms of Bosonic Variables 17112.2 Hopping Parameter Expansion of the Fermion Propagator

in an External Field 17412.3 Hopping Parameter Expansion of the Effective Action 17912.4 The HPE and the Pauli Exclusion Principle 183

13. WEAK COUPLING EXPANSION (I). THE $3-THEORY 19213.1 Introduction 19213.2 Weak Coupling Expansion of Correlation Functions in the

^-Theory 19513.3 The Power Counting Theorem of Reisz 201

Contents xv

14. WEAK COUPLING EXPANSION (II). LATTICE QED 20914.1 The Gauge Fixed Lattice Action 20914.2 Lattice Feynman Rules 21614.3 Renormalization of the Axial Vector Current in

One-Loop Order 22214.4 The ABJ Anomaly 234

15. WEAK COUPLING EXPANSION (III). LATTICE QCD 24215.1 The Link Integration Measure 24315.2 Gauge Fixing and the Faddeev-Popov Determinant 24715.3 The Gauge Field Action 25215.4 Propagators and Vertices 25715.5 Relation Between A^ and the A-Parameter of

Continuum QCD 27215.6 Universality of the Axial Anomaly in Lattice QCD 275

16. M O N T E C A R L O M E T H O D S 284

16.1 Introduction 284

16.2 Construction Principles for Algorithms. Markov chains . . . . 286

16.3 The Metropolis Method 291

16.4 The Langevin Algorithm 293

16.5 The Molecular Dynamics Method 295

16.6 The Hybrid Algorithm 301

16.7 The Hybrid Monte Carlo Algorithm 304

16.8 The Pseudofermion Method 307

16.9 Application of the Hybrid Monte Carlo Algorithm to

Systems with Fermions 313

17. S O M E RESULTS OF M O N T E CARLO C A L C U L A T I O N S 317

17.1 The String Tension and the ^-Potential in the SU(3)Gauge Theory 317

17.2 The gg-Potential in Full QCD 32417.3 Chiral Symmetry Breaking 32617.4 Glueballs 33017.5 Hadron Mass Spectrum 33617.6 Instantons 34517.7 Flux Tubes in the qq and g^-Systems 35917.8 The Dual Superconductor Picture of Confinement 36317.9 Center Vortices and Confinement 373

xvi Lattice Gauge Theories

18. P A T H - I N T E G R A L R E P R E S E N T A T I O N OF T H E

T H E R M O D Y N A M I C A L P A R T I T I O N F U N C T I O N F O R

S O M E SOLVABLE B O S O N I C A N D F E R M I O N I C S Y S T E M S 383

18.1 Introduction 383

18.2 Path-Integral Representation of the Partition Function in

Quantum Mechanics 384

18.3 Sum Rule for the Mean Energy 386

18.4 Test of the Energy Sum Rule. The Harmonic Oscillator . . . . 389

18.5 The Free Relativistic Boson Gas in the Path

Integral Appoach 394

18.6 The Photon Gas in the Path Integral Approach 398

18.7 Functional Methods for Fermions. Basics 401

18.8 Path Integral Representation of the Partition Function

for a Fermionic System valid for Arbitrary Time-Step . . . . 405

18.9 A Modified Fermion Action Leading to Fermion Doubling . . . 410

18.10 The Free Dirac Gas. Continuum Approach 413

18.11 Dirac Gas of Wilson Fermions on the Lattice 417

19. FINITE T E M P E R A T U R E P E R T U R B A T I O N T H E O R Y

OFF AND ON THE LATTICE 42419.1 Feynman Rules For Thermal Green Functions in the

A^-Theory 42419.2 Generation of a Dynamical Mass at T ^ 0 43319.3 Perturbative Expansion of the Thermodynamical Potential . . 43419.4 Feynman Rules for QED and QCD at non-vanishing

Temperature and Chemical Potential in the Continuum . . . . 44019.5 Temporal Structure of the Fermion Propagator at T ^ 0 and

H ^ 0 in the Continuum 44519.6 The Electric Screening Mass in Continuum QED in

One-Loop Order 44819.7 The Electric Screening Mass in Continuum QCD in

One-Loop Order 45219.8 Lattice Feynman Rules for QED and QCD at T ^ 0

and fi / 0 45519.9 Particle-Antiparticle Spectrum of the Fermion Propagator at

T ^ O and /i ± 0. Naive vs. Wilson Fermions 460

Contents xvii

19.10 The Electric Screening Mass for Wilson Fermions in LatticeQED to One-Loop Order 464

19.11 The Electric Screening Mass for Wilson Fermions in LatticeQCD to One-Loop Order 472

19.12 The Infrared Problem 482

20. NON-PERTURBATIVE QCD AT FINITETEMPERATURE 48520.1 Thermodynamics on the Lattice 48520.2 The Wilson Line or Polyakov Loop 49020.3 Spontaneous Breakdown of the Center Symmetry and the

Deconfinement Phase Transition 49520.4 How to Determine the Transition Temperature 49620.5 A Two-Dimensional Model. Test of Theoretical Concepts . . . 49820.6 Monte Carlo Study of the Deconfinement Phase Transition

in the Pure SU(3) Gauge Theory 51220.7 The Chiral Phase Transition 52020.8 Some Monte Carlo Results on the High Temperature

Phase of QCD 52420.9 Some Possible Signatures for Plasma Formation 532

Appendix A 542Appendix B 552Appendix C 554Appendix D 557Appendix E 560Appendix F 562Appendix G 564References 571Index 585

CHAPTER 1

INTRODUCTION

It is generally accepted that quantum field theory is the appropriate frame-work for describing the strong, electromagnetic and weak interactions betweenelementary particles. As for the electromagnetic interactions, it has been knownfor a long time that they are described by a quantum gauge field theory. But thatthe principle of gauge invariance also plays a fundamental role in the constructionof a theory for the strong and weak interactions has been recognized only muchlater. The unification of the weak and electromagnetic interactions by Glashow,Salam and Weinberg was a major breakthrough in our understanding of elementaryparticle physics. For the first time one had been able to construct a renormalizablequantum field theory describing simultaneously the weak and electromagnetic in-teractions of hadrons and leptons. The "electro-weak" theory of Glashow, Salamand Weinberg is based on a non-abelian SU(2) x U(l) gauge symmetry, which isbroken down spontaneously to the U(l) symmetry of the electromagnetic interac-tions. This breaking manifests itself in the fact that, in contrast to the masslessphoton, the particles mediating the weak interactions, i.e., the W+, W~ and Z°vector bosons, become massive. In fact they are very massive, which reflects thefact that the weak interactions are very short ranged. The detection of these par-ticles constituted one of the most beautiful tests of the Glashow-Salam-Weinbergtheory.

The fundamental fermions to which the vector bosons couple are the quarksand leptons. The quarks, which are the fundamental building blocks of hadronicmatter, come in different "flavours". There are the "up", "down", "strange","charmed", "bottom" and "top" quarks. The weak interactions can induce tran-sitions between different quark flavours. For example, a "u" quark can convertinto a "d" quark by the emission of a virtual W+ boson. The existence of thequarks has been confirmed (indirectly) by experiment. None of them have beendetected as free particles. They are permanently confined within the hadronswhich are built from the different flavoured quarks and antiquarks. The forceswhich are responsible for the confinement of the quarks are the strong interac-tions. Theoretical considerations have shown, that the "up", "down", etc., quarksshould come in three "colours". The strong interactions are flavour blind, butsensitive to colour. For this reason one calls the theory of strong interactionsQuantum Chromodynamics, or in short, QCD. It is a gauge theory based on theunbroken non-abelian SU(3)-colour group (Fritzsch and Gell-Mann, 1972; Fritzsch,

1

2 Lattice Gauge Theories

Gell-Mann and Leutwyler, 1973). The number "3" reflects the number of colourscarried by the quarks. Since there are eight generators of SU(3), there are eightmassless "gluons" carrying a colour charge which mediate the strong interactionsbetween the fundamental constituents of matter. By the emission or absorptionof a gluon, a quark can change its colour.

QCD is an asymptotically free theory (t'Hooft, 1972; Politzer, 1973; Grossand Wilczek, 1973). Asymptotic freedom tells us that the forces between quarksbecome weak for small quark separations. Because of this asymptotic freedomproperty it was possible for the first time to carry out quantitative perturbativecalculations of observables in strong interaction physics which are sensitive to theshort distance structure of QCD.* In particular it allowed one to study the Bjorkenscaling violations observed in deep inelastic lepton nucleon scattering at SLAC.QCD is the only theory we know that can account for these scaling violations.

The asymptotic freedom property of QCD is intimatley connected with thefact that it is based on a non-abelian gauge group. As a consequence of this non-abelian structure the coloured gluons, which mediate the interactions betweenquarks, can couple to themselves. These self couplings, one believes, are respon-sible for quark confinement. Since the coupling strength becomes small for smallseparations of the quarks, one can speculate that the forces may become strong forlarge separations. This could explain why these fundamental constituents of mat-ter have never been seen free in nature, and why only colour neutral hadrons areobserved. A confirmation that QCD accounts for quark confinement can howeveronly come from a non-perturbative treatment of this theory, since confinement is aconsequence of the dynamics at large distances where perturbation theory breaksdown.

Until 1974 all predictions of QCD were restricted to the perturbative regime.The breakthrough came with the lattice formulations of QCD by Kenneth Wilson(1974), which opened the way to the study of non-perturbative phenomena usingnumerical methods. By now lattice gauge theories have become a branch of parti-cle physics in its own right, and their intimate connection to statistical mechanicsmake them of interest to elementary particle physicists as well as to physicistsworking in the latter mentioned field. Hence also those readers who are not ac-quainted with quantum field theory, but are working in statistical mechanics, canprofit from a study of lattice gauge theories. Conversely, elementary particle physi-cists have profited enormously from the computational methods used in statisticalmechanics, such as the high temperature expansion, cluster expansion, mean field

* for an early review see Politzer (1974).

Introduction 3

approximation, renormalization group methods, and numerical methods.

Once the lattice formulation of QCD had been proposed by Wilson, the firstquestion that physicists were interested in answering, was whether QCD is able toaccount for quark confinement. Wilson had shown that within the strong couplingapproximation QCD confines quarks. As we shall see, however, this is not ajustified approximation when studying the continuum limit. Numerical simulationshowever confirm that QCD indeed accounts for quark confinement.

There are of course many other questions that one would like to answer: doesQCD account for the observed hadron spectrum? It has always been a dream of el-ementary particle physicists to explain why hadrons are as heavy as they are. Arethere other particles predicted by QCD which have not been observed experimen-tally? Because of the self-couplings of the gluons, one expects that the spectrumof the Hamiltonian also contains states which are built mainly from "glue". DoesQCD account for the spontaneous breakdown of chiral symmetry? It is believedthat the (light) pion is the Goldstone Boson associated with a spontaneous break-down of chiral symmetry. How do the strong interactions manifest themselves inweak decays? Can they explain the A.I = 1 / 2 rule in weak non-leptonic processes?How does hadronic matter behave at very high temperatures and/or high densi-ties? Does QCD predict a phase transition to a quark gluon plasma at sufficientlyhigh temperatures, as is expected from general theoretical considerations? Thiswould be relevant, for example, for the understanding of the early stages of theuniverse.

An answer to the above mentioned questions requires a non-perturbativetreatment of QCD. The lattice formulation provides the only possible frameworkat present to study QCD non-perturbatively.

The material in this book has been organized as follows. In the followingchapter we first discuss in some detail the path integral formalism in quantummechanics, and the path integral representation of Green functions in field theory.This formalism provides the basic framework for the lattice formulation of fieldtheories. If the reader is well acquainted with the path integral method, he canskip all the sections of this chapter, except the last. In chapters 3 and 4 we thenconsider the lattice formulation of the free scalar field and the free Dirac field.While this formulation is straight-forward for the case of the scalar field, thisis not the case for the Dirac field. There are several proposals that have beenmade in the literature for placing fermions on a space-time lattice. Of these weshall discuss in detail the Wilson and the Kogut-Susskind fermions, which havebeen widely used in numerical simulations, and introduce the reader to Ginsparg-


Wilson fermions, which have become of interest in more recent times, but whoseimplementation in numerical simulations is very time consuming. In chapters 5and 6 we then introduce abelian and non-abelian gauge fields on the lattice, anddiscuss the lattice formulation of QED and QCD.

Having established the basic theoretical framework, we then present in chap-ter 7 a very important observable: The Wilson loop, which plays a fundamentalrole for studying the confinement problem. This observable will be used in chap-ter 8 to calculate the static potential between two charges in some simple solvablemodels. The purpose of that chapter is to verify in some explicit calculations thatthe interpretation of the Wilson loop given in chapter 7, which may have left thereader with some uneasy feelings, is correct. In chapter 9 we then discuss thecontinuuum limit of QCD and show that this limit, which is realized at a criticalpoint of the theory where correlations lengths diverge, corresponds to vanishingbare coupling constant. Close to the critical point the behaviour of observablesas a function of the coupling constant can be determined from the renormaliza-tion group equation. Knowledge of this behaviour will be crucial for establishingwhether one is extracting continuum physics in numerical simulations.

Chapter 10 is devoted to the discussion of the Michael lattice action andenergy sum rules, which relate the static quark-antiquark potential to the actionand energy stored in the chromoelectric and magnetic fields of a gg-pair. These sumrules are relevant for studying the energy distribution in the flux tube connectinga quark and antiquark at large separations.

Chapters 11 to 15 are devoted to various approximation schemes. Of these,the weak coupling expansion of correlation functions in lattice QCD is the mosttechnical one. In order not to confront the reader immediately with the most com-plicated case, we have divided our presentation of the weak coupling expansioninto three chapters. The first one deals with a simple scalar field theory and merelydemonstrates the basic structure of Feynman lattice integrals. It also includes adiscussion of an important theorem proved by Reisz, which is the lattice versionof the well known power counting theorem for continuum Feynman integrals. Inthe following chapter we then increase the degree of difficulty by considering thecase of lattice quantum electrodynamics (QED). Here several new concepts willbe discussed, which are characteristic of a gauge theory. Readers having a fairbackground in the perturbative treatment of continuum QED will be able to fol-low easily the presentation. As an instructive application of lattice perturbationtheory, we include in this chapter a 1-loop computation of the renormalizationconstant for the axial vector current with Wilson fermions, departing from a lat-

Introduction 5

tice regularized Ward identity. Also included is a discussion of the AB J-anomalywithin the framework of Ginsparg-Wilson fermions. The next chapter then treatesthe case of QCD, which from the conceptional point of view is quite similar to thecase of QED, but is technically far more involved. The Feynman rules are appliedto the computation of the AB J anomaly which is shown to be independent of theform of the lattice regularized action.

At this point we leave the analytic "terrain" and discuss in chapter 16 variousalgorithms that have been used in the literature to calculate observables numeri-cally. All algorithms are based on the concept of a Markov process. We will keepthe discussion very general, and only show in the last two section of this chapter,how such algorithms are implemented in an actual calculation. Chapter 17 firstsummarizes some earlier numerical results obtained in the pioneering days. Be-cause of the ever increasing computer power the numerical data becomes alwaysmore refined, and we leave it to the reader to confer the numerous proceedingsfor more recent results. We have however also included in this chapter some im-portant newer developments which concern the vacuum structure of QCD and thedynamics of quark confinement.

The remaining part of the book is devoted to the study of field theories at finitetemperature. It has been expected for some time that QCD undergoes a phasetransition to a quark-gluon plasma, where quarks and gluons are deconfined. Inchapter 18 we consider some simple bosonic and fermionic models, and discuss indetail the path-integral representation for the thermodynamical partition function.In particular we will construct such a representation for a simple fermionic systemwhich is exact for arbitrary time step, and point out some subtle points whichare not discussed in the literature. Chapter 19 is devoted to finite temperatureperturbation theory in the continuum and on the lattice. The basic steps leading tothe finite-temperature Feynman rules are first exemplified for a scalar field theoryin the continuum. We then extend our discussion to the case of QED and QCD inthe continuum as well as on the lattice and discuss in detail the temporal structureof the free propagator for naive and Wilson fermions. The Feynman rules are thenapplied to calculate the screening mass in QED and QCD in one-loop order, offand on the lattice. These computations will at the same time illustrate the powerof frequency summation formulae, whose derivation has been relegated, in part,to two appendices.

Chapter 20 is devoted to non-perturbative aspects of QCD at finite tem-perature. The lattice formulation of this theory is the appropriate framework forstudying the deconfinement and chiral phase transitions, and deviations of thermo-


dynamical observables from the predictions of perturbation theory at temperatureswell above the phase transition. In this chapter we discuss how thermodynamicalobservables are computed on the lattice, and introduce an order parameter (theWilson line or Polyakov loop) which characterizes the phases of the pure gaugetheory. This order parameter plays a central role in a later section, where wepresent some early Monte Carlo data which gave strong support for the existenceof a deconfinement phase transition. The theoretical concepts introduced in thischapter are then implemented in a simple lattice model which also serves to il-lustrate the power of the character expansion, a technique which is used to studySU(N) gauge theories for strong coupling. The remaining part of this chapter isdevoted to the high temperature phase of QCD which, as already mentioned, isexpected to be that of a quark gluon plasma.

The material covered in this book should enable the reader to follow theextensive literature on this fascinating subject. What the reader will not havelearned, is how much work is involved in carrying out numerical simulations. Afew paragraphs in a publication will in general summarize the results obtainedby several physicists over many months of very hard work. The reader will onlybecome aware of this by speaking to physicists working in this field, or if he isinvolved himself in numerical calculations. Although much progress has been madein inventing new methods for calculating observables on a space time lattice, sometime will still pass before one has sufficiently accurate data available to ascertainthat QCD is the correct theory of strong interactions.

CHAPTER 2

THE PATH INTEGRAL APPROACH TO QUANTIZATION

Since its introduction by Feynman (1948), the path integral (PI) method hasbecome a very important tool for elementary particle physicists. Many of themodern developments in theoretical elementary particle physics are based on thismethod. One of these developments is the lattice formulation of quantum field the-ories which, as we have mentioned in the introduction, opened the gateway to anon-perturbative study of theories like QCD. Since the path integral representationof Green functions in field theory plays a fundamental role in this book, we haveincluded a chapter on the path integral method in order to make this monographself-contained. In the literature it is customary to derive the Pi-representation ofGreen functions in Minkowski space. But for the lattice formulation of field theo-ries, we shall need the corresponding representation for Green functions continuedto imaginary time. Usually a rule is given for making the transition from thereal-time to the imaginary-time formulation. This rule is not self-evident. Sincewe shall make use of it on several occasions, we will verify the rule for the case ofbosonic Green functions, by deriving directly their path integral representation forimaginary time. What concerns the fermionic Green functions, we will not derivethe Pi-representation from scratch, but shall present strong arguments in favourof it.

In the following section, we first discuss the case of non-relativistic quantummechanics.* The results we shall obtain will be relevant in section 2, where wederive the Pi-representation of bosonic Green functions which are of interest tothe lattice formulation of quantum field theories involving Bose-fields. In section3 we then discuss the transfer matrix for bosonic systems. Green functions offermionic operators are considered in section 4.

As we shall see, the Pi-representation of Green functions is only formallydefined for systems whose degrees of freedom are labeled by a continuous vari-able, as is the case in field theory. One is therefore forced to regularize the pathintegral expressions. In section 5 we discuss this problem on a qualitative level,and motivate the introduction of a space-time lattice. This, as we shall comment

* For a comprehensive discussion of the Pi-method in quantum mechanics inthe real-time formulation, the reader should confer the book by Feynman andHibbs (1965).

7


on, corresponds in perturbation theory to a particular choice of regularization ofFeynman integrals.

2.1 The Path Integral Method in Quantum Mechanics

In the Hilbert space formulation of quantum mechanics, the states of thesystem are described by vectors in a Hilbert space, and observables are representedby hermitean operators acting in this space. The time evolution of the quantummechanical system is given by the Schrodinger equation, or equivalently by*

\^(t)>=e-iH^~to)\1p(t0)>, (2.1)

where H is the Hamiltonian. Thus if we know the state of the system at time to,(2.1) determines the state at a later time t. Let q = {qa} denote collectively thecoordinate degrees of freedom of the system and \q > the simultaneous eigenstatesof the corresponding operators {Qa}, i-e-

Qa\q >= Qa\q >» a = l , . . . , n .

Then (2.1) implies the following equation for the wave function ij){q, t) =< q\ip(t) >

</>(<?', *') = / dqG(q', t'; q, t)r/>(q, t),

whereG(q',t';q,t)=<q'\e-iH^-^\q> (2.2)

is the Green function describing the propagation of the state \ip(t) >, and wherethe integration measure is given by

ndq=ndqa-a=l

A very important property of the Green function (2.2) is that it satisfies thefollowing composition law

G(t', q'; q,t) = J dq"G(t', q'; q", t")G(q", t"; q, t). (2.3)

This relation follows immediately by writing exp(—iH(t' — t)) = exp(—iH(t' —t")) exp(-iH(t"-t)) in (2.2) and introducing a complete set of intermediate states

* We set ft = 1 throughout this book.

The Path Integral Approach to Quantization 9

\q" > between the two exponentials. Using the property (2.3), Feynman deriveda path integral representation for the matrix element (2.2), which exhibits in avery transparent way the connection between the classical and quantum theory.In classical physics the time evolution of the system is given by the Lagrangeequations of motion which follow from the principle of least action. To quantize thesystem, one then constructs the Hamiltonian, and writes the equation of motionin terms of Poisson brackets. This provides the starting point for the canonicalquantization of the theory. By proceeding in this way, one has moved far awayfrom the original action principle. The path integral representation of Feynmanreestablishes the connection with the classical action principle. In the followingwe derive this representation for the Green function (2.2) continued to imaginarytime, t —» —IT, t' —» —IT', since we shall need it in the following section.

Consider the matrix element

<q',t'\q,t>=<ql\e-iH{-t'-%>, (2.4)

where\q,t>=eiHt\q>

are eigenstates of the Heisenberg operators

Qa(t) = eiHtQae-im, (2.5)

i.e.Qa(t)\q,t >= qa\q,t > .

Inserting a complete set of energy eigenstates to the right and left of the exponen-tial in (2.4), we have that

< g ' , f ' | g , t>ê- i E »< ( ' -^ n (g ' )C(«) ,n

where i/)n(q) =< q\En > is the eigenfunction of H with energy En. The sum overn extends over the discrete as well as the continuous spectrum of the Hamiltonian.This expression can now be continued to imaginary time. Making the replacementst —• —IT, t' —¥ —IT', we arrive at an expression which is dominated by the groundstate in the limit r ' — r —> oo:

< q', t'\q, t > = J2 e-*"(T'-rtyn(«')<(?) •n


The right-hand side is just the matrix element < q'\ exp(-H(r' - r))\q >. Hence,as expected from (2.4), the Green function continued to imaginary times is givenby

< q',t'\q,t> t^-ir =< q'\e-H^'-^\q > . (2.6)

To arrive at a path integral representation for the right-hand side of (2.6), we splitthe time interval* [T,T'] into N infinitesimal segments of length e = (T' — T)/N.

Let Ti, T2,..., T/v-i denote the intermediate times, i.e. r < TI < T-I < ... < T'. Thenthe imaginary-time Green function can be obtained by a sequence of infinitesimaltime steps as follows,

< g'|e~H(T'~T)|g >=< g'|e--ff(^'-^-i)e-ff(^-i-^-2) _ e-ff(n-r)|? >

= / I ] > « < g'le-^l^-1) X g^-^le-"^-2) > ... < q^\e-He\q >,•* e=i

(2.7)where

dqW=Y[dqV>.a

Here \qW > denote the complete set of eigenstates which have been introduced inthe Tth intermediate time step.

In order to evaluate the matrix elements in (2.7), we must now specify thestructure of the Hamiltonian. Let us assume it to be of the form

# = jE^+*m (2-8)where Pa are the momenta canonically conjugate to Qa. Making use of the Baker-Campbell-Hausdorff formula,

eAeB _ eA+B+\[A,B] + ...^

we conclude that exp(-i?e) can be approximated for small e by

It follows that

< qW)\e-H<\qW > « < g ( £ + 1 ) | e - i S Q ^ | g W > e-«V(,W).

* We shall henceforth refer to r as "time".


To evaluate the remaining matrix element, we introduce a complete set of momen-tum eigenstates to the right and left of exp ( - | Jâ â)- With

we have that

Substituting this expression into (2.7) we arrive at the following approximate pathintegral representation in phase space, valid for small e:

< Ae-W-'Hq >« jDqDpe*PW+l>-£%-<X(W) , (2.9a)

where

qW=q ,qW = q' , (2.96)

„ JV_I iv- l , (/)

^=nn<n^-. (*•*>0=1 l=\ 1=0

and

^(«W,PW) = E ^ a + V(,W). (2.9d)Q = l

Notice that the number of momentum integrations exceeds that of the coordinates.Actually, as the reader can readily verify, the above formula holds just as well

for any Hamiltonian of the form H(Q, P) = T(P) + V(Q), with T{P) a polynomialin the canconical momenta. For the case where T(P) has the quadratic form givenin (2.8), we can also obtain a configuration space path integral representation, bycarrying out the Gaussian integration over the momenta. The following expressionis valid for infinitessimal time slices,

< *V*<*'->|g >« / Iff ft ^ e - l T o ^ W ' ' , ^ (2.10a)J l' = la=lV/i7re

whereLE(qW,qW) = 521-q^+V(q(% (2.106)

a


JM-l) _(<).(,) ^ <?« ^ g a t ( 2 1 Q c )

and g(°) = q, q(N1 = q'. The subscript "E" on L B is to remind us that we arestudying the Green function in the "euclidean" formulation.* Let us interpret theright-hand side of (2.10a). Consider an arbitrary path in g-space connecting thespace-time points (q,r) and (?',r '), consisting of straight line segments in everyinfinitesimal time interval. Let qW denote the set of coordinates of the system attime TI (see fig. (2-1)). To emphasize this correspondence let us set

q(j} = qa(n) ,

where r0 = r, TN = T'. Then (2.10c) is the "euclidean velocity" in the time interval[n,T(+i] of a "particle" moving in an n-dimensional configuration space, and LEis the discretized version of the classical Lagrangean in the euclidean formulation(notice the "plus" sign between the kinetic term and the potential). The actionassociated with the path depicted schematically in fig. (2-1) is given by

iv-i r -i

SE[q] = £ e £ o (9"(r«))2 + V (?fa)) ' (2-n)

This is the expression appearing in the argument of the exponential in (2.10a).We therefore arrive at the following prescription for calculating the Green functionfor imaginary time:

i) Divide the interval [r, r'\ into infinitesimal segments of length e = (r' — T)/N.

ii) Consider all possible paths starting at q at time r and ending at q' at timeT'. Approximate these paths by straight-line segments as shown in fig. (2-1),and calculate the action (2.11) for each path,

iii) Weigh each path with exp(—SEWI) an<^ s u m these exponentials over all paths,by integrating over all possible values of the coordinates at intermediate times,

iv) Multiply the resulting expression with (l/\/2ne)nN, where n is the number ofcoordinate degrees of freedom and take the limit e -» 0, iV -> oo, keeping theproduct Ne = (r' - r) fixed.

* In the following chapters, where we will study the Pi-representation of fieldtheories in detail, the transition to imaginary time corresponds to formulating thetheories in euclidean space-time. We shall therefore refer in the following to theimaginary-time formulation as the euclidean formulation.


Fig . 2-1 Path connecting the space-time points (q, r ) and (q1,T')

contributing to the integral (2.11a)

The result of steps i) to iv) we formally denote by

(q',r'\q,T)= P Dqe-8^, (2.12a)Jq

where

r'SE[q}= / dr"LE(q(r"),q(r")), (2.126)

and where, for later convenience, we have introduced the short-hand notation

(q',T'\q,r)=<q'\e-H^-^\q>, (2.12c)

in analogy to (2.4). This is the path integral expression we wanted to obtain. No-tice that because the paths are weighted with exp(—SE), important contributionsto (2.12a) are expected to come from those paths for which S'sf?] takes valuesclose to the minimum, where

SSE[q] = 0.


This is the principle of least action which leads to the classical euclidean equationsof motion. Hence, within the path integral framework, the quantization of aclassical system amounts to taking into account fluctuations around the classicalpath. In the euclidean formulation these fluctuations are exponentially suppressedif SE > 0. On the other hand, in the real time formulation, an analogous procedureto the one followed above, leads to the following path integral representation of(2.4):

< q'\e-iH^'-^\q >= f Dq eiS^, (2.13)Jq

where S[q] is the action for real time. The path integral (2.13) is defined in thesame way as before (see Feynman and Hibbs, 1965), but the paths are now weightedwith an oscillating function. For this reason this path integral representation is notsuited for numerical calculations. It is, however, a useful starting point for carryingout semiclassical approximations, where one expands the action about a minimumup to terms quadratic in the coordinates. For an instructive example the readermay consult the paper by Bender et al. (1978), where the energy spectrum andeigenfunctions are calculated in the WKB approximation for a one-dimensionalperiodic potential.

An exact evaluation of the path integral (2.12) or (2.13) is only possible ina few cases. The standard example in the real time formulation is the harmonicoscillator. It is discussed in detail in the book by Feynman and Hibbs (1965).The Coulomb potential already provides a quite non-trivial example (Duru andKleinert, 1979). It therefore may appear that the path integral method is oflittle practical use. This is true for quantum mechanics, where more efficientmethods are available to calculate scattering amplitudes, bound state energiesand eigenfunctions. But in field theory, we only know how to compute Greenfunctions in perturbation theory (except for some simple models which can besolved exactly). It is here where physicists first became very interested in the pathintegral method, since it allowed one to derive the Feynman rules for gauge theorieslike QCD in a very straightforward way. This is, however, only one of the merits ofthe method. As we have already pointed out, many of the modern developments intheoretical elementary particle physics are based on the path integral formalism.In the following section we extend the above discussion to bosonic Green functionsof interest in field theory.


2.2 Path Integral Representation of Bosonic Green Functions

in Field Theory

In quantum mechanics all physical information about the quantum system

is contained in the Green function (2.2). In field theory, on the other hand, this

information is stored in an infinite set of vacuum expectation values of time-ordered

products of Heisenberg field operators. The simplest such operator is the real scalar

field (f>(x) = <f>(x,t). Its time evolution is given by

4>{x,t) = eim<j>(S,0)e-iHt,

where H is the Hamiltonian of the system. The coordinates x label the infinite

number of coordinate degrees of freedom of the system. They play the role of the

discrete index "a" labeling the Heisenberg operators Qa{t) defined in (2.5). The

Green functions of the scalar field are defined by

G{xux2,...,xt) = < n\T {cj){Xl)<t>{x2)...cf>{xe)) | n >, (2.14)

where Xi = (xi,ti), and \Q > denotes the ground state (vacuum) of the system

whose dynamics is determined by the Hamiltonian H. The time-ordering operation

"T" orders the operators from left to right according to descending time. The

analogue of (2.14) in our quantum mechanical example is evidently given by

Gaiai...at(t1,t1,...,tt) =< Eo\T(Qai(t1)Qaa(t2)...Qai(tt))\Eo> . (2.15)

Let us assume that we have ordered the operators in (2.15) according to descending

time from left to right; then

Gaaa2...Qt(t1,t1,...,te) =< Eo\Qai{ti)Qaa(t2)...Qai(tt)\Eo >, (ti > t2 > - . > tt).

(2.16)

We are interested in a path integral representation of (2.16) continued to

imaginary times, ti —¥ —iTi. It is this representation which we shall need to

formulate bosonic field theories on a lattice. The transition to imaginary times is

made by replacing the operators Qai (ti) by

Qai{n) = eHTiQa,e-Hr>. (2.17)

This corresponds to setting t — —IT in the right-hand side of (2.5). The euclidean

version of (2.16) is therefore given by

< Eo\Qai(Ti)Qai(vt)...Qai(Tt)\Eo > • (2.18)


To derive a path integral representation for this ground state expectation value we

proceed in two steps. We first show that (2.18) can be extracted from the matrix

elementsW,r'\Qai(Tl)Qa2(r2)...Qat(Te)\q,T)

= < q'\e-HT'Qai(T1)Qa2(T2)...Qae(n)eHT\q >,

by studying this expression for large positive and negative values of r ' and r.

We then demonstrate that (2.19) has a path integral representation which is an

(almost) obvious generalization of (2.12a).

We begin with the first mentioned step. Inserting a complete set of energy

eigenstates to the left and right of the operators exp(Hr) and exp(-iJr ') in (2.19),

we have that

(q',r'\Qai(r1)...Qat(re)\q,r)

= Yje~E^T'eE^^K,{ql)rM < EK.\QaArx)-Qat{rt)\EK > . (2-20)

Assuming that there exists an energy gap between the ground state and first

excited state, we therefore find that

(q',T'\Qai(T1)...Qai(Te)\q,T)

TT^, e-E^'-Tô(q'Wo(q) < E0\Qai{T1)...Qai(Tt)\E0 > • (2"21a)T—y — co

Furthermore, replacing the Qai{Ti)'s in this expression by the unit operator, we

have that

(q',r'\q,T) - » e-Eo{T'-T)M<l'Wo(q)- (2.216)T—• — OO

From (2.21a,b) we are led to the following important statement:

(q',r'\Qai(T1)...Qat(ri)\q,T) - -(a' T'\Q T) r" t> *'0\Qa1(.n)...Qat{Ti)\Eo > . (2.22)

T-»-OO

Notice that according to (2.22) we are free to choose for q = {qa} and q' = {q'a} any

values as long as < q\E0 > and < q'\E0 > are different from zero! In other words,

the ground state wave function must have non-vanishing support at q and q'. Since

(2.22) actually holds for arbitrary time, TI,...,T£, it follows that a corresponding

expression holds for the time-ordered product of the operators, i.e.

{q',T/\T(Qai{T1)...Qai(Te))\g,r) - -/ , T,I r^ T 7 ^ < E0\l {QaATiJ-Qatinjlbo > • (2.23)

r—> —oo

(2.19)


This completes the first step of our program. We now proceed with the secondstep, and construct the path integral representation for the numerator appearingon the left-hand side of (2.23). The Pi-representation for the denominator hasalready been obtained in the previous section. Our starting point, however, is notthis numerator, but the matrix element (2.19), with T\ > r ... > T(. Let us firstwrite out the time dependence explicitly, by making use of the definition (2.17):

(q',r'\Qai(Tl)Qa2(T2)...Qai(re)\q,T)

We next insert a complete set of eigenstates of {Qa} to the right and left of eachof the operators Qai. These operators are diagonal in this representation. Let usdenote the integration variables associated with Qai collectively by q^. Then

(q',r'\Qai(ri)...Qae(re)\q,r)

= /ndgW(g\r1g«,r09i1

1)(?(1),nk(2),r2)g(

22)...(?W(,W,r,|g,r).

J »=i

Inserting for {q^\ Ti\q^\ TJ), etc., the path integral expressions analogous to (2.12a)one finds that

(q'y\Qai(ri):.Qat(re)\q,T)

= f Dqqai(T1)...qat(n)e-fr' **"*"*(«'").«'•)) ( / > ^ > ^ > ^ > ^Jq

(2.24)where the path integral is calculated as follows:

i) Split the interval [r, r'] into N infinitesimal time intervals of length e =(T' - T)/N.

ii) Consider all paths starting at q at time r and ending at q' at time r' . Ap-proximate these paths by straight line segments in each infinitesimal time interval.

iii) Weigh each path with exp(—^[tf]), where Ss[q] is the action definedin (2.11), and with the product of the coordinates qai, ...,qat at times ri,...,r<,respectively. Sum the contributions over all paths by integrating over all possiblevalues of the coordinates at intermediate times.

iv) Multiply the resulting expression with (l/y/2ire)nN, where n is the numberof degrees of freedom of the system, and take the limit e —>• 0, N —>• oo, keepingthe product Ne = r ' — r fixed.


In deriving the above expression we have assumed that r ' > T\ > T2 > ... >

T( > T. Instead of (2.24) we can therefore also write

(q',r'T{Qai(rl)...Qai(Te))\q,r)

= / Dqqai{rl)...qat{Tt)e Jr w >m ».Jq

But because of the definition of the T-product, we can now write the product of

the operators Qai{Ti) in any order we wish. This symmetry under the exchange

of any two operators is reflected in the path integral, since the qa's are ordinary

commuting variables.

We are now ready to write down the path integral expression for the right-

hand side of (2.23). Inserting the expressions (2.25) and (2.12a) into the left-hand

side of (2.23), and taking the indicated limit, we find that

^F\r(n (r\ n r - ^ W ^ f D<iQaATi)-q<*i(Te)e-SB[q]

< ho\l [Qa1{T1)...Qc,t{Te)J\ho>= fj) e - 5 sM ' (2-26a)

where (2.12b) is now replaced by

/•OO

SE[q]= dTLE(q(T),q(r)). (2.266)J — oo

SE is the euclidean action associated with the path q(r). The integrals in (2.26a)

are carried out over all paths starting and ending at arbitrary points at times

T = —oo and r = +oo, respectively. This is true as long as the ground state wave

function has non-vanishing support at q and q'. In practical calculations the size of

< q\Eo > and < q'\Eo > is important. The reason is that in most cases of interest

we cannot evaluate the path integral analytically, but must recur to numerical

methods. This forces one to calculate the multiple integrals on finite time lattices.

It is then essential that the contributions to the sum in (2.20) coming from higher

energy states are suppressed as much as possible. When (2.26a) is calculated

numerically one usually imposes periodic boundary conditions, i.e., q = q', and

allows q to take arbitrary values. This choice of boundary conditions turns out to

be very convenient.

We now make some further comments about the path integral expression

(2.26a). The evaluation of the right-hand side demands that we first calculate the

multiple integrals on a time lattice with finite lattice spacing e, and then take the

(2.25)

(2.26a)


limit e —» 0. Unfortunately, one can only carry out this program for path integralsof the Gaussian type. As an example consider the following integral,

Iai...at= [fldqiqaiqa2...qaee-^'>.™'!"M"""!">, (2.27)•* i=i

where M is a real, positive definite symmetric matrix, and where the sum extendsover n,m = 1,...,N. This integral can be calculated as follows. Introduce thegenerating functional

/

N

JJê-iSnn.9-^—«">+S»J"«-. (2.28)

Then (2.27) is evidently given by

. / #Zo[J\ \Iai-ai~\dJaidJa2...dJat)J=0-

( 2 - i 9 j

We therefore need to calculate the integral (2.28). This can be easily done by per-forming an orthogonal transformation on the coordinates {<?«} which diagonalizesthe matrix M. One then finds

yde tM

where M~1 is the inverse of the matrix M, and detM is the determinant of M.This expression is very useful for carrying out a perturbative expansion of Greenfunctions in theories where the potential is a polynomial in the coordinates andcan be treated as a small perturbation. Thus suppose we want to calculate theintegral

r NKai..ae = / Y[dqiqaiqai...qaie-Slq], (2.31a)

J ! = 1

where

S[q] = - Y^ 1nMnmqm + S^q], (2.31b)n,m

with Si[q] a polynomial in the coordinates {qn}- The integral (2.31a) is given by(2.29), but with the generating functional (2.28) replaced by

Z[J}= m d « e - S M + E ™ J » « " .-' i=i

(2.30)


Expanding exp(-5j[g]), we have that

This expression can also be written in the form

] = £^(*4])V],

where Si[d/dJ] is obtained from Si[q] by making the replacements qn —>• d/dJn (n =1,...,N) in the argument of Si, and where Zo[J] is the generating functional(2.30). The above formula allows us to compute the generating functional Z[J] ofthe interacting theory in every order of 5/ . This is the first comment we wantedto make. The second comment concerns our earlier claim, that the path integralrepresentation of Green functions opens the possibility of studying field theoriesnon-perturbatively. The reason for this is the following. Consider the right-handside of (2.26a). If the action is bounded from below, then this expression has theform of a statistical ensemble average, with a Boltzmann distribution given byexp(—5B [g]). This allows us to use well-known statistical methods to calculateGreen functions in theories with a large number of degrees of freedom. The entirebook is based on this simple observation. Because of this similarity with statisticalmechanics we shall speak of the euclidean Green functions as correlation functions,and write (2.26) in the form

< qai(ri)-qat(ri) >= \ jDqqai(T1)...qae(re)e-s^\ (2.32a)

whereZ= I Dqe~SE[q]. (2.326)

We want to point out, however, that the right-hand side of (2.32a) should not

be confused with a canonical ensemble average in classical statistical mechanics.

Nevertheless, we shall refer to (2.32b) as the partition function.

For reasons mentioned at the beginning of this chapter, we have concentratedour attention on Green functions continued to imaginary times. The derivationof the path integral representation for the real-time Green functions (2.15) canbe found in the review article by Abers and Lee (1973), and in most modern textbooks on field theory. We only quote here the result:

< E0\T(Qai(ti).. .QaAw) \E0 >= f Dq eiSM ' (2.33)


Here S[q] is the action whose variation leads to the equations of motion for realtimes. In our quantum mechanical example S[q] is given by

The quantity appearing within brackets is the Lagrangean describing the dynamicsof the classical system. On the other hand, we have seen that the "euclidean"action SE[</] has the form

SE[q] = jXJr i^($J2.y + V(q(T)) , (2.35)

whose variation leads to the "euclidean" equations of motion. By comparing (2.35)and (2.34), we see that SE[Q] can be obtained from S[q] by the following formalrule: Consider the action S[q]. Replace t by — ir wherever t appears explicitly,and qa(t) by g>a(r), where the coordinates are treated in both cases as real valuedfunctions of their arguments. Then

iSM —>„ SE[q],"t-t — lT"

where "t —> —IT" stands for the above formal prescription. Of course we have onlyproved this rule for systems, where the kinetic part of the Lagrangean is quadraticin the velocities. For fermionic systems this is not the case, but the prescription isstill correct. Since we are usually given the action of the system for real times, theabove rule is useful for determining the form of the euclidean action that entersthe path integral expression (2.32a).

As we have demonstrated in this section, euclidean path integrals involve theintegration over real valued coordinates on an euclidean time lattice. A similarstatement holds for the Pi-representation of Green functions defined for real times.The difference between the two representation, merely resides in the structureof the action. Thus in the real-time formulation the paths q(t) are weightedwith a phase, while the corresponding weight in the euclidean formulation canbe interpreted as a "Boltzmann factor", if the action is a real valued functionalof the coordinates, bounded from below. This was the main point we wantedto demonstrate by deriving directly the path integral representation for Greenfunctions continued to imaginary times.

(2-34)


2.3 The Transfer Matrix

Consider the partition function (2.32b). For a system whose dynamics isdictated by the Hamiltonian (2.8), Z has the following explicit form on a finite,periodic, euclidean time-lattice:*

J £ ' = 0 / 3 V27T€

Let us write this expression in the form

/

N-l J V - 1

]^[ dg«'> n T,(<+1),(«, (2.37a)l'=0 1=0

where dq^ = fI/3 ^9,9 > a n d

Tqit+i)qW = {—J e L V ' J . (2.376)

From (2.9) we see that

Tqit+»qw =< qV+V\e-He\qW > . (2.38)

The matrix defined by (2.38) is the so-called transfer matrix. It describes theevolution of the system in an infinitesimal timestep e. Actually, the more funda-mental definition of the partition function is given by (2.37a) , with the transfermatrix defined in (2.38). This is evident from our discussion in the previous twosections, where the matrix elements of exp(—eH) played a fundamental role.

Suppose now that we were given the transfer matrix. Can we extract from itthe Hamiltonian (2.8)? Indeed, this can be done by reversing the steps which ledus from < g( £ + 1 ) |exp(-ife) |gw > to (2.9).* We now give the details. The statesq(k) > are simultaneous eigenstates of the coordinate operators Qa:

Qa\q(k)>=qik)\q

{k)>.

Let us introduce the momentum operators Pa canonically conjugate to Qa, whichsatisfy the commutation relations

[Qa,Pp] =iSap-

* i.e., q(°^ and q(N) are identified.* see e.g., Creutz (1977)

(2.36)

(2.37a)


Then exp(—i£ • P), with £ • P = 5Za£a-Pa> generates finite translations by £ =

Since

<i'\=nt(q'a-q*),a

we conclude that

< g^\e-*-r\gW >= J]%i< + 1 ) - ^ - £«)• (2.39)a

Now the matrix element (2.38) can also be written in the form

v + l V o = /"den{*(^+1) -«i? - We~*f i}e~'v (*( 0 ) .

wherer r d ^

By making use of the relation (2.39), this expression becomes

V+1,,W =< <7('+1)| [ I ^ e - ^ E . ^ + ^ P ^ J e-evW)|gW > _

Performing the Gaussian integral we therefore find that

Tqie+i)qW =< qW\e-<\Z. WM)\qW > .

By comparing this expression with (2.38), we conclude that the Hamiltonian isgiven by (2.8).

The above described procedure for constructing the Hamiltonian, given thetransfer matrix, will be relevant later on, when we discuss the lattice Hamiltonianof a gauge theory. In the lattice formulation of field theories we are given thepartition function. By writing the partition function in the form (2.37a), theidentification (2.38) will allow us to deduce the lattice Hamiltonian.

2.4 Path Integral Representation of Fermionic Green Functions

So far we have considered quantum mechanical systems involving only bosonicdegrees of freedom. But the fundamental matter fields in nature are believed to


carry spin 1/2. In contrast to the bosonic case these fields anticommute in thelimit h —>• 0, and hence become elements of a Grassmann algebra in this limit.We therefore expect that the path integral representation of Green functions builtfrom fermion fields will involve the integration over anticommuting (Grassmann)variables.* We hence begin this section with a discussion of how one differentiatesand integrates functions of Grassmann variables. The integration rules are thenapplied to calculate specific integrals, which will play an important role throughoutthis book. The results we shall obtain will give us a strong hint regarding thepath integral representation of fermionic Green functions in theories of interest forelementary particle physics. We begin our discussion with some basic definitions.

Grassmann Algebra

The elements r)i, ...,T]N are said to be the generators of a Grassmann algebra,if they anticommute among each other, i.e. if

Of*- Vj} = ViVj + mm = o, i,j = 1,..., N. (2.40)

From here it follows that

vf = 0. (2.41)

A general element of a Grassmann algebra is defined as a power series in the rji's.Because of (2.41), however, this power series has only a finite number of terms:

f(v) = fo + ^2 fiVi + Yl fWli + - + fi2...NViV2-VN- (2.42)i i^j

As an example consider the function

g(v) = e-'E"*-!™*'™.

It is defined by the usual power series expansion of the exponential. Since the termsappearing in the sum - being quadratic in the Grassmann variables - commuteamong each other, we can also write girf) as follows

g(v) = Y[e-»<A^,

* For a comprehensive discussion of the functional formalism for fermions thereader may consult the book by Berezin (1966).


or, making use of (2.41),

N

9(v)= I ] 0--T)iAijVj)-'¥•3

Next we consider the following function of a set of 2,/V-Grassmann variables which

we denote by r/i, ...TJJV, JJi, ...,f?jv:

h{V,fj) = e-^"iAiitli.

Proceeding as above, we now have that

N

Notice that in contrast to previous cases, this expression also involves diagonalelements of Aij.

Integration over Grassmann variables

We now state the Grassmann rules for calculating integrals of the form

r N

J i=i

where /(??) is a function whose general structure is given by (2.42). Since a givenGrassmann variable can at most appear to the first power in f{rj), the followingrules suffice to calculate an arbitrary integral [Berezin (1966)]:

[dTH = 0,(2.43a)

/ drurii = 1.

When computing multiple integrals one must further take into account that theintegration measures {dr)i} also anticommute among themselves, as well as withall 7/j's

{dTH,drh} = {drH,riJ} = 0, Vt.j. (2.436)

These integration rules look indeed very strange. But, as we shall see soon, theyare the appropriate ones to allow us to obtain a PI representation of fermionic


Green functions. As an example let us apply these rules to calculate the following

integral:iv

I[A) = / [ I djlidme~ 5 X - ' mAiiVj. (2.44)J e=i

We could have also denoted the Grassmann variables by r/i,..., T]2N, by setting

rjN+i = fji. But for reasons which will become clear later, we prefer the above

notation. To evaluate (2.44), we first write the integrand in the formN

e~ E ; i J niAavi = J T e-fji ]T~=1 AijVJ ^

i=l

Since fjf = 0, only the first two terms in the expansion of the exponential will

contribute. Hence

e ~ £ < - i * * ' " ' = (I - mAlhVh){l - mA2i2rH2).. .{1 - fJNANiNr]iN), (2.45)

where a summation over repeated indices it(i = 1,...,N) is understood. Now

because of the Grassmann integration rules (2.43a), the integrand of (2.44) must

involve the product of all the Grassmann variables. We therefore only need to

consider the term

K(v,rj)= ^2 ViimVi2'n2---ViNfiNAii1A2i2...ANiN, (2.46)»i , . . . , i jv

where we have set fjkVik = ~VikVk to eliminate the minus signs appearing in (2.45).

The summation clearly includes only those terms for which all the indices i\,..., iN

are different. Now, the product of Grassmann variables in (2.46) is antisymmetric

under the exchange of any pair of indices it and iy. Hence we can write expression

(2.46) in the form

K(v,V)=VlVlV2V2---VNfJN ^2 eili2-iNAli1A2i2...ANiN,ii...iN

where eili2,,,iN is the e-tensor in ./V-dimensions. Recalling the standard formula

for the determinant of a matrix A, we therefore find that

K(r],fj) = (det A)r)iT\xr)2f\2 . ..i]NfjN.

We now replace the exponential in (2.44) by this expression and obtain

I[A] = JT / dfjidrjiriifji det A = det A.i=l


Let us summarize our result for later convenience:

j D{m)e~^->"i-i=l1iiAii1'1 =de ty l ,N (2.47)

1=1

There is another important formula we shall need, which is the analog of(2.30). It will allow us to calculate integrals of the type

/i1...i<i'1...ii[^] = /D(vri)Vn--Vn% • • . ^ e " ^ - i * * ' " ' . (2.48)

Consider the following generating functional

Z[p,p] = J D{fjr1)e-^.Âiir>>+îp< + ~pir]i\ (2.49)

where all indices are understood to run from 1 to N, and where the "sources"{pi} and {pi] are now also anticommuting elements of the Grassmann algebragenerated by {r)i,fji,Pi,Pi}- To evaluate (2.49) we first rewrite the integral asfollows:

Z[p,p] = [ |z>(w)e-£..^j e£,.i*V«

where

fc

fii = vi-YlpkAkt*>k

and A~x is the inverse of the matrix A. Making use of the invariance of theintegration measure under the above transformation * and of (2.47), we find that

Z[p, p] = det Ae^".i piA^Pj. (2.50)

Notice that in contrast to the bosonic case, this generating functional is propor-tional to det A [instead of (det^)""1/2; see (2.30)].

* This is ensured by the Grassmann integration rules.


Differentiation of Grassmann Variables

We now complete our discussion on Grassmann variables by introducing theconcept of a partial derivative on the space of functions defined by (2.42). Supposewe want to differentiate f(f]) with respect to rji. Then the rules are the following:

i) If f(n) does not depend on rji, then dVif(rj) = 0ii) If f{rj) depends on rji, then the left derivative d/dr]i is performed by first

bringing the variable r]i (which never appears twice in a product!) all the wayto the left, using the anticommutation relations (2.40), and then applying therule g

Correspondingly, we obtain the right derivative d /drji by bringing the variable rjiall the way to the right and then applying the rule

Thus for example

tn""W," ~n>*

Notice that, because of the peculiar definition of Grassmann integration, we havethat

/ *»/(„) = WM.Hence integration over rji is equivalent to partial differentiation with respect tothis variable! Another property, which can be easily proved, is that

Let us apply these rules to some cases of interest. Consider the function

E{p) = eîPini,

where {%, p;} are the generators of a Grassmann algebra. If they were ordinaryc-numbers then we would have that

Ê(p) = ruElp)dpi


This result is in fact correct. To see this let us write E(p) in the form

E(p) = l[(l + PJVj)-j

Applying the rules of Grassmann differentiation, we have that

But because of the appearance of the factor rji we are now free to include the extraterm 1+ p^r/i in the above product. Hence we arrive at the above-mentioned naiveresult. It should, however, be noted, that the order of the Grassmann variables inYli PiVi w a s important. By reversing this order we get a minus sign, and the ruleis not the usual one! By a similar argument one finds that

e£^P;JL £.%ÔPi

Let us now return to the generating functional defined in (2.49). Proceedingas above one can easily show that

L 1 <•Jp=p=0

where the left-hand side has been defined in (2.48). By making use of the explicitexpression for Z[p,p] given in (2.50), one can calculate the right-hand side of(2.51). Since we shall need this expression in later chapters, we will derive it here.To this effect we first rewrite (2.50) as follows

Z[p,p] =det Ane* £ J A*'lpi = (det A){1 + PilA'Xpkl)i (2.52)

• (i + ftA^PfcJ-C1 + PuA^ktpkt)[-},

where the indices &i,..., ki are summed, and [...] stands for the remaining factorsnot involving the variables pil,...,pir The only terms which contribute to the leftderivatives in (2.51) are those involving the product pix...pit. Furthermore sincewe will eventually set all "sources" pi and pi equal to zero, we can replace [...] by 1.

(2.51)


The contribution in (2.52), which is relevant when computing (2.51), is thereforegiven by

Z[p,p] = det A Y^ PuAr1kipkl...pilAr1

ktpkt,{fci}'

where all ki& are different, and the "prime" on {hi}' indicates that the fcj's takeonly values in the set (i^,i'2>...,^), labeling the right derivatives in (2.51). Thuswe can write the above expression in the form

Z[p, p] = det A J2 AT1.,^ AT1.,^ ...A~]lpi ph p^ p^p^ ...pitpep(, (2.53a)

where the sum extends over all permutations

p . ( h H -• h \ (r, t;,.\\ l P i 1P2 ••• lpt /

Each of the products of Grassman variables appearing in the sum (2.53a) can beput into the form

Fhii::'it = PiiPilPiiPi't-PiiPi'i

by using the anticommutation rules for Grassmann variables. It follows that

z[P,p] = (detA) [ ] T ( - i r ^ ...AT*, | 4 : J ( P , P ) ,

where (—l)°"pis the signum of the permutation (2.53b). We now apply the leftand right derivatives indicated in (2.51) to the above expression and obtain thefollowing important result:

[D(fjr1)Vil...r,lef}ili...fjl,e-^f>'A'^

,-^ (2-54)= 6(detA)5>l)*Mi-i A ^ ,

p 1

where & = (—l)^*-1)/2. As a particular case of (2.54) we have that

J D(fM)riifjje- ^ mAijnj = (det A)A~jl. (2.55)

Let us define the two-point correlation function

< ViVj >= J KIUh» . 2 . 5 6J D(fJv)e~^ViAijVj


Then it follows from (2.47) and (2.55) that

riifjj =< ViVj >= A^1. (2.57)

We shall refer to (2.57) as a contraction. The generalization of (2.57) to arbitrary"correlation" functions,

<Vii-VitVi'1-Vi'e >

= fD(vv)yil...VieVl'1---Vi>ee~Ei-jfJiAijVi (2-58)

J D(rjr,)e-^.iniAijr>j

follows from (2.54) and (2.47). One finds that

1—1 *" M — ' "'-I ' \ '—' i | (2.59)where the right-hand side stands for the sum of all possible pairwise contractions(2.57) of the Grassmann variables, multiplied by a phase (—l)p, where p is thenumber of transpositions required to place the contracted variables next to eachother in the form r]fj.

This completes our discussion of Grassmann variables. Let us now answerthe question what all these exercises with Grassmann variables have to do withthe path integral representation of fermionic Green functions. To this effect letus consider the simplest type of relativistic field theory involving only fermionicfields: the free Dirac field. The corresponding action in Minkowski space is givenby*

SF[rl>,$\ = [<fix$(x)(i'fdll-M)1>(x),

where 7M are the Dirac 7-matrices, and where the Lorentz index /n, is summed. Letus write this action in the form

SF = Y1 Id4xd4yi>a(x)Ka0(x,y)My),a,0J

where

K«f>(x, y) = ( r y ^ - M)aPS^{x - y).

* We assume the reader is familiar with the quantization of the free Dirac field.


The two-point function (fermion propagator) is related to the inverse of the matrixK as follows:

{Sl\T(*a(x)*0(y))\n) = iK$(x,y),

where the time-ordering operation "T" orders the operators from left to rightaccording to descending time, treating the operators â and ^>p as elements of aGrassmann algebra. But we have just learned above how to compute the inverseof a matrix by means of Grassmann integrals. Thus a naive application of theformulae (2.56) and (2.57) leads to the following Pi-expression for iK~l{x,y):

! . _ S D$rl>)Mx)MvVSFl*^tAap(x,y)- j D^ysrfr,®

where the measure is formally defined by

Dtyl,) = l[dja(x)d1>a(x),

and where ip and ip are Grassmann-valued fields. The above observation suggeststhat the Pi-representation of Green functions involving an equal number of Diracfields of type ty and ty is given in Minkowski space by *

<n|r(*Q1(a ; i) . . .*a<(x<)*ft(yi).. .*A(w))|n)

jD^)âA^--4at{xt)^{yi)--4pM)yiSFl*'^ (2-6°)

This is certainly true for the free Dirac field, as follows from a naive applicationof formula (2.59), which is nothing but Wick's theorem. But it is also true fortheories like QED or QCD, where the fermionic contribution to the action is again abilinear function in the fields ip and ip. In these theories, this fermionic contributionalso depends on a collection of bosonic variables (the gauge potentials) and thepath integral (2.60) gives the Green function evaluated in external gauge fields.When quantum fluctuations of these fields are taken into account, the appearanceof the determinant of the matrix A in (2.55) (rather than 1/Vdet A, which ischaracteristic of the bosonic case) will play a crucial role. This will become clearlater on, when we discuss these theories in detail.

* All other functions vanish because of the Grassmann integration rules. For aderivation of the path integral expression for fermions from fundamental principles,see Berezin (1966)


As the reader will have noticed, we have not discussed the path integral repre-sentation of fermionic Green functions continued to imaginary times. We shall dothis in chapter 4, using the rule derived in section 2 for Green functions involvingbosonic variables. In the case of the free Dirac theory, the correctness of this rulecan be checked explicitly by comparing the results obtained by the Pi-methodwith those derived using conventional canonical Hilbert space methods. In fieldtheories with interactions like QED or QCD, such a comparison can be made inperturbation theory. The non-perturbative definition of the correlation functionsin these theories is assumed to be given by the PI expressions.

We close this section with a remark. In contrast to the bosonic case, we cannotcalculate numerically "ensemble averages" of products of Grassmann variablesusing statistical methods. Nevertheless, we will still be able to study theories likeQED or QCD numerically. The reason is that, as we have just mentioned, thefermionic contributions to the action in these theories is bilinear in the fields ipand V>. This allows one to perform the Grassmann integrals and to recast thepath integral expression for the euclidean correlation functions in the form of astatistical mechanical ensemble average, with a new effective action. This actiondepends in a non-local way on the bosonic fields to which the fermions are coupled.It is this non-locality that makes numerical computations of correlation functionsinvolving fermions very time-consuming.

2.5 Discretizing Space-Time. The Lattice as a Regulator of aQuantum Field Theory

As we have pointed out repeatedly in the previous sections, the path integralexpressions for Green functions have only a well-defined meaning for systems witha denumerable number of degrees of freedom. In field theory, however, where oneis dealing with an infinite number of degrees of freedom, labeled by the coordi-nates x and, in general, by some additional discrete indices, the multiple integralsare only formally defined. To give the path integrals a precise meaning, we willtherefore have to discretize not only time, but also space; i.e., we will be forced tointroduce a space-time lattice. Eventually we will have to remove again this lat-tice structure. This is a quite non-trivial task. Those readers acquainted with therenormalization program in continuum perturbation theory know that the renor-malization of Green functions first requires the regularization of the correspondingFeynman integrals in momentum space. These integrals will then depend on oneor more parameters which are introduced in the regularization process (momen-tum cut-off, Pauli-Villars masses, dimensional regularization parameter). Since


the effect of any regularization procedure is to render the momentum integrationsin Feynman integrals ultraviolet finite, let us loosely say that the first step in therenormalization program consists in the introduction of a momentum cutoff. Ifthe original Feynman integrals are divergent, then the regularized integrals will bestrongly dependent on the cutoff. The second step in the renormalization programnow consists in defining renormalized Green functions, which approach a finitelimit as the cutoff is removed. This demands that the bare parameters of the the-ory become cutoff dependent. This dependence is determined by imposing a set ofrenormalization conditions, which merely state that such quantities as the physicalcoupling strength measured at some momentum transfer, and particle masses areto be held fixed as the cutoff is removed.

The above described renormalization program is carried out on the level ofFeynman integrals in momentum space. In the lattice approach this programcan be formulated without reference to perturbation theory. The first step (reg-ularization) consists in introducing a space-time lattice at the level of the pathintegral. This regularization merely corresponds to defining what we mean by apath integral. The second step of the renormalization program then correspondsto removing the lattice structure. This amounts to studying the continuum limit.It is therefore not surprising that the bare parameters of the theory will have to betuned to the lattice spacing in a very definite way depending in general on the dy-namics, if physical observables are to become insensitive to the underlying latticestructure. Thus if the reader had some uneasy feelings about the way the infinitiesare removed in conventional perturbation theory, he will probably feel much bet-ter after having read this book. In this connection we also want to mention thatwithin the perturbative framework the introduction of a space-time lattice corre-sponds to a particular way of regularizing Feynman integrals. As we shall see, thisregularization does not amount to the naive introduction of a momentum cutoff.Although the momentum space integrals will indeed be cut off at a momentum ofthe order of the inverse lattice spacing, the integrands of Feynman integrals willnot have the usual structure, but are modified in a non-trivial way. This is oneof the reasons why lattice weak coupling perturbation theory is so difficult. Theother reason is that in the lattice formulation of gauge theories, new interactionvertices pop up, which have no analogue in the continuum formulation.

The appearance of a momentum cutoff in the lattice formulation is not sur-prising. Consider a function f(x) of a single continuous variable. If its absolute


value is square integrable, then f(x) has the following Fourier representation:

/>OO 31

f(x) = / rRk)eikx. (2.61)

On the other hand, if x is restricted to a multiple of a "lattice spacing" a, i.e.x = na with n an integer, then f(na) can be Fourier-decomposed as follows:

f(na) = / rh{k)eikna, (2.62)J-K/a 2?r

where fa(—n/a) — fa(Tr/a). Hence the "momentum" integration is now restrictedto the so-called Brillouin zone (BZ) [—ir/a,n/d\. fa(k) can be represented by aFourier series. The coefficient of exp(—ikna) is given by (2.62) multiplied by a:

oo

fa{k) = a J2 f(na)e-inka. (2.63)n = —oo

The right-hand side is just the discretized version of the expression for f(k) ob-tained by inverting (2.61). By setting f(na) = 1/2TT in (2.63), we obtain a Fourierseries representation of the ^-function in the BZ,

Sp(k) = ^J2e~inka' (2-64)n

where the subscript P stands for "periodic". It emphasizes the fact that Sp(k)has non-vanishing support at k = 0 modulo 2mr. The Dirac ^-function, 5(x — y),of course becomes the Kronecker-<5 (multiplied by I/a) on the x-lattice:

Snm = afa ±jP(n-m)a _ (2 g5)J—it/a 2 7 r

The above formulae are trivially extended to functions depending on an ar-bitrary number of variables. In particular, in four space-time dimensions, all fourcomponents of momenta will be restricted to the interval [—7r/o,7r/a]. Thus theintroduction of a lattice provides a momentum cutoff of the order of the inverselattice spacing.

We are now ready to embarque on the main task of this book, i.e. the for-mulation of field theories on a space-time lattice. As a warm-up, we begin in thefollowing chapter with a very simple field theory: the free scalar field. Althoughthe lattice formulation will be trivial in this case, we will nevertheless learn anumber of important facts by studying it in detail.

CHAPTER 3

THE FREE SCALAR FIELD ON THE LATTICE

Consider the classical field equation

(• + M2)4>(x) = 0, (3.1)

where 0 is a real field, • is the d'Alembert operator, and x stands for the space-time vector with components x^ji = 0,1,2,3). This equation of motion followsfrom an action principle, 5S = 0, where

S=-±Jdix<j>(x)(n + M2)4>(x) (3.2)

is the action associated with the Lagrangian density

c = \d^d»4> - \M24? .

In the quantum theory the coordinates qg(t) = 4>{x) and momenta ps(t) = 4>{x)become operators, $(x) and $(x), satisfying canonical commutation relations.The information about the quantum theory is contained in the Green functions

G ( i , y - ) = <n|r($(a;)$(i/)...)|n), (3.3)

where |fi) stands for the ground state of the system (physical vacuum) and Tdenotes the time-ordered product of the operators $(x). These Green functionshave a path integral representation which can be formally obtained from (2.33) bymaking the replacements Qa(t) -> $(x,t) and qa(t) -> <f>{x,t):

^{x,y, ) - j D ( f > e i S . (SA)

Here J D<f> denotes the sum over all possible field configurations <j>{x). The effectsarising from quantum fluctuations are contained in those contributions to the inte-gral (3.4) coming from field configurations which are not solutions to the classicalequation of motion (3.1) and hence do not lead to a stationary action. Now forreasons mentioned in chapter 2 we are interested in the analytic continuation of(3.4) to imaginary times, a;0 —> —ixi,y° —> —ij/4, etc. Let, from now on, x andy denote the euclidean four vectors with components £M and y^ (fi = 1,...,4).

36

The Free Scalar Field on the Lattice 37

It then follows from our discussion in chapter 2 that the Green functions (3.3)continued to imaginary times have the following path integral representation

(0(1)0(1/)...) = fDfc-sB[4,] ' ^3-5)

where we have made use of the notation for the euclidean Green functions, in-troduced in chapter 2. The euclidean action SE[] appearing in (3.5) is obtainedfrom (3.2) by i) making the replacement x0 -> -ixi where-ever x° appears explic-itly, ii) substituting for <f>(x,t) the (real valued!) field <j>{x) = (j>(x,Xi)* and iii)multiplying the resultant expression by —i. This leads to the following expressionfor SE[4>],

SE[4>] = \ l d4x<f>{x)(-a + M2)<j>(x), (3.6a)

where • denotes from now on the 4-dimensional Laplacean

4

In passing to the imaginary time formulation, the Green functions take the formof correlation functions of a statistical mechanical system defined by the partitionfunction

Z= [ D<j>e-S[<t>],

where the integration measure D4> is formally defined by

D4>= Y[d<f>(x,x4).

So far the path integral (3.5) has not been given a precise mathematical meaning.We do this now by introducing a space-time lattice with lattice spacing a. Everypoint on the lattice is then specified by four integers which we denote collectivelyby n = (711,712, 3,714). By convention the last component will denote euclideantime. The transition from the continuum to the lattice is then effected by making

* We are a bit sloppy in our notation: cp(x, X4) is not obtained from <p{x, t) bysubstituting £4 for t, but denotes a real field which is a function of the euclideanvariables xfl{^= 1,..., 4).

(3.66)


the following substitutions:

<j>{x) - > 4>(na) ,

J n (3.7a)

D</>(x) ->• — O(p(na),

D(f)-^Yl d<f>{na) ,n

where the action of the dimensionless lattice Laplacean D is defined by

Ocj)(na) = ^ (<j>(na + jia) + (f>{na - fia) - 2<f>(na)). (3.76)

Here jx = eM, where eM is a unit vector pointing along the /z-direction.

We next want to obtain a path integral expression involving dimensionlessvariables only. To this effect we scale the mass parameter M and the field cj>according to their "canonical" dimension. As seen from (3.6a) n by

<j>n = a<f>(na),(o.o)

M = aM.

With (3.7) and (3.8) expression (3.5) translates into

where5B = - i j ] 0 n 0 n + M + -(8 + M 2 ) ^ 0 n 0 n , (3.96)

n,p, n

and where the sum over fi extends over all positive and negative directions. No-tice that the lattice spacing no longer appears in these expressions! This is notparticular to the free field theory considered here, for it is merely a consequenceof the fact that, measured in units of h, the action is dimensionless, which is truefor any theory.

It is important to realize that the form of the lattice action (3.9b) is notunique, and that we have merely chosen the simplest one. Thus, a priori, the only

(3.9a)


requirement that any lattice action should fulfil is that it reproduces the correctclassical expression in the naive continuum limit*. Indeed, the scalar field is theonly case in which a simple prescription of the above type gives the correct latticeaction describing the quantum theory. Already the free Dirac theory will requirea more careful treatment!

Let us now consider the integral (3.9a) in more detail. Its structure is analo-gous to that encountered in the statistical mechanics of a spin system with nearestneighbour interactions. In the present case, however, the theory is easily solvedsince the variables <f>n are allowed to take on any real value. To carry out theintegral (3.9a) we rewrite the action (3.9b) in the form

SE = g 2j< i-Knm<Am, (3.10a)

where Knm is given by

Knm = - ^ [ < W , m + <5n_A,m - 2<5nm] + M2Snm. (3.106)

Consider the generating functional

Z0[J]= [Hdfae-^+ZJ^. (3.11)

It can be easily calculated, since the (multiple) integral is of the Gaussian type.Apart from an overall constant, which we shall always drop since it plays no rolewhen computing ensemble averages, we have that (cf. e. g. (2.28) and (2.30)

Z0[J} = =L=e*^J"K™J™ . (3.12)V det K

Here K"1 is the inverse of the matrix (3.10b), and det K is the determinant ofK. By differentiating (3.12) with respect to the sources we obtain any desiredcorrelation function. For our purpose it suffices to consider the 2-point function.From what we have learned in chapter 2, we get

<0»0m> = -K^m • (3-13)

The inverse matrix K~x is determined from the equation

Y,KntKZk=Snm, (3.14)i

* i.e. scaling the variables with a appropriately, and letting a. -+ 0.


and is easily computed by working in momentum space, where 5nm is given by

_ r d4k jfc,(n_m) .nm~]-A^Y ' ( ]

We have introduced the "hat" on £ = (k\,..., £4) to emphasize that these variablesare dimensionless. Making use of the Fourier representation (3.15), one finds that(3.10b) is given by

/

"" jit.— K0c)eik<n-m\ (3.16a)

where4 f

#(£) = 4^sin2-^+M. (3.166)

Notice that the integration in (3.16a) is restricted to the Brillouin zone (BZ),—-K < fcM < 7T. The inverse matrix (3.13) is now easily determined from (3.14) bymaking the ansatz

K-L = £ (0G(*)e*<»-™> ,and performing the sum over £ using the expression (2.64) with (o = 1) for theperiodic delta function:

f" rlAh pik-(n-m)K^ = (Mm)= ^ e—- . (3.17)

The right-hand side of this expression depends on the lattice sites n and m, andon the dimensionless mass parameter M. To make this explicit let us define

G(n,m;M) =< 4>n^m > .

Suppose we were given (3.17) and were asked to study its continuum limit in orderto extract the physical two-point correlation function, (<p(x)(f>(y)). The obviousthing to try would be to introduce the lattice spacing by rescaling <f>n and Maccording to (3.8), and to take the limit a -»• 0, holding M, <j), x = na andy = ma fixed. Hence one must know which quantities are to be held fixed asone removes the lattice structure! For example, in an interacting theory the massparameter M would in general be unphysical and cannot be held fixed as we let thelattice spacing go to zero. In the present simple case, however, the naive procedure


just described gives the correct continuum limit; i.e., we claim that the right-handside of

(0( i )0( j / )>=l im^G(^^Mo) (3.18)

approaches a finite limit, and reproduces the well-known result for the scalar two-point function. For this to be the case, G(x/a, y/a; Ma) must clearly vanish in thelimit a -* 0. From (3.17) one finds after a trivial change of integration variablesthat

/TV \ „ fn/a f)4k Pik-(x-y)G(-J;Ma)=a? 44j-^ (3.19a)

where fcM is given by

fcM = - s i n ^ . (3.196)

Since the integration in (3.19a) is restricted to the interval [— , £], the integralwill be dominated by momenta which are small compared to the inverse latticespacing; hence we may set fcM —> fcM. Taking the limit a —>• 0 we arrive at thewell-known result:

f°° d4k eik'(x~y}

M'MM-LwvTMi • (3-20)

The above discussion has made explicit use of the lattice spacing. But supposewe were not in the position of performing the continuum limit analytically, butmust rely on a numerical calculation of the path integral (3.9) where the latticespacing does not appear. What does it mean to study the continuum limit insuch a case? The idea is of course to make the lattice finer and finer with physicsremaining the same as we approach the continuum limit. Consider for examplea physical correlation length £. Decreasing the lattice spacing means increasingthe correlation length £ measured in lattice units. But £ may be controlled bythe parameters on which the theory depends! Thus, doubling £ by choosing theseparameters appropriately, we have cut down the lattice spacing by a factor of 1/2.Now what controles the correlation length £ in our example is the dimensionlessparameter M. In the continuum limit the correlation function (3.20) decays ex-ponentially for large \x — y\ with a correlation length given by the inverse mass.Hence the corresponding correlation length measured in lattice units, i.e., £ = -~^,diverges as a -> 0! Thus from the point of view of a statistical mechanical sys-tem described by the partition function (3.11) with J — 0, the continuum limitis realized for M -> 0 at a critical point of the theory! It is therefore evidentthat in any practical numerical calculation carried out on a finite-size lattice we


can never actually go to the continuum limit. How do we then decide whetherwe are extracting continuum physics? In principle the answer is very simple: Wemust ensure that our lattice is fine enough (i.e. M small enough in the presentcase) so that physical quantities become insensitive to the lattice structure. Butquantities like {(f>(x)<f>(y)) are dimensioned, and we can only calculate dimension-less objects! So we must consider dimensionless ratios of physical quantities. Inour case the simplest quantity is (4>(x)(f>(y))/M2; hence we must study the latticeratio {(j>nm)IM2 for small values of M, keeping M\n — m\ « M\x — y\ fixed. Iffor sufficiently small M this ratio becomes independent of M, then our lattice isfine enough and we are extracting continuum physics.

Admittedly the case of a free scalar field was a very simple one. Neverthelessit has served to elucidate several ideas that go into a lattice formulation. In aninteracting theory the story will be certainly more complicated. But the mainmessages of this chapter will remain.

CHAPTER 4

FERMIONS ON THE LATTICE

In the preceding chapter we have shown that the lattice formulation of the freescalar field theory poses no problems. The correct continuum limit was reached bysimply scaling all dimensionless variables appropriately with the lattice spacing a,and taking the limit a —> 0 holding physical quantities fixed. The purpose of theselectures, however, is to arrive at a lattice formulation of QCD which describesthe interaction of quarks and gluons. Hence we must learn how to deal withfermions and gauge fields on a lattice. While there is a clear-cut and elegantway of introducing gauge fields on a space-time lattice, the situation regardingfermions is not so clear. As we shall see, the difficulties arise already on the levelof the free Dirac field. From the psychological point of view it would thereforebe preferable to discuss that part of lattice gauge theories first which one believesto be well understood, and to introduce lattice fermions at a later stage, sincethey are endowed with special problems not encountered for bosonic fields. Onthe other hand, having discussed the scalar field, it is only natural to attempt asimilar naive formulation for the other kind of matter field. Thus it is interestingthat in the case of fermions the lattice forces us to deviate from the naive typeof prescription adopted in the previous chapter, in order to avoid the so-calledfermion "doubling" problem. Several proposals have been made in the literatureto get around this problem, and we shall discuss the two most popular ones.

4.1 The Doubling Problem

We begin by pointing out the difficulties one encounters when latticizing thefree Dirac field.

Consider first the Dirac equation in Minkowski space

(i-y'idli - M)i>{x) = 0,

where 7M are 4 x 4 Dirac matrices satisfying the following anticommutation rela-

tions

and ip is a 4-component field, whose components we shall label by a Greek in-dex (a, (3, etc.). The equation of motion for ip and ip(= tp^-y0) follow from theindependent variation of the action

SF[il>,$\ = f d4xip(x)(i^d>1 - M)1>(x)

43


with respect to the fields xjj and tp. In the quantum theory, tp and ip^ becomeoperators, \t and ^ satisfying the following canonical equal-time commutationrelations

{*„(£,£), **(£*)} = Ja /3^3)(f - y).

The path integral representation of the Green function *

(n\T(*ai{x1)...*ai(xt)*pM...9Pl(yt))\n)

is given by (2.60), i.e.

The corresponding representation of the Green functions continued to imaginarytimes is obtained by replacing i5^[^,^] by — SpC [ip,ip], where SpUC '' is theeuclidean action, and identifying x, y etc. with the euclidean four-vectors. Wedenote the euclidean Green functions by (if)a(x)...'i(!p(y)...). Then

fDipDi{>e-SF MM

The euclidean action can be obtained from SF by the prescription discussed insection 2 of the previous chapter. But since in euclidean space the Lorentz groupis replaced by the rotation group in four dimensions, it is convenient to express theaction in terms of a new set of 7-matrices 7^'(M = li ••-, 4), satisfying the algebra

With the hermitean choice 7^ = 7°,jf = —ijl, the euclidean action then takesthe form

4™c'-> = I dAx4>(x){rfd» + M)rP(x). (4.2)

Since from now on we shall be interested only in the euclidean formulation, weshall drop any labels reminding us of this.

So far the path integrals in (4.1) are only formally defined, since a;, y etc. arecontinuous variables. So let us introduce a space-time lattice. The fields ip and ip

* Here Xi, yi (i = 1, ...,£) denote four vectors in Minkowski space.

(4-1)

Fermions on the Lattice 45

then live on the lattice sites na, where a is the lattice spacing, and the integrationmeasure is given by

DtpDip = Y\dil>a(na) J J dipp(ma).a,n f3,m

Next we rewrite (4.1) in terms of dimensionless lattice variables, by scaling M,ip

and ip with a according to their canonical dimensions. This is achieved by makingthe replacements

M ->-M,a

a{ _. (4.3a)

1

where d^ is the antihermitean lattice derivative defined by

dMn) = 2$a(n + A) - ^ a ( " - A)] • (4.36)

Then the lattice version of (4.2) reads

SF = 2 j ô:(n)Ka0(n, m)ipf}(rn), (4.4a)n,m

where

Kap(n,m) = ^2 2^M)a/3[5m,n+A ~ <W-#] + M5mn5ap. (4.46)

With this action the lattice correlation functions are given by the followingpath integral expression

(t / ; a (n)- . -^(m)---) = J v * vay_) jpy j ^ ^f Dtp Dip e~sF

where the integration measure is defined by

Dij>DJ> = Y[dj>a{n)]ld$fl(rn). (4.56)n,a m,/3

(4.5a)


The correlation functions (4.5a) can be obtained from the generating functional

Z[V, fj]= f D^Di>e-SF+^J^{n)^{n)+în)TI''in)] (4.6)

by carrying out the appropriate differentiations with respect to the Grassmann-valued sources r)a(n) and fja(n) (see chapter 2). The integral (4.6) can be per-formed (cf. eqs. (2.49) and (2.50)) and we obtain

r?l -l j ± rs ^ „ VcMK~l(n,m)ria(m.)

Z[r),ri\ = det Ae^»,»,«j ' ««v IPK '.

Hence the two-point function is given by

(^Q(n)^(m)) = /f-/j1(n)m).

So far everything is quite analogous to the scalar case considered in the previouschapter. In particular we want to emphasize that the lattice action (4.4) wasobtained by proceeding in the most naive way possible. Such a prescription wasshown to work in the case of the free scalar field. Hence there is no a priori reasonwhy it should fail to do so in the present case. But the fact is that it fails! To seethis let us compute the physical correlation function {ipa{

x)i>p{y)) by carrying outthe continuum limit in a manner analogous to the scalar case (cf. eq. (3.18)), i.e.

(il>a(x)fo(y)) = Hm ~GaP (^, ^Ma) ,

where Gap(n,Tn,M) = K~^(n,m). The factor I/a3 arises from scaling the fieldsaccording to (4.3a). The inverse matrix K~}(n, m), defined by

^ 0 - > , < ) % ( < , m ) = Saf}5nm,

can be easily calculated by proceeding as in the previous chapter, and one obtains

where pM is given by1

Pu, = -sin(pâ). (4.7b)

For pM —>• pM the above integral would reduce to the well-known 2-point functionin the limit a —> 0. Recall that in the scalar case, we had encountered a similar

(4.7a)


situation, but with an all important difference! The argument of the sine-function

in eq. (3.19b) is only half of that in (4.7b)! This makes a big difference and is the

origin of the so-called "fermion doubling" problem. While in the case of the scalar

field we could argue that fcM in (3.19a) can be replaced by fcM in the continuum

limit, such a replacement cannot be made in the present case. The reason for

this is most clearly seen by looking at fig. (4-1) where we have plotted p^ as a

function of pM, for pM within the Brillouin zone. The straight line corresponds to

p^ = pp. Within half of the BZ the situation is analogous to that encountered

in the scalar case: near the continuum limit, the deviation from the straight line

behaviour occurs only for large momenta where p^ and p^ are both of order I/a.

Fig. 4-1 Plot of sin(pMa)/a versus p^ in the Brillouin zone. The

straight line corresponds to jL = p^. The continuum limit is determined

by the momenta in the neighbourhood of p^ = 0 and p^ = ±7r/a.

What destroys the correct continuum limit in the fermionic case are the zerosof the sine-function in (4.7b) at the edges of the BZ. Thus there exist sixteenregions of integrations in (4.7a), where pM takes a finite value in the limit a —> 0.Of these, fifteen regions involve high momentum excitations of the order of n/a(and -rr/a), which give rise to a momentum distribution function having the formresembling that of a single particle propagator. Hence in the continuum limit, theGreen function (4.7a) receives contributions from sixteen fermion-like excitationsin momentum space, of which fifteen are pure lattice artefacts having no continuumanalog. In d space-time dimensions the number would be 2d; i.e. it doubles foreach additional dimension.

The "doubler" contributions, arising from momentum excitations near thecorners of the Brillouin zone, are in fact essential for avoiding an apparent clash


with a well known result in continuum QED. For vanishing fermion mass the QED

action is invariant under the global chiral transformation

ip - > ei97B-ip ; I/J ^njjei9~/t' , (4.8)

where 6 is a parameter, and 75 = 71727374 is a hermitean matrix which anticom-

mutes with 7^ (fi = 1,2,3,4). Naively this implies the existence of a conserved

axial vector current. But because of quantum fluctuations this current has actu-

ally an anomalous divergence [Adler (1969); Bell and Jackiw (1969)]. In a lattice

regularized theory, on the other hand, such a symmetry implies that this current

is strictly conserved for any lattice spacing. The way the lattice resolves this ap-

parent puzzle, consists in generating extra excitations (—> doublers) that have no

analog in the continuum, and which cancel the anomaly of the continuum theory

arising from momentum excitations around p = 0 [Karsten and Smit (1981)].

Since the phenomenon of fermion doubling is a serious stumbling block in

constructing lattice actions involving fermions, we will look at it in more detail in

the following section.

4.2 A Closer Look at Fermion Doubling

Before we discuss the lattice Dirac propagator in more detail, it is instructive

to demonstrate the essence of the fermion doubling problem in some simple exam-

ples. In particular we want to show that the origin of fermion doubling lies in the

use of the symmetric form for the lattice derivative.

i) Example 1

Consider the following eigenvalue equation

-*£/(*) = vw •The solution is given by f\(x) = f\(0)elXx. Next consider the discretized version

of this equation, where the derivative is replaced by the right "lattice" derivative.

Let /(n) be the value of f(x) at the lattice site x = na, where a is the lattice

spacing. Then

- » [ / ( ( n + l ) ) - / ( n ) ] = A/(n),

where A is the eigenvalue measured in units of the lattice spacing, i.e. A = Xa.

The equation can be solved immediately by iteration:

/ » = e"ln(1+iX)/x(0)


In the continuum limit, which is obtained by setting n = ^, A = \a, and taking

the limit a -> 0 with A fixed, we recover the above solution.

Let us now consider a discretization which respects the hermiticity of the

operator igj . This requires the use of the symmetric lattice derivative.

— [/(("+ l ) ) - / ( ( n - l ) ) ] = A/(n). (4.9)

Thus our estimate of the derivative now involves twice the lattice spacing! As aconsequence one finds that for each eigenvalue A there exist two solutions to theeigenvalue equation (4.8). Not both of the solutions can possess a continuum limit,since the continuum eigenfunctions are non-degenerate. Indeed, equation (4.8) canbe solved with the Ansatz

f-x{n) = Ce^n ,

where p satisfies the equationsinp = A .

For a given positive (negative) eigenvalue A this equation possesses two solutionsfor p: one lying in the range 0 < p < ^ (—| < p < 0), and the other in the interval!<p<7r (—TT < p < — | ) . The corresponding eigenfunctions are given by

fY){n) = Aei{aTcsin^n , (4.10a)

ff\n) = B{-l)ne~i{-arcsin%)n , (4.106)

where

- - < arcsinA < - . (4.10c)

The solution (4.10a) possesses a continuum limit which is realized by setting n =x/a, A = Xa, and taking the limit a —> 0 with A fixed. In this limit we recover thesolution to the original continuum eigenvalue equation. On the other hand (4.10b)does not possess such a limit because of the factor (—1)™ which alternates in signas one proceeds from one lattice site to the next. Notice that the origin of the"doubler" solution (4.10b) is a consequence of having used the symmmetric formfor the lattice derivative. One might be tempted to merely ignore this solution. Inthis case, however, our eigenfunctions would no longer constitute a complete set.

ii) Example 2

Consider the Green function for the differential operator ^ + M:

[jt+M}G(t,t') = 6(t-t'). (4.11)


The general solution is given by

G(t, t') = Ae-W-*') + 6(t - t')e~M<^t-t"> . (4.12)

Next consider the dimensionless discretized version of (4.11), where the derivative

is replaced by the symmetric lattice derivative:

X ^ O W . " ' ~ -i."') + M6nn,]G(ri,m) = 5nm . (4.13)n'

Here M is the "mass" M measured in lattice units, i.e., M = Ma. The most

general solution to the homogeneous equation reads:

G{^(n,m) = Ae-(-n-m)arsinhM + g^y-me(n-m)arsinhM _ ^ ^

Thus, when discretized, the homogeneous equation has an aditional solution which,

because of the factor (—l)n~m, possesses no continuum limit. This limit is realized

for M —> 0, n, m -> oo with Mn = Mt and Mm = Mt' fixed. The homogeneous

solution to (4.11) is then seen to correspond to the first term appearing on the rhs

of (4.14).

A particular solution to the inhomogeneous equation (4.13) can be obtained

by making the Fourier Ansatz

G%art\n,rn) = f^G(p)e^-™->.

Introducing this expression into (4.13) we obtain

G%art\n,m)= fe. . . • (4-15)M J-Trîsmp + M

The integral can be easily evaluated by introducing the variable z — e1?. Then

i f ~n—mGly '(n,m) = — / dz ^ ,

M TT« Jc z2 + IMz - 1

where integration is carried in the counterclockwise sense along a unit circle in the

complex z-plane centered at z = 0. For n — m > 0 the integral is determined by

the residue of the pole at z = —M + v 1 + M2. On the other hand, for n — m < 0


we can distort the integration contour to infinity, taking into account the pole at

z = -M - \ / l + M2, located outside the unit circle. One then finds that

p — (n—m)arsinhM p(n—m)arsinhMG%art)(n,m) = 9(n - m) =- + (-l)n-m9(m - n)

V(l + M2) y/(l + M2)

(4.16)

where 8(0) — | . Note again, that only the first term on the rhs possesses a

continuum limit. For lattice spacings small compared to JJ, i.e. for small M, and

for fixed t = na and t' = ma, we have that

G%art\-, - ) « 0(t - 0 e - M ( ' - ° + (-l)^fl(t ' - t)eM^-^ . (4.17)

While the second term is not defined for a ->• 0, the first term reproduces the

inhomogeneous solution to (4.11), given by the second term in (4.12).

The expression (4.17) could also have been obtained as follows, without actu-

ally carrying out the integrals. Let us set p — pa, M = Ma, and t = na, t' = ma

in (4.15). Then

MPart)(t t ' _ fi dp e W - ' ' >

For a — 0 the relevant contributions to the integral come from momenta for which

sin pa w O(a), i.e. from i) finite (dimensioned) momenta p, and ii) momenta close

to the corners of the Brillouine zone. Let us therefore decompose the integral

(4.18) as follows:

d(Part)(t_ t' = / •* dp e»(*-''> ft dp e^C-'')

* l a ' a j J_£_2n isinpa + M J % 2TT i sinpa + M

r ^ dp e^-t")

J-iL 2TT i sin pa + M

Making the change of variables p = ~ +p' and p = - J +p' in the last two integrals,

respectively, one finds that

d(vart){t_ t' = f^ dp e'Pt*-*') | ^ j /"ft dp e^C-*')

^ a ' a j 7_^_ 2?ri sin pa + M " 7 _ ^ 2?r - i sinpa + Af '

The integrations now extend over only one half of the Brillouin zone. For a -» 0

the two integrals are therefore dominated by finite momenta p, for which sinpa is

of O(a). Hence for a —» 0 we can replace sinpa by pa and obtain

G(vart)(i t \ _^ [°° dp e**'-*') _ ^ p dpe»(«-''){a' a'aôj_00 2-Kip + M l j J^ 2TT ip - M '

(4.18)


which can be readily integrated to yield (4.17).

The simple examples we discussed show how the discretization of an equationcan lead to a doubling of solutions. The "doubler"-like solutions manifestatedthemselves in the appearance of a phase factor which changes sign as one proceedsfrom one "lattice" site to the next. This is not only peculiar to the above examples.In fact that doubler contributions to the Dirac propagator (4.7a) are expected tomanifest themselves in the same way. This we will show below.

Hi) Fermion Propagator

In the case of the fermion propagator one is confronted with matrix valuedintegrals over four dimensional momentum space. The characteristic structureof the "doublers" contributions to the Green function can nevertheless be easilyexhibited by proceeding in the way we have just described. Our starting pointis the expression (4.7a). To exhibit the effect of fermion doubling we decomposeeach of the momentum integrations into the following two regions

i) IPMI < Ya

and

n) £ < W < I - (4-19)After changes of variables similar to those we made before, the reader can easilyconvince himself that the two-point function (4.7a) can be written in the form

(4.20a)

where x = na, y = ma, and5Pti = eip" . (4.206)

The sum in (4.20a) runs over all possible sets of four-momenta p (measured inlattice units), labeling the 16 corners of the hypercube in the first quadrand inmomentum space: (0,0,0,0), (n, 0, 0, 0)---, (TT, TT, 0, 0)--, (TT, IT, IT, 0)---, (TT, TT, 7r, TT).The dots stands for all possible permutations of the components. Notice that allthe integrations in (4.20a) extend over the reduced Brillouine zone [—55, j^]- Ofthe sixteen terms in the sum, only the term corresponding to p = (0, 0,0,0) yieldsthe familiar result in the continuum limit. While the remaining 15 integrals alsopossess a continuum limit, the phase factor exp[ip-(n — m)} does not. The structureof these fifteen integrals is the same as that corresponding to p — (0,0,0,0),


except that each term is the Dirac propagator in a different representation of thegamma-matrices. Thus in each of the integrals the sign of the gamma matrix 7M

is reversed if pM = n. The new set of gamma matrices are related to the standardset by a similarity transformation. Let us denote the matrices which induce thesetransformations by Tp:

Tp^Tfl = Sp^ . (4.21)

They have a different structure depending on the number of non-vanishing compo-nents of p. Let Tp, TpV, ffiux, %uXP (all indices distinct) stand for the matrices Tpwhich induce a reversal of signs of those 7-matrices corresponding to the set of sub-scripts. Their explicit form is given by % = 7^75, %v = Iplu, %v\ = iplulMb,and TpvXp = 75, where 75 is the hermitean matrix: 75 = 71727374- By the samereasoning as that given in example ii), the propagator is found to take the followingform close to the continuum limit:

F (4.22a)= Y,Vp{x)Sf{x-y)Vp-\y),

P

whereVp(z) = eip-iTp , (4.226)

and

is the continuum Feynman propagator.

The above structure of the correlation function is intimately connected with asymmetry of the lattice action. Indeed one readily verfies that the gamma matricessatisfy the following relation,

As a consequence the lattice action

S = - ] T aîx^îx + fia) - i>{x - £a)] + M ^ ai%[>(x)ip{x)x x

is invariant under the transformation

i/>(x) -> Vp{x)i>{x) ,

(4.22c)


In momentum space the action of the operator Vp corresponds to a shift in the mo-menta by p. Corresponding to the sixteen edges of the BZ there exist sixteen suchsymmetry transformations. The fermion doubling phenomenon is a consequenceof the existence of these symmetry transformations.

Finally, let us take a look at how the doublers manifest themselves in the naivelattice fermion two-point function for a massless field with definite chirality (as isthe case for the neutrino). This field is an eigenstate of the projection operatorPL — | ( 1 — 75)- The two point function is given by

L(x)i>Uy) >= PL < ip{x)i>{y) > PR

where iphin) — P f0( n ) ; a n d PR = | ( 1 + 75). It therefore has the form

<M^L(y)>=PLJ^y4ê^^ ,

where we have made use of the fact that 75 anticommutes with the 7-matrices 7M,and that P£ = 1. On the other hand the corresponding lattice version is given by

(4.23)

Now one can easily verify that

Tp(l - 75) V 1 = (1 - ep-75) , (4.24a)

where

ep = n5PM • (4-246)

Hence we can also write (4.23) in the form

< M*)MV) >= E**-^!/* 7^^(l-^75)^^#e-(^)] V1 .(4.25)

Since there are eight momenta p for which ep = 1, and eight momenta p for whichep = — 1, it follows that the sum involves sixteen integrals which in the continuumlimit have the form of eight left handed and eight right handed correlators in


different representations of the 7-matrices. Fifteen of these integrals are however

multiplied by phase factors characteristic for the doubler contributions.

As we have already stressed, the origin of the doubling problem lies in the

use of the (antihermitean) symmetric form for the lattice derivative (4.3b). Thus

while our lattice scale is a, our estimate of the derivative involves twice the lattice

spacing. By using the right derivative

d*i>(n) = i>(n + p) - i>(n) , (4.26a)

or left derivative

dj${n) = ^(n) - i>(n - p.) , (4.266)

our estimate of the derivative would involve a distance which is just the lattice

spacing and the doubling problem would not occur. In this case the hermitean

conjugate of d^ would be —dj}. A detailed analysis however shows that in the

presence of interactions the use of the left or right derivative gives rise, for example,

to non-covariant contributions to the fermion self energy and vertex function in

QED which render the theory non-renormalizable. *

That the doubling phenomenon must occur in a lattice regularization which

respects the usual hermiticity, locality and translational invariance requirements,

follows from a theorem by Nielsen and Ninomiya [Nielson (1981)] which states

that, under the above assumptions, one cannot solve the fermion doubling problem

without breaking chiral symmetry for vanishing fermion mass.** This suggests

that one may get rid of the doubling problem at the expense of breaking chiral

symmetry explicitly on the lattice. A proposal in this direction was made originally

by Wilson (1975), and is one of the two most popular schemes dealing with the

doubling problem.

* A very detailed study of the precise form of these non-covariant contribu-

tions has been carried out by Sadooghi and Rothe (1996). Actions using one side

lattice differences have been considered in the literature before, where by a suit-

able averaging procedure the correct continuum behaviour of the quantum theory

is restored. The reader may confer the references cited in the above mentioned

paper.

** For a simple derivation of the theorem, based on the Poincare-Hopf index

theorem, we refer the reader to the book by Itzykson and Drouffe (1989).


4.3 Wilson Fermions

As we have emphasized in the last chapter, there are many different lattice

actions which have the same naive continuum limit, and we have merely chosen

the simplest one. We now exploit this ambiguity to modify the action (4.4) in

such a way that the zeros of the denominator in (4.7a) at the edges of the BZ are

lifted by an amount proportional to the inverse lattice spacing. This appears to be

a perfectly legitimate procedure. What will, however, be particular to it is that,

while one usually constructs the lattice action in accordance with the symmetries

of the classical theory, one is forced to break explicitly the chiral symmetry which

the original theory possesses for vanishing fermion mass. This is the price one

has to pay to eliminate the fermion doubling problem and to ensure the correct

continuum limit.

Let us now modify the action (4.4) by a term which vanishes in the naive

continuum limit and is not invariant under chiral transformations. As we shall see

below a second derivative term is a good candidate. Thus consider the action

S{P = SF- V-Y,kn)ai>(n), (4.27)n

where r is the Wilson parameter and D is the four-dimensional lattice Laplacean

denned by (3.7b) with a = 1. Setting ip = az/2ip and D = a2D, we see that the

additional term in (4.27) vanishes linearly with a in the naive continuum limit.

Inserting for Di/i(n) the expression analogous to (3.7b), the Wilson action can be

written in the form

4W) = J2^c,(n)K^\n,m)Mm), (4.28a)n,m

where

KS\n' m) =(M + 4r)5nm5a0

1 V^r, ^ , / N r 1 (4.286)- - 2_J.\r ~ 7/x)"/3<W+M + ( r + 'Jv,)a0(>m,n-p,\-

Notice that for r ^ 0 this expression breaks chiral symmetry even for M — 0.

The action (4.28) leads to the following two-point function of the continuum

theory

(4.29a)


where p^ is given by (4.7b) and

M{p) = M + — J2 sin2 O W 2 ) • (4-296)

Prom (4.29b) we see that for any fixed value of p^, M(p) approaches M for a ->• 0.Near the corners of the BZ, however, M(p) diverges as we let the lattice spacinggo to zero. This eliminates the fermion doubling problem, but at the expense thatthe chiral symmetry of the original action (4.4) for M — 0 has been broken. Thismakes this scheme less attractive for studying such questions as spontaneous chiralsymmetry breaking in QCD (which requires a fine tuning of the parameter M).So let us turn to the discussion of an alternative scheme for putting fermions onthe lattice, known in the literature as the staggered fermion formulation (Kogutand Susskind (1975); Susskind, 1977; Banks et al. 1977). In contradistinctionto Wilson fermions one then speaks of Susskind, Kogut-Susskind, or staggeredfermions.

4.4 Staggered Fermions

As we have seen above, the fermion doubling problem owes its existence tothe fact that the function (4.7b) vanishes at the corners of the Brillouin zone. Thissuggests the possibility of eliminating the unwanted fermion modes by reducingthe BZ, i.e. by doubling the effective lattice spacing. This could in principle beaccomplished if a) we were able to distribute the fermionic degrees of freedomover the lattice in such a way that the effective lattice spacing for each type ofGrassmann variable is twice the fundamental lattice spacing, and b) if in thenaive continuum limit the action reduces to the desired continuum form. Hencelet us first take a look at the number of degrees of freedom required to double theeffective lattice spacing. To this end consider a d-dimensional space-time latticeand subdivide it into elementary d-dimensional hypercubes of unit length. At eachsite within a given hypercube place a different degree of freedom, and repeat thisstructure periodically throughout the lattice. Then the effective lattice spacinghas been doubled for each degree of freedom. In fig. (4-2) we show the case of a2-dimensional lattice. Since there are 2d sites within a hypercube, but only 2dl2

components of a Dirac field (in even space-time dimensions) we need 2d/2 differentDirac fields to reduce the BZ by a factor of 1/2. In four space-time dimensions sucha prescription may therefore be appropriate for describing 22 different "flavoured"(i.e. "up", "down", etc.) quarks. The corresponding Dirac fields we denote byi\)'a, where / denotes the flavour and a the spinor index. The concrete realization


of the above program is, however, not as simple as it sounds. Thus the sites of anelementary hypercube will be occupied by certain linear combinations of the fieldsip^ chosen in such a way that the lattice action reduces in the naive continuumlimit to a sum of free fermion actions, one for each of the quark flavours:

S(stag) ^ I dix £ â(x)(7fld^ + M)a^(x) . (4.30)a,0,f

In a staggered-fermion formulation much of the work goes into the construction ofthe quark fields from the different one-component fields populating the lattice siteswithin an elementary hypercube, and into the study of the lattice symmetries.* Inthe following we briefly describe the main steps involved in arriving at a staggeredfermion formulation, so that the reader will be acquainted with some expressionsoccurring frequently in the literature. Our presentation will essentially follow thework of Kluberg-Stern, Morel, Napoly and Petersson (1983). The technical detailsare relegated to the next section.

Fig. 4-2 Distributing 2 degrees of freedom on a two dimensional (d=2) lattice.

Consider the naive action (4.4) for a free Dirac field on the lattice:

S=\ Z$(n)7^(n + A) - knhMn - A)] + M£^(n)^(n). (4.31)n,/i n

By making a local change of variables

i,(n) =T(n)X(n),

ij{n) =x(n)T*(n),

* See e.g., Kluberg-Stern et al. (1983), Golterman (1986a), Jolicoeur, Morel,and Petersson (1986), Kilcup and Sharpe (1987).

(4.32)


where T(n) are unitary 2d/2 x 2d/2 matrices, one can "spin diagonalize" this ex-pression by choosing the matrices T(n) in such a way that

rt(n)7 MT(n + A)=r?M(n)H, (4.33)

where ??M are c-numbers (notice that different space-time points are involved!), and11 is the unit matrix. The matrices

r(n) = 7ri7?a"-7j" (4-34)

satisfy (4.33) with the following phases r?M(n):

Vli{n) = (_i)»i+»»+-+^-i , m(n) = 1. (4.35)

Written in terms of the fields x{n) a n d x(n) the action (4.31) reads

S = 2v^(n)xa(n)dflxa{n) + M^2xa(n)Xa(n),".<* n , a

where 0M is the lattice derivative defined in (4.3b). So far, of course, we havemerely rewritten (4.31). Now comes a crucial step. Since we have got rid of theDirac matrix 7M we can in principle let a run over any number of possible values,a = 1,2, • • • ,k. The minimal choice is k = 1, so that we shall omit this index fromnow on. The corresponding action

s(stag.) = 1 £ ^ (n) mn)x{n + fl)_x(n)X(n-il)} + Mj2 x(»)xW (4-36)n,/i n

now involves only one degree of freedom per lattice site, and the only remnantsof the original Dirac structure are the phases ?7M(n). The expression (4.36) is theaction of the staggered fermion formulation in the absence of interactions. For itto be of physical relevance one must still show that in the continuum limit it maybe written in the form (4.30) where the flavoured Dirac field components ip^ aregiven as linear combinations of the one component fields living at the lattice siteswithin an elementary hypercube. Hence for finite lattice spacing the space-timecoordinates of the fields ip^ will be those labeling the position of the particularhypercube considered. Let us look at the "reconstruction" problem in somewhatmore detail.

Consider a hypercube with origin at xM = 27VM (7VM integers); then the latticecoordinates at the 2d sites within this hypercube are given by

fM = 2ATM-t-pM,


where pM = 0 or 1. This suggests the following relabeling of the fields xin)

XP(N)=X(2N + p), (4.37)

and similarly for x- Notice that N = (Ni, • • •, Nd) now labels the space-timepoints of a lattice with lattice spacing 2a, and that the (multi) index p labels the2d components of the new field X- From these components one then constructs 2d/2

flavoured Dirac fields ipf(f — 1, • • •, 2d/2) with components ipfa (a = 1, • • •, 2d/2),

by taking appropriate linear combinations:

U(N) = .M) ^(Tp)afXp(N). (4.38a)p

Here

TP = 1?1? •••!?• (4.386)

By choosing the normalization constant Mo in (4.38a) appropriately, the action

(4.36) then takes the following form in terms of the fields tp and ip^:

glHag) = £ £ > V ) ( 7 , A + MW(N) + ••-, (4.39)/ N

where 9M is now the lattice derivative on the new (blocked) lattice, and wherethe "dots" stand for terms which vanish in the naive continuum limit.* For finitelattice spacing, however, these terms are no longer invariant under the full chiralgroup. Nevertheless for M = 0, (4.39) preserves a continuous U(l) x U(l) symme-try which is a remnant of the original chiral symmetry group. Because of this, onecan use the staggered fermion formulation to study the spontaneous breakdown ofthis remaining lattice symmetry, and the associated Goldstone phenomenon (zeromass excitation, accompanying the spontaneous breakdown). This is a major ad-vantage of staggered fermions over the Wilson fermions discussed before. Of coursethe staggered formulation has its drawbacks. Thus for example it can only be thelattice version of a theory with 2d/2 degenerate quark flavours, whereas there isno restriction on the flavour number in the Wilson formulation.

We close this section with a remark. Making use of the relation (4.38), onecan calculate correlation functions of flavoured quark fields, by taking appropriate

* See the following section.


linear combinations of the ^-correlation functions. The latter are given by thefollowing path integral expression

to. W) - a . w» - J °*D**"W)''' ^.2')e"4'"". <«««)where

^ x ^ x = n ^ w n dxP'(Jv')- (4-4o&)This all sounds rather simple. Nevertheless, the details can become rather involved.Thus one must make sure that the composite fields that one constructs to study theproperties of hadronic matter carry the correct quantum numbers in the continuumlimit. In particular one wants to know what are the flavour, spin, and paritycontents of the states excited by these operators. These are the problems whichdemand a lot of effort. We shall not discuss them here. The interested reader mayconsult for example the papers by Morel and Rodrigues (1984), Golterman andSmit (1985), Golterman (1986b).

4.5 Technical Details of the Staggered Fermion Formulation

Having sketched the main ideas which go into the staggered formulation, wenow present some mathematical details. Should the reader not be interested inthe details for the moment, he may skip this and the following section and go onto the next chapter without any danger of getting lost at a later stage.

Consider the action (4.36). As we have already pointed out, the coordinatesnM of any lattice site may be written in the form nM = 2Nil + pM, where 2NfJL arethe coordinates labeling the hypercube to which the site belongs, and p is a vectorwhose components are either one or zero. Since it follows from the definition (4.35)that rj^N + p) = rjn(p), we can rewrite the action in the form

s(Hag.) = 1 £ ^ ( p ) x ( 2 J V + p)[x{2N + p + A) _ x (2Ar + p _ A)]

N'P" . _ (4-41)+ Mj2xC2N + p)X(2N + p).

We next express (4.41) in terms of the 2d-component field defined in (4.37). Tothis effect we must remember that the components of p are restricted to the valuesone or zero. Hence we must exercise some care in rewriting the difference x(2iV +p + p.) — x(2-/V + p — p.) appearing in the expression (4.41). Consider for example


x(2iV + p + p). If p + p is a vector of type "p"> i.e. if the components of p + pare either one or zero, then 27V + p + p labels a site within the hypercube withbase at r = 2/V. Hence in this case x(2/V + p + A) can be identified with xP+a(N).On the other hand if p + p is not a vector of type p, then p - p is such a vector,and x(2-/V + p + p) = Xp-p.{N + p). These conclusions can be summarized by theequation

X(2N + p + P) = ^ [ < W , P > X P < W + < W , P ' X P ' (N + A)]> (4.42a)P'

where p' is understood to be a vector whose components are either one or zero.*In a similar way one obtains that

X(2N + p - p) = Y}5P-H,P'XP> (N) + 5p+^p'Xp' {N - p)]. (4.426)P'

Inserting the expressions (4.42a,b) into (4.41), we obtain

S(stag.) = i_ £ n^xMSp+wdftxAN)"'"'"'» (4.23a)

+ 8P-p,,p*dKXp,{N)} +M^Xp(N)Xp(N),N,p

where d^ and dj} are the left and right (block) derivatives analogous to (4.26a,b),defined by

d«x(N)=x(N + p)-X(N),„_ (4.43o)

dflx(N)=x(N)-x(N-p).Before proceeding, let us digress for a moment and calculate the 2-point

correlation function {Xp(N)xP'(N')) from the path integral expression (4.40). Aquick way of proceeding is to rewrite (4.43a) in momentum space by introducingthe Fourier transforms**

v (AT) — / E_v (f^pif-N

I I- (4-44)

*»{N) = £ (SM»e~^'* It will be understood from now on, that vectors denoted by the symbol rho

(i.e. p, p' etc.) have components restricted to these values.** Here, and in the following we shall restrict ourselves to four space-time

dimensions.


into (4.43a), and performing the sum over N using eq. (2.64). A simple calculation

yields

S(sta9.) = £ r d^_UP)Kpp,mA^ (445o)

P.? J-* {2n)

where

Kpp, (p) = J ] ir^p, (p) sin + M V , (4.456)

r^ (p ) = e*-("-"()/2r^,, (4.45c)

and

r£p- = [<W,P' + < W , P ' M P ) - (4-45d)

Hence the Fourier representation of the XX~ correlation function is given by

(xP(N)xAN')) = / ' 0^K;p)ip)eiHN-N']. (4.46)

The inverse of the matrix (4.45b) can be easily calculated by making use of the

relation{r" , r "} = 25/u/Ji, (4.47)

where 11 is the unit matrix.* This relation follows from the definition (4.45d) andcan be proved by making use of the following properties of the phases (4.35):

V»(P)VI>{P + P) = -VU{P)V^(P + ')), ( M ^ V).

From (4.47) and the definition of T^p, (p) given in (4.45c), it follows that for given p,these matrices also satisfy anticommutation relations analogous to (4.47). Hencethe inverse of the matrix (4.45b) can be written down immediately:

K {r)" £ . * » * + * . •

Because of the appearance of the factor 1/2 in the argument of the sine functionin the denominator, the integral (4.46) will be dominated for M —¥ 0 (i.e. inthe continuum limit) by the momenta in the immediate neighbourhood of p = 0.Hence no doubling problem of the type discussed before arises here. The same

* The rM's therefore satisfy the same algebra as the Dirac matrices 7^.


conclusion will then also hold for the quark correlation functions, since they canbe constructed from (4.46) by taking appropriate linear combinations.

After this intermezzo, let us return now to expression (4.43a), and express theleft and right derivatives (4.43b) in terms of the symmetric first and second blockderivatives, defined by

drfpiN) = \(XP(N + A) - XP(N - A)),

%XP(N) = XP(N + A) + XP(N - A) - 2Xp(N).

In terms of these derivatives, the action (4.43a) takes the form

SF = lT, *PW £ (TPA + IKP'^) + ™5pp, xAN), (4.48a)N,p,p' L /* V l J J

where T^pl has been defined in (4.45d), and

rpp' = ( < W , P ' - SP+p.,P')vM- (4-486)

The matrices FM and F5 M satisfy the same anticommutator algebra as the directproducts 7M ® 11 and 75 ® 7M7s, respectively, where 75 = 71727374; i.e., in additionto (4.47) we have that

{r",r5i/} = o,

{r 5 " , r 5 l / } = -2(jM1/]i.This suggests that F^ and F5 M are unitarily equivalent to the above mentioneddirect products. If the second matrix in these products is interpreted to operatein flavour space, while the first matrix acts in Dirac space, then (7^ ® 11)3M wouldbe the matrix version of the kinetic term in (4.30). This suggests that the fieldsV"a a n d ii>a are related to Xp a n d XP by the unitary transformation which bringsFM and F5/i into the form 7M ® 11 and 75 ® 7^75, respectively. We now constructthis unitary transformation. Because of the way the x~fields had been introducedoriginally (cf. eqs. (4.32) and (4.34)) and the appearance of the phase ??M(p)(rather than ??M(n)) in the definitions of FM and F5/J, it is not surprising that thetransformation will involve the matrix Tp, defined by

r p = 7f172P273

P374P4- (4.49)

Thus consider the following sixteen component fields

4>ap(N) =MoY^ Ua0,pXp(N), (4.50a)p

kp(N) = M, £ Xp(N)Ulap, (4.506)p


where A/"o is a normalization constant to be determined later, and where

Ua0tP= \(Tp)a0. (4.50c)

Since in four space-time dimensions a and /3 take the values one to four, and sincep = {p\,..., pi) runs over the sixteen sites within a hypercube, we see that U is a16 x 16 square matrix whose rows are labeled by the double index a/3. Equations(4.50a,b) can be readily inverted by noting that because the trace of any productof distinct 7-matrices vanishes, Tp satisfies the following orthogonality relation

Tr{TlTp,) = A5pp,.

For the matrix U defined in (4.50c) this relation reads

(U*U)PI,=6P/,,

where (W)Piap = U*p<p = \{Tp)*af3- We hence obtain

UN) = ~J2kp(N)Ua(3tP.

Introducing these expressions into (4.48a), and making use of*

(UW)a0,a'0' — $aa' 5/3/3',

we arrive at the followingexpression for the staggered fermion action in terms ofthe "quark" fields ij) and ?/>:

0 N>°»PM » (4.51a)

+ lâ%,a-fi.%]+^MSaa.Spp.}i,a,0,(N),

* This relation can be verified by using the following representation for the7-matrices in (4.49)

/ 0 -iaA (1b 0 \7 i = ( ^ 0 ) ' T={o - l l j '

where o-j are the Pauli matrices, and 11 the 2x2 unit matrix.


where

and

Making use of the explicit representation of the 7-matrices given in the previousfootnote, one finds that

Aa/3,a</3< = (l»)<**'50/3',

Aa0,a'/3' = (75)aa ' (VsW,

where iM = 7^ and t5 = 7 5 . Since in the naive continuum limit only the firstand third term in (4.51a) contribute to the action, it follows by comparing thisexpression with (4.39) that a and j3 should be identified with the Dirac and flavourquark-degrees of freedom, respectively. In (4.38a) we had made this fact explicitby writing the field components in the form ip^, rather than tya$. Accordingly, AM

and AM can be written as the following direct products,

A" = 7M ® H,

A 5 M = 7 5 ® V 5 i

where the first matrix appearing on the right hand side acts in Dirac space, whilethe second matrix acts in flavour space.

Finally, we must determine the normalization constant No in (4.51a). Tothis effect we study the naive continuum limit of this expression by introducingthe dimensioned fields ipap and block derivatives 9M in the standard way; i.e.,ip = 6~3/2i/; and 9M = \^- Here b is the lattice spacing of the blocked lattice. Bychoosing A/o = l / \ /2 the action (4.51a) then takes the form

S(sta9.)= ^ V ^ ^ ^ J I ^

" " • " ! (4-52)

x

where M = \M and where the sum over x (= Nb) runs over all hypercubes of theblocked lattice. In the naive continuum limit (6 —1 0) this expression reduces tothe form

gistag.) J2 Itfx&ix)^!^ + M0)1>f(X)

(4.516)

(4.51c)


where Mo = 2M, and where ipap has been replaced by Va- Notice that theappearance of the mass 2M instead of M in eq. (4.52) is not surprising since wehave scaled M with the lattice spacing of the blocked lattice. Thus MQ = \M isthe dimensioned mass parameter of the original fine lattice.

Consider now the two point correlation function of the quark fields,(ipi{N)rp^,(N')), where we have introduced the more suggestive notation ip^ men-tioned above. This two-point function can either be computed from the X ~ Xcorrelation functions (4.46) by taking appropriate linear combinations, as dictatedby the eqs. (4.50a,b), i.e.

(U(N)^'(N')) = \Y1 UafAxP(N)xAN'))Uya,fl ,p,p'

and multiplying the expression with 6~3, or by inverting the matrix operatorappearing within square brackets in eq. (4.52). This inversion is easiest done bygoing to momentum space. Introducing the Fourier decompositions

MNb) = / JLP-foyp-™J-7r/b (2TT)4

ij(Nb) = [^ * * y (p)e-*-™J-n/b (2?r)4

into (4.52), and making use of the relation

N

where Sp is the periodic delta function, one finds that

s(sta9'} = / I / l ^ ^ f e l ^ ® 8inM)1 ^ (4-53)

- - (1 - cos(pM6))7s ® iM«5] + Mo U(p) .

The propagator in momentum space is given by the inverse of the matrix appearingwithin curly brackets:

E M f-*(7/. ® H)| sinpM6 + I(75 ® t t5) sin2 ^ | + Moll ® 11S{P) = L ^ 4 . 2 ^ x , , 2 • (454)

E M W S1Q 2 + M 0

(4.54)


Notice that the denominator is the same as for the scalar field discussed in chapter3. In the naive continuum limit (6 —>• 0), the above expression reduces to

Ql v ~ » E M ( 7 M ® % M + Moll ® 11

which is the correct fermion propagator describing four degenerate flavoured Diracparticles. Because the denominator in (4.54) has the same structure as for thescalar field, the two-point function ( ^ ( i ) ^ , ^ ) ) obtained by Fourier transformingthis expression and taking the limit 6 —¥ 0, is given by

(i>l(N)^{N'))= f°° ^-SaP{p)Sff.^N-N>)\

J-oo lz7rj

where

S{P) = f + MS •

Finally let us compare the expression (4.52) with the lattice action for fourflavoured Wilson fermions. Clearly, the generalization of eq. (4.27) (with SF

defined in (4.4)) to the case of four quark flavours is

s(w) = J2 a^{n) i Y, [(7M ® B)9M ~ y 01 ® fl)aM] + Mil ® 111 ^(n). (4.55)

Thus the only difference between the Wilson and the staggered fermion actions,consists in the matrix structure of the second derivative term. This term does notcontribute in the naive continuum limit, but is responsible for lifting the fermiondegeneracy. So why not choose the simpler (Wilson) version, which has the meritthat the number of quark flavours is not restricted? We only give a brief answerto this question without going into details: In the naive continuum limit bothactions (4.52) and (4.55) possess a U(4) x U{A) (chiral) symmetry for M = 0.This symmetry is broken in both cases by the second term. But whereas theaxial symmetry (involving the generator 75 in Dirac space) is completely lost forWilson fermions, the staggered fermion action preserves a non-trivial piece of thefull chiral symmetry, whose generator is 75 ® £5. Under this abelian subgroup thefields transform as follows

ip{N) -> eia(75®t5V(-/V)

•ip(N) -> ip(N)eia{'*5m^


where a is an arbitrary parameter which does not depend on N. For this reason thestaggered fermion formulation is more adequate for studying spontaneous chiralsymmetry breaking and the associated Goldstone phenomenon in theories likeQCD with massless quarks.

4.6 Staggered Fermions in Momentum Space

In the previous section the transition from a one-component field to a 2d-component field Xp w a s carried out in configuration space. We now want to con-struct an alternative lattice action by working in momentum space. As we shallsee, this action differs from (4.52) in several interesting respects. For some pa-pers which are of relevance to this section confer Sharatchandra, Thun and Weisz(1981), Van den Doel and Smit (1983), Kluberg-Stern et al. (1983) and Goltermanand Smit (1984).

Our starting point is again the action (4.36). Inserting for x(n) a n d x{n) theFourier decomposition analogous to (4.44), and writing the phase ^ ( n ) as

where5™ = (0,0,0,0), 8W = (TT, 0,0,0),

<5(3> = (7r,7r,0,0), £(4) = (7r,7T,7r,0),

the action takes the following form in momentum space

S(Sta9) = l \ ( ^ \ \ ^4*(P')M(P',P)X(P). (4.56a)

Here

M(p,p') = (2IT)4 I Y,dp](P ~P' + <5(M))«mpM + M5^\p - p')\ . (4.566)

and Sp(k) is the periodic delta function. For the following discussion it is conve-nient to extend the fields x(P) and x{p)_ periodically with period 2TT. Let us denotethe corresponding fields by (j>{p) and <j>(p), respectively. Because (4.56b) is itself27r-periodic, we can then shift the integration intervals [-n, n] by TT/2. In fig. (4-3)we show the new integration region for the case of two space-time dimensions.


Fig . 4-3 The four quadrants of the shifted integration region (solid

lines).

Next, let us divide the new Brillouin zone, BZ', into sixteen domains, whosecenters are the corners of the first quadrant in the original Brillouin zone. For twodimensions these domains are shown in fig. (4-3). Because of the appearance ofsinj3M rather than smp^/2 in (4.56b), the X~X correlation function will receive, inthe continuum limit, contributions from momenta in the immediate neighbourhoodof the above-mentioned corners. The momenta in the sixteen integration regionscan be parametrized as follows.

P» = K + *lA) ( A = l , . . . , 1 6 ) )

where-71-/2 < ky. < TT/2,

and Tr^) are constant vectors, defined by

TT(1) = (0,0,0,0) , T T ^ = (TT, 0 , 0 , 0 ) , . . . , TT(16) = (7r,7r,7r,7r).

The integral (4.56a) may then be written in the form

fir/2 J4J. Jiy -S(Sta9-] = E / 7^7CT^(*')M^B(fc',*)0fl(fe), (4.57a)

A B —71/2 V / V /

where

MAB(k', jfc) = (27r)4{^44)(fc -k' + n^ _ n(A) + ( jMje-^ ' is in jfcMM (4.576)

+ M44 )(fc-fc' + 7 r ( B ) - 7 r ^ ) } ,


and where we have introduced the 16-component (dimensionless) fields 4>A{k) asfollows

$A(k) = }{k + nw). (4.57c)

This is the analog of (4.37) in the momentum space approach. Notice that theintegral (4.57a) now extends only over half the BZ. Because k and k! are restrictedto the interval [—7r/2,7r/2], the periodic (^-functions appearing in (4.57b) can bewritten in the form

6{*\k -k' + ir^ _ ^ ) ) = 6W$ - k')5AB

«*W(jfc -k' + TT - JT<A> + *</*>) = r,1B6W(k - k')

where 5(k — k1) has only support at k = k', and where

This factor just expresses the fact that the 4-dimensional periodic ^-function onlyhas support in the integration domain given in (4.57a) if ni, —nl ' + S^ equalszero or 2n for every component v. Inserting these expressions into (4.57b), andperforming the integration over k! in (4.57a), one obtains

S(stag.) = j2j ^±^JAk)KAB{k)h{k), (4.58a)

where

KAB (A) = t £ T'XB sin k» + 5ABM, (4.586)M

and

TAB = ^ € A B - (4.58C)

These matrices satisfy the following anticommutation relations.

{f^,f"} = 2(JMWll.

Furthermore it can be shown that the f^'s are unitarily equivalent to (7^ <g> 1).Hence we can write (4.58a) in the form

4r t° f l 0 - / ' ( ^ i ~Q(P) fc(7M ® B)2i sin + 2M11 ® 111 Q(p) (4.59)


where we have rescaled the range of integration to the interval [—n, n]. This is,at first sight, a surprising result, for the action is invariant for M — 0 under thetransformations of the full U(4) ® £7(4) chiral group, and in particular under

Q -><;'<** ( 3 1 ® 7 B ) Q

~Q .+ ~Qe-i<*B(mTB)

and

for each generator TB in the -/V/-dimensional flavour space. This U(4) ® t/(4)symmetry has however been gained at the expense of giving up the locality of theaction in configuration space. Indeed, the action (4.59) is a non-local function ofthe fields Q{n) and Q(n), obtained by inverting the Fourier series

n

0(p)=^d(n)e*n.n

Substituting these expressions into (4.59), one finds that

5^> = £ <0(n) £ > M ® H)(AJnm

n,m n (4.60a)

+2M11® 111 Q(m),

where

(AM)nm = f ^-42isin^M»-™) (4.606)

connects arbitrary sites along the /^-direction. The reason for this is the appearanceof the p^/2 (instead of pM) in the argument of the sine function. If pM/2 werereplaced by pM in (4.60b), then this expression would equal <9M<5nm = (<5n+/i,m ~^n-/i,m)/2 and we would be left with an expression for the action involving onlynearest neighbour couplings of the fields.

The above discussion suggests that the fields Q(n) are related to the fieldstp(n) by a non-local transformation. This is indeed the case. Thus the fields ip{p)and Q{p) appearing in (4.53) and (4.60a) are connected by a unitary transforma-tion which depends on the momentum p (see e.g., Kluberg-Stern et al. (1983).


Because of this dependence, the field Q at the lattice site TV will be given by alinear combination of the "quark" variables ip attached to lattices sites which arearbitrarily far from n.

This concludes our discussion of staggered fermions. As the reader is proba-bly convinced by now, a thorough discussion of this subject, including all latticesymmetries, becomes quite technical. We believe, however, that the material pre-sented in the last two sections will enable the reader to follow the literature onthis subject without too much difficulties.

4.7 Ginsparg-Wilson Fermions

Of the two lattice regularizations discussed above, only the Wilson fermionsallow one to study models with an arbitrary number of quark flavours. Wilsonfermions do however break the chiral symmetry of the continuum fermion actionfor massless quarks. This makes it difficult to explore the regime of small quarkmasses in numerical simulations and to study spontaneous chiral symmetry break-ing on the lattice. In fact, as we have already pointed out, the Nielsen-Ninomiyatheorem tells us that we cannot get around breaking the symmetry (4.8) of the(massless) fermion action, unless we give up at least some important property, likee.g. locality.

But there is another way of breaking chiral symmetry on the lattice in aparticular mild and controlled way. It was proposed a long time ago by Ginspargand Wilson (1982), but has not been seriously considered for 16 years. Ginspargand Wilson had searched for a lattice Dirac operator by starting from a chirallysymmetric action and following a renormalization group blocking procedure, usinga chirally non-invariant Kernel. For Ginsparg-Wilson (GW) fermions the fermionicaction is of the form

Sferm = J2 ^(x)(D{x, y) + m6xy)^(y) , (4.61)

where the "Dirac Operator" D(x, y) is a 4 x 4 matrix in Dirac space which breaksthe standard chiral symmmetry in a very special way. Since every lattice actionmust possess the correct naive continuum limit, it follows that for a —>• 0 D(x, y)becomes the usual continuum Dirac operator. While in the continuum, or inits naively discretized form, this operator anticommutes with 75, the GW-Diracoperator satisfies the following GW-relation: *

j5D(x, y) + D(x, j/)75 = aJ2 D{x, z)j5D(z, y) . (4.62)Z

* Actually this is only the simplest version of the GW relation. A more general


We may write this expression in the more compact form, as is usually the case inthe literature,

{75, D} = aDlhD , (4.63)

where D is now a matrix whose rows and colums are labeled by a spin and space-time index. Note that the rhs is of order a. From here one trivially obtains that

{75,£>-1} = o75 , (16.64)

which shows that the anticommutator {75, D^1} breaks chiral symmetry in anultralocal way. A Dirac operator satisfying the GW relation does however notensure the absence of species doubling. Any matrix of the form

D=-{l-V), (4.65a)

where V satisfiesV^V = 1

. (4.656)V^ = J5VJ5 ,

(7! = 1) solves the GW relation. But it was only in 1998 that an explicit expres-sion for D was given (Neuberger, 1998), which is free of doublers and local in amore general sense (Hernandez, 1998). It also satisfies and exact index theorem(Hasenfratz, 1998), a lattice version of the Atiyah-Singer index theorem (Atiyah,1971). *

The Neuberger solution is obtained by choosing

V = A(AÂ)-^ (4.66a)

withA = l - aK^ , (4.666)

where K^w^ is the Wilson-Dirac operator (4.28b). In the case where the fermionsare coupled to gauge fields, K^w"> is replaced by an expression depending on thegauge potentials to be discussed in the following chapter.

version reads (in matrix notation): 75-D + .D75 = aD-y^RD, where R is a nonsingular hermitean operator which is local in position space and proportional tothe unit matrix in Dirac space.

* In the continuum the Dirac operator for massless fermions in a smooth back-ground field carrying non-vanishing topological charge Q (see section (17.6)) pos-sesses left and right-handed zero modes. The difference nt — nn in the number ofthese zero modes equals the topological charge of the backrground field.


Clearly numerical simulations with GW-fermions are much more costly thanwith Wilson fermions. So why is one so interested in GW-fermions? After all, theyalso break the standard chiral symmetry (4.8). What makes them interesting inparticular, is that the GW-action still possesses an exact chiral symmetry whichdiffers from (4.8) by O(a) lattice artefacts, as was shown by Liischer (1999a). Andthis is true for the free theory as well as interacting case. The only thing thatmatters is the bilinear structure of the action (4.61) in the fermion fields, andthat D satisfies the GW-relation. The emphasis above is on the word exact. Theexistence of an exact chiral symmetry should allow one to study not only the regimeof small quark masses, but may possibly also resolve a long standing problem ofputting chiral gauge theories (like the electroweak theory) on the lattice.

The exact symmetry referred to above corresponds to the transformations

j-yje***^-*13) , (4.67)

or infinitessimally

$ -»ip' = tp + dip , -ip -> $' = i> + Sip , (4.68a)

where*6if>(x) = ie*ys[(l - lD)i/,](x) ,

S^(x) = ie[^(l - ^D)](xh5 . (4.686)

One readily finds that the variation of the action is given by

SSferm = ieY^[F(x) + 2mi>(x)-y5iP(x) + A(x)] , (4.69a)

X

where J2X = E n fl4i a n d

F(x) = {i>D){x)l5tP{x) + IJJ(X)J5(DIP)(X) - a{i>D){x)lh{DiP){x) (4.696)

£(1) = ~[{i>D){x)^{x) + ^(aO75(£ty)(z)] • (4.69c)

* [(1 - §£>)^](a;) stands for J2y(5xy ~ ^D(x,y))ip(y). Furthermore [^(1 -


We now note that

J2 F(X) = Yl ^ ) [in*D^ - aD^D] (*> vMv) = o. (4-7°)x x,y

where we made use use of (4.63). Hence for m = 0 the action is invariant underthe global transformations (4.67), which verifies the observation made by Liischer(1998).

This is all we will say about GW-fermions at this point. We shall return tothem once more in chapter 14, when we discuss the ABJ axial anomaly.

CHAPTER 5

ABELIAN GAUGE FIELDS ON THE LATTICEAND COMPACT QED

5.1 Preliminaries

In 1971 F. Wegner studied a class of Ising models, where the global Z{2)symmetry of the Hamiltonian was promoted to a local one. Although the modelsdid not possess a local order parameter, they did exhibit a phase transition. Inconstructing the models, Wegner (1971) introduced a number of important con-cepts which turned out to play also a fundamental role in the lattice formulationof gauge field theories like QED and QCD. In particular he was led to constructa non-local gauge invariant order parameter, whose analog in QCD was later in-troduced by K.G. Wilson (1974), and provides a criterium for confinement. InQCD this order parameter is known under the name of Wilson loop, although thename " Wegner-Wilson loop" would seem more appropriate. Nevertheless we shalladhere to the general praxis and refer to it simply as Wilson-loop.

A common feature of the above mentioned theories is that they possess a localsymmetry. In the case of QED or QCD the local symmetry group is a continuousone. The action in these theories is obtained by gauging the global symmetryof the free fermionic action and adding a kinetic term for the gauge fields. Inthe continuum formulation the construction principle is well known, and we willrecapitulate it below for the case of QED. The lattice version of the action can beobtained following the same general type of reasoning, but it will differ in someimportant details from the naive discretization.

Let us briefly review how one arrives at the gauge-invariant action in contin-uum QED. The starting point is the action of the free Dirac field:

4 0 ) = f dix$(x)(i'y>ldlt - M)^(x).

This action is invariant under the transformation

1>(x) -^Gi>{x)

i>{x) -^•ip{x)G'1

where G is an element of the abelian [7(1) group, i.e.

G = eiA,

77


with A independent of x (i.e. a global transformation). The next step consistsin requiring the action to be also invariant under local C/(l) transformations withip (ip) transforming independently at different space-time points. This is accom-plished by introducing a four-vector potential A^x) and replacing the ordinaryfour-derivative <9M by the covariant derivative £>M, defined by

Dfl = dl, + ieAll. (5.1)

The resulting new action

SF = f dix^(i-yitDll - M)V, (5.2)

is then invariant under the following set of local transformations

1>{x)^G(x)1>{x),(5.3a)

i;(x)îP(x)G-1(x),

A^x) -> G{x)All(x)G-1(x) - l-G{x)dtlG-\x), (5.36)

where

G(x) = eiA^ (5.3c)

In the present case, (5.3b) is just another form of writing the familiar transfor-mation law, A^ —» Afj, — ^9MA. Since A^ and G commute, we could have writtenjust as well A^ instead of GAfiG~1. In the non-abelian case to be considered later,however, this will be the relevant structure of the gauge transformations. Thecrucial property which ensures the gauge invariance of (5.2) is that, while A^transforms inhomogeneously, the transformation law for the covariant derivative(5.1) is homogeneous:

D^^GD^G-1 .

Having ensured the local gauge invariance of the action by introducing a four-vector field J4M, we now must add to the expression (5.2) a kinetic term whichallows An to propagate. This term must again be invariant under the local trans-formations (5.3c), and is given by the familiar expression

SG = -l- I'd*xFlu/F'u'\ (5.4)

where F^ = d,j,Av - dÂ^ is the gauge invariant field strength tensor. The fullgauge invariant action describing the dynamics of ip, ip and Ay, is then given by

SQED = -\f d4xF^F^ + I dîi^D^ - M)ip. (5.5)

Abelian Gauge Fields on the Lattice and Compact QED 79

The Green functions of the corresponding quantum theory (QED) are (formally)computed from the generating functional

Z[J,ri,fj]= f DAD$Di>eiSQED+i S dixJ"A»+i $ ^ ^ ^ ^ (5.6)

by differentiating this expression with respect to the sources JM(z), 77(2;) and fj(x),where I/J, 4>, r\ and fj are Grassmann-valued fields. The integral (5.6) may be given ameaning within perturbation theory. For a non-perturbative formulation, however,we should define the generating functional on a euclidean space- time lattice. Hencelet us do this here for QED and generalize it later to the more complicated case ofa non-abelian gauge theory. Our construction procedure will parallel very closelythe one described above and is based on the following two requirements: i) Thelattice action should be invariant under local U(l) transformations and ii) reducein the naive continuum limit to the classical continuum action.

Before we carry out this program, let us first obtain the euclidean version of(5.4) and (5.2); to this effect we let x° become purely imaginary (x° -» -1x4) andreplace at the same time A0 by -\-iA\\ the prescription that 4° should be replacedby +1A4 is made plausible by considering the case where A^ is a pure gauge fieldconfiguration: A^ = d,j,A(x); thus the replacement x° —> —1x4 implies do —> +184.With this formal substitution (5.4) becomes

SG -»• %- J dfixF^F^ , (5.7)

where a sum over /i and v [fi,v = 1,2,3,4) is understood. Hence exp(i5o) goesover into an exponentially damped functional of A^, (as it should).

The transition from (5.2) to the imaginary time formulation is also carriedout immediately by substituting the euclidean derivative dM in eq. (4.2) by thecorresponding covariant derivative £>M = d^ + ieA^. * Hence the action (5.5) goesover into iSêucl\ where

Meucl) _ Meucl) Meucl) ,. Rn\SQED — SG ~r aF ' (0.00,)

withgieucl) = 1_ J d4xF^^ {5M)

* Notice that the structure of the covariant derivative remains unchanged whenmaking the transition, since both do and AQ follow the rule: do —¥ 184 and AQ —>1A4.


s(eucl) = J ^^n^ + M)ij_ (5 gc)

Here 7M (n = 1 , . . . , 4) are the euclidean 7-matrices introduced in chapter 4. Sincefrom here on we shall always work with the euclidean formulation, we shall dropin the following any labeling reminding us of this.

5.2 Lattice Formulation of QED

We start our construction program of lattice QED by considering first thelattice action for a free Dirac field. To parallel as closely as possible the stepsin the continuum formulation, we shall work with Wilson fermions, where everylattice site may be occupied by all Dirac components ipa. The corresponding actionis given by (4.28) which, after making a shift in the summation variable, can bewritten in the form

S™ = (M + 4r) J2 ^("M")1 ^ " (5-9)

- ^ W n ) ! ' 1 - 7 M M « + A) + 4>{n + jj){r + 7^)V>H]-

Here we have dropped the "hat" on the fermionic variables for simplicity. It willbe always evident from our notation, whether we are considering the dimensionlesslattice- or the dimensioned continuum formulation.

The action (5.9) is invariant under the global transformations

^(rO-n^rOG"1,

where G is an element of the U(l) group. The next step of our program consistsin requiring the theory to be invariant under local t/(l) transformations, with thegroup element G depending on the lattice site. Let us denote it by G(n). Becauseof the non-diagonal structure of the second term in eq. (5.9) (whose origin is thederivative in the continuum formulation) we are forced to introduce new degreesof freedom. Since the group elements G(n) do not act on the Dirac indices, it issufficient for the following argument to focus our attention on a typical bilinearterm, •4>{n)ip{n + ji).

In the continuum formulation it is well known how such bilinear terms shouldbe modified in order to arrive at a gauge-invariant expression. Since according to(5.3a) i>{x)ij){y) transforms as follows,

i>{x)1>{y) -> i>{x)G-\x)G(y)il>(3)),


we must include a factor depending on the gauge potential which compensates theabove gauge variation. This factor, known as the Schwinger line integral, is wellknown, and is given by

U{x,y) = eie$*dz»A»{z\ (5.10)

where the line integral is carried out along a path C connecting x and y and asummation over fi is understood. Notice that U(x,y) is an element of the [7(1)group. Under a gauge transformation, A^ —• AM — |9MA, (5.10) transforms asfollows

U(x,y)-^G(x)U(x,y)G-1(y), (5.11)

where G(x) is given by (5.3c). From the above considerations we conclude thatthe following bilinear expression in the fermion fields ip and {[> is gauge-invariant:

H(x)U(x,y)tl,(y) = i,(z)eief*d*»AMrl>(v). (5.12)

Suppose now that y = x + e. Then we conclude from (5.12) that iJ(x)ip(x + e) andip(x + e)i/j(x) must be modified as follows:

i>{x)ip{x + e) ->• $(x)U(x, x + e)ip(x + e),

ip(x + e)ip(x) ->• 4>(x + e)U^(x,x + e)tp(x),

whereU{x,x + e) =e

ieeA{x)

and f A = Y<nev.An-

The above considerations suggest that to arrive at a gauge-invariant expres-sion for the fermionic action on the lattice, we should make the following substi-tutions in (5.9)

ip{n)(r - 7M)V(n + £) - • 4>(n)(r - ltl)Un>n+(liP(n + A), (5.13a)

^(n + £)(r + 7M)VKn)^<A(n + A)(r + 7M)£/n+£,n^(ri), (5.136)

where

^n+A,n = < n + A , (5.14)

and Un>n+p. is an element of the [7(1) gauge group. It can therefore be written inthe form

0n,n+/i = e i M n ) . (5-15)


where 0M(n) is restricted to the compact domain [0, 2TT]. The right-hand side of(5.13a) and (5.13b) are now invariant under the following set of local transforma-tions

tf>(n) ^G(n)i>(n),•ijj(n) -î>{n)G-\n),

Un,n+n -^G(n)Un^n+p,G~1(n + fi),

Un+p.^ -^G{n + jl)Un+^nG~l{n),

Notice that in contrast to the matter fields discussed before, the group elementsUn,n+p. live o n the links connecting two neighbouring lattice sites; hence we will re-fer to them as link variables and sometimes simply as links. Because of (5.14) theyare directed quantities, and we shall use the following graphical representation:

Making the substitutions (5.13) in eq. (5.9), we obtain the following gauge-invariant lattice action for Wilson fermions

S £ % , $, U} = (M + 4r) J2 i>{n)i>{n)n

- I E ^ W ( r " 1»)Un,n+Mn + A) (5-17)

+ 4>{n + p)(r + 7/i)^,n+A^(n)]-

Let us pause here for a moment and forget the arguments which led us to eq.(5.17). By requiring that Un,n+p, and Un+p,yn transform according to (5.16), thisexpression represents a natural way of implementing £/(l)-gauge invariance. Thatthe link variables should be elements of the U(l) gauge group, follows from therequirement that their gauge transforms must also be elements of U(l). One thenwould have to show that in the continuum limit (5.17) can be cast into the form(5.8c) by establishing a relation between the link variables and the vector potentialApin). The procedure would then be the following. The vector potential Aîn) atthe lattice site n is real-valued and carries a Lorentz index. The same is true for</)Ai(n)' w m c h parametrizes the link variable (5.15). But < >M(n) takes only valuesin the interval [0, 2n], while the values taken by the vector potential A^x) inthe continuum theory extends over the entire real line. This is no problem, forwe must remember that A^ carries the dimension of inverse length, while 0M is

(5.16)


dimensionless. Let us therefore make the Ansatz </>M(n) = caA^n), where a is the

lattice spacing, and c is a constant to be determined. For a —*• 0 the range of A^

will hence be infinite. With this ansatz it is now a simple matter to check that by

scaling M, ip and T/S appropriately with a (i.e. M -> aM, ip -> az/2ip, ip -¥ a3/2/ip)

and replacing Untn+ii f°r small values of the lattice spacing by

Un,n+fi « 1 + icaA^n)

expression (5.17) reduces to (5.8c) in the naive continuum limit, if we choose c = e.

Because of this connection between f/n>n+/i and A^n) we shall use from here on

the more suggestive notation

£ / » s [7n,n+A = e i e a ^ n ) . (5.18)

With this identification* it is now an easy matter to verify that U^n) transforms

as follows under gauge transformations

C7M(n) -»• GiÛ^G-^n + fi) = eieaA^n\

where A^(n) is a discretized version of All(x) — i9Â(x). Hence so far the lattice

action (5.17) with A^n) defined by (5.18) satisfies the basic requirements stated

at the beginning of this chapter. To complete our construction of the lattice

action for QED, we must obtain the lattice version of (5.8b) which again should

be strictly gauge-invariant, and be a functional of the link variables only. Such

gauge-invariant functionals are easily constructed by taking the product of link

variables around closed loops on the euclidean space-time lattice. Furthermore,

because of the local structure of the integrand in (5.8b), it is clear that we should

focus our attention on the smallest possible loops. Hence we are led to consider

the product of link variables around an elementary plaquette, as shown in Fig.

(5-1). Let this plaquette lie in the \i — v plane. We then define

£ V » = U^n)Uv(n + p)Ul{n + v)Ul{n), (5.19)

where we have path-ordered the link variables. Although this path ordering is

irrelevant in the abelian case considered here, it will become important when we

study QCD. Inserting the expression (5.18) into (5.19), one finds that

Ullv{n) = eiea*F^n\ (5.20)

* Of course, this identification of A^(n) with the vector potential is only strictly

correct in the continuum limit.


Fig . 5-1 The contribution U^^n) of an elementary plaquette with base

at n lying in the /iZAplane.

where F^ (n) is a discretized version of the continuum field strength tensor:

Fîn) = -[(Au(n + £) - Av{n)) - (A^n + v) - A^n))).

It then follows immediately from (5.20) that for small lattice spacing

? E Et1 - \(U^n) + UUn))} « \"£a'F^(n)F^(n),

where the sum appearing on the left-hand side extends over the contributionscoming from all distinct plaquettes on the lattice.* Hence from now on we shallwrite the lattice action for the gauge potential in the compact form

SG[U] = ^ ^ 1 - ^ U P + UP^' <5-21)6 p l

where f/p (plaquette variable) stands for the product of link variables around theboundary of a plaquette "P" taken (say) in the counterclockwise direction.

At this point we want to mention an interesting property of the lattice for-mulation. In contrast to the continuum formulation where the coupling constant

* Notice that on the right-hand side of this equation the sum extends over all/x and v.


e enters linearly in the fermionic piece of the action (cf. eqs. (5.8c) and (5.1)),

the coupling now appears with an inverse power in the action for the gauge field!

Thus on the lattice, the strong coupling expansion turns out to be the natural one.

This completes the construction of the lattice action for QED. For Wilson

fermions it is given by

SQED[U,4>, $\ = 1 £ [ i - huP + up)]e p l

+(M + Ar) Y$(nM") ~ \ E$(nXr " )UM^(n + A) (5'22)

+ i){n + £)(r + TfM(n)rl>(n)].

The action (5.22) is to be used in a path-integral formulation, from which anycorrelation function of the fermionic and link variables can be computed. Thispath integral will involve an integration over all link variables £/M(n), which, aswe have emphasized, are elements of a unitary group. Hence the integration is tobe performed over the (compact) group manifold which in the present t/(l) caseis parametrized by a single real angular variable restricted to the range [0,27r].*Now we have made a great effort in ensuring the exact gauge invariance of theaction. Hence this gauge invariance should not be destroyed in the integrationprocess; i.e. we must define a gauge-invariant integration measure! This is quitetrivial in the present case, for under a gauge transformation the link variables aretransformed according to (5.16). But since G{n) is an element of the abelian U{1)group, a gauge transformation merely amounts to a site-dependent shift in thephase of t/M(n). With the parametrization t/^(n) = er<t>»(n\ the gauge-invariantmeasure to be used in a path integral expression is therefore given by

DU = ]Jd^(n), (5.23)

and correlation functions of the link variables and Dirac fields are computed fromthe following path integral expression

(i/>a(n)---$/,(m)---Ult(N)---) =

__ f DUDj>Dij){rj}a{n) • • • 4>p{m) • • • U^N) • • .)e-s^D (5.24)

~~ /DUDxpDtpe^sÊD

* Since the integration range is compact, one also speaks of compact QED whenreferring to the lattice formulation.


These correlation functions depend on the parameters M and e which enter theexpressions for the fermionic and gauge lattice actions (5.17) and (5.21). In theinteracting quantum theory defined by (5.24), these parameters can no longer beidentified with the physical fermion mass and charge, and must be viewed uponas bare parameters, having no direct physical meaning. To emphasize this point,one usually writes MQ and eo instead of M and e. We have not done so in thischapter, since we have merely constructed the lattice action starting from the freefermion theory. In the following chapter, where we discuss the non-abelian case,we shall, however, make use of this notation.

One other remark must be made. For the euclidean lattice action (5.22) to bea bonafide candidate for defining a quantum theory, it must satisfy the criterion ofreflection positivity. Reflection positivity is a necessary ingredient for the existenceof a non-negative hermitean Hamiltonian, with a positive transfer matrix, and aHilbert space formulation. This is an important technical detail which we onlymention here. The action (5.22) can be shown to satisfy reflection positivity,which within the continuum formulation was first discussed by Osterwalder andSchrader (1973). For details the reader may consult the book by Montvay andMinister (1994), and in particular the references quoted there.

This completes the formulation of lattice QED. As we have seen, the groupaspect has played a central role in the above discussion. In a lattice formulationit is not the vector potential which emerges naturally in the process of gaugingthe free Dirac theory, but rather the group elements Up (n) which live on the linksconnecting two neighbouring sites. Thus the connection between lattice and con-tinuum variables is much more subtle than in the case of the matter fields. In factone may easily verify that a naive lattice translation of the minimal substitutionrule dp —¥ Dp (see eq. (5.1)) will lead to a fermionic action Sj? which violatesgauge invariance in higher orders of the lattice spacing. For reasons we have al-ready mentioned, however, we have insisted on the strict gauge invariance of thelattice action.

CHAPTER 6

NON-ABELIAN GAUGE FIELDS ON THE LATTICECOMPACT QCD

The lattice gauge theory we discussed in chapter 5 can be easily extended tothe case where the abelian group U(l) is replaced by a non-abelian unitary group.Thus suppose that instead of a single free Dirac field we have N such fields tpa

(a = 1, • • •, N) of mass Mo- Then the euclidean fermionic action, replacing (5.9),is given by*

N N

SF = (Mo + 4r) Y, E r{nW{n) - - £ 5 > » ( r - 7M)V<> + £)+n a=l n,/z a=l

+4>a(n + fi)(r + jâ{n)}.(6.1)

This action is invariant under global unitary transformations in N dimensions, andin particular under the non-abelian subgroup SU(N).** Introducing the followingTV-component column and row vectors

/ ^ \f=\ : , i>=^\---,i>N), (6.2)

these transformations read

f(x) —»• Gf(x) ,

where G is an element of SU(N). We now want to generalize (5.22) to the non-

abelian case. This is straightforward. We only have to replace the Dirac fields

ip and tp by the TV-component vectors (6.2) and the link variables U^n) by the

corresponding group elements of SU(N) in the fundamental (TV-dimensional) rep-

resentation. Let us denote the matrix-valued link variables by f/M(n). They can

be written in the form

Ull(n) = e~ , (6.3)

* We omit the "hat" on the dimensionless fields ip, tjj since it is clear that weare discussing the dimensionless formulation.

** This group consists of all unitary N x N matrices with determinant equal toone.

87


where <^v(n) is a hermitean matrix belonging to the Lie algebra of SU(N). Making

the above substitutions in (5.17), we obtain the gauged version of (6.1):

n

- j Z ^ ( n X r - T/Oi^frM" + A)+ (6.4)

+ $(n + £)(r + ln)Ul(n)i>(n)}.

For the reasons stated at the end of chapter 5, we have now written Mo insteadof M. Written out explicitly, a typical term in (6.4) reads

f(n)(r-'yÛ^(n)i>(n + fi)= £) C(n)(r - T M M ^ M M W ^ + £)•

The action (6.4) is invariant under the following local transformations

ip(n) -+ G(n)i)(n),

^(rO-^fOG- 1 ^) . ( 6 '5 f l )

!7M(n) -> ^ ( ^ ^ ( ^ G - ^ n + A),. . (6.56)

Here G(n) is an element of SU(N) in the fundamental representation. It can

therefore be written in the form

«A(n)

G(n) = e~ , (6.5c)

where A(n) is a hermitean matrix belonging to the Lie algebra of SU(N).

It is now a simple matter to construct the other piece of the action analo-gous to (5.21). Clearly SG should be gauge-invariant. Since £/M(n) transforms

according to (6.5b), the simplest gauge-invariant quantity one can build from thegroup elements U^n), is the trace of the path ordered product of link variables

along the boundary of an elementary plaquette; this path-ordered product is thegeneralization of (5.19) to the non-abelian case and reads:

£W0 = VMVv{n + p)yl(n + v)UHn). (6.6)

Non-Abelian Gauge Fields on the Lattice Compact QCD 89

Notice that the trace, and the path-ordering of the links in (6.6) are importantnow, since the group elements do not commute. In analogy to the abelian casediscussed in chapter 5 we now replace (5.21) by

SG = cTr^ll ~ \{U^[n) + [ # » ) ] , (6.7)

where c is a constant which will be fixed below.

So far we have merely extended the analysis in chapter 5, to the case of a non-abelian group. Undoubtedly the lattice theory constructed in this way describesa quite non-trivial system. But does it have any relevance for physics, and inparticular for elementary particle physics? To answer this question we must seewhether it has a chance of describing in the continuum limit an interesting fieldtheory. To this effect let us first fix the gauge group that we expect to be relevantfor describing the strong interactions of quarks and gluons. It has been known fora long time that quarks (and antiquarks) must come in three "coloured" versions,(—> ipa, a = 1,2,3) and that the observed strongly interacting particles (hadrons)are colour singulets with respect to the group SU(3).* Hence we expect this groupto be the one of interest. Now any element 0 lying in the Lie algebra of SU(3)

can be written in the form8 \B

where the eight group generators \B are usually chosen to be the (3 x 3) Gell-Mannmatrices, satisfying the commutation relations**

8

[\A,\B} = 2iJ2fABc\°- (6.8)C=l

Here JABC are the completely antisymmetric structure constants of the group,corresponding to this choice of generators.

* These hadrons are built from different "flavoured" quarks (i.e. up, down,strange, charm, top, bottom). Each of these quarks comes in three colours, andthey must be combined in such a way that the hadron transforms trivially under517(3). The reader who is not acquainted with these concepts, and is interestedin learning more about it, may consult the book by Close (1979).** See e.g. the book quoted in the previous footnote.


Let us now study the naive continuum limit of the action (6.4) for the caseN = 3, proceeding in a way analogous to the abelian case. To this effect weintroduce a dimensioned matrix valued lattice field A^n) as follows,

^tj.(n) = SoaA^n). (6.9)

Here </>M(n) is defined in (6.3), a is the lattice spacing, and go is a bare couplingconstant. Again we have written g$ instead of g to emphasize that in an interactingtheory this coupling constant is one of the bare parameters on which the actiondepends. Since Aîn) is an element of the Lie algebra of SU(2>) it is of the form

yi» = ]r»)^, (6.io)B=l

where A^(n) are eight real-valued vector fields corresponding to the eight genera-tors of SU(3). Inserting (6.9) into (6.3) and expanding the exponential to leadingorder in a, one finds, after scaling Mo, i> and t/S appropriately with the lattice

spacing, that (6.4) reduces to the following continuum action for a —¥ 0:

s(cont.) = j dix^x)^^dii + igQA^) + M0)i>(x).

Next we consider the naive continuum limit of eq. (6.7). To this end we definein analogy to (5.20) the matrix-valued lattice field tensor T^ by

U^{n) =e ~ . (6.11)

Clearly, the relation between T^n) and A^n) is now much more complicated

than in the abelian case. The reason is that the link variables appearing in theproduct (6.6) are now matrices which do not commute. In order to arrive at theconnection between T^ and AM, one needs to make use of the Baker-Campbell-

Hausdorff formulaeAeB

= eA+B+i[A,B\+-t (6.12)

where the "dots" will in general involve an infinite number of terms. But because0^(n) is proportional to the lattice spacing (cf. eq. (6.9)), one only needs to


compute a few terms in the exponent of (6.12), when this formula is used tocalculate the product of link variables. By making use of such relations as

0M(n + v) K 0M(n) + adv(f>^(n) ^ « goaA^n) + g0a2dvA,j,(n) H ,

one finds thatFu.v—>Fu,v = duA,, - duAli + iga[Au, Av}. (6.13)

a—*-0~ ~ ~ ~

This is the well-known expression for the matrix-valued field tensor in continuumQCD. Since (6.13) is again an element of the Lie algebra, it can be written in theform

^ = ECf (6-14)B=l l

Making use of (6.8) and of the orthogonality relation of the Gell-Mann matrices,

Tr{XB\c) = 26BC, (6.15)

one arrives at the following connection between the eight components of F^v andA^, defined in (6.14) and (6.10), respectively:

* £ = dÂf - dvAl - gofBCDACA°. (6.16)

Having motivated the introduction of the lattice field strength tensor T^v

according to (6.11), we now compute SQ in the naive continuum limit. Approx-imating (6.11) for small lattice spacing by the first two non-trivial terms in theexpansion of the exponential and inserting this expression into (6.7), one finds that

2C v «5o q{cont.)

where

S(cont.) =l_Tr j'fap^p^ ( ( U 7 )

and a sum over /i, v is understood. This is the well-known gauge field action ofQCD. Hence we must choose c = i/g^. The continuum field strength tensor forSU(N) has the same form as given in (6.13) with A^ an element of the Lie algebra

of SU{N). Following the same procedure as above one finds that the gauge partof the lattice action is given for all N > 1 by

Sr{N))=^[l-^{UP + Ul)] (6.18a)


where27V

P=-T • (6-186)

9o

As in the abelian case, the sum in (6.18a) extends over all distinct plaquettes on

the lattice, and we have introduced the notation Up for the path-ordered product(6.6) of link variables around the boundary of a plaquette P. Both orientations forthis product are taken into account, thus ensuring the hermiticity of the action.

The action (6.18) is invariant under the local transformations (6.5b). Insertingfor U^n) the expression

igoaA^(n)Ull{n) = e " , (6.19)

one finds that (6.5b) implies the following transformation law for A^ in the con-

tinuum limit:

A^x) -> G(x)Alt(x)G-1(x) - -G{x)d^G-1(x). (6.20)go~

This is the non-abelian analog of (5.3b).

For those readers not familiar with continuum QCD we want to make thefollowing remark. Using the relation (6.15), the expression (6.17) may be writtenin the form

S{rt] = \jdAxF^F^ (6.21)

where F^v is related to the coloured gauge potentials by (6.16). Hence, in contrastto the abelian case, the pure gauge sector of QCD describes a highly non-trivialinteracting theory, which involves tripel and quartic interactions of the fields, Ajj.This is the reason why a study of the pure gauge sector of QCD is of great interest.In fact, the self-couplings of the gauge potentials are believed to be responsiblefor quark confinement. The first non-abelian gauge theory was proposed by Yangand Mills (1954), and was based on SU(2). For this reason one usually refers to(6.18), or (6.21), as the Yang-Mills action.

So far we have constructed the lattice action which possesses the desired naivecontinuum limit. We must now define the quantum theory by specifying the pathintegral expression from which correlation functions may be computed. This ex-pression will be of the form (5.24) except that now ipa, ^>a and U^.(n) will carryadditional colour indices corresponding to the three-coloured quarks lying in thefundamental representation of S[/(3). What concerns the integration measure DU


on the other hand, it will now depend on the eight real variables parametrizing thegroup elements of SU(3), and the integration is to be performed over the groupmanifold. For the same reasons mentioned before in connection with the abeliantheory, this integration measure must be gauge-invariant if quantum fluctuationsare not to destroy this important principle. Denoting by oif{A = l,---,8) thegroup parameters on which the tth. link variable depends, the corresponding in-tegration measure will be of the form*

DU = ]\j(ai){dat), (6.22a)i

where at stands for the set {o^1}, and

8

(dae) = Yl daf. (6.226)A=l

The structure of the Jacobian J(ae) in (6.22a) is determined from the requirementof gauge invariance. For our purpose it will suffice to know some of the standardintegrals involving polynomials of the link variables and we shall only list a few ofthem without proof. A general rule, however, is the following: only those integralsinvolving products of the link variables will give a non-vanishing contribution, forwhich the direct product of the corresponding representations contains the identityelement. With dUg, defined by

dUe = J(ae)(dae) ,

some useful SU(3) group integrals are:

fdUUab = 0, (6.23a)

/ dU Uab Ucd = 0, (6.236)

j dUUab{U^Yd=Uad5bc, (6.23c)

jdU uaib*Ua3b>Ua>b' = | e a i a 2 a 3 «bl6263. (6.23d)

* The integration measure DU is the so called Haar measure. We will discussthis measure in detail in chapter 15, where we shall require its explicit form inorder to perform the weak coupling expansion of lattice QCD.


Here U stands generically for any given link variable. The general rules for evalu-ating arbitrary integrals of the above type have been discussed by Creutz (1978).

An arbitrary correlation function involving the fermionic and link variablescan be computed from the following path integral expression

<V'Si(ni)"-î("»i)---C^(A:1)"-> =If- -,, J (6.24a)

= -] DUDtDMZim) • • -i$(roi) • • •U?1(k1)e-S«°'>where

Z= fDUD$D%l)e-siCD, (6.246)

and where SQCD is given by the sum of the actions (6.18) and (6.4) with N = 3.For later convenience we summarize the results for QCD below:

SQCD = SG[U] + S{P[U, i>, $ , (6.25a)

SG = - | £ [ 1 - ^ P + E )], (6-256)9o P °

4w>=(M0 + 4r)£$(»MrOn

- I E ^ n ) ( r - - ^ ( " W " + A)+ (6.25c)

+ i>{n + £)(r + ijytinWn)].

We have concentrated here on the case of Wilson fermions. The generalization ofthe free staggered fermion action (4.36) to QCD is obvious. The fields \ a n d X

become 3-component vectors in colour space and must be coupled to the matrix-valued link variables in a gauge invariant way. Each lattice site can accomodatethe three colours. Denoting by x the vector (x1)X2>X3) m colour space, we havethat

glrtag.) = 1 J2Vfl(n)x(n) (u^n)X(n + A) - U^n - (l)x(n - A))

"'" (6.26)+ M0Y,x(n)x(n).

n

For each colour, the different Dirac and flavour components of the quark fieldsare then constructed from the ^-variables at the sixteen lattice sites within ahypercube in the way described in chapter 4.

This completes our construction program for lattice QCD. In the followingchapter we introduce an important observable which will play a central role in thestudy of confinement later on.

CHAPTER 7

THE WILSON LOOPAND THE STATIC QUARK-ANTIQUARK POTENTIAL

One of the crucial tests of QCD is whether it accounts for the fact thatisolated quarks have never been seen in nature. It is generally believed that quarkconfinement is a consequence of the non-abelian nature of the gauge interactionin QCD. In contrast to QED where the field lines connecting a pair of oppositecharges are allowed to spread, one expects that the quarks within a hadron *are the sources of chromoelectric flux which is concentrated within narrow tubes(strings) connecting the constituents in the manner shown in fig. (7-1).

F ig . 7-1 (a) Picture of a meson built from a quark and antiquark which

are held together by a string-like colour electric field; (b) corresponding pic-

ture of a baryon built from three quarks.

Since the energy is not allowed to spread, the potential of a quark-antiquark(qq) pair will increase with their separation, as long as vacuum polarization effectsdo not screen their colour charge. For sufficiently large separations of the quarks,the energy stored in the string will suffice to produce real quark pairs, and thesystem will lower its energy by going over into a new hadronic state, consisting ofcolour neutral hadrons. In fig. (7-2) we give a qualitative picture of this hadroniza-tion process for the case when the quarks are bound within a meson (gq-system)or baryon (ggg-system).

* We shall often refer to the constituents of hadrons simply as quarks, withoutdistinguishing explicitly between quarks and antiquarks.

95


Fig . 7-2 Hadronization of the qq and qqq systems as the quarks are pulled apart

Once the colour charges of the quarks and antiquaries have been screened,the remaining Van der Waals type interaction between the colour neutral hadronsbecomes the short-range interaction characteristic of the known hadronic world.This picture of confinement can in principle be checked by computing, for example,the non-perturbative potential between a static quark-antiquark pair. We nowshow how this potential can be extracted from a path integral representation. Tothis effect, it will be useful to first discuss some of the ideas involved within thecontext of non-relativistic quantum mechanics, since our subsequent presentationof the field theoretical case will be quite formal.

7.1 A Look at Non-Relativistic Quantum Mechanics

Consider a particle of mass m moving in a potential V(x) in one space di-mension. Its propagation is described by the amplitude

K(x',t;x,0) = (x'\e-iHt\x), (7.1)

where H = p2/2m + V(x). Next consider the static limit of (7.1); letting m -*• oo,the kinetic term in the Hamiltonian may be dropped and H can be replaced bythe potential. Hence (7.1) takes the simple form

K(x',t;x,0) —> 5{x - x')e-iV<>x)t. (7.2)m—KX>

Continuing this expression to imaginary times (t -» —iT), we see that the potentialV{x) may be determined from the exponential decay of (7.1) as a function ofeuclidean time T. The J-function appearing in (7.2) just tells us that an infinitely

The Wilson Loop and the Static Quark-Antiquark Potential 97

massive particle does not propagate. In fact the only change in the wave functionwith time consists in the accumulation of a phase. Thus in the static limit thewave function ip{x,t) is a solution to the following equation

idtip(x,t) = V{x)i;{x,t),

which may immediately be integrated to give

i/>(z, t) = e - ^ ^ i x , 0). (7.3)

The phase exp(—iV) is just the one appearing in eq. (7.2). To substantiate theformal arguments given above, we illustrate the result (7.2) for the case of theone-dimensional harmonic oscillator whose Hamiltonian is given by H = p2/2m +KX2/2. The corresponding propagation kernel has the form*

K(x',t;x,0) = ( m" )1 /2ertf^[(*2+*'2)cos^-2,x'] ( 7 4 )

v ' ' ' ; V27risin uitl y '

where IM = ^Jn/m is the frequency of the oscillator. In order to extract thepotential from (7.4), we now take the limit m —> oo, holding « (i.e. the potential)fixed. This implies that w must vanish like 1/^/rn. It is then a trivial matter toshow that

' / , \ 1/2 , 2 "

K(x',t;x,0) —> ( — ) e"^^- e-i(v{-')+v^))t (7.5)

where e = t/m. In the limit e —> 0 (m —> oo), the factor appearing within squarebrackets just becomes S(x — x'); hence we arrive at the result (7.2).

7.2 The Wilson Loop and the Static gg-Potential in QED

We now generalize this discussion to the case of a gauge field theory. To keepthings as simple as possible, we shall restrict us for the moment to the case of anabelian U(l) gauge theory, and in particular to QED. Furthermore, we shall argueentirely within the framework of the continuum formulation where the physicalpicture is most transparent. Our presentation is based on the work by Brown andWeisberger (1979), and on the review article by Gromes (1991).

* See e.g. Feynman and Hibbs (1965) for a derivation of (7.4) within the pathintegral framework. We have set h — 1.


Consider a heavy quark (Q) and antiquark (Q), which are introduced intothe ground state of a quantum system whose dynamics is described by the action(5.5).* We want to study the energy of this (infinitely) heavy pair when it iscoupled to the gauge potential in the usual minimal way. To this effect considerthe following gauge invariant state

\<fia0(x,y)} = ¥^(x,0)U(x,0;y,0){pQ\y,0)\n), (7.6)

where \Q > denotes the ground state, and where, for arbitrary time U(x,t;y,t) isdefined by

U(x,t;y,t) = e Jz ', (7.7)

with the line integral extending along the straight line path connecting x and y.This phase ensures the gauge invariance of the state (7.6) which describes a quarkand antiquark located at x and y at time t = 0. In order to distinguish the heavyquarks, serving as test charges, from the (light) dynamical quarks responsible forthe vacuum polarization effects referred to at the beginning of this chapter, wehave attached the label "Q" to the corresponding Dirac fields. The state (7.6)is not an eigenstate of the Hamiltonean H. It serves however as a trial state toextract the energy of the lowest eigenstate of H having a non-vanishing projectionon \<j>ap). This energy will be a function of the separation of the quark andantiquark, and is the quantity that we are interested in. As in the case of ourquantum mechanical example we can extract this ground state energy by studyingthe propagation of the state (7.6). But the procedure will not completely parallelthe quantum mechanical case. The difference is that whereas the state \x), whosepropagation we have studied there, becomes an eigenstate of the HamiltonianH = p2/2m+V(x) in the infinite mass limit, this is not true for the trial state (7.6).In the field-theoretic case, where one is dealing with a system having an infinitenumber of degrees of freedom, there will be many eigenstates of H which have anon-vanishing projection on (7.6); but of all these states we are only interestedin that state having the lowest energy. In our quantum mechanical example wewould be confronted with a similar situation if we were studying the propagationof a particle of finite mass in the potential V(x). In this case the state \x) is nolonger an eigenstate of the Hamiltonian and the propagation amplitude will nolonger have the simple form (7.2). Instead, we must consider its general spectral

* Although we are studying the U(l) gauge theory, we will refer to the chargedparticles as quarks and antiquarks.


decomposition

K(x', t; x, 0) = £<a:'InXnlaOe-"3-*, (7.8)n

where En are the eigenvalues of H = p2/2m + V(x), and \n) the correspondingeigenstates. But from the structure of the right hand side of (7.8) we see thatwe can extract the contribution of the state of lowest energy by studying thepropagation amplitude for large euclidean (t —> —iT) times:

K(x',-iT;x,0) —> (2;'|0)(0|a;)e-BoT.T—too

This is the well-known Feynman-Kac formula. As an example consider the har-monic oscillator, where K(x', t; x, 0) is given by (7.4). Taking the above limit,holding the mass fixed, one readily finds that Eo = \UJ. This is of course not thequantity that we were interested in; but the example illustrates the point that ingeneral we must supplement the infinite mass limit with another limit involvingthe euclidean time. Furthermore, the order of these limits is important! To ex-tract the ground state energy of a quark-antiquark pair, we must first study thepropagation of the state (7.6) in the infinite mass limit, and then examine thebehaviour of the propagation amplitude for large euclidean times. With this inmind, consider now the following Green function describing the propagation of thestate (7.6):

Ga'P',ap(x',y';x,y;t)

= (n\T(¥0f(y\t)U(y\t-,x\t)^\x\t)¥^\x,O)U(x:O;y,O)^f\if,O))\n)(7.9)

where "T" is the time-ordering operator. Since in the limit of infinite quarkmasses*, the positions of the quark and antiquark are frozen, we expect that (7.9)will show the following behaviour

Ga,p.,aP(x',y';x,y;-iT) —• 5™ {2 - x ')6^(y - y ')Ca,p,ta0(x,y)e-EWT,1)MQ—i-oo

2)r-»oo(7.10)

where MQ is the quark (antiquark) mass, Ca'p',ap(x,y) is a function describingthe overlap of our trial state (7.6) with the ground state of the Hamiltonian in thepresence of the static pair, and E(R) is the ground state energy** of the staticpair separated by the distance R = \x — y\.

* Actually we shall keep the masses finite, in order to control any divergenciesthat might arise.

** As we shall see in the next chapter, E{R) also includes self-energy effectswhich need to be subtracted when calculating the interquark potential.


The right-hand side of (7.9) has the (formal) path integral representation

Ga.p.,cp = | f DA Dip Dj D^D^\^\y\t)...ijf\y,Q))eiS, (7.11)

where the expression within brackets stands for the quantity whose expectationvalue we are calculating, Z is the normalization constant given by the integral(7.11) omitting the above mentioned expression, and S is the action describingthe dynamics of the light and heavy quarks, and of the gauge potential:

S = SG[A] + SF[il>, & A] + SQ[V>W), i>{Q), A}.

Here SG and SF are defined in (5.4) and (5.2) respectively, and*

SQ[^Q\4>{Q\A] = /"dWQ )(*)(*7M£V -MQ)^Q){x). (7.12)

Since this action is quadratic in the fields ip^ and rp^\ one can immediatelyperform the integration over these Grassmann variables in (7.11) (see chapter 2):

Ga>0',a,p = -~ J DA Dip Dip[S00.{y, y'; A)Sa,a(x', x; A)

Sa,!3,(x',y';A)SPcx(y,x;A)\- (7-13)

•U(x, 0; y, 0)U{y ', t; x ', t) det K(Q) [A]eiSG+iS".

Here x, y, x' and y' are the following four-component vectors

x= (f,0) , y=(y,0),

x' = (x',t), y' = (y',t),

S(z,z';A) is the Green function describing the propagation of a quark in theexternal field A^,

[*7"(aM + ieA^z)) - MQ]S(z,z';A) = S^{z-z')6{t - t'), (7.15)

and det K^[A] is the determinant of the matrix

K{^y[A] = [H»(d, + ieA^x)) - MQ]a/^(x - y),

* Recall that we are still in Minkowski space. Hence 7^ (ji = 0,1, 2,3) are theusual Dirac matrices.

(7.14)


arising from the integration over the heavy quark fields. In perturbation theory the

logarithm of this determinant is given by the sum of Feynman graphs consisting

of a fermion loop with an arbitrary number of fields A^ attached to it. This

determinant approaches an (infinite) constant for MQ —>• oo which is however

canceled by a corresponding factor contained in Z. Hence in what follows we can

setdetifC?) = 1.

For the same reasons as stated above, we can of course also perform the

integration over the dynamical quark fields ijj and t/i in eq. (7.13). This gives

rise to a similar determinant; but its dependence on the gauge potential can no

longer be ignored, since these fields have finite mass. It is this determinant which

is responsible for the vacuum polarization effects mentioned at the beginning of

this chapter.

So far the result (7.13) holds for arbitrary quark mass MQ. We now want

to study this expression for MQ —» oo. Following Brown and Weisberger (1979),

we drop the spatial part of the covariant derivative, but keep its time component.

Thus gauge invariance is maintained in this approximation:

[ry°(d0 + ieA0(z)) - MQ}S{z,z';A) = 5^{z - z'). (7.16)

Here the derivative acts on z. Equation (7.16) can be easily integrated. Making

the Ansatz

S(z,z>;A) = eiefidtA°^S(z-z>) (7.17)

one finds that S(z — z') satisfies a differential equation, which does not involve the

gauge potential:

(ij°d0 - MQ)S{z - z1) = 6W(z - z'). (7.18)

This equation can be readily solved by making a Fourier ansatz for S, and leads

to the following expression for (7.17):

iS(z,z';A) = 5^(z - z')eieC° " A°im{Q(Z0 - z'o) ( ! ± ^ ) e-"M*o-*i)

+ 6(4 -zo) ( ^ ^ ) eiM*(*o-4)}.

(7.19)

This expression shows that the time evolution of the (infinitely) heavy quark fields

merely consists in the accumulation of a phase determined by AQ and the quark

mass. This is the statement analogous to (7.3) in our quantum mechanical exam-

ple. We next insert (7.19) into (7.13). Because of the appearance of the spatial


delta function, which merely tells us that an infinitely heavy quark cannot propa-

gate in space, only the first term in eq. (7.13) contributes since f / j . Recalling

the definitions of x, y, x' and y' given in eq. (7.14), one finds that

(7.20a)where

P± = ^ ( l ± 7 ° ) , (7.206)

and where the line integral extends over a closed rectangular path with spatial

and temporal extension R = \x — y\ and t, respectively, whose corners are located

at the points (7.14). The bracket ( ) denotes the ground state expectation value

in the absence of the static quark-antiquark source. It is formally given by

(Jef*»AM) = fDAD1,D$ei"fd*A»Uj*>'°{ ' J DA Dip Di> eisQen ' V-£l>

where SQED is the action defined in eq. (5.5).

Finally let us continue the expression (7.20) to imaginary times, t —¥ —iT.

This is accomplished by replacing —ISQED in (7.21) by its euclidean counterpart

(5.8), and continuing the exponentials exp(—2iMgi) and exp(ie § dz1*AM(z)) to

imaginary times. By writing out explicitely the contour integral, and recalling

that Ao must be replaced by iA^ in this continuation process, one finds that

[Ga'0<;«/J]t-HT,—> 5^(x-x')6^(y~y')(P+)a,a(P_)M,M^°° (7.22a)

•e-2M*T(WclA})eucl.,

whereWc[A] = efe/<to*j4"(*), (7.226)

and

{WclA])M . f ° " " > w ^ f , (7.22<;)J DA Dtp Dtp e~bQBD

In (7.22b) the line integral is carried out along a rectangular contour C in euclidean

space time, with corners given by (x, 0), (y, 0), (y, T) and (x, T). This is the famous

Wilson loop.

Finally, to obtain the static quark-antiquark potential we must study the

behaviour of (7.22) for large euclidean times T. Comparing this expression with


eq. (7.10), we see that the exponential factor exp(-2MQT) just accounts for thefact that the energy of the quark-antiquark system includes the rest mass of thepair. Hence we expect that (We[A]) behaves as follows for large T

W{R,T) = (WC[A\) T-^ F(R)e-EWT,

where E(R) is the interaction energy of the static quark-antiquark pair separatedby a distance R, and F(R) reflects the overlap of our state (7.6) with the groundstate of the system in the presence of this pair. Hence we conclude that this energycan be calculated as the following limit:

E(R) = - \im ±\n(Wc[A}). (7.23)T—>oo 1

We want to emphasize the formal simplicity of the above result. To computethe static interquark potential we "merely" need to calculate the expectation valueof a gauge invariant quantity built only from the gauge potential. Admittedly thederivation of the result involved a bit of handwaving. Furthermore, we have useda special trial state constructed from the quark fields and the string like operator(7.7), with the line integral taken along a straight line path connecting the quarkand antiquark. Especially, in QED, where field lines are allowed to spread allover space, we expect that there are other trial states which have a better overlapwith the ground state of the QED-Hamiltonian in the presence of a static source.But also in QCD where the field lines are expected to be squeezed into a tubeconnecting the two quarks, the use of other trial states can allow one to determinethe potential from Wilson loops with a relatively small temporal extension. *

So far we have argued entirely within the continuum formulation where thepath integral (7.22c) only has a formal meaning. To define it, we must obtain thelattice version of (7.22b) and (7.22c). This can be easily done. Thus on the latticethe exponential of the line integral in (7.22b) just corresponds to the product ofthe link variables (5.18) along the rectangular contour C shown in fig. (7-3). LetUi denote such a link variable. Then we define the Wilson loop operator by**

WC[U] = l[Ue (7.24)tec

* see e.g., Griffith, Michael and Rakow (1983)** In order not to introduce too many symbols, we use the same symbol W as

in the continuum formulation. The argument of W will tell us which formulationwe are talking about; notice also that in the abelian case the ordering of the linkvariables is irrelevant.


Its ground state expectation value

W(R,f) = (Wc[U]) (7.25a)

is given by_ JDUD*D1>Wc[U\e-s«*°V*'n

( ' ]~ J DUDi,D1>e-s**>>V<*M ( }

where in the case of Wilson fermions, SQED is given by (5.22). Notice thatW(R, T) is a function of the dimensionless ratios R = R/a and T = T/a, with athe lattice spacing.

Fig .7-3 Integration contour appearing in eq. (7.22b), relevant for calcu-

lating the static interquark potential.

On the basis of the arguments presented in this chapter, we now define the en-ergy of a static gg-pair measured in lattice units, E(R), by an expression analogousto (7.23),

E{R) = - lim X\nW{R,f), (7.26)f->oo T

where, as we have pointed out before, E(R) will still contain ^-independent self-energy contributions, which have to be substracted when calculating the interquarkpotential. Relation (7.26) will allow us to compute, at least in principle, theinterquark potential using numerical methods.

(7.256)


7.3 The Wilson Loop in QCD

The non-abelian case can be treated in a very similar manner to that discussedin the previous section. The starting point is again a state of the form (7.6), exceptthat now the Dirac fields are replaced by (6.2) with N = 3, and we must substitutefor U(x, t; y, t) the operator

ig f*dz*Ai(2,t)U(x,t;y,t) = Pe J* ~ , (7.27)

where Ai(z, t) is the matrix valued field denned in (6.10), and P denotes the path-

ordering operation. This path ordering is important to ensure the gauge invariance

of the state. Let us see why this is so.

Consider the following generalization of (7.27) to exponentials of line integralsperformed along an arbitrary path C connecting two different space-time pointsx and y:

igp dz»A^(z)U(x,y)c = Pe x ~ . (7.28)

The path ordering in (7.28) is defined as follows: divide the path C into n in-finitesimal segments and let £1,2:2,... ,xn-i denote the intermediate space-timepoints going from x to y. Furthermore define dxi = X( — xe-i, with xo and xn

identified with x and y, respectively. On each of the infinitesimal segments theexponential in (7.28) can be approximated by the first term in the Taylor expan-sion. Then (7.28) is given as the limit Sxe -> 0 of the ordered product of these(non-commuting) expressions along the path from x to y:

U(x, y)c = lim [1 + igAfl(x0)dx1]... [1 + igAfl{xn^1)dx!^]. (7.29)axe—yO

Consider now an infinitesimal gauge transformation. According to (6.20), A^

transforms asA^x) -> A^x) + i[0(x), A^x)} - -d^Jx),

where 6(x) is an infinitesimal matrix belonging to the Lie-Algebra of SU(3). Up

to terms linear in dx£ and 9{x) we have that

1 + igA^xt-^dxf -> 1 + igA^xi-^dxf

-3[?(^-i),^M(^-i)]^ (730)

- i (dîxi^)) dx$.


But (<9M0(a;£_i) j dx^ = 0(xi) - 0(a;*_i). Hence up to leading order in 0, (7.30)

can also be written in the form

1 + igAnixi-Jdxt -+ eie<-x'-^[l + igApixt-ddxfle-W*'*.

We therefore conclude that (7.29) transforms as follows under finite gauge trans-formations

U(x, y)c -> G(x)U(x, y)G~\y), (7.31)

where G(x) is an element of the gauge group. The transformation law (7.31) is the

analog of (5.11) for the non-abelian case. It guarantees that the state analogousto (7.6) is gauge invariant.

On the lattice, the path ordered exponential (7.28) is just the ordered productof link variables along a path connecting the lattice sites corresponding to x andy. Let us denote these sites by n and m, respectively, and by CL a path on thelattice connecting n and m. Then the transition from (7.28) to the lattice reads

y(x,y)c-+y(n,m)cL= Y[Ui, (path ordered), (7.32)eecL

where Ut denotes generically a link variable on CL. Since under gauge trans-

formations the link variables transform according to (6.5b), it follows that the

right-hand side of (7.32) transforms according to

U{n,m)cL ->• G{n)U{n,m)cLG~1(m).

Taking for CL the Wilson loop, we conclude that

WC[U] = Tr Yl Uc (path ordered) (7.33)eecL

is gauge invariant. This is the analog of (7.24). The corresponding expectationvalue (Wc[^1) = W(R, f), is then calculated as before according to a path integralexpression analogous to (7.25b):

WfR f) - jDUD4>D^WclU}e~s«c°V<^

where for Wilson fermions SQCD is the QCD action given by (6.25), and where DUis the gauge-invariant measure discussed in chapter 6. The interquark potential isthen computed according to (7.26).

(7.34)


The right-hand side of (7.34) can be written in a form which involves only anintegration over the link variables and hence will be suited for numerical, MonteCarlo, calculations. Indeed, the fermionic contribution to the action, given in(6.25c), is bilinear in the fermion fields and has the form*

Sf\U,i>3] = J2â(^)Knaa,ml3b[U}^{m), (7.35)

where (a, b) and (a, f3) are colour and Dirac-spinor indices, respectively. Hence wecan immediately perform the Grassmann integration (see chapter 2) and obtain

where the effective action, Seff, is given by

Seff[U] = SG[U]-lndetK[U]. (7.366)

Here K[U] is the matrix in Dirac, 5£/(3)-colour, and rn-space defined in (7.35). Inthe continuum formulation the matrix elements of K are given by **

Kax,py[A] = ( 7 ^ + igoA^) + Mo) <5(4)(z - y) .afS

The meaning of In detK[A] is well known in perturbation theory. It is givenby the sum of Feynman diagrams consisting of a single fermion loop, with anarbitrary number of external gluon fields attached to it. Hence this term gives riseto the vacuum polarization effects, referred to at the beginning of this chapter.Ignoring these effects amounts to setting detK = 1. This is the so-called quenchedapproximation. In this approximation one expects that the static gg-potentialrises with the separation of the quarks. This, as we have pointed out before, isa prerequisite for the hadronization picture mentioned at the beginning of thischapter. Hence calculating the (^-potential in the quenched approximation is animportant first check of confinement.

The above analysis does not yield any information about the spin-dependentforces between quarks since we have merely studied the static limit. To obtaininformation about the spin-dependent terms in the potential, one must allow the

* The same is true for the staggered fermion action (6.26).** For simplicity we use the same notation for the Wilson loop, fermionic matrix,

and effective action as above.

(7.36a)


quarks to propagate in space. This means that one has to take into accountthe spatial part of the covariant derivative in the Dirac equation. This programhas been first carried out by Eichten and Feinberg (1981), who treat this termperturbatively. Alternative derivations of the results obtained by these authorshave been carried out subsequently by Peskin (1983), Gromes (1984), Barchielli,Montaldi and Prosperi (1988). The reader may consult the recent review articleby Gromes (1991) for details on this subject.

Our above discussion has been quite formal. But simplified arguments basedon physical intuition often lead to the correct result. So let us verify our con-clusions at least within the framework of some simple models. After all, ourunderstanding of quark confinement will depend on our ability of calculating thenon-perturbative inter-quark potential, which - at the present state of the art -can only be determined by studying the expectation value of the Wilson loop nu-merically. Hence let us get some confidence in this procedure by studying somesolvable field theories.

CHAPTER 8

THE QQ-POTENTIAL IN SOME SIMPLE MODELS

In this chapter we study the potential of a static qq pair in two soluble modelswithin the quenched approximation: 1) QED in four space-time dimensions, and2) compact (lattice) quantum electrodynamics in two dimensions (QED2). Thelatter model will also provide us with another opportunity to study the continuumlimit and to compare our results with those obtained in a continuum calculation.

8.1 The Potential in Quenched QED

Consider the expectation value of the Wilson loop in QED. In the continuumformulation it is formally given by eq. (7.22c). Performing the fermion integration,we obtain

[DA WrlA]e-s<"M

where

Seff[A] = SG[A] - In det K[A], (8.16)

5G[A] is given by (5.8b), and K[A] is a matrix in space-time and Dirac space:

Kxa,yp[A} = [7M(dM + ieA^) + M]af}5^{x - y). (8.1c)

We now want to calculate the integral (8.1a) in the quenched approximation wherevacuum polarization effects, arising from the presence of dynamical fermions, areneglected. This, as we have seen, amounts to setting det K = 1, and hence replac-ing Seff by 5Q[^4]. The latter action can be written in the following convenientform:

SG[A] = - ^ J d^xAti{x)Q,ll,vAv{x), (8.2a)

where

^ V = <Wn ~ dv.dv, (8.26)

and D is the four-dimensional Laplacean. Hence in the quenched approximation(q.a.) the expectation value of the Wilson loop is given by

(Wc[A])q.a. = ^ ^ . (8.3)J DA e^ J dxA»n»"A"

109

(8.1a)


Since the integrands in the above expression are exponentials of quadratic forms inthe potentials, the integrals can also be carried out in the continuum formulation.Nevertheless, we cannot perform the Gaussian integration in the present form.The reason for this is well known: because of the gauge invariance of the action(8.2) the inverse of the operator Q,^ does not exist, since it annihilates all fieldconfigurations A^ which are pure gauge (i.e. of the form Ap = dÂ). This meansthat the integrands appearing in (8.3) take the same value for all field configu-rations which only differ from each other by a gauge transformation (notice thatthe closed line integral is also gauge invariant). These gauge equivalent potentialsdefine an orbit for every given field strength F^, and the integration along anysuch orbit will give rise to a divergent integral in the numerator and denominatorof (8.3). The ratio, however, will be finite. To show this, one must have a methodfor controlling this infinity. An elegant procedure has been given by Faddeev andPopov and amounts to selecting one representative field configuration from eachset of gauge-equivalent potentials.* This is done by imposing a gauge condition.Since we are computing the expectation value of a gauge-invariant quantity, thechoice of gauge is immaterial. A particularly simple choice is the so-called Feyn-man gauge and amounts to making the replacement

The Gaussian integrations in (8.3) may then be performed immediately and oneobtains

where D(z - z') is the Green's function for the operator • ,

aD(z-z')=S^{z-z') (8.46)

*>W = -hh • (S-4C)

Because the integrand in (8.4a) is proportional to 5^, the double integral will onlyreceive a contribution when z and z' are located on segments of the integrationcontour which are parallel to each other. In fig. (8-1) we show various types of

* Those readers which are not familiar with the Fadeev-Popov trick may consultthe review article by Abers and Lee (1973) or any modern field theory book. Inchapter 15 we shall demonstrate this trick for lattice QCD.

(8.4a)

The QQ Potential in Some Simple Models 111

diagrams which contribute to the exponential (8.4a). Clearly the leading contribu-tion for large (euclidean) times comes from the diagrams shown in figs. (8-la,d).Of these, however, the latter one represents the self-energy contribution to theenergy of the static qq pair. It must therefore be subtracted when computing theinter-"quark" potential. Hence the relevant diagram is that shown in fig. (8-la).The corresponding integral is easily evaluated and one obtains *

—> TV e-v^T

T-xx>where

f(R, T) = ^[arctan | - A ln(l + | j ) ] .

Since f(R,T) -» 1 for T -» oo, we find that V(R) is just the usual Coulombpotential.

F ig . 8-1 Diagrams contributing to the argument of the exponential in (8.4a)

The potential calculated from (8.3) does not include vacuum polarization ef-fects. When one takes into account dynamical fermions (i.e. fermions of finite masscoupled to the gauge potential), then one must also calculate diagrams containingvirtual fermion loops. These loops arise from the contribution of the determinantof the operator K[A] = <jf + M + ieJJL to the effective action (8.1b).** As an ex-ample let us calculate the leading order contribution of lndetK[A\. To this effectwe write lndetK[A] in the form TrlnK[A], where the trace is taken with respectto the space-time coordinates as well as Dirac spinor indices. Then

Tr In K[A] =Tr ln[($ + M)(l + -^-rjie^}

( i \ ( 8-5>=C + Trln{1 + JVMie4)'

* see e.g., Kogut (1979).** We use the standard notation ft = J2 7M^M-


where c = Tr ln(@+M) is an irrelevant constant, which drops out when calculatingthe ratio (8.1a). Expanding the logarithm in (8.5) in a formal series, one finds thatthe leading contribution is of O(e2):*

Tr In K[A] = -\TT [ ^ L ^ ^ - L ^ ] + O(e').

The right-hand side stands for the following expression

UP)2 r r r i i i (8-6)= 2j*xJ dix'TrD { < ^ I ^ ( I ' ) ( I ' I F S I ^ { 4

where Trp denotes the trace in the Dirac indices, and where

{x\j^\x') = SF{x-x')

is the (euclidean) fermion propagator. From (8.5) and (8.6) we see that

Tr In K[A] = i j cPxdPx''AÛîx - x')Av(x') + O(e4), (8.7o)

whereUîx - x') = -(ie)2TrD{7M(S'F(a; - x')-yvSF(x' - x)} (8.7b)

is the vacuum polarization tensor in one loop order. Substituting (8.7a) and (8.2a)into (8.1b) we are led to the following expression for (8.1a) in the Feynman gauge,

J DAeie§ dzÂ^z)^ J d4xdex'A^x)n^(x-x')A,(x')

' C^ 1> ~ j DAJ d*xd*x' A^xÂx-xÂ^') ' (8'8°)

where

ii^{z) = nltv{z)-^nlu,{z), (8.8b)

Slllv{z) = 6llv06W(z). (8.8c)

We now perform the Gaussian integrals in (8.8a) and obtain

(WC[A\) = e$§d*o§d*'&H*-*'). (8.9)

* Furry's theorem tells us that there are no contributions coming from an oddnumber of external photon lines.


Here fi~J is the inverse of the matrix (8.8b). It is defined by

Jd4z"n^(z - z")ti£(z" - z') = v<5(4)(* - A

and can be easily computed up to O(e2).

Cl£(x -y) = Q^(x -V)-J dAzdAz'Sl-l{x - z)Uxx, (z - z')^v{z' -y) + O(e4).

The two point correlation function of the gauge potential in O(e°) and O(e2) is

given by — f2~J and — fi^}, respectively. Hence in terms of Feynman diagrams the

right hand side of the above expression is given by:

. W W V . + I V A A I +O(e4).

Thus a typical diagram contributing to (8.9) is that depicted in fig. (8-2).

Fig. 8-2 Vacuum polarization graph arising from the fermionic determi-

nant contributing to (8.9).

By carrying out the expansion of (8.5) to higher orders, we arrive at a sum of

one-fermion loop contributions with an arbitrary number of external photon lines

attached to it.

Unfortunately we are only able to compute the effects arising from dynami-

cal quarks analytically within the framework of perturbation theory. For a non-

perturbative treatment we are forced to recur to numerical methods. This, as we

shall see later on, turns out to be a quite non-trivial task


8.2 The Potential in Quenched Compact QED2

We now perform a similar calculation of the potential between two oppositecharges but starting from a lattice formulation. For this purpose we consider thecase of compact QED in 2 space-time dimensions, which, if we neglect dynamicalfermions, may be solved in closed form. The lattice action in the pure gauge sectoris given by *

SG = PJ211-UUP + UP)1 (8-10)p z

where /3 is some parameter which, in analogy to (5.20), we shall relate to thedimensionless bare coupling e by

0 = J F ' (8-n)

and where Up is given by the product of the link variables

U^n) = ei6^

taken around an elementary plaquette "P" as discussed in chapter 5.

Next consider the expectation value of the Wilson loop (7.25b), with the con-tour C having spatial and temporal extension given by R and T. In the quenchedapproximation it is given by

fDUWc[U]e-s°M{Wc[U]) = jDUe-SaW] • (8-12)

Since the link variables are elements of the abelian group (7(1), it is evident thatWc[y] can also be written as the product of the elementary Wilson loops (—>• pla-quette variables) contained within the region Re, bounded by the square contourC, as shown in fig. (8-3).

wc[u]= n UP- (8-is)peRc

* Its structure is the same as that discussed in chapter 5 (see eq. (5.21)).** We have dropped the constant term in the action (8.10) since it cancels in the

numerator and denominator of (8.12).

(8.14)


Fig. 8-3 Writing the Wilson loop (8.13) as a product of elementary

plaquette contributions. The dashed line stands for the original product of

link variables along the contour C.

To carry out this integral it is convenient to choose a gauge where all link

variables pointing along the time direction are rotated to the unit element. This

can always be achieved by performing an appropriate gauge transformation under

which the link variables transform according to (5.16). Hence the contribution

of a particular plaquette with origin at n = (ni,ri2) will be of the form Up =

exp(i9p), where 6p is given by the difference of the phase-angles associated with

two neighbouring links lying on consecutive time slices:

dp = 0i(ni, n2) - 0i(ni, n2 + 1).

Making use of the periodic structure of the integrands in (8.14), one finds that

- . f dOp jBpeP™9"W(E-T)'MC r-.*,*— • <815)

where W(R,f) = (WC[U]). Performing the integral (8.15) one obtains

^ ' ^ ($/§)""' (8'16)where Jn(/3) are the modified Bessel functions of integer order. From (8.16) we

read off the qq potential in units of the lattice spacing,

V(R) = - lim i\nW{R,f) = aR, (8.17a)f-»oo T


where

is the so-called string tension. Thus in the lattice formulation of quenched QED2

the potential rises linearly with the separation of the qq-psdr and hence confinesthe charged pair. This is the same behaviour found in continuum QED2 andis a consequence of the two-dimensional nature of the problem. In fact, let uscompute the physical potential V(R) by taking the appropriate continuum limitof the lattice version (8.17). Since continuum QED2 is a superrenormalizabletheory we expect that a simple rescaling of the variables with the lattice spacinga will suffice. This rescaling, however, requires some care. Thus we must clarifyfirst of all which quantities must be kept fixed as we let the lattice spacing go tozero. Since the physical potential has the dimension of inverse length, we mustscale V with the inverse lattice spacing. Furthermore, R is to be replaced by R/a;we therefore consider the expression

V(R;p,a)=â(P)R. (8.18)

From (8.18) we see that if we keep (3 fixed as a —¥ 0, then V diverges like I/a2!Therefore, this cannot be the correct continuum limit. So let us take a closerlook at the meaning of the bare coupling e defined in (8.11) by studying thenaive continuum limit of (8.10) in the manner described in chapter 5. Makingthe replacement Up —> exp (iea^F^) for a plaquette "P" lying in the /it^-plane(see eq. (5.20)) and expanding the exponential in powers of the lattice spacingsquared, one finds that the coupling constant in physical units, e, is related to eby

e=-e. (8.19)a

This we could of course have guessed immediately, since the coupling constant inQED2 carrys the dimension of a mass.

From the above discussion it is evident that (8.11) is a function of the latticespacing, and that the physical potential should be calculated as the followinglimit,*

V(R, e) = lim V(R, /3(a), a) (8.20a)a—s-0

* We have assumed that (8.19) is a physical coupling constant, which is to beheld fixed when performing the continuum limit. The fact that this limit turns outto be finite, justifies a posteriori this assumption, and agrees with what is knownfrom the continuum theory.

(8.176)


where

K*)=Mi- ( 8 - 2 ° 6 )

Hence (3 diverges in the continuum limit! This makes it plausible why the latticeformulation (8.12) reproduces the correct continuum limit, as we shall see below.Thus it is evident that for large p the Boltzmann factor appearing in the integrandof (8.12) will ensure that the integral is dominated by those link configurations forwhich Up & 1. Because of (5.20), this implies that the fluctuations in eF^ aresmall compared to the inverse lattice spacing squared.

With these remarks let us now compute the continuum limit (8.20a). Insertingin (8.17b) the following asymptotic expansions for Ii((3) and Io(P), valid for large

J 1 < 0 = ^ ( 1 - 1 + • • • ) ,one finds that

V(R) = l-e*R.

This is the classical energy of a pair of opposite charges separated by a distance.ft, for electrodynamics in one space dimension.

In the special case considered here, the energy of a gg-pair is a linear functionof their separation for any coupling. In particular, in the strong coupling limit(P —> 0), the string tension (8.17b) is given by —ln(/3/2). In a four-dimensionalgauge theory (without dynamical fermions), the confining nature of the potentialobtained in the strong coupling limit is a consequence of the fact that the fluxlines connecting the quark and antiquark are squeezed into narrow tubes (strings)along the shortest path joining the qg-pair (see chapter 11). This string is notallowed to fluctuate for go —» oo. Fluctuations may, however, destroy confinement,when one studies the continuum limit. In the above two-dimensional example, thepersistence of confinement in the continuum limit (eo —• 0), is not surprising sincein one-space dimension there is no way the string can fluctuate. In QCD, however,there is no a priori reason why confinement could not be lost in the continuumlimit (which, as we shall see, is also realized at vanishing bare coupling). Shouldit persist in this limit, it must be a consequence of the non-trivial dynamics.

This completes our demonstration of how the potential of a static qq pairmay be extracted by studying the expectation value of the Wilson loop for largeeuclidean times. In the simple examples considered, the calculation could be done


exactly. In general, however, we must rely on numerical methods and the startingpoint will be the lattice version. In this respect the second case treated above ex-hibited already some interesting features which we shall meet again when studyingthe continuum limit of QCD. Thus we have seen that taking the continuum limitrequired (3 to be a function of the lattice spacing. This functional dependencewas very simple in the case considered here, and we could actually determine itfrom dimensional arguments alone. In the case of QCD, on the other hand, thisdependence will not be trivial, and will be determined from the short distancedynamics of QCD.

CHAPTER 9

THE CONTINUUM LIMIT OF LATTICE QCD

9.1 Critical Behaviour of Lattice QCD and the Continuum Limit

In chapter 6 we constructed a lattice gauge theory based on the non-abeliangroup SU(3) and have given arguments which suggest that in the continuum limitit describes QCD. These arguments were based on the observation that the latticeaction (6.25) reduces to the correct expression in the naive continuum limit. Butas we have emphasized before, there exist an infinite number of lattice actionswhich have the same naive continuum limit. We have merely chosen the simplestone, proposed originally by Wilson.* There is, however, no a priori reason why anychoice of lattice action satisfying the above mentioned requirement will ensure thatthe theory processes a continuum limit corresponding to QCD or some other fieldtheory. For this to be the case the lattice theory must exhibit first of all a criticalregion in parameter space where correlation lengths diverge. To see this, let usconsider the case of a pure SU(3) gauge theory, which in the lattice formulationresembles a statistical mechanical system described by the partition function **

Z = IDU errÛ"+Ui"\ (9.1)

Suppose that this lattice theory possesses a continuum limit, and that we wantedto extract from it the mass spectrum of the corresponding field theory by studyingthe appropriate correlation functions for large euclidean times (see chapter 16 formore details). The largest correlation length is then determined by the lowestmass in the problem. If the corresponding physical mass, m, is to be finite, thenthe mass measured in lattice units, m, must necessarily vanish in the continuumlimit. This in turn implies that the correlation length measured in lattice units, £,must diverge. Hence the continuum field theory can only be realized at a criticalpoint of the statistical mechanical system described by the partition function (9.1).

* One can make use of the ambiguity in the action to construct so-called "improvedactions", which lead to a suppression of lattice artefacts contributing to observablesfor finite lattice spacing. This allows one to extract continuum physics already forlarger lattice spacings (Symanzik, 1982 and 1983)

** We have dropped the constant term in the action (6.25b) since it is irrelevantwhen calculating expectation values.

119


This, of course, is to be expected, since only if the correlation lengths diverge does

the system loose its memory of the underlying lattice structure. It follows that if

the above system is not critical for any value of the coupling, it cannot possibly

describe QCD or any other continuum field theory.

Now studying a system near criticality means tuning the parameters accord-

ingly. In the case considered above, the only parameter is the bare coupling gQ, a

dimensionless quantity which is void of any direct physical meaning. The correla-

tion length £ measured in lattice units will depend on this parameter. Hence the

continuum limit will be realized for go —> 3o> where correlation lengths diverge:

i(go) —• oo. (9.2)9o->9o

We want to emphasize that (9.2) followed from the general requirement that phys-

ical quantities should be finite in the limit of zero lattice spacing a. To arrive

at the above conclusion we have implicitly introduced a scale from the outside, in

terms of which dimensioned observables can be measured.* This scale must clearly

be correlated with g0. The relationship between the two may in principle be de-

termined in the following way. Consider an observable 0, such as the correlation

length or the string tension a defined in (8.17a), with mass dimension rf©. Let Q

denote the corresponding lattice quantity which may in principle be determined

numerically. 0 will depend on the bare parameters of the theory (coupling, masses

etc.) which in the simple case considered here is just the dimensionless coupling

go. The existence of a continuum limit then implies that

Q(go,a)=(^J 'eKso) (9.3)

approaches a finite limit for a —> 0, if g$ is tuned with a in an appropriate way,

with go(a) approaching the critical coupling g£ defined in eq. (9.2):

e(go(a),a)—>Ophys. . (9.4)

Hence if the functional dependence of 0 on g0 is known, we can determine g0 (a)

from (9.3) for sufficiently small lattice spacing by fixing the left-hand side at its

physical value &phys.- This determines go as a function of a(0Pft,J/s)1'/de. In the

case of the free scalar field and QED in two dimensions, dimensional arguments

* There exists a priori no such scale in a pure lattice formulation!

The Continuum Limit of Lattice QCD 121

alone determined the a-dependence of the bare parameters. In the present case,however, we are faced with a quite nontrivial theory, and the answer is not sosimple.

The above discussion did not make use of any particular observable. From(9.3) and (9.4) it may appear, however, that the functional dependence of go(a)will depend on the observable considered. For finite lattice spacing this will infact be true. For sufficiently small a, however, a universal function go(a) shouldexist, which ensures the finiteness of any observable. A corresponding statement isexpected to hold if the action depends on several parameters, (e.g. bare couplingconstant and quark masses).

We want to emphasize that it is not surprising that the bare parameters willdepend in general on the lattice spacing: by making the lattice finer and finer(see fig.(9-l)), the number of lattice sites and links within a given physical volumeincreases. Hence if physics is to remain the same, the bare parameters must betuned to a in a way depending in general on the dynamics of the theory.

Fig. 9-1 Making the lattice finer by tuning the coupling with the latticespacing so as to keep physics the same.

Suppose now that the lattice theory describes some field theory in the contin-uum limit. How do we know whether it is QCD? And how do we know that we areextracting continuum physics in a numerical calculation where we shall always beforced to work on a modest-sized lattice, and hence also at finite lattice spacing?Clearly a first requirement in any numerical calculations should be that the scaleswhich are relevant to the particular problem under investigation are large com-pared to the lattice spacing, but small compared to the extension of the lattice.Thus on the one hand correlation lengths measured in lattice units should be large;but on the other hand, they are not allowed to exceed the lattice, whose size is


limited by the available computer facilities! Hence we must have some clear signalwhich tells us whether we are extracting continuum QCD, or merely performingan academic exercise.

In the following we shall show that in the case of QCD one can actuallydetermine the functional dependence of go on the lattice spacing for sufficientlysmall a. We shall restrict our discussion to the case discussed above, where theeffects of dynamical fermions are ignored and the action only depends on the barecoupling go- Having established the relation between the lattice spacing and go,the dependence of any lattice observable on the bare coupling near criticality willbe known and can be used as signal for testing the continuum in a numericalcalculation performed on a lattice of finite extent.

9.2 Dependence of the Coupling Constant on the Lattice Spacingand the Renormalization Group /3-Function

As we have pointed out above, we expect that close to the continuum limita single function go{a) will ensure the finiteness of any observable. Hence we canuse any observable to determine the functional dependence of the bare couplinggo on the lattice spacing. Consider in particular the static qq potential discussedin chapter 7. As we have seen, it can be deduced from the large time behaviourof the expectation value of the Wilson loop. Within the quenched approximationthis potential, measured in lattice units, is a function of go a n d of R = R/a, whereR is the physical separation of the quark-antiquark pair. At a finite, but smalllattice spacing the potential in physical units is then given by

V(R,go,a) = -v(-,go), (9.5)

where go must be tuned to a in such a way that for sufficiently small lattice

spacing (9.5) becomes independent of a. Hence V(R, go, a) must satisfy the so-

called renormalization group (RG) equation

a-^-p(go)-Q-\v(R,go,a) = 0 (9.6a)

where

/?G?o) = - o ^ (9.66)


is the Callan-Symanzik /^-function (Callan 1970; Symanzik 1970).* Thus if /3{g0)would be known, we could integrate (9.6b) to obtain go(a). Of course, we cannotcalculate /3(<?o) exactly, but we may determine it in perturbation theory, where(9.6a) must also hold in every order. In the continuum formulation this can e.g.be accomplished by expanding the following expression for the potential

1 iga <fi dzÂ^(z)V(R,go,a) = - lim - ln(Pe " ) (9.7)

T—>-oo _L

in powers of the coupling constant go, and inserting the expression into the RG

equation (9.6).

F ig . 9-2 Classes of diagrams contributing to the potential in order g4.

The lines connect to arbitrary points on the Wilson contour

Because of the non-abelian nature of the gauge potential and the path orderingprescription, the calculation is much more involved than in the abelian case. Infig.(9-2) we show the diagrams contributing to (9.7) in order g$. On the lattice theSU(N) potential has been computed by Kovacs (1982), and by Heller and Karsch(1985) up to O(<7o)- Because of the complicated structure of the Feynman rules(see chapter 15) these computations are quite involved. Up to the above order thepotential is given by **

V{R) * ~9~MC2{F) I1 + 3 ° ( a ) S ln(7-5015 + i ^ a ) ^ ) ] (9.8)* Suppose we determine go (a) by holding a hadron mass M fixed at its physical

value; then M = ~M(go(a)). Hence go is a function of Ma, and adga/da =f(Ma) — —/3(go(a)), where f3(go) is the lattice version of the Callan-Symanzik/3-function.

** This expression differs from an earlier approximate calculation (Susskind1976)


where C2{F) is the quadratic Casimir operator in the fundamental representation.For SU(N) it is given by C2(F) = N

2^• Next, we demonstrate how one mayuse the perturbative expression (9.8) to determine the non-perturbative relationbetween g0 and the lattice spacing a, which ensures that the full potential Vbecomes independent of the lattice spacing for sufficiently small a. To this effectwe first determine the /3-function to lowest order in go by inserting (9.8) into theRG equation (9.6). One readily finds that for SU(N)

0(9*) "-%£& (9-9)

This we expect to be a good approximation for sufficiently small bare coupling.Because ft (go) is negative in the small coupling region, we conclude from eq. (9.6b)that, when the lattice spacing is decreased, #0 will be driven towards the fixed point#o = 0, corresponding to a zero of the /3-function. Hence, if for some value of thelattice spacing, go (as determined from a fit to experimental data) turns out tobe small enough to validate (9.9), then this approximation will improve as wedecrease o, and the continuum limit will be realized at vanishing bare coupling!This is asymptotic freedom as seen on the level of the bare coupling constant: aswe make the lattice finer and finer, and hence increase the number of sites within agiven physical volume, we must decrease the coupling accordingly to keep physicsthe same.

Integrating eq. (9.6b) one now obtains a relation between go and a:*

0 = -!- e ~ ^ , (9.10a)A/,

where/3o = H-/V/487T2, (9.106)

and AL is an integration constant with the dimension of a mass.

The above derivation of the /3-function in leading order was based on theperturbative expression for the potential (9.8) and the renormalization group

* We remind the reader that we have only considered the leading term in the/3-function. This term as well as that of order g^ determines the behaviour of thetheory near the i?G-fixed point. Their structure is independent of the observablethat one uses to compute it; in the usual continuum language: The first twocoefficients of the perturbative /3-function are independent of the renormalizationscheme.


equation. An alternative procedure is to relate the bare coupling constant tothe renormalized coupling constant in perturbation theory. By holding the latterfixed, while varying the lattice cutoff, one then obtains a perturbative expressionfor adgo/da, and hence the /^-function. We feel, however, that the above ap-proach (Kogut, 1983) is more transparent, since the potential has a direct physicalmeaning.

Having obtained the /3-function, let us now use the RG-equation (9.6) toobtain an improved expression for the potential. This will not only be very in-structive to the reader, but will also serve to illustrate the basic ideas discussed inthe previous section.

Consider the potential as given by the right-hand side of (9.5). Inserting thisexpression* into (9.6), one readily arrives at the following alternative RG-equationin which the derivative d/da has been traded in favour of the physical separationof the qq-paii:

[RJ^ + P(9o)~^V(R,go,a) = -V(R,go,a).

If we define the dimensionless quantity

V{R,go,a) = RV{R,go,a), (9.11)

then V satisfies the following differential equation:

[RfR-+l3(9o)^\v(R,go,a) = O. (9.12)

This is an interesting equation, for it tells us how an infinitesimal change in R canbe compensated by a corresponding change in the bare coupling constant, keepingthe lattice spacing fixed. In other words, any change in R can be absorbed intoa (independent) redefinition of the coupling strength. Let us be more explicit.Suppose we know V(R, g0, a) for some given separation Ro of the quark-antiquarkpair. Question: can we determine the potential for separations R = XRQ?. Con-sider V{XR0,g0,a), where V has been defined in (9.11). Then (9.12) leads to thefollowing equation involving dimensionless variables only:

\x-^ + (3(go)^V(\Ro,go,a) = O. (9.13)

* Our presentation parallels closely that of Kogut (1983).


One now easily verifies that the solution to (9.13) is given by

V(XR0, go, a) = V(ROj go(X), a), (9.14)

where the "running" coupling constant </o(A) satisfies an equation analogous to(9.6b):

A f? = -^SoW), (9.15)with

<?o(l) = So-

Inserting for /3(go) the expression (9.9), one finds upon integrating (9.15) that

A = e-w[i^-^]j (9_16)

where /?o is given by (9.10b). Solving (9.16) for <?o(A), we obtain

*<A> = r = j f c <917)

Hence what concerns the dimensionless quantity (9.14), scaling Rn with a factorA is equivalent to replacing the bare coupling constant go by (9.17). The cor-responding statement for the interquark potential now follows immediately from(9.11):

V(XRo,go,a) - j^-V(\Ro,go,a)^ ° (9.18)

= -V(R0,g0(X),a).

Let us now use this relation to obtain a renormalization-group improved expressionfor the potential, replacing the perturbative expression (9.8). This expressionsuggests that we should normalize the potential as follows:

2

V(a,go,a) = C^-.A-Ka

It then follows from (9.18) that V(\a,go,a) = Cg%(\)/(4.ir\a). By choosing A =R/a, we therefore find that

V(R on a) - r^R/a)V(R,go,a)-C ^R ,


which, upon substituting for g^R/a) the expression (9.17), becomes

^».«)~i^[ i -Û(*)] ' (9-i9a)

or r iV(R,go,a) = - £ - r ^ - (9.196)

4 7 r j R 2 / 3 0 l n ( f e - ^ )

Expanding the denominator in (9.19a) to leading order in g%, we recover eq. (9.8).Furthermore, requiring V(R,go,a) to be independent of a, we arrive at the non-perturbative relation (9.10), as follows immediately from eq. (9.19b). Hence Ax,is related to the strength of the potential V by

yiT>\ _ __^_ 11 ; 4KR2PO1U(RAL)

In contrast to the lattice spacing, A^ is a physical scale in terms of which dimen-sioned quantities can be measured. Thus by construction the quantity

AL = - e~™°^ (9.20)(I

satisfies an equation analogous to (9.6) and hence is a renormalization group in-variant quantity. Solving (9.20) for go we see that the bare coupling vanishes likel/ln(aAx,) as a -» 0:

9o{a} ~ ~2(30ln{aKLY

For the sake of completeness we also give here the relation between g0 and thelattice spacing, derived from the first two (universal) coefficients in the powerseries expansion of the /^-function for Np flavours of massless quarks: *

/%o) = -A) g30 - Pigl (9.21a)

1 2 1 SH

a = j^R(9o), (9.21c)

* In one loop order the (3 -function was computed by Gross and Wilczek (1973)and by Politzer (1973); the computation to second order has been carried out byCaswell (1974), Jones (1974), and Belavin and Migdal (1974)

(9.216)


R(9o) = (A)<?o2r/3l/2/3° e ' ^ l . (9.21d)

Let us pause here for a moment. Our renormalization group arguments haveshown that the potential is of the form

V(R) = -C^f

where a(R) increases with increasing separation of the quark and antiquark:

•"*=r^ska (922)

Clearly this result can only be meaningful if R is larger than the lattice spacing, butmuch smaller than the inverse lattice cutoff. For R = A^1 the effective couplingstrength diverges! This behaviour is quite different from that encountered in QED.Thus when a charge is inserted into the vacuum of QED it will polarize the mediumin such a way that the effective charge measured at a distance R is less than theoriginal charge. In QCD, on the other hand, the opposite phenomenon takesplace. The non-abelian couplings of the gauge potentials lead to antiscreening.This suggests that for large separations of the quark and antiquark the interactionmay become strong. Unfortunately, we have no way at present to calculate the qq-potential for large separations analytically, and we must take recourse to numericalmethods.

But how do we know whether we are extracting continuum physics whenperforming calculations on finite (rather small!) lattices?. The answer to thisquestion is found in the relation (9.21c,d). This relation tells us how the barecoupling constant controls the lattice spacing. Indeed, inserting (9.21c) into (9.3),the requirement (9.4) implies that for g0 & g£ = 0, 6(<7o) must behave as follows:

e>G?o) « C'eGR(so))de (9-23)50 ->-0

where CQ is a dimensionless constant. Quantities behaving like (9.23) are said toshow "asymptotic scaling". By studying the ratio Q(go)/(R(go))d@ as a functionof go in the scaling region one then determines the constant CQ.

In an actual numerical calculation on a lattice of finite size there will existin general only a narrow region in coupling constant space where 9 (go) scalesaccording to (9.23). This region is called the "scaling window". Thus since thelattice spacing is controlled by the bare coupling according to (9.21c,d), physics


will no longer fit on the lattice if 50 (and hence a) becomes too small (-> finitesize effects). On the other hand, by increasing the bare coupling, the lattice maybecome too coarse to account for fluctuations taking place on a small scale, andwe are leaving the continuum region. This is the reason for the narrow window.

Returning now to (9.23), we find upon inserting this expression together with(9.21c) into (9.3), that the observable defined in (9.4) can be expressed in termsof the lattice scale Ax, by

Qphys. = Ce(AL)d e . (9.24)

This shows that physical quantities can be calculated in units of the undeterminedmass scale Ax,. Hence a lattice calculation can only determine dimensionless ratiosof physical quantities (e.g. ratios of particle masses).

A particularly interesting example is the string tension a which is the coef-ficient of the linearly rising part of the interquark potential. Measured in latticeunits it is only a function of the bare coupling: <T(<7O)- In physical units, however,it has the dimension of (mass)2, so that the physical string tension is given by

o-= lim -â(go(a)) .

From the above discussion we hence conclude that for go —> 0, <T(<?O) must depend

as follows on go

v(go)*CAR(go))2, (9.25)

which in view of (9.21d) tells us that IT is a non-perturbative observable.

Finally we remark that the appearance of Ax, in a theory which a priori isfree of any scale (like the case considered here) is well known from perturbativecontinuum QCD, where the necessity of renormalizing the theory also requires theintroduction of a scale AQCD- The numerical values of Ax, and AQQD are not thesame and indeed differ substantially (see chapter 15).

In the following chapter we will make use of scaling arguments of the typediscussed above, to derive a relation between the potential of a quark-antiquarkpair and the energy stored in the colour electric and magnetic field.

CHAPTER 10

LATTICE SUM RULES

In chapter 7 we have shown that the static quark-antiquark potential canbe determined from the exponential decay of the expectation value of the Wilsonloop for large euclidean times. In the pure Yang-Mills theory we expect thatthis potential rises linearly for large quark-antiquark separations, leading to quarkconfinement. As we have already pointed out, this linear rise is believed to bedue to the formation of a narrow flux tube connecting the quark-antiquark pair,in which the colour field energy is concentrated. The energy stored in the colourfields should match, after subtracting the self energy contributions of the quarkand antiquark, the interquark potential, as determined from the Wilson loop. Inorder to be able to study the distribution of the field energy surrounding the quark-antiquark pair we need a non-perturbative expression for the field energy whichis suited for Monte Carlo simulations. To this effect we shall derive an energysum rule which relates the potential to the expectation value of an operator whichcan be identified with the field energy of a quark-antiquark pair. The same lineof reasoning leads to a similar sum rule relating the mass of a glueball * to theenergy stored in the chromoelectric and magnetic fields. Such sum rules havebeen first obtained by Michael (1987), whose derivation, however, turned out tobe incomplete [Rothe (1995a,b); Michael (1996)]. Before deriving these sum rulesit is instructive to illustrate the basic idea that goes into their derivation in a simplequantum mechanical example. Although in principle we can chose any potentialfor purposes of illustration, we prefer to be specific, and consider the harmonicoscillator, where the sum rule can be checked exactly.

10.1 Energy Sum Rule for the Harmonic Oscillator

Consider the imaginary time Green function, < q'\e~Hr\q >, whose pathintegral representation has been discussed in chapter 2. By expanding \q > ina complete set of energy eigenstates one immediately concludes that this matrixelement is dominated for large r by the contribution of the ground state, i.e.,

< q'\e-HT\q > ^ V o t ? ' ) ^ ) ^ ^ , (10.1)

* Glueballs constitute the particle spectrum of the pure SU(3) Yang-Mills theory.

130

Lattice Sum Rules 131

where tpQ is the wave function with energy EQ. By setting q' — q and integratingover q we therefore have that

f dq< q\erHr\q > —> e~EaT . (10.2)J r-yco

Hence by studying the behaviour of the lhs for large euclidean times we can inprinciple extract the ground state energy. What we are interested in, however,is an expression which relates the ground state energy to an ensemble averageof the kinetic and potential energy. In the following we now illustrate the basicideas for accomplishing this program for the case of the harmonic oscillator whoseHamiltonian is given by

n2 1

The first step in deriving an energy sum rule consists in expressing the imaginarytime Green function in (10.2) as a configuration space path integral. Proceedingas in chapter two, the lhs of (10.2) is given for small time step e by

fdq < q\e~HT\q >« ( ^ ) * f f[ dqne-S^m>«'N'%N=qo , (10.3a)

where

S[q; m, «, TV, e] = £ e[\m(9"+1 ~ 9 " ) 2 + \K&] , (10.36)n=0

and we have set q — qo. The euclidean time r is given by Ne. Notice that we didnot take the continuum limit (TV —I oo, e —t 0, with Ne = T fixed) on the rhs. Forsufficiently small e, and hence large TV = ^ the rhs will be a good approximationto the Green function. How small e has to be chosen to approximate continuumphysics will depend on the values of the dimensioned parameters m and n whichdetermine the relevant scales in the problem. Now for a fixed non-vanishing valueof e, the limit r —> oo in (10.2) is realized for TV —> oo. We are therefore led to thestatement that

rN-i(JHL)f T] da e-S[q;m,K,N,eh > e-E0N f l 0 4 l

4 ^ J i}0 ° "^°° 'where Eo = eEo is the energy measured in units of the lattice spacing e. By scalingall the dimensioned variables with e according to their canonical dimension, wearrive at the following dimensionless lattice version of (10.4)

G(m,k,N) —»• e -&(* .* )" (10.5a)iV-t-oo


where

G(m,k,N) = ( f )* f n>»e-5 [*A l* '%*=* . (10.56)27F ^ n=0

andiv-i 2

5[g; m, k,N]=J2 [^MVn+i " <?»)2 + o ' ( l a 5 c )

n=0

Here rh = me and k = e3/t are dimensionless parameters. In the continuum limitrh ->• 0, « ->• 0 with ^ = ^ fixed. From (10.5) it follows that the energymeasured in lattice units can be computed as the following limit

EQ(rh,k) = - lim — lnG(m,k,N). (10.6)iV-+co TV

Let us now repeat this excercise using another (fine) discretization e' = ê. Thenthe path integral representation of the lhs of (10.2) will differ from that in (10.3a) inthat e and N are replaced by e' = ±e, and N' = £7V (recall that N'e' = Ne = T).After scaling all variables with the original lattice spacing e according to theircanonical dimensions (so that the values of the parameters m and k are the sameas before), the new path integral expression for the lhs of (10.2) reads

G{m{i\m,N') = [ ^ ] * fNl[dqne-^'^'^'N\N^0 , (10.7a)

where

N'-l

S[q, m(O> «(0, N1] = Yl hm(0(9n+i - Qn)2 + T«(0«n] - (10-76)n=0 l 2

and

m ( 0 = £ m ; A(O = | « - (10.7c)

Hence the parameters m and K have now acquired a -dependence, which is verysimple in the present case. Certainly for £ > 1 the above path integral expressionfor the Green function is at least as good as the previous one, since we have madethe lattice even finer. For JV' —> oo (10.7a) will now behave like

G(m(£),k(Z),N') —> e-.Eo(Tfc(aft«))Ar' . (10.8)N' —»oo

as follows from (10.5a). But for sufficiently small e we must have that G(rh, k, N) =G(rh(^), h(t;),l;N): where rh = TO(1), k = k(l). We hence conclude that

6Bo(ni(0. K(O) = EQ(m, ft) , (10.9)


i.e., the lhs must be independent of £. By taking the derivative of the lhs with

respect to £ we therefore arrive at the following alternative equation for the ground

state energy

E0(m,K) - -[ — Js=i . (10.10)

But according to (10.8)

ô(m(O. «(O) = " ^lirn^ ^7 lnG(m«), «(O. ^ ' ) • (10-H)

Taking the derivative of (10.7a) with respect to £, making use of (10.7c), one

therefore finds that (10.10) translates into

1 1 J V '"1 1 1E0(m,k) = - + Jim^ — ^ < - - m ( g n + i - qn)

2 + -kq2n > , (10.12a)

n=0

where generically

< m >- M f i (I0.1B)

with S the action (10.5c). The expression (10.12a) can be further simplified.

Because of the periodic boundary condition go = QN> we have compactified the

imaginary time direction. The value of the expectation value in (10.12a) will be

independent of the time slice labeled by n. Hence (10.12a) can be simplified to

read1 1 2 1

EQ(m,k) = -+ < --rh4t + ^qj > , (10.13a)

where

\i = qe+i ~ qi , (10.136)

and £ denotes some arbitrarily chosen temporal lattice site. For the ground state

energy measured in physical units, EQ = -EQ, one then obtains

E0{m, «) = —+< -~mq2(r) + -nq2{r) > , (10.14)

where all variables are now dimensionful, and where we have used the more sug-

gestive notation q(r) instead of q^.

Equation (10.14) relates the ground state energy to an ensemble average of

the kinetic and potential energy, calculated with the Boltzmann distribution e~s.

In the euclidean version, however, the kinetic term is seen to yield a negative


contribution! Although this is naively understood by recalling that the transitionfrom real to imaginary time will map \{^)2 into -\{^)2, the above result stilllooks surprising at first sight. After all, the contribution of the kinetic energyshould be positive. A detailed calculation shows that < — \(jfr)2 > is itself diver-gent in the limit e —> 0. The divergence is cancelled by the first term in (10.14)and leaves a positive contribution, which has precisely the expected form.* Thefact that the kinetic contribution diverges does not come as a surprise. Lookingat the integrand on the rhs of (10.3a) we see that the width of the distribution inqe+i — qe is only of C(y/e), so that < qj > = < {qi+\ — qe)2/e2 > is expected to beofO(±).

Since the appearance of the minus sign in front of the kinetic term is socharacteristic of the euclidean formulation, and, in fact, will pop up again whenwe discuss the SU(N) gauge theory in the following section, it is instructive toderive the above result for the kinetic contribution in an alternative way.

The energy EQ is given by the ground state expectation value of the Hamil-tonian, i.e,

E0=<0\^P2 + V(Q)\0> ,

where P and Q are the momentum and coordinate operators, and |0 > is theground state (or "physical vacuum" in the language of field theory). We areinterested, in particular, in expressing < 0|P2|0 > as a euclidean path integral.To this effect consider the (imaginary time) Heisenberg operator

Q(T) = eHTQe-HT ,

and the corresponding conjugate momentum

P(T) = eHTPe-HT .

They satisfy the canonical commutation relation

[ Q ( T ) , P ( T ) ] = » .

The equation of motion for the operator Q(T) reads

Q(T) = eHr[H,Q]e-Hr = -±P(T).ill

* The cancellation will be demonstrated in section 4 of chapter 18 for the moregeneral case of a harmonic oscillator in contact with a heat bath, where a similarenergy sum rule holds.


Since |0 > is an eigenstate of the Hamiltonian it follows that < 0|P2|0 > = <0 | P 2 ( T ) | 0 >. Consider now the ground state expectation value of the euclideantime ordered product of Q(T)Q(T'). This time ordered product has a path integralrepresentation, given by (2.26a), i.e.,

< O\e(r-r')Q(T)Q(T')+e(r'-r)Q(T')Q(T)\O > = ^ j Dqq(T)q(r')e-1 dT[im^+Viri)]

whereZ= f Dq e~ S dT{?mf{T)+V{q{-T))]

Taking the derivative * with respect to r and r ' on both sides of this equation,and making use of the fact that the operators Q(T) commute for equal times, onefinds, after setting Q(T) = — ^ P ( r ) that

-±6{T - r')[Q(r), P(r)] - - L < 0|T(P(r)F(r'))|0 >

= \ J Dqq{T)q{T')e-$ dT^m^+v^™ .

Since the operators appearing in the time ordered product commute, we can omitthe time ordering operation. Now on a discretized time lattice, the continuum^-function is replaced by (see eq. (2.65))

e J_s. 2,-K

Hence setting r — r' (i.e. n = n') implies 6(0) -» K After replacing the equaltime commutator of Q(T) and P{T) by "i", we find that

± < 0|p2|0 >= 1 - I /' Dq\mq{T)*e-fdr[im?V+vto™ .2ui zc ZJ J 2

Hence the kinetic contribution to ô has precisely the form given by the first twoterms in (10.14)

* Of course when writing q(r) within a path integral, we always mean its dis-cretized version.


10.2 The SU(N) Gauge Action on an Anisotropic Lattice

In the case of our quantum mechanical example, the dependence of the di-mensionless parameters rh and k on the asymmetry parameter £ = jr was verysimple. In lattice gauge field theories the analogous parameter is given by £ = —,where a and aT are the spatial and temporal lattice spacings. On an isotropiclattice aT = a. The spatial lattice spacing now plays the role of the reference"lattice spacing" e in our quantum mechanical example. The parameterization ofthe SU(N) gauge action on an anisotropic lattice turns out to be more subtle thanin the quantum mechanical case, and does not follow from naive considerationsalone.

Consider the continuum action (6.17) for the pure SU(N) gauge theory. Inthe temporal gauge, Af = 0, where F^ = d4Af it has the form

SG[A] = fdr fd^xWUf^^ + SjiA} ,J J B,i

where

Si[A] = \[dr[d3xY^ F?(X,T)F?{X,T) ,i,j,B

and "B" are the colour indices. In this gauge the action thus has a similar form asthat of a quantum mechanical system, except that we are dealing with a systemwith an infinite number of degrees of freedom. A naive discretization of the degreesof freedom, labeled by x, and of the euclidean time integral, yields

SG[A] = I > a 3 { ^ £ [^ f («>"4 + 1) - Af(n,ni)]2 + \Y1 F?(n)F?(n)} ,

n r i,B i,j,B

where Af (n) and F$ (n) are the potentials and colour-magnetic fields at the latticesite n = (ft, 714). The discretized colour-magnetic contribution ^ j B F^(n)F^(n)does not involve an explicit aT-dependence. Having discretized the degrees offreedom, we have thereby introduced an additional scale, given by the spatiallattice spacing. By scaling the potentials with this lattice spacing according to theircanonical dimension, i.e., introducing the dimensionless potentials Af = aAf, theabove action can be written in the form

SO = Ei l^ i f W^(n) + ^F?(n)F?(n)} ,n l 4^

where F^ = aaTF^, F^j = a2F^j, and where a sum over repeated indices isnow understood. Written in terms of the field strengths, the expression appears


to be gauge invariant. For finite lattice spacing this is however not the case.To ensure gauge invariance for arbitrary lattice spacings, we must introduce thepotentials in the form of link variables, as discussed in chapter 6. The matrixvalued fields strength tensor F^v is then related to the plaquette variables byUpvin) « expligoFpvln)]. One then readily verifies that the following SU(N)action possesses the correct naive continuum limit

2/V 1 INSQ[U] = —-ZV. + —&>T, (10.15a)

wherev» = E t 1 - v^Tr{~Up- + ukx > (1 O-1 5 6)

p.

Vr = E t 1 " ^ F T r ( ^ + Uk)} ' (1O-15C)Pr ^

Here Ups and UpT are the spatial and temporal plaquette variables which are re-lated to the colour electric and magnetic fields. Notice that the ^-dependence ofthe kinetic (electric) and potential (magnetic) term is the same as in the case ofthe harmonic oscillator. This form for the action had been proposed by Engels et.al., and by Kuti et. al. [Engels (1981b), Kuti (1981)]. The naive ^-dependence ofthe above action is however too simple. The reason is that this action describesa complicated interacting system. As a consequence of quantum fluctuations thecouplings associated with the temporal and spatial plaquettes must be separatelytuned with the spatial lattice spacing and the asymmetry parameter £ to ensurethat physical observables become independent of a and £ close to the continuumlimit. Instead of the spatial lattice spacing one can also choose the bare couplingconstant go defined on an isotropic lattice. We therefore consider the follow-ing more general form for the action [Hasenfratz and Hasenfratz (1981b); Karsch(1982)],

SG[U] = P,V. + PTPT , (10.16a)

where

9U°°N°~C (10.166)

with Ss(<?Oif) and Sr(<7o>£) satisfying the condition

ffsCflo, 1) =9T(9O, 1) = 9o • (10.16c)


Hence*

0*^0 = ^jgl .

Let us pause here for a moment and return to the case of the harmonic oscillator.There the ^-dependence of the action was absorbed into a mass and couplingparameter. This ^-dependence was given by naive arguments alone: m(£) = £rhand K(£) = jk. Hence m(£) and K(£) are functions of £ and the dimensionlessmass and K parameters m = m(l) and k = K(1). When taking the continuumlimit, these dimensionless parameters must be tuned with the lattice spacing eaccording to m = me and k = e3n, where m and K are physical parameterswhich are held fixed when taking the continuum limit. The naive £ dependenceof the parameters in the quantum mechanical case correspond to the explicit £-dependence of the couplings (10.16b), while the ^-dependence of gs(go,0 andgT(go, 0 is a consequence of quantum fluctuations. As has been shown by Karsch(1982), ps and (3T can be related in the weak coupling limit (go —» 0, or $ —>• oo)to the coupling J3 on an isotropic lattice as follows

J0T0, i) =0 + 2NcT(Z) + dip'1),

Zps(p,i)=P+2Ncs(0 + O(p-1),

where ca(l) = 0, a = S,T. The ^-dependence of the functions ca(£) have beenstudied in detail by this author. When taking the continuum limit at a fixed valueof £, the coupling constant go in (10.16b), defined on a isotropic lattice, must betuned with the spatial lattice spacing as dictated by the renormalization group.As we shall see in the following sections, the ^-dependence of the couplings ga

in (10.16b) will lead to an energy sum rule for the ^-potential, and the glueballmass, which differs in an important way from that which one would expect naively.

10.3 Sum Rules for the Static gr^-Potential

Having parametrized the 5J7(iV)-Yang-Mills action on an anisotropic lattice,we can now proceed along similar lines as in section 2, to derive an energy sum rulefor the static quark-antiquark potential, which relates this quantity to the energystored in the chromoelectric and chromomagnetic fields of a quark-antiquark pair.To guide the readers attention, let us briefly outline the general strategy we shall

* We have denoted /? with a "hat", in order not to confuse it with the renor-malization group /3-function (9.6b) which will be relevant further below

(10.17)


follow. We first derive a sum rule for the static gg-potential in the pure SU(N)gauge theory, which relates the potential to the action stored in the chromoelectricand magnetic fields. For this we will only need to know the expression of theaction on an isotropic lattice, i.e., a lattice with equal spacings in the space andtime directions. This action sum rule will play an important role in our subsequentdiscussion of the energy sum rule. For the derivation of the latter sum rule weshall need the expression for the action on an anisotropic lattice, obtained inthe previous section. As in the case of the quantum mechanical example, theenergy sum rule then follows by requiring that in the continuum limit the potentialcalculated from a Wilson loop on an anisotropic lattice should be independent ofthe anisotropy parameter £=•£-• The very same requirement, when applied to thespecial case of a confining, linearly rising, potential leads to a "coupling constantsum rule" [Karsch (1982)], which will allow us to confirm that the energy storedin the chromoelectric and magnetic field matches the interquark potential. Inparticular, it will allow us to identify the contribution to the field energy arisingfrom the trace anomaly, which will be shown to account for 1/2 of the field energystored in the flux tube.

Let us briefly state what is meant by the trace anomaly. The colour-electricand magnetic field energy density is given by TOo, where TMl, is the energy mo-mentum tensor. On the classical level this tensor is traceless and symmetric. Onthe quantum level, however, it is known from perturbation theory that quantumfluctuations give rise to the so called trace anomaly: the energy momentum tensorT^ is no longer traceless. Now T^ can be trivially decomposed into a tracelessand trace part,

V - (T^ - -Ag^T) + ±gilvT , (10.18)

where T is the trace, T = ^ M ^ N One therefore expects that in addition tothe naive field energy density (which is given by an expression analogous to thatin electrodynamics) the potential receives a contribution arising from the non-vanishing trace of the energy momentum tensor. This trace is given in Minkowskispace by [Collins (1977)]

T[x) = ^^C(x) , (10.19a)9

where£(x) = ±FAltv(x)F*(x). (10.196)

A summation over repeated indices is always understood. (3{g) — fidg/d/j, is the/3-function of the continuum SU(N)-gauge theory with fi the renormalization scale.


Hence from (10.18), we expect that the interquark potential receives an anomalouscontribution having the form

Vanom{R) = \qg= ~ < [ d3X C(x) >gg . (10.20)

In the euclidean formulation, the action density will be replaced by its euclideancounterpart \FA

liV{x)F^v{x). Since the lattice provides a non-perturbative reg-ularization of the partition function, it is the appropriate framework for derivinga non-perturbative expression for the contribution of the normal and anomalouspart of the field energy to the interquark potential. Originally it was C. Michael(1987) who derived an action and energy sum rule for the gg-potential in an SU(N)gauge theory. The derivation was however incomplete. In the following we shallderive the corrected version of these sum rules. As we shall see, the derivation isstraightforward and leads in a very natural way to a decomposition of the fieldenergy into a normal and anomalous part [Rothe (1995a,b)].

i) Action Sum Rule

Consider the ground state energy of a quark-antiquark pair separated by adistance R. As always, quantities denoted with a "hat" are understood to bemeasured in units of the lattice spacing. The energy EQ(R) can be calculatedfrom the expectation value of the Wilson loop according to

E0(R) = - lim 4 In < W(R, f) > . (10.21a)T-»oo T

where

Recall that W(R, T) is given by the trace of the path ordered product of linkvariables around a rectangular loop with R and T lattice spacings in the spatialand euclidean time directions (Our notation here deviates from that in chapter 7).On an isotropic lattice < W(R,f) > is calculated with the action (10.15) with£ = 1. This action has the form

SG = KVT + V,) (10.22)

where /3 = ^ for SU{N). The energy E0(R), defined by (10.21a), is a function

of R and (3 and includes the self-energy contributions of the quark and antiquark.

(10.216)


Since these contributions do not depend on R, they can be eliminated by consider-ing the difference EQ(R,P) — EO(RO,P), where Ro is some reference (/(/-separation.We then define the qq potential by

V(Rj) = - lim 4 [In < W(R,f) >}subtr . (10.23)f-yoo T

From here on we will always assume that such a subtraction has been carried out,and shall drop the subscript "subtr" for simplicity. Following Michael (1987) wenow take the derivative of (10.23) with respect to /? and obtain

dV^R:^ = lim 4 < VT + Vs >,,_o , (10.24a)dp f->oo T

where < O >qq--o is denned generically by

The rhs of this expression is the expectation value of the operator O in the (/(/-statemeasured relative to the vacuum. In the limit T —> oo, the rhs of (10.24a) can beapproximated by*

< Va >, ,_0 « f < T'a >,?_o, (10.25a)T->oo

where

V'° = E f 1 " ^NTr(-Up" + Uk)]". /««*> ; (CT = T, s) (10.256)

is the contribution to Va arising from plaquettes located on a fixed time slice, andwith the Wilson loop extending from n\ = — to nn = j , with T — oo. Thistime slice is conveniently chosen to be the n^ = 0 plane. Hence,

* ^ = 4<« + « > . ^P

1 (10.26)—+ a E a 3 o < E2W + B2{x) >,,_o ,

naive ^-^ 1 "

where in the last step we have taken the naive continuum limit. E2 and B2 arethe square of the euclidean (!) colour electric and magnetic fields Ef, and Bf

* See the argument given by Michael (1987). Note that < V >qq-0 is calculatedwith a Wilson loop of very large extension in euclidean time.

(10.246)


summed over i = 1, 2, 3, and colours. The expectation value | < E2(x) + B2(x) >is not to be confused with the Minkowski field energy. In fact, 0(V'T + V's) is thecontribution to the action coming from a fixed time slice.

We next make use of the renormalization group to cast the lhs of (10.26) in aform involving the potential and its derivative with respect to R. In the limit ofvanishing lattice spacing "a" we have that

l-v(-,p{a))^V{R), (10.27)

where V(R) is the interquark potential in physical units. The dependence of0(a) = 2N/gl(a) on the lattice spacing is given, close to the continuum limit, bythe renormalization group relation (9.21c,d). The invariance of the lhs of (10.27)with regard to changes in the lattice spacing leads to

dlna d/3 K ' dR

where it is understood that this relation holds close to the continuum limit. Makinguse of this expression, equation (10.26) takes the following form*

mJ) + R d ^ = £-a<V>+V's>qQ-0. (10.28)

In the case of a confining potential, V(R,(3) = a((3)R, this equation reduces to

2a0)R =-^<K+'P's >«5-o , (10.29)

while for a pure Coulomb-type potential the lhs of (10.28) vanishes. Thus in anSU(N) gauge theory, where ^ ^ ^ 0, the potential cannot be of the pure Coulombtype.

ii) Energy Sum Rule

Let us now turn to our second objective, and derive a sum rule, relating theinterquark potential to the field energy. This sum rule is obtained by requiringthat a lattice regularization involving different lattice spacings in the temporaland spatial directions should lead to the same potential as that computed from

* As was first noted by H.G. Dosch the action sum rule of Michael (1987)contained an error (private communication).


an isotropic lattice. This is analogous to the requirement which in the case ofour quantum mechanical example led us to an energy sum rule for the groundstate energy. Here it is the ground state energy of a static quark-antiquark pairinteracting via Young-Mills fields.

Consider the expectation value of a Wilson loop on an isotropic lattice. Theexpectation value is computed with the action (10.22). Next consider a Wilsonloop on an anisotropic lattice with the same physical extension in the spatial andtemporal directions. The expectation value must now be calculated with the £-dependent action (10.16a,b), which we shall denote by S(£). Hence the numberof lattice sites in the euclidean time direction is now £T. Since both Wilson loopshave the same physical extension, their expectation values must be the same, ifthe lattice is fine enough to approximate continuum physics. This leads to therequirement that

<W(R,f)>S{l)=<W(R,Sf)>S{0 .

From (10.23) it then follows that*

V(R,0) = £V(R,ft(0>/MO) - (10-30a)

where

V-(£)&(0.&(0) = - > i l-ln< W(R,f')>s(i) . (10.306)T'—>oo 1

and- - [DUW(Rf)e-&Mp'+PTMp^

<W{R,T}>m=> SDyj.mv,,MTA • HO-*)Clearly

v"(£,ft(O.A-(O)le=i = V(RJ).

Equation (10.30a) implies that

^ [^(A./MO./MO)] =° • (10-31)This equation is the basis for deriving the desired energy sum rule. From (10.30b,c)one finds that

dV ,•QQ- =< Va >,5-o; <T = T,S .

* We suppress the dependence of f3a on goi o r alternatively on a, on which thecoupling <70 depends, since it is held fixed in the analysis.


Upon carrying out the differentiation (10.31), and then returning to the isotropiclattice £ = 1, we are led to the relation

v(R,P) = - (^j < Vr >9?-_o + (^j < v> >w-_0 .

Here V(R,$) is the potential in lattice units computed on an isotropic lattice.This expression can be written in the form

V(R,0) = T?_ < -V'T + V's >qq-0 -q+ <T'T+V'S >gg-_o , (10.32a)

where

Consider the first term appearing on the rhs of (10.32a). In the continuumlimit go —> 0 (or (3 —» oo). From the weak coupling relations (10.17) one finds that

T?_ —>• J3. (10.33)/3-+oo

Hence

7?_ < -P'r + V's >,,_o r~> 0< ~K + V's >95-0/3-+oo

i (10.34)- + «]>>3^ < -S2(^) + S2(^) >,«-o ,

where we have taken the naive continuum limit in the last step. This suggests thatthe first term appearing on the rhs of (10.32a) is the (euclidean) lattice versionof the usual Minkowsky field energy of a quark-antiquark pair (measured relativeto the vacuum), coming from the traceless part of the energy momentum tensor.Notice that just as in our quantum mechanical example, the contribution arisingfrom the "kinetic term", involving the time like plaquettes (which is associatedwith the electric field energy density), carries a negative sign in the euclideanformulation.

We now show that for the case of a linearly rising (confining) potential onecan actually determine what fraction of the potential is made from "ordinary"field energy. By making use of the action sum rule (10.28), we can cast the energysum rule (10.32) in the form

V(Rj)+V+^\v(Rj) + Rd^§®]=r1-<-V'T+V's>q,-o • (10.35)op L vR J

(10.326)


The combination rj+ defined in (10.32b) has been determined non-perturbativelyby Karsch (1982), by requiring that the string tension determined either from aspace-like or time like Wilson loop on an anisotropic lattice should yield the sameresult. Karsch finds that

1 9J3 PL(9O) a / i n o K ^7?+ = ~ 7 a l — = o P' (10.36a)

4 a In a 2gowhere

PL(9O) = -a^- (10.366)

is the 0- function discussed in chapter 9. Hence 77+ (d In a/9/3) = —1/4, so that(10.32a) becomes

V(R, 0t)-\ \v(R, 0) + Rd V ^ =V-< -K + Z >9?-o (10-37)

4 | dR J

For the case of a linearly rising (confining) potential this equation reduces to

±Vconf(Rj) = T?_ < -V'T + V's >OT-_o , (10.38)

which, according to (10.34) suggests that the "normal" field energy accounts foronly one half of the interquark potential. The other half must therefore be providedby the second term in (10.32a). If the first term in (10.32a) is the lattice versionof the semiclassical field energy, then the second term should be the contributioncoming from the trace anomaly. Thus making use of (10.36a) we have that

-f]+ <V'T+r's >,,_o= \ (— < L >gg-_o') , (10.39a)4 V go )

whereL = p{V'T + V's) (10.396)

is the (dimensionless) lattice version of the euclidean continuum Lagrangian den-sity integrated over all space at a fixed time. The energy sum rule (10.32a) there-fore becomes

V(R,0)=v-< -K + V's >qg-o +\ (^ < L >qô)

The quantity (2(3L/9O)L in (10.40a) has the form of the anomalous contribution tothe Hamiltonian following from the trace anomaly as computed in lattice pertur-bation theory by Caracciolo, Menotti and Pelisetto (Caracciolo, 1992). For weak


coupling the anomalous contribution to the potential (10.40), in physical units,

takes the form

Vanom(Rj(a),a) = A f ^xl < [E^x) + B*(x)} >,,-_0 . (10.41)

The rhs is a finite, renormalization group invariant expression. It can be expressed

in terms of a renormalized coupling constant g, and renormalized squared colour

electric and magnetic fields. The form remains the same, except that /?z,(po)/<?o

is replaced by /3(g)/g, where (3(g) = /udg/d/j, is the continuum beta-function, with

\i the renormalization scale.* The right hand side of (10.41) then just becomes

1/4 of the space integral of the trace anomaly (10.19) expressed in terms of the

euclidean fields.

Finally let us return to the action sum rule in the form (10.28). From (10.36a)

and (10.39b) we see that the rhs of (10.28) is directly related to the trace anomaly.

Thus we can write the action sum rule in the form

V(H,P)+K - ^ - < L >gq-0 . (10.42)dR go

The rhs of this equation is just the trace of the energy momentum tensor summed

over the spatial lattice sites at fixed (euclidean) time. Hence for a confining po-

tential of the form Vconf = &R the second term appearing on the rhs of (10.40)

yields, as expected, just 1/2 of the potential.

10.4 Determination of the Electric, Magnetic and Anomalous

Contribution to the qq Potential

The lattice energy sum rule has been checked in lattice perturbation theory

by Feuerbacher (2003a,b) up to O{gl). The computations are very involved. Some

technical details are given in Appendix A.

The reader may wonder: why check the sum rule? Is it not exact? While

the action sum rule (10.24a) is an identity following from the definition of the qq-

potential via the Wilson loop, the derivation of the energy sum rule (10.40) relies

on a number of input informations which, although quite plausible, are not self

evident: i) The structure of the action on an anisotropic lattice is taken to be given

by (10.16). This is the standard form of the action considered in the literature;

* See e.g. Dosch et al. (1995)


ii) Close to the continuum limit the potential should become independent of theanisotropy of the lattice, £, and satisfy (10.31). This must of course be so, if it is tobe an observable. It is however not self evident that this is indeed the case; iii) Aperturbative check of the sum rule requires a perturbative expression for TJ± . Thisexpression has been given by Karsch (Karsch, 1985). Hence checking the sum rulein perturbation theory would not only confirm the presence of a contribution to thepotential arising from the trace anomaly of the energy momentum tensor, but alsoconfirm indirectly the perturbative relations obtained from other considerations inthe literature. Having checked the sum rule (at least in perturbation theory), onecan extract the electric, magnetic and anomalous contributions to the potential.To this effect let us first write the energy sum rule (10.40) in the form

V(R, $) = lim I \v- < -VT + Vs >,,_o + ^ ^ / ? < VT + Vs >gg-_ol ,T-KX3 T I ^90 J

(10.43a)where we made use of the definition (10.39b), and, in accordance with (10.25a),we have made the replacement

< P'a >= lim i < Va > . (10.436)T—>oo 1

By combining this expression with the action sum rule (10.24a) one finds forSU(N)

Velec = .lim !„_ < -Vr >„-„= \V + \9M9,)~ + . ~ f | g ,

Vmagn ^ 1^ - „ _ < Vs > „ _ „ = -V + 450 /3 , (30)^ - f,- ^ ^ ,

1 dVVanom = ~^9OPL(9O) ^~2 i

(10.44)where, according to (10.17), and (10.32b), and making use of cT(l) = cg(l) — 0(see sec. 10.2),

2/VV- = —+ N(c'T - c's) + O{gl) . (10.45)

9oThe derivatives c'a — [dca(^)/d^-i have been determined by Karsch (1985).

Fig. 10-1, taken from Feuerbacher (2003) shows the electric, magnetic andanomalous contributions to the SU(3) potential computed from (10.44) and the ex-pression (9.8). Notice that while the leading electric contribution to the potential


Fig. 10-1 The next to leading order contributions, divided by g§, of

the electric field energy (dashed), magnetic field energy (dot-dashed) and

trace anomaly (dotted) to the SU(3) qq potential (solid line). The figure

is taken from Feuerbacher (2003).

is of 0(<?o)i the magnetic contribution is of O(<?o)i a n d same is true for the anoma-lous part. Note also that in the perturbative regime the trace anomaly contributessignificantly only for small R, and that the electric and magnetic contributions areof opposite sign, while in lattice simulations carried out in the non-perturbativeregion for large quark-antiquark separations they are found to be of the same sign.The solid curve in fig. (10-1) is the full potential in 0(#o)- This curve agrees withthat obtained from a perturbative calculation of the rhs of the energy sum rule(10.43).

10.4 Sum Rules for the Glueball Mass

The interquark potential is not the only observable for which one can derive anaction and energy sum rule. In fact, the way the sum rules (10.40) and (10.42) werederived, it is evident that similar expressions will hold for any observable whichcan be determined from the exponential decay in euclidean time of the expectationvalue of some operator. In the pure SU(N) gauge theory, such observables are themasses of glueballs states, which are eigenstates of the SU(N) Hamiltonian. Innumerical simulations particle masses are determined from the exponential decayin euclidean time of correlators of operators which excite the state of interest. Inprinciple the specific form of the operator is not important, as long as it excitesstates having a non-vanishing projection onto the state of interest. In practice,


however, a judicious choice needs to be made in order to enhance the signal in

Monte Carlo calculations. The anomalous contribution to the Hamiltonian, arising

from the trace anomaly, should also manifest itself in the energy and action sum

rules for the glueball mass. To determine the mass of a glueball with a given

set of quantum numbers one can construct such correlators from combinations

of space-like Wilson loops located at times — and , averaged over all spatial

lattice sites to project out a zero-momentum state [Michael (1987)]. Let us denote

the corresponding operators by G(—T) and G(T). The lowest glueball mass with

non-vacuum quantum numbers is then given by

M = - lim 4 In < G{f)G(-f) > . (10.46)T->oo T

which is the analog of (10.21a). Proceeding in exactly the same way as for the

case of the qq potential one then arrives at the following action and energy sum

rules for the glueball mass [Rothe (1995b)],

M=^M<l>1_0, (10.47a)

M = »7_ < -V'T + V's >!_0 + ^ ^ 2 1 _ Q (10.476)

where L has been defined in (10.39b), and where the bracket < O >i_o stands

generically for the following correlator

<0>1_o=lim<G^^-^-<0>.tôo <g(T)G(-T)>

Notice that the energy sum rule (10.44b) has exactly the same form as for the

gg-potential given by (10.40). The form of the action sum rule (10.44a) however

differs from (10.42), since in the case of the glueball mass, there is no analog of the

derivative term. This manifests itself in that the contribution of the anomalous

field energy to the glueball mass accounts for only 1/4 of the glueball mass, as

follows immediately from (10.44a). This result is consistent with that obtained by

Michael (1996) for the contribution of the "normal field energy" to the glueball

mass.

As we have seen in this chapter, the lattice formulation of the pure SU(N)

gauge theory on an anisotropic lattice has allowed us to derive in a straight forward

way a non-perturbative expression for the energy stored in the chromoelectric and

magnetic fields of a static gg-pair and glueball, in which the contributions arising


from the traceless and trace part of the energy momentum tensor are clearlyexhibited. In principle these sum rules provide us with an alternative way todetermine the string tension or glueball mass. But what is more important, thesesum rules allow us to obtain detailed information about the distribution of thefield energy in a flux tube connecting a static quark and antiquark. In most MonteCarlo simulations only the action density has been measured with good precision,since it is more accessible to Monte Carlo simulations (For a recent computationsee Bali (1995)). The energy density, on the other hand, including the anomalouscontribution, has so far not been studied in such detail in the literature.

CHAPTER 11

THE STRONG COUPLING EXPANSION

In chapters 7 and 8 we have shown how the static gg-potential V(R) canbe determined by studying the expectation value of the Wilson loop for largeeuclidean times. In QCD one believes that this potential confines quarks; moreprecisely, one expects that for large separations of the quark-antiquark pair, V(R)rises linearly with R up to distances where vacuum polarization effects, due to thepresence of dynamical fermions, screen the interaction. As we have seen, such abehaviour of the potential cannot be generated within perturbation theory. Forthis reason, we have no way (at present) to calculate it analytically, and henceare forced to determine it numerically. On the other hand, analytic statementscan be made in the strong coupling region. Indeed, in the absence of dynamicalfermions the structure of the action (5.21) for QED, and (6.25b) for QCD suggestsa natural expansion in powers of the inverse coupling. This is the analog of thehigh temperature expansion in statistical mechanics. In the following section weshall concentrate on the leading strong coupling approximation to the static qqpotential, ignoring vacuum polarization effects. As was first shown by Wilson(1974), this potential confines quarks. In fact it was this observation which hasstimulated the great interest in lattice gauge theories.

11.1 The (^-Potential to Leading Order in Strong Coupling

Consider the SU(N) lattice gauge theory in the pure gauge sector *. Thecorresponding lattice action is given by

S = -(3'Y^Sp + const.,p

where

P = 2N/gl

and **Sp=^Tr(UP + UP)

* We give the formulation for SU(N), since we are interested later also in thecase of SU{2).

** SP has been normalized in such a way that Sp -> 1 if Up ->• 1, i.e. closeto the continuum limit. The links J7M(n) and plaquette variables Up lie in theAT-dimensional fundamental representation of SU(N).

151


is the contribution to the action associated with a plaquette P (see chapter 6).The corresponding partition function reads

Z= fDUel352pSp , (11.1)

and the expectation value of the Wilson loop with spatial and temporal extentionR and T, respectively, is given by

(Wc[U])- jDUUpe0Sr ' ( 1 1-2 )

where Wc[fl is denned in (7.33). We next expand the exponential in (11.2) inpowers of the coupling (3:

P i n ' JSince each plaquette in the expansion costs a factor /?, the leading contribution,for /3 —» 0 to the numerator in (11.2) is obtained by paving the inside of the Wilsonloop with the smallest number of elementary plaquettes yielding a non-vanishingvalue for the integral. Consider the case of SU(3). As is evident from the in-tegration rules (6.23), the relevant configuration is the one shown in fig. (11-1).Hence the leading term in the strong coupling expansion of the numerator is pro-portional to (3A, where A is the minimal area bounded by the rectangular contourC: A = RT. On the other hand, the leading contribution to the denominator isobtained by making the replacement exp(/3 5 p ) —> 1. Hence for small /3, (11.2)

will be proportional to ( | J

Fig. 11-1 Leading contribution to (We) in t n e strong coupling approx-imation.

(11.3)

The Strong Coupling Expansion 153

The factor multiplying this expression may also be readily calculated by notingthat according to (6.23c) the colour indices of the link variables are identified ateach of the lattice sites, and that the integration over each pair of oppositelyoriented links yields a factor of 1/3. Now the number of link integrations for agiven set of colour indices is 2RT + R + T. Hence there is a factor (l/3)2At+A+t

coming from the integrations. But there are three possible colours associated withthe (R + 1)(T + 1) lattice sites. Consequently the factor multiplying (/3/6)A isgiven by (1/3)'4~:1. We hence conclude that in leading order of strong coupling

The gg-potential in the strong coupling limit is therefore given by

V{R) = - lim 4 ln(Wc[U}) = a(go)R, (11.4a)

where

* = -ln(^) (1 L 4 6)

is the string tension measured in lattice units. Thus in the leading strong couplingapproximation, QCD confines quarks: the expectation value of the Wilson loopexhibits an area law behaviour, (We) —» exp(-aRT). If this confining propertypersists into the small coupling regime, where continuum physics is (hopefully)observed, then cr(go) must depend on go according to (9.25). In the one-loopapproximation to the /3-function this dependence on go is given by

*(ffo) « Cae~^l. (n.5)

Its explicitly known form provides us with a signal which will tell us whether weare extracting continuum physics from a numerical calculation.

We want to point out that the same calculation performed for compact QEDwould also have given a potential of the form (11.4a). But in QED we know thatthis potential is given by the Coulomb law! Hence compact QED must exhibitat least two phases. In fact it has been shown by Guth (1980), using an actionof the Villain form, that the lattice U(l) gauge theory possesses a weak couplingCoulomb phase. That the strong coupling and weak coupling regions are separatedby a phase transition, has also been verified in numerical simulations, (see e.g.,Lautrup and Nauenberg, 1980). It is therefore important to check that in the caseof QCD there exists no such phase transition to a weak coupling regime.


11.2 Beyond the Leading Approximation

Higher order corrections to the potential (11.4) can in principle be computedby expanding the exponentials appearing in (11.2) in powers of the coupling /3and performing the corresponding (Haar) integrals over the link variables. As isevident from the expansion (11.3), the contribution of order (3n will involve (beforeintegration) all possible sets consisting of n plaquettes, including also multiples ofthe same plaquette. To each oriented plaquette P in a given diagram we associatea factor (§1 Tr Up, or ( § J Tr UP, depending on its orientation, and include

a factor 1/n!, where n is the multiplicity of P. However, of all the possible setscontributing to the sum (11.3), only a certain subset of plaquette configurationswill survive the integrations in (11.2) because of the integration rules (6.23).

Thus we are faced with two problems in computing higher order correctionsto the potential: i) enumerating all the diagrams contributing in a given order,and ii) calculating the Haar integrals. As the reader can imagine, already thebookkeeping problem associated with the first step will become non-trivial as wego to higher and higher orders.

To keep the discussion as simple as possible, let us exemplify the methodfor the two dimensional abelian model discussed in chapter 8. As we shall see itwill serve to illustrate the basic point we wish to make. Consider QED2 in thequenched approximation. The partition function (11.1) has the form

Z = f DU e § E p ( ^ + ^ ) , (11.6a)

where

UP = ei9p (11.66)

is an element of the U(l) group associated with the plaquette P. The expectationvalue of the Wilson loop is then given by (8.15). Expanding the exponentialsappearing in the integrands of this expression in powers of the coupling 0, onefinds that in the next to leading order the only plaquette configurations whichcontribute to the numerator and denominator are those shown in Fig. (11-2),since diagrams containing any unpaired link will not contribute after performingthe integrals over the link variables. *

* Notice that link variables of opposite orientations must appear paired.


Fig. 11-2 Next to leading order contributions to (We) in the strong

coupling expansion. Diagrams (a-c) contribute to the numerator (11.2), while

diagram (d) contributes to the denominator.

Hence we have three types of diagrams contributing to the numerator in thisnon-leading order: i) diagrams where the inside of the Wilson loop contains atriple of the same plaquette; the number of such diagrams is given by 3RT;** ii)diagrams where a multiple plaquette is attached to the contour C, as shown inFig. (ll-2b). The number of these diagrams is Nc f» A(R + f); and finally iii)disconnected diagrams of the type shown in Fig. (ll-2c). The number of suchdiagrams is approximately given by 2(V - RT - Nc/2), where V is the "volume"of the lattice. Since in the U(l) case the group integrals are trivial, we obtainthe following expression for the numerator, including the leading contribution,

** In counting the number of each type of diagram we must take into account thevarious possible relative orientations of the multiple plaquettes. Thus the factor 3arises from the expansion of (U + U*)3 = 3UWU + • • ••


(/3/2)*f:

Consider now the denominator of (11.2). In next to leading order it is givenby the diagrams of the type shown in fig. (ll-2d). Hence

,-1+i(§)VTaking the ratio N/Z, we see that to the order of approximation considered here,we are left with the following expression for the Wilson loop:

<^>~ (§)"[>-s(§)'4Hence we are led to the following expression for the static (^-potential,

V(R) = - lim 4 \n(Wc[U}) = aR,T->oo T

where the string tension a is given in this approximation by

& = -]*£ + ¥-. (11.7)

Notice that this potential is determined solely by connected plaquette configura-tions attached to the minimal surface enclosed by the contour C.

The U(\) case discussed above is of course the simplest example we couldchoose. Nevertheless, the number of connected and disconnected diagrams con-tributing to (11.2) will rapidly increase with the order. In particular, the occur-rence of multiples of a given plaquette will complicate the bookkeeping and thecomputation of the Haar integrals in the non-abelian gauge theory. This latterdifficulty may, however, be avoided by making use of the so-called character ex-pansion of the exponential of the action. We again demonstrate the procedure forthe case of quenched QED2, where the character expansion is just the well-knownFourier-Bessel expansion of exp(/?cos#p):

00

ef(c/p+^)= J- Iv{PYv6p- (11-8)l/= — 00


7j,(/3) is the modified Bessel function and exp(ii/9p) is the character of the pla-quette variable (11.6b) in the i/'th irreducible representation of the compact t^(l)group. Hence instead of having to deal with contributions to (11.3) arising frommultiples of the same plaquettes, every plaquette will now occur only once in theexpansion (11.8), but in all possible irreducible representations of the compactU(l) group. But because of the orthogonality relations satisfied by exp(ii/6), onlya few terms in the sum (11.8) will contribute to (11.2). As an example let usderive the result (11.7) using the expansion (11.8). Making use of the fact that inthe present case the Wilson loop can be written in the form (8.15), one finds uponinserting (11.8) in (8.15), that the numerator N is given by*

N= n f fflÊ7^^} n r ^ x x ^PeRc

J-* l I J PiRaJ~n v

where Rc denotes the region enclosed by the loop C. The integrals can be per-formed immediately and one obtains

The corresponding expression for the denominator in (8.15) reads:

pev * v

Hence the volume-dependent factor cancels in the ratio N/Z and we obtain theexact result

which coincides with (8.17b). Now for small (3 we have that

M/?)«l+(f)2

*<«-(!) He i r •Inserting these expressions into (11.9), we come back to our result (11.7).

* We have normalized the integration measure so that J dU = 1.

(11.9)


The fact that we were able to solve the above problem to any order in f3 isconnected with the 2-dimensional abelian nature of the model. In four-dimensionalgauge theories more sophisticated methods are required for carrying out the strongcoupling expansion. The interested reader may consult the extensive work carriedout by Miinster (1981) who has calculated the string tension for the case of anSU(2) gauge theory up to twelvth order in f3.

11.3 The Lattice Hamiltonian in the Strong Coupling Limitand the String Picture of Confinement

We have seen in the previous sections that for strong coupling the pure SU(3)gauge theory confines quarks. The usual picture of confinement is that the chro-moelectric flux linking a quark and antiquark is squeezed within a narrow tube(string) carrying constant energy density. As a consequence, the energy of thesystem increases linearly with the separation of the quarks. The purpose of thissection is to verify the string picture of confinement in the strong coupling ap-proximation within the framework of the Hamiltonian formulation as discussedoriginally by Kogut and Susskind (1975).

In the continuum formulation, the Hamiltonian is the generator of infinites-imal time translations and is obtained by a Legendre transformation from theLagrangean of the theory. Within the framework of the lattice formulation thenatural way of introducing the Hamiltonian is via the transfer matrix. In chapter2 we defined the transfer matrix for the case of a quantum mechanical system withgeneralized coordinates qa(a = 1, ...,n) by

Tqlq = (q'\e-*H\q). (11.10)

Here H is the Hamiltonian of the system, which, in the particular example consid-ered, had the form (2.8), and e is an infinitesimal time-step. The correspondingtransfer matrix was given by (2.37b). Expressed in terms of the transfer matrix,the partition function (2.32b) took the form

z= /n^rnv+^<'> (n-u)J a,t' I

where q^ = {q^} could be interpreted as the coordinates of the system at "time"Tfc on the euclidean time lattice. We then discussed a method which allowed usto construct the Hamiltonian from the knowledge of the transfer matrix. In this


section, we want to apply this method to the case of a lattice gauge theory. Tothis effect we will have to choose a gauge. The reason for this is the following.In the lattice formulation of a gauge theory, the action is a function of the linkvariables which live on all the links of the euclidean space-time lattice. Thereforethe action not only depends on the group parameters associated with the links lyingin fixed time slices, but also on those parametrizing the link variables that live onlinks connecting sites at different times. We can, however, eliminate the latter(bothersome) variables by choosing a gauge where all time-like oriented links* arereplaced by the unit matrix. This can always be achieved by an appropriate gaugetransformation, and corresponds in the continuum formulation to choosing thetemporal gauge, A^x) — 0. This we are allowed to do, since we are computing agauge-invariant observable, namely the Hamiltonian.

Of course, setting A4(x) = 0 does not fix the gauge completely. We arestill free to perform time-independent gauge transformations. This allows us tofix a subset of the remaining link variables oriented along the space directions.The variables that can be fixed by a time-independent gauge transformation arerestricted by the requirement that there exist no closed paths on the lattice alongwhich all the degrees of freedom are "frozen" (see Fig. (11-3)). This is obvioussince the trace of the path-ordered product of link variables along closed paths isgauge-invariant.

Fig. 11-3 The temporal gauge corresponds to fixing the link variables

on the solid lines to unity. The dashed links can be fixed by time independent

gauge transformations.

That the temporal gauge is the appropriate one for constructing the Hamil-

* Sometimes we use the word "link" instead of "link variable" if it is clear fromthe context what is meant.


tonian via the transfer matrix is also strongly suggested by looking at the contin-uum action of the pure SU(3) gauge theory. This action is given by (6.21), whereFpviB = 1) •••8) n a s been denned in (6.16). Setting A4 = 0, the action takes thefollowing simple form

SG[A]= fdr fd3x ^£(ifOrV))2 + ^ i ^ V ) ^ ? ( z , r ) ,I i,B i,j,B

where we have set x4 = r and where Af denotes the "time"-derivative of Af.Notice the striking similarity between the above expression and (2.11).

The lattice version of the action in the temporal gauge is of course morecomplicated (see previous chapter). But the above observation suggests that alsothere the action will acquire a structure which will allow us to write the partitionfunction in the form (11.11).

There is another important point that must be mentioned. By eliminatingthe time component of the gauge potential in the action, we are loosing one of"Maxwell's" equations, namely Gauss's law. Consider for simplicity continuumelectrodynamics in the absence of sources. The following discussion can be readilyextended to the case of a non-abelian theory and to euclidean space-time. Let usnot fix the gauge. By varying the action with respect to the time component of thepotential, one arrives at the equation V • E(x) = 0, where E is the electric field.The components Ei are the momenta canonically conjugate to Ai. Let us denotethe corresponding operators in the quantized theory by 7Tj. Then V • T{(X) = 0is a constraint for the canonical momenta. This constraint is lost when we set^4(2;) = 0 in the action, and must therefore be implemented on the states. Inthe temporal gauge V • TT{X) is not constrained to vanish. But its time derivativestill vanishes, as follows from the time-dependent version of Ampere's law. Now itcan be easily shown that the operators V • 7r(a;) are the generators of infinitesimaltime-independent gauge transformations. Indeed, consider the following unitaryoperator

T[A] = e-ifd3xA(Z)v-x(x)_

Because V • n is independent of the time, one finds, upon using the canonicalcommutation relations of the momenta and gauge potentials, that

T[A]Ai(a;)T[A]-1 = At(x) + diA(x).

Hence imposing Gauss's law on the states is equivalent to restricting the phys-ical states to be invariant under the residual group of time-independent gauge


transformations. This observation will play an important role when we discussthe energy spectrum of the Hamiltonian. In the presence of sources V • n(x) isreplaced by V • TT(X) — p(x), where p is the charge density. This latter quantity,constructed from the fermion fields, generates the corresponding transformationson the fermionic variables.

After these remarks, we now turn to the construction of the lattice Hamil-tonian in a pure gauge theory. To keep the discussion as simple as possible, wewill consider the abelian U(l) theory discussed in chapter 5. This theory will al-ready exhibit an important feature encountered in the non-abelian case. For thederivation of the lattice Hamiltonian in the SU(3) gauge theory, using the transfermatrix approach, we refer the reader to the paper by Creutz (1977).

Our procedure for constructing the lattice Hamiltonian parallels the quantummechanical case discussed in chapter 2. The first step consists in identifying thetransfer matrix, and writing the partition function of the U(l) gauge theory ina form analogous to (11.11). This is done by working in the temporal gauge.We then obtain the Hamiltonian by studying the transfer matrix for infinitesimaltemporal lattice spacing. Since the spatial lattice spacing is to be kept fixed whiletaking the temporal lattice spacing to zero, we must first rewrite the action (5.21)on an asymmetric lattice. Following Creutz (1977) we take the U(l) action to beof the form

s°w = A - D 2 - ( ^ + uk)} + A D 2 - (^+uk)i> (1L12)L9 P P " p

where p = aT/a, and where Ps and PT denote space-like and time-like orientedplaquettes, respectively. This is the analog of (10.15) with p = l/£. This actionpossesses the correct naive continuum limit. We next choose the temporal gauge;i.e., we set all the link variables associated with time-like links equal to one. Thena plaquette variable with base (n, n.4), lying in the i4-plane, is given by

Uu(n)= £ 1 =C/i(n,n4)C/-/(n,n4 + l)_ i(ei(n,n4)-ei(n,n4 + l))

where the dashed lines stand for link variables that have been set equal to one inthe temporal gauge. The coordinate degrees of freedom of the system are labeledby the spatial lattice site n and the spatial direction of the link variables. Let uscollect these labels into a single index a = (n, i) and set n4 = I. Furthermore,to parallel our quantum mechanical example, we relabel the group parameters,


parametrizing the link variables, as follows:

Then the partition function associated with the action (11.12) can be written ina form analogous to (11.11):

Z= fDUe-s°M= [Y[d9pl[Telt+1)eW, (11.13a)

where

r^ê-înrA11^^^1, (11.136)a

with the potential V defined by

vvlih = 4- £ t1 - \VP.W + 4.w)]- (n-13c)

The sum extends over all plaquettes located on the £'th time slice. The potential(11.13c) is therefore a function of the group parameters labeling the link variableslocated on this time slice. We next introduce a set of commuting operators {0 a }and simultaneous eigenstates of these by

ea\0)=ea\e),{e'\e)=Y[5{e'a-ea).

a

The states \9) are the analogue of \q) in the quantum mechanical example discussedin chapter 2. Let Pa be the momentum canonically conjugate to 0 a :

Then exp(—i£ • P), with £ • P = Yâ £a-Pai generates translations by £:

e-*p\6) = \9 + Z).

From here, and the orthogonality of the states \9) we conclude that

(0«+i) | e-*P|0M) =Y[5(dV+V - 0(f) - ^Q).a


Hence (11.13b) can also be written as follows

Tg(t+1)ew = (0V+»\T\eW), (11.14a)

where

T = ] l / d£ae~*° P a ~^ ( 1 ~ C O B W e- O T V [ e ] . (11.146)

To obtain the expression for the lattice Hamiltonian, we must now take the tem-poral lattice spacing to zero, while keeping the spatial lattice spacing fixed. Thismeans that p = aT/a approaches zero. But for small p the integrand in (11.14b)is dominated by those values of fa for which cos£Q ss 1. We are then allowedto replace 1 — cos£a by £^/2. Performing the remaining Gaussian integral, wetherefore find that, apart from an irrelevant constant,

T = e~aA^^P"+v[e>]).

The quantity appearing within brackets is the lattice Hamiltonian we were lookingfor:

a

In the coordinate representation the canonical momenta Pa are given by

p - i d

Recall that a stands collectively for the set (n, i), where n is the spatial locationof the lattice site and i is the direction of the link with base at n* To make thisexplicit, we shall write 0j(n) instead of 6a. Then the Hamiltonian (11.15) becomes

where V[0] has the following structure:

^ ] = - i D C H O ) + const.p* (11.166)

= -—r-^T(UUUU+ h.c.) + const.

* From now on it will be understood that n and x denote spatial positions.

(11.15)

(11.16a)


Here UUUU denote the product of link variables along the boundary of a plaque-tte on the spatial lattice. These link variables are parametrized by the angularvariables Oi(n). Thus, the contribution of a plaquette lying in the ij-plane withbase at n is given by

i—<—i

-' , , =ei6i(n)ei8:i(n+ei)e-i9i(n+ej)e-iej(n)_) 'ij

Before we proceed with the discussion of (11.16), let us establish the con-nection with the Hamiltonian in the continuum formulation. To this effect weintroduce the gauge potentials in the by now familiar way:

8i(n) —> o.gAi{x), x = no.

The Hamiltonian (11.16) then takes the following form in the continuum limit

Hô-\I**Zy^)2 + nA], (11.17a)

where

VlA] = \H IJPxFijWFvix) (11.176)

is the contribution arising from the spatial plaquettes in (11.16b) and S/SA^x)denotes the functional derivative with respect to -Aj(x). Its action on Aj(y) isdefined by

^ A , ( y ) = M < 3 , ( x _ t f ) .

The right-hand side of (11.17a) is nothing but the familiar expression for the tem-poral gauge-Hamiltonian in the so-called field representation, where the potentialsAi(x) are ordinary functions of the spatial coordinates and where the componentsEi of the electric field (which are the momenta canonically conjugate to Ai) arerepresented by the functional derivative —iS/SAi. This Hamiltonian acts on thespace of "wave functions", which are functionals of the spatial components of thevector potential. We therefore see that in the case of the U(l) gauge theory thekinetic part of the lattice Hamiltonian has a structure which is similar to that ofthe continuum formulation. There is, however, an important difference. Whilein the continuum formulation the potentials can take arbitrary values, the groupparameters 6i(n) take values in the interval [0, 2TT]. The wave functions are singlevalued functions of these variables. As a consequence the eigenvalue spectrum of


the kinetic term in (11.16a) is discrete. As seen from (11.16a,b), it is this termwhich dominates the Hamiltonian in the strong coupling limit:

^ ^ - g E ^ . (11.18)

This shows that in this limit the relevant contribution to the Hamiltonian comesfrom the electric field.

Let us now study the eigenvalue spectrum of Ho. The eigenfunctions of Hoare given by

^{iV,(n)}M = n ^ ( n ) ] A r i ( n ) ' ( 1 L 1 9 )

where

Ui(n) = ei8t^

is a link variable with base at n, pointing in the i'th spatial direction, and whereNi(n) is the excitation number of the link connecting the sites n and n + e . Thesewave functions are normalized as follows:

/ IT 27T ^W(")}[g^W(m)}[g] = hNdnft.lNjim)}k,m

The energy associated with the state (11.19) is given by

2

£{iv,(n)} = f ^ £ ( A ^ ) ) 2 . (11.20)i,n

It must, however, be remembered that not all such states are physical. Thus wehave emphasized before that only gauge-invariant states belong to the physicalHilbert space. Hence only wave functions built from products of link variablesalong one or several closed loops have a physical meaning.

Let us consider a few states. The lowest energy state is the one where nolink is excited. The wave function of the next higher energy state is given byan elementary plaquette variable located anywhere on the lattice. The energyassociated with this excitation is given according to (11.20) by 4(g2/2a), sincethere are four links on the boundary of an elementary plaquette. By excitinglarger loops or several loops, we obtain states of increasing energy. In Fig. (11-4)we show various types of excitations on a two-dimensional spatial lattice.


Fig. 11-4 Two types of excitations on a two dimensional spatial lattice.

The above strong coupling picture gets modified when fermions are coupledto the link variables. Suppose that we introduce into the pure gauge mediuma very heavy pair of opposite charges at the lattice sites n and m, respectively.Then we can build a gauge-invariant state by connecting the two charges by astring built from the product of link variables. The lowest energy state of thesystem is obtained by exciting the links along the shortest path connecting the twocharges. Each excited link contributes an energy g2/2a. Consider, in particular,two charges located on a straight line path on the lattice, as shown in Fig. (ll-5a).Their energy is given by

Eo = | - L , (11.21)

where L is the separation of the pair measured in lattice units. Thus in the strongcoupling limit we have confinement already in the U(l) theory!

The above result was obtained in the strong coupling limit and in the absenceof dynamical (finite mass) fermions. The effect of the potential in (11.16) for finite(but large) coupling can be calculated in perturbation theory. Clearly the statesdiscussed above are no longer eigenstates of the Hamiltonian when the potentialis turned on. In this case the string connecting the two charges will be allowedto fluctuate. To see this consider the action of the potential on the state depictedin Fig. (ll-5a). In particular consider the effect arising from those plaquettesin the potential (11.16b) having a link in common with the string connectingthe two charges. In Fig. (ll-5b,c) we show the possibilities corresponding tooverlapping flux lines. The wave functions associated with these states are thoseconstructed from the link variables shown in Fig. (ll-5d,e), where the darkenedline corresponds to a doubly excited link. For finite coupling the lowest energystate of the charged pair will include these (and many other) excitations. Thus


• • o

(a)

• »' o • t o

O) (c)

| " { I " } + •-•-»•—I I—»—O •-+—I » I » O

(<*) (e)

Fig. 11-5 (a) Eigenstate of the Hamiltonian in the strong coupling

limit; (b,c) Configurations created by the action of the potential on the

state depicted in (a); (d,e) Corresponding link configurations from which the

wave functional (11.19) are constructed. The darkened line in (e) denotes

a doubly excited link. The dots stand for the other three possible ways of

attaching the plaquettes on the three dimensional spatial lattice.

the original rigid string begins to fluctuate when the potential is turned on.

Let us compute the change in energy AE arising from fluctuations of thetype depicted in Fig. (ll-5d,e). They can be computed by standard perturbationtheory: Let \Eo) denote the ground state of the qq-p&ir in the strong couplinglimit, depicted in Fig. (ll-5a), and \Ef) the eigenstate of Ho corresponding tolink configurations of the type shown in Fig. (ll-5d,e). Then

aB = E!<™!!, („.„,where the sum extends over all states \Ef) with the fluctuation taking place any-where along the line connecting the two charges . The energy EQ is given by(11.21). Furthermore, from (11.20) we obtain for the states of type d and e,

4d)=g(Z + 2),

4e )=g(L + 6),


irrespective of the location of the fluctuation. For all these states

(Ef\V\E0) = |nf^(°[W]iM0]1

The energy shift (11.22) is therefore given by

Hence in this (oversimplified) example, we arrive at the following expression forthe string tension measured in lattice units:

For the case of the pure SU(3) gauge theory, Kogut, Pearson, and Shigemitsu(1981) have computed the string tension up to O(g~24). These computations(which are quite non-trivial) suggest that the strong coupling expansion, whencarried out to sufficiently high orders, yields a string tension in the intermediatecoupling region which can be fitted at the low coupling end with the square ofthe function R(g) defined in (9.21d). In the strong coupling regime the flux tubeconnecting the two charges will have a finite width of the order of the latticespacing. But as the coupling is decreased, not only the width of the flux tube willincrease, but also its shape will eventually undergo strong changes. Furthermore,the number of string configurations of a given length L will increase dramaticallywith L.

O—*—•• *• O • • > O

Fig- 11-6 Breaking of the string due to pair production.


In the presence of dynamical fermions this picture is further modified. Thusfor sufficiently large separations of the two charges, the system can lower its energyby creating an oppositely charged pair connected by a string as shown in Fig. (11-6). This corresponds in the strong coupling approximation to the hadronizationprocess mentioned at the beginning of chapter 7. This concludes our discussion ofthe string picture of confinement in the strong coupling Hamiltonian formulation.We have seen that for strong coupling, the U(l) theory leads to a non-vanishingstring tension. This result is in agreement with that obtained by studying theWilson loop. In the continuum limit confinement should of course be lost. On theother hand, the SU(3) gauge theory should still exhibit confinement in this limit.We have concentrated our attention on the U(l) gauge theory since it allowed usto demonstrate in a simple way how a string picture of confinement emerges inthe strong coupling limit, if the theory is compactified. The non-abelian case is ofcourse more complicated but the basic ideas are the same. For a discussion of theSU(3) gauge theory, the reader may confer the paper by Creutz (1977).

Finally we want to mention that the construction of the lattice Hamiltonianvia the transfer matrix is not the only method. For an alternative procedure basedon canonical methods the reader may consult the review article by Kogut (1983)or the lectures of this author given at the International School of Physics, EnricoFermi (Kogut, 1984).

CHAPTER 12

THE HOPPING PARAMETER EXPANSION

The inclusion of fermions in lattice calculations is a very non-trivial problem.In the pure gauge theory correlation functions can be calculated by Monte Carlomethods (see chapter 16). Such methods cannot be applied directly to path in-tegrals involving Grassmann variables. To overcome this difficulty one must firstintegrate out the fermions. The resulting path integral expression then only in-volves bosonic variables and one can evaluate them, in principle, using statisticalmethods. But the Boltzmann distribution is now determined by an effective actionwhich is a non-local function of the link variables. Not only that. When calcu-lating fermionic correlation functions, the ensemble average must be performedover expressions which are themselves non-local functions of these variables. Forthis reason most of the numerical calculations have been performed over manyyears in the pure gauge sector, or in the so-called quenched approximation, wherethe effects of pair production processes are neglected. Computations in full QCDwere restricted to very small lattices. With the advent of the supercomputers thesituation has improved substantially, and numerical calculations with dynamicalfermions performed on larger lattices have become feasible. But the computertimes required are still astronomical.

A brute-force numerical calculation does not provide us with much insightinto the detailed dynamics. One would therefore like to have some analytic way ofestimating the effects of dynamical fermions on physical observables. The hoppingparameter expansion (HPE) allows one at least to study these effects for large barelattice-masses of the quarks (it is therefore only useful far away from the continuumlimit). Furthermore, when combined with the strong coupling expansion, it alsoprovides us with a physical picture of how hadrons propagate on a lattice, andhow pair production processes influence the observables.

The purpose of this chapter is two-fold. We first want to show how thecalculation of arbitrary correlation functions of the fermion fields and the linkvariables can be reduced to a pure bosonic problem, which can in principle behandled by numerical methods. This will be the subject of the following section.In section 2 we then discuss the hopping parameter expansion of the two-pointfermion correlation function, and show how hadrons propagate on the lattice in acombined HPE and strong coupling expansion. Section 3 is devoted to the hoppingparameter expansion of the effective action. This allows one to include the effects of

170

The Hopping Parameter Expansion 171

pair production processes. Finally, in section 4 we demonstrate in some exampleshow the HPE expansion respects the Pauli exclusion principle, which forbids thattwo identical quarks (or antiquarks) can occupy the same lattice site.

12.1 Path Integra] Representation of Correlation Functionsin Terms of Bosonic Variables

In the following we shall consider Wilson fermions of a single flavour. LetA, B, C... be collective indices for the colour and Dirac degrees of freedomlabeling the quark fields. Thus tpA, i>B will stand for VS a n d i>bp, respectively.The path integral representation of a general correlation function of the fermionicand link variables is then given by

(tl>Al (m) ' • • 1>AN (nN)TpBl W - i>BN {mN)U^ (h) • • • U'f (*,)>

f DUD{^)Âl (m) • • • $Bl (mi) • • • U£dl (fti) • • • e - ! W E W . f l (12.1a)~ / DUD(tpip)e-sQCD[u^^]

where *

D(W) = E[ di>A(n)dipA(n), (12.16)A,n

and where U£b(n) denotes a matrix element of the link variables C/M(n). Notice

that (12.1a) contains the same number of tp and ip fields since all other correlationfunctions vanish because of the Grassmann integration rules discussed in chaptertwo.

Consider the fermionic contribution to SQCD- For Wilson fermions it is givenby (6.25c). Let us write this contribution in the form

S™ = ^J2f(n)?-m[U]f(m), (12.2a)

where

K = — (12.26)8r + 2M0

* For later convenience we have chosen to write the measure in this form, ratherthan DtpDip. This has of course no influence on the ratio (12.1a), but eliminatessome unpleasant minus signs which would otherwise arise in intermediate steps ofthe calculation.


is the so-called hopping parameter (for reasons which will become clear below),and Knm are matrices in Dirac and colour space:

Knm[U] = Snm-H-KJ2[(r-7Ûll(n)Sn+iitm + (r+jfi)Ul(n-fi)5n^m}. (12.2c)

Thus Knm is of the form

Knm = Snml - KMnm[U], (12.3a)

where the only non-vanishing matrices Mnm are those connecting neighbouring

lattice sites:M n n + A [f / ]=(r - 7 | l )C/ M (n) ,

(12.36)

M B B - A [ ^ = ( r + 7 ( 1 ) ^ ( n - A ) .

The corresponding expressions for the matrix elements read:

{Mn,n+p)aa,l3b ={r - -y^)a0(U^{n))ab,(12.3c)

(Mnin-p,)aatl3b =( r + -y,j,)ap{Ul(n - p,))ab.

In the literature it is customary to eliminate the factor 1/2/t appearing in (12.2a)byscaling the fermion fields with \[2k : ip —>• y/îp, i> —>• -\/2/«/>. The correspondingJacobian in the fermion integration measure drops out in the ratio (12.1a). Henceby eliminating the factor 1/2K in (12.2a),the path integral expression (12.1) yieldsthe original correlation function multiplied by (1/2K)J V . With this in mind, theaction we shall be working with is given by

SQCD = SG[U] + S™[u, $, t/i], (12.4a)

where SQ has been defined in (6.25b), and where

4 * ° = Â(n)KnA,mB[U]i>B(m), (12.46)A,B

with

KnA,mB[U] = SnmSAB ~ K(Mnm)AB. (12.4c)

Here we have made use of the collective notation for the Dirac and colour indices

introduced above. Thus 6AB = SapSab, and {Mnm)AB = {Mnm)aa,0b- Using the

fact that (see chapter 2),

det K[U]= I D{^)e-S^)[u^'^\ (12.5)


one readily verifies that (12.1a) can also be written in the form

_ / DU(ipAl (ni) • • • 4>Bl (mi) • • -)gFUc^ (h) • • • det K[[/]e-s»WfDUdetK[U]e-s°M

(12.6a)

where (î^ (ni) • • • ^>BX (m i ) ' ' ')sF is the purely fermionic correlation function in

the "external" field defined by the link variable configuration {[/^(n)}; i.e.

- / W ) ^ (TH) •••jBl (mQ • - • e - 4 w ) [ ^ . ^< ^ i ( « l ) • • • </>!?! ( m i ) • • -)SF = 1 ,w)

fD(ipt/;)e-SF '[u,i>M

(12.66)

In the so-called quenched approximation (q.a.), where detif[C7] is replaced by a

constant, the correlation function (12.6a) is just the ensemble average of the ex-

ternal field correlation function (12.6b) multiplied by the link variables appearing

on the left hand side of eq.(12.6a), and averaged with the Boltzmann distribution

of the pure gauge theory, exp(—SG[U\).

In chapter 2 we have shown that path integrals of the type (12.6b) can be

expressed in terms of products of the two-point correlation functions (2.57) ac-

cording to (2.59). A simple translation of these expressions to the path integral

(12.6b) leads to the following result:

(V'Ai ( « i ) •••il'Ai {jii)i)Bl (mi)--- ipBe (me))sF

= J2 î(rii)"-^K)^B1(m1)---ViB,(m^). (12'7)contr.

The right-hand side is meant to be the sum over all possible complete contractions

of ip — •tp pairs,* where a contraction is defined generically by

Mj^Bim) = K-\imB[U\. (12.8)

Hence in the quenched approximation we "only" need to compute the ensemble

average of products involving the two-point function (12.8) and the link variables

with a Boltzmann distribution corresponding to that of the pure gauge theory:

(Â1(ni)--^B1(mi)U^(k1)---)q.a.

= ( J2 « M i ( m ) •••$B1 (mi) • • -)UC^ (fci) • • -)sG, ( 1 2 - 9 )

contr.

* For the precise meaning of the right-hand side of (12-7) see the discussion in

chapter 2 after eq. (2.59).


where ( )sa denotes the ensemble average taken with the Boltzmann factor of thepure gauge theory. But neither the integral (12.6a) with detK = 1, nor K"1^]alone, can be calculated in closed form, and we are forced in general to computethese quantities numerically. We may, however, calculate (12.8) within the so-called hopping parameter expansion. This we will do in the following section.

12.2 Hopping Parameter Expansion of the Fermion Propagatorin an External Field

Consider the matrix K defined in (12.4c).

K[U] = 1 - KM[U] (12.10)

Its inverse is the fermion propagator for a given link-variable configuration. Forsmall K (i.e., large bare quark mass Mo) * we can expand K~l in powers of thehopping parameter as follows:

oo

K~l = (1 - KM)-1 = ^ V M £ .1=0

The corresponding expression for the matrix elements of K~x readsoo

KnA,mBlU] = ^nmÂB + K(Mnm)AB + ] T Kl ^ ( M n n i M n i n 2 • • • Mn(_im)AB,i=2 {„,}

(12.11)

where the non-vanishing matrices Mnm have been given in (12.3b). It follows from

(12.11) and the fact that M ^ connects only nearest neighbours on the lattice,

that the contributions to K~\ mB [U] in order K1 can be computed according tothe following rules:

i) Consider all possible paths of length I on the lattice starting at the latticesite n occupied by Vu (open circles) and terminating at the site m occupied byrpB (black blobs). As shown in fig. (12-1), these paths may intersect each other,and turn back on themselves, creating for example appendix-like structures.

ii)Associate with each link with base at k, and pointing in the ±/i direction,the matrices

Mfcifc+A = (r-7M)i7M(fc)

* We are thus necessarily away from the continuum limit, which is realized forMo -»0.


Mfc,fc_A = (r + 7 ^ t ( f c _ / i )

iii) Take the ordered product of all these matrices along a path following thearrow pointing from n to m, and take the AB-matrix element of this expression

iv) Sum over all possible paths leading from n to m.The number of diagrams reduces drastically if we choose the Wilson parameter

to be r = 1, for in this case diagrams with appendices, such as shown in fig. (12-lc), will not contribute. This is a consequence of the fact that (1 —7/J(l+7/i) = 0.In the following we shall choose r = 1.

Fig. 12-1 Diagrams contributing to K~^ m #[ f ] in the hopping pa-rameter expansion.

As an example let us calculate in order K4 the correlation function describingthe propagation of a colour-singlet scalar particle, consisting of a local gg-pair, be-tween two neighbouring lattice sites n, and n + /2. Let us denote the correspondingcomposite field by 4>(n):

a,a

The correlation function of <f> in the quenched approximation is given by:

(<P(n + (i)4>{n))q.a = X)(ffi(" + £ ) $ ( » + fl)ffi(")C("))5o

+ 52@p(n + fiWpjn + fj)fa(n)^{n))sG .


Consider the first term on the right-hand side which is the contribution describingthe propagation of a quark (and antiquark) between two neighbouring lattice sites:

((4>(m)4>(n)))q.a. = - Y,<KnlA,mB[U]K^nA[U])sa- (12-12)A,B

Here m •= n + p,, and A and B stand for the set (a, a) and (b, /?), respectively. Tocompute the contribution in order K4 of (12.12), we must expand K~x up to orderK3. In fig. (12-2) we show, for the case of a two-dimensional lattice, the variouspaths Ci, C2, C3 of interest.

Fig. 12-2 Diagrams contributing to K~A n+pjg[U] up to order K3.

The corresponding expressions obtained by applying the general rules statedabove with r = 1, then read

1=1

whererC l =(1 - 7M)

UCl =C^(n)

andTc2 = ( 1 - 7 ^ ( 1 - 7 M ) ( 1 + 7 , ) ,

Uc2 =Uu{n)U^n + u)Ut{n + A),

Ic, =(l+7l,)(l-7/1)(l-7,,)1

yc3=Ul{n-v)U»{n-v)Uv{n + iL-v) .


Fig. 12-3 Diagrams contributing in order K to the external field

correlation function (12.12).

Fig. 12-4 (a) Diagrams contributing to an external field 4-point corre-

lation function; (b) contribution to the correlation function in the leading

strong coupling approximation; (c) type of diagrams contributing to (12.12)

in the limit (3 —> 0.

Hence the relevant diagrams associated with the correlation function (12.12)in order K4 are those shown in fig. (12-3). Their contribution is obtained by takingthe product of the individual propagators K~^ making up the diagram.

So far we have considered quark-correlation functions evaluated for a givenconfiguration of the link variables. In the quenched approximation (det K[U] —¥const,), these correlation functions must still be weighted with a Boltzmann dis-


tribution of the pure gauge theory, i.e. with

eSa[U]PG[U] = fDUe-ScM • d2-1 3)

It is instructive to calculate the ensemble average of some simple correlation func-tions in leading order of the strong coupling approximation. Consider for examplethe contribution of O(ns) to the correlation function (12.12) depicted in fig. (12-4a). This contribution must be averaged with the Boltzmann distribution (12-13).Expanding the action So according to (11.3), we see that the leading contributionto the ensemble average is obtained by paving the minimal area enclosed by thequark paths with plaquettes, each of which contributes a factor /?. This is shownin fig. (12-4b). Hence in the strong coupling limit (/? -» 0), the quark-antiquarkpair can only propagate as a local unit, as shown in fig. (12-4c).

Fig. 12-5 Diagram contributing to the gauge invariant correlationfunction < ip(m)U...Uip(n)ip(n')U...U'ip(m') > in order K6, for thecase where n and m (n and m ) are neighbouring lattice sites.

On the other hand consider the propagation of an extended meson. Then wemust study correlation functions of a gauge-invariant composite field having thefollowing schematic structure

4>qg = i>{m)U---U^(n),


where U •••U denotes the matrix product of link variables along a path connecting

the lattice sites n and m. In fig. (12-5) we show a diagram contributing to the"meson" propagator

T = {i>(m)U • • • Uip(n)i>(n')U • • • Uip{m')}

in O(K6) of the HPE, and O(/33) of the strong coupling expansion, for the casewhere m and n (ml and n') are neighbouring lattice sites. The dashed linesstand for the product of link variables appearing in the above correlation function.Notice that the "strings" connecting the quarks and antiquarks (dashed lines) areessential, since the group integral over a single link variable vanishes. For thesame reason a single quark cannot propagate in any order of the HPE. This canbe viewed as another statement of confinement.

12.3 Hopping Parameter Expansion of the Effective Action

As we have just seen, we can in principle compute a general correlation func-tion in the quenched approximation by calculating the ensemble average of prod-ucts of fermionic two-point correlation functions and the link variables, with aBoltzmann distribution given by (12.13). In full QCD, however, we must also in-clude the fermionic determinant, det K[U], in (12.6a). This is equivalent to takingthe ensemble average with the following probability distribution

eSeff[U}

PW = fDUe-8.,,m • (12-14fl)

where the effective action, Seff, is defined by

Seff[U] = Sa{U] - IndetK[U}. (12.14b)

Correspondingly, (12.9) is now replaced by

(^1(ni)..^B1(m1)..t^;dl(fci)»> = ( E ( V ' A x K ) . . ^ ^ ) . . ) ^ 1 ^ ! ) - ) ^ .contr.

We remark that it is not obvious that (12.14a) can be interpreted as a probabilitydistribution. For this to be the case one must show that Indet K[U] is real andbounded from above. That det K[U] is real can be shown rather easily. We firstnotice that K[U] is not a hermitean matrix. From the definition (12.2c) and thehermiticity of the 7-matrices, it follows that

KmB,nA = 5nmSap6ab-Kj^j\(r + 'Ylj,)ap(Uli(n))ab6n+ji,mM

+ {r~ l»)ap{Ul{n - fl))abSn-p,,m\ ,


i.e.,

^mfl.nA / KnA,mB-

In fact the hermitian adjoint of the matrix K is obtained from K by replacing 7M

by -7 M . This observation can be stated in the form

K^ = 75-^75 ,

where 75 is a matrix in Dirac space which anticommutes with all 7M's, and whosesquare is the unit matrix. Since det K^ = det 75^75 = det K, it follows that det Kis real. Furthermore it has been shown by Seiler (1982) that for K < 1/8 (which isrealized for the usual choice of Wilson parameter r = 1) 0 < det K[U] < 1. Hencewe can interpret the expression (12.14a) as a probability distribution.

In contrast to SG[U], the effective action (12.14b) is a non-local function ofthe link variables, and its numerical calculation demands an enormous amount ofcomputer time. Before the advent of supercomputers, one has therefore mainlyconcentrated on calculating fermionic correlation functions in the quenched ap-proximation. This may be a reasonable approximation to estimate such quantitiesas hadron masses. On the other hand, there are problems where vacuum polar-ization effects play a crucial role. Thus for example the screening of the quark-antiquark potential at large distances is due to processes involving the creation ofquark-antiquark pairs. These effects arise from the determinant of the fermionicmatrix K[U\.

We now derive graphical rules for computing In det K[U] in the hoppingparameter expansion. To this effect we first rewrite In det K as follows

\n det K[U] = Tr In K[U],

where "Tr" denotes the trace in the internal space as well as in space-time. Sub-stituting for K[U] the expression (12.10) and expanding the logarithm in powersof KM, we have that*

0 0 1

Tr\nK[U} = -Y,jTr(Me). (12.15)1=1

* Actually, there is no contribution to the trace coming from t — 1, since Mnm

connects different lattice sites.


To get a diagramatic representation of the sum in (12.15), we write out the space-time trace explicitly:

TrlnK{U] = - E T E i r ^ M n ^ - ^ , (12.16)

where "tr" now stands for the trace in the internal space. The non-vanishingmatrices Mnm have been defined in (12.3b). From the structure (12.16) it follows

immediately that the contributions of order K1 can be associated with closed pathsof length £ on the lattice with an arbitrary sense of circulation. These contributionsare calculated as follows. Consider a given closed geometrical contour Ct0 on thelattice, with £$ the perimeter of the contour measured in lattice units. Becauseof the space-time trace in (12.15) a path winding around the contour one or moretimes can start at any one of the 0 lattice sites on Ce0 • Hence each path is weightedwith a factor £Q • A path tracing out the contour Cg.Q n times will however contributein order Kne°. Hence the contribution to (12.16) associated with all possible pathson the closed contour Ce0 is given by

°° Knt0

n = l

where Mct is the path ordered product of the matrices (12.3b) along the geo-

metrical contour Cea. The rhs of (12.16) is now obtained by summing the above

expression over all possible geometrical contours of arbitrary shape, and arbitrary

perimeter:

00 (Kto)nTr In K[U] = \ndet K[u) = - ^ E E ^~^tr¥cto • (12-17)

4) {C,0}n=l

From here we can further obtain an elegant expression for the determinant itself[Stamatescu (1982, 1992)]. Thus (12.17) can also be written in the form

\ndetK[U] = E E i r l n ( 1 ~ Ki°¥ct0) •to {Ce0}

It therefore follows that

det K[U] = I I I I d e t ( 1 " Kl°¥ct0) • (12.18)to {Cta}


Note that the determinant appearing in this expression is that of a finite matrixin colour and Dirac space.

In fig. (12-6) we show some diagrams contributing to (12.17). As in the caseof the external field Green's function discussed in section 2, they also include pathspassing through a given lattice site an arbitrary number of times, as well as pathshaving appendix-like structures. As before these appendices do not contribute ifthe Wilson parameter is chosen to be r = 1.*

Fig. 12-6 Diagrams contributing to In detK in the hopping parameter expansion.

The above hopping parameter expansion for the effective action (12.14b) canbe combined with the corresponding hopping expansion for the fermionic corre-lation function discussed in section 2. In those regions of parameter space wherelndet K and K"1 can be approximated by a few terms in the expansion (i.e. smallhopping parameter, or large Mo), the computational effort is thereby drasticallyreduced. In practice, however, where one is interested in working in the scalingregion with small quark masses, non-perturbative methods for computing K~l

and det K are required.

The hopping parameter expansion can be used to obtain a qualitative pictureof the screening of the static quark-antiquark potential due to "pair-production"processes (see e.g. Joos and Montvay, 1983). The following picture is very crude.Within the framework of the strong coupling expansion, the Wilson loop will bepaved not only with the plaquettes arising from the action SQ\U], but also withclosed loops built from link variables, arising from the fermionic determinant. Inthe HPE, these latter contributions come with a power of n, determined by the

* In the continuum limit physics should be independent of the choice for r, aslong as r 7 0.


Fig . 12-7 Contribution to (We) coming from the fermionic deter-

minant (inside loop) which screens the qq -potential in the strong coupling

limit and in lowest order of the hopping parameter expansion.

length of the loop. These different loop contributions will compete with each otherin determining the interquark potential. In particular, in the strong couplinglimit, P —> 0, the Wilson loop will be paved with loops arising only from thefermionic determinant. In fig. (12-7) we show a diagram which corresponds to adynamical quark and antiquark being created at the positions of the (infinitely)heavy antiquark and quark, respectively. The contribution of this diagram is ofthe form

( W c [ t / ] ) « « 2 ( * + f ) = e ( « + f > l n K \

which shows that the dynamical qq-pair leads to screening of the heavy quark-antiquark potential, which in this crude approximation is just a constant: Vqq =- I n * 2 .

12.4 The HPE and the Pauli Exclusion Principle

In the previous sections we obtained graphical rules for calculating the two-point fermion correlation function and the effective action in any order of thehopping parameter expansion. In particular the rules for calculating the quarkpropagators in a background field, corresponding to a given link-variable config-uration, where based on eq. (12.7), derived in chapter 2 using the concept of agenerating functional. We now want to rederive the results obtained above byapplying the Grassmann integration rules (discussed in chapter 2) directly to thepath integral expressions for the external field correlation function (12.6b) and thedeterminant of K[U]. In this way we shall demonstrate explicitly that the Pauli


exclusion principle, which forbids that the same lattice site can be occupied bytwo identical quarks or antiquarks, is satisfied.

Consider the exponential, exp(—SF), in (12.6b) where SF is given by (12.4b).Inserting for K the expression (12.10) and using the fact that bilinear expressionsin Grassmann variables behave like c-numbers under commutation, we have that

e ~ s F = I T e—4>A(n)i>A{n) TT eKxpB(m)MmB,m+AB/<AB/ (m+A) (12 19)

n,A m,B,B'A

where MmginBi = (Mmn)BB' • Here, and in the following, the unit vector jl can

point in the positive and negative directions. Now, since products of a given Grass-mann variable vanish, only the first two terms in the expansion of the exponentials(12.19) will contribute; hence

eSr =JJ[i + mA(n)} J ] [l + KdBB,(m,ii)], (12.20a)n,A m.B.B'

where*mA(n) = tf>A(n)ifjA(n), (12.206)

dBB'(m,p-) - i>B(m)MmBim+fiB>[U]i>B'(m + p.). (12.20c)

For convenience we shall refer to (12.20b) and (12.20c) as "monomers" and "dimers",respectively. These names have also been used elsewhere in the literature, althoughin a somewhat different way, and we apologize for borrowing these suggestivenames for our purpose.**

According to (12.20), the basic elements appearing in the integrand of thenumerator and denominator of (12.6b) in any order of the hopping parameterexpansion are:

i) The quark (antiquark) fields ipAl(ni) • • • ( ^ ( m i ) • • •) of the correlationfunction to be calculated. We denote these graphically by an open circle (if> —»• o)and an extended dot (V» —> •) and shall refer to them as the external fields.

ii) Monomers, consisting of a quark and an antiquark located at the samelattice site.

* Notice that we have interchanged the order of the Grassmann variables in(12.20b). This gives rise to a minus sign.** See e.g., Gruber and Kunz (1971); Rossi and Wolff (1984); Burkitt, Mutter

and Uberholz (1987).


iii) Dimers, consisting of a quark and antiquark located at neighbouring lattice

sites and joined by a string bit C/M(n). We shall denote these graphically by

• ) o : dBB' (n, A)n,B n+fi,B'

Hence the contributions to the integrand of (12.6b) of order K* will consist of the

external fields mentioned in i) and a system of monomers and £ dimers distributed

on the lattice. Of these only a subset of configurations will survive the fermion

integration. Thus it follows immediately from the structure of the integration

measure (12.1b) and the fact that the only non-vanishing Grassmann integral is

of the form

f dtpdipipip = - I dipdipTptp = 1 , (12.21)

that every lattice site must be occupied by one, and only one, quark-antiquark pair

for every internal degree of freedom. This is the manifestation of the Pauli exclu-

sion principle! Since the internal indices of the 95-pairs associated with monomers

are automatically paired, it then follows that the degrees of freedom of the quark

and antiquarks associated with the external fields and dimers must also combine

into pairs at every lattice site. Thus the maximum number of such pairs at any

site cannot exceed the total number of degrees of freedom of the quarks. What

concerns the monomers, we can ignore them from now on, since their role is merely

to fill in those degrees of freedom which are not supplied by the external fields and

dimers;** hence they are the "filling material" which ensures that the Grassmann

integrals do no vanish, and we shall not exhibit them in any diagram. (Notice

that according to the definition (12.20b) of rriA each monomer merely contributes

a factor +1, as follows from the integration rule (12.21)). What concerns the de-

nominator in (12.6b), the same general criteria as discussed above apply, except

that now the integrand only receives contributions from monomers and dimers.

Since this is the simplest quantity to study, we shall discuss it first.

Consider the integral (12.5). In order ne the only relevant contributions to

the integrand involve ^-dimers distributed on the lattice in accordance with the

principles mentioned above. In figs. (12-8) and (12.9) we show some typical

configurations in order K4 and K8. Consider first the diagram of fig. (12-8a). Since

each lattice site is only occupied by a single <j<j-pair, there is no restriction on the

** Recall that monomers (and dimers) act like C-numbers under commutation

with any Grassmann variable.


internal degrees of freedom labeling the pair. For a fixed choice of Aly ...A4, thediagram gives the following contribution to det K[U\.

f 4

ÂI-.A* = I ndtpA^n^dipA^m) ipA1(n1)\(Mnin2)AlA2tpA2(n2)'tpA2{'n'2)J s = l

•••i>AAni)4>Ai{ni){Mnini)AiAî)Al(nl),

Except for the two fields, i[)Al(n{) and ipAl(ni), appearing on the right and leftof the square bracket, all others are grouped in the form ipij>. Upon integrationeach of these pairs yields a factor +1. The remaining integral gives rise to thewell-known minus sign associated with a fermion loop. Hence

JAI..A4 = -(^n1n2)A1A2(Mn 2 n 3)A 2 A 3.. .(Mn 4 n i)A 4 A l .

Upon summing over all possible values of the internal degrees of freedom, we areled to the diagram in fig. (12-8b), whose contribution is obtained by taking thetrace of the product of matrices (12.3b) along the closed path, and including aminus sign for the fermion loop.

Fig. 12-8 (a) Dimer configuration contributing to det K in order ft4;(b) corresponding diagram obtained by taking the trace.

Let us now take a look at the more complicated diagrams shown in fig. (12-9a,b). Here we notice that the path to be followed is not unique. Thus with thediagram in fig. (12-9a) we can associate the paths shown in fig. (12-10), while forthe diagram in fig. (12-9b) the possible paths are those depicted in fig. (12-11).The possibility of associating different paths with a given diagram of dimers isclearly connected with our ability to match the degrees of freedom of the qq pairsat lattice sites occupied by more than one gg-pair in different ways. Notice alsothat the various connected and disconnected paths associated with a given set ofdimers differ in the number of independent quark loops.


Fig. 12-9 Dimer configurations contributing to det K in order K8.

Fig. 12-10 The two possible paths which can be associated with the

dimer configuration depiced in fig.l2-9a.

Fig. 12-11 The two possible paths which can be associated with the

dimer configuration depicted in fig. (12.9b).

If we consider the contribution of any one of these closed paths obtained by

taking the trace of the ordered product of the matrices Mmin±p, along the path

with a given sense of circulation, then we are clearly violating the Pauli exclusion


principle, since several quarks (and antiquaries) at a given lattice site would beallowed to carry the same quantum numbers. But if we add the contributions ofthe paths shown in fig. (12-10) (or Fig. (12-11)), Pauli's principle is restored!Thus consider for example the two paths depicted in fig. (12-10a,b). For both ofthem, the contributions violating Pauli's principle correspond to the following qqconfigurations at the lattice site denoted by a cross:

But because the diagram in fig. (12-10b) contains an extra fermion loop, it willreceive an extra minus sign. Hence the configuration shown above will not con-tribute in the sum. In general, diagrams contributing to the determinant consistof a single loop, or of several closed loops (in contrast to In det K which, as wehave seen, only involves simple closed paths). The contribution of each loop is cal-culated by taking the trace of the ordered product of the matrices (12.3b) alongthe path with a given sense of circulation and including a minus sign for everyfermion loop.

Let us now carry through a similar analysis for the numerator in (12.6b). Fordefiniteness sake we shall demonstrate the general ideas by studying the followingfour-point correlation function:

Âl,A2;BuB2{n^n2\mi,m2;U) - (l>pA1{ni)'>pA2(n2)^B1(

mi)'>pB2(m2))sF (12.22)

The relevant dimer configurations contributing to the numerator of the correspond-ing path integral expression are given by the rules stated earlier in this section.In fig. (12-12) we show some typical diagrams built from the external quark-antiquark fields appearing in (12.22) and from a set of dimers. Depending onthe way the internal degrees of freedom of the qq-p&iis are matched* we generatedifferent sets of diagrams consisting of paths connecting the external quarks andantiquarks, and of closed loops which may, or may not, have links in common withthese paths, as shown in fig. (12-13). Once we have generated all possible sets ofpaths of a given total length, there is no longer any need for restricting the sumover internal indices.

* or equivalently, the way we follow the arrows through the diagram.


Fig. 12-12 Two possible dimer configurations which contribute to the

correlation function (12.22).

Fig. 12-13 Diagrams which can be associated with the dimer configu-

rations in fig. (12-12).

The cancellation of those contributions violating Pauli's principle in individual


diagrams, takes place between diagrams consisting of the same set of oriented links.As we have seen, all these diagrams originate from the same dimer configuration.

Notice that the contributions shown in Figs. (12-13a) and (12-13c) factorizeinto a contribution of O(K6), calculated according to the rules stated in section 2,and a closed loop. This loop is cancelled by a corresponding contribution arisingfrom the denominator, i.e. from the fermionic determinant. In this way we recoverthe results of section 2.

Fig. 12-14 Two diagrams generated from the same dimer configurationshown in fig. 12-15, and which taken together ensure the Pauli principle.

There is another point which must be mentioned. Not only the minus signassociated with fermion loops but also the signum of the permutation associatedwith the contractions in (12.7) plays an important role in ensuring Pauli's prin-ciple. Thus consider e.g. the diagrams in fig. (12-14) generated from the dimerconfiguration depicted in fig. (12-15). None of these diagrams contains a closedloop. Symbolically their contributions are of the form

® M---M © © M---M ©

and© M---M © ® M---M ©

where the circles indicate the position of the external quark (antiquark) fields;hence, according to our integration rules, one of the two diagrams receives anextra minus sign ensuring the cancellation of Pauli-forbidden contributions at thelattice site denoted by a cross.

For examples where the hopping parameter expansion has been used in aMonte Carlo simulation see Hasenfratz and Hasenfratz (1981a), and Hasenfratz,Karsch and Stamatescu (1983).


Fig. 12-15 Dimer configuration which gives rise to the paths depicted

in fig. 12-14.

This concludes our discussion of the hopping parameter expansion. For a moredetailed study of the fermionic determinant in the HPE see Stamatescu (1982). Inthe following three chapters we now study the lattice analog of the weak couplingexpansion in continuum field theory.

CHAPTER 13

WEAK COUPLING EXPANSION (I)THE $3-THEORY

13.1 Introduction

Weak coupling perturbation theory can be used in continuum QCD to inves-tigate its short distance properties. The reason is that QCD is asymptotically free.On the other hand, lattice QCD was invented to study non-perturbative phenom-ena, like quark confinement, hadron masses, hadronic weak matrix elements, etc..So why should we be interested in studying weak coupling perturbation theory inlattice QCD which, as the reader might expect, is much more involved than inthe continuum formulation? Because of the specific gauge invariant regularizationwith a lattice cutoff, the lattice action will include so-called irrelevant vertices,which have no analog in the continuum. Although these vertices vanish in thecontinuum limit, a finite number of them will actually contribute in a give or-der of perturbation theory to the Green functions in the limit of vanishing latticespacing. Hence the number of Feynman diagrams that need to be considered islarger than in the continuum formulation. But also the integrands of Feynmanintegrals are now periodic functions of the momenta, and the usual power countingtheorems of the continuum formulation, needed for formulating a renormalizationprogram, do not apply. It is therefore not a priori clear that lattice gauge theoriesare renormalizable.

Let us come back to the question of why we are interested in a perturbativetreatment of lattice gauge theories. By having introduced a space-time latticeto regularize the quantum theory, one has manifestly broken Poincare invariance.It is therefore important to investigate whether this symmetry is restored in thecontinuum limit. A non-perturbative analysis is very difficult. We can, however,get an answer to this question within the framework of perturbation theory. Onthe other hand there are symmetries of the classical continuum action which areknown to be broken on the quantum level. In this case one speaks of an anomaly.Clearly a lattice formulation should correctly reproduce the anomalous behaviourof the quantum theory. A typical example is the famous Adler-Bell-Jackiw (ABJ)anomaly in continuum QED. For vanishing fermion masses the action possessesa (chiral) symmetry which, according to the Noether theorem, implies the con-servation of the vector and axial vector currents on the classical level. But on

192

Weak Coupling Expansion (I). The $3-Theory 193

the quantum level it has been shown that there exists no regularization schemein which both currents are conserved. If one insists on the requirement that thecurrent coupled to the gauge field (i.e. the vector current) remains conserved onquantum level (we do not want gauge invariance to be broken), then the axialcurrent will no longer be conserved. Its divergence can be calculated and is re-ferred to in the literature as the Adler-Bell-Jackiw anomaly (Adler, 1969; Bell andJackiw, 1969). The question then arises whether the lattice formulation of QEDreproduces this result. At first sight there appears to be a contradiction with whatis known from the continuum formulation. Thus the action with naive masslessfermions is invariant under chiral transformations for any finite lattice spacing.Hence one expects that the vector and axial vector currents remain conserved alsofor a -> 0. This is indeed the case. Nevertheless there is no contradiction withthe result obtained from continuum perturbative theory, for we have seen that theaction with naive fermions actually describes sixteen fermion species. As has beenshown by Karsten and Smit (1981), these sixteen fermions give rise to contribu-tions to the divergence of the axial vector current which alternate in sign, andthe theory exhibits not anomaly. On the other hand, by using Wilson fermions,chiral symmetry is broken explicitly for any finite lattice spacing. Although thissymmetry is broken on the lattice by an "irrelevant" term of the action, this termplays a crucial role when studying the continuum limit, and ensures that the ABJ-anomaly is correctly reproduced. In continuum QCD there exist a similar anomalyin the axial flavour singlet current, which must be reproduced correctly by latticeperturbation theory.

But there are also other questions for which one wants to have an answer, andwhich are not connected with any symmetries. For example, the short distanceproperties of QCD, which can be studied in continuum perturbation theory, involvea scale, A, with the dimension of a mass, which determines the rate with which therenormalized effective coupling constant decreases with decreasing separation ofthe quarks. This scale can be measured in deep inelastic scattering experiments.On the other hand we have seen that one encounters a similar mass-scale AL,when studying the continuum limit of observables in the lattice formulation. Thetwo scales, which have been introduced in very different ways, are expected to berelated. In this case this would lead us to the interesting perspective that, givenA and the quark masses, QCD should be able to predict all physical phenomena.

The above examples do not exhaust the list of applications of weak couplingperturbation theory on the lattice, which, for vanishing lattice spacing, shouldcorrectly reproduce all the perturbative results of the continuum theory. For finite


lattice spacing, however, the continuum results will be modified by lattice artefacts.By making use of the large freedom one has in choosing the lattice action (which,a priori, is only required to reduce to its continuum form in the naive continuumlimit), one can construct so-called improved actions (Symanzik 1982, 1983) forwhich these artefacts are reduced*. Non-perturbative calculations using these"improved actions" are then expected to allow one to extract continuum physicsalready for larger lattice spacings. This is important since in praxis one is forcedto perform numerical claculations on rather small lattices.

In this and the following two chapters we want to to provide the readerwith the basic framework for carrying out perturbative calculations on the lat-tice, stressing the differences between continuum and lattice perturbation theory.We will not discuss renormalization theory. Renormalization of lattice gauge the-ories is based on the same general ideas familiar from the continuum formulation.But their practical implementation is much more involved. The interested readermay consult the work of Reisz (1988a,b), where the renormalization program forlattice gauge theories has been studied in detail.

In this chapter we begin with a discussion of a scalar field theory. Our objec-tive is to demonstrate how the integrands of Feynman integrals are modified bylattice artefacts. The ^>3-theory is the simplest non-trivial example of such a fieldtheory. Our discussion, however, applies to any lattice field theory relevant to ele-mentary particle physics. Irrelevant vertices will play no role in our discussion. Inthe next chapter we then consider the more complicated case of an abelian latticegauge theory. There we shall also have to deal with Feynman diagrams involvingirrelevant vertices. Finally, in chapter 15, we consider the case of lattice QCD.The weak coupling expansion of Green functions in this theory is far more com-plicated than in the </>3-theory and lattice QED because of its non-abelian nature.But having prepared the ground in this and the following chapter, the reader willhopefully be able to follow the material on the weak coupling expansion in latticeQCD, which we will present in great detail.

* Confer also the lectures by Liischer (1984). For an important example wherethe program of Symanzik has been applied in the SU(2) and SU(3) gauge theories,see e.g., Bernreuther and Wetzel (1983)


13.2 Weak Coupling Expansion of Correlation Functionsin the 03-Theory

Consider the following euclidean continuum action for a real scalar field:

S[4>] = \JdAx4>{x){-0 + M2)<Kz) + | j d4x(cl>(x))3.

Here • is the four-dimensional Laplacean, M is the bare mass, and #0 is the barecoupling constant carrying the dimension of a mass. As usual, the combinatorialfactor 3! has been introduced to simplify the Feynman rules. Introducing a spacetime lattice, and scaling <j>, D, M and go with the lattice spacing a according totheir canonical dimension*, the lattice action takes the form

n,m n

where K is the matrix defined in (3.10b), and where n and m are 4-componentvectors labeling the lattice sites. Correlation functions of the fields 0n can becomputed from the generating functional

by differentiating this expression with respect to the currents Jn:**

<^1k2...kl>=^rA,fdlZ[J}f \ • (13.1)

Z[0\ { dJni... dJni J J=Q

As has been shown in section 2 of the second chapter, Z[J] can be computed inperturbation theory as follows:

Z[J] =e-Si"t[fj]Z0[J}

A 1 / d \N - (13.2a)

* i.e., we define the dimensionless quantities 4> = a<j>, M = aM, D = o2D, andgo = ago-

** Notice that the subscript on rij labels different four-component vectors, andis not to be confused with the i'th coordinate of a lattice site.


where

Sint$ = | X > n , (13.26)n

and Sint[d/dJ] is obtained from (13.2b) by making the replacement <j>n -> d/dJn.Zo[J] is the generating functional of the free theory, given by eq. (3.12), i.e.,

Zo[j] = , 1 e1' Sn,m J « K » i J - . (13.2c)

From (13.1) and (13.2) one derives the Feynman rules in the standard way. In everygiven order of perturbation theory the contribution to the correlation function(13.1) can be represented by a set of Feynman diagrams built from the interactionvertices with coupling — go and propagators A n m = K~^, represented graphicallyas follows:

There are two types of lines (propagators) associated with a general Feynmandiagram: a) lines that connect one of the external lattice sites appearing in thecorrelation function with an interaction vertex, and b) lines connecting two ver-tices. We shall refer to the former as external lines, and to the latter as internallines.

Fig 13-1 (a) Diagram contributing to {4>n4'm) ln O{g0); (b) Lines em-anating from the two vertices which must be contracted to form the diagramin (a).

As an example consider the contribution of order g$ to the two-point corre-


lation function < 4>n4>m > shown in fig. (13-la). It is given by*

< 4>n4>m >(a)= ^0 Yl ^nltill'^Vm, (13.3a)1,1'

where

flw = (Aw)2. (13.36)

The "symmetry factor" of 1/2 multiplying (13.3a) arises as follows. Let us count

the number of ways that the six lines emanating from the two vertices shown in fig.

(13-lb) can be connected to make up the diagram depicted in fig. (13-la). There

are six ways of choosing the endpoint n. This leaves us with three possibilities

for choosing m. The number of possibilities to connect the four remaining lines is

two. Hence there are 6 • 3 • 2 — (3!)2 ways of building up the diagram. Now each

vertex yields a factor 1/3!. This leaves us with a factor of 1/2! arising from the

second order term in (13.2a). Let us now write the right-hand side of (13.3a) in

momentum space.

As we have learned in chapter 3, the free propagator A n m has the following

Fourier representation (cf. eq. (3.17))

- r d4k e ^ ( " - m )A n m = / JTT-TJ^ — , 13.4a

J ^ (2TT)4 fc2 + M 2

where4

F = J>2, (13.46)

and

fc^ = 2 s i n ^ . (13.4c)

Notice that while feM denotes the momentum measured in lattice units, k» (with

the extended "hat") denotes the dimensionless periodic function (13.4c). Because

of the appearance of the square of fcM in the integral (13.4a), the integrand is a

periodic function of k^ with periodicity 2TT. Inserting the expression (13.4) into

(13.3) and performing the sum over I and V using the representation (2.64) for the

periodic 5-function (with a = 1) one finds that

< Mm >= / j^-u j^G(k, A'; M)eik-n~ik m, (13.5a)JBZ Vz 7 r / \Z7T)

* As always, we denote dimensionless (lattice) variables and functions with a

small hat.


where

G{k;k';M) = = ^ n(fc, Jfc';M)^=^ , (13.56)k2 + M2 k,2 + M 2

and

-2 r 2 J 4 /

2 -i

n(fc,fc';M) = |y^n^[(27r)T^4)(fc-/i-4)^4)(fc/-/1-/2)n /f2 + jg2y

(13.5c)Here Zj denote the "line" momenta carried by the internal lines of the momentumspace diagram corresponding to fig. (13.1a). Notice that the general structure ofthe expressions (13.5) is the same as that of the continuum formulation if fcM isreplaced by k^, except that all the variables are expressed in lattice units. Butbecause of the appearance of the periodic 5-functions, momenta are conserved atthe vertices modulo 2mr (n an integer). This is important, since, e.g., for certainvalues of k and l\ in the (first) BZ, the argument of 8p'(k — l\ — l2) will vanishonly for Z2 in the next BZ. Consider, for example, the integration in (13.5c) over /2.Then /2 is fixed to be Z2 — k —1\ + 2Nn, where the integer N is determined by therequirement that Z2 lies within the integration interval. But since the integrandis itself a periodic function of the momenta, integrating over /2 is equivalent tosetting l2 = k — h- In other words, we can implement momentum conservation inthe way familiar from the continuum formulation. We are therefore left with thefollowing expression for (13.5c):

U(k, k'; M) = (2n)i5{p\k - k')tl(k, M),

where

2 JBZ (2TT)4 [ 2 + M 2 ] p - g)2 + M2]'

and

(k -q)u, = 2 sin[-(fc- g)M].

Let us next compute the contribution of the diagram in fig. (13-la) to thephysical correlation function (<j){x)4>{y)) by scaling all lattice variables appropri-ately with a. Proceeding as in chapter 3, we introduce the dimensioned variables4> = (f>/a, x = na, y = ma, M = M/a, k = k/a, k' = k'/a, q = q/a, as wellas the dimensioned coupling constant go = go/a, and study the behaviour of theintegral as the lattice spacing is decreased, keeping x, y, M and go fixed. One finds

Weak Coupling Expansion (I). The *3-Theory 199

that formally*

Wfw-*G{k>k';M>a)eikXe~ik '"• (13-6a)

where

G(k, A'; M, a) = (27r)444)(fc - fc')[ir^-—n(fc; M, a)j^—], (13.66)Ar + M2 kz + M2

and

n(*;M,a) = f / ^ ^ L ^ . (13.6c)

Here the dimensioned variables denoted with a "tilde" are defined generically by

p^sin^ (13.7a)

The graphical representation of the quantity appearing within square brackets in(13.6b) is given in fig. (13-2).

Fig. 13-2 Contribution of O{g0) to the propagator in momentum space

Let us summarize the important properties of the Fourier transform of thecorrelation function (13.6a):

i) The general structure of G(k,k';M,a) is the same as in the continuum for-mulation, except that propagators are replaced by their lattice analogues,

A^ = whr> • (13-8)* Actually this limit does not exist and one must invoke renormalization before

taking it. We shall come back to this point later; here we are only interested indiscussing some formal aspects.

(13.76)


and that the momentum integrations are carried out over the Brillouin zone:[~a> a)4- Hence apart from these modifications the Feynman rules are thesame as those of the continuum formulation,

ii) In the limit a —> 0 the lattice propagators reduce to those of the continuumtheory,

iii) G(k, k'; M, a) is a periodic function in each of the components of the momenta,with periodicity 2?r/a.

iv) The integrand of the lattice Feynman integral (13.6c) is a periodic function

of the loop momentum q, with periodicity 2TT/Q. Furthermore, it possesses afinite continuum limit,

v) If the integrand of the lattice Feynman integral (13.6c) is replaced by itsnaive continuum limit, then the resulting integral is given by the continuumFeynman rules with a momentum cutoff TT/O.

Although we have only discussed a particular example, these properties holdfor any Feynman diagram if we choose an appropriate set of loop integrationvariables. A natural set of integration variables is obtained by identifying thesewith a subset of the line momenta.* This is the choice we shall make in thisand the following two chapters. In fig. (13-3) we show such a natural choice ofintegration variable for a diagram contributing to the two-point function in 0(#g).

Fig. 13-3 A natural choice for the loop integration variables

Given a lattice Feynman integral having the above properties, we now wantto learn something about its continuum limit. In general this limit cannot bebe calculated by first evaluating the integrals for finite lattice spacing and thentaking the limit a —» 0. But since the integrands are finite in this limit, onemay expect that under certain conditions, one can replace the integrands by theirnaive continuum limit, and hence get rid of the periodic structure of the integrands

* See Reisz (1988c) for a discussion of more general integration variables.


which complicate enormously the computations. The power counting theorem ofReisz (1988c), which we discuss in the following section, will tell us when this canbe done. It also plays a central role for formulating a renormalization program.*

13.3 The Power Counting Theorem of Reisz

Consider a general lattice Feynman integral in the scalar 4>3 theory. Weassume that we have scaled all variables with the lattice spacing a in an appropriateway so that this integral is the lattice-regulated version of a continuum Feynmanintegral**. Furthermore we assume that

a) all trivial integrations asociated with the conservation of energy and mo-menta at the vertices have been performed, and

b) that the loop integration variables qi(i = 1, ...,L) have been chosen in sucha way that the integrand is a periodic function in each component of <&, withperiodicity 2n/a. The domain of integration is the first BZ.

Let k and I denote collectively the set of momenta associated with the externaland internal lines of the diagram, respectively, and q the collection of independentloop integration variables. A general Feynman integral in the 03-theory then hasthe following structure,

where the integrand is given by a product of the propagators (13.8) associatedwith the internal lines of the diagram:

/D(k,q;M,a) = H(lUk,q)+M2).

i= l

Here / is the number of internal lines, and the /u'th component of f; is definedby an expression analogous to (13.7a). With a natural choice of loop integration

* A summary can be found in the lectures by Liischer at Les Houches (1988).** In lattice gauge theories we shall also have to deal with Feynman integrals

which have no continuum analogue.


variables li will then be of the form*

L

h(k, q) = ^Cijqj + Qj(k),j=i

where Cy are either ±1 or 0. We will be interested, however, in integrals of amore general structure. The reason is the following: take for example the integral(13.6c). It actually diverges logarithmically for a —> 0. But by subtracting fromit its contribution at k = 0, we arrive at an expression which possesses a finitecontinuum limit. We therefore decompose (13.6c) as follows:

n(k; M, a) = 11(0, M, a) + Yl(k, M, a) (13.9a)

where

The first term appearing on the right-hand side of (13.9a) diverges in the limita —¥ 0. This divergent constant can be absorbed into the bare mass parameter Min the way familiar from continuum perturbation theory. The remaining integralwill be shown to possess a finite continuum limit. This integral has the followingform

fl(k- Ma)- 9o f dQ N(k>«M>a) /13 1 O a)mMâ)-YjBZJ^D(k,q-M,ar (13-10a)

where**N(k, q; M, a) = f - (q~k)2, (13.106)

andD(k, q; M, a) = (q2 + M2)2[ ( ?~A)2 + M2}. (13.10c)

But this is not the only motivation for studying integrals of a more general struc-ture. In lattice QED or QCD the integrals associated with Feynman diagrams willbe already of the type

I (hMa)- f U diQl N(-k>*M>a) rig i n

* We want to emphasize that it is important that the loop momenta are chosenin such a way that the coefficients Cy are integers. Only then does the powercounting theorem of Reisz apply for Feynman integrals involving an arbitrarynumber of loop integrations. Such a choice is always possible

** In the present case the numerator function N does not depend on M.

(13.%)

Weak Coupling Expansion (I). The -Theory 203

before invoking any renormalization procedure.* We therefore are interested inan answer to the following questions: a) When does the integral (13.11) possess afinite continuum limit, and b) if so, what can we say about this limit?

These questions have been answered by Reisz (1988c), who proved a powercounting theorem for lattice theories, analogous to that familiar from continuumperturbation theory. This theorem applies to integrals of the type (13.11) with Nand D satisfying the following requirements:

i) There exists an integer n such that **

N(k, q; M, a) = a~KN(ka, qa; Ma) (13.12a)

where N is a smooth function of the variables ka, qa, Ma. Furthermore, N isperiodic in each component of the dimensionless loop momenta q = qa withperiodicity 2TT, and a polynomial in Ma.

ii) The continuum limit of N(k, q; M, a) exists. We shall denote it by P(k, q; M):

lim N\k,q;M,a) = P(k,q;M) (13.126)a—• ()

iii) The denominator D(k, q; M, a) is of the form

i

D(k, q; M,a) = Y[ Di(k{k, q); Mit a). (13.12c)i=i

Furthermore, there exists a smooth function FJ(/J; Mi), which is periodic in Z$ withperiodicity 2ir and a polynomial in Mi, such that

Di{li\Mi,a) = \piiUa\Mia). (13.12d)

iv) The continuum limit of £)»(/*; Mi, a) exists and is given by

YunDi{li\Mi,a) = l2i+Mf. (13.12e)

a—>0

v) There exist positive constants oo and K such that

IA&; Mi, a)| >*• (£ +Af?) (13.12/)

* If IF depends on several masses, then M stands collectively for all of them.** Recall that k and q denote collective variables. To be general we also include

the case where several masses {Mt} = M are involved.


for every a < oo, and Z, in the BZ.

Notice that this condition is automatically satisfied for scalar particles. Butfor naive fermions it is violated for momenta £i at the edges of the BZ. On theother hand, for Wilson fermions, the denominator function appearing under theintegral in (4.29a) is given by

D{1; M, o) = 5 3 i sin2(/Ma) + M(l)2,

where M(l) has been defined in (4.29b). Expressed in terms of the momental^, D(l; M, a) takes the following form for r = 1:

D(l; M, a) = (P + M2) + MaP + £ £ iff.

Hence D(l,M,a) > (I2 + M2) and (13.12f) is satisfied. Now if the integrand in(13.11) has the properties (13.12) and if it satisfies the power counting theorem ofReisz (which we discuss below), then a) the integral (13.11) possesses a finite con-tinuum limit, and b) this limit coincides with the expression obtained by replacingthe integrand by its continuum limit, and sending the cutoff ir/a to infinity; i.e.

lim IF(k;M,a)= /"" f] ,% f ^ M ) , (13.13)

where P(k,q;M) has been defined in (13.12b). This is a very nice result, for inthis case we get rid of the periodic structure of the integrand in (13.11) and areleft with an ordinary continuum Feynman integral.

We want to emphasize that the conditions i)—iv) are rather weak. In fact weknow of no example when they are not satisfied. On the other hand, condition v)imposes a non-trivial constraint on the structure of the denominator for it impliesthat for momenta /, lying at the edges of the BZ, Di(li,Mi,a) must diverge likeI/a2 in the continuum limit. Hence naive fermions are excluded, while Wilsonfermions satisfy condition (13.12f). The following example illustrates the roleplayed by this condition.

Consider the one-dimensional integral

(13.14)

Weak Coupling Expansion (I). The *3-Theory 205

where N = 1 or 2. For N = 2 the integrand is the one-dimensional analogue of the

scalar propagator (13.8). On the other hand for N = 1 the denominator in (13.14)

is the analogue of that encountered for naive fermions. Consider first the integral

which is obtained from (13.14) by replacing the integrand by its continuum limit

and sending the cutoff to infinity. Independent of the choice of N, the result is

TT/M. Next let us calculate the integral exactly in the limit a -» 0. To this end

we make use of the following integral representation for the integrand

x2+M2-yo dpe

where x2 = ^ - sin2 ^ = | ^ ( l - c o s ^ ) . Substituting this expression into (13.14),

one finds that for N = 1 or 2, the integral can be written in the form

Ma) = ^f/0 dye-(1+2^Îo(y), (13.15)

where Io(y) is the modified Bessel function. For large arguments, Io(y) behaves

as follows:

J0(y) —> -i=ey. (13.16)y->°° \/2ny

Hence for any finite a the integral (13.15) exist. For a —> 0, however, the integral

diverges, as it must, if (13.15) is to possess a finite limit. We can therefore calculate

this limit by substituting the right-hand side of (13.16) into (13.15). The resulting

integral can be immediately performed, and one obtains

Ma) ô I (£) •Hence the result coincides with the one obtained in the naive approach for TV = 2,

while it is twice as large for N = 1. This does not come as a surprise, since for

N = 1, and k within the BZ, sinfea not only vanishes for k = 0, but also at the

corner of the Brillouin zone. This is the analog of the familiar doubling problem!

The above example shows that even if a lattice integral, and its naive ap-

proximation, both possess a finite limit for a —>• 0, their continuum limits will not

necessarily coincide. In fact for N = 1 the denominator appearing in the integrand

of (13.14) violates condition (13.12f).

After these preliminaries, we are now in the position to discuss the power

counting theorem of Reisz, which applies to integrals having the form (13.11),

with the integrand satisfying conditions (13.12a-f). In order to establish the exis-

tence, or non-existence, of the continuum limit, we need a definition for the lattice


degree of divergence (LDD) analogous to that introduced in continuum perturba-tion theory to study the convergence of Feynman integrals. But in contrast to thecontinuum formulation, the LDD not only refers to the behaviour of the integrandfor large loop momenta, but it characterizes its behaviour under a simultaneousscale transformation of the loop momenta and lattice spacing. This is connectedwith the periodic structure of the integrand. Let us first introduce the lattice de-gree of divergence for an integral involving a single loop momentum, and generalizethe concept afterwards.

Consider a function W{k, q\ M, a) depending on a single loop momentum q.Then according to Reisz (1988), the LDD is given by the exponent a defined by

W{k,\q;M,a/X) « W0{k,q;M,a)Xa + O(Xa-1). (13.17)A-+oo

To make it explicit that the LDD is obtained by scaling q and a in such a waythat the product q = qa is held fixed, we shall use the following notation:

degi^W = a.

Applying this definition to the numerator and denominator functions, TV and D,of a one loop integral of the type (13.11) with L = 1, one finds that the LDDof the integrand is given by degr TV — degi^D. The LDD of the correspondingintegral (which includes the behaviour of the integration measure under the scaletransformation q —)• \q) is then defined by degr/p = 4 + degr TV — degr^Z). Thepower counting theorem of Reisz then states that if degr/jr < 0, then the con-tinuum limit of the lattice integral exists and coincides with its naive continuumlimit.

As an example consider the integral (13.10). A simple calculation shows thatdegr^JV = 1 and degr^L) = 6, so that degr^ft = - 1 . Hence the integral convergesfor a -» 0 to

- g$ f°° _A 2q-k-k2

ll(fc; M,a)-^- J^ (27r)4(g2 + M2)2[(<7-Jfe)2 + M2]"

Consider next Feynman integrals involving L loop momenta qi, • • -,qi, whereL > 1. The overall LDD is obtained by scaling all these momenta with the samefactor A, while reducing at the same time the lattice spacing by a factor I/A. Buteven if this LDD is negative, the Feynman integral need not converge. The reasonis that divergencies may arise from integration regions, where only a subset of

Weak Coupling Expansion (I). The ^-Theory 207

the line momenta becomes large, while the others are kept finite. Consider forexample the diagram depicted in fig. (13-3) for a fixed external momentum k. Bykeeping q, or q', or q-q' fixed, large momenta are only allowed to flow in the loopsof figs. (13-4a,b,c) denoted by a solid line. In the remaining diagram, none ofthese momenta are kept fixed.

Fig. 13-4 Diagrams displaying the four Zimmermann subspaces corre-

sponding to holding (a) q, (b) q', (c) q-q' fixed or (d) none of these in fig.

13-3.. The solid lines can carry arbitrarily large loop momenta

This defines four Zimmermann subspaces. To each of these subspaces we canassociate an LDD by studying the behaviour of the integral associated with thediagrams obtained by omitting the propagators denoted by the dashed lines infig. (13-4), and fixing the momenta carried by these lines. Consider for examplethe diagram depicted in fig. (13-4c). Let u be the momentum in the dashed linewhich is held fixed. Then the relevant integral corresponding to this Zimmermannsubspace H is given by

(H) JBZ (2TT)4 (p + M2)[(q - u)2 + M2][(q - k)2 + M2][{q - k - u)2 + M2}'

The LDD corresponding to the subspace H is obtained by scaling the loop vari-able q and lattice spacing a with A and I/A, respectively. This LDD is given bydegrH!(N) = - 4 .

The generalization of these ideas to Feynman integrals involving any numberof loop integration variables is straight-forward: Given a Zimmermann subspace,


we write the line momenta as a function of those momenta which are to be heldfixed and a set of loop momenta which are scaled with a common factor A. Atthe same time the lattice spacing is multiplied by A"1. One then determines theLDD of the integrand according to (13.17) where q now stands collectively for allmomenta that are integrated over the BZ. The LDD of the integral is obtained byincluding the behaviour of the integration measure under the scale transformation?->• Xq.

Let us summarize the results of this section. For lattice integrals satisfyingthe conditions (13.12a-f), the power counting theorem of Reisz makes the followingassertion:

Theorem

Let Ip be a lattice Feynman integral of the form (13.11), which satisfies conditions(13.12a-f). If the lattice degree of divergence for all Zimmermann subspaces isnegative, then the integral possesses a finite continuum limit given by (13.13).Furthermore the right-hand side of (13.13) is absolutely convergent.

Let us apply this theorem to the integral (13.14). For N = 1 this theoremdoes not apply since the integrand violates (13.12f). For N — 2 this condition isfulfilled. Furthermore the degree of divergence of the denominator is 2. Hence theLDD of /2(a) is -1, and the continuum limit is given by

f°° 1lim/2(a)=/ dk .

This agrees with our earlier observation.

For a proof of the above theorem we refer the reader to the work of Reisz(1988c). As we have already mentioned, this theorem plays a central role indeveloping a renormalization program for lattice field theories. But it is clearly alsovery useful for studying such problems as we have mentioned in the introduction,since it allows us to replace all lattice integrals which satisfy the conditions of thetheorem by ordinary Feynman integrals whose symmetries are manifest.

This concludes our discussion of some of the basic lattice concepts which arerelevant to the perturbative study of any lattice field theory. In the following twochapters we discuss lattice gauge theories which will be burdened by a number ofadditional problems.

CHAPTER 14

WEAK COUPLING EXPANSION (II)LATTICE QED

The scalar </>3-theory we discussed in the previous chapter was a good labora-tory for introducing a number of important concepts in weak coupling perturbationtheory which are relevant to all lattice field theories of interest to elementary par-ticle physics. We now extend our discussion to the case of lattice gauge theories,which present some problems of their own. Since the perturbative treatment oflattice QCD involves a number of technicalities arising from its non-abelian struc-ture, we will begin with a discussion of lattice QED, where the Feynman rules canbe easily derived.

As we shall see, the lattice regularization of a gauge field theory gives rise toan infinite number of so-called "irrelevant" interaction vertices which vanish in thelimit of zero lattice spacing. Nevertheless, some of these vertices can contribute tocorrelation functions in the continuum limit through divergent loop corrections.For QED in a linear covariant gauge, these vertices originate only from the lat-tice regulated action. The purpose of this chapter is to demonstrate i) how thestructure of the interaction vertex in the continuum formulation is modified bythe lattice regularization, and ii) to elucidate the role played by irrelevant verticesin cancelling ultra-violet divergencies in lattice Feynman integrals, which cannotbe removed by renormalization.

14.1 The Gauge Fixed Lattice Action

In lattice QED, the link variables are elements of the abelian U(l) group.Their parametrization in terms of a single angular variable is given by C/M(n) =exp(i0A1(n)). Correlation functions of the link variables and the fermion fields arecomputed according to (5.24), where, because of the abelian nature of the linkvariables the integration measure DU has the simple form (5.23).

Since we are dealing with a gauge theory, we will eventually be interested instudying the ground state expectation value of gauge invariant functionals of thedimensionless fields (j>, 4>, and V>. We denote these functionals by F ^ ; ^ , ^ ] . Theground state expectation value of F is given by

< F >= ^ ID<t>D4)D4>r[(f>,ijj,i>}e-siED^'J>\ (U.la)

209


where

Z = I D^D^Dê-^"0^'^ (14.16)

is the "partition" function for lattice QED, and SQED[4>>iJ,'4l\ 1S the_ gauge invari-ant action expressed in terms of the dimensionless fields </>, ip, and ij). For Wilsonfermions it is given by (5.22),* with the link and plaquette variables expressed interms of {0M(n)}- One readily verifies that

SQEDMA] =So[<l>] + S^r)[<l>M], (14.2a)

where **

S G W = ^ 2 E t1 " COS<M«)L (14-26)

4>^(n) = 0*&,(n) - d?<t>^n), (14.2c)

andS^tf, 4>, j>] =(Mo + 4r) Yî>{n)^{n)

n

- \ £$(n) ( r - ^)eiMnH(n + A) (14-M)

+ ^(n + A)(r + 7M)e-i^C")^(n)].

The action of the right lattice derivative, djf, appearing in eq. (14.2c), is definedby an expression analogous to (4.43b).

Because the coupling constant eo occurs with an inverse power in (14.2b), one

naively expects that the integral (14.1) is dominated for small coupling by those

configurations <j> lying in the immediate neighbourhood of the classical minimum

of SQ- *** This minimum is realized for link configurations which are pure gauge,

* In chapter 5 we had supressed the "hat" on the Dirac fields, since we wereonly interested in the dimensionless formulation

** Here, and in the following, we shall use the same symbols for SQED, SG

and Sp , irrespective of whether they are considered to be functions of the linkvariables U^n), or the angular variables </>M(n) parametrizing these. The factor1/2 multiplying the sum in (14.2b) takes account of the fact that we are summingover all values of n and v. Notice that there is no contribution coming from fi = v.*** We want to emphasize that this argument is only formal. Large quantumfluctuations in the fields could turn out to play an important role, invalidatingperturbation theory.

Weak Coupling Expansion (II). Lattice QED 211

and therefore is degenerate. This degeneracy must be removed before performingthe weak coupling expansion, since otherwise one cannot define the free propagatorof the photon. In continuum perturbation theory this is a well-known fact. Thatit is also true in the lattice formulation can be seen immediately by expanding(14.2b) in powers of 0 ^ , and looking at the quadratic contribution:

SG[<P] = -^2 Yl <?Wn)<?w«) + • • • •

Except for the factor l/e^, this contribution has a structure analogous to thatencountered in the continuum formulation. Hence the free propagator of the 0M-field cannot be defined, since cp^ vanishes for all configurations <f> which are puregauge: 0M(n) = dj}A(n). The solution to this problem is well-known: we have tointroduce a gauge condition in (14.1) which selects from each gauge orbit a singlerepresentative. Along each such orbit the integrands appearing in the numeratorand denominator of (14.1) are constants. This must be done in such a way thatgauge invariant correlation functions are not affected by the gauge fixing procedurefor any finite lattice spacing. An elegant way of introducing a gauge condition wasproposed by Faddeev and Popov, and is referred to in the literature as the Faddeev-Popov trick.* Since we shall demonstrate this trick for a generalized Lorentz gaugelater on when we discuss the non-abelian theory (where the computations are non-trivial), we shall only state here the result, which in the abelian case is very simple.Consider the following generalized Lorentz gauge**

^n [</>; x] = dfa (n) - x(n) = 0, (14.3)

where \ 1S some given arbitrary field, d^ is the left lattice derivative defined byan expression analogous to (4.26b). The reason for having introduced the leftderivative will become clear later on. Applying the Faddeev-Popov trick, onefinds that the above gauge condition can be implemented by merely introducing aset of (5-functions in the integrands of (14.1a,b) which ensure that only those fieldconfigurations </>M contribute to the integrals which satisfy (14.3). Hence (14.1a)can also be written in the form

jD4>D^DJjY\n5{Fn{frx})e-sQe°^<^

* If the reader is not familiar with this trick in continuum field theory, he mayconsult the review article by Abers and Lee (1973), or any modern field theorybook.

** It will be understood from now on that repeated Lorentz indices are summed.

(14.4)


Next we perform one further standard trick to get rid of the 5-functions. Sincewe are calculating a gauge invariant correlation function, the choice of x in (14-3)is immaterial. We can therefore average the numerator and denominator in (14.4)over x with a Gaussian weight factor exp (—^j S n (x ( n ) ) 2 ) - The resulting expres-sion then takes the form

/ D4>Dtj,D$e-sQa*k+-*J]

where the "total" action SQ°ED is given by

SgfLfo, ^ $\ = Salt] + S{F

W)[4>, $, $ + SGF[4>], (14-56)

5GF[0] = ^ E ( ^ M ^ ( « ) ) 2 - (14.5C)n

The subscript "GF" stands for "gauge fixing".

So far the coupling constant occurs in that piece of the action depending onlyon the link variables. Furthermore, it appears with an inverse power, which ispeculiar to the lattice formulation. In the continuum formulation, on the otherhand, this coupling constant enters linearly in the fermionic part of the actionand not at all in the kinetic term for the gauge field. To establish the connectionbetween the lattice and continuum action we must introduce a lattice scale a, anda set of dimensioned gauge potentials and fermion fields. This is done in a wayanalogous to that described in chapters 4 and 5. But since we want our discussionto parallel as much as possible the continuum case, we shall use a slightly modifiednotation. Let x^ = nâ be the coordinates of the lattice sites and ij){x), i>{x) andA^x) the dimensioned fermion fields and gauge potentials evaluated at these sites.The action (14.2a) can be written as a functional of these fields by making thefollowing substitutions: i>(n) —t a?/2ip(x), i>(n) —> a3^2/tp{x), <^M(n) -> egaA^x).Furthermore, we define a dimensioned bare mass parameter Mo by Mo = aMoand introduce the following short-hand notation

x n

In the continuum limit J2X So e s o v e r m t o f d4x. With these replacements (14.5b)becomes*

S{QED[A, V, V>] = SG[A] + S{P[A, 1>, $ + SGF[A], (14.6a)

* In order not to introduce new symbols, we keep the old notation for the various

contributions to the action.

(14.5a)


where

SG[A] = ^ J - j £ [1 - cos(e0a2F^(x))}, (14.66)

0 x,n,v

S™ =(M0 + -)Y,ii(xMx)X

~ — Z ) fa(a;)(r - 7M)eie°aj4"(a;)V'(^ + a£) (14-6c)

+ ^ ( i + a£)(r + 7M)e"ieoaA"(x)V'(^)] ,

Here we have introduced the arbitrary parameter ao by setting a = eâo- The

lattice field strength tensor appearing in (14.6b) is given by

F^{x) = d*Av{x) - d?A»(x), (14.7)

and the action of the dimensioned "right" and "left" derivatives in (14.7) and

(14.6d) are defined by

9*/(s) = - ( / ( * + <*£)-/(*)), (14-8o)QJ

djtf(x) = ±(f(x)-f(x-ap.)). (14.86)

The next step consists in expanding the action in powers of the bare coupling.

This gives rise to an infinite number of interaction terms contributing to SG and

Sp • Of these only those terms survive in the naive continuum limit which are

characteristic of the continuum formulation. Nevertheless, as we have pointed out,

we cannot simply ignore the irrelevant contributions when performing the weak

coupling expansion in a lattice regulated theory. In the following we include only

those "irrelevant" interaction vertices which vanish linearly with a in the naive

continuum limit.

Consider first the contribution to the action depending only on the link vari-

ables, i.e. SQ- Expanding (14.6b) in powers of the lattice spacing one finds that

5G = \ E F^x)FM + °("4)- (14-9)X

(14.6d)


By making use of the relation*

$>*/(*))$(*) = - E f(x^9(x) , (14.10)x a:

we can rewrite (14.6b) up to terms vanishing like a4 as follows:

5<? ~ ~\ X > M ( * ) ( < W ° - dj!dZ)Av(x). (14.11a)a;

Here

is the hermitean lattice Laplacean in 4 space-time dimensions:

D -^) = E ^ + a® + f(x " a® - 2f(x)].

Next, consider that piece of the action arising from gauge fixing 5-function, i.e.(14.6d). Using again relation (14.10) we have that

SGF = -^-Y,A^)dÂv{x). (14.12)

Notice that because (14.6d) involves the left lattice derivative, the tensor structureof (14.12) is the same as that appearing in (14.11a). Hence by combining (14.11a)and (14.12), we arrive at the following contribution to 5^° quadratic in the gaugepotentials

SG\A] = lJ2Mx)^(x,y)My), (i4.i3a)

where

svfoy) = (-<Wn + (i - -^-oKd») SP](X - y)- (14-136)

Here SP (z) is the periodic 5-function

S(PHZ)= [ ^-Aeik-\ z = na, (14.14)JBZ (2?rJ

* This relation follows immediately by introducing (14.8a) into the left-handside of (14.10), and making a shift in the summation variable. We assume herethat we are dealing with an infinite lattice, or with a finite lattice, with f(x) andg(x) satisfying periodic boundary conditions.

(14.116)


where from now on BZ stands for the dimensioned Brillouin zone [—IT/a, ft/a}.

Finally, consider the fermionic contribution (14.6c). Expanding the link vari-

ables Up = expfeoaAfj) up to terms quadratic in the coupling, one finds that

S™ = 4°) + SW + <#) + 0(a% (14.15a)

where

S£° =(M + ^)^2^(x)^p(x)x x (14.156)

is the free fermion action expressed in terms of the dimensioned variables, and

s(i) = _ ie^> ^(X)(r - 7 ^ , ( 1 ) ^ ( 1 + oA)2 tH (14.15c)

- tjj(x + ap){r + -yÂîpix)],

SF] =ejaY,mx)(r-^)Al(xW(x + aj>)4 x,n (14.15.d)

+ 4>{x + afi)(r + -rÂl{x)^{x)}.

Collecting our results, we therefore find that the total action (14.6a) is given by

S%$D[A, iP, i>] = S{^[A} + 4 0 ) [ ^ , i>] + Sint[A, t, 4,} + O(a2), (14.16a)

where2

e=i

is the contribution arising from the fermion-gauge-field interaction. From (14.15)we see that Sint describes not only the interaction of a single photon with theDirac field, but also includes a contribution involving the coupling of two photonsto the fermions. Whereas the former contribution reduces in the naive continuumlimit to the familiar interaction term,

SW -> ie0 f dix$(x)'YllA»(x)il>(x),

the latter contribution, 5 ^ , has no analog in the continuum and in fact vanishesfor o —> 0. Nevertheless, as we shall see, it plays an important role in canceling

(14.166)


divergent contributions to the vacuum polarization, which cannot be eliminated

by renormalizing the fields and bare parameters. This is not surprising, since these

vertices are a consequence of the lattice regularization which provides us with a

gauge invariant cutoff.*

14.2 Lattice Feynman Rules

Assuming that SQ^'D can be approximated by (14.16), and that the integra-

tion range of A^ in the path-integral expression for the correlation functions can

be extended to infinity,** the perturbative expansion of any correlation function

involving a product of the fermion fields ip and ip and gauge potentials A^ is ob-

tained in the way familiar from continuum perturbation theory. In any given order

of the coupling, the contributions to the correlation function can be represented

in terms of Feynman diagrams built from the free propagators of the gauge po-

tential and the fermion field, and from the interaction vertices. Their momentum

space representations can be easily deduced by writing the action in momentum

space. Consider first the contribution (14.13). Introducing the following Fourier

decomposition of the fields

JBZ (27r)

iM*) = fBz ^-Mp)eipx' (* = n a ) (14-17)

JBZ (27r)

and making use of the relation (2.64), one readily finds that it can be written in

the form

40 ) = \fBz -0^0j-Mk>) [e-ifc**<W*/,*)e-ifc-'*] Av(k), (14.18a)

where

Q^(k',k) = (2n)'l6{p)(k + kl)Q^(k), (14.186)

and

Ci^(k) = [S^k2 - (1 - — ) kJiA • (14.18c)

* In continuum perturbation theory it is well-known that Feynman integralsmust be regularized in a gauge invariant way.

** We know of no rigorous proof that this is a legitimate procedure.


Here fcM is defined by an expression analogous to (13.7a). Notice that becauseof the appearance of the phase factors in the integrand of (14.18a), the quantitywithin square brackets is a periodic function in each component of the momentawith periodicity 2n/a.

Next, consider the contribution Sp defined in (14.15b). Its decomposition inmomentum space is given by

40) = JBZ ~^~yk{p')KaP(p',P)Mp), (14.19a)

whereKa0{p',p) = (27r)4^4)(p - p')Kal3(p), (14.196)

and

Kap{p) = - ^ ( 7 M ) « / 3 sinpMa + M(p)6a/3. (14.19c)

The momentum dependent mass M(p) has been defined in (4.29b).

Finally, one easily verifies that (14.15c) and (14.15d) have the following momentispace decomposition:

(14.20a)

ÎAP'^P' k) = (2*)4tp(p-p' + QvVpfp+p1), (14.206)

^%(?) = -*co [(Û cos 9-f - ir8a0 sin ?f] ; (14.20c)

.[e^-îr^;a/3(p',p,k',k)], (14.21a)

rJ.l);a/J(p,p', k,k') - (27r)444)(p -p' + k + k')vWa0(p + p') (14.216)

C U ( 9 ) = -eoa<W [rS*f> cos q-f - t(7p)a/3 sin ^ ] . (14.21c)


Notice again, that because of the appearance of the phases, the quantities appear-ing within square brackets in (14.20a) and (14.21a) are periodic functions in allcomponents of the momenta with periodicity 2n/a. Hence momentum conserva-tion can be implemented in the usual way. Accordingly, we can replace the phasefactors in (14.20a) and (14.21a) by exp(-ifeMa/2) and exp(-i(fc + fc')AJo/2), respec-tively. These phases can now be absorbed in the Fourier transforms of the gaugepotentials, which amounts to redefining the potentials at the midpoints of the linksconnecting two neighbouring lattice sites. Although we could have avoided thesephases right from the start by Fourier decomposing A^x) as follows

MX) = Lz -0^~AêiHX+aN2^ ( 1 4 ' 2 2 )we have nevertheless preferred to carry them along in order to exhibit the 2?r/o-periodic structure of the above mentioned expressions. But when computing thecontribution of a particular Feynman diagram one finds that the phases associatedwith the interaction vertices, and the photon propagator (deduced from (14.18a))cancel at each interaction vertex. The only phases that remain are those associatedwith the gauge potentials appearing in the correlation function. Hence by Fourierdecomposing the correlation functions as follows,

<1pax (Zi) . . . 1pan (in)^ft (yi) ••4fin (yn) AM1 (Zl) . . . A w {zt) >

=/ft (SF ft 0 n l^, . .^ , . .^ , . ,^} , «>, {*,»

(14.23)we can calculate the contribution of a Feynman diagram to the correlation functionin momentum space, F a i . . . w (pi,... k(), using the propagators and vertices deducedfrom (14.18) to (14.21) ignoring the phases factors. The propagators of the gaugepotential and of the fermion field are given by the inverse of the matrices (14.18c)and (14.19c), respectively, while the vertices are given by (14.20c) and (14.21c).In the continuum limit V^l'.a/3(q) vanishes, and VM1/;a/g(g) reduces to the vertexfunction of the continuum theory, i.e., — «eo(7M)Q/a.

Except for the fact that on the lattice we must also take into account "irrele-vant" interaction vertices, the rules for computing the contribution of a particularFeynman diagram are the same as in the continuum formulation. For finite latticespacing the corresponding Feynman integrals are however much more complicatedthan those encountered in continuum perturbation theory, where the integrals are


regularized a posteriori, and do not follow from a space-time regulated generatingfunctional.

We now summarize the rules for calculating the contribution of a Feynmandiagram to the correlation function rQ l . . .w(pi, . . . , ke) defined in (14.23).

i) To an internal fermion or photon line associate the propagators

D,u{k) = Qrf{k) = l2 (s^ - (1 - ao)^\

ii) For the vertices insert the following expressions:

-feo(2*)«#>(P-P' + *) [(7,)tt/3cos(^±|V) _ir^sin(^(£±|Vj]

-el{2,)^\P-P'+k+k')aS^ [rSa0coS (<P±£t±") - l{lli)Q0sin ( ^ ± f ^ ) ]

iii) For every closed fermion loop include a minus sign.

iv) Contract all Dirac indices following the fermion lines, and all Lorentz indicesfollowing the photon lines.


v) Integrate the internal momenta over the Brillouin zone with integration mea-sures having the generic form d4k/(2Tr)4.

As an example consider the vacuum polarization tensor TTMI, in order eg. Inthe lattice regularized theory there are two diagrams that contribute. They aredepicted in fig. (14-1).

Fig. 14-1 Diagrams contributing to the vacuum polarization tensor H/ll,(k)

Applying the above Feynman rules, and performing the trivial integrationsassociated with the (periodic) J-functions, one finds that

IWife, k') = {2-n)48{p\k + fc')IV(*), (14.24a)

whereiWfc) = nW(*) + n $ , (i4.246)

n<Z>(*) = - / T^fjTr \V?\2P - k)SF(p)vU(2P - k)SF(p - k)] , (14.24c)JBZ \L-n) L J

n<5 = - j ^ -0^-Jr {v$>(2p)SF(p)} • (14.24d)

Here V^ and Vfiy are the matrices in Dirac space whose matrix elements are givenby (14.20c) and (14.21c). The minus sign in (14.24c,d) takes into account rule iii)given above. Applying the power counting theorem of Reisz discussed in chapter13, we conclude that the lattice degree of divergence of both integrals is 2. Hencethey diverge like I/a2 in the continuum limit. If this divergence would persist aftercombining the two integrals, then the theory would not be renormalizable, sincethere is no mass-counterterm available to cancel this divergence. In continuumQED, where only the diagram shown in fig. (14-la) contributes, the correspondingFeynman integral is also superficially quadratically divergent. But because of the


Ward identities, it actually only diverges logarithmically. This divergence canbe eliminated by wave function renormalization. On the other hand, the Wardidentities, following from gauge invariance on the lattice, are only satisfied afterincluding the contribution coming from the diagram depicted in fig. (14-lb); henceonly then do we expect a cancellation of unwanted divergencies. For vanishing barefermion mass and Wilson parameter r, this cancellation can be easily demonstrated(Kawai, Nakayama, and Seo, 1981). We first isolate that part of n ^ and n j jdiverging like I/a? for a —» 0 by decomposing these quantities as follows:

n$(*) = n«(o) + [n$(*) - n«(o)],

where i = a or b. The quadratically divergent part is contained entirely in TY^lfO).

Performing the trace in (14.24c) one finds that

n W f m - - ^ r -^Lr * s i nPAsinp, COBPM cos & f l 4 25a)a J_n (Zir) ( E C T

s m V<T)

whereW = Tr{Wlvlp) ( i 4 2 5 6 )

= 4(5MA^r/p - S^v5\p + S^pSvx).

By carrying out the summation over the Lorentz indices one easily verifies that(14.25a) is proportional to 5^ and can be written in the form

»A)~ a 2 ^ / _ 7 r ( 2 7 r ) 4 E A s i n 2 ^ + 2 S m ( 2 ^ ) ^ E , s i n 2 p J_4e§. f rf4p sin2pM

(14'2b)

- a*b»v]_A^Yi:xSm2px

On the other hand, for r = Mo = 0, (14.24d) becomes

^ - a2 7_, (27r)4Tr [^ Epsin2pp j S m ^ '

which, upon making use of the relation tr^^v) = 43^, reduces to the negative of(14.26). This simple example demonstrates the important role played by irrelevantvertices in canceling divergences that cannot be removed by renormalization.

Let us summarize the lesson we have learned. The lattice provides us with agauge invariant regularization scheme. Although this gauge invariance is brokenby the gauge condition, the Faddeev-Popov procedure will leave gauge invariant

(14.25a)


correlation functions unchanged, if we include the contributions of all irrelevantvertices. This is true for finite lattice spacing. But when studying the continuumlimit of correlation functions in a given order of weak coupling perturbation theory,only a subset of these vertices need to be taken into account. These vertices willensure the renormalizability of the theory and the restauration of the continuumspace-time symmetries in the limit of vanishing lattice spacing.

14.3 Renormalization of the Axial Vector Current in One-Loop Order

When computing decays like ir~ —>• e + z/e, one needs to calculate the matrixelement < 0|j5M(0)|TT~(P) >= pM/ff, where /„. is the pion decay constant, andJ5/j,(x) is the renormalized axial vector current. We must therefore know how thelattice regularized bare current is renormalized. Here Ward identities will serveas guidelines for determining the renormalization constants. It is instructive tocompute these constants in one-loop order perturbation theory.

Let us first exemplify the main idea for QED in the continuum. In the treegraph approximation the axial vector current is js^ltree = ^(z;)7/i75V'(a;)- The oneloop correction to this current is given by the diagrams depicted in fig. (14-2).In the limit of vanishing fermion mass the QED action is invariant under global75-transformations,

i>(x) -> eiei5tp(x) ,

i>(z) -> i){x)eie^ . (14.27)

By performing an infinitessimal local 75-transformation of the fermion fieldsin the generating functional of Green functions (i.e., e becomes ^-dependent), andmaking use of the invariance of the measure, one easily derives a Ward identitiy foran n-point vertex function with an the insertion of the divergence of the axial vectorcurrent. In the continuum formulation of the path integral the Ward identity isonly formally defined. In a lattice regularization of the path integral, this Wardidentity is an exact statement, but needs, in general, to be renormalized as thecutoff (lattice spacing) is removed. It is instructive to first derive this naive identityin continuum QED in one-loop order directly for the diagram shown in fig. (14-2a).In momentum space its divergence is given by

iq^(p,p') = i(-ie)2q»Y,J AL [7A5O(P'+^)7M755O(P + O 7 A ] ^ . (14-28)

where q = p — p', and So is the free fermion propagator So = (iy^q^ + mo)"1; mois the bare fermion mass. Making use of the trivial identity

»(P - P%7M75 = 2mO75 - S ^ V h s - JsSîp) , (14.29)


Fig . 14-2 Diagrams contributing in O{e ) to the axial vector (pseu-

doscalar) current in the continuum formulation. The cross stands for the

insertion of 7M7s (75).

which is valid for arbitrary momentum I, one immediately verifies that

i<ln^{p,p') = 2m0A5{p,p') + 75S(p) + S(p')7s • (14.30)

Here E(p) is the self energy, and A5 is given by

MP,P)=i(-ie)2^2J-0ji [yxSo(p' + £h5So(p + e)lx\^ . (14.31)

The self energy is divergent and requires mass as well as wave function renormal-

ization. Mass and wave function renormalization as is dicated by QED in fact

suffices to render all the terms in (14.30) finite. This can be easily checked in the

Pauli-Villars, or dimensional regularization. The renormalized Ward identity for

the corresponding vertex function takes the form

iq^Sn(p,p')R = 2mr5(p,p')R - S ^ V J H T S - 75S£1(p)ji , (14-32)

where Sp(q)R is the renormalized fermion propagator to one loop order, i.e., (17-

p + m-'ER(p))-1, and

^(p,p')R = Z2T5li(p,p') ,

r5(p,p')R = ZPT5{p,p') ,


TOO = Z^{m + 5m) . (14.33)

In the one-loop approximation one finds that*

• 7 i e 2 i r m \Z 2 - 1 = 8^ l n ( A } '

Z p - 1 = &1^' ( 1 4 - 3 4 )

5 m = m 2 ^ l n ( A ) '

where A is a momentum cutoff. Equation (14.32) is the generalization of the treelevel Ward identity (14.29).

On the lattice the Ward identity (14.32) will be modified by terms which arenaively irrelevant, i.e., which vanish in the naive continuum limit. We shall referto them in the following simply as "irrelevant". In particular it will involve acontribution arising from a chiral symmetry breaking term in the action, whichensures the absence of fermion doubling. For concreteness sake we will considerthe case of Wilson fermions. As we shall see, the irrelevant contribution referredto above leads to an additional finite renormalization of the axial vector current(apart from the QED wave function renormalization) in the continuuum limit.This problem has been first discussed by Karsten and Smit (1981). ** Here wedicuss it in some detail.

To derive the lattice regularized Ward identity analogous to (14.32) we con-sider the following lattice integral,

Zo = f DUDI/JD^ O(ij,^)e-S , (14.35a)

where

O = i>a(y)ip0(z) , (14.356)

and S is the lattice action for QED with Wilson fermions, (5.22). The fermioniccontribution can be written in the form

Sferm = £J{x){l-ltl{D*[U] + D$[U\) + TO0} + AS , (14.36a)x

* We are neglecting finite contributions. It therefore corresponds to a minimalsubstraction scheme.

** The renormalization of the axial vector current for different lattice realizationsof the current has been considered by Meyer (1983).


whereAS = \aY,^)D^[U]D^[U\^{x) (14.366)

X

is the chiral symmetry breaking Wilson term which vanishes for a —>• 0, and

D*[V\1>{x) = llU^xMx + afi) ~ 4,{x)} ,

DZ[U]1>(x) = i[V(*) - Ul(x - a(i)^{x - ap)] , (14.37)

are the covariant right and left lattice derivatives, r is the Wilson Parameter.Making use of the invariance of the partition function (14.35a) under the followinginfinitessimal local axial transformation of the fermion fields (i.e. a change ofvariables),*

8i/)(x) = it{x)~jr,%}>{x) ,

8i>{x) = ie(x)ip{x)j5 , (14.38)

where 75 = 171727374, and {7^, 75} = 0 for all n, one is led to the identity

< OSS - 5O >= 0 . (14.39)

One then readily verifies that for Wilson fermions

SS = i ^ £ ( I ) [ - ^ ( j ) + 2moj5(x) + A(x)} , (14.40)X

where J2X ~ S n fl4> xv = nva (nM G )> &% denotes the left lattice derivative,

h»(x) = 2$(X)WMXMX + aA) + fa + afiWsUl(x)1>{x)] (14.41)

is the gauge invariant axial vector current, and

js(x) = i> (x)-Y5i> {x) (14.42)

is the pseudoscalar current. Furthermore

~ Ya Yyt>(x - atiU»(x - °A) + $(x + ap)Ul(x) - 24>{x))-y5i>(x) .

(14.43)

* The integration measure is invariant under this transformation


is the operator originating from the chiral symmetry breaking Wilson term (14.36b)in the action. Finally, the variation SO in (14.39), with O given by (14.35b), isgiven by

x 5

Since e(x) in (14.40) is an arbitrary infmitessimal function, one is led to the Wardidentity

< d^j5t,(x)i>a(y)iji0(z) > = 2m0 < h{x)ipa{y)i>p{z) > + < A(x)ipa(y)ipp(z) >

- ^26xy(75)05 < •4>5(y)-4>a(z) >

6

- ^2$xz < 1pa(y)il>6(z) > (75)5/3 •

(14.44)Notice that A is an "irrelevant" operator which vanishes in the classical continuumlimit. It could however (and in fact does) play a relevant role on quantum level.In the following we will consider (14.44) up to O(e2). To this order A is given by

2 2

A(x) « - ^ Y, {$(xh5 [l + ieaA^x + a£/2) - ^-(A^x + a£/2))2]^(:c + a£)

1 - ieaA^x - ap./2) —(A^x - ap,/2))2\ip(x - a/x)

} 2> _+ — 2Z^(x)l5ip(x) •

(14.45)Hence up to 0(e2) ,

A(x) = -ar- Y,$(x)îjj(x) ~a^J2^(^^(x) + X{x) , (14.46)

where x(x) ls a n "irrelevant" operator involving one and two photon fields. Inmomentum space the sum of the first two terms on the rhs of (14.46) becomeproportional to Mr(p; a) + Mr(p'; a), where

Mr(q;a) = -Y^q-f (14-47)

is the r-dependent part of the Wilson mass (4.29b).


With the decomposition (14.46), the Ward identity (14.44) implies the follow-

ing relation between the axial vector and pseudoscalar vertex functions in momen-

tum space

iq^5^(p,p';mo,a) = 2[M(p; mo,a) + M(p'; m0, a)]T5(p,p';mo,a) + F^x)(p,p';mo,i

- i5SF1(p;m0,a) - Sp1(p';m0,a)'y5 ,

(14.48)

where F ^ is the two-point fermion vertex function with an insertion of the \-

operator. M(q;mo,a) is the "Wilson" mass

M(q; mo, a) = m0 + Mr(q, a) , (14.49)

and Sp1 is the inverse of the lattice regularized fermion propagator

Sîp; m0, a) = ij^Pn + M(p; m0, a) - E(p; m0, a) , (14.50a)

where

Pn = -sinpâ . (14.506)

In one-loop order, FsM and Fs are given by the sum of Feynman diagrams

of the form a — f and a,e,f, respectively, shown in fig. (14-3). The diagrams

contributing to the self energy E(p) are those labeled by g and h. The Ward

Identity (14.48) is nothing but the generalization of the following tree-level lattice

identity analogous to (14.29):

sin [(^y^-)Ma] r5M(P.P'; m o , a)tree = 2[M(p; m0, a) + M(p'; m0, a)]j5

- ^Sîp; m0, a) - S^lp'; m0, a)-y5 ,(14.51a)

where

T5fl\tree = COs( ~)lia^^5 . (14.516)

In O(e2) we have the following relation between the one particle irreducible vertex

functions

«ZMA5M(P,P'; mo, a) = 2[M(p; m0, a) + M(p'; m0, a)]A5(p,p'; m0, a)

+ A{^\p,p'; m0, a) + 75E(p; m0, a) + S(p'; m0, a)j5 ,

(14.52)

which is the lattice version of (14.30). Apart from the appearance of the Wilson

mass, it involves a naively "irrelevant" contribution Ag . In the following we will


Fig. 14-3 Lattice diagrams contributing in O(e2) to the Ward identity

(14.48). Diagrams b, c, d and h have no counterpart in the continuum.

Diagrams a — f contribute to T^^. In this case the big dots stand for the

insertion of the axial vector vertices defined via the expansion of (14.41) in

powers of the gauge field. Diagrams contributing to F 5 are shown in figs,

b, c and d, where the big "dot" now stands for the insertion of the vertices

defined via (14.45) and (14.46). Only diagrams a, e, f contribute in one loop

order to Tc, (p, p1; a) because of the ultralocality of the pseudoscalar current

(14.42).

study in detail the renormalized version of the bare lattice Ward identitiy (14.48)to one-loop order. Our emphasis will be placed on the role played by the abovementioned irrelevant contribution. The determination of the renormalization con-stants involve some tedious, but straight forward calculations. They are extractedby Taylor expanding the expressions for the vertex functions around vanishingmomenta. Only first order polynomials in the momenta need to be considered.The coefficients in the expansion are in general complicated integral expressions,


whose behaviour for a —» 0 can however be determined rather easily. We leavethese computations as a lengthy exercise for the reader. In Appendix B we havesummarized the relevant vertices in momentum space required for carrying outthe computations.

1/9

Consider the lattice regularized Ward identity (14.48). Let Z2 be the QEDwave function renormalization constant for the fermion fields. Multiplying (14.48)by Z2 we have that

iqp,Z2r5lJ,(p,p';mo,a) - 2m0Z2T5(p,p';m0,a) - Z2T{5A)(p,p';m0,a)

= —y5Z2Sp1(p;m0,a) - Z2Sp1(p';mo,a)'y5 ,

where

Z2r ( A )(p,p ' ;m0,a) = 2Z2[Mr(p;m0,a) + Mr(p';m0,a)]r5(p,p';m0,a)1 > (14.54)

+ Z2r^>(p,p';m0,a) .

Diagrams contributing to Igx' are labeld by b, c, d in fig. (14-3). For the rhs of(14.53) to be finite we must also perform a mass renormalization:

m0 = Z^l{m + 5m) . (14.55)

Here m is a renormalized mass. With the definitions (14.47), (14.49), (14.50a) and(14.55), we then have to one loop order that

lim Z2Sp1(p;m0,a) = ij^p^ + m- EH(p,m) , (14.56)

whereSfl(p; m) = lim[S(p; m, a) - 5m - i(Z2 - l)7uPM] (14.57)

a—•()

is the renormalized self energy. Z2 and 5m are chosen such as to the render thisexpression finite.

The in the limit a —> 0 divergent parts of 5m and Z2 can be computed byTaylor expanding E(p; m, a) up to first order in the momentum. From the diagramsg and h in fig. (14-3) one finds after some lenghty, but straight forward algebrathat the divergent parts are given by

5m = —— + —-mln(ma) , (14.58a)

e2

Z2 - 1 = ^ 2 ln(ma) , (14.586)

(14.53)


where

C(ma) = re2 Jûz^ ^ , (14.58c)

with

V(!) =I2-2YJ sin2 | 5 > ° s 2 ( y ) - r-2 sin2(^)] - 4[? + M2(£, ma)] (14.58d)

and

t = 2 s i n ^ ,

Ia = sinia. (14.58e)

M(£,ma) is the Wilson mass (14.49), with m0 -» m, measured in lattice units(Quantities with a "hat" are measured in lattice units), and C(0) is a finite con-stant. Hence to one loop order and a —> 0 (14.55) is given by

C(0) 3e2

rn0 = m + — ^ + —- ln(ma) . (14.59)

In contrast to the continuum formulation, mo involves a linearly divergent contri-bution proportional to the Wilson parameter. As we shall see this linear divergencewill be elliminated by a corresponding divergent contribution of the naively irrele-vant term Tl*' in the Ward identity. The logarithmic divergent expressions Z2 - 1and 8m are the lattice analog of the expressions in the continuum (14.34).

Consider now the lhs of (14.53). Since with the above choice of Z2 and 5mthe rhs is finite for a —> 0, so is the lhs. Consider first the contribution i^Zj^hti-Since the highest "lattice degree of divergence" (LDD; see sec. 13.3) of the oneloop diagrams contributing to FsM is zero, this expression is at most logarithmicallydivergent. In one loop order we have that

iqlj.Z2T5li(p,P>; m0,a) = iq^^s + igM[A5^(p,p';m,a) + (Z2 - 1)7M7S] • (14.60)

As^ denotes the contribution of the one-loop diagrams a — d shown in fig. (14-3).Of all these diagrams only the triangle diagrams turns out to be (logarithmically)divergent. We can extract the divergent part by studying the small a behaviourof the corresponding lattice Feynman integral at zero external momenta. Aftersome lengthy, but straighforward algebra one finds that the quantity appearingwithin square brackets in (14.60) is finite for a —» 0. Consequently the remaining(pseudoscalar) terms on the lhs of (14.53) are also finite, since the Ward Identity


holds for arbitrary a. Consider first the contribution of 2[Mr(p, a)+Mr(p', a)]F5 in(14.54). Since the one-loop contribution to F5 has LDD = 0, and hence diverges atmost logarithmically, while Mr vanishes linearly with the lattice spacing, this termwill make no contribution in the continuuum limit. Next consider the contributionof Z2T

<£i) in (14.54). To O(e2) Z2r^x) is just F5x).* The corresponding Feynman

diagrams 6, c, d in fig. (14-3) have LDD = 1. We therefore decompose this termfollows:

r<*> = ror£x) + (i\ - T O ) F ^ + (I - r ^ r ^ , (14.61)

where Tn denotes the Taylor expansion in the momenta around p = p' = 0 upto n'th order. The last term on the rhs has negative lattice degree of divergenceand hence possesses a finite continuum limit. This limit may be calculated bymaking use of the Reisz theorem (see sec. 13.3). Since Fg is a naively irrelevantcontribution, this term vanishes in the continuum limit. For ToT§ ' and (T\ —TcOFg one then finds after some lengthy algebra that

T0T^=2mz^5,

where

p ma J-*Wf[P + M2]

and zÂ ' is finite. After a rather lengthy calculation its expression is found to be

zA (rn) - e / ^ _ _ +e z^z I IJ-n V*) £ [I? + M2{l,m)] •*-" l ' £ [£2 + M2(Am)]2

(14.62c)where rh = ma,

U(i) = -2rMr(l) + 2r2 cos4 sin2 y + 2rM(l, rh) sin2 y , (14.62d)

ga(£) = 2Mr (I) \r,a (I) + r2 £ ) sin2 ^ ] (r sin2 4 - M(£, m) cos la)x (14.62e)

+ r cos £ff (sin2 4 - ^ sin2 £A)] ,A

* Recall that x is itself an operator of O(e). It arises from the expansion of thefermionic contribution to the lattice action in powers of the gauge potentials, andis a lattice artefact.

(14.62a)

(14.626)


with

M(l m) = m + 2r J^sin2 y , (14.62/)A

and

Va(l) = 2 cos2 ^ - £ c o s 2 T • (U-629)

As always, quantities with a "hat" are measured in lattice units. The continuumlimit corresponds to vanishing m = ma. The above integrals are finite in thislimit.

Summarizing we have that Z2^5 in (14.53) is given by

Z i T f ] (p,p') = 2 m z Px ) j 5 + i z M q p W i + ••• , (14.63)

where the "dots" stand for terms vanishing in the continuum limit. Note that thisexpression is completely local and has the form of possible counterterms for thepseudoscalar and axial vector current. This will in fact be just the case.

Finally consider the contribution 2moZ2T5(p,p';mo,a) in (14.53), where mois defined in terms of the renormalized mass m by (14.55). In the one-loop ap-proximation it is given by

2m0Z2T5(p,p'; m0, a) = 277175 + 2m[A5(p,p'; m, a) H 75] , (14.64)TTt

where the diagram contributing to A5 is shown in fig. (14-3a). Since As(p,p'; m, a)has LDD = 0, it diverges at most logarithmically for a —> 0. But 8m has a linearlydivergent part, so that (14.64) diverges even linearly for a —> 0. Now comesan important role played by the A-term in (14.53). By combining (14.64) with(14.63), and making use of (14.58a) and (14.58b), we have that for a -> 0

2moZ2T5(p,p'-mo,a) + Z2T{A)(p,p') « 2m[T5{p,p';m,a) + 75] + i z^VvYs ,

(14.65a)where

zP = z^ + — = ^ ln(ma) . (14.656)TYl ATT

This is precisely the lattice analog of the renormalization constant zp = Zp — 1 in(14.34). Hence, after wave function and mass renormalization, the Ward identity(14.53) reads as follows in the continuum limit

iq^^{p,p')R = 2mT5{p,p')R + iz^q^Js ~ Sp\p) - Sp\p')lb , (14.66)


where z^ = z^(0) is a finite constant determined from (14.62c), and Sp is therenormalized propagator. The extra term iz£ qîpls can be removed by a finiterenormalization of the axial vector current, so that the final form of the Wardidentity becomes

«fcf ^{P,P')R = 2mr5(p,P')R - 7s5j:1(p) - ^ ( p ' h s , (14.67a)

whereF5»{P,P')R = lim Z%rlZ2(ma)V^{p,p';ma,a) , (14.676)

a—>0

^5(P,P')R - lim ZP{ma)r5(p,p';ma,a) , (14.67c)a—>0

and Z]£ — 1 + *A i %P = 1 + zp. Zi and Zp are logarithmically divergentrenormalization constants having a form completely analogous to (14.34) of thecontinuum formulation. Note that the linear divergent contribution T0I5 ' in(14.61) to the Ward identity (14.53) played a crucial role in rendering the expres-sion finite. This term cancelled the linear divergent contribution arising from 6m,which is a consequence of the lattice regularization, and has no counterpart in thecontinuum. What is new on the lattice, is that the axial vector current requiresan additional finite renormalization, in order that the Ward identity retains itsnaive structure after renormalization. Thus QED renormalization alone does notsuffice.

We have considered above the Ward identity relevant for studying the renor-malization of the axial vector current. Ward identities involving the insertion ofthe divergence of the axial vector current in more general Green functions can alsobe readily be obtained. They can be best summarized by making use of functionalmethods. Thus consider the generating functional of Green functions,

Z[r), rj,J]= I DUD^Di) eS[U^M+sSOUTCe ^ (14.68a)

where

Ssource = Y^Mx)i>(x) + i>Wv(x) + MX)U»(.X)} • (14.686)X

Making the infinitessimal change of variables ip(x) —>• rp(x)+6rp(x), 4>(x) —> ip(x) +Stp(x), with Sip and and 6ip given by (14.38), under which the measure is invariant,one imediately concludes that

< 5S - SSsource >„,, , j= 0 , (14.69a)


where, as before, SS is given by (14.40), and

5SSOUrce = i ] P e(x) [fj(x)j5ip(x) + ip{x)j5rj(x)] . (14.696)

Expression (14.69a) can also be written as follows

JDUD^Di, [ - a J5M(x) + 2m0j5(x) + A(x)

or compactly

37 f)7< -d^h^x) + 2moj5(x) + A{x) >v,n,j= fj{x)j5-^7-T - ^-r^75r?(a;) . (14.70)

By differentiating this expression with respect to the sources we generate Wardidentities involving the insertion of the divergence of the axial vector current inarbitrary Green functions. The Ward identity we have discussed above followsby differentiating (14.70) with respect to t\ and fj and setting thereafter r\ = fj =Jft — 0. A similar analysis as the one discussed above shows that Ward identitiesinvolving the insertion of the divergence of the axial vector currrent in higher n-point functions involving rip > 2 external fermion lines and UA > 1 gauge fieldsare finite and non-anomalous upon QED renormalization.

14.4 The ABJ Anomaly

In continuum QED or QCD it is well known that Ward identities followingfrom gauge invariance play a very important role in securing the renormalizabiltyof these theories. In general, Ward identities relating different unrenormalizedGreen functions are derived by considering local transformations of the fields in thegenerating functional. If the "naive" form of these identities retain their structureafter renormalization, then we say that the Ward identities are non-anomalous.If their structure is not preserved (on account of quantum fluctuations) then onespeaks of anomalous Ward identities. Such a breakdown on quantum level canbe desastrous, for it may not allow the quantization of the theory. Thus e.g. inthe case of the electroweak theory it is important that the chiral symmetry ofthe classical action, in the massless quark limit, remains unbroken on quantumlevel, since the gauge fields are coupled to chiral currents which are the sourcesfor the fields. An example of a harmeless anomaly is the ABJ-anomaly (Adler


1969, Bell and Jackiw 1969), which plays an important role in the description ofthe electromagnetic decay TT° —• 27. The anomaly is harmless because it manifestsitself in the divergence of a current (the axial vector current) which is not thesource for the gauge fields in QED or QCD. How does this anomaly arise withinthe framework of a lattice regularized gauge theory?

Let me first remind the reader of how the axial anomaly arises in continuumQED. Consider first the partition function for continuum QED in an externalgauge field,

Z[A] = f Dip Dip e-5/e,m[A,V,*] 5 (14.71)

where 5 / e r m is the fermionic contribution to the action (5.8c). In the limit ofvanishing fermion mass this action is invariant under the global transformations4){x) —> exp{iaryz)ip{x) and 4>(x) -4 ip(x)exp(ia^z) 1 where a is an ^-independentparameter. Next consider an infinitessimal local transformation with a -> c(x).The fermion measure in (14.71) is, at least formally, invariant under this transfor-mation.* Let 5Sferm be the corresponding change of the action. Since the abovelocal transformations of the fermion fields just correspond to a change of variables,we (naively) conclude that

< 5Sferm >A= I f D1PD1P SSferme-s'"-™ = 0 . (14.72)

< SSferm >A is the expectation value of SSferm in a background gauge field. Onereadily finds (after a partial integration) that

3Sferm = / dAx e(x)\2mi/j(x)75^(x) - <9M(^(37)7^75^))] •

Since e(x) is an arbitrary function, it follows from (14.72) that

< dnhn{x) >A= 2m < j5(x) >A , (naive) , (14.73)

where j 5 M = •07M75V'. But actually this equation is violated because of quantumfluctuations, as is demonstrated in any book on quantum field theory. The vi-olation is induced by the triangle graphs shown in fig. (14-4), each of which is

* Actually, the integration measure needs to be regularized. As has been shownby Fujikawa (Fujikawa, 1979) the ABJ anomaly can be viewed as a consequenceof having to regularize this measure.


Fig . 14-4 Triangle diagrams which give rise to the ABJ anomaly in

the continuum.

linearly divergent and therefore must be regularized. One then finds that (14.73)is modified by an additional term as follows

e2

< d^j5^(x) >A= 2m < h{x) >A +YQ^F^F^ ' (14-74)

where F^ is the dual field strength tensor, F^ — \e^vxpFxp-On the lattice the regularization is introduced already on the level of the

partition function. Hence any considerations of the above type will leave us withequations which are exact. One then may be left with an anomaly when the cut-offis removed, i.e. upon taking the continuum limit.

In the continuum formulation of QCD (or QED) it is well known that differentgauge invariant regularization schemes all yield the same expression for the axialanomaly. Any candidate for a lattice discretization of QED or QCD should alsoreproduce the correct axial anomaly in the continuum limit. As we have seen inchapter 2, a naive discretization of the fermionic action which is local, hermitean,chirally symmetric for vanishing fermion mass, and having the correct continuumlimit, will necessarily lead to the fermion doubling problem. That this must be sois a consequence of the Nielsen-Ninomyia theorem (1981). To avoid the problemof species doubling, the chiral symmetry must be broken explicitely, if one refrainsfrom abandoning at least some of the other properties. As we have seen in chapter5, a simple way to accomplish this is to add to the naively discretized fermionaction a "Wilson term", leading to (5.17), which ensures that in the limit ofvanishing lattice spacing the unwanted fermion modes acquire an infinite massand hence decouple. For Wilson fermions the axial anomaly has been first studiedby Karsten and Smit (Karsten, 1981). These authors showed that the origin ofthe anomaly was an irrelevant term in the lattice Ward identity. The anomaly wasalso studied by Rothe and Sadooghi (Rothe, 1998), using the small-a-expansion


scheme of Wetzel (1985). In the former reference it was shown that, in the limit ofvanishing lattice spacing a, this naively irrelevant contribution is indeed given bythe D —> 4 limit of the dimensionally regulated continuum triangle graph. Thesecomputations are very involved and we do not present them here. Subsequently,the anomaly was studied within a more general framework by Reisz and Rothe(Reisz 1999), where the action is not assumed to have the form proposed by Wilson.

As we have pointed out in section (4.7), there is an even milder way of breakingchiral symmetry on the lattice, proposed a long time ago by Ginsparg and Wilson(1982). For Ginsparg-Wilson (GW) fermions the fermionic action is of the form

Sferm = J2 ^(X)(D(^ V) + x,y)Hv) , (14-75)%,y

where the "Dirac Operator" D(x, y) is a 4 x 4 matrix in Dirac space which breakschiral symmmetry in a very special way. While the Dirac operator in the con-tinuum, or its naively discretized version, anticommutes with 75, the GW-Diracoperator satisfies the following GW-relation:

{J5,D} = aDlhD , (14.76)

where D is a matrix whose rows and columns are labeled by the a spin and space-time index. It is a function of the link variables and can be expanded in terms ofthe gauge potentials,

D(x,y)= Yl -[D{£...^{x,y\xl---xn)Alll{x1)---All,n{xn) , (14.77)

where x denotes a lattice site. For Wilson fermions D(x,y) is a strictly localoperator connecting only neighbouring lattice sites, and with the gauge poten-tials living on the corresponding links. On the other hand, the Dirac operatorD(x, y) for GW-fermions is non local in the sense that it connects arbitrary latticesites and does not involve only the gauge potentials at sites close to x. For theNeuberger solution to (14.76), given by (4.65a) and (4.66), the non-locality arisesfrom the inverse of (AÂ)1^2. Nevertheless, as has been shown by Hernandez etal. (Hernandez, 1999), Neuberger's Dirac operator is still local in the more generalsense, that the Dirac operator decays exponentially at large distances, with a de-cay rate proportional to I/a. In the naive continuum limit we have of course thatD(x,y) —> 7MDM[J4], where -DM[A] is the covariant derivative in the continuum.

In the following we shall discuss two alternative points of view of how theABJ anomaly is generated in the case of GW fermions. In the first approach


we will make use of the observation made by Liischer (1998), which was provenin sec. 4.7, that the GW-action possesses an exact global axial-type symmetryfor vanishing fermion mass. As we have seen in sec. 4.7, this symmetry differesfrom the standard one (4.8) by lattice artefacts. Associated with this symmetryis an axial vector current which is conserved on classical level. Following a similarprocedure as in the continuum, we then derive a Ward identity and identify theanomalous contribution (ABJ anomaly). Within this approach the anomaly willarise from the non-invariance of the fermionic integration measure under the non-standard local axial transformations. In the second approach we then consider theWard identity derived for standard local axial transformations under which themeasure is invariant, but the action is not. The ABJ anomaly now originates fromthe explicit chiral symmetry breaking in the GW-action, and hence parallells theapproach to the anomaly in the case of Wilson fermions.

Approach 1

Consider the action (14.75), where D satisfies the GW-relation (14.76). TheGW-Dirac operator is a function of the link variables. This action is invariantfor m — 0 under the global infinitessimal transformations (4.68). Consider thecase where e in (4.68b) is a function of the space-time coordinates, i.e., a localtransformation.:

rf> -> ip' = ip + Sip , i> -> ip' = $ + 5$ , (14.78a)

5rl>(x) = ie(x)K [(1 - \D)^ (X) , (14.786)

8$(x) = ie(x) [^(1 - | D ) ] (X)75 • (14.78c)

One then readily verifies that the variation of the action is given by

SSferm = i J2 <x) [F(x) + 2mi,(x)75^(x) + A(x)] , (14.79)X

where "£x = E w « 4 - F(x) and A (a;) are defined in (4.69b) and (4.69c), with F(x)satisfying (4.70). Now according to the Poincare Lemma on the lattice (Liischer(II), 1999), (14.70) implies that there exists an axial vector current J5^(x) suchthat

F(x) = -dL^{x) , (14.80)

where d^ is the left lattice derivative (the proof of this lemma is quite involved).Since the GW-Dirac operator has the correct continuum limit, it follows that for


a -» 0, J5^{x) -¥ $(x)'yn'y5tl)(x). With (14.80) we conclude that (14.79) is given

by8Sferm = i £e(x)[-d^jBli(x) + 2m^(x)l5rP(x) + A(x)} . (14.81)

X

In contrast to the case of Wilson fermions, where js^x) is given by (14.41), wedo not know the explicit expression for the axial vector current. The last term onthe rhs is not responsible for the anomaly. In fact its external field expectationvalue vanishes in the continuum limit. The anomaly arises from the non-invarianceof the fermionic measure under the variations (14.78). This leads to a Jacobian.Thus from (14.78) we have that

The determinant of this matrix is given by

detB = eTrlnB &l + TrlnB .

Here the trace is carried out in Dirac, as well as in coordinate space. Hence theJacobian of the transformation is

J & = detB = l-iê(x)trD(y6D)(x,x) ,^ X

where trn denotes the trace in Dirac space. A corresponding expression holdsfor J[<5f/>'/5t/>]. Consider now the partition function in a background link variableconfiguration,

Z[U] = ID1>D$ e-S/.rm[*.tf,u] _ ( 1 4 8 2 )

Making the infinitessimal change of variables (14.78) leaves this expression invari-ant. Hence we conclude that (taking into account the Jacobian of the transforma-tion) that

< <55/erm >u -ia^2e(x)trD{-y5D(x,x)) = 0 . (14.83)X

To arrive at the final form of the Ward identity we rewrite the contribution <A(x) >u to < SSferm > in (14.81) as follows: from (4.69c), < A(x) >u is givenin matrix notation by *

< &{x) >u= ytrD [(D + m)-1D75 + l5D(D + m)"1] (x,x) ,

* We have dropped the explicit dependence of the Dirac operator on the linkvariables, for simplicity.


where we have made use of

< rl>a(x)$fi(v) >u= (D + m)-fa,y) (14.84)

Making use of (D + m)~1D — 1 — m(D + m)~x and trcy^ = 0, we can further

write

< A(x) >u= -m2atrD('y5(D + m)~1)(x, x) = m2a < tp(x)ip(x) > .

Hence the Ward identity (14.83) takes the form

< d^j5^(x) >u= 2m < j5(x) >u -atrD(-y5D(x, x)) , (14.85a)

where

Mx) = (1 + ~)ip(x)75ij(x) . (14.856)

The last term on the rhs of (14.85a) is an anomalous contribution and yields in

the continuum limit the ABJ anomaly (Hasenfratz, 1998: Luscher (I), 1998).

We now proceed to derive the above Ward identity from a more conventional

point of view which parallels the case of Wilson fermions.

Approach 2

Consider once again the action (14.75) and the partition function (14.82).

Let us carry out a change of variables (14.78a) induced by the standard local axial

transformations, where Stp and dip are given by (14.38). Under this transformation

the integration measure is invariant since tr£>y5 = 0. The change in the action is

easily computed and now given by

SSferm = i^2e(x)^p(x)y6(D^)(x) + (^D){x)yB^(x)\ +2im'ê{x)i,(x)1^{x) .X X

(14.86)

We now make use of the definition (4.69b) and of (14.80) to write this expression

in the form

SSferm = »£e(x)[-d^jSll(x) + 2m^(x)j5^{x) + a(^D)(x)75(D^)(x)} (14.87)x

Since the fermionic measure in (14.82) is invariant under the above local axial

transformations, the Ward identity now reads

< 5Sferm >u= 0 . (14.88)


Making use of (14.84), the expectation value of the last term in (14.87) can bewritten in the form

a < (tpD)(x)-y5{Dip)(x) >= -atrD(-y5D(x.x)) - m2atrD U5(D + m)~l(x, x)] ,

(14.89)where the last term is just m2a < 4>(x)j5ip(x) >u- Thus one arrives once againat the anomalous Ward Identity (14.85).

We shall not discuss the anomalous contribution any further, since it has beenshown by Reisz and Rothe (Reisz 1999) that any lattice action satisfying some verygeneral conditions (which also hold for GW-fermions) will necessarily reproducethe correct anomaly in the continuum limit. For the more complicated case ofQCD we will present a proof in section 6 of the following chapter.

CHAPTER 15

WEAK COUPLING EXPANSION (III)LATTICE QCD

Weak coupling perturbation theory in lattice QCD is much more involved thanin the U(l) gauge theory discussed in the previous section. The reasons for thisare the following: a) The lattice action is a complicated functional of the colouredgauge potential; b) the gauge invariant integration measure associated with thelink variables depends non-trivially on the gauge fields, and c) the generalizedcovariant gauge, analogous to that discussed in the abelian case, can no longer beimplemented in a trivial way. This latter feature is of course also true in continuumQCD. But whereas there the Faddeev-Popov determinant, which emerges whenthe gauge is fixed, can be represented in terms of an effective ghost-gauge fieldinteraction which is linear in the gauge potential, this is no longer true in latticeQCD.

The complexity of the expressions in perturbative lattice QCD, is a conse-quence of the gauge invariant lattice regularization, which, as in the U(l) case,leads to an infinite number of interaction vertices. But because of the non-abelianstructure of the theory, most of these vertices have a very complicated structure.Although in the naive continuum limit only those vertices survive which are char-acteristic of the continuum formulation, irrelevant contributions to the action doplay an important role when studying the continuum limit of Feynman integrals.Hence one must exercise great care in including all lattice artefacts in the action,which contribute to the correlation functions in this limit. In this connection let usrecall that the whole point of the lattice formulation was that it provides a regu-larization scheme, where gauge invariance is ensured for any finite lattice spacing.Only by including all lattice artefacts we will therefore be ensured that for any fi-nite lattice spacing gauge invariant correlation functions will be independent of thechoice of gauge, and that the gauge fixed theory will possess a BRS-type symmetry,reflecting the original gauge invariance of the theory before fixing the gauge. Thisis important since this symmetry leads to Ward-identities which play an impor-tant role in developing a renormalization program. One therefore should abstainfrom making any approximation when expressing the link-integration measure andFaddeev-Popov determinant in terms of the gauge potentials before removing thelattice cutoff.

In the following we shall set up the generating functional in a form which

242

Weak Coupling Expansion (III). Lattice QCD 243

is suited for performing the weak coupling expansion in lattice QCD. We thenderive lattice expressions for the propagators and vertices relevant for low orderperturbative calculations. We will however not discuss lattice Ward identities, norrenormalization in this book. A detailed discussion of these topics can be foundin the work by Reisz (1988a,b).

15.1 The Link Integration Measure

In lattice QCD, the link variables are elements of SU(3) in the fundamentalrepresentation.* They hence can be written in the form

U^n) = e ^ ( n ) (15.1a)

where </>M(n) is an element of the Lie-algebra of SU(3):

0 » = £ ti(")TA- (1 5-1 &).4=1

Here TA (A = 1 , . . . , 8) are the generators of the group in the fundamental repre-sentation. We chose them to be given by

rpA _ ^_2 '

where XA are the Gell-Mann matrices introduced in chapter 6. Prom (6.8), and(6.15) we have that

[TA,TB] = iYJfABcTc, (15.2a)c

Tr{TATB) = ±5AB. (15.26)

Under a gauge transformation, the link variables transform according to

U^n) -»• gWU^g-^n + A),

* In contrast to the notation used in chapter 6, we shall not underline matricesin colour space with a "twidle". Except for the generators, all quantities carryinga colour index will be c-numbers. Quantities without a colour index are matrix-valued.


where g(n) and g(n + fi) are elements of SU(3). Correlation functions involving

the product of link variables and coloured quark fields are computed according to

(6.24), where DU is the gauge invariant measure associated with the link variables.

Our present objective is to express this integration measure in terms of the group

parameters 4>A{n) defined in (15.1b). To this effect we first construct the invariant

measure associated with a single link variable, following Kawai et al. (1981).

Let U be an element of 5(7(3) with Lie algebra L. Then U can be written

in the form U = exp(icf>), with <p £ L. Consider the following bilinear differential

form

d2s = Tr(dU'rdU), (15.3)

where dll — U(tp + dtp) - U(tp). It is invariant under left or right multiplication

of U with a group element of SU(3). Expressed in terms of the coordinates {<f>A},

parametrizing <p, (15.3) will have the form

d2s = YJ9AB{<t>)d<pAdcj)B. (15.4)

A,B

This defines a metric, gAB{4>) on the group manifold. The gauge invariant inte-

gration measure (Haar measure) is then given by

dn((t>) = y/detgWUdt*, (15.5)A

where g(tp) is the matrix constructed from the elements gAB(<P)- To calculate g(tp)

let us rewrite (15.3) as follows (Kawai et. al. (1981)):

d2s = Tr{(U-1dU)HU~ldU)}. (15.6)

As is shown in appendix A, U~ldU is an element of the Lie-algebra of SU(3) and

has the form

U~ldU = iYJTAMAB(cj))dcj)B, (15.7a)

A,B

where

MAB{<1>) = I1——) , (15.76)\ l v / AB

8

§ = Y, <t>AtA> (l5-7c)A=l


and tA are the generators of SU(3) in the adjoint representation*. Their matrix

elements are given by

tic = ^fABC, (15-8)

where JABC are the structure constants of the group appearing in (15.2a). The

generators tA satisfy the following orthogonality relation:

Tr{tAtB) = 35AB. (15.9)

Inserting the expression (15.7a) into (15.6), and making use of (15.2b), one finds

that the metric QAB(</>), defined in (15.4) is given by

9AB{4>) =1(M\4>)M(<P))AB

= /1-ccSN (15-10>

V *2 JAB'

Hence g(<j>) has the following power series expansion in terms of {</>}:

where $ has been defined in (15.7c). This is the expression we were looking for.Notice that by construction, g{(f>) is a non-negative hermitian matrix. Hence itsdeterminant is real and non-negative.

The invariant integration measure associated with a single link variable U^ (n)is obtained from (15.5) and (15.10) by replacing <pA by 4>A{n). Taking the productof these measures we arrive at the desired expression for DU:

DU =\ I I Jdet \\MHMn))M(Mn))] | D4>, (15.12a)

where

M^n)) = w^' (15-12fo)

A

* Capital letters always run from one to eight.

(15.11)

(15.12c)


and

D = n ^ M W - ( I 5 - I M )n,A,p

We next rewrite (15.12) in a way which is convenient for later computation. Sincethe determinant of g((f>) can also be written in the form exp(Tr Ing{4>)), we obtainthe following alternative expression for DU:

DU = e - s » » ' W ^ , (15.13a)

where, apart from an irrelevant additive constant, Smeas[(j)] is given by

SmeasM = -\ETrln\2{1-;?(^n))}. (15.136)

The quantity appearing within square brackets is a polynomial in the matrix $M.Smeas.^] can also be written in the form

Smeas.[<t>] = ~\ £ T r M l + JV(^(n))], (15.14a)

where

Ar( (n)) = 2^^^ I ($ M )^ , (15.146)

with <J>M denned in (15.12c). Notice the difference in the structure of the integration

measure (15.13) and its abelian U{\) counterpart, where DU is given by (5.23).

Indeed in the abelian case U~ldU = id(f>, where <j) ls the single real variable

parametrizing U. Hence according to (15.6), the right-hand side is just given by

W>)2-Consider now the ground state expectation_ value of a gauge invariant func-

tional of the dimensionless fields <p^(n),tp^{n),ip^(n).* We denote this functional

by T[(f>, 4>,ip]- For the action we will take the standard Wilson form given in eqs.

(6.25), except that now we shall write S Q C D ^ , V*, V'] instead of SQCD[U,I/),'4>] to

emphasize that SQCD should be expressed in terms of the fields <j>^. Writing the

* Recall that while the capital letters A, B,... run from one to eight, smallLatin letters run from one to three, since the quark fields transform under thefundamental representation of SU(3).

(15.12d)


link integration measure in the form (15.13a), the ground state expectation value

of T[cf>, ifi, ip] is given by

- = SD^D^T[Û]e-^*'*^-s^*\ (15 J5)/2ty£ty2tye-SgcD[*,W]-Sm...M

This expression is not yet suited for carrying out the weak coupling expansion. As

in the (7(1) case considered in the previous chapter, we must still fix the gauge.*

15.2 Gauge Fixing and the Faddeev—Popov Determinant

A popular choice for a local gauge condition which is linear in the fields

(j>A, and respects the discrete lattice symmetries, is the following generalization of

(14.3),**

7$[b X] = %<tf(n) - XA(n) = 0, (15.16)

where xA(n)> A=l,...8, are some arbitrary given fields. We want to introduce thisgauge condition into the functional integral (15.15). This must be done in such away that expectation values of gauge-invariant observables are not affected by thegauge-fixing procedure. Following the prescription given by Faddeev and Popovfor the continuum formulation, we consider the integral

A^M i X] = / Dg f ] 5(^m, x]), (15.17)

where 9(f> denotes collectively the gauge transform of the group parameters {4>A(n)}which parametrize the link variables J7M(n). The above integral is carried out overthe gauge group manifold, with the integration measure Dg being given by theproduct of the invariant Haar measures on SU(3) at every lattice site,

Dg = Y[dfi{gn).n

Because by definition of the Haar measure dfj.(gg') = dfi(g), it follows that App[4>; x]is gauge invariant:

AFPN;X] = A F P [ 0 ; X ] .

* Within the framework of continuum perturbation theory, this is an obviousrequirement, since otherwise the gluon propagator cannot be defined. In a latticeformulation Smeas[4>] includes a term quadratic in the fields 4>A which is not gaugeinvariant. But since it depends on the coupling, it must be treated as part of theinteraction.

** It will be always understood that repeated Lorentz indices are summed.


We next introduce the identity

1 = AFP[d>;x} [Dgl[6(J*[°<l>;x])J n,A

into the integrands of (15.15). Using the fact that T,_SQCD, AFP, as well as the

integration measures DU = exp(-Smeas.)D(j) and DipDip are gauge invariant, we

can replace the fields </>, t/J, and ip in these expressions by their gauge transformsg(j>, 9tp and 3ip, respectively. A simple redefinition of the integration variables then

leads to the following alternative expression for (15.15):

< rfo, j>, 3>]>=j [ D</>D$DJ>AFP[; x] n 5^n [0; xD

J n,A (15.18a)

where the normalization constant Z is given by

Z= />Z30JDÂFp[0;x]n(5^n[0,x])e"SoCD[^'*1"5meas [01- (15-186)J n,A

We must now compute Afp[<f>;x]- But because of the gauge fixing (^-function

appearing in (15.18), we only need -to know AFP[<J>;X] f°r field configurations <f>

satisfying the gauge condition (15.16). Hence it suffices to calculate the integrand

of (15.17) for 94> in the infinitesimal neighbourhood of g — 1 with <f> restricted by

(15.16). Accordingly, we must compute the change in the fields ^(n) induced by

an infinitesimal gauge transformation. In the continuum formulation the corre-

sponding change in the potentials A^ is a linear functional of the gauge fields. In

a lattice regulated theory this is no longer true. Hence also App[(t>', x] w^ acquire

a non-trivial structure.

The response of <t>^(n) to an infinitesimal gauge transformation has been

calculated in appendix B. Let S^(j>^(n) denote the change in (j>^(n) induced by

the transformation, i.e.

jj^n) _> eie (n )f/M(n)e-ie (n+ ' i ) = e^W+SuM"))

where e(m) are elements of the Lie algebra of SU(3) in the fundamental represen-

tation:

e(m)=J2TAeA(m).A


Then it has been shown in appendix B that

he)4>*(n) = -Y,DMABeB(n) , (15.19a)B

whereDM[0] = M - 1 ( ^ ( n ) ) ^ + i$M(n). (15.1%)

M " 1 (</>M(n)) is the inverse of the matrix (15.12b), and $M(n) has been definedin (15.12c). The first few terms in the expansion of M~x in powers of 0M(n) aregiven by

M"1 (0») = 1 + i $ » - ^ ($»)2 + . . . . (15.19c)

The non-linear response of <pA (n) to an infinitesimal gauge transformation is dueto lattice artefacts. Indeed, making the replacement

<t>B(n) ^ goaAB{x)

in (15.19), and using the more suggestive (continuum) notation eB(x) instead ofeB(n), one finds that

go5(e)A^x) ^Q-YJDM]Bcec{x) ,a^ c

where

B

is the matrix valued covariant derivative of the continuum formulation.

Let us now calculate the function J^[9<l>,x] m (15.17) for g in the neighbor-hood of the identity, and for fields 4>^{n) satisfying the gauge condition (15.16).Making use of (15.19) one finds that

^\94>\ X] %~J2 LnA,mBWB{m) , (15.20a)9~ m,B

where

LnA,mB[4>\ = dZbMAB6nm (15.20b)

is the analog of the matrix d^D^ÂBîx - y) in the continuum formulation.Note that according to (15.19b), Dfi{4>] is a local function of the matrix valuedfield $M(n), and that all lattice derivatives act on the lattice site "n". We next


introduce the expression (15.20a) into (15.17). Since for fields 4>A(n) which satisfythe gauge condition (15.16) the integral only receives a contribution for groupelements g in the immediate neighbourhood of the identity, we may replace thegroup integration measure by

Dg^Y[deA(n).n,A

The integral (15.17) can now be immediately performed and one finds that forfield configurations satisfying the gauge condition, AFp is independent of x andgiven by

AFP{(j>] = det(-L[</>])

where L[cj>] is the matrix defined in (15.20b). AFp is referred to in the literatureas the Faddeev-Popov determinant.

In principle we could incorporate the effect of AFp[<f>] into an effective actionby setting AFp[(j>] = exp(Trln(-L[(/>])). We had adopted such a procedure inconnection with the link integration measure. But whereas Smeas.[4>] is given by asum over local products of the fields 4>A(n), this is not the case for Tr ln(—L[<f>]).*To derive the Feynman rules, however, we want to start from an action where thefields are coupled locally. Hence we cannot incorporate the effect of the Faddeev-Popov determinant into an effective action in the above mentioned way. Using astandard trick however, we can circumvent this difficulty. Thus making use of theformula (2.47) we can write det L[<j>] in the form

AetL[<j>] = f Y[d~cA{n)dcA{n)e-SFPÂl\ (15.21a)J A,n

where

SFP[<l>,c,c\ = - Yl &A(n)df;DMABCB(n). (15.216)A,B,n

The Grassmann valued fields cA(n) and c (n) (A = 1 , . . . , 8) carry a colour butno Dirac-index. They transform according to the adjoint representation of SU(3),and are the lattice analogs of the famous Faddeev-Popov ghost fields.

Finally, we must get rid of the gauge fixing ^-functions in (15.18a,b). We dothis in the way described in chapter 14. Since the numerator and denominator in

* This non-local property is of course not peculiar to the lattice formulation.


(15.18a) do not depend on the choice of the fields xA(n), w e c a n average these ex-

pressions over xA(n) with a Gaussian weight factor exp — ^ J2n A XA(n)xA(n) •

Collecting the results obtained so far, we therefore find that (15.18) can be written

in the form*

< T[<f>, i>, ip\ > = * r — — - , (15.22a)f D<t>DipD4>DcDce~sQ°°['t''i''tp'c'c]

where the "total" action, SQQD) is given by

SQ°CD = SG[<fi] + SFW)[4>, i>, 4>] + SGF[<t>} + Smeas.[<£\ + SFP[<P, c, I]. (15.226)

Here SG[4>] and S^lfai),^] are given by (6.25b) and (6.24c) with the link vari-ables J7M(n) replaced by exp(«</y(n)), and SGF[<P]

1S t n e non-abelian analog of(14.5c):

SGF[} = ^ E ( ^ M < ( " ) ) 2 • (15-22c)n,A

The expression (15.22a) provides the appropriate starting point for performing aweak coupling expansion analogous to that described for the abelian case consid-ered in the previous chapter. We have gone quite a way to arrive at this expression.And we still have to do some work to derive the Feynman rules from the generatingfunctional

Z[J,V,fl,i,i\ = f D</>D4>DipD6Dce~sQc°['t''i'>3''e3

E [ # <-n)K (")+% (n)V-S W+K (n)vaa (n)+~t (n)cA {n)+lA (n)£A (n)l

• e L J

The reason is that the contribution of SG[4>] to SQCD, is a complicated functionalof the fields </>•£ and their derivatives. This is a consequence of the non-abeliancharacter of the link variables. In the continuum formulation this piece of theaction gives rise to triple and quartic interactions of the gluon fields. In the latticeformulation, on the other hand, not only do these vertices get modified by latticeartefacts, but there are also an infinite number of additional interaction verticeswhich contribute to the correlation functions for finite lattice spacing. Only afinite number of these vertices, however, contribute in the continuum limit in agiven order of perturbation theory.

* In chapter 2 we had omitted the "hat" on ip and ip for convenience.


15.3 The Gauge Field Action

Consider the action (6.25b). It can be written in the form

SG[cf\ = \Tr V (11 - Uîn)), (15.23)9o „,,,,„

where "11" denotes the 3 x 3 unit matrix, and £/MI,(n) is given by the orderedproduct of link matrices around an elementary plaquette lying in the /w-plane:

Uîn) = ei<M")eiMn+£)e-»'Mn+*)e-i<M»). (15.24)

Clearly,

UlAn) = VvM- (15-25)

Since {/^(n) is an element of SU(3) in the fundamental representation, it can bewritten in the form

U^n) = e{*^n\ (15.26)

where 0MI,(n) is an element of the Lie algebra of SU(3) in the above mentionedrepresentation:

^w = ECw^- (15-27)A

Prom (15.25) it follows that

Consider the trace of (11 — U^). Expanding (15.26) in powers of (j)^, and recalling

that TrTA = 0, we have that

Tr(11 - CV) = Tr | ^ ( ^ ) 2 + ^ ( ^ , ) 3 - ^ W v ) 4 + • • • } •

Because cf)^ — —<t>vn, it follows that the cubic term will not contribute to theaction. Hence

SG[4>] = \ J2 r r { | ( ^ ( n ) ) 2 - ^j(^(n))4 + ...} • (15.28)


The trace can be easily evaluated by making use of the following relations*

Tr(TATB) = l-5AB,

Tr{TATBTc) = \{dABC+ifABc),

Tr(TATBTCTD) = ±-5AB5cD ~ \fABElcDB + \dABEdcDE

n

+ nifABEdcDE + fcDEdABE),o

which can be easily derived using the commutation relations (15.2a) and the fol-

lowing expression for the anticommutator of TA and TB:

{TA,TB} = \sABI + dABCTc.

Written in terms of the fields 0 ^ , . . . , < ^ , (15.28) becomes

s*w = 72 E fiE(^)2 - iE«) 2 (O 2

° w L A A'B (15.29)~ T o o E dABEdCDEA

v<t>^v ^1v<^.v + • • • ] .i y J A,B,...,E

where we have suppressed the dependence of the fields on the lattice site. Next we

express the right-hand side of this expression in terms of the fields {<f>A (n)} which,

apart from a factor go, are the lattice analogues of the coloured gauge potentials.

To this effect we first derive a relation between </>Ml/ (n) and the matrix valued fields

</v(ra), 4>v{n+ji), ^M(n+/>) and (j>v{n) appearing in the product (15.24). We do this

by making repeated use of the Campbell-Baker-Hausdorff (CBH) formula, which

states the following: Let G be a Lie group with Lie algebra L, and let Bi and B2

be elements of L. Consider the product exp^x) exp(i?2). It can be written in the

form

eBieB,=ec(B1,B,)t (15.30a)

with C{B1,B2) £ L. According to CBH, C(BUB2) is given by

OO

C{BU B2) = Y^, Cn(B1, B2), (15.306)n=l

* The tensor dABC is completely symmetric in the indices. Its components aregiven by d118 = d228 = d338 = -d888 = 1/V3; dX56 = d157 = -d247 = d256 =d344 = d 3 5 5 = —^366 = —d37r — 1/2; d 4 4 8 = ^553 = ^668 = ^778 = ~2~73'


where the contributions Cn(Bi,B2) are determined by the following recursionrelations:

C1(B1,B2) = B1+B2,

(n+l)Cn+1(BuB2) = i[fl! - B2,Cn(BuB2)]

+ I > 2 * E [Cm1(Bi,B2),[...,[Cm2p(B1,B2),Bl + B2]...]].p>l mi,.. . ,m2p >0

2p<n mi + ... + m,2p = n

(15.30c)Here k2p are rational, and &2p(2.p)! are the Bernoulli numbers. We now makerepeated use of this formula to calculate the product

eBleB2eB3eBi=eM(B)

where B stands collectively for Bi,...,Bi. Clearly

4

»=i

Consider the case where exp(Bi) are elements of SU(N) in some matrix represen-tation. Since M(B) is at least of O(B), and since Tr M = 0, it follows that if wewant to calculate Tr (11 — exp(M)) up to fourth order in the Bj's,* we only needto know M(B) up to O(B3). Hence we will also only need to know (15.30b) up tothis order. Prom (15.30c) one obtains

C(Bi,Bj)=Bi + Bj+^[Bi,Bj]

+ ±([Bi,[Bi,Bj]] + [Bj,[Bj,Bi]]) + ... .

Making repeated use of this expression and of the Jacobi identity, one finds aftersome algebra that M(B) can be written in the form

M(B) =J2B> + \ Hi13*' BA + y2 E &> P - B&(15.31)

+ g £ {[BuiB^B^ + lB^lBjî]}} + O(B*).i<j<k

* This will suffice for our purposes.


We now apply this formula to the matrix product (15.24), and obtain an expressionfor (15.27), correct up to third order in the fields </>M:

4>v,=Y,0i + \5><,Oj] - ^ E f t - ft-^\ i<j iJ (15.32a)

- g E { f t . [ M * ] ] + [ < W M « ] ] } + --- •

Here the Lie-algebra valued fields 0,- are given by

01 (n) = < £ »

02 («) = &,(n + £) = &,(n) + S*<Mn)* D (15.o2o)

6»3(n) = -</y(n + i>) = - ^ ( n ) - 9 ^ ^ ( n )#4 H = -<f>v{n).

For simplicity we have suppressed the dependence of </>M(, and &i in (15.32a) on thelattice sites.

Consider for example the contribution of the first two terms appearing on theright-hand side of (15.32a). Inserting the definitions (15.32b) one finds that

+ ^([^,^]-[^,^M]) (15.33)

-\[d^,d?h] + O{<f).

Introducing the dimensioned (matrix valued) gauge potentials A^x), and the fieldstrength tensor F^ in the familiar way, i.e.,

<?W (n) ~^9oa2F^ (x), (^ = na)

we see that only the first two terms in (15.33) contribute to F^ in the continuumlimit. Thus

F^(x) = d*Av(x) - SiÂyix) + ig0 [A^x), Au{x)} + ...,

where d^ is the dimensioned right lattice derivative, and where the dots stand forterms which vanish for a —» 0. Hence in the continuum limit F^v coincides withthe field strength tensor in QCD.


We now return to eq.(15.32a). Decomposing the matrices 0; as follows

0i = J2e?TA>A

and making use of the relations (15.2), one finds that the colour components ofthe field (f)^, defined in (15.27), are given by

< » = EÂ(n) - \ E**» + Y2 XX-(»)' i<j ij (15.34a)

i<j<k

where0.1(«) = £/ABC«f(n)0?(n)

B'° (15.346)Ofjk(n)= E fABEfcDEe?(n)0f(n)9°(n),

B,C,D,Eand where Of-,...,Of are related to the fields <f>^,4>f a n d their derivatives byexpressions analogous to (15.32b):

< ( n ) = ^ ( n )

. . -„ * (15.34c)

0£(n) =-tfin) - dUfr)9A{n) = -flj(n).

Inserting (15.34a) into (15.29) one arrives at the following expression for the action,correct up to fourth order in the fields {Of-}:*

+ i6 2ê;A< + ^ L ^ -feiv« ' • • » • • *

+ ^ E ^ (#« + < ) (15.35)

^ ^ ^ Y~* QAQBOCQD~ J^dABEacDE 2î Vi °i Vk Vl

i,j,k,l

* From now on it will be understood that also repeated colour indices aresummed.


Here a sum over repeated colour indices is understood. The dependence on n, fi,and v of the quantity appearing within curly brackets is given implicitely by therelations (15.32b).

Clearly, (15.35) is a complicated function of the fields {(f>^) and their deriva-tives. Most of the contributions are however lattice artefacts and do not survivein the continuum limit.* This is easlily demonstrated. Introducing the definitions(15.34b,c) into (15.35), and the dimensioned colour components of the vector po-tentials according to

(j>f{n)^goaA%(x), x = na, (15.36)

one has that

X>?(n)->floa2/£(*),i

whereC(x) = tfA?(x)-d?AJ!(x).

Hence the sum £^ Of actually vanishes with the second power of the lattice spac-ing. It is therefore evident, that in the naive continuum limit only the first threeterms appearing on the right-hand side of (15.35) survive. For a —> 0 their contri-bution can be easily evaluated and yields the usual expression for the continuumaction

Sa^\ J d*xF%,(x)F»,(x),

where Fû(x) is the non-abelian field tensor defined in (6.16). On the other hand,for finite lattice spacing, only the contributions quadratic and cubic in Of- have asimple form. We shall treat them in detail below.

15.4 Propagators and Vertices

i) The gluon propagator

Consider first the contributions to (15.22b) arising from Sa and SGF which arequadratic in the fields 0^. Expressed in terms of the dimensioned gauge potentialsone readily finds that**

S(G\A} = lj2tf(x)n™(x,y)AC(y), (15 3?a)*,y

* This is true for the classical action. But on the quantum level, we cannotsimply ignore all these contributions, as we have pointed out repeatedly.

** We use the notation Y^x — 2na4, introduced in chapter 14. The procedure

for casting SG into the form (15.37) is the same as that described in this chapter.


where

n%?(x,y) = 6Bcn»,(x,v), (15.376)

and Q,^ is defined in (14.13b). Following the procedure discussed in chapter 14, wecan immediately write down the propagator for the gauge potential in momentumspace. To avoid the appearance of any superfluous phases, which will eventuallycancel in the Feynman rules,* we Fourier decompose the fields A^(x) as follows

* » = [ ÂBAk)eikx+ik^2. (15.38)JBZ \Z7r)

Then the gluon propagator in fc-space is given by

k 1 / ,fcMfc"\ r

i G ~ S S = P [^ ~(1 - a o ) -^ ) 5BC>where k^ is the lattice momentum defined by (13.7a).

ii) The two-gluon vertex

Consider next the contribution of Smeas[<j>] quadratic in the gauge potential.From (15.15) one finds, using (15.9), that

S£L[A] = 5 j | ^ X > ? ( s ) ^ ( * ) - (15-39)X

This contribution is proportional to g% and hence should be considered as partof the interaction. What is striking about this contribution is, that it divergeslike I/a2 in the continuum limit!** In fact it has the typical structure of a mass-counter term. This is indeed the role it plays in lattice perturbation theory whereit serves to eliminate quadratic "ultraviolet'" (o —> 0) divergences in Feynmanintegrals contributing in O(gl) to the gluon self-energy. In this way the latticeprovides its own counterterms to ensure the renormalizability of the theory. Thisdemonstrates in a particularily drastic way, how important it is to include theeffects of the lattice cut-off in the integration measure.

The two gluon vertex arising from (15.39) is given in momentum space by

* See the discussion in chapter 14.** Since ^2X = J2 n

a*' ^ e contribution to Smeas arising from the Haar measureassociated with an individual link variable actually vanishes for a —> 0.


where 5^p\k + k') is the periodic <5-function. Notice that we did not include thefactor 1/2! appearing in (15.39) in the definition of the vertex, conforming to usualconventions.

iii) The three-gluon vertex

The contribution to SG involving three gauge fields is obtained from the sec-ond term appearing on the right-hand side of (15.35).* Making the substitutions(15.34c) and (15.36), and using the antisymmetry of the structure constants /ABCunder exchange of any pair of indices, one finds that

4 3 ) = So £ / A B C (Afr) + \d*A${x)) (d*A*(x)) Acx{x)5Xv.

x

Notice that this expression includes lattice artefacts vanishing linearly with a. Itcan be written in a more symmetric form by making use of the antisymmetryand cyclic symmetry of the structure constants }ABC in A,B,C. Using the firstmentioned property we have that

4S) = " f £ W ((l + ±8*) Afr)) (A5(x)%A?(x)) 6XV,

where the action of dj} is defined by

g(x)d*f{x) = g(x)tff(x) - {d*g(x)) f(x).

Next we cyclically permute the pair of indices (A, /i), (B, v) and (C, A) and obtain

43) = - | £ / A B C {((1 + \d*)A${x)) (x)d^(x)^ 5Xv +cycl.perm.\ .

(15.40)In this way, we have made manifest the Bose symmetry under the exchange ofcolour and Lorentz indices. Notice that by having cyclically permuted simultane-ously the Lorentz and colour indices, the gauge potentials appearing in all threeterms within the curly bracket in (15.40) are labeled by the pairs (A,n), (B,v)and (C,A).

* Notice that Smeas[(f>] does not contribute in this order.


Let us calculate explicitly the 3-point gluon vertex, F ^ f , in momentumspace. This vertex function is defined as follows

4s)M = -JFE/^|^|^^(*)^(*')A?(*'')r^(*,Jfc',Jfc'')xei{k+k'+k")-x

(15.41)We represent r ^ c ( £ , k', k") by the following diagram

Consider the explicit expression displayed on the right-hand side of (15.40).

In momentum space the operators 1 + | c ^ and dj? act multiplicatively as follows

(1 + -d*)eikx -»• (eife"f cos ^ ) e ' f c x

2 o ___ 2 (15.42)

eik-xdReiq-x _^ {(q _ kêi(k+g).xei(k+qî ;

where fc^ and ^ are defined by an expression analogous to (13.7a). When thephase factors appearing in (15.42) are combined with the overall phase exp[i(fcM +k'v + k'l)a/2], arising from the definition of the Fourier transform (15.38), anduse is made of the fact that - because of the summation over x in (15.41) - thesum of the momenta flowing into the vertex vanishes, one finds after some trivialtrigonometric algebra that

r£fAc(fc, *', k") =tffO(2ir)4JJ,4)(fc + k' + k")fABC [ (fc / r r* ' )M cos ^6vX

+ (k~k")v cos ^ 5^ + {k~k)x cos ^ j j ,

_ (15.43)where (p - q)^ stands for

O.~,),-|-»[^.].


Hence in the continuum limit, (15.43) reduces to the familiar expression of the

continuum formulation

r£?Ac(fe, k\ *")—> igo(2n)45^(k + k' + k")fABC [(*" - k%SvX

+(k - k")v8^ + (k1 - k)^).

iv) The four-gluon vertex

The computation of the four-gluon vertex (see fig. on page 245) from thefourth-order contribution in {9f} to the action (given by the last five terms in(15.35)) is quite tedious.* Making use of the definition (15.34b) and of £V 9f =djj,(t>v ~ dv4>^ following from (15.34c), as well as of the antisymmetry of fcDEunder C <-> D, this contribution takes the form

(SG)Ai = S{J] + 4 d ) , (15.44a)

where

4 f l = ^ E E E fABBfcDB{{&?tf-B*<lt)['Zlofefo?' y ° n ti,v ABCDE j^k

+ * E W * P + <W>] (15 446)j<k<l

+1 E(W - W)( »P - « ) }i<lk<l

and

SG = ~Jm2~^2^2^2 E \T;(SABSCD + SACSDB + 8ADSBC)+\ '> U0 n p,v ABCDE

+ {dABEdcDE + dACEdDBE + dADEdBCE)\

(15.44c)where {0A} have been defined in terms of the gauge potentials in (15.34c). Wehave suppressed the dependence of the fields on the lattice site n. The expression(15.44c) has been written in a manifestly symmetric form under the exchange ofany pair of colour indices.

* The following derivation has also been carried out independently by P. Kaste.


Consider first the expression (15.44b). We need to express the r.h.s. in termsof the fields {9^} and their derivatives according to (15.34c). This clearly generatesan anormous amount of terms which can be readily obtained by using a programlike "Mathematica". The next step consists in classifying these terms according tothe number of identical Lorentz indices of the gauge potentials. A large number ofthese terms can be combined by making use of the antisymmetry of the structureconstants under the exchange of any pair of indices, and of the invariance of theindividual contributions to the action under the exchange of fi and v. Let usdenote the contributions to SQ involving two, three, and four gauge potentialswith identical Lorentz indices by SQ',SQ and SQ , respectively.

Consider for example the contribution S^2\ After exploiting the above-mentioned symmetries one finds at the end of the day that the (hundreds!) ofterms combine to

4 / 2 ) 4 E E fABEfcDE{hK%<fi)(tf%tf)^° n/i,i/ ABCDE

- (WxWxWxW)}-This expression can be written in an even more convenient form:

4 / 2 ) = A E £ fABEfcDE{{[{l + \d?)Km + \^)€]^° n,n,v ABCDE

- \(dUt)0Uc,)} [[(i + \&?)tf][(i + \d?)tf] - ±( W ) ( W ) ]

(15.45)Next we introduce the dimensional gauge potentials through the identification4>B = goaA^, and Fourier decompose the fields AB according to (15.38). Letk, q, r and s denote the incoming momenta associated with the gauge potentialscarrying colour indices A,B,C and D, respectively. Making use of (15.42), onereadily finds, after carrying out the sum over n (which yields an energy momentum


conserving 5-function), that

Q(h) _ 9o v ^ V^ f dik d41 d^r d*s t tG = ^ h A ™ J ™ ^ ^ ^ ^ E

x {<W<,P [cos -a(k - r)v cos -a(q - s)M + â2{k - r)v{q - s)M - — a^q^f^s^

-^^5Apa2(g^J:)A(S~r)M}i^(fc)iS(g)I^(r)i^(S)

(15.46a)where, generically,

2 1kp = - sin -akp. (15.466)

The term proportional to S^xûp can be written in a more symmetric form by mak-ing use of the antisymmetry of /CDE under the exchange of C and D. The contri-bution proportional to S^Sxp is already invariant under the relabeling (C, A, r) o(D,p,s). Adding to (15.46a) the expression with (C, \,r) and (D,p,s) inter-changed, and dividing the result by 2, we obtain

c(/2) _ 1 \ ^ V^ /" ^4^ ^4<7 d^r d4s

° ~~~\hPAhJBZ^W^W (15.47a)xr{

lf^BCD(k,q,r,s)AA(k)A^(q)A^(r)Af(s)

where

1 iiu\p (K,q,r,s) —

-flo yZfABEfcDE |<W<W cos -a(fc - r ) J / c o s - a ( ? - s)M - — a4fcj,gMf^SMl

S^pS^x [cos -a(fc - s)v cos -a(g - r)M - — a^k^f^sJ j

(15.476)and

T^\lD{k,q,r,s) = -go ^ / A B E / C D S J T <5MA<5I/pa2(A; - r ) ^ ( g - s)M

- ^ M P ^ A a 2 ( ^ ' ^ ' s ) I / ( g ~ r ) M (15-47c)

-^ ( 5^5APa2(9-fc)A(s ' : : :V)M| .


Let a, /?, 7 denote collectively the sets

a = (A, fi, k); 0 = (B, v, q); 7 = (C, A, r); 5 = (D, p, s). (15.48)

Then the expression (15.47b) is symmetric under the relabelings i) a o 0, ii)7 -H- 6, iii) a -H- 7, 0 <-»• d and iv) a -H- <5, /3 o 7. The full Bose symmetry of thecontribution SQ to the 4-gluon vertex function can now be incorporated if weadd to (15.47b) the expressions obtained by carrying out the permutations /? «-» 7,and p -H- S, and dividing the result by 3. Then

4 ! h P ^ B Z (27T)4 (27T)4 (27T)4 (2^)4 t 1 " " ^ ^ ' ? ' r ' S J

(15.49)Having written SQ in this symmetrized form, one finds, upon making use of

2_}iÂBEfcDE + fACEfDBE + JADEIBCE] = 0 ,E

following from the Jacobi identity for double commutators of the generators of thegroup, that the last term in (15.47b), together with the permutations in (15.49)does not contribute to (15.49)

Consider next the contributions SQ' and SQ , with three and four Lorentzindices of the gauge potentials identified. Making again use of the symmetries ofIABEICDB one finds after a fair amount of work that they can be reduced to thesimple forms

f = - ^ E E E fABEfcDE(tfO*tf)(d?4>t)[(l + ld?)tf]^° n n,v ABCDE

(15.50a)and

4/4) = " i E E E fABEfcDE(ffiU»C/UZ)- (15.506)g° n n,v ABCDE

Going over to momentum space, one finds, making again use of (15.42) that

o(/s) , c(h) _ _ J L V" V f rf4fc d4g d4r d4s

° ° 'KXMDJB'WWWW (15.51a)xTl%MABCD(k,q,r,s)A^k)A^q)A^(r)Af(S),


where

r$tpMABCD(k,q,r,s) = - g2

0 Y, fABEfcDE[Mvx6V(>a2(8~r)ltkvcos Ê

(15.516)While the second term in this expression is symmetric under the permutations: i)7 «-> S, ii) a <-> /3, iii) a O 7, /? O 5 and iv) a f> (5, /? U 7, where a, (3,7, <5 standfor the collection of indices (15.48), the first term is only symmetric under theexchange 7 «-)•<$. We therefore add to it the corresponding expressions obtainedby implementing the permutations ii) - iv), making use of the symmetries of thestructure constants, and divide the result by 4.

x I -6v\6vpa2{s - r)^ cos -agM

- 7Â^w a 2 ( s ~ r)^ c o s 2afcl/

+ g^^MPa2(9 ~ k)^p c o s 2aSx

- TÎ/<SAÔ2(9 - k)p§\ cos -arpD A

(15.52)The expression appearing within curly brackets is still not symmetric under allpermutations of the collective variables (15.48). To exhibit the full Bose symmetrywe add to it the corresponding expression obtained by the permutations appearingin (15.49), and divide the result by a factor 3.

The remaining contribution (15.44c) to the action, expressed in Fourier space,is obtained in a straightforward way. Combining it with that obtained above oneis then led to the following expression for the four-gluon contribution to the action

, . __1_ f d4k dAq d4r d4s[ G'Ai ~ 4! JBZ (27T)* (2n)4 ( 2 ^ ( 2 ^ 5 3

x £ Y, rZT(k,q,r,s)At(k)A*(q)Acx(r)A?(s) .

livXp ABCD

(15.53a)


where*

r # £ D ( * . «,»•,*) =~9o [2_^fABEfcDE{8ii\Si,p[cos -a(q - s)Mcos -a(k - r)v - —kuqIJ.fl/sIJ,}

E1 i 4

- <^P<Wcos -a(q - r)Mcos -a(k - s)v - — Kq^f^s^

+ -x$u\5vPa2{s - r) Jiv cos(-a?M)1 ^ ^

- g<W<W°2(s - r)I,9MC0S(2afc'/)2 - 2

+ g<W<Wa2(9 - *0A^>cos(2aSA)

- g V ^ " 2 ! ? ~ fc)pSACOs(-arp)

+ — S^S^xSâ2 J ] (9 - /c)CT(s - r)CT|cr

+ (B «-> C, ^ -H- A, 9 -H- r) + (B -H- £>, 1 -H- p, q -H- s)l

+ j | a 4 j g (Âs5C£> + <5AC5BD + &ADSBC)

+ 2_^{dABEdcDE + dACEd.BDE + dADEdBCE) }E

x { 'W 'W'W'X^ Q<rfasa - S^S^xkpqpfpSâ

- S^S^pkxqxSxf^ ~ S^xSupk^f^s^q^ - S^xS^pq^f^s^k^

+ S^Sxpkxqxr^Sn + Sûpkvf^q^s^ + 5^p5uxkvsvqIJ,fhli \

(15.536)

This concludes our discussion of the pure gluonic sector. We now proceed to theanalysis of the fermionic and ghost sectors.

iv) Fermionic and ghost contributions

Consider first the contribution to (15.22b) arising from SF [A,^,^]. ForWilson fermions it is given by (6.4), where the link variables are replaced by(15.1a). Expanding these in powers of </>M, and introducing the dimensional fermionfields and gauge potentials according to (4.3a) and (15.36), one obtains up to

* This is a corrected version of the expression given by Kawai et al. (1981).


second order in the gauge potentials A^(x):

SF[il>,$,A] = SP[4,ri>] + S$)[A,rP,ii>] + S$\A,4,i>], (15.54a)

where

40 )hM = (M + 7 ) 5>O(*WO(*)^ ' X

~ ^ 5Z [â(a;)(r - ^)i>a(x + ap) + 4>a(x + afi){r + W{x)} ,

(15.546)

2 tt (15.54c)-ija(x + afi)(r + lti)ij

b(x)}A^(x),

x [r(x)(r - 7M)V6(X + oA) + V(x + aA)(r + 7 M ) / W ] A ^ ( I ) ^ ( I ) .(15.54d)

A summation over the "quark" and gluon colour indices is understood*. Apartfrom some group theoretical factors, and the fact that the fields now carry colourindices, the structure of Sp[A, ip,i>] is quite similar to that discussed in the abelianU(l) case. Hence except for some obvious modifications, the structure of thefermion propagator and gluon-fermion vertices will also be the same as thoseobtained in the abelian case.

Finally, consider the contribution of the ghost fields to the action. It is givenby (15.21b), where JDM[0] has been defined in (15.19b). By expanding the matrixM-1(<^M) up to terms linear in </>M (cf. eq. (15.19c)), we include lattice artefactsvanishing linearly with a in the naive continuum limit. Next we introduce thedimensioned Faddeev-Popov ghost fields cA(x) and cA(x) according to **

cA(n)âcA{x),

c (n) -» ac (x).

* Recall that small latin letters run from one to three, while capital letters runfrom one to eight.

** They carry the same dimension as the scalar field discussed in chapter 3.


Then (15.21b) becomes

SFP[A,c,c] = -J^cA(x)5ABQcB{x)X

-g0J2fABCCA(x)d^(x)(l + ^d*)cB(x)}X

- ^ ^ E ^ { * < 7 . * I > } A f l ( ^ W ) ( ^ c B ( x ) ) ^ ( a : ) ^ ( x ) + ...)

(15.55)where we have made use of (15.8). By Fourier decomposing the quark and ghostfields in a way analogous to (14.17), one readily derives from (15.54b-d) the prop-agators and interaction vertices in momentum space. Except for some colourmatrices, the quark propagator and the gluon-quark interaction vertices have thesame structure as the corresponding expressions in the U(l)-gauge theory. Fur-thermore, from the quadratic contribution to (15.55) we see immediately that theghost propagator is given by &AB/k2 • The ghost-gluon interaction vertices in mo-mentum space can also be read off immediately from (15.55) by making use of theproperty (15.42). Below we summarize the lattice propagators and vertices.

i) Propagators and vertices which possess a non-vanishing continuum limit*

1 / k k \

V*EM \iy.sinPMfl + M(P)) ap

ab

¥ 5 A B

* For the definition of M(p) confer eq. (4.29b).


igo(2n)4S^)(k + k' + fc")/ABc[Â(A"~fc')Mcos ^ . a.— I . ^

+ <W& - *;")„ cos -k'xa + 5^(k' - k)x cos -k'â]

igo(27r)4S^(k + p- V')UBCV'» cos(pMa/2)

(see eq. (15.53b))


ii) Vertices which have no continuum analog

- §SO 2 (2T) 4 4 4 ) (* + k'+P-P')5^{TA,TB}ab

.[rCos(^)-^sin(^)]^

^g2oa

2(2n)H{pi)(k + k1 + p - p'){tc, tD}ABS^p,

Note that the factor -^ multiplying the contributions to the action involvingcouplings with n-gluons have not been included in the expressions for the vertices.The symmetry factor multiplying a Feynman integral is computed in the sameway familiar from continuum perturbation theory.

In contradistinction to continuum perturbation theory, there are, in general,more diagrams to be considered when calculating a vertex function in a givenorder of the coupling. Thus consider for example the diagrams contributing tothe gluon self energy in one-loop order. They are depicted in fig. (15-1). Theircontributions have been calculated by Kawai, Nakayama and Seo (1981).


Fig . (15-1) Diagrams contributing to the gluon self- energy in one loop

order

Each diagram contributes a quadratically divergent mass term. But whenthe graphs are summed these divergencies are found to cancel! There is anotherremarkable cancellation that occurs. When performing the calculation one en-counters non-covariant terms of the type p^<5MJ/. But after summing all the con-tributions, these terms cancel out and one is left with a transverse expression forthe gluon self energy, reflecting the gauge invariance of the theory!

Because of the complexity of lattice QCD Feynman rules (note that we haveonly expanded the action up to O(g%)), and the periodic structure of the in-tegrands of Feynman integrals, perturbative calculations of more than one loopcontributions become prohibitively difficult. In this connection the power countingtheorem of Reisz discussed in chapter 13 is of great help, for it allows one at leastto take the naive continuum limit in those cases where Feynman integrals satisfythe conditions for which the theorem applies.

In the following two sections we will apply the knowledge we have acquired sofar to the computation of two important quantites: the ratio of two renormalizationgroup invariant scales, and the ABJ anomaly of the axial vector current within theframework of QCD. As you shall see, the structure of this anomaly is, under verygeneral conditions on the action, independent of the lattice regularization. Whatconcerns the first mentioned quantity, we will not dwell on any technical details,but merely discuss the problem on a qualitative level.


15.5 Relation between AL and the A-Parameter of Continuum QCD

In chapter 9 we have seen that in QCD with massless fermions, dimensioned

physical quantities, such as a hadron mass, or the string tension, can be calculated

in units of a lattice scale parameter AL, which determines the rate at which the

bare coupling constant go approaches the fixed point g^ = 0 with decreasing lat-

tice spacing. A similar renormalization group invariant scale, AQCD-I a l s o occurs

in continuum QCD. But there, AQCD determines how the renormalized coupling

constant g, denned, for example, as the value of the three or four-gluon vertex

function at some momentum scale n, changes with /i. The connection between g

and fi, which ensures that physics does not depend on the choice of the renormal-

ization point fj,, can be obtained by studying the response of g to an infinitesimal

change in /Lt. This response is measured by the following /3-function

P(9) = Hj-- (15-56)

In two-loop order this /3-function is given by

P = - P o 9 3 ~ P i 9 5 + •••, (15.57)

where fto and /3i have the same values as those appearing in the two-loop expansion

of the /3-function considered in chapter 9 (cf. equation (9.21b)). Because (5Q > 0

the renormalized coupling constant is driven to g -^ g* = 0 a s ^ i — » o o . This

is the statement of asymptotic freedom. In the one loop approximation to the

/3-function, given by the first term on the right-hand side of (15.57), integration

of (15.56) leads to the following relation between g and ji:

- = e~w^. (15.58)M AQCD

The value of the integration constant, AQCD, depends on the definition of the

renormalized coupling constant. Because of the asymptotic freedom property of

QCD this scale can be measured in deep inelastic scattering processes, where the

short distance dynamics can be described by renormalization group improved per-

turbation theory. Its value is found to be of the order of 200 MeV. On the other

hand we have seen in chapter 9 that the connection between the bare coupling

go and the lattice spacing a, which ensures that physical observables remain un-

changed as we remove the lattice structure, is given in the one-loop approximation

to the /3-function by

a = - 3 - e ~ 3 ^ . (15.59)AL


The lattice scale Ax,, determined, for example, from a Monte Carlo calculation of

the string tension is of the order of a few MeV. To confirm that QCD involves only

a single scale which describes the large distance physics at quark separations of the

order of 1 fm, as well as the short distance physics taking place at separations of

0.1 fm or less, one must check whether A^, as determined from a measurement of,

say, the string tension, corresponds to the value of AQCD as obtained from deep

inelastic scattering data.

The first calculation relating the two scales has been performed by Hasenfratz

and Hasenfratz (1980) in the pure gauge theory. This calculation has been subse-

quently extended to the case of QCD with massless quarks by Kawai, Nakayama

and Seo (1981). The basic idea underlying these computations is the following.

Suppose we calculate an observable O in QCD with massless fermions. In the

lattice regulated theory the value of this observable depends on the bare coupling

constant go and the lattice spacing a:*

O = O((fo, a,...). (15.60)

Now for sufficiently small lattice spacing, go can be tuned to a in such a way that

O remains fixed as we remove the lattice structure. In the two-loop approximation

to the /^-function this dependence is given by (9.21c,d). Hence go is a function of

the product aAi, : go = go{o-h.i). On the other hand, we can also eliminate the

cut-off dependence by introducing a renormalized coupling constant g in the way

familiar from continuum perturbation theory. This coupling constant depends on

go, the renormalization scale //, and the cut-off a. Since g is dimensionless, it will

depend on /J, and a only through the product /ia:

9 = 9(9o, /ia). (15.61)

Solving this equation for go, one obtains the bare coupling constant as a function

of g and fia:

9o = goigâ). (15.62)

Upon inserting (15.62) into (15.60) one arrives at an expression in which the

dependence on the lattice spacing has again been eliminated

O(go(g, (Mi), a;...) ~ O{g, //;...). (15.63)

* The "dots" stand for possible dependences on kinematical variables such as

momenta; for example, O could be a scattering matrix element.


Now g can also be tuned to fi in such a way that the right-hand side of (15.63)

remains fixed as we change the renormalization point /i. In the one-loop approx-

imation to the /3-function (15.56) the dependence of g on \x is given by (15.58);

hence g = g{ji/ AQCD)- But both, the dependence of go on aA^, and of the renor-

malized coupling constant on H/AQCD, are determined from a single equation,

namely from (15.61), or its inverse (15.62). Thus holding g0 and a fixed, we deter-

mine the fi dependence of g. Alternatively, holding g and \i fixed we determine the

a-dependence of go. This shows that Ax, and AQCD must be related. To obtain

this relation one calculates the right-hand side of (15.61) in perturbation theory.

In the one-loop approximation one obtains an expression of the form

g2 = g20 - 2#)<ri \n{fia/c) + O(g6

0), (15.64)

where c is a constant. From here one readily verifies the one-loop expressions for

the /3-functions (9.6b) and (15.56). Now to this order AQCD/AL, as determined

from (15.58) and (15.59) is given by

AQCD wîJ2~-k)

AL

But from (15.64) it follows that

-2 - -2 = 2#> HfJ-a/c) + O{g20).

9 do

We therefore conclude thatAQCC _

Hence the purpose of a perturbative computation consists in calculating the con-

stant c in (15.64). In the momentum subtraction scheme (MOM), Hasenfratz and

Hasenfratz (1980) have calculated the ratio AMOM/A-L for the pure SU(3) gauge

theory. After a lengthy calculation they found that

AMOMIAL = 83.5 (pure SU(3)).

In full QCD with massless quarks this value was found to change as follows (Kawai

et al., 1981)AMOM/AL = 105.7 (3 flavours)

AMOM/AL = 117.0 (4 flavours)

These ratios are consistent with those determined from non-perturbative lattice

calculations (A^) and from the deep inelastic scattering data (AMOM)-


15.6 Universality of the Axial Anomaly in Lattice QCD

In this section we study the ABJ anomaly in the divergence of the coloursinglet axial vector current in QCD, and show that not only for Wilson or Ginsparg-Wilson fermions the well known anomaly is reproduced in the continuum limit,but that the same result is obtained for any discretization of the action satisfyingsome very general conditions. In the case of a U(l) gauge theory, this has beenfirst shown to be the case by Reisz and Rothe (1993). Here we discuss the case ofSU(S) which has been studied subsequently by Frewer and Rothe (2001). Fromthe analysis it will be evident that the same proof goes through for any SU(N)gauge theory. The main steps we will follow are the following: we first discuss thegeneral form of the lattice axial vector Ward identity. As you will see, its precisestructure, which depends on the particular discretization of the lattice action, neednot be known to compute the anomaly. Only very general properties thereof arerequired. In fact, the entire ambiguity in the lattice Ward identity, arising fromdifferent discretizations, will reside in a contribution which vanishes in the naivecontinuum limit. Although its structure depends on the way one has discretizedthe action, we will only make use of quite general properties thereof to generatethe anomaly.

General Structure of the Ward Identity

In the following all expressions will be written in terms of dimensioned vari-ables. We are interested in computing the anomalous contribution to the di-vergence of the colour singlet axial vector current iMs(x) = xp(x)/y5ÎJ,tp(x) in anexternal gauge field, where ip(x) are 3-component fields in colour space.

Consider the fermionic contribution to the lattice action for QCD. It is of theform (4.61), i.e.

Sferm = 2${x){Du{x,y) + m5xy)ip{y) , (15.65)x,y

where Du(x,y) is the Dirac operator (a matrix in Dirac-spin and colour space)depending on the matrix valued link variables which we denote collectively by U.As always ^ = YLn

ai> where n labels the lattice sites. SU(3) Colour indiceswill be denoted in the following by small latin letters. The action is assumed to begauge invariant, and to possess the discrete symmetries of the continuum theory.* The Dirac operator, which is a function of the link variables, can be expanded

* The Dirac operator can always be decomposed into a chirally symmetric, and


in the gauge potentials in the form

Du(x,y)= J2 ^D^.%:a"(x,y\xl---xn)Al\(x1)---All(xn). (15.66)Til

The next step consists in deriving a lattice Ward identity for the divergence ofthe singlet axial vector current. This is achieved by performing in the partitionfunction the infinitessimal colour blind local axial transformation of the fermionfields analogous to (14.38). Note that, since we do not specify the Dirac operator,we have no other alternative to define a sensible axial transformation. Since themeasure is invariant under this transformation one is led again to the statement(14.88). In the case of Wilson fermions the variation 5Sferm is given by (14.81),where A(a;) is an irrelevant operator vanishing in the continuum limit. For differentlattice discretizations, jz^x) and A will differ from (14.41) and (14.46) by termswhich vanish in the naive continuum limit. What concerns the axial anomaly,however, the precise form of the various terms in (14.81) need not be known. Thisis quite remarkable. In fact, as we shall see, any lattice discretization of the action5 with the following properties:

i) S has the correct continuum limitii) S is gauge invariantiii) The Dirac operator is localiv) Absence of species doubling

reproduces the axial anomaly in the continuum limit. This anomaly arises froman "irrelevant" (A) term in the Ward identity, which, in view of what has beensaid above, will necessarily have the form

< dpJ5n{x) >u= 2m < j5(x) >u + <A>u , (15.67)

where J5fl(x) and j$(x) possess the correct continuum limit. Thus for Wilsonfermions, the Ward identity is of the above form (Rothe, 1998). Any other dis-cretization of the action will differ only by lattice artefacts which can, in principle,be absorbed into the A-term. The reader may ask: is there no way of avoidingsuch a term? The answer is "no" as will be clear from our analysis.

a chiral symmetry breaking (sb) part as follows, D(x, y) = D(x, y)sym + D(x, y)sb,where D(x,y)sym = ^[ATsfrs , and D(x,y)sb - i{£>, 75)75 . Any candidate fora lattice action should possess the correct continuum limit. It therefore followsthat for a -+ 0, D(x,y)sym -> j^DÂ], where DÂ] is the covariant derivative,while D(x,y)Sb vanishes in the continuum limit. It therefore also follows that fora —> 0 D(x,y) —> 7p.D[A], where -DM[A] is the covariant derivative.


Before proving the above assertion it is convenient to rewrite the Ward identity(15.67) in terms of correlators in momentum space. Let O{x) stand for any ofthe operators appearing in (15.67). Then < O{x) >u has the following formalexpansion in the gauge potentials

<r r)(v\ s , , - Y ^ —- V ^ r ( ° ) a i ' ' a " M < r i -ro • • • r ) Aai (r-, ) • • • Aa" IT \<. U\X) >u— / j — / J J-Mi'-Mn \X\X1J X2i i • c n ; / 1

M l {^lj Ap.n \xn) >

(15.68)where, because of the assumed symmetries of the action, the sum over n startswith n = 2. The correlation functions F^..?^ 'an(x\xi,X2, • • -,xn) are symmetricunder the exchange of any pair of collective labels (xi, fii, a,). Defining the Fouriertransform of F^.a^n"°" (x\%i, x2, • • -,xn) by

r(O)O l-an, | \__ [" d41 -iq-xTT d4ki iki-Xif(O)ai-<>-n( \u . . . j . \1 MI •••*»» l ^ F i i ,Xn) — / / -o_^4e 1 1 ('o^^4 1 c i - c « (.ylrci! i K n ; >

(15.69a)where, by translational invariance,

n

f {°^'a"(9\ku • • ; K) = % - X) fci)r&«"°B(fei, • • •. *n) , (15.696)»=i

the Ward identity (15.67) translates as follows to momentum space,

- n ^ ^ - , n (*!>•• •, kn) = 2mf---»rMB (fci, • • •, fen) ? 0p(A)ai-an(-, , v (, • aJ

where^ = e

i S ^ ^ 8 i n ^ . (15.706)

As we shall see further below, gauge invariance implies that every term in (15.67)possesses a finite continuum limit. If in this limit the A-contribution is differentfrom zero, we are faced with an anomaly, since the divergence of the axial vectorcurrent would not vanish in the chiral limit m —> 0 (as it would, if the axialsymmetry would be implemented on quantum level). We now show that under theconditions i)-iv) this is indeed the case, and moreover, that this limit is universal.

Let us first study the implications of the assumptions i)-iv). Clearly thefirst assumption is a "must" and needs no further elaboration. Consider nextassumption ii):

ii) Gauge invariance


Gauge invariance has strong implications. In fact the renormalizability ofQED or QCD relies heavily on gauge invariance. Gauge invariance tells us that ifO(A, ip, tp) is a gauge invariant operator, then its external field expectation valuesatisfies

FO[AU] = FO[A] , (15.71a)

wherefDj,DJO[A,il,,i>]e-slA>*M

F°[A] = JWDJe-W,*,*] ' ( 1 5 7 1 6 )

with Au the gauge transformed potential. This can be readily verified by makinguse of the gauge invariance of the fermionic measure. On the lattice the variationof the gauge potentials induced by an infinitessimal gauge transformation is givenby (15.19) with 4>l = gaA^, i.e., *

6A%{x) = [gfabcAfa) - M-^gaA^x^yix) , (15.72)

where fabc a r e the structure constants of S[/(3), d^ is the dimensioned right latticederivative, and the matrix M is given by (15.12b), or the expansion (15.19c).Because of the structure of the rhs of (15.68), the F's are symmetric functionsunder permutations of the labels 1, • • -,n. Hence the variation can be written inthe form

SUFO[A] = E 5 £ ^°-X'a"(x\xux2, • • ;Xn)6A%(Xl) • • • AH(xn) .

Inserting for SA^\ the expression (15.72), and considering in turn the coefficientsof O(A2) and O{A3), one finds that (15.71a) implies that

d^°la2^(x\xux2)=Q, (15.73a)

and furthermore

dZA°il%T2(x\^^xz) = 9ofaiatbT^{x\x2,x3)8XlXl

- |soa/«l0a69£ ( ^ r ^ M * ! ^ , ^ ) ) + (2 o 3) .(15.736)

These relations can be readily translated to momentum space. Defining the Fouriertransform by (15.69), one finds that (15.73a) takes the form

CkD^lTHkukJÔ, (15.74)

* In the following we prefer to use small letters for the colour indices.


where * denotes complex conjugation, and where k^ is defined in an analogousway to ( 15.70b). Because of Bose symmetry a corresponding statement holdsfor (fci)Ml replaced by (fa)^- The second identity (15.73b) reads as follows inmomentum space,

- io | / . i a a c (fcl)wf £ £ * ( * ! + fc2' **) (15.75)

+ (2 *+ 3) .

Further relations connecting higher and lower order correlation functions followfrom the requirement of gauge invariance. We will however not require them tocalculate the anomaly.

iii) Locality

Gauge invariance alone will of course not allow us to compute the axialanomaly. But when combined with iii) and iv) (see p. 276) this will be possi-ble. The point is that if iii) and iv) hold, then the correlation functions will beanalytic in the momenta around vanishing momenta, and hence possess a Taylorexpansion. This has important consequences. Thus consider (15.74). Analyticityaround {hi = 0} tells us that

f £ J S l O * ( * i , ft2) = f £ £ l O J ( o , 0) + c f ^ k i + & ° > T K + •••

Because of Bose symmetry under the exchange k\ <-> fc2,Mi M2>ai +•> «2 wemust have that

But since k^ and &2 are independent variables, it then follows from (15.74) that

fSr2(o,o)=o,

cfilT1 =°'and therefore also

/-i(O)a2<ii /-,

Summarizing we therefore conclude that

Tifi°1fca*(k1,k2) = O, (17.76)


where Ti denotes the Taylor expansion around zero momenta up to first order.Consider next eq. (15.75) evaluated for ki = &3 = 0, and small k\. Because

of (15.76) the rhs is of O{k\). It therefore follows that

fS:M32O3(0'0'°) = 0 - (15-77)

Relations (15.76) and (15.77) are weaker than those obtained from gauge invariancein the case of QED (Reisz. 1999), where an analogous statement to (15.76) holdsfor vertex functions involving an arbitrary number of gauge fields.

The reason that (15.76) and (15.77) play an important role for the compu-tation of the anomaly, is that we shall derive an expression for the renormalizedWard identity in the continuum limit using the momentum-subtraction schemewith finite lattice spacing. We then have to show that the continuum limit of thepotentially anomalous contribution to the divergence of the axial vector current isuniversal and given by the well known continuum expression. For this we will needto make use of the power counting theorem of Reisz (see section (13.3)), whichrequires not only that the lattice degree of divergence of all Zimmermann spacesof a Feynman integral is negative but also that the free fermion propagator is freeof doublers. This is just our condition iv).

Universality of the ABJ Anomaly

We are now ready to prove our assertion. Our attention will be focusedon the "irrelevant" contribution of the A-term in (15.67), which should generatethe anomaly in the continuum limit. Because the expressions considered in thefollowing are lattice regularized, all operations are well defined. Furthermore, sincewe are studying the Ward identity in an external background colour field, the onlyFeynman graphs which contribute are diagrams involving an arbitrary number ofexternal gauge potentials attached to a fermion loop.

Consider the axial vector Ward identity (15.67). By power counting the ul-traviolet lattice degree of divergence (LDD) of f^ and I5 is given by 3 — n, wheren is the number of external gluon fields, while the LDD of HA) is 4 — n. Hence theLDD of Feynman integrals contributing to F ' A ' is negative for graphs involvingmore than 4 external gauge fields. Their contributions thus vanish by the Reisztheorem (see sec. 13.3) in the continuum limit, since A is an irrelevant opera-tor. We therefore only need to consider the correlation functions i'/i1l^

ia' (ki, £2),


T<£l%7a3(ki,k2,k3), and f^la2lZ7ai(ki,k2,k3,k4) with LDD = 2,1 and 0, re-

spectively. Let us decompose these vertex functions as follows

r^lTHh, k2) = (i - T^lrnh, k2)+ Tif^la

2^ (kx,k2) + (T2 - TOf i a 2 (fci, k2) ,

f [£lZT* (ki, h, k3) = (1 - Ti)f £)™3(fci, fc2, k3) + Tof X 0 3 (fci, *2, h)

+ (T1-T0)Tfi£)Za>(k1,k2,k3) ,(15.79)

and

•p(A)aiO2a3a4/ji u l.o l-.'l — Ci _ J> \ f (A)aia2a3a4 /JL IL JL r. \iMiM2M3/»4 t.'Clj K2, «3, «4j — 1.-1 - t0jJ-M l M 2 W M 4 l « l , «2, «3, K4J (15 80)

+ -'0-1 MlM2M3M4 i ^ i , «2, «3, «4; ,

where Tn denotes the Taylor expansion around vanishing momenta up to n'thorder. By the Reisz theorem the first term appearing on the rhs of (15.78) to(15.80), vanishes in the continuum limit, since it has negative LDD and A is anirrelevant operator. The respective second terms in (15.78) and (15.79) vanish bygauge invariance (cf. eqs (15.76) and (15.77)). Hence we conclude that

Umf^°loa(fei,A:2) = Um(T2 - T O f ^ - ' t f c i . f c ) , (15.81a)

hm f £>X a 3 (* i . fc2< h) = limm - T0)f £)-£*«3(fel) fc2j h) , (15.816)

lim f ( , f ^ ^ ( f c i , fc2, k3, h) = Kmjotfila2^

ar4(ki, k2, h, k4) . (15.81c)

Hence the contributions (15.81a) to (15.81c) are of second, first and zeroth order inthe gluon momenta, respectively. As we now show, the Ward identity allows us tocalculate the limits from the corresponding Taylor terms of continuum one-loop-fermion diagrams involving two and one external gluon fields, with a 75 insertion.It then follows that the anomalous contribution is necessarily universal, i.e., itdoes not depend on the particular form of the irrelevant term in the lattice Wardidentity.

Consider first the rhs of (15.78). Applying the operation (T2 — Ti) to the(lattice regularized) axial vector Ward identity (15.70) with n = 2, and makinguse of (15.74), following from gauge invariance, we conclude that

(T2 - T^ffil*^ (kuk2) = -2m(T2 - Tjttlfi^ (ku k2) . (15.82)

The expression appearing on the rhs has negative LDD; hence its continuum limitis given, according to the Reisz theorem, by applying the (T2 — T±) operation to

(15.78)


the integrand of the corresponding continuum Feynman integral, i.e. the trianglediagram. One then readily finds that (15.81a) is given by

limf^"lOa(*i,fc2) = - ^ W ^ ^ A i U A * ) , • (15.83)

Consider next the rhs of (15.81b). Applying the (Ti — To) operation to the Wardidentity (15.70) with n = 3, and making use of (15.77) (following again from gaugeinvariance) one obtains

(Tj - T0)f ££},«'«' (fci, k2, k3) = -2m{Tx - T 0 ) f ™ M 3 (*i, k3, k3) . (15.84)

Since the LDD of the rhs is negative, its continuum limit is given by applyingthe (7\ — To) operation to the integrand of the continuum box Feynman diagram,involving three vector vertices and a 75 insertion. After some lengthy but straightforward calculation one finds that

lim f(,f^«»(fc1) k2, k3) = i ^ f t W ( / , M ! t , (15.85)

where q = ki + k2 + k3. Finally, consider (15.81c). Applying To to the Wardidentity (15.70) we obtain

^mJ0t^laX^(kuk2,k3,k,) = -2mUmTof^^/M(fc1,fc2)fc3,fc4) •(15.86)

Since the LDD of the vertex function on the rhs is again negative, the limit a —» 0can be calculated by applying To to the continuum pentagon graph involving a 75insertion. A simple calculation shows that (15.81c) is given by

*im,iMiM2M3M4 *> l ! 2> 3 ' 4' ~ 4 2IeMiM2M3M4J-r\J- 1 1 J- )+perm.\,

(15.87)where Ta are the SU(3) generators in the fundamental representation. Notice thatwe have made constant use of the Ward Identity (15.70) which we know is truefor arbitrary finite lattice spacing.

The (continuum) correlation functions in coordinate space, corresponding to(15.83), (15.85) and (15.87), are obtained by performing the Fourier transforma-tion (15.69), where the limits of integration now extend to infinity. The anomalouscontribution is then given by (15.68) with O —> A, where the sum over the coor-dinates are replaced by integrals. Only the two and three gluon vertex functions


actually contribute to the anomaly, as can be shown by making use of the Ja-cobi identity. One then easily verifies that the anomalous contribution to the(euclidean) axial vector Ward identity takes the well known form

2

A(x) = -JL-ê^F^Fîx) , (15.88a)

where

*%,{*) = dMx) - dvAl(x) - gfabcAl(x)Al(x) (15.886)

is the non-abelian field strength tensor.We have seen above that the A-term in (15.67), although superficially linearly

divergent in the continuum limit, is actually finite in this limit, and a 4'th orderpolynomial in gluon field. The other two terms in this gauge identity are also finite.The reason is, that the latter have LDD == 1 and 0 forn = 2 and n = 3, respectively,and negative LDD for n > 4. Gauge invariance and locality of the Dirac operator,as embodied in the statements (15.76) and (15.77), allows us to replace the n = 2and n = 3 vertex functions by the corresponding Taylor subtracted forms withnegative LDD. These possess a finite continuum limit.

This completes our discussion of lattice perturbation theory. All analyticmethods discussed in the last five chapters have their own merits, but they cannotbe used to calculate non-perturbative observables in continuum QCD. One istherefore forced to compute these observables numerically. In the following chapterwe now introduce the reader to various numerical methods that have been proposedin the literature.

CHAPTER 16

MONTE CARLO METHODS

16.1 Introduction

When one computes the expectation value of an observable in lattice QCD,one is faced with the problem of having to perform a tremendously large numberof integrations. Thus suppose we want to compute the following ensemble average

fDUO[U}e-sM{O) = fDUe-sm ( 16 '1}

where S[U] is a real functional of the link variables bounded from below. Using a104 space-time lattice, the number of link variables is approximately 4 x 104. Forthe case of SC/(3), each of these link variables is a function of 8 real parameters;hence there are 320000 integrations to be done. Using a mesh of only 10 points perintegration, it follows that the multiple integral will be approximated by a sum of10320000 terms. This example shows that we must use statistical methods to eval-uate the ensemble average (16.1). In fact, since most of the link configurations willhave an action which is very large, only a small fraction of them will make a signif-icant contribution to the integral (16.1). Hence an efficient way of computing theensemble average would consist in generating a sequence of link variable configu-rations with a probability distribution given by the Boltzmann factor exp(—S[U}).This is the technique of "importance sampling". If the sequence generated consti-tutes a representative set of configurations, then the ensemble average (O) will beapproximately given by the following sum

(o>4EoHyM (16-2)where {U}i (i = 1 , . . . , N) denote the link configurations generated.

For a few systems there is a simple method of generating field configurationson the lattice with the desired probability. This method is called the heat-bathalgorithm. Let us demonstrate the basic idea of this algorithm by considering aone-dimensional integral

(F) = / dxF(x)P(x) (16.3a)

Ja

284

Monte Carlo Methods 285

where P(x) is a normalized probability distribution

rb/ dxP{x) = 1 . (16.36)

Ja

Making the change of variable

y = f dzP(z) , (16.4)J a

(16.3) can be written in the form

{F) = f dyF{y) ,Jo

whereF(y) = F(x(y)) .

Hence the variable (16.4) is now distributed uniformly in the interval [0,1], and wecan generate for y a representative set of values with the help of a random numbergenerator. The crux of the whole matter is, of course, that one must be able toevaluate analytically the integral (16.4) and to obtain x(y).

In praxis the right-hand side of (16.3a) is replaced by a multiple integral overthe degrees of freedom of the physical system of interest. In this case one usuallyapplies the above method to each variable in turn, holding all the others fixed;i.e., each variable is brought to equilibrium with a "heat bath reservoir". For thisreason it is called the "heat bath algorithm". Unfortunately, the method can beapplied only to a few systems of physical interest (for example, to the pure SU(2)gauge theory). We shall therefore not dwell on it any further, and proceed toa discussion of other algorithms which can be applied to general systems. Butbefore we go into details, let us first explain what the general strategy will be tocalculate the expectation value (16.1). We first need to construct an algorithm,that will generate a sequence of configurations which will eventually be distributedwith the desired probability. Once the system has reached equilibrium we may usethe thermalized configurations to measure our observable. In praxis the numberof such configurations will be finite. But if the sequence generated by the algo-rithm constitutes a representative set, then the ensemble average (16.1) will begiven approximately by the sum (16.2). Furthermore, if the N measurements arestatistically independent, then the error in the mean will be of order 1/y/N. Inpraxis, however, the configurations we generate sequentially are not all statistically


independent. Hence when computing the statistical error, one must make sure toselect only configurations which are uncorrelated.

Summarizing, the procedure for calculating the ensemble average (16.1) is thefollowing: given a rule for generating a sequence of configurations, one first up-dates the configurations a sufficiently large number of times until thermalizationis achieved. The number of steps required for thermalization will depend on 1)the algorithm used, 2) the observable one is studying, and 3) on the values ofthe bare parameters. Thus it becomes increasingly more difficult to generate athermalized ensemble as one approaches the continuum limit, where, as we haveseen in chapter 9, the theory must exhibit critical behaviour. Indeed, the numberof steps required increases with a power of the correlation length. This is knownas critical slowing down. A first check of whether the system has reached thermalequilibrium, consists in choosing different starting configurations for generatingthe sequence. In particular, we can start from a disordered (hot start) or an or-dered (cold start) configuration. Once the system is thermalized, it must have lostall memory of the configuration we have started with, and expectation values ofobservables must approach the same values independent of the starting configura-tion. Again for a finite set of configurations this will of course not be exactly true.But if the measurements of various observables for different runs agree within theestimated statistical errors, one has a good chance that the effects of the startingconfiguration have been eliminated.

16.2 Construction Principles for Algorithms. Markov Chains

In this section we will discuss some general principles for constructing algo-rithms which ensure that the sequence of configurations generated will constitute arepresentative ensemble which can be used to measure the observables. In partic-ular, we shall be interested in the case where the above mentioned configurationsare the elements of a Markov chain generated by a Markov process. So let us firstdiscuss some elementary notions about Markov chains.* For simplicity we shallassume that the configurations can be labeled by a discrete index. If they arelabeled by continuous variables then the summations will be replaced by integrals,with appropriately chosen measures, and the probability of occurrence of a givenconfiguration must be replaced by a probability density.

* For a more detailed discussion, the reader may confer the books by Hammer-sley and Handscomb (1975), Clarke and Disney (1985).


Markov chains

Let C\, C2, • • • denote a countable set of states of the system. For example,in a pure lattice gauge theory, Q(i = 1,2,...) would correspond to different linkvariable configurations on the lattice. Consider now a stochastic process in whicha finite set of configurations is generated one after the other according to sometransition probability P(Ci -> Cj) = P%j for going from configuration C, to Cj.Let CTl, CT2... be the configurations generated sequentially in this way. Becauseof the probabilistic element built into this procedure, the state of the system at anygiven (simulation) time will be a random variable, whose distribution will dependonly on the state preceding it, if the transition probability is independent of allstates except for its immediate predecessor (i.e., if it does not involve any pasthistory). This defines a Markov chain whose elements are the random variablesmentioned above. Now given a set of configurations {CTi} generated by the Markovprocess and an observable O evaluated on this set of states, we can define a "time"average of O taken over TV elements of the Markov chain by

{O)N=^YJO{CT%). (16.5)i=i

It is this quantity which we compute in praxis, and which we want to equal theensemble average corresponding to a given Boltzmann distribution. With this inmind we shall therefore restrict the following discussion to so-called irreducible,aperiodic Markov chains whose states are positive, since there exist importanttheorems about such chains which are relevant for our problem. Let us first explainthe terminology.

i) A chain is called irreducible if, starting from an arbitrary configuration Cj,there exists a finite probability of reaching any other configuration Cj aftera finite number of Markov steps. In other words there exists a finite N suchthat

PiP = £ P«ip*ifc • • • P ^ - i i ^ °- (16-6){i*}

ii) A Markov chain is called aperiodic if pk ' / 0 for any N.iii) A state is called positive if its mean recurrence time is finite. If p£' is the

probability to get from Ci to Cj in n-steps of the Markov chain, withoutreaching this configuration at any intermediate step, then the mean recurrencetime of Cj is given by

00

n=l


We now state some important results which hold for Markov chains satisfyingi) to iii):Theorem 1

If the chain is irreducible and the states are positive and aperiodic, then the limitTV —t oo of (16.6) exists, and is unique; in particular one can show that *

lim P,(N) = TT7- (16.7)

where {itj} are numbers which satisfy the following equations:

nj>0, ^ > , - = l, (16.8a)3

ir^YjKiPij- (16-86)i

Let us interprete these results. First of all eq. (16.7) states that the limit N —> ooof P>j ' is independent of the configuration used to start the Markov process. Fur-thermore, eq. (16.8) states that the set {?ri} is left unchanged when we updatethese numbers with the transition probabilities {Pij}- But this, together with(16.8a), is just the condition that the system is in equilibrium, with TTJ the proba-bility of finding the configuration Ct.Theorem 2

If the chain is irreducible and its states are positive, and if

71 = 1

then the time average (16.5) approaches the ensemble average

{O) = J2*iO(Ci) (16.9)i

with a statistical uncertainty of order O(—?=).

The above theorems provide the basis for using a Markov process to calculatethe ensemble average (16.1). We must now determine the transition probabil-ities that generate a Markov chain of configurations, which will eventually bedistributed with a given probability.

* See e.g. Hammersley and Handscomb (1975)


Since we shall be interested in the case where the configurations are labeledby a set of continuous variables we will drop from now on the discrete subscript onCi and use capital letters to denote an arbitrary configuration. Thus, for example(16.8b) and (16.9) will be replaced by*

Peq(C) = Y/Peq(C')P(C'^C), (16.10a)c

(O) = J2Peq(C)O(C), (16.106)c

where Peq(C) now denotes the probability density for finding the configuration Cat equilibrium, and P(C —> C") is the transition probability density for going fromC to C. We will now show that for a Markov process to sample the distributionexp(—S(C)), it is sufficient to require that the transition probability (density)satisfies detailed balance:

e - s ( C ) p ( c _,. c^ = e-s{C)p^c, _> c ) (16 u )

for every pair C and C". Notice that the transition probability P(C —> C") is nota function of the variables labeling the configurations C and C", but stands for arule which tells us how to select the next configuration in a Markov chain giventhe configuration immediately preceeding it.

Consider the following probability density which we want to generate:

eS(C)

P^C) = £ c e - 5 ( C ) - (16-12)

Making use of the normalization condition

5^P(C->C") = 1, (16.13)c

and assuming that P(C —> C) satisfies (16.11), we see that (16.10a) is also satis-fied. Hence if the conditions of Theorem 1 are satisfied, then (16.12) is the uniqueequilibrium distribution generated by the Markov process. In fact, it may be easilyshown that the deviation from equilibrium decreases with each Markov step. Thefollowing proof is taken from Creutz (1983a).

* Actually, the sums must also be replaced by integrals with an appropriatemeasure. But for the following discussion it is convenient to keep the summationsign.


Let PN{C) be the probability of finding the configuration C at the TV'th stepof a Markov chain. If Co is the configuration with which we start the Markovprocess, then PN{C) is given by*

PN(C) = Y p(°o -+ Ci)P(Ci -»• C2)... P{CN.i -»• C), (16.14)

where each sum now runs over all possible configurations of the system. A usefulquantity which measures the deviation of PN{C) from equilibrium is

aN = Y \PN(C) - Peq(C)\. (16.15)c

Consider the deviation <TJV+I- Since according to (16.14) the probability distribu-tion at the (N + l) ' th step of the Markov chain is given by

PN+I(C) = YPN{C')P{C -> C) (16.16)c

we find, upon substituting (16.16) into the expression for CTN+I, and making use ofthe normalization condition (16.13), as well as detailed balance, eq. (16.11), that

aN+1 = Y I E (PN(C')P(C' -> C) - Peq(C)) |c c

= E I E (PN(C')P(C' -><?)- Peq(C)P(C -> C')) Ic c ' (16.17a)

= E l E ^ ^ ' ) - P«,(C'))P(C -v C)|c c

^ E E ip^(c') - ^(c')i^(c' -»• c),c c

or

<7JV+I < ffjv- (16.176)

Consider the equality sign. Then it follows from the last two lines in in eq. (16.17a)

that

E i E w c ' ) - p^'))p{c -+C)\ = Y \pN{C) - peq(C)\p(c -> c).c c c,c<

* For finite N,P^(C) will depend on the initial configuration Co- We havesuppressed this dependence here.


But if for all C and C" P{C —>• C") / 0 (i.e. the Markov chain is aperiodic), then

this equation can only be satisfied if PN{C) = Peg(C) for all C, since both PN(C)

and Peg(C) are normalized distributions. We therefore conclude that the devia-

tion <TJV decreases with each Markov step until we reach the desired equilibrium

distribution (16.12). This does not mean, however, that for every configuration C,

P/v+i(C) is closer to Peq{C) than it was PN(C), as is evident from the definition

(16.15).

The requirement of detailed balance does not determine the transition prob-

ability uniquely, and one can use this freedom to invent "efficient" algorithms

adapted to the problem one is studying. In the following section we will give an

example of an algorithm which is quite effective for studying systems whose action

is a local function of the configuration space variables.

16.3 The Metropolis Method

This method was originally proposed by Metropolis et al. (1953) and is in

principle applicable to any system. Let us first state the rule for generating the

configurations in a sequence, and then prove that it satisfies detailed balance.

The rule: Let C be any configuration which is to be updated. We then suggest

a new configuration C" with a transition probability PQ{C —> C") which only

satisfies the following microreversibility requirement:

P0(C -»• C) = P0(C ->• C). (16.18)

As an example consider the pure U(l) gauge theory. A particular configuration

C is then specified by the values of the link variables U^n) = exp(i8M(n)) for

H = 1, • • •, 4 and for all lattice sites n. We can then suggest a new configuration

by just choosing one of these link variables, and multiplying it by exp(ix), where x

is a random number between — •K and IT.* Then (16.18) is clearly satisfied. Having

suggested a configuration C", we now must decide whether it should be accepted.

Clearly the answer must depend on the actions S(C) and S(C), if the transition

probability P(C -> C") is to satisfy (16.11). The decision is made as follows:

If exp(-S(C')) > exp(-S(C)), i.e. if the action has been lowered, then the

configuration C is accepted. On the other hand, if the action has increased, one

* Of course we can also suggest a new configuration which differs from the

previous one in several link variables. But the usual procedure consists in updating

only one variable at a time, sweeping through the lattice systematically, or choosing

the variables in a more or less random way.


accepts the trial configuration only with probability exp(—S(C'))/exp(—S(C)).To this effect one generates a random number R in the interval [0,1] and takes C"as the new configuration if

K - e-s(c) •

Otherwise C" is rejected, and we keep the old configuration. Notice that it isthis conditional acceptance which allows the system to increase its action. Thuswhile the classical configurations correspond to minima of the action, the quantumsystem is allowed to move away from the classical configuration. In this way thealgorithm builds in the quantum fluctuations.

We now show that this algorithm satisfies detailed balance. Making use of thefact that the probability for the transition C —> C is the product of the probabilityPQ{C —¥ C") for suggesting C" as the new configuration and the probability ofaccepting it, we see that the Metropolis algorithm for P(C —> C") implies thefollowing statements:

If exp(-S(C")) > exp(-S(C)), then

p(C -> C") = P0{C -»• C"),

ande~S(C)

P(c'^c) = po(c'^c)^Wn.Since P0{C -> C) = P0(C ->• C"), we see that (16.11) is satisfied. On the otherhand:

If exp(-S(C")) < exp(-S'(C)), then

P(C^C')=Po(C^C'f-^:

andP(C -> C) = P0(C ->• C),

and detailed balance holds again.

As we have already mentioned, this algorithm is used in general to update asingle variable at a time. The reader may ask, why not change all lattice variablesat once? The reason is that such an updating procedure would involve in generallarge changes in the action. Hence the acceptance rate for those configurationswhere the action has increased, will be very small, and the system will move onlyslowly through configuration space. But the algorithm also becomes very slow for


single variable updating, if the action depends non-locally on the coordinates, since

in this case the ratio exp(-S(C'))/exp(-S(C)) will no longer be determined by

the nearest neighbour interactions. This is the problem we shall be confronted with

when taking fermions into account. Hence we shall need more efficient algorithms

to handle such situations, and which allow one to update the entire lattice at once.

In the following sections we will discuss some of the new algorithms which have

been proposed in the literature. They are arranged in chronological order, so that

the reader can see how these algorithms have improved in the past years, leading

up to the so-called hybrid Monte Carlo algorithm which is free of systematic errors,

and suited for handling systems described by a non-local action.

16.4 The Langevin Algorithm

The Langevin algorithm has been originally introduced by Parisi and Wu

(1981), and was proposed as an updating method in numerical simulations in full

QCD by Fukugita and Ukawa (1985) and Batrouni et al. (1985). In the following

we discuss this algorithm for a bosonic system consisting of a finite number of

degrees of freedom. Our presentation is based on the lectures of Toussaint (1988)

and Negele (1988).

Let qi (i = 1, • • • TV) denote the coordinates of a system with action S[q\. In a

pure gauge theory these are the link variables on the space-time lattice. We want

to obtain a rule for updating these coordinates in a numerical simulation which

satisfies the requirements of ergodicity and detailed balance. So let us think of

these coordinates as depending on a new time variable r which, when discretized,

labels the elements of a Markov chain. Consider the following difference equation

relating the variables at "time" r n + i = (re + l)e£ to those at rn = nej_,.*

qi{rn+1) = 9i(Tn) + eL ( - ^ W , + Vi(rn)) • (16.19)

Here {%(rn)} is a set of independent Gaussian distributed random variables which

probability distribution given by

P(Mrn)}) = II M™Pl-ei^(Tnn (16.20)i

In the limit e^ —> 0, eq. (16.19) becomes the Langevin equation:

£~^W>- <-»* ez, is called the Langevin time step for reasons that will become clear below.


Since the width of the distribution (16.20) is of order 1/y/ti,, the Gaussian noise

term in (16.19) is actually of order y/eZ- We can make this explicit by introducing

the new random variable %(rn) = \JeL/2\ 7?j(rn). Then (16.19) take the form

qi(rn+i) = ft(rn) - {^L)\^^ + V^m(Tn), (16.22)

where {fji(Tn)} are now random variables distributed according to

HIM) = ( n ^ K E ' Hi(T")2 (16-23)with variance

< Vi(Tn)rjj{Tm) > = 5ij6nm.

This form of the difference equation will be useful when discussing the hybridmolecular dynamics algorithm later on.

Consider eq. (16.22).*It states that the probability for making the transitionfrom qi at "time" rn to q[ at "time" rn+1 is the probability for % (i — 1,2,...)to equal \/2e.L\~Q^ + (li — Qi)/V^L- Hence we conclude that the transitionprobability P(q -> q') is given by

p(^^ = N°^{-\T.[^ + \ ^ ] ' } ' (1624)

where NQ is a normalization constant. We now claim that in the limit ££, —> 0 this

transition probability satisfies detailed balance, i.e. that

e-S(i)p(q -> q') = e-s^P(q' -»• q). (16.25)

Indeed, since q' - q is of 0(v/eL) , it follows from (16.24) that

p(i -» g') ^ g-E.W-fl'Jas/a*

P(g' ->• q) m o>. e-[S(g')-S(«)] .

We therefore conclude that in the limit ex, —> 0 the transition amplitude (16.24)satisfies (16.25). For finite Langevin time steps, however, this algorithm leadsto systematic errors which need to be controled. In praxis this means that the

* The following argument is based on the lectures by Negele (1988).


data obtained using this algorithm must be extrapolated to ex, -» 0. Hence oneis forced to generate configurations for different Langevin time steps. This is verytime-consuming. On the other hand it is a simple algorithm which can be usedto update all variables at once without having to worry about acceptance rates,since it does not involve any Metropolis acceptance test. The Langevin algorithmis therefore useful for treating systems with a non-local action, like QCD withdynamical fermions.

This concludes our discussion of the Langevin algorithm which, like the Metro-polis method, is applicable to arbitrary systems. Its main drawback is the system-atic error which is introduced by the finite Langevin-time step, ex,, which in praxisone would like to choose as large as possible to ensure that the system coversmuch of the configuration space in a reasonable number of updating steps. A lotof effort has been invested in the past few years to invent new algorithms whichare more effective and less subject to systematic errors. Most of these algorithmsare modifications of the so-called molecular dynamics (or microcanonical) methodof Callaway and Rahman (1982 and 1983). So let us first discuss this method.*

16.5 The Molecular Dynamics Method

The basic idea of the molecular dynamics method is that the euclidean pathintegral associated with a quantum theory can be written in the form of a partitionfunction for a classical statistical mechanical system in four spatial dimensionswith a canonical Hamiltonian that governs the dynamics in a new "time" variable(the simulation time). Invoking ergodicity, and making use of the fact that inthe thermodynamic limit the canonical ensemble average can be obtained fromthe microcanonical ensemble average, computed at an energy determined by theparameters of the system**, one can calculate the expectation values of observablesas time averages over "classical" trajectories. Hence, in the molecular dynamicsapproach, configurations are generated in a deterministic way, and the complexityof the motion in four space dimensions gives rise to the quantum fluctuations inthe original theory of interest.

We now give the details. To keep the discussion as elementary as possible, wewill consider a scalar field theory, with an action S[4>; (3} depending on the scalarfield 4> and on some parameter p (e.g. a coupling constant).*** Consider the

* The microcanonical ensemble approach was also proposed by Creutz (1983b)** In conventional statistical mechanics this energy would be fixed by the tem-

perature of the system.*** See Callaway and Rahman (1982) and (1983).


expectation value of an observable

< O >= | : />JD</.O[0]e-sî/5], (16.26a)

where

Z = f D<pe-Ste'®. (16.266)

We assume that we have introduced a space-time lattice to define the integral(16.26). Hence the degrees of freedom of the system will be labeled by the coordi-nates of the lattice sites which we denote collectively by "i". Although the integral(16.26) resembles that encountered in statistical mechanics, Z is not the partitionfunction of a classical Hamiltonian system. But one can cast it into such a formby introducing a set of canonically conjugate momenta TTJ. Thus (16.26) can alsobe written in the form

< O >= i [ DtDnOlfle-"^''®, (16.27a)

where

H[4>, 7T-P} = Y1 \*2i + S& ft, (16.276)

i

Z= f Dc/tDTve-11^^. (16.27c)

and the integration measure is defined as usual:

Dcf>Dn = Y[d(j)idTTi.i

Clearly (16.27) coincides with the expression (16.26), since O does not depend onthe momenta. Hence, as advertised above, the quantum theory in four-dimensionaleuclidean space-time resembles to a classical canonical ensemble in four spatialdimensions in contact with a heat reservoir. Notice that (16.27) is not to beconfused with the usual phase space representation of the path integral (16.26),whose structure is completely different, and involves the Hamiltonian of the systemin three space dimensions.

Having formulated our problem in terms of a classical statistical mechani-cal system, we can now make use of known results to calculate the expectationvalue (16.26a). In particular it is well known in statistical mechanics that in thethermodynamic limit (where the number of degrees of freedom become infinite),the canonical ensemble average can be replaced by the microcanonical ensemble


average evaluated on an "energy" surface, which is determined by the parametersof the system. We have put "energy" into quotation marks to emphasize that, inthe present case, it is the energy of a statistical mechanical system in four spatialdimensions. Hence, according to (16.27b), the action of the quantum mechanicalsystem is allowed to fluctuate! This is the way how quantum fluctuations are takeninto account. Let us recall how one arrives at this result. At the same time, thiswill give us the connection between the parameter /3 and the "energy" at whichthe microcanonical ensemble average is to be calculated.

Consider the expectation value (16.27). It can be written in the form

sn^ (R\ / D4>D*O{<t>] I dES(H[</,, n; 0\ - E)e~E

< O >can (/?) - jD(j)D7rjdES{H[(l)j7r.p]_E)e-E ' (16-28)

where we have introduced the subscript "can" to emphasize that we are calculatinga canonical ensemble average. This quantity depends on /?.

On the other hand, the microcanonical ensemble average < O >TOjC, evaluatedfor a given (3 on the energy surface H = E, is given by

< O >mic (E, /?) = ) [D<f>DKO[4>}6(H[cf>, TT; 0]-E), (16.29a)

whereZmic(E, p) = J D<f>Dn5(H[cf>, n; 0\ - E) (16.296)

is the phase space volume at energy E. We now want to establish a connectionbetween the two ensemble averages in the thermodynamic limit. Define the "en-tropy" of the system by

s{E,0) = lnZmic(E,0). (16.30)

Introducing the definitions (16.29) and (16.30) into (16.28), one finds that thisexpression can be written in the form

< O >can (/?) f dEe-iE-w)) • (16-31)

Now comes the standard argument: when the number of degrees of freedom of thesystem becomes very large, the exponentials appearing in the integrands of (16.31)are strongly peaked about an energy E given implicitly by the equation

(ds(E,P)\ _{-dE-J^-1- (1632)


Hence in the thermodynamic limit we are led to the connection

{0)Can{/3) = [(O)mic]E=m, (16.33)

where /3 and E are related by (16.32). This relation is, however, not a useful onefor doing numerical calculations. A more adequate expression can be obtained bymaking use of the equipartition theorem, which tells that the following equationholds for each momentum TT;:

dH_ _ 1dirt os/oE

Inserting for H the expression (16.27b) and making use of (16.32), one finds that

N[< Tkin >mic}E=E(f3) = y> (16.34)

where Tîn — S i "f/2 is the kinetic energy, and N is the number of degrees offreedom of the system. We have hence succeeded in rewriting (16.32) in terms of amicrocanonical ensemble average. But we are still left with the problem that givenP, E is only implicitly determined by (16.34). Fortunately this does not turn outto be a major stumbling block; E can be adjusted to any prescribed value of /? bythe heating/cooling procedure described by Callaway and Rhaman (1983).

A more familiar form of eq. (16.34) is obtained when the dependence of theaction on the parameter P is multiplicative, i.e. if*

S[0,/3] = /?V[0].

In this case the expectation value (16.26) can be written as follows

< O >= 4 f DÎhrO[4>]e-pH[*>ir],Zi J

where

i

andZ= DtDire-PWrt.

* This is the case in gauge theories within the pure gauge field sector, wherethe action is of the form S = PV[U].


By following the arguments presented above, one arrives at the following connec-tion between the canonical and microcanonical ensemble averages

< O >can 03) =< O >mic {E),

where

< O >mic (E) = _ , _ . / D4>DirO[4>]5(H[<l>,*] ~ E),

Zmic{E) = f Dcf>Dn5{H[4>,n} - E),

and f3 is related to E by

X7j —< Tkin >mic {E) •

This is the familiar form of the equipartition theorem with (3 playing the role ofthe inverse temperature.

With these remarks let us return to expression (16.33), and rewrite the right-hand side in a convenient form for numerical calculations. Assuming that themotion generated by the Hamiltonian (16.27b) is ergodic, we can use the Hamiltonequations of motion

k = 9Jf^ , (16.35a)

to generate a representative ensemble of phase-space configurations with constantenergy. Since the observables, whose expectation value we want to calculate,only depend on the coordinates, the microcanonical ensemble average can then bereplaced by a time average over 4>{T). We are therefore led to the following chainof equations valid for ergodic systems in the thermodynamic limit,

< O >can. {(3) —• [< O >mic.}E=EW) —», l i m ^ [ dTO{{<Pi{r)}),therm, lira. "Wi ergOd. J Jo

(16.36)where the trajectory {4>I(T)} is a solution to

dHi _ dS[4>]

1^ ~ —dJT' (16-37)

subject to the initial conditions that it carries energy E*

* Because the motion is assumed to be ergodic, one can use any starting con-figuration with this energy to integrate (16.37).

(16.356)


Notice the difference in the structure of (16.37) and the Langevin equation(16.21). While (16.37) is a deterministic set of coupled second order differentialequations, the Langevin equation is of first order an involves a stochastic variable.It is instructive to compare the two equations when the times are discretized.* Inthe Langevin scheme the corresponding difference equations are given by (16.22).Consider the naive discretization of (16.37). Using the symmetric version for thediscretized second derivative, 4>i{rn) = [<fo(Tn+1) + </>i(rn_i) - 2^>i(rn)]/(Ar)2, wefind that

<t>i{Tn+i) = <t>i{rn) - e 2 ^ y \ +CTi(rn), (16.38a)2d<j>i{Tn)

where e = A T = Tn+i — rn is the microcanonical time step, and

*i(*n) = ^ ( & ( r n + i ) - &(rn_!)). (16.386)

Notice the striking similiarity between (16.38a) and (16.22)! The discretized ver-sion of the Langevin equation is obtained from (16.38a) by promoting the momenta7Tj to random variables distributed according to (16.23), and identifying ti, withe2/2.**

In practical calculations, one integrates the Hamilton equations of motion(16.35) using the so called "leapfrog" method (see section 6). Let us compare the"distances" travelled in configuration space for the above-mentioned algorithmsfor ei = e2/2. In the molecular dynamics case the distance covered after n stepsis of O(ne), while in the Langevin approach it is only O(y/ne), because of itsstochastic nature. Hence the classical algorithm moves the system faster throughconfiguration space. But for this algorithm to be valid, a number of conditions hadto be satisfied. For one thing, eq. (16.33) only holds for systems with an infinitenumber of degrees of freedom. This is clearly not the case in practical calculationswhere one is dealing with rather small lattices. Furthermore, the last step in thechain (16.36) requires the motion to be ergodic. But, in contrast to the Langevinapproach, ergodicity is not explicitly built into the molecular dynamics algorithm.This suggests that one should look for a new algorithm which combines the goodfeatures of the two methods.

* see e.g., Toussaint (1988).** The fact that the Langevin time step is the square of the microcanonical time

step is not surprising, since the Langevin equation is of first order, while (16.37)involves the second time derivative.


16.6 The Hybrid Algorithm

A possible way of combining the stochastic Langevin approach and the mi-crocanonical simulation into a new algorithm which takes into account ergodicitywas proposed by Duane (1985). This so-called hybrid algorithm is suggested byconsidering the molecular dynamics algorithm in the form (16.38a), which - as wehave pointed out before - becomes the discretized Langevin equation with time stepex, = e2/2, if the momenta {TTJ} are replaced by random variables with probabilitydensity

p«*'»=(n^)«-E'w- (i6-39)Notice that this is precisely the distribution of the momenta appearing in (16.27a)!In fact, if the motion generated by the Hamiltonian (16.27b) is ergodic, then themomenta would be eventually distributed according to (16.39).

This suggests that instead of integrating the Hamilton equations of motion(16.35) along a single trajectory with given energy E, we interrupt this integrationprocess once in a while and choose a new set of momenta with a probabilitydensity given by (16.39). This corresponds to performing a Langevin step, which ifrepeated many times builds in the ergodicity we want. At the same time one takesadvantage of the fact that because of the molecular dynamics steps the system willmove faster through configuration space than in the Langevin approach. Hence thehybrid algorithm contains a new parameter: the frequency with which momentaare refreshed. This parameter can be tuned to give the best results. In fact Duaneand Kogut (1986) have suggested that at each step of the updating procedure arandom choice is made between a molecular dynamics update with probability a(which corresponds to relating the momenta in (16.38a) to <j> according to (16.38b))and a Langevin update with probability 1 —a (which corresponds to replacing {71-j}by Gaussian distributed random numbers). If a is close to one, then the systemwill move almost as fast through configuration space as in the molecular dynamicscase, but ergodicity may not be fully realized. On the other hand if a is close tozero, ergodicity is ensured, but the system will move slower through configurationspace. The optimal choice for a will lie between these two extremes, and has tobe determined numerically.

After these qualitative remarks, we now demonstrate how this algorithm isimplemented in praxis. For simplicity we shall again study the case of a scalarfield theory.

Consider the equations of motion in their Hamiltonian form (16.35). We want


to discretize them in the time with time step e. To this effect let us expand 0j(r+e)

and 7Tj(r + e) in a Taylor series up to order e2:

Mr + e) = Ur) + 4i{r) + ^ & ( T ) + O(e3),^ (16.40)

7Ti(r + e) = TTi(T) + e^(r) + y ^ ( r ) + O(e3).

But from the Hamilton equations of motion we have that <j>i = TTJ(T), and <j>i{r) =

TC^T) = -8S/d<j>i(T), so that

^T) = -^dMr)dMrfi{T)- (16-41)

In leading order the right hand-side of (16.41) is given by

•sp d2S 1 / dS d S \

^ * ( r ) ^ ( r ) " j ( T ) " e [d^r + e) ~~ d^r)) + W "

Introducing the above expressions into (16.40) we find after some rearrangements

that

Mr + e) = Mr) + £ (W) - | ^ | f r ) + O(e3),

( . . e dS \ / . . e dS \ dS „. 3.

(16.42)

But the quantities appearing within brackets are the momenta evaluated at the

midpoint of the time intervals; hence up to O(e3), eqs. (16.42) are equivalent to

the following pair

cf>i{T + e) = <t>i(T) + £7Ti(r + e/2) , (16.43a)

*&+ h = «« + %-< e$$hry (16-436)

which, when iterated, amounts to integrating the Hamilton equations of motion

(16.35) according to the so-called leapfrog scheme: while the coordinates are eval-

uated at times rn = ne, momenta (i.e. derivatives of cfri) are calculated at the

midpoint of the time intervals.

In the hybrid algorithm the momenta are refreshed at the beginning of every

molecular dynamics chain. Suppose we want to start the leapfrog integration at


time T. Let {4>i(r)} be the coordinates at this time; then the momenta {fti{r)} are

chosen from a Gaussian ensemble as described before. But to begin the iteration

process (16.43), we need to know 7T;(T + e/2). We calculate this quantity by

performing a half-time step as follows

^(T + e/2) = Mr) ~ | | * + O(e% (16.44)

which means that we are making an error of 0(e2). But this error is made only

once for each molecular dynamics trajectory generated, whose accumulated error

after 1/e steps is also of order e2. Hence measurements of the observables will be

correct up to this order.

Before we summarize the steps required for updating the configurations {fa}

according to the hybrid algorithm, let us rewrite eqs. (16.43) and (16.44) in a

way appropriate for numerical calculations. Denoting by {(pi(n)} and {^(n)} the

momenta at time step ne, and by {^(rc)} the momenta at time (n + l/2)e, eqs.

(16.43) and (16.44) take the following form:

fa(n + l) = fa{n)+eni(n), (16.45a)

.l(n+l) = 7ri (n)-e^L, (16.456)

where

*'W = *<<n>-|«L- (16.45c)

In a lattice formulation, all quantities appearing in (16.45) will be dimensionless.

The hybrid algorithm is then implemented by the following steps:

i) Choose the coordinate {fa} in some arbitrary way.

ii) Choose a set of momenta {TT;} from the Gaussian ensemble (16.39).

iii) Perform the half step (16.45c) to start the integration process,

iv) Iterate eqs. (16.45a,b) for several time steps, and store the configurations

generated in this way.

v) Go back to ii) and repeat the steps ii) to iv).

As has been shown by Duane and Kogut (1986), this algorithm generates

a sequence of configurations {fa} which are eventually distributed according to

exp(-S[0]).

Suppose now that instead of refreshing the momenta only once in a while we

refresh them at each time step. In this case eq. (16.45b) plays no longer any role


in generating the coordinates {4>i}, which are now updated according to

*(„ + !) = * ( „ ) - £ ! & + «*(„),where {?y;(n)}are independent Gaussian distributed random numbers. But this isnothing but the Langevin equation with time step e2/2. Hence the hybrid methoddiffers only from the Langevin approach if the number of molecular dynamics stepsis larger than one.

This concludes our general discussion of the hybrid algorithm. For moredetails the reader may confer to the lectures by Toussaint (1988).

16.7 The Hybrid Monte Carlo Algorithm

Although the hybrid algorithm is certainly an improvement over the previ-ous two we have discussed, there remains the problem of having to control thesystematic error introduced by the finite time step. In fact, of all the algorithmsdiscussed so far, only the Metropolis method was free of systematic errors. Butas we have already mentioned, this algorithm becomes very slow when updatingconfigurations with an action depending non-locally on the fields. On the otherhand, the hybrid algorithm allows one to update all variables at once, and henceis better suited for studying systems with a non-local action (as is the case forQCD when fermions are included). This suggests that one should try to eliminatethe systematic error in the hybrid algorithm by combining it with a Metropolisacceptance test.

Originally, this idea was proposed by Scalettar, Scalapino, and Sugar (1986)in connection with the Langevin approach, and later applied to full QCD by Got-tlieb et al. (1987a). Subsequently Duane, Kennedy, Pendleton, and Roweth (1987)suggested a similar modification of the hybrid algorithm. This modification con-sists in introducing a Metropolis acceptance test between step iv) and v) of thehybrid method discussed in the previous section. In other words, the phase spaceconfigurations generated at the end of each molecular dynamics chain, are usedas trial configurations in a Metropolis test with the Hamiltonian (16.27b), whichplays the role of the action in our discussion in section 16.3. For this reason thisalgorithm is called the hybrid Monte Carlo algorithm. We now state the rulesfor updating the configurations {fa} and then show that they generate a Markovchain satisfying detailed balance:

i) Choose the coordinates {fa} in some arbitrary way.


ii) Choose the momenta {TT;} from the Gaussian ensemble (16.39).

iii) Calculate n according to (16.45c).

iv) Iterate eqs. (16.45a,b) for some number of time steps. Let {$i,7rj} be the last

configuration generated in the molecular dynamics chain, where TT is obtained

from TTj- according to (16.45c).

v) Accept {4>'i,^'i\ as the new configuration with probability

p = m J n { l , e - ^ ' " ' V e - H 1 W } ,

where H is the Hamiltonian (16.27b).*

vi) If the configuration {< , 7r } is not accepted, keep the old configuration {fa, iti},

and repeat the steps starting from (ii). Otherwise use the coordinates {</>';}

to generate a new configuration beginning with the second step.

Notice that if the Hamiltonian equations of motion could be integrated ex-

actly, then H would be constant along a molecular dynamics trajectory and the

configuration would always be accepted by the Metropolis test. But for finite

e, 5H ^ 0 and the Metropolis test eliminates the systematic error introduced by

the finite time step. Indeed, as has been shown by Duane et al. (1987), this al-

gorithm leads to a transition probability P(<j> —> 4>') which satisfies the detailed

balance equation (16.11) for an arbitrary time step e. Since the proof is rather

simple, we will present it here.

Let {fa} be the coordinates at the beginning of a molecular dynamics chain.

Consider in turn the various steps required for updating this configuration accord-

ing to the hybrid algorithm.

Step 1

Choose a set of momenta {TT,} from a Gaussian ensemble with mean zero and

unit variance. The corresponding probability density is given by

PG(7r)=./Voe-iX>.2, (16.46)

where No is a normalization constant.

Step 2

Let the phase space configuration** (</>, 7r) evolve deterministically for N time

steps according to (16.45), including the half-time steps at the beginning and the

* We have suppressed here the dependence of H on the paramter (3.** For simplicity we suppress the subscript "i" on fa and TTJ.


end of the molecular dynamics chain. Denote the configuration generated in thisway by ((f>(N\n(N^). Here <fi(N1 and TT^' are functions of the starting configuration(</>, TT). But because (16.37) involves the second time derivative, the motion from<j> —> (j)^ can be reversed by changing the sign of all momenta in the initialand final phase-space configurations. Hence the probability for the transition(</>, TT) —» (<£>', TT') is the same as that for going from (</>', — TT') to (<j>, —TT): i.e.

PM{(4>, *) ->• (<A', *')) = ^M((<A', -TT') -»• (0, -TT)). (16.47)

The subscript "M" stands for "molecular dynamics". Actually, since the systemevolves deterministically, the state (ft, TT') generated by the molecular dynamicschain will occur with unit probability. Hence, PM is a 5 -function, ensuring that(</>', TT') evolved from (<j>,n).

Step 3

Accept the configuration (<£', TT') with probability

PA{(4>, T) -> (</>', T ' ) ) = "»" (1 . e-^^'^ 'Ve-^t*'"1). (16.48)

By taking the product of the probability densities for these three steps, and inte-grating over the momenta, we obtain the transition probability density P(<f> —> ft):

P(<t>^4>') = f DTTDn'PG{jr)PM((<t>,ir) -»• ($', rr'))PA ((</>, n) -> (</»', TT')) .

(16.49)Consider next the product e~s^P(4> —> <j>') corresponding to the left-hand

side of eq. (16.11). Introducing the expression (16.46) into (16.49), this productcan be written in the form

e " S W P ( H 0') = JD7rD7r'e-H^PM((<j>,7r) -> {<t>'y))PA{{,n) ~* (^'.TT')).

(16.50)But from the definition (16.48) it follows that

e-«I*.*]pA((^)7r) ->• (0',7r')) = e- H ^ '^P A ((0 ' ,7 r ' ) -»• (^.TT)), (16.51)

which is the statement of detailed balance in phase space for the Metropolis ac-ceptance test. Inserting this expression into (16.50), and making use of (16.47)and of the fact that H[(j>, TT] — H[<f>, —TT], one finds that

e-s^P{4> -> </,') =e~s^ I DTTDAPG^PMW,*') -»• (</>,*))

xPA{(4>', TT1) ^ (0, TT))] = e-WpW -> 0).


This is the relation we wanted to prove. The hybrid MC algorithm thereforegenerates a Markov chain of configurations {$;} distributed eventually accordingto exp{-S[4>}).

Since according to this algorithm entire field configurations are updated atonce, one may wonder whether one is not running into the same "acceptance"problems mentioned in section 4. This could indeed be the case if e is not chosensufficiently small and/or the number of molecular dynamics steps is too large. Thenthe acceptance probability (16.48) will in general be small, and the algorithm willmove the system only slowly through configuration space. Fortunately, numerical"experiments" show that even for small e, of the order of the time step needed inthe hybrid algorithm for extrapolating the data to e —» 0, the system moves atleast as fast through configuration space as with the hybrid algorithm. But thegreat advantage of the hybrid Monte Carlo method is that it is free of systematicerrors arising from the finite time step. Hence no extrapolation of the data isrequired, which for one thing would be time-consuming, and furthermore a sourceof errors.

16.8 The Pseudofermion Method

The algorithms we have discussed in the previous sections can only be used tocalculate ensemble averages of functionals depending on c-number variables. Theycan therefore not be applied directly to systems involving Grassmann fields. Forthis reason, one must first integrate out the fermionic degrees of freedom beforeperforming a Monte Carlo calculation in QED or QCD. As we have pointed outin chapter 12, this can always be done for any correlation function involving theproduct of an arbitrary number of fermion fields, since the action is quadratic inthe Grassmann variables. In the following we will restrict our discussion to thecase of Wilson fermions. Then the ensemble average of O[U, ip, i>\ is given by

_ !DU(O)s,e-s.f'W{ }~ fDUe-s'ttV>] ' ( l b ' 5 2 a j

where

\"/SF = TSM r , (lb.5/0)

can be expressed in terms of Green functions calculated in a background fieldU = {f/i(«)}, and where

Se/f[U] = SG[U) - In det K[U]. (16.53)


Here SQ is the action of the pure gauge theory, and -ft^f] is the matrix appearing

in the fermionic contribution to the action. For the case of QCD with Nf flavours

of mass-degenerate quarks, the matrix elements of K[U] are given by

Knaaf',m0bf[U] = Knaa,tm0b[U]6ffi , (16.54a)

where Knaarn0t[U] is denned implicitly by (6.25c), i.e.,*

Knaa,mf3b[U] = ( M 0 + 4r)5nmSal35ab

~2 ^ ^ ~ ^)«/3(C/M(n))a6(5n+A,m (16.546)

+ {r + j^,)a0(Ul(m))abSn-^,m}.

The index / ( / ' ) in (16.54a) labels the flavour degrees of freedom, and n, a, a (m, P, b)

stand for the lattice site, and Dirac and colour degrees of freedom, respectively.

Let us denote the latter triple by a single collective index i. Then the fermionic

contribution to the action reads

where Kij[U] is flavour-independent. It therefore follows that

detK[U] = (detK[U))Nt. (16.55)

An important property of the matrix K[U] is that its determinant is real , i.e.,

det K[U] = det fCt[[/], and that det K[U] > 0 for values of the hopping parameter

K = l/(8r + 2M0) less than 1/8 (Seiler, 1982). We can therefore also write detK

in the form

det K[U] = ^ det Q[U], (16.56a)

where

Q[U] = K\U]K[U] (16.566)

is now a positive, hermitean matrix. This will be important for the following

discussion. It follows from (16.55) and (16.56a) that (16.52a) is given by the

ensemble average of {O)sF, defined in (16.52b), calculated with the Boltzmann

factor exp(—Seff), where

SeffP] = SG[U] - ^ lndetQ[f/]. (16.57)

* In chapter 6 we only considered the case of a single flavour.


Suppose now that we use the Metropolis algorithm to update the link-variable con-

figurations. Then the Metropolis acceptance/rejection test requires us to calculate

the difference Se/f[U'] - Seff[U], where U denotes the original configuration to

be updated, and U' is the new suggested configuration. Hence we must calculate

the ratio

Although Q[U] is a sparse (but large) matrix, since it is constructed from K[U]

which only couples nearest neighbours, the exact computation of p(U, U') is too

slow to be useful for updating link configurations on large lattices, and one must

look for a faster method for computing (16.58). Clever suggestions in this direction

were made in 1981 by Fucito, Marinary, Parisi and Rebbi (1981) and by Petcher

and Weingarten (1981).

The proposal of Fucito et al. was the following. Let 6U denote the difference

between the suggested link-variable configuration and the old configuration, i.e.,

5U = {<5!7M(n)}, where SU^ = U'^n) — t/M(n). The corresponding change in the

matrix Q[U] we denote by 5Q: SQ = Q[U'} - Q[U]. Then

p(U, V) = det{^QQ) = det(l + Q-HQ). (16.59)

For small matrices Q~16Q, this expression can be linearized in 8Q:

p{U, [/') ~ 1 + TrQ-lSQ. (16.60)

This is a reasonable approximation if the link-variables are updated one by one,

and if only small changes 5U are allowed. Now the computation of the trace is easy.

But Q~l must be evaluated at each updating step. This is very time consuming.

The usual procedure is therefore to use the same matrix Q~l\U] for an entire sweep

through the lattice. This is consistent with the linearized approximation (16.60),

but may lead to large systematic errors, since the errors introduced at each updat-

ing step can accumulate during a sweep. Even with this approximation the exact

evaluation of Q~l on the link-variable configurations generated after each sweep

through the lattice is not feasable. But one may determine Q~x approximately by

noting that its matrix elements are given by the following ensemble average

Q7* = <0i0P = J % s-o „ • (16-61)

(16.58)


Here <f> and </>* are the so-called pseudofermionic (bosonic!) variables.* BecauseQ is a positive hermitean matrix, the exponential appearing in (16.61) can beinterpreted as a Boltzmann factor. This is the reason why we have expresseddet K[U] in the form (16.56a).

According to (16.61), one can calculate the inverse matrix Q~l by a) generat-ing a complex set of pseudofermionic variables with probability density exp(—(f>*Q(f>),b) measuring the products <j>i<j>^ on each configuration generated, and c) taking theensemble average. Hence the main work consists in generating the configurations.

At the same time when Fucito et al. made their proposal, Petcher and Wein-garten (1981) made an alternative suggestion based on similar ideas. These au-thors made use of the fact that the determinant of Q is given by the following pathintegral expression over pseudofermionic variables

As seen from (16.57), this is the relevant determinant for the case of two quarkflavours. The corresponding partition function can therefore be written in theform

Z= f DU det Q[U]e-Sa[u]

r ^ . (16-62)= [ DUDPD</*-SolU]-5:<.iMJ™.

This shows that to compute ensemble averages, one must generate configurationswith a non-local action. Hence the effectiveness of this method depends on theefficiency of the algorithm for calculating the inverse of Q[U].

Following the work of Fucito et al., Scalapino and Sugar (1981) pointed outthat in a local updating procedure, the computation of (16.58) is reduced to theevaluation of the determinant of a small matrix. This can be easily seen. Considerthe logarithm of the ratio det K'/ det K = det K[U']/ det K[U):

oo , . .NAM-1

* To every fermion degree of freedom there corresponds a pair of complex(bosonic) variables.


NowTr{K~l6K)N = TriK-HKK-HK... K~l5K}.

If only a single link-variable is changed in the updating procedure, then 5K hasonly non-vanishing matrix elements between two fixed nearest neighbour sites.Because of the trace we only need to know K~x in the subspace defined by 5K / 0.Let us denote this matrix by K~1. Then

N=l

= Trln(l + ic='lSK)

= det{l + if:'lSK).

If the suggested configuration U + SU is accepted by the Metropolis test, then thenext updating step requires the knowledge of \K{U + 5U))~1 — (K + SK)~1 in theappropriate new subspace. The inverse of the matrix K + 6K can be calculatedas follows:

oo

= ^2(-l)lf(K-15K)NK-1

N=0

= K'1 - K^SKK'1 + K-xbKK-xbKK-x + ...,

or(K + 5K)-1 = K~l - K'^SK - SKK^SK + .. .]K~K

The matrix appearing within square brackets has only non-vanishing support inthe subspace mentioned before. Define

SM = SK - SKK^SK + 5KK-15KK~l5K + ...

Then

(K + 5K)-1 =K~l -K-i-SMK-1, (16.63)

andSM = 5K{\ - K-l5K + K-^5KK~l5K + ...)

= SK(1 + if^SK)-1.

Hence one never has to calculate directly determinants or inverses of large matrices.However, because after every change in a link variable the whole large inverse


matrix must be updated according to (16.63), this algorithm is too slow to beuseful in Monte Carlo calculations, except when studying field theories in twospace-time dimensions, or four-dimensional field theories on very small lattices.But it can serve to check the efficiency of other approximate algorithms, such asthose mentioned above, and for checking the results obtained by other approximatemethods.

Finally, we want to mention an interesting modification of the algorithm ofFucito et al., proposed by Bhanot, Heller and Stamatescu (1983), which is notburdened with the systematic errors arising from the neglect of higher order cor-rections in SU. These authors suggested that one should compute the ratio (16.58)directly using the pseudofermionic method. Since

this ratio is given by the following expression

where 5Q = Q' - Q, with Q' = Q[U'], Q = Q[U}. To simplify the expressionwe have used matrix notation. According to (16.64), p is given by the inverseof the ensemble average of exp(—4>*5Q<f)), calculated with the Boltzmann factorexp(—SQ), where SQ = <$Q4> is a local expression of the link variables:

P= (eMJsQ4>))sQ- (16'65a)

Alternatively, p is also given by

p={exp(tfSQ<t>))sQl, (16.656)

where SQ> is the pseudofermion action for the suggested link-variable configurationU'. How does one recover the result of Fucito et al.? Consider the expression(16.65b). For small arguments of the exponential we can replace p by

p « 1 + Y^(^j)sQSQij,

or making use of (16.61)pfvl + TrQ-^SQ.

(16.64)


This coincides with the approximation (16.60).

The advantages of the method of Bhanot et al. are that i) one is not limitedto small changes in the link-variables; ii) the only errors introduced are statisticaland iii) the accuracy of the method can be estimated by computing the ratio ofthe determinants either from (16.65a) or (16.65b).

This is all we want to say about the pseudofermion method. For details ofhow this method is implemented in praxis the reader should confer the publishedliterature.

16.9 Application of the Hybrid Monte Carlo Algorithm toSystems with Fermions

We conclude this chapter by giving an example of how the hybrid Monte Carlo(HMC) algorithm of Duane et al. (1987) is implemented in gauge theories withfermions. To keep the discussion simple, we will restrict ourselves to lattice QEDwith two flavours of mass-degenerate Wilson fermions.

Before the HMC algorithm can applied to calculate a correlation function onemust first write the path integral expressions in the form of an ensemble averageover bosonic variables. This is done by proceeding in the way described in theprevious section. The relevant partition function is given by (16.62), where Q isthe positive hermitean matrix defined in (16.56b).

The next step consists in rewriting the partition function as a path integral inphase space. To this effect one introduces the momenta TT* , TTJ and P; canonicallyconjugate to (pi,(p* and Ai, where Ai is the angular variable parametrizing a linklabeled by the collective index I* The corresponding link variable we denote byUi. Defining the "Hamiltonian"

H = \Y.P? + Z)<7r* + SGM + SPF[U, </>, p), (16.66)I i

where SFP is the pseudofermionic action

SPF = X>;Qyi[ [%. , (16-67)

* I stands for the coordinates of the lattice site n and the direction \i labelinga particular link variable, and Ue and A( are related by U^ = exp(iAt).


and SG is the action of the pure U(l) gauge theory defined in (5.21), the partitionfunction (16.62) can be written in phase space as follows

Z= f DUD</>D4>*DPDirDTr*e-H.

Recall that the Hamilton equation of motion derived from (16.66) describe thedynamics of a system in a new time r, which is identified with the simulation timein a Monte Carlo calculation. These equations are given by

& ( T ) =7Ti(T), (16.68a)

*iM = - Z ) 0 « 1 [ £ / W T ) > (16-686)3

MT) =PI(T), (16.68C)

fl''l = - ^ ^ « " W * < " (16*4

Expressed in terms of the link variables, (16.68c) becomes

U(T) = iPi(T)U(T).

The derivative of Q~l appearing in (16-68d) can be written in a more convenientform by making use of the fact that Q is given by the product (16.56b), and thatd{K-lK)/dAi = d({Ki-lKi)/dAi = 0. One readily verifies that

dQ-1 _ xdQ x

~ÂT~Q dAiQ •

Introducing this expression into (16.68d), we see that the equations (16.68) areequivalent to the following set:

Vi=Y^Q^[U]<f>j, (16.69a)3

4>i =7ru (16.696)

K i = - J]i, (16.69c)

Ui =iPiUh (16.69d)

In principle these equations are to be used in the molecular dynamics part of theHMC algorithm (see section 7). But this is not what one does in practice, for there

(16.69e)


exists a faster way of generating a set of configurations in the pseudofermionic andlink variables. Thus consider the pseudofermionic action (16.67). The inverseof the matrix Q is given by the product K~1(K^)~1. Hence this action can bewritten in the form SPF = Yî £»*&> where the vector £ = (£1, £2> •••) is constructedfrom 4> = {<f>i,(}>2: •••) a s follows

£=(*-t[ tf]- i)0.

Now one can easily generate configurations in these new variables which are dis-tributed according to exp(£t£). By calculating (f> — iff [[/]£, one then obtains, forfixed U = {Ui}, an ensemble of configurations in the pseudofermionic variablesdistributed according to exp(—4>^Q~1<p). Each of these configurations can be usedas an background field in a molecular dynamics chain for the link variables. Thisis what one does in practice. Hence the implementation of the HMC algorithmgoes as follows:

i) Choose a starting link variable configuration.

ii) Choose P; from a Gaussian ensemble with Boltzmann factor exp(— | J^; Pf).iii) Choose £ to be a field of Gaussian noise.iv) Calculate

v) Allow the link variables and the canonical momenta Pi to evolve determinis-tically according to (16.69d,e), where r\ = Q~1[U]cf>, with 0 held fixed,

vi) Accept the new configuation (A',P') generated by the molecular dynamicschain with probability

/ e-H[A',P']\p = min 1, = ,F y ' e-H[A,p] J '

where H is the Hamiltonian governing the dynamics of the link variables inthe presence of a background field cj>:

vii) Store the new configuration generated, or the old configuration, as dictatedby the Metropolis test,

viii) Return to ii).


The step v) is realized using the leapfrog method, described in section 6. Thus(16.69e) is written in the form

where the first half step is calculated as described in section 6. On the other handthe naive discretization of (16.69d) reads:

U,(T + e) = UI(T) + ieP^r + ^)U,(T).

Link variables updated in this way-would, however, not be elements of the 1/(1)group. For this reason one discretizes (16.69d) as follows:

C/,(r + e) = e i e P '(T+f)^(r),

which for infinitesimal time steps is equivalent to the naive discretization.

The Hybrid Monte Carlo algorithm is the best algorithm available at presentfor simulating dynamical fermions. It allows one to choose large time steps withoutintroducing any systematic errors. This is guaranteed by the Metropolis test. Sincein step v) the system is allowed to evolve according to the Hamilton equations ofmotion, the only change in H will be due to the finite time-step approximation.Hence most of the configurations generated will be accepted by the Metropolistest, if the time step is not too large. This means that the system is movingfast through configuration space. The algorithm is however rather slow, since itrequires the computation of the vector r\ = Q~1[U]<j> in step v). Furthermore, it isnot clear how to implement it for an odd number of quark flavours.

CHAPTER 17

SOME RESULTS OF MONTE CARLO CALCULATIONS

We begin this chapter with an apology. As the reader can imagine, there hasbeen an enormous amount of activity within the computational sector of latticegauge theories since their invention in 1974. Most of the pioneering work has beenperformed on small lattices. Since then many calculations have been improvedby working on larger and finer lattices, and by developing new computationaltechniques. Clearly we cannot possibly do justice to all the different groups ofphysicists who have made important contributions to this field. What we intendto do, is to acquaint the reader with some of the questions that physicists have triedto get an answer to, and to present some lattice calculations which illustrate theresults one has obtained, placing special emphasis on pioneering work. Thus anycomments should always be seen in the light of the Monte Carlo data obtained atthat time. For a critical analysis of the data we present the reader should consultthe original articles we cite. References to other contributions can be found in theproceedings to the numerous lattice conferences.

17.1 The String Tension and the gg-Potential in the SU(3)Gauge Theory

The string tension was defined in chapter 8 as the coefficient a of the linearlyrising part of the potential for large separations of a quark-antiquark pair. It is theforce between a static quark and antiquark at infinite separation in the absenceof pair production processes. In a numerical calculation this force can only bemeasured for relatively small ^-separations (in lattice units). Hence if the scaleon which this force is seen in nature is determined by the size of hadronic matterthen one must choose the lattice to be sufficiently coarse (i.e., the coupling gosufficiently large) for the hadron, represented by the qq system, to fit on it. Butif the lattice is made too coarse, one cannot expect to extract continuum physics,and the dimensionless string tension era2 will not exhibit the behaviour predictedby the renormalization group. Thus one is faced in practice with a precarioussituation: if one chooses too small a lattice spacing (or coupling constant), thenone cannot expect to see the asymptotic form of the force on a lattice of smallextent. On the other hand, if the lattice spacing is too large, then one cannotexpect to see continuum physics. Thus at best one can hope to measure the

317


physical string tension in a narrow range of the coupling constant. This is theso-called "scaling window".

Let us now see how things look in practice. To this effect recall first of all thatthe static (/g-potential in lattice units can in principle be determined by calculatingthe following limit:

f 1 1V(R) = - lim \^lnW(R,T)\ ,

f-HX [T J

where W(R, T) is the expectation value of the Wilson loop with spatial and tem-poral extension R and T, respectively. Since this expectation value is determinedin general from a MC calculation where neither R nor T are very large, V(R) is notexpected to be of the form (8.17a). In fact, already the self-energy contributionsto In W (R, T) will be proportional to the perimeter R + T, and will compete withthe area term aRT, for finite T and R. Effects arising from terms proportional tothe perimeter can, however, be easily eliminated. Assuming that W(R, T) has theform

Wt(R, f) = e-wtf-atf+r)-^ ( 1 7 - 1 )

we can isolate the string tension a by studying the Creutz ratios

\W(R,T-1)W(R-1,T)J

If the Wilson loop depends on R and T in the way given by (17.1), then x(-R, T) willbe independent of these variables, and will coincide with the string tension. In fig.(17-1) we show the first MC data for the SU(2) string tension (black dots) obtainedby Creutz (1980) together with the results of the strong coupling expansion carriedout by Miinster (1981) up to twelvth order in @ = 4/gg. For comparison the lowestorder result for a in strong coupling has also been included. Notice that in thenarrow "window" 2.2 < (3 < 2.5 the MC data follow an exponential curve predictedby the renormalization group; indeed, for the case of SU(2), the relation betweenthe lattice spacing a and the coupling go analogous to (9.20) is found to read asfollows:

a=—R{g0)

where

R(go)*e-^2W

Some Results of Monte Carlo Calculations 319

and*

Fig . 17-1 The SU(2) string tension in lattice units as a function of

A/gl. The dots are MC results of Creutz (1980). The solid curve is the

result of the cluster expansion carried out by Miinster (1981).

Since the physical string tension a has the dimension of (mass)2, cr(go) mustbehave as follows in the scaling region

a2a = a(g0) « Ca[R{g*)]2. (17.2)

This gives yfa in units of the (dimensioned) lattice parameter Ax,:

^=\[6akL. (17.3)

r~x—

The constant v Ca can be determined from the weak coupling fit to the MonteCarlo data.

* Recall that for SU(N), (3 was defined as (5 = 2N/gQ. Ax, is of course not to beidentified with the corresponding lattice parameter in the SU(3) gauge theory.


We want to point out that a lattice calculation can only determine a physi-cal observable in units of the lattice parameter A ,; hence one can only calculatedimensionless ratios of physical quantities. Alternatively one may use the experi-mentally measured value for an observable (such as a hadron mass, string tension,etc.) to determine the scale A^ or the lattice spacing. For example solving eq.(17.2) for a we get

a=J^R(g0). (17.4)V cr

Since Ca can be determined from a lattice calculation, (17.4) determines a inphysical units, once a is known. Alternatively one can also use the mass of ahadron to determine a as a function of go. Let MJJ be some physical hadron mass(proton, pion, etc.); then from dimensional arguments alone we know that in thescaling region

MH = MHa = CH R(go),

where CH can again be determined from a lattice calculation; solving for a we get

fl=(|)%) •

Fig. 17-2 Creutz ratios as a function of /3 = 6/gfg for the SU(3) gauge

theory. The figure is taken from Creutz and Moriarty (1982). The solid curve

is the string tension in the leading strong coupling approximation.


Once the lattice spacing has been determined, one knows the physical ex-tension of the lattice used in a MC calculation performed at a fixed value of thecoupling. This gives us a handle of accessing the relevance of the lattice employedin studying a particular phenomenon.

Similar calculations to the one just described have been performed subse-quently by Creutz and other authors for the case of SU(3). In fig. (17-2) we showthe MC results of Creutz and Moriarty (1982) for the Creutz ratios constructedfrom different Wilson loops at various values of /?, together with the strong cou-pling result in leading order. This calculation was carried out on a 64 lattice.From the figure it appears that asymptotic scaling sets in for values of j3 slightlybelow 6.0. Subsequent studies performed on larger lattices, however, indicatedthat scaling probably sets in for (3 > 6.2.

So far we have only considered the string tension, i.e. the linearly rising pieceof the potential. For a proper determination of the string tension one should,however, also include in the fit the behaviour of the potential at short distances.For small separations of the qq-paix, the potential is expected to exhibit a Coulombtype behaviour with a "running" coupling constant a depending logarithmicallyon the distance scale. This is predicted by asymptotic freedom.

Fig. 17-3 MC data of Stack (1984) for the static (^-potential at variousqq separations. V and R are measured essentially in units of -y/cr and l/s/a.The solid curve is a fit to the data based on the ansatz (17.5).

In fig. (17-3) we show the results of a MC calculation performed by Stack(1984) for the case of SU{3) on a 83 x 12 lattice, with j3 values (/? - Q/gl) lying


within the observed scaling window (6.0 < (3 < 7.0). The potential and theseparation of the qq pair are measured essentially in units of \fa and l/y/a. Thesolid line is a fit to the data based on a linear combination of a Coulomb and linearpotential of the form

V(R) = -^+*R. (17.5)

A large scale simulation was carried out later on by de Forcrand (1986) on a243 x 38 lattice at (3 = 6.3. Fig. (17-4) shows the results of the MC calculation,together with a fit based on the form (17.5) for the potential. Similar results havebeen obtained by the same author at f3 = 6.0 on a 174 lattice.

Fig. 17-4 MC data for the (/(/-potential in the pure SU(3) gauge theoryobtained by de Forcrand (1986).

What is noteworthy is that a violation of asymptotic scaling was observed,and that the string tension measured in units of Az, (cf. eq. (17.3)) was muchlower than that obtained in previous calculations (y/a/AL « 92 at (3 = 6.0, andT/CT/AL « 79 at (3 = 6.3). In fact, since the original determination of a by Creutzand Moriarty, the string tension measured in units of the lattice parameter AL, haskept decreasing. The original value for vW-A-z, was y/a/'AL = 167. At the Berkeleyconference (see Hasenfratz, 1986) the value had decreased to about 90. This valuewas extracted from measurements performed at rather small gg-separations (up to8 lattice sites). In a more recent calculation of the static gg-potential performedwith high statistics (Ding, Baillie and Fox, 1990) and for large separations ofthe quark and antiquark (up to R = 12a) an even lower value was obtained:sfajAj, « 77. Also the value of the coupling a in eq. (17.5) was found to be quite


different from that obtained in previous simulations (a = 0.58 as compared toa « 0.3). As Ding et al. point out, their values compare favourably with those ofEichten et al. (1980) obtained by fitting the (heavy quark) charmonium data witha Coulomb plus linear potential ansatz. Since charmonium is built from heavyquarks, it is reasonable to describe this system with a potential model. The valuesfor the string tension y/a and coupling a estimated by Eichten et al. were 420MeV and 0.52, respectively. In a further paper Ding (1990) also finds that thedimensionless ratio V/y/a, which according to to (17.5) should be a function ofy/aR = \/&R, is independent of the couplings used (/3 = 6.0,6.1 and 6.2), as mustbe the case if one is in the scaling region. This is shown in Fig. (17-5).

Fig. 17-5 MC data of Ding et al. for the gg-potential obtained atvarious couplings /3 = 6/<?Q. All the points are seen to lie on a single curve,indicating that one is in the scaling region. The figure is taken from Ding(1990).

A potential which rises linearly with the separation of the quarks is what isexpected from a flux-tube picture of confinement. As we have mentioned in theintroduction of chapter 7, it is generally believed that the non-abelian nature ofthe 5(7(3) gauge theory causes the flux, linking the quark and antiquark, to besqueezed into a narrow tube. It is therefore of great interest to study the spatialdistribution of the energy density between two static sources, immersed into a puregluonic medium, in a Monte Carlo "experiment". Early work in this direction hasbeen carried out by Fukugita and Niuya (1983) and by Flower and Otto (1985).


More complete studies of this problem have been carried out by Sommer (1987 and1988) for ^-separations up to four lattice units. For the case of the SU(2) gaugetheory, detailed Monte Carlo studies of the flux distribution for a quark-antiquarkpair have been reported by Haymaker et al. (1987-1992) and Bali et al. (1995).The authors find that the energy and action density are indeed confined to a verynarrow region around the qq-axis. For more details the reader may consult section17.7.

17.2 The ^-potential in full QCD

So far we have neglected the effects arising from pair production. Theseeffects arise from the fermionic determinant which must be included when updatingthe link variable configurations. In the presence of dynamical quarks, quark-paircreation can take place, which causes the flux tube connecting the static quarkand antiquark to break when the sources are pulled far enough apart. This isexpected to occur when the energy stored in the string becomes larger than 2M,where M is the constituent quark mass. Thus the potential should flatten for largeseparations of the quark-antiquark pair. But to observe this flattening, one needsto study large Wilson loops, which is very difficult. Since simulations in full QCDare extremely time-consuming, one is restricted to relatively small lattices, whichdo not allow one to study the potential for very large separations of the quark andantiquark (this is especially true for Wilson fermions since they involve a largernumber of degrees of freedom than Kogut-Susskind fermions). Hence to calculatethe potential at large distances, the lattice spacing must be chosen sufficientlylarge.

To identify screening effects due to quark-pair production processes, the fullQCD data should, however, not be compared with the quenched data at the samevalue of the coupling. The reason is that part of the effects seen in numericalcalculations can be interpreted in terms of a shift in the gauge coupling. Such ashift is expected, since when quarks are coupled to the gauge potential, vacuumpolarization effects will lower the effective coupling. The amount by which thecoupling is decreased will depend on the number of quark flavours. Also therelation between gauge coupling and lattice spacing is changed when dynamicalfermions are included. To compare the full QCD results with the quenched dataone should therefore first get an estimate of how the coupling is renormalized byvacuum polarization effects. The way that this is usually done in praxis is todetermine the coupling in the quenched theory for which the expectation value of


a Wilson loop or a plaquette variable agrees with that obtained in the full theory

for a fixed choice of coupling.

The first calculation of the interquark potential with Wilson fermions whichshowed an impressive flattening of the potential for larger separations of the quark-antiquark pair was carried out by de Forcrand and Stamatescu (1986).

Fig. 17-6 MC data of Born et al. (1989) for the gg-potential in full

QCD with Kogut-Susskind fermions. The data has been fitted with the

screened potential (17.6). The neighbouring dotted lines show the shift in

the potential if the lattice spacing is changed by 10%.

The lattice used was however very small, and it could not be ruled out thatpart of the screening observed was due to finite size (or temperature) effects.Kogut-Susskind fermions allow one to use larger lattices. Fig. (17-6) shows thepotential in physical units obtained by Born et al. (1989) using Kogut-Susskindfermions. The calculation was performed on a 123 x 24 lattice for two couplings,&/QQ = 5.35 and 5.20. The lattice spacing was determined by measuring the p massin the same simulation. Hence a = Mp/Mp, where Mp is the p mass in lattice unitsand Mp = 770 MeV is the physical p mass. These authors have parametrized thedata according to

V(R)=(-%+irR)izgl, (17.6)


where the string tension was fixed to be yfo = 400 MeV. The coupling and thescreening mass fj, were then determined from the fit to be a = 0.21 ± 0.01 and/ i" 1 = 0.9 ±0.2 fermi. Although the data appears to indicate that the qq potentialhas been screened, this need not be the case. The conclusion is very sensitive tothe lattice spacing. Thus by chosing a different lattice spacing, the data could alsobe fitted with an unscreened potential. The authors point out, however, that thechange in lattice spacing required would be inconsistent with that determined fromthe measurement of the p-mass in the same simulation. But since the measurementof the p mass requires an extrapolation, no conclusive statement can be made.

We have mentioned two simulations which suggest that the potential is screenedat "large" distances due to the creation of 95-pairs from the vacuum. Not all thesimulations that have been carried out, however, confirm this screening picture(see e.g. Gupta, 1989). Clearly to get an unambiguous signal for string-breakingat large distances, one must be sure that the screening seen is not a short distanceor finite volume effect. For this the simulations have to be carried out on muchlarger lattices, and at a smaller lattice spacing. In this case the observation of aflat potential for large separations of the quark-antiquark pair would provide uswith conclusive evidence for string breaking.

17.3 Chiral Symmetry Breaking

A symmetry of the action is said to be spontaneously broken if it is notrespected by the ground state. If the transformations leaving the action invariantform a continuous group, then this spontaneous breakdown is accompanied bythe appearance of massless particles, the so-called Goldstone bosons, which canpropagate over large distances, giving rise to long-range correlations. The numberof such Goldstone bosons equals the number of generators associated with thebroken part of the symmetry group.*

In the limit of vanishing quark masses the pions observed in nature are be-lieved to be the Goldstone bosons associated with a spontaneous breakdown ofchiral symmetry. Let us illustrate what we mean by chiral symmetry for the caseof an abelian U{1) gauge theory. The continuum action for vanishing fermion masshas the form

S = i fcfixF^F^ + /"d4a^7M(dM + ieA^t}) .

* In the case of a gauge theory, there exist important cases where these Gold-stone bosons do not appear, but are absorbed into the longitudinal degrees offreedom of the gauge field (Higgs mechanism).


Now the field ip can always be decomposed into "left"- and "right"-handed partsas follows:

wheretpR =P+ip, ipL = P-ip ,

F±=i(i±7 5); Pl = P±-

With

$R = ipP-, i>L = jp+,

the fermionic part of the action becomes

SF= ^xû^dfj, + igAîpL

+ / ^xtpR-y^d,, + igAîpR .

This action is invariant under the following transformations

V>L —•> eiai>L, i>L —> e~iai>L ,

i>R —> eiA^fl, V>ii —> e-iAî? ,

where a and A are arbitrary parameters. These transformations are elements of aUR(1) x UL(1) group: the chiral group. Consider now the ground state expectationvalue oitpRtpL, i.e., < tpRipL >• i'R'^L transforms as follows under (17.7),

4>R"4>L > e?K$Ri>L ,

where A = a — A. If the transformations (17.7) are implemented by a unitaryoperator, and if the ground state is left invariant under the action of this operator,then < 4>RII>L > must vanish. The same is true for {ipLipR)- This need howevernot be the case in a quantum theory involving an infinite number of degrees offreedom, where the ground state may not be chirally invariant. Hence (i>Riph)may in fact be different from zero, implying that chiral symmetry has been brokenspontaneously.

As we have just demonstrated, a good quantity for testing the spontaneousbreakdown of chiral symmetry is (i>RipL). Alternatively, {ipip} = {4>RII)L) + {'4>L'4>R)

will also do the job, and is the quantity usually studied in the literature. Since {piphas the dimension of (length)"3, its expectation value in lattice units must havethe following dependence on the bare coupling constant in the scaling region

(U) = C^[R(g0)}3, (17.8)

(17.7)


where R(go) is given by eq. (9.21d). The constant C^ can be determined from a

MC calculation. In physical units (17.8) then reads

<^> = C^k\

where A.£ is the lattice scale appearing in (9.21c).

Fig. 17-7 MC data of Barkai, Moriarty and Rebbi (1985) for the

chiral condensate {tpip) measured in lattice units, for various values of the

dimensionless bare quark mass. The solid curve is a fit to the data, and has

been extrapolated to the zero quark mass. The dashed line is a fit obtained

by leaving out the point denoted by a cross which is rather sensitive to finite

volume effects.

In fig. (17-7) we show the quenched data for {'tptp) in lattice units as a functionof the bare quark mass mq, obtained by Barkai, Moriarty and Rebbi (1985) ona 163 x 32 lattice at (3 = 6.0. The non-vanishing extrapolated value of (ifii/j) atmq = 0 is a sign of spontaneous symmetry breaking. If in the limit of vanishingquark mass the pion becomes the Goldstone boson associated with the breakdownof chiral symmetry, then its mass should vanish linearly with ^/mg. In fig. (17-8) we show the MC data of the above mentioned authors for Mv as a function of^ffiiq. The linear fit extrapolates indeed to Mn — 0 for vanishing quark mass. Thecalculation was performed using staggered fermions. Staggered fermions have beenused also in most of the other computations of (ip'ip) in the literature. The reason isthat, as we have seen in chapter 4, the action for Wilson fermions, (6.25c), breaks


chiral symmetry explicitly also for Mo -> 0. Hence we cannot identify Mo witha bare quark mass. A possible definition of the quark mass is however suggestedby the above made observation that Mv should vanish for mq ->• 0. This willoccur for some critical value KC of the hopping parameter, corresponding to some

Mo = Mc:

8r + 2MC

This suggests the definition fhq = Mo - Mc, or equivalently

F i g . 17-8 MC data of Barkai, Moriarty and Rebbi (1985) for the pion

mass as a function of the square root of the bare quark mass. Both masses

are measured in lattice units.

Hence in a formulation using Wilson fermions, one must first determine thecritical value of the hopping parameter where the pion mass vanishes, and thenstudy the "chiral condensate" (tptp) in the limit n -» KC. In fig. (17-9) we showa somewhat later calculation of the pion mass (in lattice units) as a function of^Jihq, performed by Bowler et al. (1988) on a 163 x 24 lattice, at (5 = 6.15.For a quark mass of less than 0.01 the data deviate from the linear behaviour.The above mentioned authors argue that this is probably due to finite-time effectswhich becomes increasingly important as the quark mass is lowered.

(17.9)


Fig. 17-9 Monte Carlo data of Bowler et al. (1988) for the pion mass

in lattice units as a function of the square root of the quark mass rhq.

17A Glueballs

As we have pointed out in chapter 6, QCD in the pure gauge sector is a verynon trivial theory. We therefore expect that it posesses its own spectrum of boundstates built from the gluon fields. These bound states are called glueballs. Ifthe theory confines colour, then the glueballs must be colour singlets. Of course arealisitic calculation, which can be confronted with experiment, should include theeffects arising from quarks. But one may hope that by studying the pure gaugesector one obtains -a first approximation to the true glueball bound state spectrum.If glueballs are indeed predicted by QCD, then they must found experimentally, orQCD is ruled out as the theory of strong interactions. It is therefore very importantto calculate the mass spectrum of the glueball states. The lattice formulation ofQCD provides us with this possibility.

The basic idea that goes into the computation of the glueball mass spectrumis very simple. One first constructs a gauge invariant functional of the link vari-ables located on a fixed time slice of the four-dimensional lattice, carrying thequantum numbers of the state one wishes to investigate. This functional is buildfrom the trace of the product of link matrices along closed loops on the lattice.From these functionals one constructs zero momentum operators by summing thecontributions obtained by translating the loops over the entire spatial lattice. Fur-thermore, since on the lattice we have only a cubic symmetry, but no rotationalsymmetry, the zero momentum operator should transform under an irreducible


representation of the cubic group. These representations couple only to certain

angular momenta in the continuum limit. *. The lowest glueball state is expected

to be that carrying the quantum numbers of the vacuum (Jpc = 0+ +) , where P

and C stands for parity and charge conjugation. An example of a zero momentum

operator coupling to the spin zero state in the continuum limit is given by

x,orie.nt.

where the square denotes an elementary plaquette variable with base at (x, r),

and where the sum is carried out over all the positions and space orientations of

the plaquette located in a fixed time slice. This is the simplest operator one can

use to study a zero angular momentum glueball. In most calculations, however,

other more complicated loops have been considered as well. Let us assume that

we have chosen an operator O(T) to study a particular glueball state. The lowest

mass of a glueball carrying the quantum numbers of O can then, in principle, be

determined from the behaviour of the correlation function < O(T)O(0) > for large

euclidean times. Indeed, recalling that O(T) = exp(JfT)O(0) exp(-ifr), where H

is the Hamiltonian, one has that

C(T) = ^\(n\O)\2e-E"T , (17.10)n

where \O > denotes the state created by the operator O = O(0) from the vacuum,

and En is the energy of the n'th eigenstate of H (measured relative to the vac-

uum.) In the large euclidean time limit, C(r) is dominated by the lowest energy

state carrying the quantum numbers of O . If these quantum numbers happen

to coincide with that of the vacuum, then one is interested in the next higher

energy state. Hence one must first subtract the vacuum contribution in (17.10)

before taking the large euclidean time limit. This is accomplished by replacing

O(T) by O{T)— < O(T) >, or equivalently, subtracting < O(T) >< O > from the

correlation function. This yields the connected correlation function. Then

C(T)conn. — • \{G\O)\2e~E°T , (17.11)r—»oo

* For a detailed discussion of this problem the reader may confer the lectures

of Berg (Cargese 1983)


where \G) denotes the lowest glueball state above the vacuum with the quantumnumbers of O. By studying the exponential decay of the correlation function forlarge euclidean times, one then extracts the glueball mass of interest. This soundsvery simple indeed. But in praxis one is confronted with a number of seriousproblems. In fact, measuring the glueball masses is one of the most difficult tasksin numerical simulations of the SU(3) gauge theory. The ideal situation would bethat the measured values of C(T) would fall on a simple exponential curve, andthat the lattice is fine enough for the gluon mass to exhibit the scaling behaviourpredicted by the renormalization group. In particular, close to the continuumlimit, the ratios of different glueball masses should become independent of thecoupling. To see a single exponential would however require that we either areable to measure the correlation function for very large times, or that \O) has onlya projection on the particular state of interest. This is certainly not the casein praxis. In the early days of glueball computations the operator O was builtfrom simple plaquettes or from a linear combination of more complicated loopsof different shapes and orientations. * The difficulty one encountered was thatthe signal was drowned within the statistical noise for times beyond two latticespacings, even for the lightest glueball. One of the problems was that the overlapof the glueball states with \O) was too small. This overlap gets worse and worseas the lattice spacing is reduced. This is intuitively obvious since the physicalextension of the glueball remains fixed, while the local operator O , constructedfrom small loops , probes an ever smaller region of the glue ball wave function asone decreases the lattice spacing. For this reason it is only possible to obtain areasonable signal on coarse lattices, where the state \O) has a reasonable overlapwith the glueball wave function. But even if the overlap is enhanced for largerlattice spacings, the glueball mass measured in lattice units is also increased, andhence the exponential in (17.11) leads to a stronger supression of the correlationfunction. Clearly what one needs are operators which have a strong overlap withthe glueball state of interest for small lattice spacings. This would solve bothproblems mentioned above, and at the same time allow one to probe the scalingregion. How important this is, is made evident by the observation that for localoperators the overlap decreases with the fifth power of the lattice spacing.** Thisis clearly a disaster and makes it very hard to study continuum physics with

* for a review of early work see Berg (1983). For the earliest glueball calculationssee Berg (1980); Bhanot and Rebbi (1981); Engels et al. (1981a).

** See the talk by Schierholz at Lattice 87 (Schierholz,1988), and the review talkby Baal and Kronfeld at Lattice 88 (1989).


local operators. After 1986 there have been several proposals for constructing nonlocal operators for which C(r)/C(0) « a (Berg and Billoire, 1986; Teper, 1986a;Albanese et. al., 1987; Kronfeld, Moriarty and Schierholz, 1988). The use ofsuch (non-local, smeared, or fuzzy) operators improves the signal to noise ratiodramatically, and allows one to measure the correlation function of the 0 + + and2 + + states for temporal separations much larger than than in previous calculations.As an example we show in fig. (17-10) the 0 + + and 2 + + correlation functionsobtained by Brandstaeter et al. (Schierholz, 1989) at 6/pg = 6.0 on a 164 lattice.A clear signal is obtained up to t = 6 for the 0 + + state, and up to t = 5 for the2++ state.*

Fig. 17-10 MC data of Brandstaeter et al. for the (a) 0++ and (b)2 + + correlation function. The figure is taken from Schierholz (1989).

A particularity efficient prescription for constructing non local operators isthat proposed by Teper (1986a). Since this prescription is very simple and intu-itive, we describe it here. Teper constructs by an iterative method very complexnon local operators O consisting of an enormous number of elementary paths. The

* For details regarding the operator used, see Schierholz (1988).


iterative scheme involves only the link variables located in a fixed time slice, andgoes as follows. First one constructs so called fuzzy link variables associated witha path of length 2 along an arbitrary /i direction and with base at an arbitrarypoint n, by adding the contributions of the direct path and the spatial "staples"(as Teper calls them), as shown below:

? f * f n

i - « + I •- LI *-* A ' — > v (17.12)i ±V, V * 4 v -1

At this level the contributions appearing on the right hand side are calculatedfrom the matrix products of the usual link variables. The result is the fuzzy"path" shown on the left hand side. Let us denote (with Teper) the usual linksvariables by Up (n) and the new fuzzy variable by Up (n); then

UP{n)=U<P\n)UM{ri + ijL)

+ Uio){n)U^\n + v)U^\n + v + fi)U^ (n + 2/i).

In the next step of the iteration processes one replaces the link variables appearingon the right hand side of (17.12) by the fuzzy links obtained in the first step. Thenew fuzzy links Up (n) are now associated with a path joining the lattice site nwith n + 4/i . At each step of the iteration process the length of the link increasesby a factor of 2. The N'th step of the iteration process is depicted in the followingfigure, taken from Teper (1986):

uHn+2LN_l(i)

Here IN = 2N. Having generated at the N'th step the fuzzy "links" Up (n), onecan e.g. construct from these superplaquettes in a manner completely analogousto the usual elementary plaquette:

Q = [/^(n) = ulN\n)u\N\n + lNi)Ur\n + lNj)U^\n)

(17.13)


Hence loops of fuzzy links are now complicated objects when expressed in terms ofelementary paths. Note that the matrizes U^ (ji) are not group elements. Onlyin the case of SU(2) these are proportional to a unitary matrix, with the pro-portionality constant given by the determinant. But this is not of relevance here,since the aim is to merely construct sufficiently complex operators, to ensure thatthey create states having a large overlap with the glueball candidate of interest.

Because of the simplicity of the iterative scheme, one is now in the positionto increase the number of iteration steps with decreasing coupling in such a way,that the size of the superplaquettes (or more complicated versions thereof) canbe adjusted to the increasing volume (in lattice units) occupied by the glueballon the lattice. In this way one can achieve a large overlap of the operator withthe glueball wave function even for small lattice spacing. Michael and Teper(1989) have applied the fuzzying prescription to the SU(3) gauge theory, and havecalculated the glueball masses for all the JPC states, using couplings 6/<?Q rangingbetween 5.9 and 6.2, and spatial lattices with volumes in the range from 103 to203. They find, for example, that their fuzzy 0 + + glueball operators have about90% overlap with the corresponding glueball ground state, and this at a coupling6/#Q as large as 6.2.

Fig. 17-11 Glueball masses in units of the square root of the stringtension calculated by Michael and Teper (1989). The masses in physicalunits are given on the right hand scale and have been obtained using thevalue of 440 MeV for the string tension.

Fig. (17-11) shows the mass spectrum obtained by these authors. The masses inphysical units are shown on the right hand scale. They have been obtained by


using the value \fa = 440MeV for the string tension.The 0 + + and 2 + + glueball masses have so far been measured with greatest

confidence. They can be extracted by studying the correlation function of thefollowing operators: (see e.g., Ishikawa et al., 1983)

0o++(r) =ReTr^2(U12(x,T) + U23(X,T) + U31(X,T)),X

4>2++(r) =ReTr^2(Ui2(x,T) - U13(x,r)).X

Here the "plaquette" variables Uij can be ordinary plaquette variables or fuzzysuperplaquette variables. Monte Carlo calculations performed with large latticevolumes by several groups (see the review of Michael, 1990) agree that the massratio m(2 + + ) /m(0 + + ) is about 1.5, while early calculations based on the study ofcorrelation functions of local operators, and 6/<?g < 6.0, suggested that this ratiois of the order of one or less. The physical values of the glueball masses can beexpressed in terms of the string tension. Assuming a value of 420 MeV for thesquare root of the string tension, taken from potential models for heavy quarkbound states (Eichten et al., 1980) the 0 + + and 2 + + masses reported by Baal andKronfeld at Lattice 88 (Baal and Kronfeld, 1989) were

TO(0++) =1370 ± 90MeV

m(2++) =2115 ± 125Mey

The original value obtained by Berg and Billoire (1982) for the 0 + + mass was920 ± 310MeV. This calculation had been carried out on a 43 x 8 lattice.

This concludes our discussion of the glueball mass spectrum. We now turn tothe discussion of the Monte Carlo simulations of the hadron mass spectrum.

17.5 Hadron Mass Spectrum

The lattice formulation of QCD gives us the possibility of answering one of themost fundamental questions in elementary particle physics: what is the origin ofthe masses of the strongly interacting particles observed in nature? QCD shouldbe able to predict the bound state spectrum of hadrons build from quarks andgluons, which are permanently confined within the hadrons. The determinationof the masses of the lowest bound states with a given set of quantum numbers isbased on the same ideas as discussed in the previous section; i.e., the masses areextracted from the large (euclidean) time behaviour of correlation functions for


zero momentum operators carrying the appropriate quantum numbers to createthe hadron of interest. For local operators constructed from Wilson fermions thegeneric form is given by

Om(r) =J2r{rBi>A(S,T)i>B(x,T), (mesons)X

°b(T) = Yl SABCÂ(X, r)ipB(x, T)I/)C(X, T) , (baryons)x

where we have used a continuum notation for convenience, and a summation overthe collective indices (Dirac, colour and flavour) is understood. The coefficientsF^g and SABC are chosen in such a way that the operators are colour neutral*(XAB C* â*> f°r mesons, and SABC oc eabc for baryons), and carry the quantumnumbers (spin, parity, etc.) of the hadron of interest. Thus, for example, operatorsthat can be used for studying the n+ ,p+, and proton are given by

On+ = £ dalsua- Op+ = Y, da1ua; (Op)a = £ ) eabcuaa[ub

0(Cl5)0sdc5}

X X X

where u and d are the "up" and "down" quark fields, C is the charge conjuga-tion matrix. The construction of the corresponding operators for Kogut-Susskindfermions is more complicated and we will not consider them here. **

For example, meson masses can be extracted by studying the behaviour ofthe correlation functions

rm(r) = (Ol(r)Om(0))

for large euclidean times. This correlation function can be written as a sum ofexpectation values of products of two external field quark propagators calculatedwith a Boltzmann distribution exp(—Seff[U]), where Seff[U] has been defined in(12.14b). In particular the pion correlation function for degenerate quark masses,describing the propagation of a quark-antiquark from x to y, is given by

C*(T) = ^Tr(75Kzl.gTl5Kzl.So))Scff (17.14)

where K~x is the quark propagator, i.e., the inverse of the matrix (12.2c). Forbaryons the correlation function analogous to (17.14) will involve the product ofthree quark propagators.

* Here a, b, c are the colour indices of the Dirac fields.** See e.g., Morel and Rodrigues (1984); Golterman and Smit (1985); Golterman

(1986b).


The procedure for calculating a correlation function is the following. Let usassume that one has decided on a set of values for the parameters appearing inthe action, i.e., gauge coupling and quark masses in lattice units. The latticesize should in principle be chosen large enough to easily accommodate the hadronwhose mass one wants to calculate. In particular, the extension of the lattice inthe euclidean time direction must be large enough to allow one to study correlationfunctions for large euclidean times. This is important, since one doesn't want themeasurements to be contaminated by contributions coming from higher excitedstates carrying the same quantum numbers of the hadron of interest. A priori,we do not know how large the lattice must (at least) be chosen, since we do notknow the lattice spacing. In principle, this lattice spacing, which is controled bythe gauge coupling, should be small enough to allow one to extract continuumphysics. The values for the coupling constant and quark masses, and the linearextensions of the lattice used in numerical simulations, will be limited by the avail-able computer facilities. Having fixed all these quantities, the next step consistsin generating a set of link configurations which, in the quenched approximation,are distributed according to exp(—SG[U]), and in full QCD are generated withthe probability density detK(U) exp(—SQ[U]). For each configuration one thencomputes the propagator K~1(U). It is this step which makes the computationsof fermionic correlation functions - already in the quenched approximation - time-consuming. The algorithm most widely used to calculate K~1(U) is the conjugategradient method. Once one has calculated the external field quark propagators,one constructs from these the hadron propagators, and averages the expression overthe ensemble of field configurations. For mesons the ensemble average is carriedout over products of two quark propagators, while for baryons the average is takenover products of three quark propagators. Only uncorrelated field configurationsshould be used for calculating the ensemble averages and the statistical errors.Thus for example in a local updating procedure, successive configurations gener-ated by a Markov chain will be highly correlated, since one is changing only one,or a few links, at each updating step. Hence one must generate many more con-figurations than are actually used to calculate the ensemble average. In practice,only configurations separated by a fair number of sweeps through the lattice areused for measuring the observable. Finally the lowest hadron mass is extractedby studying the behaviour of the correlation function for large euclidean times.This yields the mass in lattice units. If one is working close to the continuumlimit, then the ratios of different particle masses measured in lattice units shouldbe independent of the coupling (or equivalently the lattice spacing) and can be


identified with the corresponding ratios of the masses measured in physical units.In an actual Monte Carlo simulation these ratios will in general differ substantiallyfrom the experimentally measured values. The reason is that most calculations areperformed at unrealistically high quark masses, of the order of the strange quarkmass. Statistical fluctuations in the hadron propagators, which increase with de-creasing quark masses, do not allow one to simulate such hadrons as the it, p andnucleon, for realistic dynamical bare (up and down) quark masses of the order ofa few MeV.

Most earlier Monte Carlo simulations have been performed with degeneratequark masses. Hence the output of the calculation depends on the values used forthe bare coupling constant, and on a single dimensionless quark mass. The quarkmass in physical units at which the MC calculation has been performed can thenbe determined as follows. One calculates the masses of some hadrons such as thepion and the p for several values of the bare quark mass rh. One then fits thedata for Mn and Mp using the following ansatz which is motivated from currentalgebra:

Ml =\«m, (17.15a)

Mp=\pm + Pp. (17.156)

F ig . 17-12 Example of a fit to the MC data for the pion and rho masswith the ansatz (17.15). The figure is taken from Born et al. (1989).

In the case of Wilson fermions rh is related to the hopping parameter by(17.9). Fig. (17-12) , taken from Born et al. (1989), gives an example of a fit tothe MC data based on (17.15). From the fit one determines the coefficients X^, Xp


and Pp. The lattice spacing can then be determined by recalling that Mn = Mna,and Mp = Mpa. Thus by inserting for Mv and Mp their physical values, 140 MeVand 770 MeV, one calculates from (17.15) the lattice spacing and the quark mass.This information can be used to extract other hadron masses from Monte Carlocalculations. Assuming that the lattice spacing is only a function of the bare cou-pling (which was held fixed in the simulations performed at different quark masses)one then also knows the quark masses in physical units at which the MC calcula-tions have been performed. The above way of proceeding involves an uncertaintyin the extrapolation procedure. Furthermore it is only justified in simulations offull QCD, and not in the quenched approximation, since one doesn't know whatthe pion-rho mass ratio should be when the effects of dynamical fermions are ig-nored. Although the coefficients AT,Ap and (5P in (17.15) are dependent on thevalue of the bare coupling constant used, the ratio of hadrons masses should beindependent of the coupling if one is working in the scaling region. In order tobe able to compare directly the results obtained by various groups for Wilson andKogut-Susskind fermions, and for different couplings and quark masses, the Ed-inburgh group (Bowler et al. (1985)) have suggested that one should exhibit, forexample, the nucleon, pion, rho mass data as a plot of the nucleon-rho mass ratioversus the pion-rho mass ratio. In the scaling region all the data should then fallon a universal curve. This is the so called Edinburgh plot. In this way one canstudy the general trend of the nucleon-rho mass ratio as Mn/Mp approaches thephysical value.

The Quenched Approximation

The quenched approximation is not the real world, but it is important toobtain reliable results, since only then can one determine how dynamical quarksinfluence the mass spectrum. It may turn out that the effects of pair creation(and annihilation) do not influence the hadron spectrum significantly. In fact, thesuccess of potential models in describing the spectrum of heavy quark bound statessuggest that at least for such bound states the effects arising from dynamical quarkscan be absorbed into a renormalization of the coupling constant. Unfortunately,only a calculation in full QCD can provide us with an answer of how the quenchedspectrum is modified by the dynamical quarks.

The earliest MC calculation of the quenched hadron spectrum in the SU(3)gauge theory date back to 1981 (Hamber and Parisi, 1981). Since then manygroups have performed such calculations on larger lattices and with smaller latticespacings using Wilson and Kogut-Susskind fermions. Up to 1988 the most am-bitious calculation with Wilson fermions had been carried out by de Forcrand et


al. (1988) on a 243 x 48 lattice. These authors used special blocking techniquesin order to reduce the large number of degrees of freedom. Recall that for Wilsonfermions every lattice site can accommodate all internal degrees of freedom. Thismakes these computations more demanding than those with staggered fermions.Fig. (17-13) taken from the above mentioned reference, illustrates the general formof meson correlation functions extracted from a Monte Carlo calculation performedon a periodic lattice in the time direction. The propagators displayed in Fig. (17-13) have been fitted in the interval 15 < t < 24 with single mass exponentialssymmetrized about t = 24.

Fig. 17-13 MC data for the pion and rho propagators, obtained byForcrand et al. (1988). The data has been symmetrized about t = 24.

The use of staggered fermions for computing the hadron mass spectrum posesa problem, since for finite lattice spacing the staggered fermion action brakesflavour symmetry (while Wilson fermions do not; see chapter 4). Already for thisreason one does not expect that results obtained with Wilson and Kogut-Susskindfermions will agree, unless one is close to the continuum limit. And in fact, theydid not agree for a long time. This situation has improved substantially and thecalculations using Wilson and staggered fermions have given comparable results


for Q/g$ > 6.O.*

Figure (17-14), taken from Yoshie, Iwasaki and Sakai (1990), shows thequenched results for the nucleon-rho mass ratio versus M^/Mp reported prior to,or at the Capri lattice conference in 1989. The solid (open) points correspond tocalculations performed with Wilson (Kogut-Susskind) fermions. The solid curve isobtained from a phenomenological mass formula (Ono, 1978). The open point atthe left lower end of the curve corresponds to the experimental mass ratios. Thesedata are from Barkai et al. (1985), Gupta et al. (1987), Hahn et al. (1987), Bowleret al. (1988), Bacilieri et al. (1988) and Iwasaki et al. (1989). The quark mass inall these calculations is, however, unrealistically large.

As seen from the figure there is a general tendency for the nucleon-rho massratio to drop as M^/Mp decreases. The data of Iwasaki et al. agrees extremelywell with the phenomenological curve down to Mn/Mp w 0.7. This calculation wasperformed with Wilson fermions on a 163 x 48 lattice at 6/<?Q = 5.85. At lattice89, Yoshie, Iwasaki and Sakai (1990) presented new data with increased statistics,including a calculation on a 243 x 60 lattice. They are shown in Fig. (17-15).

Fig. 17-14 Plot of MN/Mp versus M^/Mp in quenched QCD.The solid (open) points correspond to calculations performed with Wilson(Kogut-Susskind) fermions. The figure, taken from Yoshie, Iwasaki and Sakai(1990), summarizes the results reported prior to, or at the Capri lattice con-ference.

The data are seen to be consistent with the phenomenological curve down toMn/Mp = 0.52. One does however not expect that the quenched data (once it hassettled down) follows this curve all the way down to the physical mass ratios. Atsome point the effects of light dynamical quarks must start to show up.

* see the plenary talk by Gupta at lattice 89 (Gupta, 1990).


Fig . 17-15 MC data of Yoshie, Iwasaki and Sakai (1990) for MN/Mp

versus M^jMp.

We have only briefly discussed a few earlier measurements in quenched hadronspectroscopy. As we have pointed out, the calculation of the hadron spectrumis a difficult task, even in the quenched approximation. But even with precisenumerical data available, one has no way of comparing it with experimental values,since we don't know what the result of a quenched calculation should be. Onlywhen the effects of dynamical fermion are included, will we be able to answerthe question whether QCD has survived this important test for being the correcttheory of strong interactions. So let us take a look at some data that has beenobtained including dynamical quarks. We will again restrict ourselves to earlierpioneering work, and leave it to the reader to confer the numerous proceedings formore recent data.

Hadrons in Full QCD

The quenched calculation of the hadron spectrum is already very time consum-ing since one needs to evaluate the quark propagators on the ensemble of link con-figurations distributed according to exp(—Sa[U}). In full QCD, however, one mustgenerate these configurations with the probability density det K[U] exp(-SG[£/]).Because of the appearance of the fermionic determinant, the times required fornumerical simulations of the hadron spectrum in full QCD are very (!) muchlarger than in the quenched theory. For this reason the lattice volumes used aremuch smaller than in the quenched case. To avoid strong finite size effects one istherefore forced to work with lattice spacings which are much larger than thoseused in quenched simulations. By 1990 one was still far away from being able tocompute numbers which can be confronted with experiment. Most of the calcula-


tions performed have served to test the various algorithms for handling dynamicalfermions and to get a qualitative idea of the effects arising from the presence ofquark loops.

Fig. 17-16 (a) The pion (b) and p correlation functions obtained byFukugita et al. (1988) in full QCD with Wilson fermions, for various valuesof the hopping parameter.

Calculations of the hadron mass spectrum in full QCD have been performedwith Wilson and Kogut Susskind fermions. The most widely used algorithms forgenerating the link configurations have been the pseudofermion, Langevin, hybridmolecular dynamics and hybrid Monte Carlo (see chapter 16). Of these algorithmsonly the hybrid Monte Carlo is free of any systematic step size errors. As wehave seen in chapter 16, this is achieved by subjecting a suggested configurationgenerated in a microcanonical step to a Metropolis acceptance/rejection test whicheliminates the systematic errors introduced by the finite time step. There aremany technical details that need to be discussed when simulating full QCD. Weshall not discuss them here, and refer the reader to the proceedings of latticeconferences, and the literature cited there. Fig. (17-16), taken from Fukugita et.al. (1988), gives a nice example of the form of the correlation functions extractedin a numerical simulation with Wilson fermions. The correlation functions werecomputed on a 93 x 36 lattice using the Langevin algorithm and show a clearexponential decay at larger euclidean times. What concerns the Edinburgh plot,Fig. (17-17), taken from Laermann et al. (1990), exhibits the results obtained


in some recent calculations. The dark points are those of Learmann et. al. Thetriangles, squares and inverted triangles are the results of Gottlieb et al. (1988),Hamber (1989), and Gupta et al. (1989), respectively.

Fig. 17-17 Edinburgh plot showing the data of Gottlieb et al. (1988)

(triangles), Hamber (1989) (squares), Gupta et al. (1989) (inverted trian-

gles), and of Learmann et al. (1990) (dark points). The figure has been

taken from Learmann et al. (1990)

Notice that this plot includes several measurements of the hadron ratios whichare much closer to the experimental values (denoted by the cross) than in thequenched case. Clearly the data is still much too inacurate to allow one to esti-mate the effects of the dynamical quarks. So far there had been no definite signsthat quark loops modify the quenched results in a significant way (see also Bitaret al., 1990; Campostrini et al., 1990). Even today the problem of determiningthe hadron mass spectrum from lattice calculations in full QCD remains a verychallenging problem. Nevertheless, there has been substantial progress made, espe-cially because of the available computer power. The effects of dynamical fermionsappear to be small. But the quark masses used in the simulations are still toolarge.

17.6 Instantons

There is a general consensus among physicists that QCD accounts for quarkconfinement. Although there exists no analytic proof of this assertion, latticesimulations have demonstrated in a quite convincing manner that QCD confines


quarks, and that the relevant degrees are to be identified within the pure Yang-Mills sector. The role played by dynamical quarks seems to be mainly that ofallowing for string breaking, once the creation of a quark-antiquark pair is ener-getically favored at some large enough separation of the quarks, where hadroniza-tion sets in. What concerns the mechanism for quark confinement, however, thisis a problem which is still under intensive study. It is generally believed thatthe mechanism is to be found in special Yang-Mills field configurations populat-ing the QCD vacuum. Candidates for non-trivial field configurations which havebeen studied intensively in the past years are instantons, and finite temperatureversions thereof, the calorons, as well as abelian magnetic monopoles and centervortices. Although the relevance (if any) of instantons for quark confinement isquite unclear, there is no doubt that they play an important role in determiningthe structure of the QCD vacuum. Furthermore instantons provide a natural ex-panation for the observed chiral symmetry breaking associated with confinement,and also provide a solution of the so called [/(l)-problem.* Because of this itappears plausible that they may, in one way or another, play some role in thedynamics of quark confinement.

The instanton is a non-perturbative solution to the classical euclidean SU(2)Yang-Mills equations of motion, carrying one unit of topological charge, and thuscorresponds to a special field configuration associated with a subgroup of 5f/(3).In fact this subgroup plays a distinguished role, as emphasized e.g. by Shifman(1999). The instanton solution was constructed by Belavin, Polyakov, Schwarz andTyupkin (Belavin 1975), and is refered to in the literature as the BPST instanton

Consider the SU(2) euclidean Yang-Mills action,**

SB = \ f d4x F^(x)F^(x)

= -Tr I d4x F^F^ > 0 ,

where F^ is the matrix valued SU(2) field strength tensor defined as in (6.13).Consider further the following integral,

i± = \l^x (F^ ± F;V){F;V ± F;J , (17.17)

* For more detailed discussions of instantons we refer the reader to Coleman(1985), Polyakov (1987) and Shifman (1999).** Sums over repeated indices are always understood. For SU(2) the "color"

indices a run from 1 to 3.

(17.16)


where F£v is the dual field strength tensor, F^v = ê^xpF^. Clearly

I± > 0 . (17.18)

Since F*VF«V = F^F^, it follows that

o 2

SE = ^^-Q + I±, (17.19a)92

where

Q = 3^2 / d'x *%>*%> =w^Jd4x Tr(F^p^ • (17-1%)

Since I± > 0 and SE > 0, we have that

s E > ^ r l Q I • (17-2o)

What makes this inequality interesting is that Q is a topologial invariant takinginteger values 0, ±1,±2, • • • for all gauge fields with finite action. Finiteness of(17.16) requires that the integrand vanishes faster than l/|a;|4 for \x\ -> oo, orthat the leading contribution to A^x) is pure gauge (cf. (6.20)),

A^x) —»• --G{x)dltG-l{x) . (17.21)|x|->oo g

In the case of 5(7(2), the unitary unimodular group elements G(x) have the form

G{x) = a4(x) + id{x) -a ; a\ + a2 = 1 , (17.22)

where <Ti are the Pauli matrices. Field configurations satisfying (17.21) can beclassified by a topological invariant taking positive and negative integer values, aswe will comment in more detail below. Configurations characterized by differentvalues of Q are not homotopic to each other, i.e. they cannot be continuouslydeformed into each other without violating the finiteness of the action. In eachsector characterized by a given integer Q = n, the action satisfies

5 B > ^ r | n | . (17.23)

As follows from (17.18) the lower bound in (17.23) corresponds to self dual, oranti-selfdual field configurations,

F;u{x) = F^(x), (selfdual) ,


F£v(x) = -F^v{x). (anti - self dual) . (17.24)

Since the action (17.16) is non-negative, it follows from (17.19a) that for self-dualsolutions (i.e. /_ = 0) Q is positive. For Q = 1 the solution is referred to in theliterature as an instanton. On the other hand for I+ = 0, Q must be negative.The corresponding Q = - 1 solution is referred to as an anti-instanton.

Before constructing the so called BPST instanton let us first demonstratewhy (17.19b) is an integer. There are different ways of showing this. The waywe shall proceed, will at the same time shed light on the relevance of instantonconfigurations for the vacuum structure of the Yang-Mills theory.

Consider Q defined in (17.19b). One first shows that the integrand can bewritten as a divergence:*

TriF^F^) = d^K^x) , (17.25a)

whereK^ = 2e^XpTr \AudxAp + iÂÂxA^ . (17.256)

From here it follows that, if K^ is regular within the domain of integration, then

Q = ~^ J 3 d3a n^K^x) , (17.26)

where dza is the 3-dimensional surface element of a sphere in four dimensions withradius R—>oo and normal n^{x). Now (17.25a) is gauge invariant. A particularityuseful gauge for discussing the vacuum structure of the Yang-Mills theory is the"temporal" gauge ^4|(a;) — 0. The reason is, that in this gauge the Yang-MillsHamiltonian takes a form quite similar to that familiar from Quantum Mechanics:

H = Jd3x [ivrX + \F^] , (17.27)

* This can be shown by introducing the definition of FMJ/ given in (6.13) into(17.19b), and making use of the antisymmetry of the e-tensor, and of the invarianceof the trace under cyclic permutations. Making further use of

e^xpTrÂÂxApj = e^XpTr Âv{dliAx)Apj = e^^TrÂuA^dÂp)^

= ^(AUAXAP)

one then arrives at (17.25a)


where the canonical momenta TT?(X) = F4i(x) are subject to the Gauss law con-straint,

Vab • *b = 0 . (17.28)

Here V is the covariant derivative for SU(2),

{Vt)ab = Sahdi + igtabcAt . (17.29)

Zero energy configurations (i.e., vacuum configurations) correspond to time inde-pendent gauge transformations of the trivial vacuum configuration Ab^ = 0, or inmatrix form

Aîx) = --G{x)diG-l{x) . (17.30)

In the temporal gauge these are the type of configurations which are approachedfor infinite times, ensuring the finiteness of the energy.

Consider now once more the topological charge (17.26), where S^ is nowreplaced by the three dimensional surface of a box in four dimensions. In the"temporal" gauge A\ = 0, the integrals over the surfaces with normal along thespacial directions will not contribute because of the e- tensor in (17.25b) and thetime independence of the asymptotic (vacuum) fields. Only the surface integralswith positive and negative oriented normals along the euclidean time direction willcontribute, so that

Q = q+ - q. = - | - 2 { [ d3x K4(x) - [ d3x K4(x)\ . (17.31)

For field configurations approaching vacuuum configurations (17.30) for x4 —¥ ±ooone has that

Q± = 1 & / dZaKi{x)

Ifor Jx^±oo ( 1 7 3 2 )= 24^Jd3x ^TrYG±diGll){G±djGl1){G±dkG-±

1)] .

This can be readily shown. Thus consider, more generally, (17.25b) evaluated ona vacuum configuration (17.21):

g2K^ = -2eM1,Aprr[(G^G-1)9A(G9pG-1) + ^{Gd^-^GdxG-^GdpG-1)] .

(17.33)Now

dxiGdpG-1) = (dxGWpG-1) + GdxdpG'1 .


Because of the e-tensor in (17.33) the second term does not contribute. Writing

(dxGWpG-1) = (d^G^GidpG-1)

and making use of the identity

0 = dxiGG-1) = {dxG^-1 + G(dxG~l)

one obtains

92K» = ^xpTr^GdvG-îGdxG-'KGdpG-1)} (17.34)

Setting n = 4 we thus arrive at (17.32). If the group elements G{x) are identifiedat spacial infinity, then (17.32) is an integer (the so called winding number").The reason is that in this case, the three dimensional space becomes topologicallyequivalent to the three dimensional surface of a sphere, i.e. to S3. On the otherhand, the group elements G(x) are parametrized by (17.22). Hence G{x) definesa map S3 —>• S3. Homotopy theory tells us that such maps can be classified by atopological invariant taking on integer values n — 0, ±1 , ±2, • • •, where \n\ gives thenumber of times S3 in group space is covered as we sweep over S3 in coordinatespace. The corresponding expression is given by (17.32).* A Q = 1 instantonwill therefore connect two vacuum configurations at X4 — ±00 carrying differentwinding numbers, which cannot be deformed continuously into each another. Forthis reason the instanton is interpreted as describing tunneling between classicalinequivalent vacua. In a semiclassical approximation, the ground state of thequantum theory is then expected to be a linear combination of the infinite numberof possible vacua carrying arbitrary winding numbers.**

Let us now briefly discuss the construction of the BPST instanton. *** Thetemporal gauge is not the most convenient one for constructing the self dual solu-tion. Thus it is easier not to fix the gauge and to return to the expression (17.26)for the topological charge, which is written as a surface integral over S3. Insertingthe asymptotic behaviour for the potentials (17.21), and making use of (17.34), Qtakes the form

Q = 2 4 ^ / s d V r iMÂprr{(G^G-1)(GaAG-1)(G'5pG-1)} . (17.35)

* We refer the reader to the literature quoted earlier for more details.** For a discussion of the connection between instantons and tunneling in

Minkowsky space the reader may confer the work of Bitar and Chang (Bitar 1978).*** For further details see Belavin et. al (1975), Shifman (1999), Actor (1979).


As we have just learned this quantity is an integer. The BPST instanton corre-sponds to a group element G{x) which maps compactified R4 to the group element(17.22) in a one-to-one way:

. x4 + ix -a .G{x) = — (17.36)

where \x\ — ^Jx^x^. Thus aM in (17.22) is identified with x^. From (17.21) and

the definition A^ = YLb tfiTv o n e ^ e n ^n(^s that for \x\ —>• 00

2 x•"•fiX^) ^ ~'llativ 2 ' (17.o7)

9 %

where the r\ailv are the so called 't Hooft symbols,

£a/j,v ; A4 = v = 1 ) 2 , 3

lap, = Sau I » = 4 ( ^ 3 g )

SafM ; v = 4

0 ; fj, = J/ = 4

Having constructed the asymptotic form for the gauge potentials, we now makethe following Ansatz for the Q = 1 instanton:

A%{x) = -f(x2)Tja^ (17.39)g x

with the condition that f(x2) -> 1 for |x| -»• 00, and /(x2) « x2 for \x\ ->• 0. Thelatter condition insures that the field is non singular at the origen. The self dual(BPST) solution is found to be

The corresponding field strength reads

F^=--g^{{x-S+P^> {i7Ai)

and the topological density is given by

6 o4

Q{x) = -^T, ^ T.r . (17.42)7T2 ((X - X0)

2 + p2)4 V y

(17.40)


Note that the solution (17.40) couples space-time and SU(2) colour indizes. Toactually see that the instanton solution connects different vacuum configurationsof the Hamiltonian (17.27) carrying different winding numbers, one must of coursemake a gauge transformation to the temporal gauge.

Multi-instanton configurations in the A% = 0 gauge with topological numberlarger than one connect vacua at X4 —>• ±00 whose winding number differ bymore than one unit. These vacuum configurations can be easily constructed (seee.g. Coleman, 1985). Approximate solutions corresponding to widely separatedinstantons, or instanton-antiinstantons, can of course be written down immediatelyas a linear superposition.

This completes our brief summary of the main concepts related to instantonsin the continuum formulation. One obvious question now arises: can we "see"these instanton configurations on the lattice, and if so: what is their size dis-tribution, and their possible relevance for confinement. To see the instantons isnot so easy, since in a Monte Carlo simulation their structure will be blurred byquantum fluctuations. Having generated a set of link-variable configurations wemust therefore strip off these fluctuations to see the underlying classical structure.This is called Cooling (Berg, 1981; Teper, 1986a; Ingelfritz, 1986). The idea isthe following: with a given configuration {U^{x)j generated by a MC algorithmthere is an associated euclidean action. In the continuum this field configurationwould belong to a given topological sector, i.e, it would be characterized by a giveninteger, e.g. Q = 1. It would however not in general correspond to a minimumof the action, but rather be a field configuration which is a deformation of anunderlying instanton configuration. To actually see this underlying instanton onemust smoothen the short range quantum fluctuations, while keeping the long rangephysics unchanged. A way to proceed is to lower the action of a configuration ina systematic way until it takes the value $\-. Once the action is minimized in theQ — 1 sector we are left in principle with a stable pure instanton configuration.In praxis this is of course not true, since we have discretized the action, and theabove continuum arguments do not really apply. Because of lattice artefacts, lat-tice instantons will never be true instantons. In particular lattice instantons breakin general scale invariance, i.e., the action depends on the size p of the instanton(see eq. (17.41)). The lattice manifests itself in that by cooling the configurationstoo long, the system will eventually end up in the trivial vacuum, i.e., the instan-ton has disappeared. One therefore needs to cool the system the right amount tosee the instanton. The situation can be improved by working with so called " im-proved actions" where lattice artefacts have been eliminated up to higher orders


in the lattice spacing. An important criterium for the identification of a Q — 1instanton is that the topological charge of the cooled configuration should be unityup to lattice artefacts, and that the charge density and action density should beproportional with a proportionality factor given by f~j. Multiinstanton config-urations are identified by fitting the topological charge and action density to asuperposition of instantons and anti-instantons.

How is the cooling performed in praxis? Suppose we have generated a par-ticular configuration {U^x)}. Let SE[U] be the corresponding euclidean action.Consider the variation of a single link SU^x) such that the action is lowered. InSU(2) a link variable U is parametrized as in (17.22). The action SE is given by asum over plaquettes. A particular link variable U appears in all plaquettes havingthis link in comon. Hence the action (6.18a) is of the form

S E = - / 3 T r ( U W ) + ••• , (17.43a)

where 0 = | £ for SU(N),U = 0,4 +iff-a , (17.436)

and where the "dots" stand for contributions not containing U. W is a 2 x 2matrix consisting of a sum of unitary matrices (staples). Hence W is not unitary.For SU(2) we can however define a unitary matrix W by

W = . 1 W . (17.44)Vdet W

Then the action takes the form

S E = - P V d e t W T r ( U W ) + ••• . (17.45)

Keeping all link variables other than U fixed, and hence also W, we can minimizethe action by choosing

U = W'1 . (17.46)

since UW is an element of SU(2), whose trace is bounded from above by the traceof the unit matrix. One now repeats the procedure by starting from this newconfiguration and changes the value of another link variable such as to lower theaction even further. Sweeping in this way through the lattice one thus generatesa new configuration. The system can be further cooled by sweeping the lattice anumber of times. Each sweep correspond to a complete cooling step.

An expression for the topological charge density can be obtained e.g. bytransfering the expression Tr{FtlvFlll/) = \e,iVp\Tr{FIJiVFilv), in (17.19b), to the


lattice. A very natural definition was given by Peskin (1978). The most naiveexpression which reproduces (17.19b) in the continuum limit is given by

G = -3^2 E E ^"PTr{U^n)Uap{n)) .

With this definition, however, the charge density does not possess a definite parity.This deficiency can be easily corrected for, by symmetrizing the above expressionwith respect to fj, —> — /j,, v —> —u,, etc. (Di Vecchia, 1981)

1 ± 4

Q = -24,327r2E E W P ^ ( ^ W ^ W ) - (17-47)n {/J,uap}=±l

Fig. 17-18 Wilson loop involving two orthogonal planes, as introduced

by Peskin (1978)

Fig. (17-18) shows the corresponding Wilson loop in Peskins definition of thetopological charge. In praxis one takes for both, the action density and topologicalcharge density, improved expressions whose lattice artefacts start only in O(a6).This can be accomplished by constructing the lattice F^ operator from linearcombinations of (gauge invariant!) 1 x 1 , 1 x 2 and higher Wilson loop operators(de Forcrand, 1997; Garcia Perez, 1999). In fig. (17-19) we show an example of aQ = 1 configuration as generated by a Monte Carlo algorithm with an improvedaction, and the corresponding cooled configuation. For an instanton the actiondensity, after rescaling, should equal the topological charge density. The particularshape of the cooled configuration is a consequence of the periodic structure. *

* Actually there exists no self-dual configurations on a hypertorus. To allowfor a stable Q = 1 configuration one must impose twisted boundary condtions ('tHooft, 1981a). The reason that the Q = 1 instanton configuration neverthelessappears, is that its size is fairly small compared to the extension of the lattice,and hence is not very sensitive to the choice of boundary conditions.


Fig . 17-19 Two-dimensional slice of the action density (top) and topo-

logical charge density (bottom) on a 24 lattice for a Q = 1 (a) uncooled

and (b) cooled configuration after 500 improved cooling steps. The figure is

taken from Wantz (2003).

In fig. (17-20) we show the MC data for the profile of the action density (circles)and topological charge density (squares) of an instanton configuration as a functionof euclidean time, obtained by summing the action and charge density over thespacial lattice sites,

S(T) = ^ 5 ( f , r ) , (17.48a)X

Q(T) = ^Q(X,T), (17.486)X

Notice that the action density (17.48a), after rescaling, coincides with the topo-logical charge density (17.48b), as expected. The solid curve in fig. (a) is the(periodic) action and charge density for a Q = 1 instanton in the continuum. *

A similar plot for a two Instanton configuration is shown in fig. (17-21a). Theplot (b) in the same figure exhibits the integrated topological charge density, andrescaled action as a function of the number of cooling steps. Special problems are

* We are very grateful to I.O. Stamatescu for providing us with this and also thefollowing unpublished plots, extracted from the MC data of de Forcrand, Garcia-Perez and Stamatescu (de Forcrand 1997).


Fig. 17-20 MC data for the (periodic) action density (circles) and

topological charge density (squares) for a Q — 1 instanton, summed over

the spacial lattice sites, as a function of euclidean time. The solid curve is

the corresponding continuum expression for an instanton width of p = 4,

based on a periodic version of (17.42). The dashed curve is obtained directly

from (17.42).

Fig. 17-21 (a) Similar plot as in fig. 17-20 but for a two instanton

configuration. Fig. (b) gives the action and topological charge as a function

of the number of cooling steps. Action and charge are seen to agree after

150 steps. The figure was provided to us by 1.0. Stamatescu.

encountered when studying instanton-antiinstanton configurations on the lattice.Performing the same number of cooling steps as in the one instanton case destroysthese configurations. This is not surprising since they carry the topological chargeof the trivial vacuum configuration, whose action is lower than that correspondingto a superposition of an instanton and antiinstanton. The latter is a metastablestate which upon cooling the system too long will decay into the trivial vacuum.


This is exhibited beautifully in fig. (17-22). The plots have been extracted from

the data of Garcia-Perez et. al. (1999).

Fig. 17-22 Euclidean time profile of the topological charge density

(17.48b) of an instanton-antiinstanton configuration for different number of

cooling steps. The dashed and solid lines are fits with a periodic Ansatz

consisting of a superposition of an instanton and antiinstanton. After 300

cooling steps the instanton-antiinstanton pair has dissappeared (boxes).

While the r-profile of the charge density is fully consistent with an instanton-antiinstanton configuration up to 100 cooling steps, the topological charge densityQ{T) is seen to vanish for 300 cooling steps, indicating that one is left with atrivial vacuum configuration. To keep the configuration alive for a larger numberof cooling steps, Garcia-Perez et. al. (1999) have introduced a physical criterionwhich essentially defines a cut off for the number of cooling steps, depending onthe scale of fluctuations to be smoothened by cooling.

As we have already emphasized, instantons are an important ingredient forthe structure of the Yang-Mills vacuum, and are believed to be the driving mech-anism for chiral symmetry breaking*. But do instantons have anything to do withconfinement? The instanton (17.40) is a field configuration at zero temperaturewhere also confinement holds. In fact, as we shall see in chapter 20, Monte Carlosimulations show that quarks are actually confined up to high temperatures of theorder of 1012 degrees! As we shall also see in chapter 18, field theory at finitetemperature T is implemented in the path integral by integrating over the fields

* See e.g. the review by Schafer and Shuryak (Schafer, 1998)


on a space time manifold compactified in the euclidean time direction, with thebosonic (fermionic) fields satisfying periodic (anti-periodic) boundary conditions.Field configurations carrying a topological charge, and corresponding to self oranti-selfdual solutions to the euclidean SU{2) Yang-Mills equations can also beconstructed at finite temperature. These are the so called Calorons. The sim-plest caloron, obtained as a classical periodic solution (in euclidean time) to theequations of motion, was first discussed by Harrington and Shepard (Harrington,1978). New caloron solutions (corresponding to modified boundary conditions —>•"non-trivial holonomy") have been constructed (Lee, 1998; Kraan, 1998). An im-portant ingredient in the construction is the Nahm transformation (Nahm, 1984)and the Atiyah-Drinfeld-Hitchin-Manin (Atiyah, 1978) construction. **

Fig. 17-23 Two dimensional profile of the action density of a 5(7(2)

caloron at three different temperatures, increasing from left to right. The

figure is taken from Bruckmann et. al. (2003).

The discovery of these new calorons has given renewed impetus to the ideathat instantons may, after all, play some role in the understanding of confinement.The new calorons have been the subject of intensive numerical investigations inthe past years. Close to the phase transition, they are found to be much moreabundant in the confined than in the deconfined phase (Grattinger, 2004), and toexhibit a composite structure (Lee, 1998; Kraan, 1998). SU{N) calorons are madeup of N so called BPS-monopoles (Prasad, 1975), carrying fractional topologicalcharge. Conversely, monopoles can be looked upon as a periodic array of calorons(van Baal, 1998). In fig. (17-23) we show Monte Carlo data of Bruckmann et. al.(2003) for the action density of an SU{2) caloron at three different temperaturesT = 1/J3. Thus the QCD vacuum appears to have a very rich structure, beingpopulated by various types of topological excitations: instantons, monopoles, and

** See van Baal (1999) for a summary.


- as we will see in section 17.9 - vortices. The question whether instantons playat all a role for confinement is still waiting for a definite answer. This is all wewill say about calorons. The details are fairly complex. The interest reader shouldconfer the cited literature, and conference proceedings.

17.7 Flux Tubes in qq and qqq-Systems

In section 17.1 we have seen that Monte Carlo simulations of the pure SU(2)and SU(3) gauge theories confirm the long standing expectation that the qq-potential rises linearly with the separation of the quarks for large separations.For small separations, asymptotic freedom predicts that the colour electric andmagnetic fields should spread out in space in a similar way as for a dipole fieldin QED, leading to a Coulomb-like potential with a logarithmic dependence ofthe coupling constant on the qq separation (see chapter 9). The absence of freequarks, and the fact that meson resonances lie approximately on Regge trajecto-ries, i.e., that there exists a linear relation between the angular momentum andthe mass- squared, can be explained if one assumes that a quark-antiquark pairis connnected by a string (Goddard, 1973), with a constant string tension a thatis related to the slope of the Regge trajectory a' by I/a' = 2ira. This suggeststhat the non-perturbative dynamics at large distances squeezes the chromoelectricand magnetic fields into narrow flux tubes connecting the quark-antiquark pair.Whether this is indeed the case can in principle be checked by studying the dis-tribution of the field energy in a Monte Carlo simulation. Some early analyticalwork on the flux tube behaviour has been carried out by Liischer, Miinster andWeisz (Liischer, 1981), and Adler (1983).

On the lattice the flux tube problem has been first studied by Fukugita andNiuya (1983), followed by Flower and Otto (1985), Sommer (1987) and by Wosiekand Haymaker (1987). Since then the numerical data has improved substantiallyand there are now good indications that a flux tube is indeed formed for largeseparations of the quark-antiquark pair. To study the evolution of the flux tube asthe qq separation is increased is a major challenge, since it requires large lattices,and special techniques for reducing fluctuations, and enhancing the projection ontothe ground state. Because of limited computer power the earlier calculations havebeen restricted to rather small lattices. The energy (and action) density profilesfor SU(2) that were obtained e.g. by Haymaker et al. (1991, 1992) are consistentwith those obtained more recently by Bali et al. (1995), who have studied, inparticular, the distribution of the action density for quark-antiquark separations upto about 1.9 fm. The electric and magnetic contributions to the action and energy


densities are determined by studying correlators of electric (space-time) plaquettes,and magnetic (space-space) plaquettes with a Wilson loop. The correlators havethe generic form (10.24b). All authors find that the action density along the axisconnecting the quark-antiquark pair is much larger than the energy density, andthat they fall off fairly rapidly in the transverse direction. Fig. (17-24) showsthis fall-off in a plane perpendicular to the axis of the gg-pair at the midpoint, asmeasured by Haymaker et al. (1992).*

Fig. 17-24 (a) Energy density and (b) action density distribution in aplane midway between the qq-pair (Haymaker et al, 1992)

In fig. (17-25) we show, as an example, the distribution of the energy andaction density between a qq-pair, obtained respectively by Haymaker et al (1992)and Bali et al. (1995). The peaks are the self energy contributions of the quarkand antiquark.

In contrast to the gg-potential, there have been only a few investigations of thethree quark potential before 1999 (Sommer, 1984; Flower, 1986; Thacker, 1888).Possible flux-tube configurations that have been envisaged for the three quarksystem are the A and Y-type flux tube configurations. The Y-tube configurationis illustrated in diagram (b) of fig. (7-1). Within the string picture the classicalground state of the three quarks is envisaged to be given by three strings ema-nating from the quarks and meeting at a point whose position corresponds to theminmimal length of the three strings. Thus one may expect to "see" a Y-type fluxtube configuration in a Monte Carlo simulation.

As in the case of the qq-potential, the S^-potential can be extracted by study-ing the propagation in euclidean time of the corresponding gauge invariant 3-quarkstate

\qqq >= W2a3 U?* U^ Uftb 'qtftfi, 0)qif(x2, O)qg\£3, 0)|0 > , (17.49)

* See also Haymaker et al. (1996).


Fig. 17-25 (a) Energy density distribution between a qq pair, obtained

by Haymaker et al. (1992); (b) Action density distribution, measured in units

of the string tension, for a qq pair separated by about 1.35fm, at /? = 2.5

(Bali et al. (1995))

where the unitary matrices Urt are the path ordered product of link variablesalong the paths F^, i = 1, 2,3 shown in fig. (17-26), connecting the quarks to the"center" £ of the triangle with corners x\, x2 and 2J3, for which the sum of therespective distances is a minimum.

Fig. 17-26 The 3-quark state at r = 0. The heavy dots denote the

quarks, and the lines stand for the ordered product of link variables which

are tied together at the "center" by the e-tensor.

Specifically, one considers the following correlator*

G(X1,X2, X3; 1/1, £/2, 2/3) = falO2O3eb1b2b3

< 0|«£>WqgMqg(V3)U*}* (yu Y)U*?b>(y2, Y)U*f>(y3, Y)

x U^{X,xz)U^\X,x2)U^{X,Xl)qf}{x3)q^{^)q^{x^ > .(17.50)

* This the lattice analog of the continuum Green function studied by Brambillaet. al. (1995)


Here the 4-vectors x; (i = 1,2,3) and X have vanishing euclidean time components.

Furthermore, yt = £; with the time components of j/j and Y all equal T. Y't are

the paths obtained by a translation of the paths 1^ in the euclidean time direction

by T. By proceeding in a similar way as in the case of the QQ-potential, one is

then led to the following expression for the three quark potential

V3Q{XU X2, X3) = - lim — In < W3Q > , (17.51a)1 —K5O ±

where the gauge invariant 3-quark Wilson loop operator is given by

W3Q = yeabcea.blc,Uf?'ugug (17.51a)

Here Ura, Urb and Urc are the path ordered product of the link variables along

the paths shown in fig. (17-27).

i •*

Fig. 17-27 The 3Q Wilson loop operator. A 3Q state created at time

T = 0 propagates to T = T, where it is annihilated.

The extraction of the potential via (17.51a) requires the evaluation of <

W3Q > for large euclidean times T. This poses of course the usual problems.

Since the signal is suppressed exponentially with T, it is important to enhance the

projection onto the ground state using a smearing technique, as described in sec.

17.4. Although the question whether the flux tube structure is of the A or Y-type

is not yet settled, newer data obtained by Takahashi et. al. (2001-2003), and by

Ichie et. al. (2003) support the Y-type flux tube picture.

In fig. (17-28) we show the action density in the presence of three quarks,

obtained by Takahashi et. al. in a MC simulation performed at ft = 6.0 in

quenched QCD. The potential was fitted to the following conjectured Y-ansatz

with a deviation of only 1%:

V3q{rl, r2, r3) = -A3Q ^ — — + a3QLmin + C3Q , (17.52)


Fig. 17-28 Action density in the presence of 3 quarks, measured in a

MC simulation on a 163 X 32 lattice at /? = 6.0 for SU(3). The figure is

taken from Takahashi et al. (2004)

where Lmin is the minimum length of the 3 strings. The authors also find that&3Q ~ o, where a is the two-body string tension, and that AJ,Q « \AQQ- A recentcomputation of the three quark potential in full QCD has been performed by Ichieet. al. (2003) in the "maximal abelian gauge" (see next section), and a similarflux tube profile was obtained.

17.8 The Dual Superconductor Picture of Confinement

Having obtained good indications that a flux tube is formed as the gg-separationis increased, the next question one would like to have an answer to, concerns thedynamics responsible for the formation of the flux tube. It has been suggested along time ago by Nielsen and Olesen (1973), and by Kogut and Susskind (1974)that confinement could be explained in a natural way if the QCD vacuum reactedto the application of a colour electric field, due to a quark-antiquark pair, in muchthe same way as a superconductor reacts to the application of a magnetic field.This could be achieved by adding to the gauge field an elementary charged scalar(Higgs) field, which has however not been detected so far.* The dual supercon-ductor mechanism of 't Hooft (1976) and Mandelstam (1976) does not require theintroduction of such a field, but assumes that dynamically generated topologicalexcitations provide the persistent screening currents. Consider first a type I su-perconductor. Its ground state corresponds to a condensation of Bose particles

* The Higgs theory is the four-dimensional generalization of the Ginzburg-Landau theory.


(Cooper pairs). When an external magnetic field is applied, these Cooper pairsorganize themselves to form persistent currents that expell the magnetic field,which is only allowed to penetrate the superconducting material a distance givenby the London penetration depth (-> Meissner-Ochsenfeld effect). This is the caseif the applied magnetic field does not exceed a critical value. Beyond this criticalvalue superconductivity breaks down. In the superconducting state the currentsonly exist in a thin surface layer determined by the London penetration depth.A quantitative description of the relation between the surface currents and mag-netic field is given by the London equations, * which in a stationary state, can besummarized in the Coulomb gauge by the following relation between the vectorpotential and the current density,

A = ~ — J, (17.53)

where A = m*/n*e*2, with m*, n* and e* the mass, density and charge of theCooper pairs. By taking the curl of (17.53), assuming A to be constant, oneobtains **

V x J + -^-B = 0 . (17.54)A

Combining this equation with Ampere's law, V x B = J/c leads to the equa-

tion V2B = j?B, which implies that the magnetic field decays in the interior of

the superconductor with a skin depth A (London penetration depth). Since the

Cooper-pair density is a function of the temperature, the same applies to A. As one

approaches the critical temperature A increases strongly, and the magnetic field

begins to penetrate more and more the superconductor. Clearly a superconductor

of type I can have no analog in the ground state of a Yang-Mills theory with the

colour electric field of a quark-antiquark pair squeezed into a narrow tube. In fact,

* The London equations relating the electric, magnetic fields and the current

in a type I superconductor have the form

** Our presentation is only qualitative. For a comprehensive discussion see e.g.the book by Tinkham (1975).


such property of the ground state (if true) suggests that it might actually be thedual analog of a superconductor of type II. A type II superconductor also exhibitsa Meissner phase below a critical field strength BCl. The superconductivity ishowever not destroyed as the applied field is increased beyond this critical value.Instead a new phase appears (Shubnikov phase), in which the material is dividedinto normal and superconducting regions. Magnetic flux can now penetrate thematerial (mixed state), but only in narrow flux tubes (—> Abrikosov flux tubes)whose separation decreases with increasing external field. The condition for this tohappen (which implies that the new state is favored energetically) is determined bythe ratio of the London penetration length A to a new scale, the coherence length £.The coherence length is a measure of how strongly the Cooper-pair density variesin space, and has been introduced as a new parameter by Ginsburg and Landau,whose theory is an extension of the London theory. Abrikosov (1957) found asolution to these equations in which the flux tubes are arranged in a regular ar-ray, with each tube carrying one unit of flux $o = hc/e*. The energetically mostfavorable configuration turns out to be a triangular array. The relevant quantitywhich characterizes a superconductor of type I or II is

A

We have the following simple criterion:

« < "y= I (type I)

K>V2 '' {TyPB H)

For K close to l / \ /2 the system is in a mixed state consisting of the Meissnerand Shubnikov phase. In the Shubnikov phase, the flux tubes are trapped bycirculating persistent currents. This phase persists up to a critical magnetic field.BC2 > BCl. Away from the superconducting-normal boundary, between the fluxtubes, equation (17.54) again holds. This equation can be modified to take intoaccount the presence of the core (Tinkham, 1975). If C is a closed curve encyclingthe flux tube, then

f A2

/ dS-(B+ — V x j ) = n$ 0 , (17.55)Js c

Note the appearance of Planck's constant. Indeed, the magnetic flux within thevortices, i.e., $ = § ds • A must be a multiple of $ 0 in order that the wave function


of the Cooper pairs be single valued. For an extreme type II superconductor theregion of normal material consists of very thin filaments. The solution of theGinzburg-Landau equations found be Abrikosov corresponds to one unit of fluxconcentrated in each of the vortices. Hence in the limit where the radius of theflux tube (taken along the z-direction) goes to zero one has that

A2

B+—V x J = <f>05(xT)ez . (17.56)

By analogy, if the QCD vacuum behaved like the dual version of a type II super-conductor, one expects that the ground state consists of colour magnetic chargeswhich in the presence of a quark-antiquark pair would organize themselves intopersisting circulating magnetic currents which confine the colour- electric flux intonarrow (dual Abrikosov)-flux tubes, 't Hooft (1981) has conjectured that the rele-vant degrees of freedom responsible for the confinement are actually U(l) degreesof freedom defined in the so called " maximal abelian gauge", and that the con-densate in non-abelian gauge theories consists of U(l) Dirac monopoles. Since theU(l) lattice gauge theory is also known to confine charges for strong coupling, it isof interest to first check the above picture for the confinement mechanism in thistheory. If the dual superconductor picture is correct, then for strong coupling theground state should be strongly populated by Dirac magnetic monopoles. As thecoupling is reduced this theory is known to undergo a transition to the Coulombphase. In this phase the ground state should no longer show a condensation ofmagnetic charges. This has indeed been confirmed in numerical simulations byDeGrand and Toussaint (1980). These authors showed that on the lattice thereare objects that can be naturally identified with Dirac monopoles. If a monopoleis located inside an elementary spatial cube on the lattice, then the enclosed mag-netic charge can be determined by measuring the total magnetic flux through thesurface of this cube. The magnetic flux through the surface of a plaquette lyingin the ij-plane is directly related to the phase of the plaquette variable

UPii{n) = eiea2Fi^n) = ei<t>ij{n) ,

where Ftj is the component of the magnetic field in the direction perpendicularto the y-plane. The measurement of this flux however involves a subtle point,resulting from the fact that Upi:j{n) is a periodic function of 4>ij{n)- As we haveseen in chapter 5 the plaquette variable is given in the U(l) gauge theory by theproduct of the oriented link variables around the boundary of the plaquette. Alink variable associated with a link pointing in the /^-direction, with base located


at the lattice site n, is given by f/M(n) = ei4>»{n\ where -TT < <?!>M(n) < n. Hence

the product of such variables around the boundary of a plaquette lying in the

/Z7/-plane is

UP^{n) = e^" , (17.57a)

where

-4TT < (j)^ < 4?r . (17.576)

For small angles fy^v we can identify the space-space components of this quantity

with ex (the magnetic flux through the surface of the plaquette). On the other

hand, for large angles, the physical flux should be identified with 0M,, (mod 2nn),

since the plaquette value remains unchanged by shifting </>Ml, by a multiple of 2TT.

The electric flux in the i'th direction through an elementary plaquette can be

determined in a similar way from the phase of the plaquette variable Upu.

Let us denote in the following the angle associated with a given plaquette P

simply by (f>p. We then decompose this angle as follows (DeGrand, 1980)

(j)P = $P + 2-nnp , (17.58a)

where

- 7 T < 4>P < TV , (17.586)

and np are integers. Because of (17.57b), we then have that np = 0, ±1, ±2. Now

it is clear that if we add up the plaquette angles 4>p of the six plaquettes bounding

an elementary cube we will get a vanishing result, since each link is common to

two plaquettes, and gives rise to the sum of two phases, equal in magnitude, but

of opposite sign. It therefore follows that the magnetic flux | ^ p 4>p through the

closed surface S bounding the elementary cube is given by

M=J2-tr = --Y,np> (17-59)res e e pes

where M is the magnetic charge enclosed by the surface. This charge is therefore

a multiple of ^f* If a magnetic monopole is located in an elementary volume,

then at least one of the plaquette angles must be larger in magnitude than 7r,

so that there is a Dirac string crossing the corresponding surface. By making a

"large" gauge transformation (e.g., a particular link variable is mapped out of

the principle domain [—n, TT}) a Dirac string can be moved around. But the net

* We set % = c = 1, as is appropriate to the lattice formulation.


number of such strings leaving the elementary volume will not be affected by thegauge transformation. Furthermore, it is clear, that the number of monopolescontained in a volume V is given by the sum of the monopole numbers of theelementary boxes making up the volume. The lattice simulations of DeGrand andToussaint (DeGrand, 1980) have shown that the majority of the configurationsin the confined phase of the U(\) theory correspond to pairs of monopoles andantimonopoles located in adjacent boxes. Such a pair corresponds to the situationwhere only the plaquette angle associated with the common face exceeds TT (inabsolute value).

The observation that monopoles are abundant in the confined phase of theU(l) lattice gauge theory, and scarce in the deconfined phase, fulfills the firstprerequisite for a test of the dual type II superconductor picture of confinement.* But if this picture is correct then one would also expect to "see" that when anoppositely charged pair is introduced into the vacuum these monopoles organizethemselves to form persistent currents which squeeze the electric field into a narrowAbrikosov flux tube connecting the pair. Let the two charges be located on thez-axis. Then the magnetic supercurrents are expected to satisfy the dual versionof the London equation for an Abrikosov vortex (we have set h — c = 1),

Ez - A2(V x j M ) z = n$0<5(£r) (17.60)

where $o is obtained from $o by replacing the electric charge of the Cooper pairby the monopole charge qm—^f- The fluxoid density given by the left hand sidevanishes everywhere except at the vortex. Consider a surface perpendicular tothe z-axis on which the two charges are located. If this axis passes through thesurface, then the electric flux through this surface is related to the circulation ofthe magnetic current around the boundary of the surface by

fdS-E-X2 [ d£-jM = n$0. (17.61)Js Jas

If the surface is an elementary plaquette, then the line integral is roughly given bythe z-component of the curl of the magnetic current multiplied by the area of thesurface.

For the C/(l) lattice gauge theory this equation has been first studied in detailby Singh, Haymaker and Browne (Singh, 1993a), where A has been considered to

* Di Giacomo et al. (2000) proposed a monopole creation operator, which servesas an order parameter for studying the deconfinement phase transition.


be a free parameter. These authors have chosen to define the i'th component ofthe electric field by a2Ei = ÎmUp4i, which takes account of the fact that thephysical flux is determined from the phase of the plaquette variable, modulo 2mr.To check (17.61) one also needs a lattice expression for the components of themagnetic current. They are defined (DeGrand, 1980) in terms of the dual fieldtensor F^ = ê^xPF\P by

• (mag) _ a p .

Consider for example the first component of the current. It can be written in theform

j(mag) = y . , (17 6 2)

where Mi = (F43, F2i, F32), and V = (d2, d3, d4). Integrating j[ma9) over the vol-ume of an elementary cube with edges along the 2, 3 and 4 directions, is equivalentto computing the flux of Mi through the surface of this cube. Hence to computethe three components of JM one needs to calculate the flux through the surfacesof 3 cubes, having one edge directed along the time-axis. This computation iscarried out in a completely analogous way as described before for the magneticflux. Hence by construction the three components of the current, when measuredin lattice units, will be multiples of ^ . To obtain V xj(mag) one performs the dis-crete line integral around the elementary square shown by the dotted lines in fig.(17-29), where the time direction of the 3-volumes determining the components ofthe currents has been suppressed.

Fig. 17-29 The four elementary plaquettes shown in solid lines repre-

sent the four 3-volumes from which the components of the magnetic current

are determined. The-time direction has been suppressed. The discrete line

integral is performed along the dashed lines bounding the elementary square

shown by the dotted lines. The fig. is taken from Singh et al. (1993b).

Singh et. al. (1993a) have measured the z-component of the electric field,and of the curl of the magnetic current, in the presence of a qq pair, by correlating


these quantities with a Wilson loop located in the z — r plane. The measurementswere carried out in a plane perpendicular to the z-axis located at the midpointbetween the two charges.

Fig. 17-30 Dependence of (a) the electric flux, (b) the curl of the

monopole current, and (c) of the fluxoid in the U(l) gauge theory at /3 =

0.95 as a function of the perpendicular distance from the axis of the qq pair.

The dashed line is the expected electrical flux obtained from the continuum

equation (17.28) and the dual version of Ampere's law: cV X E = —JM-

The fig. is taken from Singh et al. (1993a)

Fig. (17-30) shows the dependence of these quantities on the distance from theaxis. From their analysis the authors conclude that (17.60) is indeed satisfied witha London penetration depth, given for (3 = ^ = 0.95, by A = 0.482 ± 0.008 (inlattice units), and one unit of electric flux.

The above result is encouraging. But what one really wants to study is thecorresponding problem for the case of a non-abelian gauge theory. In the non-abelian case the situation is far less clear. As we have already mentioned at thebeginning of this section, it is believed that the relevant degrees of freedom areactually U(l) degrees of freedom. It has been suggested by 't Hooft (1981) thatthese U(l) degrees of freedom can be isolated by going to the so-called MaximalAbelian Gauge (MAG). On the lattice the MAG has been first discussed by Kro-nfeld et. al (1987). Consider for example the SU(2) gauge theory. An SU(2) link


variable can be written in the form

UJn) = V r— (17.63)V -r?M(n)e-^(") y'l - ^ ( n ) e " ^ W ^

with ?7M(n) > 0. Let {t//i(«)} be some given link-variable configuration. We wantto make a gauge transformation such that for all link variables the matrix is asdiagonal as possible in the mean. Let us denote the transformed parameters witha "tilde". The transformed variables are given by U^{n) = G{n)Ull{n)G~1{n +ft). These variables will of course again be of the form (17.63). Then the abovecondition is equivalent to maximizing the quantity

R = Y1 Tr[azilli{n)azUl{n)\ . (17.64)

The gauge transformation which maximizes this expression is not unique, since nostatement is made about the phases. Thus the value of (17.64) is left unchangedby gauge transformations generated by the group elements

/ P*7(n) n \G ( n)=( 0 e-"(n)j • (17-65)

We can therefore restrict ourselves to gauge transformations of the type

C/M(n) -> G W C / ^ n J G " 1 ^ + A) ,

where_ / v / l - « 2 ( n ) «(n)e«("> \

G[n) ~ \-n(n)e-iS^ y/l - K*(n) ) '

which is of the form G(n) = go{n)l + igi{ri)ai + ig2(n)<J2, with Ui the Paulimatrices. In the maximal abelian gauge the link variables take again the form(17.63), or equivalently

fr (n)-(Vl-\Un)\2 U») \ (u^n) 0 \

U^n)-\ -C»(n) y r p f i l 0 u;(n)) 'where u^n) = exp[i^>fi(n)]. Under the residual gauge transformations, inducedby (17.65), £M(n) and MM(n) transform as follows

£M(n) ->• e2'T(")^(n) , (17.66a)


u^n) -+ jiWupWe-^+M . (17.666)

Notice that the variables wM(n) transform like U(l) link variables under this gaugetransformations, while the transformation of fA(rc) is local. Having obtained thenew link-variables from a given configuration in the maximal abelian gauge, therelevant U(l) gauge degrees of freedom are identified with uM(n), which, as wehave just seen, transform in the desired way under the remaining U(l) gaugegroup. Monopoles are now identified by calculating the magnetic flux throughthe surface of a 3-volume bounded by plaquette variables constructed from thesevariables. A similar statement holds for the components of the magnetic current.These quantities are then correlated with a Wilson loop constructed from theabelian link variables.

Fig. 17-31 Profile of (a) the electric flux, (b) the curl of the monopole

current as a function of the perpendicular distance from the axis of the qq

pair at (3 = 2.4. The fig. is taken from Singh et al. (1993b)

Singh, Haymaker and Browne (Singh, 1993b) have measured the monopolecurrent and the electric field midway between an " quark-antiquark" pair and haveshown that the results are consistent with the dual Ginzburg-Landau model, whichis a generalization of the London theory that allows the magnitude of the conden-sate density to vary in space. The results for the flux and curl of the current areshown in fig. (17-31). These results have been confirmed by Matsubara et. al.(1994) with better statistics. These authors have also studied the same problemin the SU(3) gauge theory.

If monopoles defined in the MAG are the relevant degrees of freedom forconfinement, then they must also account for the string tension in the full non-abelian theory. For SU{2) this question has been first studied by Suzuki et. al.(1990, 1994).* A high precision MC measurement of the projected abelian string

* A more detailed analysis of isolating the monopole content in the light of the


tension in SU(2) was carried out by Bali et.al. (1996). The authors find that theabelian monopoles reproduce the full string tension within 5%. This is referred toin the literature as abelian dominance.

17.9 Center Vortices and Confinement

As we have just seen, Monte Carlo simulations suggest that the mechanismof confinement is the condensation of U(l) magnetic monopoles defined in theMaximal Abelian Gauge. The abelian degrees of freedom in the MAG seem tocontain almost all the information regarding the long distance properties of theYang-Mills theory relevant to confinement. It seems, however, that the relevantdegrees of freedom for confinement can be reduced even further. When studyingthe deconfinement phase transition at high temperatures (see chapter 20) one findsthat this transition goes along with a breakdown of the center symmetry.* Not onlythat: chiral symmetry (see sec. 17.3) is restored at a critical temperature which inMonte Carlo simulations is found to coincide with the deconfinement transition.Hence the relevant degrees of freedom may be just be the center elements of thegroup. In the perturbative regime the link variables are close to the unit element.The relevant group is therefore SU(3)/Z(3). On the other hand the center elementscorrespond to large fluctuations of the gauge fields of the order of l/ga.

A possible mechanism of confinement, which has been conjectured already along time ago ('t Hooft, 1979; Mack, 1979; Ambjorn, 1980), and which has beendiscussed intensively in the past years, is the condensation of center vortices.** Anexample of a vortex in three spacial dimensions is a closed thin tube of magneticflux. Such a vortex can be thought of as the magnetic field generated by a toroidalsolenoid, in the limit of vanishing cross section. What is characteristic of such anarrangement, is that the vector potential is pure gauge outside the torus and hasthe form

A(x) = -£-Vfi(C;£) , (17.67)

De Grand-Toussaint prescription, discussed earlier, has been carried out by Shibaand Suzuki (1994), and by Stack et. al. (1994).

* The action is invariant under local SU(Z) transformations, and also underthe multiplication of the link variables on a fixed time slice by a center elementof SU(3). As we will see in chapter 20 this center symmetry is broken in theconfined phase. For SU(2) and SU(3) the center elements are given by (1, -1 )and (exp(-27r«'/3)l, 1, exp(27™/3)l), respectively, where 1 is the unit matrix.

** For a recent review see e.g. Greensite, 2003.


where F is the magnetic flux in the torus, and fi(C; x) is the solid angle subtendedby the contour C of the infinitely thin torus at the observation point x. Thissolid angle is a multivalued function, which changes by 4n along a closed contourpiercing the surface E bounded by C. Thus the vortex (17.67) is introduced bymaking an a-periodic gauge transformation on the trivial vacuum configurationA(x) = 0:

A(x) = iG^^VGîx)-1 , (17.68a)

whereGW(x) = z ^ , (17.686)

withz = eiF . (17.68c)

If a loop C", parametrized by x(s) (0 < s < 1), winds through the hole of thetorus, then

GW(X(1)) = ZGW(X(0)) . (17.69)

In the above discussion ft is a multivalued function of x. Alternatively onecan define a function Q(x) which is regular and single valued everywhere excepton a surface E whose boundary is the loop C. As x crosses E, fi jumps by ±4?r.This function is given by

n(E-,x)= [ 4ffj(f). T £z |Lh I*"*! (17.70)

where df is the differential surface element, and n(x') is the unit normal to thesurface E at the point x!. When n(x') and x — x' form an acute angle, this isnothing but the standard integral representation of the solid angle subtended byC at x. This integral is discontinuous across the surface E, where it jumps by±4TT. * Now comes a subtle point. If one computes the rhs of (17.67) replacing Q.by £1, this will not yield a gauge potential whose curl is the toroidal magnetic field.In fact, in this case, the corresponding expression is gauge equivalent to the trivial

* As a simple example the reader can consider the case of a circular loop in thex-y plane, and an observation point lying on the z-axis. Writing the integrand of(17.70) in cylindrical coordinates one readily finds the following dependence of thesolid angle on z: £l(z) = -j==^-2ne(z), where e(z) = 1 for z > 0, and e(z) = - 1for z < 0.


configuration A = 0. Indeed, the discontinuity of fi(a?) can be smoothened acrossthe surface E. In this case the rhs of (17.67) with ft —> fi is a bonafide pure gaugeconfiguration, corresponding to a vanishing magnetic field everywhere. In order toobtain the toroidal magnetic field the gradient in (17.67) or (17.68a) should notoperate on the discontinuity at the surface E. The following discussion is basedon work of Engelhardt and Reinhardt (1999), who considered vortices within thecontinuum formulation of QCD.

The main goal is to isolate the contribution to Vfi arising from the abovementioned discontinuity. Consider

*n="l«l^' (17-?1)where n' = n(x'), and, as always, sums over repeated indices are undertood.Making the decomposition

d^ = S^V2 - (SijV2 - didj) ,

and use of the identity

SijV2 - didj = eikmejemdkde ,

as well as of

^j^-T, =-4TT5(X-X'),\x — x'\

one finds that

ditt = 4TT f dfriîx -x')+ [ df'n' • (V' x £ ) (17.72a)JE •/£

where

(Fi)m = -J^^rnkd'k^-^ . (17.726)k x-x\

Application of Stokes theorem leads to*

-^din = Ai(X;x)-ai{dY,,x) (17.73o)

where

A(E; x) = f df'n'iS(x - x") (17.736)

* This is the 3-dimensional analog of the decomposition carried out by Engel-hardt and Reinhard (1999) for the case of SU(N) gauge theories.


and

ai(dE,x)= dx'k eMd'tD{x - x1) , (17.73c)JdS

withD{x - x') = } (17.73d)

4n]x - x'\satisfying

V2D(x - x') = -6{x - x') . (17.73e)

Note that Ai(x) has only a support on E. It is the contribution arising fromthe discontinuity mentioned above. Accordingly, ignoring this contribution to(17.73a), the relevant vortex vector potential is given by

Avortex{x) = -FS{dH, £) , (17.74)

which is now determined from the boundary of E, i.e. the location of the vortex. Itis referred to in the literature at the thin vortex. Let us compute the line integralof the vortex field along a closed path C piercing the surface S bounded by Cn-times. A simple calculation yields

f dx • Avortex(x) = -FL(C, C) , (17.75a)Jc

where

w^hL^L^^ww (17-75b)is the Gaussian linking number. This is, of course, the expected result for a toroidalfield configuration. As one can also verify, the magnetic field computed from thecurl of (17.74) is localized on C.

The configuration (17.74) is actually gauge equivalent to the so called idealvortex field (17.73b), which only has non-vanishing support on the surface of dis-continuity (Engelhardt, 1999). This can be seen as follows. Consider a gaugetransformation generated by the inverse of the group element (17.68) with a singlevalued, but discontinuous, angular variable ii(x). As already mentioned, one canimagine the discontinuity across E to be smeared out over an small interval, sothat fi is single valued and differentiable everywhere. After having performed thedifferentiation (17.68a) this interval is taken to vanish. A (bonafide!) gauge trans-formation on (17.74) carried out with this group element transforms the vortexfield into the gauge equivalent potential A'vortex = —FA{Y,,x), which now onlyhas support on the surface of discontinuity. In this way one has eliminated thepotential everywhere in space, except at points located on this surface.


Let us now leave the continuum and turn to the lattice. As we have al-ready pointed out, it has been speculated for some time, that the relevant degreesof freedom for confinement in QCD may be Z(3) degrees of freedom, and thatconfinement is intimately connected with the condensation of Z(3) vortices. Toexpose the relevant Z(3) content on the lattice one needs to carry out a "cen-ter projection". The idea is the same as for the abelian projection. One fixes agauge, called the maximal center gauge, or MCG, where the link variables take aform as close as possible to Z(3) elements, and then replaces these variables bythe respective center elements. This is called "direct maximal center projection"or DMCP (Del Debbio, 1998). In terms of a concrete prescription this two stepprocess reads as follows for SU(2), where the link variables can be parametrized inthe form (17.22). One first makes a gauge transformation on a given link variableconfiguration U —>• U9 which maximizes the quantity

F[U] = £ ( T V E / » ) 2 = 2 ^ a 42 ( n , p ) ,

where |o4(n,/i)| < 1; i.e., one computes

max{g}J2(TrU°(n))2 , (17.76)

where U^{n) is the gauge transform of £/M(n). This leaves one with configurationswhich (on the average) are as close as possible to center elements. So far physics(which resides in gauge invariant quantities) has not been changed. The approxi-mation comes with the second step: the gauge fixed link configurations {UM} arenow projected to Z(2) elements according to

U^n) -» zM(n) = sgnlTrll^n)} . (17.77)

Del Debbio et al. (1997) have also proposed another way of performing thecenter projection, where the maximal center gauge itself is reached in a two stepprocess. The first step consist in going into the maximal abelian gauge (MAG), andperforming the abelian projection. The remaining U(l) gauge symmetry is thenfurther reduced by performing an additional gauge transformation which bringsthe link variables as close as possible to center elements. Center projection is thencarried out by replacing the link variables by the respective center elements. ForSU(2), for example, the U(l) matrix valued link variables, after abelian projectionin the MAG, take the diagonal form diag{eie^n\e"ie>i{-n'1). Under the residual


gauge transformation (17.66b) we have that 0M(n) -> O^n) + a^(n). By a gaugetransformation, the configurations are now brought as close as possible to centerelements by maximizing the lattice average of Y2n «

c o s 2 ^ ( n ) f This yields a setof t/(l)-link variables t/M(n) parametrized by {#p(n)}. Finally one carries out thecenter projection by making the replacement

U^n) -J- sgn{cos9^{n)) (17.78)

This procedure is called indirect maximal center projection (IMCP) The IMCPhas the merit that one can identify the U(l) monopoles in the MAG, and studytheir possible correlations with the center projected link variable configurations.

In either way of carrying out the center projection one is finally left with aset of pure Z{2) link variable configurations, i.e. one has stripped off all quantumfluctuations around the center elements.

How does one identify a so called P-vortex on the lattice? Consider againSU(2) for simplicity. Consider also for simplicity a three dimensional lattice withlink configurations consisting only of Z(2) elements. Then a plaquette can onlytake the values ±1. On a three dimensional spacial lattice this would determine theZ(2)-magnetic flux (or center flux) through the surface of the plaquette. Considernow the dual lattice consisting of the sites located at the center of the elementarycubes. Then a non-vanishing flux through a plaquette on the original lattice(corresponding to a center element —1), can be associated with a non-trivial Z{2)link on the dual lattice, with the base located at the center of a cube and piercingthis plaquette. In the Z{2) scenario there is no arrow that can be attached to thislink. Let Zp denote the value of a plaquette P on the original lattice, and let Vbe the volume of an elementary hypercube bounded by 6 elementary plaquettes.It then follows trivially that

n z ? = i •pedv

This is just a special case of the Bianchi identity, Ylpedv aP = 1> van<i f°r (1)-Here op are the values of the plaquette variables bounding the volume V, computedfrom the product of U(l) link variables with a sense of circulation determined, e.g.,by the outward normal to dV. The links on the dual lattice, taking non-trivialvalues in the center of SU(2) must therefore form closed loops in 3 dimensions. *.

* In four dimensions the P-vortices are closed two dimensional surfaces on thedual lattice. The existence of such Z(2)-vortices are believed to signal also theexistence of "thick" vortices. Thick vortices are closed extended structures. They


To elucidate some of the above ideas let us consider again SU(2) and a lat-tice in two space time dimensions. In two dimensions vortices are pointlike, andassociated with sites on the dual lattice, located at the center of a plaquette withZP = - 1 .

Fig. 17-32 Two vortices (heavy dots) on a two- dimensional lattice.

The figure shows a possible link variable Z(2) configuration. Links denoted

by heavy lines take the value — 1. Correspondingly, shaded plaquettes take

the value -1. All other plaquettes take the trivial value +1.

In fig. (17-32) we show two vortices located at A and B. Dotted (solid)lines correspond to link variables taking the value +1 (-1). The plaquettes at Aand B take both the value -1, while all other plaquettes take the value +1. Thenon-trivial links between the two vortices can be shifted around by performing anappropriate center gauge transformation. Their location have therefore no physicalsignificance. Any Wilson loop which links with one of these vortices clearly yieldsa non trivial center element -1. If both vortices pierce the W-loop, then its valuewill be +1. * The value of the Wilson loop functional (7.24) on a particular Z(2)configuration can only take the values (—1)", where n is the number of vorticespiercing the area of the Wilson loop bounded by C. Hence the potential can becalculated from the expectation value of this sign fluctuation.

Simulations of the SU(2) string tension using the Z{2) projected field configu-rations for the Wilson loop show that the full string tension is well reproduced. Infig. (17-33) we show the gg-potential calculated by Kovacs and Tomboulis (1998)from the unprojected configurations, and from the sign average of the Wilson loop.

are expected to be the relevant configurations that scale in the continuum limit,and to lead to an area law behaviour for the Wilson loop.

* This is the analog of the observation made earlier in conjuction with thetoroidal Aharonov-Bohm arrangement, except that there the vortex flux has adirection, and vortices piercing the W-loop in opposite directions will not affectthe Wilson loop functional. Furthermore the center element is replaced by theU(\) flux dependent element exp(iF).


Fig. 17-33 The static (^-potential for SU(2) computed from unpro-

jected configurations at j3 = 2.4 (squares), and from the sign average of the

Wilson loop. The figure is taken from Kovacs (1998)

If center vortices are indeed the relevant degrees of freedom for confinement,then confinement should be lost when one removes configurations from the originalensemble, which give rise to vortices after center projection. This may not be aneasy task to implement. In praxis it is easier to proceed as follows. After choosingthe maximal center gauge the field configuration is written in the form

t / » = Z M ( n ) t / » ,

where Z^n) is the center element after projection. By calculating the W-loopwith the set of link variables U'^(n) one effectively throws out the center vortices.The result of following such a procedure for SU(2) is shown in fig. (17-34), takenfrom Langfeld (2004).

As seen from the figure, the string tension vanishes upon removal of the centervortices. On the other hand the author finds that only 62% of the full SU(3) stringtension is recovered. Although many of the results obtained so far seem to point inthe right direction, looking at the detailed numerical simulations show that thereare many subtle problems such as e.g., the Gribov copy problem, the Casimirscaling problem, and the dependence of the results on the gauge in which theconfigurations are center projected. In this connection we only mention here, thatcenter projected vortices in the so called "Laplacian gauge" reproduce not onlythe SU(3) string tension, but are also free of Gribov copies (Alexandrou, 2000;de Forcrand, 2001). The question now arises: what is the quantum mechanicalcreation operator for a vortex? Such an operator was introduced by 't Hooft.To understand 't Hooft's formulation it is useful to go back to our Aharonov —


Fig. 1T-34 The static QQ-potential for SU(2) computed from unpro-

jected, and center projected configurations, as well as from ensembles with

the vortices removed. The figure is taken from Langefeld (2004)

Bohm arrangement of an infinitely thin toroidal solenoid, and construct a unitaryoperator which generates the gauge transformation

A{x) -»• AQ{x) = A{x) + Avortex{x) . (17.79)

where Avortex{x) is given by (17.74). For this we must treat Ai as an operator.We shall denote it in the following with a "hat". Let TXi{x) be the momentumcanonically conjugate to Ai(x). Making use of the operator relations

e-°Bec = B + [B,C],

andeBeC = e[B,C]eCeB ^

which are valid if [B, C] is a c-number, one readily verifies that

V{C)Ai{x)V-\C) = A{x) + Avortex(x) , (17.80a)

wherev(C) = e

l! d*x *•<>««&•*& . (17.80&)

Consider now a Wilson loop. The W-loop functional (7.22b) evaluated on thegauge transformed field configuration (17.80a) is given by

V{C)W{C')V-l(C) = e~iFfo d*-a&W(C) . (17.81)


or, inserting for a the explicit expression (17.73c) one readily verifies that

V(C)W(C) = z^c'c">W{C')V(C) , (17.82)

where z = exp(iF), and L(C, C") is the Gausssian linking number of the two loops

C = <9E and C", i.e. (17.70). When V(C) is applied to an eigenstate \A{x) > of

A(x) then it induces a "translation" of the field A(x) by (17.74) i.e.

V(C)\A(x) >= \A(x) + Avortex > , (17.83)

showing that a vortex has been created.In 1978 't Hooft introduced a vortex creation operator V(C) defined in an

analogous way to (17.82), except that z is an element of the center of SU(N) ('tHooft, 1978). As was shown by 't Hooft, the expectation value of this operatorserves as an order parameter for confinement. The area (perimeter) law for con-finement (deconfinement) of the temporal Wilson loop is replaced by a perimeter(area) law for the 't Hooft loop operator, respectively. The corresponding vor-tex creation operator has been constructed by Reinhardt (2002) along the linesdescribed above.

Let me close with some remarks. Monte Carlo calculations have shown thatboth, U(l) monopoles, and center vortices can account for most part of the stringtension in an appropriate center gauge. Independent of the choice of gauge onefinds that if either the abelian monopoles or vortices are removed from the ensem-ble of configurations, confinement is lost. Not only that: chiral symmetry breakingis also lost. Although the various proposed mechanisms discussed in this and theprevious section seem to account for quark confinement, a clear picture unifyingall these observations is still lacking. Of all the proposed mechanisms, the dualsuperconductor picture of confinement is probably the closest to a dynamical pic-ture leading to flux tube formation. The vortex and monopole pictures may justbe alternative views of one and the same fundamental dynamics. Thus numericalsimulations show that vortices and C/(l) monopole currents extracted in the indi-rect maximal center gauge are correlated (Kovalenko, 2004). With these remarkswe conclude our discussion of this fascinating subject.

CHAPTER 18

PATH-INTEGRAL REPRESENTATION OF THETHERMODYNAMICAL PARTITION FUNCTION FOR

SOME SOLVABLE BOSONIC AND FERMIONIC SYSTEMS

18.1 Introduction

So far we have studied the properties of hadronic matter at zero tempera-ture. As we have seen, numerical calculations have given strong support for quarkconfinement. Thus QCD accounts for the fact that isolated quarks have neverbeen seen in experiments performed in a normal environment. A natural questionthen arises whether quark confinement persists when one is dealing with hadronicmatter under extreme conditions, such as at very high temperature or density.

It has been speculated already in the late seventies (Polyakov, 1978; Susskind,1979) that there exists a phase transition from the low temperature regime, wherequarks and gluons are confined and chiral symmetry is broken, to a chirally sym-metric phase, consisting of a quark-gluon plasma in which the colour charge ofquarks and gluons are Debye-screened. If such a plasma phase exists, then oneshould be able to detect it in high energy ion collisions. Such laboratory experi-ments could provide further tests of QCD as being the correct theory describingthe strong interactions. The study of QCD at high temperatures is also of cos-mological interest, for hadronic matter at high temperatures and densities wassurely present in the early stages of the universe. Hence the study of QCD at veryhigh temperatures and/or densities is important for constructing models of theuniverse.

Using renormalization group arguments, Collins and Perry (1975) have pointedout that at high densities the effective coupling of quarks and gluons should besmall, and a perturbative description should be possible. The reason is that, whenhadrons overlap, the separation between quarks will be small, and their dynamicswill be determined by the asymptotic freedom property of QCD. But renormal-ization group arguments also suggest that the quark-gluon coupling constant de-creases with increasing temperature. Hence at sufficiently large temperatures onemight expect that thermodynamical observables can be computed in perturbationtheory.

Actually, physics at high temperatures turned out to be more complex thanoriginally expected. It was pointed out by Linde (1980) that the thermodynam-

383


ics of massless Yang-Mills fields involves a severe infrared problem which leadsto a breakdown of perturbation theory beyond a given order, which depends onthe observable considered. For the thermodynamical potential the breakdown isexpected to occur in 0(<?6), where g is the QCD coupling constant. This failureof perturbation theory is closely related to the fact that, in the presence of aheat bath, a screening mass of O(<72) is generated in the magnetic sector of QCD.Inspite of this "infrared problem", low-order perturbative calculations may stillprovide an adequate description of the thermodynamics of QCD at sufficientlylarge temperatures. At present, however, this can only be checked by calculatingthermodynamical observables numerically, within the lattice formulation of QCD,and comparing the results with the perturbative predictions. One must thereforelearn how to compute Feynman diagrams in finite temperature QCD.

Finite temperature continuum field theory has a long history. The pioneeringwork was carried out by Matsubara (1955) within a non-relativistic context. Thedevelopment of perturbative methods for relativistic gauge field theories is, how-ever, fairly new (Bernard, 1974; Weinberg, 1974; Dolan and Jackiw, 1974). Weshall discuss these methods in the following chapter. The purpose of this chapteris to introduce the reader to a finite temperature formalism which will allow one tostudy the thermodynamics of a relativistic field theory like QCD by Monte Carlomethods.

The central object which is of interest when studying the thermodynamicsof a system is the partition function. For simple systems, such as discussed instandard lectures on statistical mechanics, this partition function can be computedexactly. In the case of interacting field theories this is no longer possible and onehas to recur to perturbation theory. The starting point for such computationsis usually the path integral representation of the partition function. For non-interacting systems we can write down such a path integral representation whichcan be calculated exactly, and hence be checked against the expression obtainedby other well established methods. In this chapter we shall study such modelsinvolving bosonic as well as fermionic degrees of freedom in quite some detail. Thiswill give the reader some confidence in the path integral approach to studying thethermodynamics of relativistic quantum-mechanical systems.

18.2 Path-Integral Representation of the Partition Function inQuantum Mechanics

In chapter 2 we had derived a path integral representation for the imaginarytime Green function in quantum mechanics. This allows us to write down im-

Path-Integral Representation of ... 385

mediately a corresponding path-integral representation for the thermodynamical

partition function,Z = Tr e~m . (18.1)

Here H is the Hamiltonian, (3 = j ^ , with T the temperature, and fcs the Boltz-mann constant. For convenience we shall set ks equal to one in the following. Letn be the number of degrees of freedom of the system, and \q) = |<?i, 92, • • -qn) de-note the simultaneous eigenstates of the coordinate operators Qi with eigenvaluesqt. Then (18.1) is given by

Z= f f[dqa(q\e-0H\q) . (18.2)

The integrand has the phase-space pathintegral representation (2.9) with r ' — Treplaced by j5, and with the coordinates at "time" T = 0 and T = (3 identified.The partition function is obtained by integrating the expression over q (which wedenote by q^ in the following)

Z = Ujm f DqDp c*[«.r ie-^""o1 *Htow,Pw)\qlN)=qm , (18.3a)

where

£=0 a

and

^9.P] = E E ^ ( ^ + 1 ) - ?i£)) • (18.3c)£=0 a

The corresponding (formal) continuum form of this expression is given by

Z= f Dqf Dp e-/ofl*Ea^M«<.W-«(«W."W) , (18.4)

J per J

where the subscript "per" (—¥ periodic) is to remind the reader that the coordinatesat "time" r = 0 and r = (3 are to be identified. Notice that only the coordinatedegrees of freedom are required to satisfy periodic boundary conditions.

There are three features which distinguish the above expression from the classicalpartition function:

i) The phase space measure involves coordinate and momentum variables associ-ated with every euclidean time support on the discretized interval [0,/?].

(18.36)


ii) The usual Boltzman factor is replaced by exp(—(3H), where H is the following"time" average of the Hamiltonians defined at the discrete time supports: H —

jr^l'o1 H(?W,P«).

iii) The phase-space measure is multiplied by a phase factor which depends onthe coordinates and momenta, and couples the coordinates at neighbouring latticesites on the time (temperature) axis..

Since the integrand in (18.3a) involves the phase el^q'p\ it cannot be inter-preted as a probability distibution in phase space. It is for this reason that thephase-space path integral representation of thermodynamical observables (whichcan be obtained from the derivatives of Z in the usual way) does not lend itself toMonte Carlo simulations. But if the Hamiltonian is of the form (2.8), then by per-forming the Gaussian integration over the momenta we are led to a configurationspace-path integral expression which is suited for such simulations:

Z= f [dq]e-$° dTLE(qA), (18.5a)Jper

whereJ V - l n , (I)

M = n n ^ . (i8-5&)and LE is given by (2.10b), with q = q(°\ Hence within the path-integral frame-work, temperature is introduced by merely restricting the euclidean time to thefinite interval [0, /?], and imposing periodic boundary conditions on the coordinatedegrees of freedom. This is truly a remarkable simple result. When (18.5) is gen-eralized to field theories, this representation will provide the basis for computingthermodynamical observables using well known numerical methods in statisticalmechanics.

18.3 Sum Rule for the Mean Energy

We now make use of the path integral representation of the partition functionto obtain an expression for the mean energy which is suitable for Monte Carlosimulations. Such simulations are carried out on finite lattices. In quantum me-chanics this lattice is one dimensional, with the lattice sites labeled by the discreteeuclidean time supports on the finite compactified time interval [0,/?]. Let N bethe number of time steps of length e, with Ne = j3. To compute the mean energywe have to differentiate the partition function with respect to temperature. We


must therefore be able to vary the temperature in a continuous way. This canbe easily done. By keeping the number of lattice points in the euclidean timedirection fixed, the temperature can be varied by changing the lattice spacing e.Hence we can calculate the mean energy by first computing

<E>=-±\£.\nz\ . (18.6)N l9e INfixed

and then taking the limit N -r oo, e -> 0, with Ne = P fixed. To keep ourdiscussion as simple as possible, we shall restrict ourselves in the following tosystems with only one coordinate degree of freedom. The generalization to severaldegrees of freedom will be obvious.

Consider first the phase-space path-integral representation of the partitionfunction (18.3), for the case where the Hamiltonian is of the iovmH(q,p) = ^P2 +V(q):

Z= f Dq {Dp e^n[iPn{q^-q")-e{^+V{q"))] , (18.7a)Jper J

Here "n" now labels the discrete times on the interval [0,/3] (i.e., it plays the roleof the superscript £ in (18.3)), and

DqDp = Nf[^. (18.76)

The sum in the exponential extends from n = 0 to n = JV — 1. In principlewe could immediately perform the Gaussian integration over the momenta andcalculate the mean energy according to (18.6). It is however instructive to keep,for the moment, the partition function in the form (18.7). Notice that the phase-space integration measure (18.7b) does not depend on e, which only appears inthe exponential, multiplying the Hamiltonian.* Performing the differentiation in(18.6) we obtain

1 N~X 1< E >= - ]T « — p\ + V(qt) » , (18.8)

where << O(q£,p£) » stands generically for

(18.9)

* In the configuration space representation, on the other hand, the measuredoes depend on e.


We have used a double-bracket notation, since the rhs of (18.9) does not have theform of an ensemble average. For this reason the above expression is not suitedfor numerical simulations using statistical methods, as we have already pointedout. A convenient expression for the mean energy can however be obtained byexpressing the rhs of (18.8) in terms of configurations space path-integrals. Tothis effect let us define the generating functional

Z[J] = f Dq fDp eE>»<«»+i-««>-«(!!+''<««)>+'•«•'«] . (18.10)Jper J

The Gaussian integration over the momenta can be performed immediately, andyields

Z\J]=(f-)* / n1^- j : - e [ 4 m (*"-J j" ) 1 + v ( f c ) i iw=*. (i8-iia)Zn€ J n=0

where

qn = (?„+! - <Zn)A • (18.116)

Notice that the normalization factor is e-dependent, and hence contributes to themean energy. For J = 0 this is the partition function in the configuration spacepath integral representation:

z=& / nV-E-^*^'-^1!,*-. (i8-i2)Z 7 r t •> n = o

From (18.10), or equivalently from (18.11), we can calculate e.g. << pe >>:

< < W > > = i ( ^ Z [ J ] ) '= ° -Using for Z the expression (18.11) we obtain

« pe »= im<qt> , (18.13a)

wheref Dq qte-sM<4e> = z J £ g M ' (18'136)

with

S[q] = E e ^ n + Viin)} • (18.13c)n=0


Notice that <> now has the form of a statistical average, since e~s/ f Dqe~s

can be interpreted as a probability distribution. Notice also that (18.13a) is therelation one would have guessed naively, by starting from the continuum relationp = m^j- in Minkowski space, and going over to imaginary times, by replacing ^by i-jjjp- In a similar way one finds

«pj» = h~z[j})J=0z dJe (18.14)

= m < qf > .

Hence the first term appearing on the right hand side of (18.8) is given by

ff«£»=h-f£<l^>- <«•«>Notice the minus sign in the second term on the right hand side! The contributionto the average energy arising from the kinetic part of the Hamiltonian is thereforegiven by the configuration space path integral expression

while the contribution arising from the interaction is given by

< Evot > - j y L f Dqe-S[q\ • <18-17)1=0 JPer y

Of course the expression for the total mean energy < E > = < Ekin > + < Epot >could also have been derived immediately from the partition function written in theform (18.12), by performing the differentiation (18.6). For the case of the harmonicoscillator it has the same structure as that derived for the ground state energyin chapter 10 (cf. eq. (10.14)), except that here the brackets denote ensembleaverages calculated at finite temperature, i.e., on a finite periodic euclidean-timelattice.

18.4 Test of the Energy Sum Rule. The Harmonic Oscillator

In this section we want to clarify the role played by the (in the limit e ->• 0)divergent contribution ^ in the energy sum rule (18.16), and the minus sign

(18.16)


multiplying the " velocity" term, to ensure that the rhs of this expression gives afinite positive result. We cannot prove this assertion in general, but we can at leastcheck it in a solvable quantum mechanical system. We will do this in this sectionfor the case of the harmonic oscillator in contact with a heat bath, for which wehave just shown that an analogous sum rule holds for the mean energy.

In the case of the harmonic oscillator V(q) = \nq2. * For the computationof the partition function it is convenient to introduce the potential in the pathintegral expression as an average over two neighbouring time slices; i.e. we makethe replacement**

V(qn) -»• \[V(qn) + V(qn+1)} = -^{q2n+l + q2

n) .

The partition function is then given, according to (18.12), by the following config-uration space path integral expression:

Z = O f / n>»e"SWU=« . (18-18a)J n=0

where

S[q] = £ e[im(^±^)2 + Wn+1 + ql)] . (18.186)n=0

Although the evaluation of the integral (18.18a) can be found in text books, wewill, for completeness sake, present it here, following the method of Roepstorff(1994). *** Since q^ and go are identified in (18.18a), we can write the actionS[q] in the form

JV-l N-2

js[<t] = £ £<7n ~ £ 2 ( ? n + l 9 n + ^ n + i ) + Sio - <7O(<ZAT-I + ?i) , (18.19o)n = l n = l

whereA = ^ ; ^ l + e ^ . (18.196)

* For later purposes it is convenient not to set K = mu2 at this stage.** In deriving the path integral representation we had chosen to approximate

e~cH by e~e%ê~cV(q\ We could have also chosen to write this product in reverseorder.*** See, e.g. also W. Ditrich and M. Reuter, "Classical and Quantum Dynamics,Springer Verlag (1992)


This action can be written in matrix form as follows

\s[q} = \qTQq-qTb+^bTb, (18.20)

where Q is an (N — 1) x (N — 1) matrix with the non-vanishing entries given by

/ 2 £ - 1 \- 1 2£ - 1

- 1 2£ - 1

Q= ~* 2 f ~* (18.21)

- 1 2£ - 1V - 1 2£ /

and where <? and 6 are the N — 1 component vectors

« = ( * ! , ft,*," • , « * - ! ) ,

6 = ( « o , 0 , 0 , - - - , g 0 ) •

Completing squares, (18.20) can be cast into the form

\s[q} = \qTQq + \bT{Z - Q'^b , (18.23a)

where

?-r-rQ->. (18236)

Inserting the expression for the action in (18.18a), and performing the integrationover the N — 1-dimensional vectors q we obtain

*=\IHiBah •-****'*• (I8M)

We therefore must compute the determinant of Q and its inverse. We first computethe determinant.

Let Dn be the determinant of an n x n matrix having the form (18.21). Forn=l , D\ = 2£. Define Do = 1. One then readily verifies that the followingrecursion relation holds for n > 1:

£>„+! = 2£Dn - £>„_! .

(18.22)


The difference equation, with Do = 1 and Di — 2f is solved by

Dn = cenf> + de-np ,

where

p = arcosh£ . (18.25)

andeP , e'P

c = ; d = .2 sinh p 2 sinh p

For the determinant of Q one therefore finds that

detQ = DN^1 = S^P. (18.26)sinh p

The other quantity we need to know is bTQ~1b. This can be easily calculated as

follows. Let Q- 16 = a. We can compute the vector a by solving the inhomogeneous

equation Qa = b, where a = (ai, •, •, •, CJJV-I) and 6 is the vector (18.22). Note that

except for the first and last rows, the structure of the remaining rows of the matrix

(18.21) is always the same. Hence Qa = 6 is equivalent to solving the following

set of equations

2£ai - a2 = q0

-an+2 + 2£an+i - an - 0 , (n = 1, • • -, iV - 3)

-OJV-2 + 2£ajv-i = go

The second set of (recursion) relations can again be solved with the Ansatz

an = Cenp + De~np ,

where p is given by (18.25). The remaining two equations fix the coeffiencients C

and D. One finds thatM-e-iVp-l

G " UsmhTVpj q° '

D=-[j^hNp-\qo-Now bTQ~1b = bTa, with 6 = {qo,0, • • -,0,go)- Hence bTQ~1b = qo(ai + a jy-i) .

One then finds that

bTQ-1b= , 2^° [sinh(JV-l)p + sinh;9] .


Since £bTb = 2£<?o, we finally obtain

bT{£ - Q'l)b = 2q2Q{v - w) , (18.27a)

where

tanh Np v '

W = - 5 ^ - . (18.27c)sinh Np y '

Inserting (18.27) in (18.24), and noting that according to (18.26) and (18.27c)detQ = w~l, on finds, after performing the Gaussian integration over qo, that

Z=Î^ = 7^ôShNp-l- (18-28)

Let us first verify that in the continuum limit N —>• oo, e —> 0, Ne = /3 fixed,we obtain the correct expression for the partition function. Since p is given by(18.25) with £ denned in (18.19b), we have that for e -4 0, p « ey/%. Hencecosh Np —> cosh/3w, where w = ^ / ^ is the frequency of the oscillator, and wehave set /? = Ne. We therefore arrive at the correct expression for the partitionfunction,

e-i** [IT1 - e-P" V m

From here we obtain the total mean energy

Let us now verify that the contributions to < E > arising from the kinetic andpotential term of the Hamiltonian are indeed given by (18.16) and (18.17). From(18.18) it follows that

n=0 L J Kfixed (18 29")

Using for Z the expression (18.28), one finds, upon taking the derivative withrespect to m (for fixed n) that (18.29) reduces in the continuum limit to

1 ^ 1 .2 1 1

n=0

(18.276)


Note that the right hand side involves a piece which diverges like ^ for e —» 0,and a finite term which is the negative of the mean kinetic energy of the harmonicoscillator at temperature T. From (18.16) we therefore conclude that < Ekin >=\ < E >. Notice that the minus sign appearing in the integral expression on theright hand side of (18.16) was crucial for obtaining the correct expression for themean kinetic energy. The contribution of the potential (18.17) is obtained from(18.18) according to

< Epot >= ^ E I < (ti+i +<?«) >= -We^lnZ .n=0

Making again use of the explicit expression (18.28) one then finds that in thecontinuum limit < Epot >= | < E >, as expected.

The origin of the strange looking relation (18.16) is of course that the ther-modynamic partition function is given by a euclidean path integral representation.Hence < q\ > is the ensemble average of the euclidean velocity squared. Thisensemble average is necessarily positive. The positivity is however only ensuredby the term ^ j , which diverges in the continuum limit. This divergence is a con-sequence of the fact that the width of the probability distribution in {qn+\ — qn)

2

appearing in the configuration space path integral (18.18) is only of the order ofe, so that < (g"ti-fr>)2 > w i l l b e o f O ( I )

18.5 The Free Relativistic Boson Gas in the Path Integral Approach

In section 2 we have shown that for a quantum mechanical system, whosedynamics is governed by a Hamiltonian of the form (2.8), the partition functionis given by the path integral expression (18.5), where the integral includes allpaths satisfying the periodic boundary conditions qa{0) = qa(0)- The integrationmeasure Dq, defined in (18.5b), is seen to be dependent on the temperature, which,for fixed N is controled by the temporal lattice spacing e. The formal translationof this expression to the field theory of a free real scalar field <j>(x, r) is immediate:qa(r) is replaced by <fe(r) =: C/)(X,T), and the euclidean Lagrangean LE is nowgiven by

LE[<j>,<j>] = I d3xCE{<l>,dv<t>) , (18.30a)

where CE is the Lagrangian density

CE = \d^d^ + \M2CJ>2 . (18.306)


Hence the path integral expression for the partition function (18.5) goes over to

Z0=uf Ztye-ZoW"8*^'*), (18.31a)Jper

where formally

D(f> = Y[d4>(x,T) , (18.316)

and M is a dimensioned normalization factor * playing the role of [} ]nN in

(18.5b). The fields <J>(X,T) are required to satisfy the periodic boundary condition

<{>(x,0) = ct>(x,p). (18.32)

Hence the partition function is a weighted sum over all field configurations which

live on a euclidean space-time surface compactified along the time direction. In

two space-time dimensions this surface can be viewed as a cylinder with its axis

along the spatial direction. The radius of this cylinder becomes infinite in the limit

of vanishing temperature.

As we now show, the expression (18.31) reproduces the well known expression

for the grand canonical partition function of a gas of neutral bosons. Introducing

the dimensionless variables </> = /3<j>, f = â;4, & = ^Xi and M = (3M, (18.31)

becomes

Z0 =MJDh-ôd^d^{-Q+A2)\ (18.33)

where N is now dimensionless, and where we have made a partial integration in

the expression for the action. The fields cf>(£, f) now satisfy the periodic boundary

conditions

kt,O) = $(t,l). (18.34)

The Gaussian integral (18.33) can be immediately performed and yields the fol-

lowing formal expression**

Z0=7VIdet(-n + M 2 )] - 1 / 2 , (18.35)

where the determinant of (—D + M2) is defined as the product of the eigenvalues of

the operator in the space of functions satisfying the periodic boundary conditions

* Since the partition function is dimensionless, N carries the inverse dimension

of Dcj>

** Actually M differs from the normalization constant appearing in (18.33) by

an irrelevant numerical factor


(18.34). Hence \nZ0 will involve the sum of the logarithms of the eigenvalues. Tocalculate this sum it is convenient to enclose the system into a box of linear dimen-sion L and to impose also periodic boundary conditions in the spatial directions.The normalized eigenfunctions of (—D + M2) then have the form

Vvwhere L = =|, V = L3, and n and m* (i = 1, 2, 3) are integers. The eigenvalues of(—• + M2) are given by

An,^ = (27r)2(n2 + ^ ) + M 2 .

For the logarithm of (18.35) we therefore obtain

i ^°=-^££ i 4" 2 +h? i n + i n ^ ' (i8-36a)" k { L -I J

whereE(k) = \]k2 + M2 , (18.366)

and where the sum over k extends over the discrete values

k = ~m . (18.36c)

Going back to the quantum mechanical case discussed in section 2, where thepartition function is given by (18.5), and scaling the integration variables accordingto their canonical dimension with /?, one finds that the contribution analogous toln.AA is proportional to the number of degrees of freedom. Hence we expect thatin the present case lnJV" = const, x V J T^S- Since such a term merely gives riseto a constant shift in the pressure, we shall drop it from here on.

Let us next replace the sum over discrete momenta in (18.36a) by an integralin the standard way

?W<0' (i8-37)k

which is valid in the large volume limit. Then

(18.38)


Consider the sum over n of the logarithm *. Consider the following (also divergent)sum

oo

Y^ ln(n2 + x2) = \ux2 + 2g(x) , (18.39a)n = — oo

whereoo

g{x) = ^ ln{n2 + x2), (18.396)n=l

and

x = (5E/2-K. (18.39c)

Whereas the right-hand side of (18.39b) diverges, its derivative

T = £ -rh ( 1 8-4°)ax *-^ nz + x1

is finite. The sum can be evaluated in closed form:**

^ 2x 1 2TT

71=1

By integrating (18.40) we can determine g(x) up to a (divergent) constant. In-serting the result into (18.39a) one then finds that

£ ^(»2 + (^f)2) = ^ + 2/n(l-e-^) + C

where C is an integration constant. For (18.38) we therefore obtain (recall thatwe dropped the last term in (18.36a))

The second and third term merely gives rise to a constant shift in the mean energydensity and pressure. Dropping these (irrelevant) terms we therefore find that the

* We follow here the method of Dolan and Jackiw (1974). For an alternativeprocedure see Kapusta (1989).

** See e.g. "Table of Integrals, Series, and Products", by I. S. Gradshteyn andI. M. Ryzhik, Academic Press, New York and London (1965).


thermodynamical potential fi = — -^ In Z of an ideal gas of scalar neutral particlesis given by

^=-wlnZ=lIw^H1-e~0E)- (18-41)In the zero mass limit this expression reduces to

" = - ^ T 4 , (M = 0) (18.42)

so that the energy density e = d(/3tt)/d(3 and pressure p of an ideal gas of masslessscalar particles are given by the Stefan-Boltzmann law:

301

P=r.

18.6 The Photon Gas in the Path Integral Approach

Let us now consider the thermodynamics of a free photon gas within the path-integral framework. The euclidean finite temperature action, replacing (5.8b), isgiven by

S = I /" dr I ^xFlu,{x)Fia,{x) , (18.43)

where Fy,v = dy,Av — dvA^ is the euclidean field strength tensor expressed in termsof the gauge potentials. Naively one would write down the following path integralexpression for the partition function

Z = NJ M-J/'*/^WW'). (18.44a)J per

where the gauge potentials are subject to the periodic boundary conditions

AM(2,0) = AM(£,/3). (18.446)

Written in this form the integral (18.44a) diverges, since gauge field configurationswhich are related by a gauge transformation are weighted with the same exponen-tial factor. The divergence can be isolated following the well known Fadeeev-Popovprocedure, by which a gauge condition is introduced into the path integral. Thefollowing arguments are very formal, and we refer the reader to text books on fieldtheory for more details.


Let us first write the partition function (18.44a) in the form

Z = xf êUW^^XVn-^VU*) ( (18.45)J per

where we have made a partial integration in the action (18.43) expressed in termsof the gauge potentials. Next define the following gauge invariant functional A [.A]

A[IJ = / Dk II WMvi?) ~ dM*)) - /<*)]. (18-46)

where f(x) is some arbitrary periodic function. Note that A^(x) - dMA(a:) is justa gauge transform of the potential A^x). We now rewrite the 5-function in theform*

\[5[dM» - Â) - /] = IpK-Q)(A - ^ d ^ - /))]X X

Performing the functional integral over A in (18.46) we therefore have that

A[A] = det ( - • ) .

A[A] is the so called Faddeev-Popov determinant, which, although in the presentcase is independent of the gauge potentials, cannot be ignored. In fact at finitetemperature, however, the determinant, which is given by the product of theeigenvalues of —D, with the eigenfunctions satisfying periodic boundary conditionsanalogous to (18.44b), is temperature dependent and hence is relevant for thethermodynamics of the system. This is the main message we wanted to convey.

The next step consists in introducing the identity

1 = A[A] [DAHSidÂ^x) - aMA(i)) - /(*)]J X

into the path integral (18.45). After making a change of variables A'^ — A^ — 9MA,and using the fact that the action as well as integration measure are invariantunder this transformation, one finds that

Z=tf[[DA][ DA det(-D) I ] SldÂ^x) - /(*)]J Jper x

2 So dT I d3xAAx){&^~9^)A"(x)

* We have chosen to factor out a — Q since the eigenvalues of this operator arenon negative.


Here we have dropped the "prime" on A'^. The remaining steps leading to acovariant gauge fixed expression for the partition function are standard. Sincethe integral does not depend on the choice of f(x) we can multiply (18.46) byexp[- 2^ / d4xf2(x)\ and then functionally integrate the expression over f(x) withthe measure Df = Y\x df(x). After dropping the factor Af[f DA], since it shouldnot affect the thermodynamics, we are left with the expression

Z= f DA det(-a)eU"dT/d3^(*)(Vn-(i-i)W^(*) _J per

Choosing the Feynman gauge (a = 1), we therefore have that*

Z = det(-D) \tf [ DA e-ijyrJdÂ^)(-â)AAx)]L J per J

The path integral factorizes into four path integrals having the form characteristicof a zero mass scalar neutral field. Hence we immediately conclude that

-i 4

lnZ = ln . +lndet(-D)[A/det(-D)J

= 21n[det(-D)]- =

Hence In Z is just twice the corresponding expression for a massless scalar neutral

field. The factor two accounts for the fact that the photon has two transverse

polarizations. We therefore conclude that the partition function for the photon

gas is given by

^ 3 l n ( l - e - " l f c " ) , (18.47)

* The determinant could in principle be incorporated into an effective actionby introducing a set of Grassman fields (ghosts) c(x) and c(x). Then det(—•) =

/ DcDc eJo xc^x' c^x'. Since the determinant is the product of the eigenvalues ofthe operator — D with the eigenfunctions satisfying periodic boundary conditionsin euclidean time, this integral is to be evaluated with the ghost fields subject tothe same boundary conditions. The relevance of the Faddeev-Popov determinantin obtaining the correct form for the partition function has been discussed in detailby Bernard (1974). The reader should consult this reference for a more generaldiscussion of the gauge-fixing problem for the thermodynamical partition function.


and that the thermodynamical potential is just twice the expression (18.42), i.e.

45

Summarizing, we have seen that the contribution of the Faddeev-Popov de-

terminant was essential in deriving this result. Omission of the ghost contribution

would have yielded a result which is twice as large. Furthermore it was important

that the ghost fields obeyed the same boundary conditions as the photon fields

which allowed for the cancellation of the unphysical degrees of freedom of the

gauge potentials.

18.7 Functional Methods for Fermions. Basics

Having discussed in detail the path integral representation of the thermo-

dynamical partition function in some simple bosonic theories, we now want to

extend our discussion to the case of fermionic systems. A functional formalism

which allows one to derive a path integral representation of the partition function

for fermions is that of Berezin (1966).* In the following section we shall apply this

formalism to a simple non-relativistic fermionic system, which allows us to derive

a path integral representation for the partition function which is exact for an ar-

bitrary choice of time step. This example will illustrate some important points

which will be relevant when studying the thermodydnamical properties of rela-

tivistic field theories involving fermions on a lattice. In this section we will present

the basic formulae that we shall need, and check them in a simple model. For a

detailed discussion of the general functional formalism the reader may consult the

book by Berezin (1966).

Consider first a fermionic system whose Hilbert space only consists of the

vacuum state, |0), and the "one particle" state |1) = a+|0), where |0) is annihilated

by a. The operators a*, and a satisfy the anticommutation relation {&,&} = 1.

All other anticommutators vanish. An general operator A, acting on this space,

then has the form

A = Koo + K10a) + K01a + Kutfa . (18.48)

Note that the coefficients Kij are not the matrix elements of the operator in the

above mentioned basis. We have therefore denoted them with the symbol K,

What we will need is an expression for the trace of an operator, and of a product

* See also Soper (1978)


of operators, in terms of an integral over Grassmann variables. This can be easilyachieved in our example. We first associate with the operator (18.48) the so callednormal-form

A{a*, a) = Km + Kloa* + KQla + K1Ya*a , (18.49)

where a and a* are the generators of a Grassmann algebra, i.e., {a, a} = {a*, a*} ={a, a*} = 0. From the normal form, we construct the so called matrix-form

A(a*,a) = ea"aA(a*,a) . (18.50)

Since ea a = 1 + a*a, one readily verifies that

A(a*,a) = Aoo + Awa* + Aoxa + Ana*a ,

where Aij are now the matrix elements of the operator A in the basis |0) and|1) = &t|0). Making use of the Grassmann integration rules

I da*a* = / daa = 1; / da - f da* = 0 (18.51)

one finds that TrA = AQO + J4.II is given by the following Grassmann integral

TrA= I'dada*ea"aA{a*,a) . (18.52)

To calculate the trace of a product of operators C = AB we need an expression forthe matrix form of the operator C, in terms of the matrix forms of the operatorsA and B. One easily verfies for our simple example that

C(a*, o) = I da*dae-a'aA(a*, a)B(a*, a) , (18.53)

where a, a*, a, a*, da, da*, are all anticommuting variables.

All the above expression have been written in a form which generalize to thecase of a fermionic system with an arbitrary finite number of degrees of freedom.We summarize the main expressions we shall need in the following sections, andrefer the reader to the book by Berezin (1966) for details.

Let hi and a\ (i — 1,2, •• -,k) be operators satisfying the anticommuationrelations

{ai,a]} = Sij ,

{ai,aJ-} = {at,at} = 0 . (18.54)

(18.49)


A general function of these operators can then be written in normal ordered form:

n."»{»*}>{jik}

where the coefficients i"Q"™ ... j n . ^ j 2 ... ,• are separately antisymmetric in the in-dices ii,22, • • -in and ji,J2, • • 'jm- By definition, n = 0, or m = 0 is understoodto imply that the corresponding contribution to the sum (18.55) does not containoperators of the type a\ or &i, respectively. A general state on which this operatoracts is given by a linear combination of the vacuum state |0), which is annihilatedby all the &i's, and the states a^ • • • a\n |0) , where 1 < n < k.

We now associate with (18.55) a normal form, obtained by replacing the oper-ators a\ and o, by (anticommuting) Grassmann variables o* and a,, respectively:

i(a*,a) = £ £ K%?l..t^,..jm<<---<*h«»---*irn' (18-56)n>m{ik},Uk}

The Grassmann variables a and a* appearing in the argument of A stand for thecollection of variables {a*} and {aj}, respectively. The coefficients -fQ"™ ...j1 j2

are not the matrix elements of the operator A in the above mentioned basis. Fromthe normal form, we can construct the matrix form in a way analogous to (18.50):

A(a*, a) = e S , a'a> A(a*, a) . (18.57)

Given the matrix forms of two operators A and B, the matrix form associatedwith the product AB = C can be computed as follows:

C(a*,a)= fY[da*daje-â:aiA(a*:a)B(a*,a), (18.58)J j

where a*, a,, a^, a are now generators of an extended Grassmann algebra.

Knowledge of the matrix form of an operator allows us to compute its trace ac-cording to

TrA= [Yldajda*eE»a'iaiA(a*,a) , (18.59)J J

which is the generalization of (18.52). This expression can also be written in theform

TrA= j Y[da*daje-îa"'aiA{a*,-a) . (18.60)


Furthermore, using the Grassmann integration rules (18.51), one verifies that

[Udai !\{da*eâ'^-b')F{a,c)=F{b,c) ,J i J j

where again a, b, c stand for the collection of Grassmann variables {a^}, {&;}, {c;}.Hence

6{a,b)= I \[da*eîaâ'-bi) . (18.61)J j

acts as a 5-function.

As a simple application of the functional formalism, let us compute the grandcanonical partition function for a system whose dynamics is governed by the Hamil-tonian

H = Y2 Eia\a,i , (18.62a)i

with the chemical potential /J. coupled to the number operator

N = ^2a}a,i. (18.626)i

The partition function for this system is given by

E(P, fi) = TrCl , (18.63a)

whereCl = e-0(*-"tf> = g - ^ E i ^ - " ) ^ 4 * . (18.636)

The exponential factorizes. Making use of the anticommutation relations (18.54)one finds after some algebra that

k

£l = l + Y^ Yl Vh • • • Vieal • • • altait • • • ai, ,1=1 ii<i2<-<il

whereVi = e-flSi-M) - 1 ,

and where k is the number of degrees of freedom. The normal form associatedwith Cl is obtained by replacing the operators a\ and a^ by the correspondingGrassmann variables a* and aj. Making use of the fact that the square of aGrassmann variable vanishes, it can be written in the form

fi(a*,a) = e X > - < a i .

Path-Integral Representation of .. • 405

The corresponding matrix form is then obtained according to (18.57):

n(a*,a) = eê~^'~")a-a' . (18.64)

Finally, (18.63a) is calculated from (18.59). For lnS one obtains

lnS(/3,/i) = 5 3 H i + e-PVSi-ri] , (18.65)i

which is the correct expression for the partition function.

18.8 Path Integral Represenation of the Partition Functionfor a Fermionic System valid for Arbitrary Time Step

We now use the functional formalism discussed in the previous section toderive a path integral expression for the partition function (18.63), which holdsfor an arbitrary finite "euclidean time"-step.

Consider the model whose Hamiltonian is given by (18.62a). To obtain apath integral representation of the grand canonical partition function we split theinterval [0,0\ into N intervals of length e = fi/N, and write the partition functionin the form

H = Tr(Cle)N ,

where

The matrix form of tle is given by (18.64) with @ replaced by e,

ne(a*, a) = eî «-(Bi-M)«.T«< , (18.66)

where, for notational reasons which will become clear below, we have denoted thesets of Grassman variables {a*} and {a;} by a* and a, respectively. We nextcalculate the matrix form of Ci^ by making repeated use of the product formula(18.58). Let us label the integration variables with an extra index, which will beinterpreted later as labeling different time slices. Thus in the following an denotesa set of k Grassmann variables <zi>n, i = 1,2, • • -,k, where k is the number ofdegrees of freedom of the system. A similar statement holds for a^. Then thematrix form of (Cle)

N is given by

. JV-i

n(a*N,aN) = / I ] I I da^daê-3*--8-J i n=\

x[n£(a^-,ojv_i)fie(o^r_1,ojv_2)...n£(a5;,ao)]ob=aAf ,


The trace of {fle)N is calculated according to (18.60):

TrCl= I Y[dalNdaitNe-s"SNn(a*N,-aN) .

Writing out the components of an and a* explicitely we are led to the followingexact path integral representation for the partition function,

Tre-0(H-,Q} = f j j j | 4 . A / - S ( K J , K , , } ) | B i ] O = _ M , (18.67a)J n= l i

where

N

S(W,B}, {*,„}) = £ £ « > * , « " «i,n-l] + (1 - e"^'^))a* nai>n_1} .n = l i

(18.67b)Notice that the path integral (18.67a) is to be calculated subject to the antiperiodicboundary condition aifi — -aitN. While the index "i" labels the degrees of freedomof the system, the index "n" labels the different (euclidean) time slices.

The partition function can now be calculated by diagonalizing (18.67b). Tothis end we introduce a new set of variables Oi,n and a* n by the following unitarytransformation (we take N to be even),

AT/2-1 i V / 2 - 1

ai,n - J ^ cmai,e ; a*n = ^ c*nea*e , (18.68a)e=-N/2 I=-N/2

where

and QJI are the Matsubara frequencies,

11 + 1wi = (———)?r ; [fermions) (18.69)

The coefficients (18.68b) satisfy the relations

JV/2-1

l=-N/2

N

J2cnec"*' = ( 5 « ' • (187°)n = l

(18.686)


Notice that the antiperiodic boundary condition in (18.67a) has been incorporatedin the expansion (18.68a). Upon inserting (16.68a) into (18.67b), and performingthe sum over n by making use of (18.70) one finds that

J i « = - *

— - l

= 11 II {l-e-^-ê-***} .* fc-f

We therefore see that the chemical potential is introduced into the partition func-tion for fj, = 0 by the simple substitution rule

UJI ->• 0>t + ip, , (18.71)

where jl = e/i is the chemical potential measured in lattice units. Combining thepositive and negative frequency parts, TrCl = E can be written in the form

f-is = nn t i + i i - 2 a ; i c o s ^} = n^ 1 +* .* ) • (18.72a)

i (=0 i

where

Xi = e-<Bt-ri. (18.726)Upon setting eN = f3, one recovers the result (18.65).

The path-integral representation (18.67) exhibits several interesting featureswhich deserve a comment:

(i) As we have already mentioned, it is the exact path-integral expression forthe partition function (18.63) for every choice of e = (3/N.

(ii) An alternative expression for the path integral (which will turn out tobe convenient when we compare it with the lattice actions employed in numeri-cal simulations) is obtained by making the redefinitions a,in —> <Zj,n+i and a\n-+e~tfia*n. This is always possible, since the a's and the a*'s are independent Grass-mann variables. One can then easily show that (18.67) can also be written in theform *

3 = [Ue^ I ft nd<«daiine-s(K,n},{a-})|Qi]JV+1=_a,a , (18.73a)i J n=l i

* Care must be taken of the fact that the integration measure for Grassmannvariables does not transform in the way one is used to from integrations overc-number variables. Thus / daeXa = X J dbeb


where

N

S(« n } ,Rn}) = X)Z)Kn[e"eMOi,n+i-Oi,n] + (l-e-eE')o;>nai,n} . (18.736)n=l i

The only subtle point in obtaining this result concerns the integration measure.The reader can convince himself that it can be still written in the form given in(18.67a), by modifying the boundary conditions in the way exhibited in (18.73a).

iii) Still another form of the path-integral expression (18.73) can be obtainedby introducing the variables

<n = e-^X» .

< n = e«"Xn • (18-74)

which leaves the measure unchanged. In this way one can eliminate the /i-dependencein the action, and incorporate the chemical potential into the boundary condition.Dropping the "prime" one then obtains that

- -M fT\Y\da* da- \e-T,i,Jai.n<.ai^+^-a^)+<.l-e"Bi)aln^,nh" ~~ / 11 \_Laai,naaiMe Jai|N+1=-e-'3Mai,1

•' n = l i

(18.75a)where

./V=[JJe^]. (18.756)i

Note that the chemical potential now enters in the form of a boundary conditionin the combination Pfi, which is independent of the chosen discretization. Byexpanding e~eEi in (18.75) up to leading order in e we can obtain an expressionfor the partition function, valid in the continuum limit, which can be generalizedto the case where the Hamiltonian no longer has the simple form (18.62a),

r Nr^ i r T / TTTT 7 * T r — > {a,* _ (a; n+i— a; n)+€H(a" „ ,a, „)}-,- = N h™0 / l l l l d a i , n d a » , n [ e ^ . ^ "»l ''"+1 ' '•"• '•"'% N+1 = -e-P»a,

(18.76a)or equivalently

AT

= = A/ lim / [[ [[daindaiin[e ^-.» ]Oi,N+i=-a,

fc-7 J "=1 *(18.766)


whereU[a\in,ai,n) = Eialn,aitn (18.76c)

is the Hamiltonian "density" defined on the n'th time slice. In obtaining (18.76b)we have again made a change of variables analogous to (18.74) in (18.76a). Ex-pression (18.76b) has a form which most closely resembles that used in latticecalculation. Except for the overall factor multiplying the integral, the chemicalpotential is introduced in the bilinear terms coupling neighbouring lattice sites inthe euclidean time direction in the way proposed by Kogut et.al. [Kogut (1983c)],and Hasenfratz and Karsch [Hasenfratz (1983)]:

a*inai:n+i ->• a*ne"Aai )n+i . (18.77)

where /} is the chemical potential measured in "lattice" units. As we have seen inthis section, this prescription follows naturally from the functional formalism.

Let us verify that the partition function (18.76b) yields the correct answer forthe mean energy, < E > and mean particle number < N >. By diagonalizing theaction as before one finds that

S= lim Y[ I I eiC"[l-{l-eEi)e-i£"+£l'}

Taking into account that the product includes as many positive as well as negativefrequencies, this expression can again be written in the form (18.72a) with xi =(1 - eEê^. Hence

lnS = ]Tln[l + (l-e£i)JV3'i] .

i

Upon setting e = -ft, and taking the limit N —t oo, we recover the result (18.65).

For fixed N, the mean energy is given by (recall that /?/x is to be held fixed),

< E >= - lim ^- h r - I n S | (18.78)

Hence

<E>= lim Y-^ s

„•!& Y l - ^ , ( 1 - §jEi)-Ne-to + 1" - " (18.79a)

E EjePiEi-ix) + i •

i


A similar calculation for < N > yields

<N>= lim — — ^ 3 = ^ - ^ r (18.796)

As the reader will have noticed, the factor (18.75b) multiplying the integral in(18.76a) was important for obtaining the correct answer. Ignoring it would notinfluence the result for the mean energy, but would modify the expression (18.79b)by the potentially divergent sum Jî = nf, where n/ is the number of degrees offreedom.

18.9 A Modified Fermion Action Leading to Fermion Doubling

We have seen in the previous section that the functional formalism for fermionstells us that the kinetic (time-derivative) contribution to the action is discretizedusing the right lattice derivative. The dependence on the chemical potential thenappears in the form of a /x-dependent boundary condition. Alternatively, thechemical potential can be introduced into the action at (i — 0 according to therule (18.77). In this case the Grassmann variables satisfy antiperiodic boundaryconditions. Although the action in the latter formulation resembles that used inlattice simulations, it actually differs from it in an essential way. When simulatingrelativistic field theories involving Dirac fermions on the lattice, one wants theaction to exhibit a hypercubic symmetry, which is the lattice remnant of the 0(4)symmetry in the (euclidean) continuum formulation. The fermionic part of theaction is therefore disretized using the symmetric form for the temporal (as wellas spatial) lattice derivative. In chapter 3, where we discussed the fermion prop-agator, we have seen that using such a discretization leads to a serious problem:the fermion doubling problem. Of course we also expect to see the contribution ofthe doublers in the partition function. In fact, as we now show, the doubler con-tributions which arise from the use of a symmetric lattice tame-derivative manifestthemselves in a non-trivial way. This can be most clearly demonstrated for thenon-relativistic fermionic system considered in the previous sections. The manifes-tation of the doubers in the relativistic case is more subtle and has been discussedby Bender et.al. [Bender (1993)].

The following partition function is a modification of (18.76b), where the ki-netic term in the action (i.e., the "time"-derivative term) is modified in a waywhich resembles that used in actual Monte Carlo simulations of lattice gauge field


theories:

1 r N

3(/3,/x)= lim f ] ^ / II II <»*>*,«*y-/ff.<i L * -1 i n = 1 (18.80)

x e - s : . 1 « E < < . . t ' " < l " - + r ' " - i )+ « w . . . . < - ) ] | .(i01._.4iN .

ai,N + l — -ai,l

Here we have introduced the chemical potential into the symmetric time-derivativeat \i = 0 according to the Kogut-Hasenfratz-Karsch prescription [Kogut (1983c);Hasenfratz (1983)]

ai,nai,n+l ~^ a i ,ne~M ai ,n+l i

a*nai]n_i -> a*neAai)n_i , (18.81)

where ft = e/i is the chemical potential in lattice units. Because of the modifiedkinetic term the anti-periodic boundary conditions in (18.76b) have now beenreplaced by aij0 = -a^jv, and a^jv+i = -at,i- The integral can be performed bydiagonalizing the action in the manner described earlier. Making the change ofvariables (18.68), which incorporate the above antiperiodic boundary conditions,one finds that

s = Y i n eA[* s m ( ^ + * A ) + e E i \ •

From here we obtain for the mean energy

r -1 —-1

<E>=~h ^r = - ^ E ^ f E/^'^)]> (18-82a)L -1 Np, fixed i l=-%

where

/(£/,î) = • • , , \ ^ , P • (18.826)2sm(w^ + i/z) + eEi

Note that the frequency sum is performed over the finite interval

- 7 r + - ^ < ^ < 7 r - - ^ , (18.82c)

i.e., over frequencies lying within in the first Brillouin zone. This frequency sumcan be written as a contour integral by making use of the formula

(18.83)


where the contour C encloses the poles of the integrand in the complex a)-plane,lying within the interval (18.82c), arising from the zeros of the denominator. Theintegration is carried out in the counterclockwise sense. These poles are located atu> = <2>£, where Qit has been defined in (18.69). The corresponding residues of theintegrand are given by Ri — JJF{<2>I). If F(£J) does not possess any singularitieson the real axis, then the above expression can also be written in the form

1 £ Wi) = - - r ^ ^ ~ + - P ^ 4 ^ - • (18-84)N ^ y ei 2nJ_7t_ie e ^ + l 2TT J_^+it e*N« + 1 V ;

A more convenient form can be obtained by introducing the integration variablez = eia. Then

i t ! , , If _, F(z) 1 f J F(z)- ^ FM = - _ j ^ d,^^ + _ J ^ dz-(^T) ,

(18.85)where F(elu)) = F(w), and where the integrations are performed on circles in thecomplex z-plane, with radii \z\ = l±e , in the counterclockwise direction. If F(z) isa meromorphic function of z, satisfying \z\~NF(z) -» 0 for \z\ —¥ oo, then we candistort the integration contour in the first integral to infinity, taking into accountthe contribution of the poles of F(z) for \z\ > 1. The second integral is just 2ititimes the sum of the residues of —^ located inside the unit circle. Hence

jf E Wt) - E J^T • (18.86a)

where

R(zi) = ResZi ]-^- . (18.866)

z

We now use this expression to calculate the frequency sum in (18.82a). Theposition of the poles of (18.82b) in the z-plane are given by

Z± = [-eEi ± y/e*E? + l]ee" = ±e^rsinheEi+e,j.

Making use of (18.86) one then finds that

_ y ^ Ei 1 1 ,•^ > ~ L^, /e2jp;2 i j *-eN{a.TsinheEi+eti) _|_ J e-N(arsinheEi-e/i) -|- 1 ' '


Upon taking the limits e ->• 0, N -» oo, with Ne = (3 fixed, and making use of theidentity l/(ex + 1) + l/(e~x + 1) = 1, one obtains

i i

This is an interesting result, for it deviates in a drastic way from (18.79a). Apartfrom a potentially diverent additive contribution, the remaining two contributionsresemble those of a gas of particles and antiparticles! Hence (18.80) is not the pathintegral representation of the partition function (18.63b). This example clearlydemonstrates that modifications in the action which are inconsistent with the gen-eral form dictated by the functional formalism, can give rise to spurious unphysicalcontributions. This is the case, although the actions appearing in (18.76b) and(18.80) differ formally only by terms of C(e). Nevertheless they give rise to verydifferent expressions for the partition function. We had already been confrontedwith a similar situation when we discussed the propagator for Wilson fermions inchapter 3. There the action was modified by a so called irrelevant term, vanishingin the naive continuum limit. When introduced into the path integral, however,the fermion doubling problem disappears.

Whereas the doubler contributions to the partition function do possess a con-tinuum limit, this is not true for Green functions, as we have already seen inchapter 3, and will be further demonstrated in section 9 of the following chapter,where we discuss in detail the Dirac propagator for naive (and Wilson) fermionsat finite temperature and chemical potential.

18.10 The Free Dirac Gas. Continuum Approach

We have discussed in great detail the path integral representation of the par-tition function for a simple fermionic system, in order to point out some subtlepoints. In this and the following section we now extend our discussion to therelativistic case. In text books on finite temperature field theory the thermody-namical partition function for a free Dirac gas is usually derived from its pathintegral representation in the continuum, where such concepts as the right or left-derivative do not appear. Nor does the expression for the partition function reflectthe fact that in the discretized version the chemical potential is introduced intothe bilinear terms in the Grassmann variables coupling neighbouring sites alongthe euclidean time axis. In lattice regularized gauge field theories, however, thechemical potential should be introduced in the exponentiated form, discussed inthe previous section, in order to ensure the renormalizabilty of the theory. For

(18.87)


completeness sake, we will include the discussion of the free Dirac gas within thecontinuum formulation in this section. In the following section we then study theDirac gas within the framework of the lattice regularization.

Consider the Hamiltonian for a free Dirac field

H = / (Pxîxhibjdj + m]ij){x) , (18.88)

where 7M are the euclidean gamma matrices introduced in chapter 3, satisfyingthe anticommutation relations {7M,7,,} = 2<5M,,. The charge operator is

Q = f d3xip^{x)ip(x) . (18.89)

Within the continuum approach the (formal) path integral representation of thegrand canonical partition function

E = Tr{e-^H-^Y (18.90)

is given by

Z=N f £ty«£tye-/0W^*[*'<*)W+™W*)-M**(*)*(*)] ( 1 8 9 1 )J antip.

where AT is a dimensioned normalization factor. The subscript "antip." is toremind the reader that the Grassmann integration is to be performed with thefields ip satisfying antiperiodic boundary conditions in "time". Notice that thisform for the partition function is formally obtained from (18.76b) by i) expandinge~£fi to leading order in e, ii) replacing the Grassmann variables a;]n and a\n

by* tpa(x,T) and ip^(x,r), and approximating CHIP*{X,T)II){X,T + e) by the localform e/j,ipa(x, T)i(ja(x, T). This amounts to taking the naive continuum limit of thediscretized action.

The definition (18.91) is the usual starting point for discussing the thermo-dynamics of a Dirac gas within the path integral framework (see e.g. Kapusta(1989)). We now proceed as in the case of the free scalar field, and rewrite thepath integral in terms of dimensionless variables, by scaling the fields, space-timecoordinates, and the mass m with an appropriate power of /3 according to theircanonical dimensions. Thus introducing the dimensionless variables ty = /33/<2i/>,

* Recall that (x, a) label the degrees of freedom.


ij)* = /32 / 3^*, f = T/{3, Xi - Xi/P, fi = Pn,ih = /3m, the expression (18.91) takes

the form

E = Sf[ rty'Dfa-ti**!***'^-^*™ , (18.92a)J antip.

where•H = a • V +-y4m , (18.92b)

with a = 747, is the Hamiltonian density measured in units of /3"1 , and N isnow a dimensionless normalization factor. Since the integral (18.92a) is of the"Gaussian" type, it can be performed immediately. As we have seen in chapter 2,the result is just the determinant of the operator appearing within square bracketsin (18.92a). Hence

In H = In J\f + In det (94 -p. + fi)

= inAf + lndet[/J(0r - n + H)],

where Ti = a • V + 74m is the dimensioned Hamiltonian density. The deter-minant is given by the product of the eigenvalues of the operator acting in thespace of functions satisfying antiperiodic boundary conditions on the compactifiedeuclidean time interval [0,0\. The eigenfunctions of dT — /n + % have the formexp(iu>£T) exp(ip- x)W(p), where

C = ^ (18.93)

are the dimensionful Matsubara frequencies corresponding to (18.69), and whereW(p) are eigenvectors of the operator id • p + 74m. The eigenvalues are given by±E(p), where

E[p) = Vp^ + m2 , (18.94)

and are two-fold degenerate, corresponding to the two possible spin projections.Hence the eigenvalues of dT — + % are given by

Xfj=tu>t-H±E(0), (18.95)

For the logarithm of the partition function we therefore obtain

In2 = lntf-H 2££{ln[/?(tW < - M + E(f))} + (E -> -E)} ,v i


where t runs over all possible integers. As in the case of the scalar field, wehave enclosed the system into a box of finite volume to discretize the momenta.Collecting the positive and negative frequency contributions we can write thisexpression in an explicit real form:

oo

In 3 = l n ^ + 2 £ £(ln{[(2£ + l)^]2 + [p(E - M)]2} + (E -> -E)) .P e=o

The frequency sum diverges. To extract the finite temperature and chemical po-tential dependent part let us write the logarithm in the form*

H/3(£-M)]2 -,

H[{2l+l)n?+[f3{E-n)]2} = / dy2 ' 2+ln[(2m)V+l] .

Dropping the irrelevant constant contribution, and performing the sum over £ bymaking use of

^Q{2i+ iyn* + y i - 4ytaUh2 '

One then finds that

r ,3 J ( } (18-96)+ 2V - i^{ ln( l + e-«B-">)+ln(l + e-«B+">)} ,

where we have replaced the formal sum over momenta by an integral accordingto (18.37). Apart from the first two terms, this is the expression familiar fromstatistical mechanics. The second term gives rise to a constant energy densityshift and is just the contribution of the negative energy states in the Dirac sea. Inthis connection, recall that within the path integral approach, there is no normalordering prescription. Of course we also would expect to see a corresponding shiftin the mean charge density. In fact, comparison of the formal path integral expres-sion (18.92a) with (18.76a), where the normalization factor is given by (18.75b),suggests that InN is given by ^ C J / ^ J where i labels the (infinite) number of de-grees of freedom. Since J2i ~ 2V f d3p/(2n)3, one then easily verifies that thefirst term in (18.96) gives rise to the expected contribution to the mean charge<Q>— Tj^jf^, arising from the Dirac sea.

* We follow here the method of Kapusta (1989)


18.11 Dirac Gas of Wilson Fermions on the Lattice

Let us now consider a lattice regularized version of the partition function forthe Dirac field. Since we are interested in computing thermodynamical observables,we must be able to vary the temperature and volume in a continuous way. Hencewe need an expression for the partition function formulated on an anisotropicspace-time lattice, i.e., with different temporal and spatial lattice spacings, aT

and a, respectively. The partition function will then be given by a path integralover Grassmann variables living on the space-time lattice sites n = (n, 114), wheren labels the coordinate degrees of freedom and takes all possible integer values,while 714 runs over a finite number of lattice sites, 1 < 714 < NT, where NTaT is theinverse temperature /?. Since NT is the inverse temperature measured in latticeunits, we will use the more suggestive notation /? in the following. The discretizedversion of the action having the correct naive continuum limit is not unique. Weshall take it to be of a form which is consistent with the fermionic actions usuallyemployed in Monte Carlo simulations of gauge theories. This means that, first ofall, the kinetic term has a structure analogous to that appearing in the action of theexpression (18.80). Secondly, the spatial derivative appearing in the Hamiltonian(18.88) will also be discretized using the symmetric lattice derivative, in order thatthe Hamiltonian be hermitean. With this prescription the action appearing in theexponential in (18.80) translates into

SF= £ £ a W n ) - [ e - A ^ + e4)-e^(n-e4)]ri4 = l n I • /

+ aTij)i(n)[atj • dj + 74m]V'(n)} ,

where a summation over repeated indices is understood, and where

dii>(n) = i - ty (n + it) - 1>{n - !<)] . (18.976)

is the symmetric lattice derivative of ip(n), oij = 747^, with e$ a unit vectorpointing in the i'th-direction. The action (18.97) can be written in an explicitdimensionless form by scaling the fields tp and ip\ as well as the spatial derivativesand fermion mass, with the spatial lattice spacing a according to their canonicaldimension. The corresponding dimensionless quantities will be denoted as usualwith a "hat". One is then led to the following path integral representation of thepartition function

r 0

3 = ti / H I ] d&(n)#a(fi) e-s*\ ^ . n , ^ , ^ , , (18.98a)J 3 i a n 4 = i ^(fl,4+D=-*(fl,i)


where

S F = ^ ^ { ^ t ( n ) - [ e - ^ ( n + e 4 ) - e ^ ( n - e 4 ) ] + -V; t (n) [ a j - .4+74^(71)} ,

(18.986)and £ = ^- is the anisotropy parameter. The (dimensionless) lattice derivative djis defined by (18.97b) with a = 1. For a fixed number of temporal lattice sites thetemperature is now controled by the parameter £.

The action (18.98b) is expected to lead to fermion doubling. In particular,the kinetic term does not have the form dictated by the functional formalism. Insection 9 we had demonstrated in a non-relativistic model that such a choice forthe kinetic term leads to a non-trivial modification of the partition function. Thisis also the case for the relativistic Dirac gas, as has been discussed by Bender etal. [Bender (1993)].

The fermion doubling problem can be avoided by introducing a Wilson term.For vanishing chemical potential and zero temperature, where the extension ofthe lattice in the time direction is infinite, this Wilson term is given in (4.28),and exhibits a hypercubic symmetry. This symmetry is is the remnant of the0(4) symmetry in the continuum, and is broken down to a spatial cubic sym-metry at finite temperature due to the presence of the heat bath, as is evidentfrom (18.98b). The Wilson parameters multiplying the contributions involvingthe second time-and space derivatives need therefore not be equal. The finite tem-perature parametrization of the Wilson term is ambiguous. The reason is thatthis (so called irrelevant) term vanishes linearly with the lattice spacing in thenaive continuum limit. Expression (18.98b) however suggests a parametrization,where the ratio of the Wilson parameters associated with the temporal and spa-tial derivative contributions is given by the assymmetry parameter £. Thus forvanishing chemical potential we will take the Wilson term to be of the form

n ' n,j

where

dffln) = i>(n + eM) + 4>(n - eM) - 2ip(n) (18.996)

is the discretized second derivative of V>, and where it is understood from now onthat ^2n = ^2- X2n4=o- By expressing (18.99a) again in terms of the dimensioned

(18.99a)


fields and second derivatives, one easily verifies that such a parametrization im-plies that the temporal and spatial derivative contributions vanish in the naivecontinuum limit linearly with aT and a, respectively.

The chemical potential is now introduced into (18.99) according to the Hasen-fratz Karsch-Kogut prescription [Hasenfratz (1983); Kogut (1983)]

$(n)%i){n + e4) -> •0t(n)e~/iV>(n + e4) ,

ft(n)4>(n - e4) ->• ^ t(n)eA^(n - e4) , (18.100)

which is the analog of (18.81). Then the fermion action (18.98b), supplemented

by the Wilson term (18.99a), takes the form:

4 W ) = {r+1-(m + 3r)] ]T^(n)74<Kn)' n

- 2 E ft(nhi[{r - 74)e"AV-(n + e4) + (r + -y^^n - e4)] (18.101)n

- ^7 y2^(n)j4[(r - 7i)i>(n + ii) + (r + ji)rp(n - e,-)] •^ n,i

This is the generalization of (4.28) to finite temperature and chemical potential(recall that %j) = 74 ) • Notice that our finite temperature, finite chemical potentialparametrization has preserved the r ±7M structure of the T = fi = 0 Wilson action(4.28). We now diagonalize this action by Fourier decomposing the fields ip(n) andip1 (n) as follows (we take $ = NT to be even)

^7" T (18.102)

^w 4 E Y 7r^^ttf.^)e-i**t*<B4).where w are the dimensionless Matsubara frequencies for fermions (18.69), withN identified with /3, i.e.,

*t = {^f^ . (18.103)

Notice that both, the frequencies and momenta, are restricted to the first Brillouinzone. Making use of the relation

1 0

J E E e^-P'ê^'^"^ = 6u,(2n)35^0-P') , (18.104)


where 6p (p—p') is the three dimensional version of the periodic J-function (2.64),the action takes the following form for r = 1,

P ,_ a J-* ^n>

where

K{p, (bt, £, £) = i sin(u^ + ip,) + -H{p, ue, 4, p) ,

H(p, Q}t, £,p) = Yîaj sinpj + KM0, wt, £, p.) ,i

andM{l<ht,Z,p) = M{p) + 2tism2(^±^) ,

M{p) = m + 2y"sin2^- . (18.105)j

The matrix K can be further diagonalized in Dirac space. Then the partitionfunction is given by the product of the eigenvalues (which are two-fold degenerate)

A± =ism{wi + ip)± -z£(]>,Cji,£,p) ,

where

£(p,£/,£.£) = jY,^2Pj + M2{p,Cjt,i,p) • (18.106)

Hence, formally, the logarithm of the partition function (ignoring an additive con-tribution arising from the normalization factor in (18.98a)), which should notinfluence the thermodynamics, is given by

x 4 - iIn3 = 2 ^ J2 H s i n 2 ( ^ + i/x) + — £2(p,w/,C,£)] , (18.107)

where the factor 2 accounts for the two spin degrees of freedom. Making thereplacement

tÊM" (i8io8)P=-7T


valid in the large volume limit (V" is the volume measured in lattice units), we

obtain

lnS = 2V V f' TTTI In[sin2(o;, + ip.) + ~£20, Ct, £, A)] . (18.109)

*— 2

Prom here we compute the mean energy (in lattice units) and mean charge

Consider for example the mean energy. Performing the indicated differentiationone finds after some algebra that

- I-1

< E >= 2j Y, f ( ^ 3 F ^ + iix^] ' (18.111a)

where

F(a* + iA,$) = - 2 . ^ + M(g)[l-coB(a)< + iA)] ( 1 8 . l n 6 )

^2(p) + 2[1 + M(p)][l - cosfa + v ) ]

with

S( ) = JY^^pj+M^p) , (18.111c)

and M(p) given by (18.105). The frequency sum of F(w + ifi,p) in (18.111a) cannow be easily carried out by making use of either the summation formula (18.86)or, even more conveniently, with the help of the summation formula (D.3) derivedin appendix D. The function analogous to g(el^+l^\ {Pi}) in (D.I) is now definedby g(el(u+lil\p) = F(ui + ip.,p). Correspondingly, g(z) in (D.3) is now given by

9{z) = [z-JJit-z-]'where

f(z) = ?-^[Mz2 - 2(E2 + M)z + M] ,(1 + M)

(18.110a)

(18.1106)


z± = [K(3) ± \l^(3) - 1 ] ,and

K0) = 1 + E ^K . (18.112)2(1+ M(p))

The rest of the calculation is straight forward. After performing the frequencysum in (18.111a) one finds for the energy density measured in lattice units that

1 f* dzr>T < E >= 2 / j~p(p)E(p)[r)F(p) + fjF(p)] + constant , (18.113a)

where

P ( S = ' + *<*>/* m m(1 + M$))yjl + M0) + (E0)/2)2

is a function which approaches unity in the continuum limit, rjpip) and fjplp) arelattice versions of the Dirac distribution functions for particles and antiparticles:

VF(P) = —~ 7=^= , (18.114a)

fjF(p) = — • = , (18.1146)

where K has been defined in (18.112). The continuum limit is realized by settingm = ma, ft = fia, /3 = f j and taking a —>• 0. In this limit we can replace K2

—*2

by 1 + i(p + m2),* One then finds that, apart from a temperature and chemicalpotential independent contribution, the mean energy density in physical unitsreduces to

v<E>=2£ikm [^rn + pôn] • (18-115)A similar calculation of the mean charge density yields

*<Q>=2LWT [ ^ T T " PcôTi] • (1 8 '1 1 6)We close this section with a remark. If we had chosen to work with naive

fermions (i.e., no Wilson term) then the integrals (18.115) and (18.116) would have

* High momentum excitations, corresponding to finite p do not contribute to(18.113a) in the continuum limit, as can be easily verfied.


been multiplied my an additional factor 24, arising from the doubler contributions.Thus by only looking at the partition function, the "doublers" manifest themselvesin a rather trivial way. Our discussion in section 9 however suggests that this isonly so, because the Dirac field excites particle and antiparticle states. In fact bystudying separately the positive and negative energy contributions to the partitionfunction, one finds that in both, the positive as well as negative energy sectors,the partition function resembles that of a gas of particles and antiparticles, exceptfor an overall factor 23 arising from the "doublers" associated with high-three-momentum excitations at the corners of the Brillouine zone [Bender (1993)]. Thisagrees with our findings in section 9, where we discussed the doubling problem ina simple non-relativistic model.

CHAPTER 19

FINITE TEMPERATURE PERTURBATION THEORYOFF AND ON THE LATTICE

In the previous chapter we have studied in detail the path-integral represen-tation of the partition function for some non-interacting bosonic, and fermionicsystems. Because of the Gaussian nature of the integrations we were able to calcu-late the path integrals explicitely. In the presence of interactions this is in generalno longer possible and one has to recur either to perturbation theory, or evaluatethermodynamical observables within the framework of a lattice regularized the-ory by Monte Carlo methods. In this chapter we shall show how the euclideanFeynman rules at zero temperature are modified when a relativistic system of in-teracting fields is placed in contact with a heat bath. We will first demonstratethis for the case of the Xcjft theory in the continuum formulation. As we shallsee, the prescription for making the transition from the zero-temperature, zero-chemical potential Feynman rules to the T ^ 0, fi =£ 0 rules turns out to be verysimple. The A</>4-theory is considered in detail in the book by Kapusta (1989).We shall therefore only discuss some elementary aspects of this theory, since itprovides a simple laboratory for studying the effects arising from the presence of aheat bath. The remaining part of this chapter will then be devoted to gauge fieldtheories at finite temperature and chemical potential, both in the continuum aswell as on the lattice. Although the prescription for making the transition fromthe zero-temperature, zero-chemical potential lattice Feynman rules to the 1 ^ 0 ,(i / 0 rules turns out again to be very simple, the actual computation of latticeFeynman integrals at finite temperature and chemical potential is more involvedthan in the continuum, as we shall demonstrate in some sample calculations.

19.1 Feynman Rules for Thermal Green Functions in the A<64-Theory

Let <f>(x) be a real scalar field whose dynamics is governed by a HamiltonianH. At zero temperature all physical information about the system is contained inthe ground state expectation value of time ordered products of the field operators$(x). When the system is placed in contact with a heat bath, all possible energyeigenstates of H are excited with a probabilty given by the Boltzmann factor. Thethermal correlation functions are then defined by

< (p(x1)...<p(xn) >/3= Tr e_pH , (19.1)

424

Finite Temperature Perturbation Theory Off and On the Lattice 425

where Xi = (a?i,Tj), and $(x) are the field operators whose euclidean time de-pendence is given by (2.17). By introducing a complete set of eigenstates of theHamiltonian, one readily verifies that in the limit of vanishing temperature (i.e.,(3 — |T -> oo) the rhs of (19.1) reduces to the correlation function (2.14). It isnow easy to derive a path-integral expression for (19.1) using the by now familiartechniques. We leave the details to the reader and only quote here the result,which is valid if the Hamiltonian density is given by the sum of a kinetic term,quadratic in the canonical momentum, and an interaction depending only on thefields (f>{x):

J Dcj> <l>(x1)...<Kxn)e-s"')M

Jper X^V/C

HererP r

5 < « M = / dr d3xCE((l>,d^) (19.26)

is the finite temperature action. The path integral is to be evaluated with thescalar field satisfying the periodic boundary condition (18.32). The correlationfunctions can be obtained from the generating functional

Z[J]= [ D^-S^W+S^ dr f ^xJ{g,r)4,(g,r) ^ g}

Jper

in the usual way, by functionally differentiating this expression with respect to thesources J(X,T). If the interacting part of S^\ which we denote by Sj [</>], is apolynomial in the fields, we can also write (19.3) in the form

Z[J] = e-s>'3)^Z0[J], (19.4a)

whereZ0[J] = JD(h-Sl'i)M+fo

0*rf*xJ(*,T)4,(3,r) {194b)

is a path integral of the Gaussian type. Once ZQ[J] is known, Z[J] can be computedperturbatively by expanding the exponential in powers of the interaction.

For the (/>4-theory the finite temperature action is given by

^ M = 4/3)[0] + 5 f ) M , (19.5a)

where

S^W =\ I d4z<Hz)(-D + M2)<t>(x) , (19.56)1 J0

(19.2a)


sPM^^fdixMx))4, (19.5c)

and x = (X,T). Here we have introduced the short-hand notation

f dix = f dr f d3x . (19.5d)Jp Jo J

The factor 1/4! in (19.5c) has been introduced for later convenience. Since cj>(x)satisfies periodic boundary conditions, it has the following Fourier decomposition

where oj( are the Matsubara frequencies

UJI - —I . (bosons) (19.66)

Here t takes all possible integers values. The factor 1//3 has been inserted so that<j> carries the same dimension as the corresponding field in the T = 0 formulation.Since

/ dTe*"'-"'1* = P6w , (19.7)./o

we can invert (19.6a):

/•" r4>(we,k) = / dr / d3x<f>(x,T)e-iuJlT-ikS. (19.8)

From (19.6a) and (19.8), we see that the finite temperature expressions follow fromthose at T = 0 by making the substitutions

&4 —> u>£

f dk4 1 (19.9a)

andI dix-+ f d4x . (19.96)

Consider now the generating functional of the free theory defined in (19.4b). Sincethe exponent in the integrand is quadratic in the fields, we can perform the inte-gration and obtain

zo[j] = zo[o]e^dH^ ' eW'WM , (19.10)

(19.6a)


where A1-^ (x — y) is the inverse of the operator — • + M2 (i.e., Green function) onthe space of functions satisfying the periodic boundary conditions (18.32). HenceA^\z) is periodic in the euclidean time direction,

A(«(z,0) = A ( « ( ^

and therefore has the following Fourier expansion

A(/3)W = \ E / ( 0 A ( / 3 ) ^ ' 3«W4**- (IS-")

The propagator in momentum space, A^(ui, k), can be determined by insertingthis expression into the equation

(-D + M2)A^(z) = 5(z)8P(r),

where 5p(r) is the periodic <5-function

One finds that

A ^ W ) =-;—i—— . (19.13)uJ\ + k2 + M2

The corresponding expression for A^(z) reads

Comparing (19.13) and (19.14) with their zero temperature counterparts

A w = vTW2 ' (19-15fl)

/J4L, pik-z

where k = (fc, fc4), and k2 = k2 + k2, we see that they are related by the rules(19.9a).

The temperature dependence of (19.14) is contained in the infinite sum overthe temperature dependent Matsubara frequencies (19.6b). The frequency sum-mation formula we shall derive below allows us to decompose A^'(z) into a zero

(19.12)

(19.14)

(19.156)


and a finite temperature contribution arising from the presence of the heat bath.Suppose we want to calculate the sum

F(K) = ^f^t,K), (19.16)

where K stands collectively for all remaining variables on which the function /may depend. Let us promote u>e to a continuous variable u> and define the functionf(ui,K). Suppose that f(u>,K) has no singularities on the real axis. Consider thefollowing function of the complex variable w,

h^ = ^ S n • (19-17)It has poles located at w = ^jt with unit residue. Then we can write the sum(19.16) as follows,

F{K) = ^Jdu>h(u>)f{u>,K), (19.18)

where C is the contour shown in fig. (19-1).

^

c_

Fig. 19-1 Upper and lower branches of the contour C in eq. (19.18).

In the lower half plane (19.17) decreases exponentially for large imaginaryparts of u. In the upper half plane, on the other hand, h(ui) is finite for Im(w) —>•oo. Since we eventually want to close the contours at infinity, we shall make useof the following alternative form for (19.17) on the upper C+-branch of C:

Then (19.18) can also be written as follows:

2TT 7_o o + i e e-/*" - 1

(19.19)


If f{u),K) is sufficiently well behaved for |w| —¥ oo, so that we can close theintegration contours in the second and third integrals in the lower and upper halfplanes at infinity, and if /(CJ, K) is a meromorphic function of ui, then

D /,-, \ i (19.20)

eH3u>i _ 1/raSi<0

where Rf(u>i) are the residues of f(u)i,K) at the poles whose positions we havedenoted by £>*.

As an application of (19.19) we derive the expression for the propagator inMinkowski space, as the analytic continuation of the euclidean propagator (19.14)to real times. An alternative derivation, has been given by Dolan and Jackiw(1974).

Consider the finite temperature expression (19.14) for the euclidean propa-

gator. The function f(w,K) is now given by exp(iwr)/(w2 + k2 + M2). Since

T\ is restricted to the interval [O,0\, we can close the contours in the second and

third integrals in (19.19) in the lower and upper half of the complex oi-plane,

respectively, and obtain

(2ffc2 + M 2 (19.21a)

/d3k 1 -

where

E= \Jk2 + M2 ,and

^ ) = ^ F T T (19-21fc)is the Bose-Einstein distribution function. In (19.21a) we have set ui = k$. Notethat the first integral is just the propagator for vanishing temperature.

We next continue this expression to real time by setting r = it. Performingthe usual Wick rotation (£4 —> —ik°) in the first integral one has that*

W T=it-lJ (2TT)4 k2 - M2 + ie +

+27TI -0^^v(E)(S(k° -E) + 5(k° + E))e~ik-',

* The ie-prescription is that of the T = 0 field theory.


where z = (z, z°), k • z = k°z° - k • z. Making use of the relation

5(k2 - M2) = ^r-[5(k° + E) + 5(k° - E)] (19.22)

we finally arrive at the following expression for the propagator in Minkowski space,

AgUto = / (^SW^"*"' (19.23a)

where

The second term can be interpreted as the contribution arising from on-mass shellparticles in the heat bath.

The reader must be warned that (19.23) is not the propagator appearing inthe Feynman rules for real-time Green functions in an interacting theory. Thereal-time Feynman rules are more complicated and require a doubling of degreesof freedom. *.

Having discussed in detail the generating functional of the free theory, wenow obtain the Feynman rules for the thermal Green functions (19-1) in the stan-dard way from the generating functional (19.4). This generating functional differsfrom that at zero temperature in that the zero temperature propagator (19.15b)is replaced by (19.14), and that the euclidean time integration is restricted to theinterval [0,/?]. It is therefore evident that the T / 0 Feynman rules in coordi-nate space are obtained from those at zero temperature by merely making thereplacements

A(z)->A<«(z) ,

/ dr ->• dr .J-oo JO

The corresponding rules in frequency-momentum space can also easily be obtainedby making use of the Fourier decomposition (19.14) and of the orthogonality re-lation (19.7), which is the analog of f_oo dr exp(ip4r) — 2-K5(JP±) at zero tempera-ture. Thus the integration over the space-time coordinates of a vertex leads to theappearance of a factor (3 and a Kronecker-5 which enforces that the sum over Mat-subara frequencies flowing into the vertex equals the sum over frequencies flowing

* See e.g. Niemi and Semenov (1984); Landsman and Weert (1987)

(19.236)


out of the vertex. A thermal correlation function then has the form

_, x V - 1 /" d3h ^ 1 f d3k2

G(Z1,Z2,...)-^- ~^-J j ^ - - .h l* (19.24)

x G K , h;ujt2, fc2; • • •) I ]e^T l + i* i - ? i ,i

where Z{ = (ii,Tj), and where w^ and ft; are the incoming frequencies and mo-menta associated with the external lines of a Feynman diagram. The KernelG(w(.x, fci; u>e2, k2; • • •) has the following structure

G(uilX^t3^---) = P(2n)3Sj2,uttio^3H^^)g{uii1M]u)ta,k2\---) (19-25)i

and is computed with the following Feynman rules:

i) To each line of the diagram, which we label by an index i (i = 1,2,...),associate a propagator

A w k . t i ) = 2 I *,2 • ^19-26)w\. + k} + M2

where ki is the momentum carried by the «'th line, and u)^ is the correspondingdiscrete energy, 2itl.il(1.

ii) To each vertex assign the factor

-^(2^3<K£)/?«5W,O , (19.27)

where w and K is the sum of the Matsubara frequencies and momenta flowing intothe vertex.

iii) Sum over all discrete energies of the internal lines, and integrate over themomenta carried by these lines according to

iv) Multiply the resulting expression with the factor, resulting from the num-ber of distinct ways one can build the diagram from the given number n of vertices,and a factor ^ arising from the n'th order term in the expansion of exp(—Si).

(19.28)


x zVl /^ y

Fig . 19-2 Two loop contribution to the propagator.

As an example consider the second order contribution to the propagator shownin fig. (19-2).

D^\x,y) = ^ I d4z I d4z'A^(x - z) [A^(Z - z')]* A^{z' - y). (19.29)

The factor 6 arises as follows. There are eight possibilities to identify an externalline emanating from the two ^4-vertices with the space-time coordinates x. Thisleaves us with 4 possibilities for choosing y. The remaining lines can then be tiedtogether in 3! ways. From the expansion of exp(—51} ) to second order we geta factor 1/2!. This leaves us with a factor (4!)2/6. Finally we must multiply theexpression with (A/4!)2.

Applying the above Feynman rules, the corresponding momentum-space rep-resentation of (19.29) is given by

ft® (wtl, h; ue,, k2) = p(27r)35U(i + ^ fi5^ (h + k2)D^ (wtl, h) (19.30a)

where

w?! +kl + M2 J\x + k\ + M2

and where the self energy is given by

1X [(u< - wv - w r ) 2 + {k- k' - fc7')2 + M2}

(19.30c)Notice that this latter expression could have been immediately obtained from thecorresponding zero temperature expression according to the rules (19.9a). Thecorresponding expression in coordinate space is then given according to (19.24).

(19.306)


19.2 Generation of a Dynamical Mass at T / 0

Having obtained the finite temperature Feynman rules, we are now want tostudy some features of the A04-theory which arise from the presence of the heatbath.* The first phenomenon we will consider is the generation of a dynamicalmass at T / 0. As we shall see below, a temperature-dependent mass is generatedalready on the one-loop level. To this order, only mass renormalization needs to betaken into account. The relevant euclidean action, including the mass counterterm,is therefore given by

S[<l>] = I [ d4x (f>(x)(-0 + M2)<f>(x) + A / d*x [4>{x)]A + l-5M2 [ d 4 z [<f>{x)}2 .* J(3 Jf3 2 Jp

(19.31)We have not included the factor ^ in the interaction term, since the formulaewe shall obtain then take a simpler form. The term proportional to 8M2 is thecontribution of the mass counterterm. In lowest order perturbation theory therenormalized inverse propagator is given in momentum space by

[A{ut, k)}-1 = wj + k2 + M2 + 7r(^(T) , (19.32)

where ix(R' (T) is the renormalized self-energy in the one loop approximation:

•K<£\T) = -12 _ Q _ +5M2. (19.33)

We will determine 5M2 below in such a way that M is the mass of the <f> field atzero temperature. The factor 12 multiplying the one-loop contribution arises asfollows: there are 6 ways of contracting two lines emanating from a 04-vertex toform a loop, and 2 possibilities for connecting the remaining lines to two points xand y in a propagator. Hence the diagram is weighted with a factor 12. With theFeynman rules given in the previous section, we otain for (19.33)

4«m = m £ i / | ^ - F - ^ + ™ > (1,M)

The integral is infrared (IR.) finite for M / 0 , but ultraviolet (UV) divergent. Thisdivergence is entirely contained in the zero-temperature contribution.** Indeed,

* Much of our discussion in this and the following section closely parallels thatof Kapusta (1989). We have included it here, since the problems we address to arealso of relevance in QCD.

** Since the presence of the heat bath should not affect the short distancebehaviour of the correlation functions, we expect that thermal Green functions willbe ultraviolet finite once the theory has been renormalized at zero temperature.See Norton and Cornwall (1975); Kislinger and Morley (1976, 1979).


carrying out the frequency sum, making use of the summation formula (19.20), we

can write (19.34) in the form

*£\T) = 41}(0) + 12A / ~ - ^ + *M\

where

is the unrenormalized self energy in one-loop order for vanishing temperature, and

E = yk2 + M2. The second term appearing on the rhs is UV finite because of the

appearance of the Bose-Einstein distribution function, and vanishes for P -4 oo,

i.e., for T = 0. We therefore see that the UV-divergence is entirely contained

in the T = 0 contribution, which is renormalized in the standard way. We first

regulate the divergent integral by introducing a momentum cutoff A. By choosing

/

A H^h 1

KW (19-35)

we eliminate the above UV-divergence, and ensure that M is the physical mass

of the 0-field in one loop order at zero temperature. With this renormalization

prescription, we are thus led to the following expression for ir^'(T):

•SPm-'w/l^j^rT- 09.36)

In the limit M —> 0, this expression reduces to

K%\T) = Xr2, (Af = 0 ) . (19.37)

Hence at T ^ 0 a dynamical mass is generated in one loop order.

19.3 Perturbative Expansion of the Thermodynamical Potential

Using the finite temperature Feynman rules derived in section 1 we can now

calculate perturbative corrections to the thermodynamical potential

n = --^lnZ (19.38)

of a free gas of neutral spinless bosons. The partition function is given by

Z = Aff £>0e-SoW>]-S/[0] (19.39)J periodic


where 5o[ >] is the finite temperature action of the free field, and £/[</>] is theinteraction term. Here we have dropped for simplicity the superscript (3 in theaction S^[</>]. The normalization factor J\f carries the dimension of the inverseintegration measure.

The perturbative espansion of Z is obtained by expanding the exponentialexp(—Si) in (19.39) in powers of Si* One easily verifies that

Z = Z0 i + Y/±f-((SI)l)So\, (19.40a)

1=1 J

whereZo = M j D0e - S o W , (19.406)

and, f DHStWe-™ ( 1 9 4 0 c l

{Sl)s° JD^SOW ( 1 9 - 4 ° C )

is the expectation value of Slj[4>] calculated with the Boltzmann distribution of thefree theory. Prom (19.40a) we have that In Z is given by

mz = \nZ0 + J2Ut- ESr-^ft • (is-41)n=l n Ll=l '• J

The expectation value {S\)s0, which can be computed from the free generatingfunctional (19.4b), receives contributions from connected and disconnected Feyn-man diagrams with no external lines. But only the former ones actually contributeto the sum. This is expected since the free energy is an extensive quantity. Infact, as we now show, (19.41) can also be written in the form

In Z = In Zo + f ] ^ ( 5 } ) cS o , (19.42)

where the superscript "c" stands for "connected". The proof of (19.42) goes asfollows.

Consider the ensemble average of a power of 5/, i.e., (S\)s0- It is givenby the sum over all possible pairwise contractions (i.e., propagators) of all thefields appearing in S\. Hence (Sj) can be decomposed into a sum of products

* Our following presentation closely parallels that of Kapusta (1989). We nev-ertheless include it here for completeness sake.


of connected components, each of which consists of one or more vertices. Thecontribution of a connected component, consisting of n vertices, we denote by(5™)so. Let an be the number of times that this connected component appears ina particular product. Since there are a total of / vertices, the products of connectedcomponents have the form

(SîS^-iS^ , (19.43a)

where

ox + 2a2 + h kak = / , (19.436)

and where for simplicity of notation we have now written (S")c instead of (5™)|o.There are however, in general, several terms appearing in the decomposition of< 5} >50 yielding identical contributions (19.43a), since it does not matter whichcollection of n vertices are used to make up a particular product of connectedcomponents (19.43a). Hence (S|) will be of the form

(Si) = E E C^-aASiWiS])? • • • (S*)ack6ai+2a2+...+kak^0 , (19.44)

where the 5 -function ensures that we pick up the contribution of order I. We nowcompute the combinatorial factor Caiai-ak- To this effect we first numerate the /vertices making up 5} from 1 to I. Consider a particular partition of the / verticesinto connected components. By permuting the I vertices we generate /! sets ofconnected components, all of which yield the same contribution to {S\). Thesesets are however not all distinct, for the V. permutations also include those whichmerely permute the an connected components of the type < Sf >%0- These mustnot be counted as distinct. We therefore must divide l\ by a\\a,2\...afc!. But thereis another factor which must be divided out. Recall, that the definition, {S™)C

SQ

includes all possible pairwise contractions of the fields. Hence permutations ofthe vertices within a connected component of order n are already included in thedefinition of (5?)|Q. We therefore must also divide by (l!)Ol(2!)Q2...(fc!)a*. Hencewe conclude that the combinatorial factor in (19.44) is given by

Inserting this expression into (19.44), and carrying out the sum over / by making


use of the (^-function, we conclude that

1=0 n=l \ a n = 0 n L J /

where we have now again introduced the subscript SQ. Hence, according to(19.40a), \nZ is given by (19.42), which is the result we wanted to prove. Thethermodynamical potential (19.38) is therefore given by

n = V0-w^[-^-{S?ySB. (19.45)

Let us now calculate the lowest perturbative correction of O(\) to the ideal gasformula (18.41), i.e.,

for the case of the A04-theory, where Si[(f>] (including the mass counter term) isgiven by the last two terms in (19.31). The expectation value (Si)s0 is then givenby

<Sj)So = f d4x {3A[A^(0)]2 + ]-6M2AW{0)} . (19.46)Jp l

Here A^'(0) is the finite temperature propagator (19.14) evaluated for vanishingargument. The factor 3 takes into account the three distinct ways in which thefour fields emanating from the vertex can be contracted to form a double loop.Because the integrand in (19.46) does not depend on x, (Si)s0

ls proportional to

<5/>s0 = /?V[3A(A(/3)(0))2 + |*M2A<fl(0)] . (19.47)

Let us next isolate the zero-temperature contribution to A ^ ( 0 ) . According to(19.21),

A<«(0) = i(0) + / ^ 3 i p s L - [ , (19.48)

where A(0) is given by (19.15b) with z = 0. Introducing the decomposition(19.48) into (19.47), and recalling that we had already determined SM2 to be


given by (19.35), one finds that, apart from an irrelevant additive constant, theO(A)-correction to the thermodynamical potential is given by

n^ = w{Sl)=sx[Iwr^^^\ . (19.49)M->O 48

Combining this result for M = 0 with (18.42) we are therefore left with thefollowing expression valid up to O(X):

fi^ = - ( S - i A ) r 4 ; <M=°)-From lnZ = — (3Vil we can calculate the mean energy < E > and pressureaccording to

<E> = --^-\nZ

x% (19-50)

One then verifies that the mean energy density and pressure are lowered in O(A).In higher orders of perturbation theory one is in general faced, in the zero

mass limit, with infrared divergent integrals. By a partial resummation of higherorder Feynman diagrams with self energy insertions, the propagators in a diagramof given order are replaced by massive propagators with a temperature dependentmass, which in lowest order is given by (19.37). This eliminates the infrared di-vergencies, but leads to a non-analytic behaviour in the coupling constant. Thereader may consult the book by Kapusta (1989), where the self energy and ther-modynamical potential is discussed in the so called "ring"-approximation. Herewe shall illustrate the problems one encounters by considering the contribution tothe thermodynamical potential arising from a class of Feynman graphs which arenot of the ring type.

Consider the class of diagrams shown in fig. (19-3).

Suppose the diagram has N vertices. The number of propagators is 2N. Becauseof energy (frequency)-momentum conservation there are only N + 1 independentloop-momenta, and N + 1 frequency sums. In the limit M -» 0 the dominantinfrared divergence arises from that term in the sum over Matsubara frequencieswhere all frequencies vanish. The corresponding contribution has the form

n(u,=O) „ XNTN+, / dZki d3kN+i p j l_ ( ( 1 9 5 1 )

"* 1=1 qi


Fig. 19-3 Generic diagram contributing to the thermodynamical po-

tential, which is not of the "ring" type.

where " i" labels the the frequencies and 3-momenta of the internal lines, and wherethe gz's are homogeneous linear combinations of the N + 1 integration variables.The lowest order diagram corresponds to N = 3, and, by naive power counting,is infrared convergent for M —> 0. For N > 3 we are however faced with infrareddivergent integrals. We can eliminate the IR-divergencies by summing diagramswith the same skeleton-structure as that depicted in fig. (19-3) , but with an arbi-trary number of (renormalized) self energy insertions. This amounts to replacingthe bare propagators by, i.e., by

A{uni,ki) = — f2 ,^li + ki + KR

where the self energy ITR depends in general not only on the temperature, butalso on the frequency and momentum carried by the line. Inserting for -KR theexpression (19.37), valid in lowest order perturbation theory, and setting wn. = 0,one therefore has that (19.51) is replaced by

0 ^ ~ A*T™ I ^ , . A + 1 _[[ ^ ^ y , (19.52)

where m(T) — y/XT. This cures the infrared divergence. By scaling the momentawith m(T) one is therefore led to the conclusion that O receives a contribution ofthe form

~x3T\MT)J ' ( A r-3 ) {19-53)

This is an interesting result, for it tells us that for m(T) = %/AT1, the contributionto the thermodynamical potential is non-analytic in the coupling constant for


N — 3 ^ 2k (k a positive integer), and that it is of lower order than A^. Notice

that a naive perturbative expansion of the propagators in (19.52) would lead one

to conclude that the leading order contribution is of O(XN). This conclusion is

however incorrect, since each term in the expansion is IR-divergent for N > 3, with

the divergence getting worse as one proceeds to higher orders. The resummation

of these divergent contributions has thus led us to the above result.

Finally we remark that if the mass generated at T ^ 0 had been of O(\T)

(rather than O(\f\T)), then we would be faced with the problem that there are

an infinite number of non-trivial diagrams which contribute in O(A3). As has been

pointed out by Linde (1980) such a computational barrier arises in QCD, and is

known there as the " infrared problem".

19.4 Feynman Rules for QED and QCD at non-vanishing

Temperature and Chemical Potential in the Continuum

In section 1 we have shown for the 04-theory, that the transition from the T = 0

Feynman rules to the rules at finite temperature is effected in a very simple way. A

temperature was introduced by merely compactifying the euclidean time direction,

and imposing periodic boundary conditions on the fields. Hence the prescriptions

given in section 1 apply to any bosonic theory. In the case of QED or QCD, where

the gauge potentials are coupled to the fermion fields, these prescriptions must be

supplemented by corresponding ones for the fermionic degrees of freedom. The

ghost degrees of freedom are subject to the same rules as the gauge potentials, as

we have already pointed out in chapter 18.

To keep our discussion general, we will allow also for a non-vanishing chemical

potential n coupled to the fermion charge density. The corresponding actions for

QED and QCD have the form

S = SG+ f d4a;^(x)[74(94 - n) + 7i<% + m]ip{x)h (19.54)

+ ie ^xtpix^Â^xpix) + SGF + Sghost,hwhere SG is the action for the gauge fields, L has been defined in (19.5d), and

SQF and Sghost are the gauge fixing term and ghost contributions arising from the

Faddeev-Popov gauge fixing procedure. For QCD ij) and ip are three-component

colour vectors, and A^ is the matrix valued field defined in (6.10). The fields

A^x) satisfy the boundary condition

A»(x,f3) = Afi(x,0). (19.55)


The same is true for the ghost fields. The fermion fields, on the other hand,satisfy antiperiodic boundary conditions in euclidean time, as we had shown inchapter 18. This leads to a discretization of the fourth component of the fermionmomentum, P4 —> ui^ = g(2£ + 1)TT, where wj are the Matsubara frequencies forfermions (18.93). We have denoted them here with a superscript "-" in order todistinguish them from the Matsubara frequencies (19.6b) for bosons, which willhenceforth be denoted by w£. Prom (19.54) we see that the chemical potentialonly appears in the kinetic term of the fermionic fields in the combination 84 — /x,which in frequency space takes the form i(uj" + i/i). It is thus evident that thechemical potential will only manifest itself in the fermion propagator.

From the above discussion it is evident that the euclidean continuum Feynmanrules for QED and QCD at T =£ 0, /J, ^ 0 in frequency-momentum space followimmediately from those at T — \i = 0 by the following simple prescriptions:

i) Consider the euclidean expressions for the propagators and vertices at T =fi = 0 in momentum space. They are obtained by taking the naive continuumlimit of the lattice expressions given in chapter 14 and 15. In particular for QCDthey are given by*

Fermion propagator

• I • r-i-YnPy + mP, b a , a p2 + m2

Gluon propagator

* k v 1 KK

* In the case of QED in a linear gauge, the vertices coupling three and fourgauge potentials, as well as the vertex involving the ghost fields are absent. Theremaining expressions for the propagators and photon-fermion vertex are obtainedby omitting the colour factors. The general prescription ii)-vi) given below there-fore also hold for QED.


Ghost propagator

¥SAB

Fermion-gluon vertex

Ghost-gluon vertex

*g(27r)4j(4> (fc + p - p ' ) / A B c ( p %

Three-gluon vertex

-ig(2n)45^(kA + kB + kc)fABC [<Wfl(*U - ke)^ + c.p]

Four-gluon vertex

-g2(27r)i5^(kA + kB + kc + kD)

*[fABEfcDE{b»Alj,cb»BHD ~ ^tiAliD^^Bt^c)

+ fACBfBDE{SfiAliBS^ctlD — ^HAHDÎIBHC)

+fADEfcBE(SfiAncSVBHD ~ ^ M u ^ D M c ) ]

ii) Replace the fourth component of the momentum associated with a photonor ghost line as follows

* 4 - > w + ; w / = ^ f > (19-56)


where I is an integer.

iii) Replace the fourth component of the momentum associated with a fermionline as follows

p 4 - > w , - + v ; ^ - = ( 2 ^ 1 ) 7 r . (19.57)

iv) Integrations over the fourth component of momenta become infinite sumsover Matsubara frequencies

/ f^/(*4; •••)-> \ E / H + ; • • •); (&osons) >, d 1 ^ _ (19-58)

I ~2^^Pi] ' " ^ ~B^ Û* + *^;' ' ') ; {fermions) .

v) At each vertex implement energy momentum conservation by *

0(2*)36fcuri+J2vti)tl3)(Eiti>+ £,&>) , (19.59)

i j i j

where i and j label the different three-momenta and Matsubara frequencies flowinginto the vertex.

vi) For every fermion or ghost loop include a minus sign.

The euclidean correlation functions are then given by

/

n

-) i = l

UWi II^G«1-/'1-Ml-(te}!{^}!{^})e^-J)lX-+i^^^+Ei*^

(19.60)

* To emphasize the analogy with the T = 0 formulation, we have also written^(Si w ^ + Y^j^tj) instead of < y> W~+V w+)o- Notice that since at each ver-tex two fermion lines are coupled to the gauge potential, the sum over fermionicMatsubara frequencies will add to a Matsubara frequency of the bosonic type.


where we have suppressed the colour indices on the fields and the momentum-spaceGreen function (so that the expression also holds for QED), and have introducedthe compact notations

Pi = (pi,u£) ; h = (£i,w£+) , (19.61a)

G!ai"-0i-"/i1---({pj}, {Pjh {kj}) is the correlation function in frequency-momentumspace calculated with the above Feynman rules.

The Feynman integrals in QED or QCD at finite temperature will involvein general sums over bosonic as well as fermionic Matsubara frequencies. Theevaluation of such sums for bosons has been discussed in section 1. We now derivethe analog of (19.19) and (19.20) for the case of fermions.

Consider the function

This function possesses simple poles with unit residue located at u> = wj. As-suming that f(u) has no singularities on the real axis one finds, proceeding in ananalogous way as in the bosonic case, that

1 f°°+« /(u>)Z7r J-oo+ie e + 1

The first integral is just the zero temperature ((3 —> oo) limit of the lhs. If /(w)

is also a meromorphic function, and is sufficiently well behaved for \u\ —> oo, so

that we can close the integration contours in the last two integrals in the lower

and upper complex w-planes, then (19.62) reduces to

e-i/3w; + 1 '

ImOi>0

where Rf(u>i) are the residues of f(w) at the poles, whose location we have denotedby iDj. This expression allows us, for example, to decompose one-loop Feynman

(19.616)

(19.62)

(19.63)


integrals into a zero-temperature contribution and a contribution arising from thepresence of the heat bath.

19.5 Temporal Structure of the Fermion Propagator at T / 0 and

H / 0 in the Continuum

Consider the fermion Feynman propagator for QED in Minkowski space atvanishing temperature and chemical potential,

sfink\*) = [ £ * Tp*t?•*-*'' (19-64)e J (2TT)4 p2 - m2 -\- le

where 7M are the usual Dirac gamma-matrices satisfying the anticommutation rela-tions {7M,7^} = 2gliv, and p-z = p°z° —p-z. SF{Z) propagates positive (negative)energy states in the forward (backward) direction in time. In the Dirac hole theorythe absence of a negative energy state is interpreted as an antiparticle with positiveenergy. The correspondence between positive (negative) energy states and forward(backward) propagation in time follows immediately from (19.64) by integratingthis expression over p°. The integrand possesses poles located at p° — ±E whereE = ^Jp2 + m2. Depending on the sign of t = z°, we can close the integrationcontour at infinity in the upper or lower half of the complex p°-plane. One thenfinds that

S(F

Mink\z,t) = I -^S{F

Mink){p,ty^ , (19.65a)

where~s(Mink)^t) = A + ( p - ) A ( + ) ( £ i ) + A_(j5)A(->(p,t) , (19.656)

and

A± (P) = 27E [±ry°E + 7JPi + ™\ » (19.65c)

A(±) (p, t) = 9{±t)ezfiBt . (19.65d)

E= x/p^ + M2 .

The (±) superscripts denote the positive and negative energy contributions aris-ing from the poles at p° = ±E. A similar decomposition holds for the euclideanfermion propagator.* The purpose of this section is to study how the temporal

* Although this correlation function no longer describes propagation, we nev-ertheless shall refer to it as a propagator.


structure of this propagator is modified at finite temperature and chemical po-tential. Since the chemical potential controls the particle-antiparticle content, itseffect on the temporal structure will be different for the positive and negative en-ergy contributions. In section 9 we will repeat this exercise within the frameworkof the lattice regularization for naive as well as Wilson fermions. Comparison withthe continuum results obtained below will shed further light on the nature of the"doubler" contributions which we discussed in chapter 4.

The euclidean fermion propagator at finite temperature and chemical poten-tial can be immediately written down using the prescriptions given in the previoussection:

S{F

M fr T) = j ^ ~ A M (P- *)<?** > (19-66a)

where

and

*"«»«>- --^w^Br" • ""•"'The 7-matrices have been denoted here with a superscript 'E' (—> euclidean) inorder to distinguish them from the 7 matrices appearing in (19.64). They are the7 matrices which we have been working with in the euclidean formulation.

The frequency sum (19.66b) can be carried out by making use of the sum-mation formula (19.62). An even more convenient summation formula is given by(C.4) in Appendix C, where the first term on the rhs corresponds to the T = fj, = 0contribution. * One then finds that

S{F

M (p, r) = T+ (?)AgJ.) (p, T) + T_ (p)A\;l} (p, T) , (19.67a)

where

r±(p) = ^ [ ± 7 f E - i-yfpj + m] , (19.676)

is the euclidean analog of A± in (19.65c), and where A L ' , ( p , r ) is given by

* The reader can easily convince himself that this formula can also be appliedto the sum (19.66b) for \r\ e [0,j3].


Here

W ^ ' H ) = e0{E-\) + 1 ' ^D{E,H) = e0iJ,) + l (19-67d)

are the Fermi-Dirac distribution functions for particles and antiparticles. For \i = 0and (3 —> oo the expressions (19.67) reduce to the euclidean analog of (19.65).

Consider the particular case fi = 0, where particles and antiparticles con-tribute with the same weight to the propagator. Then (19.66a) can be written inthe form

o(/3.<% T) _ f d4P H ^ + m) ^SF (Z,T)-J (27r)4 p2 + m2 e

+ J J^VFDiE)^ {-ffEsinh(Er) + (nfPj - m)cosh(ET)} ,

(19.68a)where

VFD(B) = -^jjr^ . (19.686)

The first integral is just the T = 0 contribution, and is the euclidean versionof (19.64). Expression (19.68a) is the analog of (19.21a). Consider its analyticcontinuation to real times, T —¥ it. Performing a Wick rotation, p4 —>• —ipo, onefinds that*

F w J (2TT)4 p2 - m2 + it

+1 WfVFD{E)E' {^°Esin(Ei) + (7'Pi " ™) cos(Et)} ,

where we have now introduced the 7-matrices in Minkowski space, 70 = 7457* =i%, satisfying {7^,7^} = 2gl±u. Expressing the trigonometric functions in terms ofexponentials, and making use of (19.22) and of the relation

5(p0 -E)- 6(p0 + E) = 2E(p)e(Po)S{p2 - m2) ,

where e(p°) = 6(p°) — 0(—p°), and p2 = pMpM, one finds, after making the changeof variable p^ —¥ —pM (in order to conform to usual conventions), that

sW,Mink) {z) = I J*PgW,mnk) {p)e-ip.z ^ {1QMa)

* Note that, after making the change of variables pM ->• -p M , the continuationto real times of the T = jx = 0 contribution to the correlation function differs from(19.64) by a factor T.


where

^'Mmfc)(rf = J^TTe - Jri^ + mW ~ (19-696)

and j> — Yin 7MiV This is however not the propagator which appears in the real-time Feynman rules in an interacting theory. As in the bosonic case, the Feynmanrules for computing finite temperature real-time correlation functions are far morecomplicated in the presence of interactions, and requires a doubling of degrees offreedom [Landsman and Weert (1987), Niemi and Semenoff (1984)]. The Diracpropagator in momentum space becomes a 2 x 2 matrix in the extended space, ofwhich the first diagonal component is given by (19.69b).

19.6 The Electric Screening Mass in Continuum QED in

One-Loop Order

As a less trivial application of the finite temperature, finite chemical potentialformalism, we derive an expression for the electric screening mass to one looporder in QED. The result will be compared in section 10 with the screening masscomputed in lattice perturbation theory. Let us first state what is meant by theelectric screening mass.

When an external static charge Q is introduced into an electrically neutralQED plasma in thermal equillibrium, polarization of the medium will screen theCoulomb potential of the charge, leading to a short range potential with a Debyescreening length given by the inverse screening mass. If the introduction of theexternal charge can be treated as a small perturbation, then one can use the theoryof linear response (see e.g. Kapusta (1989)), to show that the screened Coulombpotential is given by

/

j3f, pikr

where II44 (0, k) = Ilf^(u = 0, k) is the 44-component of the vacuum polariza-tion tensor at finite temperature and chemical potential, evaluated for vanishingMatsubara frequency. It is defined here as the negative of the one particle irre-ducible diagrams with two external photon lines. At large distances the behaviourof the integral is determined by contributions of momenta \k\ of C( l / r ) . Definingthe electric screening mass as the static infrared limit of II44'M'(0, k),

m2el= \imU44{0,k) , (19.71)

(19.70)


we have that

f4* rA = f- • (19.72)(2TT)3 jfc2 + m 2 ; 4n r

Hence the static Coulomb potential of an external charge is screened exponentiallyin the presence of a plasma.

We only mention here that the electric screening mass can also be computedin a different way. As has been shown by Pradkin (1965) it can be related tothe second derivative of the pressure with respect to the electric charge chemicalpotential:

This is an interesting relation, since it allows one to compute m2el for a neutral QED

plasma up to order e5, since the pressure is known to O(e3). For computations ofthe electric screening mass using this relation, see Kapusta (1992).

In the following we compute the electric screening mass according to (19.71)in one-loop order. Our presentation is adapted to be easily compared with thelattice calculation given in section 10.

In the euclidean continuum formulation of QED, the vacuum polarizationtensor 11^,, at finite temperature and chemical potential is given, using the rulesdiscussed in section 4, in one-loop order by (see fig. (19-4))

(19.73a)where

k4 = u>~[k ; pi = UJJ + ifj, . (19.736)

Fig . 19-4 Diagram contributing to the vacuum polarization tensor in

O(g2)-


Performing the trace by making use of the relations

2 V ( 7 M 7 , ) = ^ ( i 9 7 4 )

71r(7^7!/7p7A) = AiS^Spx - S^S^x + Sûp)

expression (19.73a) takes the form

n ^ ik,uth)-e p2^J (2^(p2 + m2)[(p+k)2 + m2]}' (19-75a)

where

V^(k,p) = 8pMpv + 4 ( 2 ^ + A^p,,) - 4(m2 + p2 + p • k)5^ . (19.756)

For vanishing photon frequency this expression reduces to

(19.76a)

where

E2=f + m2, F2 = (p+k)2 + m2, G2 = p-(p + k)+ m2 . (19.766)

To perform the frequency sum we can in principle make use of the summationformula (19.63), where the integral appearing on the rhs would however yield thecontribution at finite chemical potential and vanishing temperature. It is thereforemore convenient to use an alternative formula in which the T = /j, = 0 (vacuum)contribution is separated from the temperature and chemical potential dependentpart. Only the T = /x = 0 contribution will then need to be renormalized in thestandard way. The relevant summation formula is given by (C.4) in Appendix C.The function /(w) in (C.I) has the following form in the present case (we onlyexhibit the dependence on w):

û>- (cji + E2)(u2 + F2) '

A straight forward calculation yields

i£;/fr*-+w-££/M€== — OO

G 2 + E 2 r i i i ( 1 9 . 7 7 )

2E(F2 - E2) [e^+M) + 1 e0(B-") + l j1 G2 + F2 1 1 I

2F(F2 - E2) [ePiF+ri + 1 eW-ri + l j '


The first term on the rhs is the T = /i = 0 (vacuum) contribution to 1144(0, k).It is ultraviolet divergent and is renormalized in the standard way, yielding nocontribution to the screening mass, as follows from Lorentz and gauge invariance.Hence we shall concentrate on the matter part, given by the last two terms. Thedependence on the photon momentum k is contained in the functions G and F. Tocompute the screening mass we must take the limit k —> 0. In this limit F —> E,so that the denominators in (19.77) vanish. Setting F = E + e, and taking thelimit e -»• 0, one finds that

2 _ 2 f d3P / e^-ri eP(E+n) •»"*«' ~2ePJ (2^)3 \ [e/3(£-M) + 1]2 + [e0(E+fi) + 1]2 / • ( 1 9 7 8 )

After carrying out the angular integration, this expression can be written in theform

e 2 t-ca Qmli = - ^ I dP PVP2 + m2~[r]FD(E, fi) + fjFD(E, (j,)] .

where rjp£>(E, JJ) and fJFD(E, AO are the Fermi-Dirac distribution functions (19.67d).After a partial integration we finally arrive at

e2 f°° 2v2 + m2mli = -2 / 0 IVFDiE,fi) + fjFD(E,n)] . (19.79)

For vanishing temperature and finite chemical potential this expression reduces to

TÔ IS* Jo ^/p2 _|_ m2

Carrying out the integral one obtains

e2

m2el(T = 0) = - j i x V M 2 - m2 for | / j | > m

— 0 for \fx\ < m

Hence for \/x\ > m there is also an electric screening mass generated at zero tem-perature.

We close this section with a remark. In order to parallel the lattice calcula-tion in section 10, we have taken the limit k —> 0 before performing the angularintegration. This yields the form (19.78) for the screening mass, which has a directanalog on the lattice. By performing first the angular integration (which cannotbe carried out analytically on the lattice) one finds that

n&M)(o, t) = 4 I" dp rf—^hFD(E,M) + fjFD{E,M)]

x - 1 + 4(p2 + m 2 ) - P 1 / 2 p ± | g \ l

4p\k\ \2p-\k\J\


where p = \p\. For |fc| -> 0, the logarithm behaves like \k\/p. Hence one is led to

(19.79).

19.7 The Electric Screening Mass in Continuum QCD in

One-Loop Order

The computation of the electric screening mass in QCD is more involved than

in QED. In one loop order the vacuum polarization tensor receives contributions

from the four diagrams shown in fig. (19-5). The quark loop contribution has the

same form as in QED exept that there is an additional colour factor Tr(~^-) =

\6AB-, arising from the gluon-quark vertices. Hence we immediately conclude that

the contribution of diagram (a) is related to that in QED (c.f. eq. (19.75)) by

n£?(c4,%) - \sAB^v{uj+,k)\QED • (19.80)

where we have dropped the superscript (/3, fi) for notational reasons. The remain-

ing diagrams (b,c,d) involve summations over bosonic Matsubara frequencies and

hence are expected to involve the Bose-Einstein distribution function. They do

not depend on the chemical potential. Hence by dimensional arguments their con-

tribution to the screening mass will be proportional to gT. Our objective is to

compute the constant multiplying gT. In the following we discuss in turn the

contributions of diagrams b,c, and d (Kaste, 1997).

Fig. 19-5 Diagrams contributing to the vacuum polarization tensor.

Consider the diagram (b). The combinatorial factor which multiplies the

finite temperature Feynman integral, obtained with the prescriptions stated in

section 4, is calculated as follows: Consider the six lines emanating from the two

vertices before they are contracted to form the loop. Label the lines from 1 ty 6.

Then there are 6 • 3 possibilites to label the external lines of the diagram, and 2


possibilites to contract the remaining lines to form the loop. Thies yields a factor

(3!)2. Each vertex receives a factor ^ arising from the symmetrization of the

3-gluon interaction. Finally there is a factor ^ associated with the second order

contribution, arising from the expansion of the exponential of the action. After

some algebra one finds that in the Feynman gauge

n- ( ^ ) ( b ) = £fACDfBcD±zJ ( g i ^ r f - (19-81«)

where

T$ {q, k) = 2k,,.kv - h{q^K + Kqv) - IQq^(ly.oio)

-5lllJ{2q2 + 2q-k + 5k2) ,

and a summation over repeated indices is understood. Here q2, k2 and q • k denote

the four dimensional euclidean scalar products, and it is always understood from

now on that

k4 = u}+ ; q4 = wj . (19.82)

Next consider the contribution of the diagram (c). The factor multiplying

the Feynman integral is given as follows: Take the 4 lines emanating from the

vertex and label them from 1 to 4. Then there are 4 • 3 possibilities to label the

external lines of the diagram. The remaining lines are then contracted to form the

loop. Finally there is a factor y arising from the symmetrization of the four-gluon

contribution to the action. One then readily verifies that in the Feynman gauge

n£?(W+,fc)(c) =3gX,fACDfBCD^J2J ( ^ ^ • (19-83)

Finally consider the contribution of the diagram (d). Since the ghost fields

are Grassmann valued, there is a minus sign associated with the loop. There is

only one possible way to contract the ghost-lines emanating from two vertices to

form the loop. Hence the combinatorial factor multiplying the Feynman integral

is determined by the two possibilities to label the external lines, and a factor ^

associated with the second order diagram. Application of the Feynman rules then

yields

TTAB/ + A 2f f 1 V^ f d3q {q + k)nqv

"I


This expression can be cast into a symmetric form in fi and v by noting that achange of variables q —>• —q — k interchanges fi and v. Hence

TTAB, + A 1 2T f 1 v ^ f d3q {q + fy^ + iq + k)^nM, ("„,*)(<£) = ~29 fACDfBCDpl^J ^ p q^qTW2 '

(19.84)Combining the contributions (19.81), (19.83) and (19.84), and making use of therelation

/J fAC of BCD = 35ABC,D

one finds that

n£?(w+, k){b+c+d) = 6ABUIU/(U+, k){b+c+d) , (19.85a)

where

and*

W^(q, k) = 2klikv - 4(q^kv + qvk^j - 8q^qv + SÂq2 + k2 + Wq • k) (19.85c)

As always, q± and k^ are understood to be given by (19.82).

In the following we now restrict ourselves to the computation of the electricscreening mass,

m2el = lim 1144(0, k) ,

where n44(u;+,fc) is defined in (19.85a). For fi — v — 4 and vanishing gluon-frequency, (19.85b) takes the form

n m f t _ 3 21 v /• d\ ^ + P + log • k - 4uf1 1 4 4 ( 0 , k) {b+c+d) - - g -gl^ TÂSr + 2 ^ _ . + 2 , . - , ^ 2 1 •

The frequency sum can be calculated with the bosonic summation formula (19.20).For the same reason as discussed in the previous section, we only need to considerthe contribution arising from the presence of the heat bath, i.e. the last two terms

* The expression for W^ differs from that obtained by Gross et al. [Gross(1981)] by a term which can be shown not to contribute to the integral (19.85b).

(19.856)


in (19.20). The T = 0 (vacuum) integral does not contribute to the screeningmass. Performing the frequency sum in the above expression one then finds, aftercarrying out the angular integration, that this contribution, which we denote byII^P, is given by

(19.86a)

where q = \q\, and where

VBE(q) = ^ T J (19.866)

is the Bose-Einstein distribution function. The corresponding contribution to theelectric screening mass is obtained by taking the limit k —> 0:

(m2el){b+c+d) = ~~ dqq riBE{q) = g2T2 . (19.87)

7r Jo

Combining this result with the contribution to the electric screening mass arisingfrom the diagram (a) in fig. (19-4), which, according to (19.80), is given by onehalf of the screening mass (19.79) in QED, we conclude that

(m2el)

QCD = g2T2 + ^ dp *t±l!L[r,FD(E,ri + fiME,ri] , (19.88)

where T]FD{E, H) and TJFD(E, /I) are the Fermi-Dirac distribution functions (19.67d).We therefore see that an electric screening mass is also generated in the pure glu-onic sector, i.e. in the absence of quarks. For SU(N), £ ] c D fACDfBCD = NSAB,

so that the contribution of the pure gauge part to the screening mass is given by

(m2el)gauge = \Ng2T2 SU{N)

This concludes our discussion of QED and QCD at finite temperature andchemical potential in the continuum formulation. In the remaining sections of thischapter we now consider these theories within the framework of lattice regularizedperturbation theory.

19.8 Lattice Feynman Rules for QED and QCD at T ^ 0 and \x 0

As we have seen, there is a simple prescription for making the transition fromthe continuum Feynman rules at vanishing temperature and chemical potential, to


the T / 0, [i 0 rules. We now show that the finite temperature, finite chemical

potential lattice Feynman rules can be obtained from those given in chapters 14

and 15 by the same prescription, except that momentum integrations and sums

over Matsubara frequencies are restricted to the first Brillouin zone. This is not

completely obvious. In fact the simplicity of the prescription relies heavily on the

way the chemical potential is introduced in the lattice action. As will become clear

from the following arguments, it suffices to consider QED. The same prescriptions

then apply to QCD.

At zero temperature and chemical potential the lattice U(l) action for Wilson

fermions has the form (5.22), where the sum over plaquettes and lattice sites extend

over a lattice of infinite extent. At finite temperature the lattice is compactified

in the euclidean time direction, with the link variables satisfying the boundary

condition

Ull(nJ) = Ull(fi,0) , (19.89)

where /3 is the inverse temperature measured in lattice units. The fermion fields

satisfy antiperiodic boundary conditions. Introducing the chemical potential into

the fermionic contribution to the action according to the Hasenfratz-Karsch-Kogut

prescription (18.100), the U(l) action, written in dimensionless variables, takes the

form*

SQED = JrEf1" \VP + UP)1 + (™ + 4r) E E ^ WP 714 = 1 n

- 2 E E ^ ( " ) ( r ~ 74)e"AC/4(n)^(n + e4) + i>{n + e4)(r + 74)eAC/4t(n)i/i(n)]

"4 = 1 n

i *~ 2 E E $ ( " ) ( r " li)Ui{n)^{n + et) + fj>(n + c4)(r + 7i)U}(n)i,(n)} .

(19.90)

By introducing dimensioned parameters and fields according to p, = fia, rh = ma,

J3 = (3/a, ip = a3/2ip, if) = a3/2ip, A^ = aA^, and taking the naive continuumlimit, one recovers the continuum action, where the dependence on the chemical

potential only appears in the kinetic term of the fermionic action in the form given

in (19.54).

* Recall that this prescription was suggested by our analysis in section 8 of

chapter 18


In weak coupling perturbation theory (7M(n) is expanded as follows

f/M(n) = 1 + ieA^n) - ^ e 2 i ^ ( n ) + . . . , (19.91)

where A^ is the gauge potential in lattice units, i.e., A^ = aA^. From the struc-ture of the lattice action (19.90) it is evident that the vertices generated by thisexpansion will depend, for finite lattice spacing, on the chemical potential. Our ob-jective is to find out just how the chemical potential enters in the momentum-spaceFeynman rules. Although the answer is almost obvious, the following discussionwill nevertheless be helpful.

To derive the Feynman rules in momentum space, the fermion fields are Fourierdecomposed according to (18.102), while the Fourier expansion of the gauge po-tentials, replacing (14.22), now reads

M") = J E / ^ ( f Â^K f c ( n + W 2 ) . (19-92a)

where

k4 := CJ+ = -^- . (19.926)

are the Matsubara frequencies (19.56) measured in units of the lattice spacing. Forreasons which we discussed in chapter 14, the gauge potentials have been definedat the midpoints of the links connecting the lattice sites n and n + e^. Since thechemical potential only appears in the fermionic contribution to the action, weonly need to consider the last three terms in (19.90).

Consider first the contribution of O(e°). Making use of the completenessrelation (18.104), it can be written as follows in frequency-momentum space,

£-1

[Sfj&ow = i E / " S^M^)K^\lCoj)M^) , (19.93a)t ~ 2

where CJJ are the Matsubara frequencies for fermions (18.69) with N = (3, i.e.,

CJ = {2l+$1)n , (19.936)

here denoted with the superscript "-", and where

K{p>rt(p,CJJ) = «74sin(w7 + ip.) + i^ smp, + M(p,CJJ + ifi) , (19.93c)


M($, CJJ + ip) = M{p) + 2r sin2(^ ****) , (19.93d)z

M(p) = m + 2rJ2 sin2 ^ . (19.93e)3

A summation over repeated indices is always understood. The fermion propagatorin frequency-momentum space is given by the inverse of the matrix K^>IJ^(p, /2 .-.-) _ ~»74 sinjuij- + ip) - t 7 j sinpj + M{j>, GsJ + ifi)

sin {oji + ifi) + 2_^j sin p^ + Mz{p, u>e +t^)

From here we see that a finite temperature and chemical potential is introducedinto the T = jl = 0 propagator, discussed in chapter 4, according to the substitu-tion rule p4 —> QJJ + ip,.

Consider next the contribution of O(eN). It involves N gauge potentialscoupled to the fermion fields. By Fourier expanding the fields as before, onereadily verifies that the contribution involving the fourth component of the gaugepotential is given by*

[ e f e ^ E E / rfv/ *pf UJkA^M^Kj714 = 1 n (~) (~> '/(+)i=l '

x i,a(p')Mp)A4(k) • • • A4(kNytt-P'+R)n ,(19.95a)

where

Ag^.*,**) = \ [e'<* + *-H» - (-iê-'W-^H (74)a/3rL \ (19-956)

_ I \ei<*4 + %-+ifi) + (_i)ive-i(p4-^+iA) Saf) ^

and where we have used the compact notation

I-1

j{±) A / ( s , ?4; - ) = E J £ (^/ («> - ) • ) • (19-95c)* Vertices involving the spatial components of the gauge potentials are clearly

not affected by the chemical potential

(19.94)


which is the lattice analog of (19.61b). The spatial and temporal components ofthe four-component vector K are given by the sum of the N photon momenta, andMatsubara frequencies, respectively. Carrying out the summation over the latticesites, by making use of (18.104), we can write (19.95) as follows

NrcOVhA, _ * f J 4 - / f T4- f TT J4JL r(/3,M)j"-> ^ f>\lbferm\o(e»)-Jp d P dP LLd ** L aP lP'P'Ai

• J(-) J(-) J(+) i—i

â{P')ip{i(P)Ai{h) • • • Ai(kN) ,

where

*p' ep *-K £

and

Vlp'îv) = (74)a/3[(liv cosq + i£N sinq) - r5ap(£N cosq + i^N sing)] ,

SN = \[1 + (-1)N] \tN=\[l-{-l)N]-

We therefore see that for finite lattice spacing the vertices are /i-dependent, andfurthermore, that the chemical potential always appears in the combination CJJ +ifi,, and GiJt + ijl, as was the case in the continuum formulation. We emphasizethat the simplicity of this result is a consequence of the fact that the chemicalpotential was introduced in the form e±iX into those terms of the fermionic actionat fj, = 0, coupling neighbouring lattice sites along the time direction.

In our discussion we have chosen to work with a manifestly dimensionlessaction, where the lattice spacing does not appear, since this is the form of theaction used in Monte Carlo simulations. The Feynman rules given in chapters14 and 15 involve explicitely the lattice spacing. From the above analysis it isevident that the corresponding Feynman rules at finite temperature and chemicalpotential are obtained by the same prescriptions given in section 4, except thatintegrals over three-momenta, and sums of Matsubara frequencies are restrictedto the first Brillouin zone. Since this result is a direct consequence of the periodicboundary conditions satisfied by the link variables and fermion fields, and the factthat the chemical potential was introduced in exponential form, it also applies tolattice QCD.


19.9 Particle-Antiparticle Spectrum of the Fermion Propagator atT / 0 and /i 0. Naive vs. Wilson Fermions

In section 5 we have studied in detail the fermion two-point correlation func-tion at finite temperature and chemical potential in the continuum. We now carryout a similar analysis on the lattice for naive as well as Wilson fermions. In thecase of Wilson fermions we of course expect to recover the results of section 5in the limit of vanishing lattice spacing. For naive fermions, on the other hand,we will be confronted with the fermion doubling problem. In chapter 18 we haddemonstrated in a simple model that the "doublers" modify the correct partitionfunction in a quite non trivial way. Their unphysical nature is exposed most clearlyby looking at the correlation function [Rothe (1995)], as we now show.

i) naive fermions

For naive fermions, (i.e., vanishing Wilson parameter r) the two-point fermioncorrelation function at finite temperature and chemical potential is given by (19.94)with M. replaced by rh, i.e.,

$$•»&&;) = - ^ s i n ( ^ - + ^ ) - ^ s i n p , + m ^ ^ ^sin2(u^ +i£) + E2(p)

where

E0)= / ^ s i n ^ + m2 (19.966)

is the energy measured in lattice units. The (dimensionless) lattice analog of

(19.66a) thus reads

£<?•">(n, n4) = f -^S<j?-">(fc n 4 ) e * s , (19.97a)

where

1 S"1

S(F

M(P,n4) = 4 ]T S(F

M{lGjJ)e^n* . (19.976)Pe=-i

Note that the momentum integrals and sum over Matsubara frequencies are nowrestricted to the first Brillouin zone. The expression (19.97b) can be decomposedas follows

S(F

M (P, n4) = f + (3)^) iP, "4) + f _ (pjxfoj,, (P, "4) , (19.98a)


wheref ± 0) = - i - [±74-E(l) - hj sinpj + m] (19.986)

2E(p)is the lattice analog of (19.67b), and

X>,(±\(iU4) = ± i Y 6^— —. (19.98c)

The finite frequency sum can be evaluated by making use of the summation formula(18.85). As an example consider V)pK(p,n4). Defining z = eluJ, the function F(z)in (18.85) now has the form

2 e A z n , + l

V ; z2 + 2Ee»z - e2A '

Hence the frequency sum in (19.98c) is given by

tW"'=1+« (^ + D ( — + ) ( — ) (19.99a)

eM r zn*+ ^J\z\ = l-e * {Zf> + 1)(Z - Z+){z - Z-) '

where2+ = e-(£"-« ; z_ = -e(f"+« ,

f = ar sinhS(p) . (19.996)

The two circles \z\ = 1 ± e enclose the zeros of z13 + 1, located at the Matsubarafrequencies. The integrations are carried out in the counterclockwise sense. ForI "41 < $ (recall that euclidean time has been compactified), we can distort thecontour in the first integral to infinity, taking into account the poles located outsideof the unit circle. Hence the rhs of (19.99a) is twice the sum of the residues ofthe integrand at the poles lying inside and outside of the unit circle, multiplied bye'2. For n4 < 0 these poles also include a pole of order |n4|. One then finds that^X8/i)(Pin4) c a n b e written in the form

Vwi) 6 n4) = A g ^ 0, n4) + (-1)"4 A ^ j 0, n4) , (19.100a)

where

A£n)(P."4) = [0(n4) - fiFD(£, fi]±—JL?L , (19.100b)1 W Vl + E2


A(/3i)(P'n4) = [0(-n4) - W * , £ ) ] - - = = = , (19.100c)

and

e [ + 1 (19.100d)

W £ , A ) = e/5(f-+A) + 1 >

are the lattice analogs of the Fermi-Dirac distribution functions (19.67d). For«4 = 0, 0(0) := | . In a similar way one obtains

Viiil)&"4) = ^l)(3,"4) + (-1)"4A[+)M)(^n4) . (19.101)

Introducing the dimensioned variables r = an^ p = | , /3 = a/3 and ^ = ^, onefinds that (19.98b) and (19.100b-d) approach the continuum expressions (19.67b)and (19.67c,d). We therefore see that the terms proportional to (-1)"4 impedethe propagator from having the correct continuum limit. These terms are the con-tributions of the doublers. They arise from the poles of the integrand in (19.99a)located on the negative z-axis, and are pure lattice artefacts having no continuumanalog. The factor (—I)™4, which changes sign as one proceeds from one lattice siteto the next in the euclidean time direction, is typical for the doubler contributions,as we have already seen in chapter 4. From (19.100b,c) we see that

AS)M)fen4) = Agi

)_M )(l ,-n4).

Making use of this relation, "D,g \{p, n^) can also be written in the form

which shows that, apart from the (non-trivial!) factor (-1)"4, the admixture ofthe doublers involves a reversal of the sign of the chemical potential as well as ofeuclidean time. It is thus not surprising that when one computes the logarithm ofthe partition function for a Dirac gas of naive fermions, one finds that it resemblesthat of a gas of particles and antiparticles in both, the positive as well as negativeenergy sectors [Bender et.al. (1993)].

ii) Wilson Fermions

Let us now see how the above results are modified for Wilson fermions. Inthe following we will choose r = 1 for the Wilson parameter. The propagator in


frequency-momentum space is given by (19.94) with r = 1 in (19.93d,e). Hence(19.97b) is now replaced by

S{F

M(P,n4) = \ Y , F{3,u>J;nt) , (19.102a)

I— 2

whereF(p, £>j; n4) = S ^ (p, ^ K ^ " 4 , (19.1026)

g(M(p, QJ) = ~»74sin(d>7 + ift) - in sinpj +M{j>,QJ + ip) ^ ( 1 9 1 0 2 c )

sin2(wf + ip) + 2 j sin2 pj + M2{p, (bj + ip)

and

M{1,&7 + ip) = M(p) + 2 s i n 2 ( ^ y ^ ) ,

Y - 2 Pi ( 1 9 ' 1 0 2 d )

M(p) = m + 22^sin2^- .3

The frequency sum in (19.102a) can again be performed by making use of (18.85).Introducing the variable z = etw, the function replacing F{z) takes the form

- ^ I7 4( z2e-2A _1)+ i ( z e - A _ 1 ) 2 + {il. s-np. _ M(p))ze-> ni

e-*»[l + M(p)](z-*+)(*-*-)

where

z±=e±W,

£ = \n[K + \lK2 - 1] = ar cosh K (19.103a)

and

K{^ = 1 + or/'SvUn ' ^ ^ = * Ij2sm2pj + M*(p). (19.1036)2[1 + M(p)} y -y

Proceeding as before one then finds after some algebra that (19.102a) can bewritten as follows

S{F

M (3, n4) = f + (M^ (P, n4) + f _ i3)%i) (P, n4) , (19.104a)


where- 1 _ _r±(p) = „ . . J±74 sinh£ - ijj sinpj + rh(p)} ,

2 sinh ££. I (19.1046)

m{p) = m + 2 V sin2 ^ - 2 sinh2 - ,

^ ( f c ^ ) = [6(n4) -f)FD(£,fi)]- ~1+

rff} (19.104c)

^ j ( P ' n 4 ) = [^(-n4) -^ D (^ ,A) ] r - ^ ,1 + M(p)

andf)FD{£.,ft) = ,

6 / 3 ( f 7 ) + 1 (19.104,*)

are the lattice Fermi-Dirac distribution function for Wilson fermions. Notice thatin the case of Wilson fermions no terms proportional to (—I)"4 appear! In thecontinuum limit we have that M(p) —> m, \/K2 — 1 —• E(p), £ —> E(p), andsinpj —> pj for finite physical momenta. For Wilson fermions these are in fact theonly relevant momenta contributing to the integral (19.97a). This can be readilyseen. In the continuum limit (3 goes to infinity. Hence the integral only receivescontributions from momenta for which £ goes to zero in this limit. Because ofthe momentum dependent Wilson mass M(p), only (dimensionless) momenta p inthe immediate neighbourhood of p = 0 contribute to the integral. Hence in thecontinuum limit the expressions (19.104a-d) approach (19.67a-d), with the usualcorrespondence holding between particles (antiparticles) and forward (backward)propagation in time, translated into the euclidean language.

19.10 The Electric Screening Mass for Wilson Fermions in LatticeQED to One-Loop Order

In section 6 we had computed the electric screening mass in continuum QEDto one-loop order. We now repeat this calculation within the framework of latticeregularized perturbation theory. The computation will be carried out for Wilsonfermions, where we expect to recover the result of section 6 in the continuum limit.The calculations in this section are based on work carried out together with R.Pietig (Pietig, 1994). A similar computation for naive fermions, carried out bythis author can be found in Appendix G.


The electric screening mass has been defined in (19.71) as the infrared limitof the 44-component of the vacuum polarization tensor evaluated for vanishingphoton frequency. In lattice perturbation theory the vacuum polarization tensorin one-loop order receives contributions from the two diagrams shown in fig. (19-6).Diagram (b) has no continuum analog but is required by gauge invariance, which isimplemented on the lattice for arbitrary lattice spacing. In the following we shallchoose r = 1 for the Wilson parameter. This leads to a drastic simplification of thecomputations. For finite lattice spacing the result for the screening mass will ofcourse depend on r, while in the continuum limit it is expected to be independentof this parameter. The following computation is carried out in the dimensionlessformulation, i.e., all quantities are measured in lattice units.

qp + k y*^.

P

(a) (b)

Fig.19-6 Diagrams contributing to the vacuum polarization tensor.

For r = I the fermion propagator in frequency-momentum space has the form(19.102c,d). For vanishing temperature and chemical potential the vertices arethose given in chapter 14 with the lattice spacing a, and the Wilson parameter rreplaced by unity. The transition to finite temperature and finite chemical poten-tial is effected in the by now familiar way: the fourth component of the photonmomentum is replaced by <2/£, defined in (19.92b), and the fourth component ofthe fermion momenta by <2~ + ifi, with CJ~ defined in (19.93b). Finally, integralsover the fourth component of momenta at T — 0 are replaced by finite frequencysums.

Consider first the contribution of the diagram (a) shown in fig. (19-6). Forvanishing photon frequency it contributes as follows to tif^{O,k),

| _ i

n!£M)(0, fc)(o) = He)2 Y, \ f 7|^Tr{[74cos(w7 + ifi) - isin(^ + ifi)}

xSr^'M)(^w7)[74COs(a>7 + ifi) ~ ism{ut + ifi)]^^{CJJ,f, + k)}

(19.105)


where Sp (q,uj) is given by (19.102c,d). Upon carrying out the trace in Diracspace, making use of (19.74), and expressing the trigonometric functions in termsof exponentials, one finds after some algebra that the above expression can bewritten in the form

I"1

tUd'^HoAa) = -*2\ E f ^4-3f(a)(e^'+i^P,k), (19.106a)

where

/W(2;?,i) , W + D-W + .HW*rUiO*-^)

and1 1

<n = 1 -—

[1 + M(p)] [i + M($+h '1 l V ;J (19.106c)

[1 + M(^][l + Mfe^] '

Q = 1 + ^ s i n p j sin(p + k)j • (19.106d)3

The position of the poles of f^(z;p, k) are given by

zx = e*; z2 = eT4"* «, ( 1 9 - 1 0 6 e )

with

^ - (19.106/)i> = £{p + k) ,

with £( | ) denned in (19.103a,b). Note that

cj) i—> _i/> , (19.107)

while i}, £ and Q are invariant under the transformation p —> —p — k. This will beimportant further below.

The frequency sum in (19.106a) can in principle be again performed withthe help of the summation formula (18.85). A more convenient formula can be

(19.106b)


derived however, which is based on the observation that CJJ always appears in thecombination &J + ifl. The following formula is derived in Appendix D.

where Res^^p- are the residues at the poles of g(z)/z, whose position we havedenoted by z~i- We now apply this formula to calculate the (finite) frequency sumin (19.106a). The function g(z) in (19.108) is now replaced (19.106b). It hassimple poles in z located at (19.106e) and a pole at z = 0. The evaluation of theresidues of f^(z;p,k)/z is straightforward. One finds

ff{a)(z)\ ( ft'aHz)\ResZ3 V—Y1) = -tes* ( —^ ) = H1>,,TI,Z,G) .

whereK<P^^G)=C^^^±^. (19.209)

smh </>(cosh $ - cosh ip)

For simplicity we have suppressed the dependence of fâ\z;p,k) on p and k.Application of (19.108) then yields

(19.110)To obtain fl^'^Ô; fc)(o), we must integrate this expression overp, withp e [—n,ir].

Noting that -q, t; and Q are invariant under the transformation p —> —p — k, andmaking use of (19.107), as well as of the fact that the integrand in (19.106a) is aperiodic function in pi and ki, we can combine the last two contributions on ther.h.s. of (19.110) and obtain

nla4)(o,fc) = 2e2 f ^L[h{,i>,T,,z,g)-i]

r % (19-llla)

- 2e2 / j-^h(4>, i,, V, £, g)[fjFD(d>) + f)FD{<l>)}.J —It \L~)

(19.108)


whereVFDW = . . . . . — - ,

e W " f + X (19.1116)

are the lattice Fermi-Dirac distribution functions for particles and antiparticles.Note that with the definition of (j>, given in (19.106f), they coincide with those in(19.104d).

We next compute the contribution to II44 '(0,ft) of the Feynman diagram(b) depicted in Fig. (19-6). This diagram has no analog in the continuum. It isgiven by

e ~ ~ 2

(19.112)The factor 2 arises from the two possibilities to label the two external photonlines emanating from the vertex, and the factor ^ from the expansion of the linkvariables to second order in the gauge potentials. The first factor in the argumentof the trace is the finite temperature, finite chemical potential version of the vertexgiven in chapter 14, measured in lattice units. Performing the trace and expressingthe trigonometric functions in terms of exponentials, one finds that

I-1

n ^ ( 0 , t)(6, = -2e2i J2 J[ 0 / W (e'W-H» £) (19.113a)

where

' " " • f t - / - ; £ - ; . ) • (mi36)

'=TTM- (19U3c)

and where z\ and z2 have been defined in (19.106e). Making again use of thefrequency summation formula (19.108), one verifies that

e^.-v/;^^-^-.) (i9n4)+2e! £ 0 (coth *~ i sb )»" - w + «" W 1


Combining this expression with (19.111a) one therefore finds that

n ^ ( o , k ) = n£r>(o,*) + 2 e 2 f' w^H^^P>**>WVFDW + ~VFD(4>)}

(19.115a)where

H{<l>,xl),p,ri,Z,g) = cot}i<l>-^--h(<l>,,l>,ri,Z,g). (19.1156)

and

ftirC)(0, %) = -2e2 f J^H^, 1>,P,ri, Z, 9) (19.115c)

is the T = fi = 0 contribution. As we now show, n ^ ^ O , k) vanishes in the limit

k -> 0, and hence does not contribute to the screening mass.

Consider the function h((f>,ijj,ri,^,Q) defined in (19.109). It is singular for

k —> 0, since in this limit tp —> 4>. The singularity is however integrable. This can

be seen as follows. Since according to (19.107), and the statement following it

h{4>,i/,,T,,$,g) _ —»• fh(i>,,r,,t,g) (19.116)p-t-fi-k

we can also write (19.115c) in the form

nsr»(o, i)=-^ £ H (cch * - - ^ ) + s £ ^-M ,„,«),(19.117a)

where

H<l>, V-. V, Z, g) = H4, *l>, V, £, Q) + h(il>, 0, r;, f, 5 ) • (19.1176)

Although each term in this last expression is singular for k —» 0 (•0—^0), the sumpossesses a finite limit. Thus setting ip = (fi + e and taking the limit k —>• 0 (e —>• 0),one verifies that

1lim h(<f>,ip,r),£,g) = 5—{ - coth</>(cosh2(/> - 2pcosh(f> + p2Qo)kô suih 0

+ 2(sinh2^> - psinh^)}

where g0 = G(p,0), with Q{p,k) denned in (19.106d). From the definition of 0given in (19.106f), with £ defined in (19.103a,b), one finds that

Qo = G{P,O) = ^ ( 2 p c o s h 0 - l ) .


HenceUmA(0,^,»7)f)$) = 2(coth0--r4-T) • (19.118)

Prom (19.117a) we therefore conclude that

UmUâc)(O,k) = O.fc-s-0

This result is not unexpected, since for vanishing temperature and chemical po-tential it is well known in the continuum formulation, that Lorentz and gaugeinvariance protects the photon from acquiring a mass. The screening mass istherefore determined by the finite temperature (f.T.), finite chemical potentialcontribution, given by the integral in (19.115a). By making again use of the factthat <f> <r* ip, when p —> —p — k, while 77, £ and Q remain invariant under thischange of variables, we can write this contribution in the form

6 ^ ( 0 , k)f.T. = 2e2 f ^ ( c o t h * - - J L ^ f o . ^ ) + f)FD(</>)}

-e2 y ' ( 0 j { M * . I > , V , Z , G)[VFD(<P) + ~fiFDm (19-119)

+h(rl>,<l>,Ti,S,g)[riFD(il>) + fjFD(il>)]} .

Consider the second integral. It can be rewritten as follows

where h has been defined in (19.117b), and

&f)FD = [VFD(4>) - VFDW] + [VFDW ~ VFDW] •

According to (19.118), h approaches a finite limit for k —) 0. Hence the contribu-tion proportional to h is cancelled by the first integral in (10.119). We thereforeconclude that

Umn^M)(0,fc) = - e 2 l im f ^h(<f>,ip,V,Z,g)AfiFD(<j>,iP) . (19.120)

We have now dropped the subscript "f.T.", since in this limit only (19.119) con-tributes to the screening mass. To calculate this limit we proceed as before andset ip = 4> + e. One then finds that

AfiFD = e(3NFD{<t>) + O(e2) , (19.121a)


whereJ(4>+p.) J(4>-P-)

MFD{4>) = — + — • (19.1216)

From the definition of rj and £ given in (19.106c) it follows that for k —> 0, r\ —> 2p,

£ —> p2. One then verifies that for small e (or k —> 0)

h((j)^ + e,r,^,g)^--+O(e0).

Inserting this expression and (19.121a) into (19.120) one therefore has that

, „ „ , r* d3p r e W + ^ e4(0-A) •,m2

el{P,fi,m) = 2e2p r ^ l - ^ +—^ [• (19.122)

Notice the similarity in structure of this expression with its continuum counterpart(19.78). For the same reason as discussed in section 9, only momenta p in theimmediate neighbourhood of p = 0 contribute to the integral in the continuumlimit. But in this limit

U0) = -P£(pa) —> /V?2 + ™2 ,a a->o

where £(p) has been denned in (19.103a,b). Hence

VFD O-»0 e^(V^+m2+^) + 1 '

IHFD(4I) —> ,

The screening mass in physical units is now obtained as the following limit

ml, - lim —m2el{-,iia,ma) . (19.123)

a-»o a-* a

Introducing in (19.122) the dimensioned momenta p as new integration variablesone then verifies that one recovers the expression (19.78), or equivalently (19.79).

For naive fermions, on the other hand, one finds that (see Appendix E)

(-2, 2A r d3p j eW-A) e^+V \{mel)naive - 4e pj_^ ( 2 7 r ) 3 | ^_fi) + i ] 2 + [ e / . ( . + .} + ^ 2 | ,


wherecf> = ar sinh. E(p) .

Notice that here <j) is just the function £(p) we encountered when studying thelattice fermion propagator for naive fermions (cf. eq. (19.99b)). The above ex-pression for the screening mass differs in two important respects from that forWilson fermions: First of all there is an extra factor 2 multiplying the integral.This factor arises from the doubler contributions associated with frequency exci-tations lying in the second half of the Brillouin zone, ^ < CJ~[ < ir. Secondly, thefunction <j> now vanishes not only for p = 0, but also at the corners of the Brillouinzone, where one or more components of p take the values ±TT. Hence the integralreceives in the continuum limit 23 identical contributions having the form of the"normal" contribution arising from finite physical momenta, for which p = pa isof O(a). Hence in the continuum limit

{mli)naive = 16"^,

where m^ is given by (19.79).

19.11 The Electric Screening Mass for Wilson Fermions in LatticeQCD to One-Loop Order

The computation of the electric screening mass in lattice QCD is more in-volved than in QED. The following discussion is based on work carried out togetherwith P. Kaste (Kaste, 1997). In O(g2) the vacuum polarization tensor receives con-tributions from the seven diagrams shown in fig. (19-7). The last three diagramshave no continuum analog but are required by gauge invariance. Diagram (g) isthe contribution arising from the non-abelian compact integration measure. TheFeynman integrals can be easily written down using the finite temperature, finitechemical potential Feynman rules discussed in section 8. As in the continuumcase, the contribution of diagrams (a) and (e) to the vacuuum polarization tensoris related to that in QED by the relation (19.80). We therefore "only" need toconsider the diagrams involving gluon and ghost lines only, which do not dependon the chemical potential. Furthermore, they only involve sums over Matsubarafrequencies of the bosonic type. Since in the continuum limit the only dimensionedscale is the temperature, their contribution to the screening mass will be of theform const x gT. For finite lattice spacing, however, the temperature dependencewill of course be modified by lattice artefacts. In the following we first considerdiagrams (b-d) which have an analog in the continuum. In all the expressions


it will be understood that the fourth component of the momenta are Matsubarafrequencies. The combinatorial factors associated with each diagram are of coursethe same as those in the continuum formulation.

Fig . l9-T Diagrams contributing to the vacuum polarization tensor in

lattice QCD in O(g2).

Contribution of diagram (b)

Using the Feynman rules discussed in section 8 one finds, after making use of

J2C,D IACDIBCD = 3SAB, that

n£f(#, *) W = ~\ÂB [ <tqTp&k\ , (19.124a)

where J,d4q has been defined in (19.95c); p^ and and p are defined generically

^ = 2 s i n ^ ; ~p2=^p~l, (19.1246)

and

T${q, k)={(g- k)^ + 2k)v cos(^(9 + k)v) cos ^

- (a - ^)M(2g + k)v cos(-(g + fc)M) cos -k,,

'— - — 1 1 - 1- (29 + k)^ + 2k)u cos -qu cos -kv + (/x -H- u)j (19.124c)

r — 2 1 — 2 1 I+ <W{(«-*) cos2(-(g + fc)M) + (g + 2/c) c o s 2 - ^ ]

+ (2q + *)M (2g + *)„ ^ cos2 i A, .


To compute the corresponding contribution to the electric screening mass we onlyneed to know Il44(A:)(6) for vanishing gluon frequency. It is given by

n4A

4B((U%) = \?*ABJ £ J[ (03/ ( 6 )(e^+;!,£). (19-125«)

where

and

a(£) = ] T c o s 2 | ,

r ' ~ 2 ~ 2i ( 1 9 J 2 5 c )

The zeros of the denominator in (19.125b) are located at

. _ . (19.126a)

where

^ = arcosh if (q) ,

4> = arcosh ff(|+ fc) , (19.1266)

^ ) = l + 2 ^ B i n 2 | .

The frequency sum can be calculated by making use of (F.5) in appendix F. Aftersome straight forward algebra one finds that

nff(o, % ) =6g2sAB / 75-T3 W> ^ a< WBEW

3 ( " f* d3a 1+ ^g25AB U{k) + J j^[h(t,1>,a,b) + h{rl>,$,a,b)n ,

(19.127a)

where. - ~ —a sinh2, V, a, b) = , ~9 y ^—^ , (19.1276)smh(p[cosh <—> î> (19.128)

under the variable transformation q —¥ —q — k, while a(k) and b(q, k) are invariantunder this transformation. Note that the function (19.127b) is singular for k —> 0,since in this limit 0 —> tp. The singularity is however integrable as can be seen bymaking use of (19.128) to write (19.127a) in the form

(19.129a)where

A W < £ , Vs) = VBE(4) - VBEW • (19.129b)

The limit k —>• 0 can now be easily be taken and one obtains the following contri-bution to the electric screening mass

where

^^^ = PÎp • (19-1306)

Contribution of diagram (c)

This diagram involves the 4-gluon vertex (15.53b), which consists of types ofterms differing in the colour structure: terms involving the structure constants/ABC, and terms involving the completely symmetric colour couplings CIABC-

We denote the corresponding contributions to the vacuum polarization tensor by

(19.130a)


[nAB(k)(c)][f] and [UAB(k)){c)}[d], respectively. Consider first [UAB{k)){c)}[f]. Itis given by

[«?(*)«]„ - l^lÎ^kn, (19.131a)where

[T$& *)] = ^ ^ { 2 4 c o s ^ Y,cosk°a

- 12cos2(i(g+ fc)M) - 12cos 2 ( i (g- fc)M)

+ KlK - Yl K) + 4[(« - *)M + (q + fc)M]i cos !*„ .(7

+ 3 {[(9 + fc)M ~{q- fy^kv cos - g M + (/z -H-1/) |

(19.1316)For vanishing gluon frequency we have that

[fl^(0.*)(c,][/] = ^ 2 ^ i E f_ 705 [/W(e^J9,*)] , (19.132a)c ~ 2

where

andc(fc) = l + 2^cosij ,

i

d^) = 1 ~ 3 E ^ ' ' (19.132c)j

P(|, k) = 2 + i El(<H-\2 + («'-'*)'] •

Performing the frequency sum in (19.132a) one finds that

[nf4>, £)(«,)][/] = ^ 2 ^ B { - C ( * ) + d(*)

r a3a i i - ^ ( 1 9 - 1 3 3 )

+ / 7^[(c-d)coth0+(d-ip)-i: i][l + 2^(0)]} •y_,r (2TT)J / smh</> J

(19.1326)


Taking the limit k —> 0 one obtains the following expression for the screening mass

(^)g = [-9 + \ £ I f , [ coth * + ^ ] [1 + ^BEm . (19.134)

Consider next the contribution [nAB(fc)(c)][d]. It is given by

[n^(*)(c)]f , = -\f?*AB I d 4 5 ^ % * , (19.135a)

where

-2 " ~ -2 ~ ~ "I

Here d(A) is defined by

8

/ ] [ÂFfidsFB + &BFE&AFE + dFFEdABE\ = d(A)5AB •E,F=1

For vanishing gluon frequency expression (19.135a) can be written in the form

where

if [Z'9) fc)J M ~ 12 U d ( A ) J [ - i x p - f2] ' (19-1366)

and

^ -2 (19.136c)

j

Performing the frequency sum one obtains

[*iB(0,i)w]M=g2^Bi(f+<J(A))

(19.1356)

(19.136a)


Since K(k) and L(q, k) vanish for k —> 0, it does not contribute to the screeningmass, i.e.,

[{mel)c}w = 0 . (19.137)

Contribution of diagram (d)

The only other diagram possessing an analog in the continuum is the ghostloop shown in fig. (19-7d). Its contribution is given by

n£?(*)((0 = 1925AB [ d*q T ^ k \ , (19.138a)

where

T$& k) = l^q + k)v cos(i(9 + fc)M) cos ±qv + (/i ^ v) . (19.1386)

For vanishing gluon frequency we have that

I-1

n^B(0,*)(d) = \925AB\ E r 7fl3/(d)(^;?,fc) , (19.139a)

where

/W(z;|,I) = -1 y ~ y . (19.1396)

Performing the frequency sum one finds that

£v (19-140a)

where

g&$)= fh\7- (19.1406)cosh 0. To compute the limit we proceed asdiscussed earlier and write (19.140a) in the form

n£f(o,!)(d) = lg*6AB{-i+ f A[g&iP) + 9(4>,4>m + 2VBEW]

^ 3 5 ( 0 , V ) A 7 ? B £ ( 0 , ^ ) ,


where AfJBE(4>) has been defined in (19.129b). Taking the infrared limit oneobtains for the contribution of diagram (d) to the electric screening mass:

(™e<)(d) = ^ 2 { - 1 + / T |4J [1 +2^jB(^)]coth0* ^ 3

j (19.141)

with MBE{<J>) denned in (19.130b). Combining the results (19.130a), (19.137),(19.134), and (19.141), we therefore find that those diagrams possessing a con-tinuum analog yield the following contribution to the electric screening mass forfinite lattice spacing

( ^ W + W H 4 / | ^ (coth^-^)[l + 2 ]}

+Gg2PJ -^MBE{4>) •

As we now show, the remaining diagrams (f) and (g), which are a consequence ofthe lattice regularization, precisely cancel the first term in (19.142).

Contribution of Diagram (})

This contribution is given by

~2

[n£f](/) = \92dAB^v f d*q% . (19.143)

For vanishing gluon frequency and fi = v = 4 this expression reduces to

(titfh)=\?**B\ E £ (0/ ( / ) (^+) ' (i9-i44«)P——S. n

where

^'"-[,-V-u- <I91446)

The corresponding contribution to the screening mass is found to be

(«';)(/) = \g2{l - I 7 ^ [ c o t h ^ - -r\-j][l + 2fJBE(4>)]} • (19.145)

(19.142)


Contribution of diagram (g)Finally, the contribution of diagram (g) to the screening mass (arising from

the link-integration measure) is trivially given by

(™e«)(s) = X~/ (19-146)

Prom (19.145) and (19.146) we see that their sum just cancels the first term ap-pearing on the rhs of (19.142) leaving us with the following simple expression forthe contribution to the screening mass arsing from diagrams containing only gluonand ghost lines:

{rhl)G = Qg2pJ" 0-3AfBE(4>) . (19.147)

The corresponding expression for the dimensioned screening mass squared is givenin the continuum limit (19.123) by

(mel)G = hm6, Pj^ — ^ ^ ^

= -^Jo dqqw^w

where q = \q\. After a partial integration this expression takes the form

Making use of the formula

roc T a - 1/ dx?— = r(a)C(a) ,

Jo e — i

where F(a) is the Euler Gamma function, and ((a) the Riemann Zeta-function,we finally obtain

(m2el)G = 92T2 . (19.148)

Including the contribution of diagrams (a) and (e) to the screening mass (which arejust one-half of the expressions obtained in the previous section for the screeningmass in QED) we therefore have that

2~ T d3p f e^+V eW-A) ,

^(AM,m)=^y_ 7 r ^ i { [ e . ( 0 + A ) + i]2 + [e-w_.) + i ] 2}^ - (j.y. i4y j

r d3a e13*+ 6g2P / 7—v? — (lattice)

y ^ }_* (2TT)3 [eW - l]2 v ;


where <f> and cf> have been defined in (19.106f) and (19.126b). In the continuumlimit this expression reduces to

meJ = T~2 / dp / o o iVFDiE,A*) + r?Fr>(£,A*)] + 52T2 {continuum)l^ Jo \/p2 + m2

(19.150)In lattice simulations of the pure gauge theory the electric screening mass is

extracted from correlators of Polyakov loops or from the long distance behaviour ofthe gluon propagator [see e.g. Irback (1991), Heller (1995)]. In these simulationsthe number of lattice sites NT = /3 is fixed and the temperature is varied byvarying the lattice spacing a = YTF' ^ the lattice expression for the (dimensioned)screening mass is to approximate the continuum, then the lattice spacing must besmall compared to all physical length scales in the problem. Hence we must havethat a << i , a << ^ and a « ^. This implies, that ma = ( f 1 ) ^ <<1, or NT » Y- Similarily we must have that NT » i±. For NT fixed, theelectric screening mass in physical units, divided by the temperature, is given by[mei]iatt/T = NTmei{NT,{^lT)N-1 ^m/^N-1), while in the continuum limit(NT —>• oo) this ratio is just a function of m/T and fi/T. In fig. (19-8) we haveplotted the ratio (mei)iatt/(mei)con as a function of ?p and ^ for 7VT = 8,16. Inthe parameter range considered the deviation from unity is seen to be at most1.7% for Nr = 16, and 10% for NT = 8.

Fig. 19-8 Dependence of [mei]tatt/[mei]cont on ^ and ^ for (a)Nr = 8, and (b) NT = 16

The deviations are mainly due to the fermion loop contribution as seen fromfig. (19-9), where we have plotted [mel]latt/[mel]cont for the pure SU(3) gaugetheory for various values of NT. The solid line is drawn to guide the eye. Thisratio only depends on the number of lattice sites NT. As seen from the figure thedeviation from the continuum is very small for NT > 8. For NT = 8, and NT = 16,it is about 2%, and 0.4%, respectively.


Fig . 19-9 Dependence of the pure gluonic contribution to [mei]iatt/[mei]con

on the number of lattice sites A T. The solid line is drawn to guide the eye.

19.12 The Infrared Problem

When we discussed the Xcf>4-theory we have seen that a mass of O(\) was gen-erated in the presence of a heat bath. As we have just seen a similar phenomenon isencountered in QCD, except that there the situation is more complex. Because theheat bath singles out a preferred reference frame the vacuum polarization tensoris no longer Lorentz covariant, but only covariant under spatial rotations. Hencedifferent screening masses can be generated in the electric and magnetic sectors.

The generation of a magnetic screening mass turns out to pose a serious prob-lem, although the appearance of an infrared cut-off in the magnetic sector is inprinciple quite welcome. The problem is that the contribution to the magneticscreening mass of O(gT) is found to vanish. The higher-order perturbative con-tributions are infrared divergent, and one is therefore forced to sum an infiniteset of diagrams to calculate the mass in lowest order. In principle this could giverise to a magnetic screening mass of O(gaT) where 1 < a < 2. In fact, a com-putation carried out by Kajantie and Kapusta (1982) based on the self-consistentsolution of an approximated Schwinger-Dyson equation for the gluon self-energyin the temporal (A§ = 0) gauge, suggested that gluons acquire a magnetic scree-ing mass-squared of order g3T2. This could have cured the IR-problem pointedout by Linde (1980). The non-analytic structure in g2 reflects the original IR-divergences in the perturbative expansion. The above mentioned work, however,has been subsequently questioned by several authors (Baker and Li, 1983; Toimela,1985). If the magnetic screening mass is of O{g2T) then one is stuck with a seriouscomputational impass. Consider, for example, the contribution to the free energyarising from diagrams involving only four-gluon couplings. Since the four-gluoninteraction vertex is dimensionless the same general argument as given for the caseof the (f>4 theory leads to an expression similar to (19.53), except for an importantdifference: the four-gluon coupling is of order g2. Hence the contribution to the


thermodynamical potential arising from a diagram containing N such vertices isnow given by (19.53) with A replaced by g2:

If we substitute for m(T) the magnetic screening mass m,M(T), which we assumeto be of O(g2T), then an infinite set of diagrams contribute to O(g6). Even if wecould calculate the contributions from the individual diagrams (which we cannot)their sum could very well diverge. But the situation is even worse. For a similarreason the magnetic screening mass itself cannot be computed even in lowest order.(Gross, Pisarski and Yaffe, 1981).

What we have just described is the notorious infrared problem in QCD. Be-ing unable to estimate higher order corrections to the free energy, one could takethe pessimistic point of view that perturbation theory can tell us nothing aboutthe thermodynamical properties of QCD, even at high temperatures, where renor-malization group arguments suggest that the coupling becomes small (Collins andPerry, 1975). It may however turn out that MC calculations confirm the early ex-pectations that at sufficiently high temperatures the non-perturbative correctionsto thermodynamical observables are small. In section 8 of chapter 20 we shallpresent some numerical results which give us some insight into this problem.

We now briefly summarize the results obtained in the literature for the con-tributions of O(g2) and 0{g3) to the thermodynamical potential fi of an SU(N)gauge theory with Nf flavours and at zero chemical potential.

The correction of O(g2) requires the calculation of a set of two-loop diagramsinvolving gluon, quark, and ghost fields (Kapusta, 1979). The zero temperaturerenormalization prescriptions are sufficient to eliminate all ultraviolet divergen-cies. After subtracting the vacuum contributions, these diagrams make a finitecontribution to f2:

^2=^N2-^N+\Nf)-


In fourth and higher orders of the coupling one encounters infrared divergen-cies. These become more and more severe with increasing number of loops. Bysumming the infrared divergent contributions one therefore expects to encountercorrections of lower order than g4. Indeed, summing the ring diagrams of thetype depicted in fig. (19-10), where the shaded blobs stand for the gluon self-energy in O(g2), Kapusta (1979) finds that there is an O(g3) correction to thethermodynamical potential:

^ - ^ - l ^ + f ) ] ^

This is the so-called plasmon term. It arises from those interactions which givethe gluon an electric screening mass at one loop level.

Fig. 19-10 Ring diagrams which when summed give rise to a contri-bution of O(g3) to the thermodynamical potential.

Concluding, we are left with the following somewhat disappointing situationin QCD: while the electric screening mass is of O(gT), the magnetic screeningmass is expected to be of O(g2T), but cannot be computed. As a consequence,the thermodynamical potential can at best be calculated up to fifth order. If oneis lucky, non-perturbative effects do not play an important role, and the potentialobtained by analytic means suffices to describe the thermodynamics of QCD atvery high temperatures, where the coupling is expected to be small. The only wayto test this is to compute thermodynamical observables numerically by startingfrom the lattice formulation of QCD.

CHAPTER 20

NON-PERTURBATIVE QCD AT FINITE TEMPERATURE

In the remaining part of the book we want to study the behaviour of hadronicmatter at finite temperature as predicted by QCD. Here are some questions thatone would like to have an aswer to: i) Does QCD predict a phase transition froma low temperature confining phase, where chiral symmetry is broken, to a hightemperature phase where quarks and gluons are deconfmed, and chiral symmetryis restored? ii) If so, what is the critical transition temperature and the nature ofthe phase transition? Is it of first or second order? iii) What is the nature of thehigh temperature phase? Is it that of a quark-gluon plasma where the interactionsof quarks are Debye-screened? iv) Does perturbation theory provide an adequatedescription of the thermodynamical properties at very high temperatures? v)What is a possible signal for plasma formation in heavy ion collisions?

The non-perturbative framework provided by the lattice formulation of QCDallows us, at least in principle, to obtain an answer to the above questions. Inthe following four sections we will set up the theoretical framework which providesthe basis for studying a possible deconfining phase transition. The theoreticalideas introduced will be exemplified in a simple lattice model in section 5. Insections 6-8 we then present some Monte Carlo results on this transition, and onthe high temperature phase of QCD. As always, we shall restrict ourselves to earlypioneering work, and we leave it to the reader to confer the numerous proceedingsfor more recent results. Finally in section 9, we discuss some possible signaturesfor plasma formation in heavy ion collisions.

20.1 Thermodynamics on the Lattice

For the computation of thermodynamical observables, like the mean energy,pressure and entropy, we need a non-perturbative expression for the partitionfunction. From what we have learned in the previous chapters we immediatelyconjecture that the partition function is given by

ZQCD = [DUDijDê-SQon[u^^} ( 2 0 ^

For Wilson fermions, SQ^Û, ip, ip] is the finite-temperature, finite chemical po-tential action having the form (19.90), where ij> and C/M are vectors and matrices

485

(20.1)


in colour space, and is replaced by -%. The link variables and Dirac fields aresubjected to periodic and antiperiodic boundary conditions, respectively.

Consider for example the implementation of the antiperiodic boundary con-ditions on the fermionic degrees of freedom, i.e.,

•4>(n,l) = -4,(nJ+l) .

The only contribution to the action which is sensitive to these boundary conditions,is the fermionic contribution involving the link variables pointing along the timedirection. By implementing these conditions, this piece of the action becomes afunction of the Grassmann variables located on the time slices n± = 1, • • -J3, andcan be written in matrix form as follows,

0

J14,7714 = 1 n

where M{n) is a $ x $ matrix with the non-vanishing entries given by

/ C V(n,l) ~y(nJ)\y(n,l) C 2/(n,2)

\-V(nJ) '> : '• y(n,3-l) C /

HereC,=rh + Ar ,

andy(n,n4) = --(r - î)e~'J-Ui{n) ,

y{n,n4) = _ _ ( r + 7 4 ) ^ ( 0 ) .

The fermionic contribution to the action can therefore be written in the form

0

nt,m,4=ln,m

and the Grassmann integration in (20.1) extends over the independent variablesV>(n, rii) and V>(n, 714) with 04 = 1, •••,/?. The Grassmann integral just yields thedeterminant of the matrix K, so that

Z= f DUe-SaM+lndetKM . (20.3)Jper

Non-Perturbative QCD at Finite Temperature 487

Note that we did not bother to fix a gauge. The integral will therefore includefield-configurations which are related by a gauge transformation. Since the group-volume for a compact group is finite, (20.3) is a well defined expression for anyfinite lattice as is used in Monte Carlo simulations.*

Given the partition function, one can proceed to study the behaviour of ther-modynamical observables as a function of the temperature, and to determine thecritical properties of the theory. For example, the existence of a discontinuity inthe energy density (latent heat) or a jump in an order parameter would tell usthat the transition is first order. Of course, in practice all calculations are carriedout on lattices having finite spatial extensions. Hence one can only hope to seea rapid change in these quantities near the transition temperature. Looking formetastable states and hysteresis effects will further help one to determine the orderof the transition.

In the following we will restrict ourselves to the pure SU(N) Yang-Mills the-ory. Then the partition function has the form

Z= I DUe~Sa[u] . (20.4)J per

Because of the non-abelian structure of the action, this theory is quite non-trivial.In fact, as we have seen in chapter 17, the pure SU(3) Yang-Mills theory con-fines a static quark-antiquark pair at low temperatures. The question is whetherconfinement persists as the temperature is raised, or whether there exists a criti-cal temperature where deconfinement sets in. To answer this question we need anon-perturbative expression for such observables as the mean energy density, andpressure.

The energy density and pressure are obtained from the partition function inthe usual way:

* Recall that in perturbation theory we were forced to fix the gauge. The gaugecondition was implemented in the path integral by introducing a judisciously cho-sen factor 1. One was then led to a gauge fixed path-integral expression where thefield configurations are weighted with an effective action involving the logarithmof the Faddeev-Popov determinant. Apart from an overall group-volume factor,which plays no role for the thermodynamics, the gauge fixed partition functionwas completely equivalent to the non-gauge fixed expression.


These expressions must be translated on the lattice into expectation values ofgauge-invariant expressions constructed from the link variables. Since accordingto (20.5) we are to keep the physical volume or temperature fixed while varying,respectively, the temperature or volume, we must be able to vary independentlythe extension of the lattice in the time and space directions. For a given lattice,this can be done by choosing different lattice spacings a and aT along the space andtime directions. In the continuum limit physics should of course be independentof the lattice regularization. Here we shall only use the anisotropic formulation tocalculate the derivatives in (20.5). The resulting expressions are then evaluatedfor a = aT = a.

The action on an anisotropic lattice has already been constructed in chapter10. It is given by (10.16a,b), with Vs and VT denned in (10.15b,c). At finitetemperature the lattice has a finite physical extension in the euclidean time direc-tion, and the link variables satisfy periodic boundary conditions. Note that thecouplings associated with the time-like and space-like plaquettes depend on thespatial lattice spacing, as well as on the anisotropy parameter. This dependenceis a consequence of quantum fluctuations, while the explicit dependence of theaction on £ is that dictated by naive arguments alone. If aT is the lattice spacingin the temporal direction, then the inverse temperature is given by NTar, whereNT is the number of lattice sites along the temporal direction. Of course in anumerical simulation the number of lattice sites in the spatial direction, Ns, willalso be finite. The spatial extension of the lattice in physical units should howeverbe large enough for finite volume effects to be negligible. In particular it shouldbe large compared with the largest correlation length which is determined by thelowest glueball mass. A finite temperature then implies that the extension of thelattice in the temporal direction will be necessarily smaller than its linear spatialextensions. Given the action on an anisotropic lattice, we can now vary the tem-perature by keeping NT fixed, and varying the temporal lattice spacing. Hencethe mean energy density and pressure can be calculated according to

_ <E> _ 1 rainZ(a,aT)1£ " V NTNfa* [ daT J a r = a '

1 ld\nZ(a,aT)~\p-3NTN?a3[ d~a \ar=a '

where we have returned to an isotropic lattice after having performed the differ-entiation. Here In Z is considered to be a function of the spatial and temporal


lattice spacings a and aT. These expressions can also be written in the form

1 dlnZ(a,£).

1 [ ainZ(o,Q . rd\nZ(a,Q]P~3a*NTNf[a da +^ d£ Jt=1 '

where the partition function is now considered to be a function of the spatial latticespacing a and the asymmetry parameter £. Prom (20.4) it then follows that

£=N^<-W>^' ( 2 ° - 6 a )

The derivatives can be readily calculated from the expression (10.16) for the action,where go is the coupling defined on an isotropic lattice, whose dependence on thespatial lattice spacing is dictated by (9.21c,d). For SU(3) one finds that [Engelset al. (1982)]

1 R£ = jVrJV33fl4 -tf(aj[< P°-VT> "So(«) < csVs + cTVT >] , (20.7a)

where Vs and VT have been denned in (10.15a,b), and where

For the difference e - 3p one obtains

In obtaining the above expressions, we have made use of the relations

9a(a, 1) = go(a),

fdga(a,Q\ = dgo(a) (20.9)

Notice that (20.8) tells us that the difference e — 3j» is proportional to the derivativeof the bare coupling with respect to the lattice spacing. For a noninteracting ideal

(20.66)

(20.76)

(20.8)


gas of massless vector particles the right-hand side of (20.8) therefore vanishes,since in the absence of interactions go does not depend on a.

Let us now return to expression (20.8). It may be used to obtain a formulafor the latent heat produced in a first order deconfinement phase transition (whichis expected to be seen in the pure SU(3) gauge theory). The reason is that thepressure should vary continuously in the transition region, while, in the limit ofinfinite spatial lattice volume, e will exhibit a discontinuity at the critical temper-ature. On a finite lattice this discontinuity will of course be smoothed out, andtherefore only manifest itself as a rapid change in e. But if the pressure variescontinuously in the transition region, then the discontinuity in e (latent heat) isdetermined from (20.8) to be

Ae=^NjN-T7oP{90)A{V)> ( 2 ° - 1 0 )

where V =VS+VT, and P(g0) = ~adgo/da is the /3-function (9.6b). For smallcoupling f3(g0) is given by (9.21a,b) with NF — 0. Inserting this expression into(20.10) we are left with the following formula for Ae:

Hence Ae, measured in units of a~4 can be obtained directly in a MC-simulation.To obtain Ae measured in physical units, we must eliminate the lattice spacing o infavour of the physical lattice scale A ,, which is a renormalization group invariant.The relevant relation is given by (9.21c,d). Inserting the expression for a into(20.11), we obtain

Hence by measuring the discontinuity in (V) at the (first order) phase transition,for a given value of the bare coupling, one can determine Ae in units of the latticeparameter. The above relation holds of course only close to the continuum limit.

20.2 The Wilson Line or Polyakov Loop

At zero temperature, the potential of a static quark-antiquark pair can bedetermined by studying the ground state expectation value of the Wilson loop forlarge euclidean times. At finite temperatures the lattice has a finite extension in

(20.11)

(20.12)


the time direction, and the Wilson loop no longer plays this role. The questiontherefore arises which is the corresponding object to be considered when studyingQCD at finite temperature. To this effect we notice that, because of the periodicstructure of the lattice, we can also construct gauge-invariant quantities by takingthe trace of the product of link variables along topologically non-trivial loopswinding around the time direction. In fig. (20-1) we show a two-dimensionalpicture of the simplest loop we can construct, located at some spatial lattice siten. Consider the following expression, constructed from the link variables locatedon this loop,

PL(n) = Tr Yl UA{n,nA), (20.13)

This expression is invariant under periodic gauge transformations:

Ui{n,ni) —¥ G{n,ni)Ui{n,ni)G^l{fi,ni + 1)

G(n,l)=G(n,0+l).

L(n) is referred to in the literature as the Wilson line, or Polyakov loop. As we nowshow, its expectation value has a simple physical interpretation. The argumentgiven below is only qualitative. For a more detailed discussion we refer the readerto the work of McLerran and Svetitsky (1981). To keep the presentation as simpleas possible, we will consider the U(l) gauge theory in the absence of dynamicalfermions. Furthermore we shall use the continuum formulation.

Fig. 20-1 Two dimensional picture of a loop winding around the time

direction.

Consider the partition function of the system consisting of an infinitely heavyquark coupled to a fluctuating gauge potential.

Z = '$2(s\e-PH\s). (20.14)S


The sum over s extends over all states of the form ^(x,0)\s'), where ^(x,0)creates a quark located at x and at time X4 = 0 when acting on the states \s')which do not contain the heavy quark. Using the fact that exp(—/3H) generateseuclidean time translations by /?, we can write (20.14) in the form

Z =N^2(s'\^(x, 0)e-/3H^(x, 0)|s')s'

=iV^( s ' | e -^*( f , /3 )* t ( f i 0 ) | S ' ) , ( 2 ° ' 1 5

where N takes into account the normalization of the quark state. But in the staticlimit the dependence of (x, 0) on f3 is determined by*

(9T-ie^4(f,r))*(f,T) = 0.

Hencew(x,p) = e Jo 4V 'ty(x, 0).

Inserting this result into (20.15), we are left with an expression involving the op-erator ^(x, 0)^(x, 0). This operator describes the creation and destruction of aheavy quark at the same space time point. It merely gives rise to an (infinite) con-stant which is cancelled by the normalization factor N. A more careful treatment,similar to that presented in chapter 7, shows that (20.15) can be replaced by

Z = Tr{e~pHL{x)) , (20.16a)

whereL{x) = eieh dTA*(S'T) (20.166)

is the U(l) continuum analogue of the Wilson line (20.13), and where the traceis performed over the states of the pure gauge theory. The corresponding pathintegral representation of (20.16a) is therefore given by

Z= I DA L{x)e~soÂ\ (20.17)

where 5G[A] is the finite temperature gauge field action, and where the integrationextends over all fields A^x) satisfying periodic boundary conditions. Notice that

* This corresponds to neglecting the contribution of the kinetic term and of thevector potential in the Dirac equation. We have dropped the mass term since itgives merely rise to an exponential factor.

Non-Periurbative QCD at Finite Temperature 493

L(x) is invariant under periodic gauge transformations. In the non-abelian case

the potential A4 is a Lie algebra-valued matrix, and (20.16b) must be replaced

by the trace in colour space of the corresponding time-ordered expression. From

(20.17) we conclude that the free energy of the system with a single heavy quark,

measured relative to that in the absence of the quark is given by

e~^ = (L) = i £(£(£)). (20.18)X

where V is the spatial volume of the lattice. In writing the last equality we have

made use of the translational invariance of the vacuum. On the lattice, the left-

hand side of (20.18) is replaced by exp(-(3Fq), where /3 and Fq are the inverse

temperature and the free energy measured in lattice units.

The Polyakov loop resembles the world line of a static quark in a Wilson

loop. This suggests that the free energy of a static quark and antiquark located

at x = na and y = ma, with a the lattice spacing, can be obtained from the

correlation function of two such loops with base at n and m, and having opposite

orientations (see fig. (20-2)):

T(n, m) = (L{n)L\m)). (20.19)

Fig. 20-2 Two Polyalov loops, winding around the time direction, used

to measure the gg-potential.

Indeed, by the same type of arguments which led to the connection between the

Wilson loop and the static gg-potential in chapter 7, but with T now replaced by

ft, one can show that (20.19) is related to the free energy Fqq(n,m) of a static

quark-antiquark pair, measured relative to that in the absence of the qq pair, as

follows:

T(n,rh) = e-/3^(".™). (20.20)


Assuming that T(n, rh) satisfies clustering, we have that

(L(n)Lt(m)) —> \(L}\2. (20.21)\n—m\—KX)

From here we conclude that if (L) = 0, then the free energy increases for large |n —rh\ with the separation of the quarks. We interpret this as signalizing confinement.

(L) = 0 (confinement). (20.22)

On the other hand, if (L) ^ 0, then the free energy of a static quark-antiquarkpair approaches a constant for large separations. In the absence of vacuum po-larization effects, arising from dynamical quark, we interpret this as signalizingdeconfmement:

(L) 7 0, (deconfinement in the pure gauge theory) (20.23)

The qq potential in the deconfined phase is obtained by dividing (20.20) by |(£)|2,which removes the self-energy contributions of the individual quarks, i.e.,

e - | ( L ) | 2 , (i0.24)

where R = \n — rh\.

The above connection between the expectation value of the Wilson line andthe free energy of a heavy quark was "derived" assuming that no quarks of finitemass coupled to the gauge potential. But this connection also holds when weinclude dynamical fermions. This is borne out by a more careful treatment, whereone first considers the partition function including "light" and heavy quarks, andthen takes the infinite mass limit for the heavy quarks. In this way one findsthat the free energy Fq is again related to (L) by (20.18), except that now theexpectation value is calculated with the finite temperature action involving thedynamical fermions [McLerran and Svetitsky (1981)].

According to (20.22) and (20.23), the expectation value of the Wilson lineevaluated in a pure gluonic medium serves as an order parameter for distinguishinga confined from a deconfined phase in the pure SU(3) gauge theory. In statisticalmechanics, phase transitions are usually associated with a breakdown of a globalsymmetry. We now show that this is also expected to be true in the SU(3) gaugetheory.


20.3 Spontaneous Breakdown of the Center Symmetry and the

Deconfinement Phase Transition

The lattice action of the pure SU(3) gauge theory is not only invariant under

periodic gauge transformations, but also possesses a further symmetry which is not

shared by the Wilson line. Consider the elements of 5(7(3) belonging to the center

C of the group.* They are given by exp(2iril/3) e Z(3) where / = 0,1,2. Clearly

the action (10.16) is invariant if we multiply all time-like oriented link variables

C/4 between two neighbouring spatial sections of the lattice by an element of the

center

Ui(n,n4) -»• zU4(n,n4),

where z £ C. But the Wilson line is not invariant since it contains one link variable

which transforms non-trivially:

L(n)^zL(n).

Now if the ground state of the quantum system respects the symmetry of the

classical action, then link configurations related by the center symmetry will occur

with the same probability. Thus the same number of configurations will yield the

values U = e2izil/3L {I = 0,1,2) for the Wilson line. Since J2i exp(2wil/3) = 0, it

follows that the expectation value of the Wilson line must vanish. But this we had

interpreted as signalizing confinement. Hence we expect that the center symmetry

is realized in the low temperature, confining phase of the pure SU(3) gauge theory.

On the other hand, if (L) / 0, then the center symmetry is necessarily broken. We

hence expect that a deconfinement phase transition is accompanied by a breakdown

of the center symmetry and that the phases of the Polyakov loops cluster around

any one of the Z(3) roots. As we shall see in section 6, this is borne out by

MC calculations. That the Z(3) symmetry plays a crucial role in the confinement

problem has been suggested some time ago by Svetitsky and Yaffe (1982). These

authors argued that the critical behaviour of the SU(3) gauge theory is that of an

Z(3) spin model. Since this model exhibits a first-order phase transition, one also

expects that a deconfinement phase transition in the pure SU(3) gauge theory, is

of first order.

The above considerations applied only to the pure gauge sector, or equiva-

lently in the infinite quark mass limit. The case of infinitely heavy quarks is of

* The center C of a group G consists of all elements z for which zgz"1 = g, with

geG.


course unphysical. Nevertheless the study of the pure SU(3) gauge theory pro-vides us with important information regarding the role played by the non-abeliangauge field for quark confinement. As we have pointed out before, one generallybelieves that quark confinement is a consequence of the non-abelian self-couplingsof the gauge potential. At low temperatures, this non-abelian coupling is expectedto lead to the formation of a flux tube (-* string) connecting the quark and an-tiquark. The energy stored in the string is proportional to its length which is ofthe order of the separation of the quark and antiquark pair. As the temperatureis increased, the string begins to fluctuate more and more, its length now beinglarger than the separation between the quarks. At the same time the numberof possible string configurations with a given length will increase. That is, theentropy will start competing with the energy carried by the string. At sufficientlylarge temperature, the system may then prefer to form networks of flux tubes towhich the quark and antiquark are attached to. Thus above some critical temper-ature a new phase may be formed, consisting of huge networks of flux tubes (Patel,1984). The energy of such a configuration will be roughly the same, no matterwhere the quark and antiquark are hooked onto a network. This corresponds todeconfinement. When dynamical quarks are coupled to the gauge potentials, thenflux tubes can break. The probability for this to occur increases with decreasingquark mass. This is also true for the single flux tube connecting the quark andantiquark at low temperatures. In both cases the forces between the quarks getscreened. But whereas the screening at low temperatures is due to pair produc-tion processes along the single string connecting the two quarks, the breakdown ofthe network in the presence of light dynamical quarks resembles more the Debyescreening in a plasma.

Finally we want to mention that a fermionic contribution to the action breaksthe center symmetry explicitly. Hence the expectation value of the Wilson lineneed not vanish in the confining phase. In this case the free energy of a qq pairapproaches a finite constant for infinite separations, which is consistent with theexpectation that the force between two quarks is screened by vacuum polarizationeffects.

20.4 How to determine the Transition Temperature

When studying QCD at finite temperature in a Monte Carlo simulation onemust first decide on the spatial and temporal extension of the lattice to be used. Inprinciple, the spatial volume should be chosen large enough so that finite volumeeffects do not play an important role. For a given set of lattice sites the linear


physical extension of the lattice will depend on the lattice spacing, and henceon the value of the bare coupling constant. Again, in principle, this couplingshould be taken small enough to approximate continuum physics. Let Ns bethe number of lattice sites along any one of the three spatial directions. It thenfollows from the euclidean space-time symmetry of the path integral formulationthat for lattices with temporal extent NT > Ns physics should be insensitive to theperiodic boundary conditions in the time direction - if it is not sensitive to the finitespatial volume. Hence such a choice of lattice is appropriate for studying QCD at"zero" temperatures.* On the other hand, to study QCD at high temperatures,the temporal extension of the lattice must be much smaller than in the spacedirections. This is the reason why Monte Carlo computations are carried out onlattices with NT < Ns. For a given lattice spacing a the physical temperature isgiven by

T = .NTa

Thus the temperature can be varied by either changing NT (then the temperaturevaries in a discontinuous way), or by varying the lattice spacing, which can bedone by changing the coupling go. In principle the best way of proceeding wouldbe to introduce two types of couplings which allow one to vary the lattice spacingin the temporal direction without changing the spatial volume of the lattice. Butif this volume is large enough for finite volume effects to be negligible (which theyare not in general) then a single coupling constant should suffice.

To determine the critical temperature at a phase transition one studies thetemperature dependence of an order parameter, and of such thermodynamicalquantities as the energy density, specific heat, etc.. If the QCD coupling 6/gg islarge enough, then the lattice spacing is related to the bare coupling constant by(9.21c,d) and the temperature is given by

T= AL

NTR(go)'

Let gc be the critical coupling at which the phase transition takes place. Then thecritical temperature is given by

T= = K (2025)

* Actually, the temperature is never really zero on a lattice with finite temporalextension.


To test whether one is really determining a physical transition temperature, oneshould repeat the simulation for lattices of different temporal extensions, determinegc in each case (gc will of course depend on the choice of NT) and check that

NTR{gc(NT)) = const. (20.26)

Another way of obtaining the transition temperature in physical units is to measuresome other observable, such as a hadron mass. Since the critical temperature andhadron mass in lattice units are given by Tc = Tca and rhn = Tin a, where m# isthe physical hadron mass, it follows that Tc/mjj = Tc/rriH- Hence the temperaturein physical units is given by

Tc = (•?£-) mH.\mHJAs we shall see in section 6 there exists strong evidence that QCD undergoes adeconfmement phase transition at high temperatures. If so, this would be an inter-esting prediction of this theory which, if confirmed by experiment, would constitutean important test of QCD as being the correct theory of strong interactions.

20.5 A Two-Dimensional Model. Test of Theoretical Concepts

In the previous sections we have introduced a number of important conceptswhich are relevant for studying the thermodynamics of a pure gauge theory. Sinceour discussion has been in part quite formal, it is instructive to check the theoret-ical ideas in a solvable model. The simplest lattice model one can consider is theU(l) gauge theory in one time and one space dimension. This theory will confinea (/g-pair, since in one space dimension the field energy cannot spread out in space.In this section we will use the compact U(l) gauge theory as a toy model to cal-culate the following quantities: i) the expectation value of the Polyakov loop, ii)the expectation value of two oppositely oriented Polyakov loops (which is relatedto the free energy of a static gg-pair) , and iii) the ensemble average of the electricfield energy density stored in the string connecting an oppositely charged pair. Wewill compute these quantities using the character expansion, in order to illustrateat the same time a technique which is relevant for strong coupling expansions inSU(N) lattice gauge theories. * For the simple model we consider the calculationscould actually be carried out without making use of this sophisticated method.

* See e.g. Migdal (1975); Drouffe and Zuber (1983); Munster (1981). Thecharacter expansion has been used by Rusakov (1990) to compute observables inU(N) gauge theories on arbitrary two-dimensional manifolds.


The basic idea of the character expansion is the following. In a lattice gaugetheory the pure gauge part of the action has the form

SG[U] = KJ2F(UP),P

where Up are the plaquette variables, and F(Up) is a real valued function of theplaquette variables which is invariant under gauge transformations, F(g~1Upg) =F(Up), with g an element of the unitary gauge group G. We have denoted thecoupling by k, instead of J3, which we reserve for the inverse temperature measuredin lattice units. Since exp(—kF{Up)) is a class function on G it can be expandedin terms of irreducible characters (traces of irreducible representations) of Up asfollows

exp(-kF(UP)) = ^2dv\v(k)xu(UP) , (20.27)V

where v labels the irreducible representations of G, \v is the character of Up inthe z/th irreducible representation, and dv is the dimension of the representation.For a real class function F(U), conjugate representations Xv a n d Xv contributewith the same weight in (20.27). The plaquette variables Up must be chosen witha conventional orientation. The convention chosen is immaterial, for changing theconvention merely replaces Up by Up, and hence the representation ' V by itsconjugate " p". Below we list some important formulae which we will need for ourdiscussion.

Let V, W, Vi, fii, etc. be elements of a compact unitary group G, and dVthe normalized Haar measure on G (i.e., J DV = 1), satisfying

dV = dV~l = d(VW) ; W e G . (20.28)

Then the following relations hold:*

f dVxv{V)xv'{V*) = 8VV, , (20.29a)

f dVX,AViV)Xv2(V^V2) = J-5UlU2Xl/1(V1V2) , (20.296)J dVl

f dVxAVSliViSli) = ^-Xv(toi)Xv(n2) . (20.29c)J dv

* Making use of (20.28), these formulae are seen to hold also if V denotes aproduct of link variables and dV, the corresponding product of Haar measures.


From the character expansion (20.27), and the orthogonality relation (20.29a) itfollows that

K(k) = ^JdUXv(U*)e-kFW . (20.30)

In the case where G — U(l), the irreducible representations are one dimensional,and the characters are given by* Xu{U) = eiue, where v is an integer. The U(l)action has the form (see (8.10))

SG[U] = ft£[l - \(UP + UP)] = « £ [ 1 - cosOp] , (20.31)p p

where k = p-, with e the charge e measured in lattice units, i.e., e = ea. dp isthe sum of the phases of the directed link variables around a plaquette P. TheHaar measure, normalized to unity, is given by dU = d9/2-K. When calculat-ing the expectation value of an observable we can replace the Boltzmann factorexp(—KSG[U]) by Yip exp(kcos9p). Then

fDUOiUJYlpe^^- fDUYlpeWr) ' [M-62a)

wheref(UP) = cos 9p . (20.326)

Accordingly, eq. (20.27) is replaced in the U(l) gauge theory by

e-nup) ^Y^h^xAUp) , (20.33)

where the coefficients have been calculated from (20.30) with kF(U), dU andXv{U) replaced by -k cos 9, d9/2n and exp(ii/0), respectively. Iv(k) is the modifiedBessel function of order v.

Consider two plaquettes P\ and P2 having a link in common. With each pla-quette there is a factor exp(kf(Up)) asssociated with it. The character expansionallows us to perform the integration over the common link-variable, which we de-note by V. This is also the case for non-abelian gauge theories, where this method

* For SU(2) the link variables in the fundamental representation can be parametrized in form U = cos | + ia • nsin | , where oi are the Pauli matrices and

n a unit vector. The Haar measure and characters are given in this case bydU = gij sin2 f d9dfl and Xu{U) = s i n ( ^V 2 ) e , where j takes integer or half integer

values.


acquires its real power. For the U(l) gauge theory the integral of interest is given

by

jdVeVViW^ = ^ / ^ ( K ) J ^ ( K ) fdV XVAUPJXVAUP,) •J VlV2

J

Let fii and Q2 denote the path ordered products* of link variables along the solidand dottet paths shown in fig (20-3). Then UPl = fliV and Up2 = V^2

Fig. 20-3 Two plaquettes which are glued together at the common link.

Making use of (20.29b) with du = 1, one obtains

where Uap12 is the "path ordered" product of link variables around the boundaryof the area covered by the two plaquettes. This area, measured in lattice units,equals the power of the coefficient /,/(«). Proceeding in this way, one can carryout successively the integration over all link variables lying inside of a simplyconnected two-dimensional lattice of area A, bounded by a simple closed contourdA. The result is the K-functional [Migdal (1975)],

KA{UgA) = ^[/.(«)] V ( £ W , (20.34)

where UQA = SloQ'QfjQ is the path ordered product of link variables around theboundary of the lattice shown in fig. (20-4), and A is the number of plaquettesin the domain bounded by dA. This expression holds for a lattice bounded by anarbitrarily shaped contour.The only other K-functional we shall need is that associated with a lattice with ahole, depicted in fig. (20-5a). We shall denote this functional by KA,0, where Ais the number of plaquettes bounded by the two closed curves. It is obtained by

* Of course in the abelian case path ordering is irrelevant. We shall neverthelessuse this language and always write the product of link variables in the path orderedform, as would be required in the non-abelian case.


Fig . 20-4 All link variables except for those on the boundary of the lat-

tice have been integrated out. The corresponding K-functional only depends

on the path ordered product of the link-variables along the boundary

glueing together the K-functionals for the two sufaces without holes, shown in fig.(20-5b), along the common links. Hence

KA,o = IdV1dV2KAl(U9Al)KA2(U9A2) .

Fig . 20-5 Two surfaces without holes which are glued together along

the common links

Written out explicitely we have that

KA,O = 52[IVlW]Al[I*(k)]A' IdV1dV2XuAîV2ulVl)xV2(Vtw2V}to2) .1/1 V2

Making use of (20.29b) with dv = 1, we can perform the integral over V\ andobtain

KA,O = J ^ O ^ / dV2X»(tolV2U>lW2Vln2) •v J

This integral is of the form (20.29c).* Hence

KAtO(to, w) = ^£[L(K)}AxAn)xA") , (20.35)v

* In the non-abelian case one makes use of the invariance of the character (trace)under cyclic permutations of the matrix valued group elements.


where ft and UJ are the products of the link variables along the closed contoursshown in fig. (20-5a). As we shall see, all other K-functionals which we will needcan be obtained from (20.34) and (20.35). These K-functionals will depend on i)the link variables living on the boundary of the lattice, and ii) the link variableson which the observable depends whose expectation value we want to calculate.

At finite temperature the lattice has a finite extension in the euclidean timedirection, given by the inverse temperature /3, and the link variables at timesT — 0 and r = (3 are identified. For the time-like oriented link variables onthe spatial boundary of the lattice one can, for example, impose periodic, or freeboundary conditions. Let TL denote the set of unconstrained variables on theboundary of the lattice, and T the set of variables on which the observable Odepends. The corresponding finite temperature K-functional we denote genericallyby K ( / 3 ) ( r L , r ) . Then the expectation value (20.32a) will be given by

fprLDro\r]KW(rL,r)<0>= fDTLKW(TL) ' ( 2 ° - 3 6 a )

where

K^(TL) = I DTKW(TL,r) . (20.36b)

Having set up the general framework, we now proceed to some concrete calcula-tions.

(i) The Polyakov Loop

Consider the expectation value of a Polyakov loop (or Wilson line) in thev = 1 representation, i.e.,

< L >=< XI{UT) > ,

where Ur denotes the (path ordered) product of link variables, along a closedloop T winding around the compactified surface along the temperature axis. Tocalculate this expectation value we must evaluate the numerator and denominatorin (20.32a).

Consider first the denominator. The relevant finite-temperature K-functionalis obtained from (20.34) by identifying fijj with fi0 in UQA = tto^'ftpQ (see fig.(20-4)), and integrating the expression over Qo, making use of (20.29c) with dv = 1.One then finds that

K^(Sl, Q') = Y,\lA*)]AXv(U)Xv{V) . (20.37)


Fig . 20-6 Lattice which is compactified along the temperature axis.

The coresponding finite temperature K-functional only depends on the path

ordered products of link variables ft and ft'.

This is the K-functional associated with the surface shown in fig. (20-6).

We next must decide what type of boundary conditions we want to impose

on the link variables at the ends of the cylinder. Imposing periodic boundary

conditions, i.e., identifying ft and ft , and integrating the expression over ft, fig.

(20-6) is glued together to the torus. Making use of (20.29a) the corresponding

K-functional is given by

KW,torus) = z^rus) = Y}lv(k)]A , (20.38)

which is just the denominator of (20.32a).

If instead of toroidal boundary conditions we impose free spatial boundary

conditions, then integrating (20.37) over ft and ft' projects out the trivial repre-

sentation. HenceK(fi.free) = ^ = ^ . ^ _ ( 2 Q 3 9 )

Zfree is just the denominator of (20.32a) for free spatial boundary conditions.

Consider next the numerator of (20.32a), or of (20.36a) with O[T] replaced by

Xi{Ur)- The relevant K-functional is that associated with fig. (20-7a). It is given

by the product of the K-functionals associated with the two surfaces shown in fig.

(20-7b):

K{A\ = K%?{n, UT)KW(UI ft') . (20.40)

Imposing the periodic boundary condition ft = ft, and integrating over ft, we

obtain

K$£ut\uT,ub = Y,[iu{k))Axv{VT)xM).

where A is the total area of the lattice, measured in lattice units. The expectation

value of the Polyakov loop < L >=< Xi(Ur) > is n o w given by

< L >(torus)= -^— fDUrXi(Ur)K^XUS)(Ur,Ul) .^(torus) J


\J 1 J 1 J \ J J~k/jj

(a) (6)

Fig. 20-7 (a) All link variables except for those living on the closed

contours are integrated out. The corresponding K-functional is given by

product of the K-functionals associated with the two surfaces shown in (b).

A pictorial representation of the numerator is given in fig. (20-8). Evaluation ofthe integral yields

< L >torus= ^ — Y\!v{K)]AVxv ,4 (torus) v

where the Wigner coefficient P*r, is defined by

Vkrr, = I dUXr(U)Xr>(U)Xk(tf) 4 i )

= 6k,r+r. (for 17(1))

Hence

< L >torus= 0 ,

which according to the criterium (20.22) implies confinement. This is of courseexpected to be the case in one space dimension, as we have pointed out at thebeginning of this section. As the reader can readily verify, the same result isobtained by imposing free boundary conditions.

\xi("r)

Fig. 20-8 Fig. (20-7b) glued together to a torus. Also shown is theinsertion of a Polyakov loop.

(20.41)


In order to keep the following disussion as simple as possible, we will choosefree boundary conditions from now on.

(ii) The free energy of static qq-pair

Acording to (20.20) the free energy of a qq pair, measured relative to thevacuum, is related to the expectation value of two oppositely oriented Wilson linesby

Fqg(R, /3) = - \ In < LnLl > , (20.42a)

where R = \n — n'\, and

Ln = Xi(Ur) ; Ll=Xi(.u£,). (20.426)

Here [/p and Ur, denote the product of link variables along the two closed pathsshown in fig. (20.9a), located at the spatial lattice sites n and n' of the two charges.

Fig. 20-9 (a) Two oppositely oriented Polyakov loops winding around

the compactifled lattice; (b) The relevant K-functional is given by the prod-

uct of the K-functionals associated with the 3 surfaces.

For free spatial boundary conditions the relevant K-functional is given by theproduct of the K-functionals associated with the three surfaces in fig. (20-9b),integrated over fi and f2',

K%A£ = fdndaKW(n,ur)KW(ul,ur.)KW(ul,,a).

Making use of the expression (20.37), the integral can be readily performed andone obtains

V


The expectation value in (20.42a) is then given by

< LnLl >= — — [dUrdUr,xi(Ur)xi(ul,)K{£{le£(Ul,UTI) .6{free) J

The integral can be evaluated immediately by making use of the orthogonality

relation (20.29a) and of (20.39):

where A.2 is the number of plaquettes enclosed by the two Polyakov loops. Ac-

cording to (20.42a), the free energy of a static oppositely charge pair is therefore

given in lattice units by

Fq9(R,k)= (in ! [ § ) £ • (2°-44)

The right hand side coincides with the expression we derived in chapter 8 for the

qq- potential at zero temperature (cf. eq. (8.17)) . This is not surprising, since in

one space dimension the string connecting the charged pair cannot fluctuate. From

our discussion in chapter 8 it therefore follows immediately that in the continuum

limit (k = js = ; j ^ —> oo) the free energy, measured in physical units, is given by

Fq,(R) = \e2R • (20.45)

Hence in this model the gg-system is confined,

iii) Energy Sum Rule

In the following we first derive an energy sum rule which relates the mean

energy of a static quark-antiquark pair to the field energy stored in the string.

The sum rule is then evaluated using the character expansion.

Consider the partition function for a static qq pair:

Zqn = jDULnLl,e-s° =Z0< Lnh\, >

where ZQ = f DUe~s°. The mean energy, measured relative to the vacuum, is

then given by

< £ > = - — In <LnLl > .

(20.43)


To evaluate the rhs we must be able to vary the temperature in a continuous way.Since P = (3aT, where aT is the lattice spacing in the euclidean time direction, wecan vary the temperature by varying aT, keeping the number of lattice spacings $in the temporal direction fixed. To calculate the /3-derivative we must thereforefirst evaluate the expectation value < LnL*n, > on an anisotropic lattice. Fromour discussion in section 2 of chapter 10 it is evident that the finite temperatureaction on an anisotropic lattice is given for our U(l) model by

S<« [U;£} = ktJ2ll- ReUp] , (20.46)p

where £ = ^-, with a the spatial lattice spacing. Since the naive continuum limitof this action is that of a free theory, we do not expect that the coupling k mustbe tuned with the lattice spacings a and aT when taking the continuum limit (aswas the case for the non-abelian theory). The mean energy, measured in latticeunits is therefore given by

< E >= U§-\n < LnLl >) . (20.47a)

wheref Dill ft p-Êpt1-^^]

In (20.47a) we have returned to an isotropic lattice after differentiation. Perform-ing the differentiation in £ we obtain

< E >= * < -V >q5_0 , (20.48a)

where

V = Y^l1 - Rajp\ > (20.486)

p

and

< V >qQ-o= <L"Ln'V>- < V > . (20.48c)< LnLn, >

In the naive continuum limit V -» \ J cPxF^F^- In one space dimension thefield tensor F^v has only one non-vanishing component, JFI2, corresponding to theelectric field. One of the objectives of this calculation will be to confirm that theeuclidean formulation, with this minus sign in (20.48a), gives the expected answer

(20.476)


for the mean energy. A similar expression had been obtained in chapter 10 for the

contribution of the electric field to the energy sum rule in the pure SU(3) gauge

theory.

Because of translational invariance in the euclidean time direction, we can

also write (20.48a) in the form

< E >= k < -V >gg-_0 , (20.49)

where V denotes the contribution to (20.48b) arising from plaquettes located ona fixed time slice. From here we infer that the energy density, as probed by aplaquette, is given in lattice units by

= k\<LnLl,ReUP>_ < ReUp i

L < LnUn, > J

The expectation values are calculated on an isotropic lattice according to (20.32).For free spatial boundary conditions < LnI,nl > is given by (20.43). The relevantK-functional for evaluating < Lnh\,ReUp > before imposing free spatial boundaryconditions, is given by the product of i) the K-functionals associated with thesurfaces with areas A\ and A3 in fig. (20-10), ii) the K-functional associatedwith the surface A2, where the window has the size of a single plaquette, and iii)the K-functional of a single plaquette, whose Fourier Bessel expansion is givenby (20.33). The finite-temperature K-functional associated with the surface A2 isobtained from (20.35) in the by now familiar way. One readily finds that

KAIO = Y2^kXA2X»(Ur)xAUr)xAUiP) • (20.51)

Fig. 20-10 The K-functional relevant for the computation of < Lnl)n,ReUp >

is given, before imposing spatial boundary conditions, by the product of the

K-functionals associated with the 3 surfaces with areas A\, A2, and A3,

and the K-functional for a single plaquette

(20.50)


Hence the K-functional associated with fig. (20-10), integrated over fi and ft' (->•free boundary conditions) is given by

x ^2 IVi (k)xu4 (Up) .

This K-functional has to be folded with

LnLlReUp = Xi(Ur)xi(u}.,)^\xi(Up) + Xi(UP)] .

One then finds that

where the Wigner coefficients have been defined in (20.41). Hence

This result is also valid if the plaquette touches a Wilson line. Dividing thisexpression by (20.43), with A2 replaced by A2 + 1, (i.e., the area enclosed by thetwo Polyakov loops, including the area of the window), we obtain

<LnLl\{UP + UP)> = I0{k) + h(k)

<LnLt> 2 J X ( « ) • { • '

The reader will have noticed that in obtaining this expression we have probed thefield energy with a plaquette placed in between the Wilson lines. It can be shownthat the corresponding expectation value with a plaquette placed outside the areabounded by the two Polyakov loops vanishes.

Finally, let us compute the expectation value of ReUp in (20.50). The relevantK- functional is given by the product of i) the K-functional (20.51) with J42 = A—I,integrated over f r and UT> (—> free boundary conditions), and ii) the K-functionalfor a single plaquette is given by (20.33). One then finds that

< ReUp >= ^ [ / O ^ E T F ! (DUP\[XI{UP)+XI{UP)}XV{UP)6 free „ M«) 1 2


or

Here we have made use of the fact that Xv(U^) = x~v(U), and I-V(k) = /„(«).Taking the difference of (20.52) and (20.53), we obtain the mean energy densityin lattice units

= k'oW + W - ^ - (20.54)2/i (ft) 70(ft)

;

In the continuum limit « —>• oo. Expanding the ratio of modified Bessel functionsin powers of i one finds that each of the two terms in the above expression issingular in this limit. The singular parts however cancel in the difference, andafter setting h = ~^?i o n e finds that the mean energy per unit length, measuredin physical units, i.e. S = ^ , is given by e2/2, or

<E>=\*R.

Notice that the mean energy coincides with the free energy (20.45). This is par-ticular to the simple model we have considered, where the free energy does notdepend on the temperature.

The expression (20.54) could also have been derived directly from (20.48a).Thus on an anisotropic lattice < LnL'n, > is given by (20.43) with k replaced by £k(as follows from the form of the action (20.46)). Performing the differentiation in(20.47a) one readily verifies that one recovers the result (20.54). But our work hasnot been in vain, for it served to test some subtle points we mentioned in chapter10. Thus it not only served to test the energy sum rule, but in particular theminus sign in (20.49a), which is typical for the euclidean formulation, as we havealready seen in chapter 10. There is another lesson we have learned. As we havementioned above both terms in (20.54) diverge in the continuum limit, but theirdifference is finite. We had encountered a similar situation in chapter 18, when wediscussed the energy sum rule for the harmonic oscillator. Hence the computationof the electric field energy density from correlators of plaquette variables with twoWilson lines, and averaged plaquette variables, involves, close to the continuumlimit, taking the difference of two large numbers of the same order of magnitude.This may be a serious stumbling block for numerical simulations. These are themain messages we wanted to convey.

(20.53)


20.6 Monte Carlo Study of the Deconflnement Phase Transitionin the Pure SU(3) Gauge Theory

Numerical simulations of the pure SU(3) gauge theory at finite temperaturehave been carried out since the early eighties. * Since then much effort has beeninvested in studying the deconfinement phase transition, which is expected to beassociated with a breakdown of the Z(3) center symmetry, and which had beenpredicted by Polyakov (1978) and Susskind (1979). Svetitsky and Yaffe (1982) thenconjectured that the phase transition should be first order. The order parameterwhich distinguishes the two phases is the expectation value of the Wilson line (orPolyakov loop). When quarks are coupled to the gauge fields the action is nolonger Z{2>) symmetric and the Wilson line no longer plays the role of an orderparameter. But at least for large quark masses, such as those of the "charmed","bottom" and "top" quarks, it is still expected to show a rapid variation acrossthe transition region. QCD with only heavy quarks is however not the theoryone is ultimately interested in. In the real world we also have the light "up" and"down" quark, and a heavier "strange" quark. Of these, at least the "up" and"down" quarks, are expected to influence the phase transition in a decisive way.Thus in the limit of vanishing quark masses, the continuum action possesses achiral symmetry. This symmetry is broken in the low temperature phase and isexpected to be restored at sufficiently high temperatures (Pisarski and Wilczek,1984). The order parameter which characterizes the two phases in the zero quarkmass limit is the chiral condensate < xpij) >. It now replaces the Wilson line whichtests the center symmetry in the pure gauge theory.

In this section we present some numerical results of lattice calculations whichstrongly support the above mentioned expectations that the deconfinement phasetransition is of first order. Clearly, it is impossible to discuss the numerous con-tributions made in the literature , and to present a critical analysis of the results.But this is also not the purpose of this section. Our objective is to stimulatethe readers interest in this subject. To this end we shall select a few representa-tive results of early Monte Carlo simulations. Hence the figures we present, donot represent the best numerical data available today, and we apologize to themany physicists that have made important contributions in this field, and whomwe do not mention here explicitely. The reader should confer the proceedings for

* Of course many calculations have also been performed in the pure SU(2)gauge theory, which is much easier to simulate.


more recent results, and the original articles cited for a critical assessment of thenumerical results presented here.

After these general remarks, let us now first take a look at what Monte Carlo"experiments" tell us about the phase transition in the pure SU(3) gauge theory,i.e., in the infinite quark mass limit. In particular we are interested in establishingthe nature of the transition (if it exists), and the critical temperature at which thephase transition takes place.

In the infinite volume limit a first order deconfinement phase transition shouldshow up as a discontinuity in the Wilson line and energy density. The latent heatassociated with the transition tells us how much energy has to be pumped into thesystem to produce the new state of matter. On a finite lattice any discontinuousbehaviour of an observable will be smoothed out, but a rapid variation across thetransition region should still be seen. Such a variation would however not excludethe possibility that the transition is second order. There are several characteristicfeatures of a first order phase transition: i) the coexistence of phases at the criticaltemperature. In a Monte Carlo calculation this should manifest itself in the moreor less frequent flip of the system between the "ordered" and "disordered" phases.The frequency with which these flipps occur will depend on the lattice volume used,decreasing with increasing volume, since the system will tend to remain in eitherof the two metastable states for a longer simulation time; ii) On a finite lattice thecritical coupling 6/<?Q at which the phase transition occurs should show a specificfinite size scaling behaviour. In particular, for a first order phase transition, thelocation of the transition is expected to be shifted by an amount proportional to1/V , where V is the spatial volume of the lattice.; iii) additional informationabout the nature of the transition can be obtained from the finite size scalinganalysis of the peak and width in the specific heat or the susceptibility of thePolyakov loop, which should show a specific dependence on the volume.* Thus,if the transition is first order, then the height and width in the susceptibility ofthe order parameter is expected to increase linearly with the volume and shrinklike 1/V, respectively.** Studying the response of thermodynamical observablesto a change in lattice volume is probably the best method to establish the orderof the phase transition. But it is very time consuming. In most simulationsone has therefore looked for signs of metastable states. Such states are however

* The susceptibility of the Polyakov loop is defined by % — V(< (ReL)2 >-< ReL >2).** The scaling behaviour of first-order phase transitions has been discussed e.g.

by Imry (1980); Fisher and Berker (1982); Binder and Landau (1984).


not easily detected, especially if one is working on small lattices, since the flipsbetween different thermodynamical states cannot be clearly disentangled from thestatistical fluctuations.

What concerns the measurement of the transition temperature Tc, there aredifferent ways one can proceed. If the transition is first order, and the spatiallattice volume large enough, then localizing the (smoothed out) discontinuity in theenergy density or order parameter should suffice to determine Tc. Alternatively,one may test the Z(3) symmetry directly by looking at the distribution of thereal and imaginary parts of Polyakov loops, measured on a large number of linkconfigurations, as a function of the temperature. In the Z(3) symmetric phase,configurations related by the Z(3) symmetry should occur with equal probability.On the other hand, in the Z(3) broken phase the system will spend substantialsimulation time in one of the three vacua, before tunnelling between the vacua willrestore the Z(3) symmetry.

After these general remarks, let us now take a look at some specific examplesof early Monte Carlo calculations.

Fig. 20-11 MC data of Celic et al. (1983) showing (a) the hysteresispattern for the order parameter, and (b) the temperature dependence of theenergy density in the phase transition region.

First evidence for the existence of a first order phase transition in the pureSU(3) gauge theory came from computer simulations dating back to 1981 (Ka-jantie, Montonen and Pietarinen, 1981).* In the first few years most of the cal-

* For a review of early calculations see Cleymans et al. (1986).


culations were performed on lattices of small spatial volume. The observation ofhysteresis effects, of coexisting states, and of rapid changes in the energy densityand Polyakov loop suggested that the transition is of first order. In fig. (20-11)we show data obtained by Celic, Engels and Satz (1983) exibiting the hystere-sis pattern for the order parameter, and the energy density as a function of thetemperature measured on a 83 x 3 lattice. These data are suggestive of a firstorder phase transition. The latent heat was found to be Ae = (3.75 ± 0.25)T^.Using Monte Carlo data for the string tension available at that time, the criticaltemperature in physical units was determined to be Tc = 208 ± 20MeV.*

As another example we show in fig. (20-12) later data obtained by Kogut etal. (1985) for the averaged Wilson line and the energy density on a 63 x 2 lattice.Notice that both quantities exhibit a very steep variation in the transition regionat the same value of the coupling (and therefore also temperature).

Fig. 20-12 (a) The Wilson line and (b) the gluon energy density as a

function of the coupling 6/gg. The figure is taken from Kogut et al. (1985).

* For other early calculations of the latent heat see e.g. Kogut et al. (1983b);Svetitsky and Fucito (1983).


In both examples the lattices are however still too small to allow one to besure that one is seeing continuum physics. Strong evidence that the SU(3) gaugetheory does indeed exhibit a first order phase transition came from numericalcalculations performed by Gottlieb et al. (1985) on lattices with varying temporalextent, ranging between 7VT = 8 and NT = 16. These authors performed longMonte Carlo runs at different couplings, and measured the real and imaginaryparts of the Polyakov loop averaged over the spatial lattice for each configurationgenerated after every ten sweeps. The results for the largest lattice they have used(193 x 14) are shown in fig. (20-13a).

Fig. 20-13 (a) Scatter plots of the Polyakov loop (raw data) obtained

by Gottlieb et al. (1985) for 6/gg = 6.45, 6.475 and 6.5, exhibiting the

transition from the confined to the deconfined phase; (b) Scatter plot of the

same data as in (a), where always five successive measurements have been

averaged.

The averaged Polyakov loop is denoted by P. Fig. (20-13b) shows the same data,where always five successive measurements have been averaged. In the figures


labeled by (a) the system is in the deconfined phase, with the data distributedmore or less uniformely around the origin. The figures labeled by (/?) show thecoexistence of the deconfined and confined phases, while the figures labeled by(7) correspond to the Z(3) broken phase, with the data clustering around a non-vanishing expectation value of the Polyakov loop. Notice that the transition fromthe Z(3) symmetric to the Z(3) broken phase occurs within the narrow interval6.45 < 6/<?Q < 6.5.* For another choice of lattice the critical coupling will ofcourse be different. To test whether one is extracting continuum physics, onemust check whether relation (20.26) holds. If so, the critical temperature in unitsof the renormalization group invariant scale, A ,, is given by (20.25).

Fig. 20-14 The critical temperature measured in units of the lat-

tice parameter calculated on lattices with temporal extensions NT =2,4,6

(Kennedy et al. 1985) and NT = 8,10, 12 and 14 (Gottlieb et al. 1985).

Asymptotic scaling appears to set in for 6/g2 > 6.15.

Fig. (20-14) shows the results for TC/AL obtained by these authors for variouschoices of NT ranging between NT = 8 and NT = 14. The data at NT = 2,4,and 6 (first three points) are earlier results obtained by Kennedy et al. (1985).The plateau observed for 6/^Q > 6.15 strongly suggests that one is extractingcontinuum physics.** This need however not be so. Just like the first two points

* Recall that varying the coupling corresponds to varying the lattice spacingand hence the temperature.

** In fact, Gottlieb et al. used the measurement of Tc to determine the coupling


in the figure, when considered by themselves, could have suggested that scaling

sets in already for 6/g% = 5.1, it could happen that the plateau observed in the

range 6.15 < %/gl < 6.5 does not correspond to scaling but is followed by a scaling

violating behaviour similar to that observed for 5.1 < 6/#Q < 5.7. To obtain the

temperature in physical units one must eliminate the lattice scale parameter. This

can be done by measuring another physical quantity such as the string tension.

Then Tc = (Tc/V&)y/a, where fc and a are the critical temperature and string

tension measured in lattice units. Using the value 400 MeV for the square root of

the string tension one finds that Tc = 250MeV.Further evidence for a first order phase transition came from measurements

of the latent heat on large spatial lattices, and from the observation of metastablestates. A nice example is provided by a Monte Carlo simulation of Brown et al.(1988) performed on a 243 x 4 lattice.

Fig. 20-15 MC data of Brown et al. (1988) for the sum of the energydensity and pressure measured in units of T4 plotted versus 6/<7Q f°r a

24 X 4 lattice. The solid points have been obtained by dividing the eventsin a single MC run, showing flip-flop behaviour, by hand into confined anddeconfined parts.

One of the quantities which has been measured by these authors is the sum of theenergy density and pressure measured in units of T4. Since the pressure should

at which scaling sets in.


vary continuously across the transition region, a measurement of the discontinuityin (e + p)/Ti yields directly the latent heat in units of T4.

Fig. (20-15) shows the behaviour of (e + p)/T4 in the transition region, asobtained by Brown et al. on a 243 x 4 lattice. In the same Monte Carlo experimentthe coexistence of the two phases at the critical temperature was also clearly seen.

Fig. 20-16 Evolution of (e + p ) / T 4 (a) and the argument of the

Polyakov loop (b) on a 243 X 4 lattice as a function of the simulation time.

The solid horizontal lines are MC data showing that the system resides in

one of the three vacua with broken Z(3) symmetry. In the remaining time

intervals the system is in the Z(3) symmetric confining phase. The figure is

taken from Brown et al. (1988).

In Figs. (20-16a,b) taken from Brown et al. (1988), we show the evolution of(e+p)/T4 and of the argument of the Polyakov loop as a function of the simulationtime at the phase transition. The values for (e + p)/T4 obtained by dividing theevents in a single Monte Carlo run -showing flip-flop behaviour- by hand into con-fined and deconfined parts are displayed in fig (20-15) by the dark dots. As seenfrom fig. (20-16b), the "time" evolution of the argument of the Polyakov loop alsodisplays the Z(3) symmetric phase as well as the three vacua with broken Z(S)symmetry in which the system remains for longer simulation times. The observa-tion of coexisting phases allows one to determine the discontinuity indirectly fromfig. (20-15). The authors find that Ae/T4 = 2.54 ± 0.12.

Fukugita, Okawa and Ukawa (1989) have analyzed in a very large-scale simu-lation the finite size dependence of the flip-flop behaviour at the phase transition,and of the peak and width in the susceptibility of the Polyakov loop on lattices


for spatial volumes ranging between 83 and 363 , and fixed temporal extension,-/Vr = 4. They found, in particular, that on a 243 x 4 lattice and at a coupling6/^0 = 5.6925, the system "appears to stay in one phase over (1 — 3) x 104 sweepsbefore flipping to the other phase". The average duration between flip-flops wasfound to increase strongly with the lattice volume and to be consistent with thatexpected for a first order phase transition. Furthermore the volume dependenceof the height and width of the susceptibility of the Polyakov loop also turned outto be consistent with the predictions for a first order phase transition, increasinglinearly with the volume, and shrinking like 1/V, respectively.

The detailed studies that have been carried out so far strongly support thatthe deconfinement phase transition in the pure SU(3) gauge theory is of firstorder.*. This is a very nice result. But the SU(3) gauge theory is not the realworld. The next step therefore consists in including the effect of dynamical (finitemass) quarks.

20.7 The Chiral Phase Transition

Let us now ask what happens to the phase transition when we couple dy-namical quarks to the gauge potentials. For sufficiently large quark masses it isreasonable to expect that the presence of quarks will not influence the results ob-tained in the pure gauge theory very much. A realistic simulation, however, shouldbe performed with "up" and "down" quarks with a bare mass of the order of a fewMeV, and a heavier " strange" quark with a bare mass about twenty times larger.The influence of the heavy "charmed", "top", and "bottom" quarks on the phasetransition can very likely be ignored. This may also to be true for the "strange"quark. Hence as a first approximation one would like to study the case of QCDwith two light, roughly degenerate, quarks. This case is interesting, since the chi-ral phase transition is expected to be driven by the light quarks. But also thecase of four degenerate low mass quarks is interesting from the theoretical point ofview, since there exist general arguments (Pisarski and Wilczek, 1984) predictinga first order chiral symmetry restoring phase transition for three or more flavoursof massless quarks.

The majority of the Monte Carlo simulations have been carried out withKogut-Susskind fermions. The reason for this is the following. As we have seenin chapter four, the staggered fermion action possesses a continuous axial flavour

* There had been some doubts regarding this raised by the APE collaboration(Bacilieri et al., 1989)


symmetry in the limit of vanishing quark masses. This allows one to study sponta-neous chiral symmetry breaking and the associated Goldstone phenomenon with-out having to tune any parameters, as would be required in the case for Wilsonfermions. Thus in the case of staggered fermions the chiral limit just correspondsto setting the quark masses to zero. On the other hand, for Wilson fermions chi-ral symmetry is broken explicitely by the fermionic action, and the chiral limit isrealized at a critical value of the hopping parameter K, corresponding to vanishingpion mass (see section 3 of chapter 17). This critical value can however only be de-termined by generating ensembles of gauge field configurations at different valuesof K, and extrapolating the data to Kcrn.. This turns out to be a quite non-trivialand time consuming problem. In addition one is faced with the problem that thenumber of degrees of freedom for Wilson fermions is much larger than for Kogut-Susskind fermions. Consequently, simulations with Wilson fermions have beenperformed on much smaller lattices than those employed for staggered fermions.For these reasons, staggered fermions are more convenient for studying the chiralphase transition.

By 1991, numerous calculations with dynamical fermions had already beenperformed on lattices with a spatial extension up to 16 lattice sites, and temporalextention NT = 4,6,8. In most simulations the quarks have been taken to have thesame mass. The general scenario that emerged was the following. As the quarkmass is decreased from rh = oo, the first-order deconfmement phase transitionweakens,* and even seems to disappear for intermediate quark masses. As thequark masses are decreased further, the transition gathers eventually again instrength and, in the case of three or four quark flavours, exhibits characteristicsof a first order phase transition for sufficiently small quark masses. Clear signalsof metastable states have been observed. ** In the case of two quark flavours thetransition turned out to be much weaker.

Let us now look at some Monte Carlo simulations. As in the case of the pureSU(3) gauge theory we have selected only a few representative early exampleswhich will hopefully stimulate the reader to further reading.

In the early days, before the advent of the supercomputers and the develop-ment of more refined algorithms for including the effects of the fermionic determi-

* See for example Hasenfratz, Karsch, and Stamatescu (1983); Fukugita andUkawa (1986).

** See for example: Gavai, Potvin and Sanielevici (1987), and Gottlieb et al.(1987b).


nant in numerical simulations, physicists have studied the temperature dependenceof the chiral condensate < •tjjip > and of the quark and gluon internal energies inthe quenched approximation. In fig. (20-17) we show early results obtained byKogut et al. (1983a) for the chiral condensate and the Wilson line. The chi-ral condensate was computed from the fermion propagator by taking the limit ofvanishing bare fermion mass.

Fig. 20-17 MC data of Kogut et al. (1983a) for the chiral order

parameter and the expectation value of the Wilson line as a function of

6/(7Q in quenched QCD. The calculations were performed on a (a) 83 X 2

and (b) 83 X 4 lattice.

Notice that there exists only a single transition region across which both parame-ters show a very rapid variation. Since the fermionic determinant was neglected inthis calculation, the sharp rise in the Wilson line corresponds to the deconfinementphase transition in a pure gluonic medium. A similar strong variation has beenobserved at the same transition temperature separately in the fermion and gluoninternal energies (see e.g., Kogut et al., 1983b).

Until the mid-eighties, simulations including dynamical fermions were still inan exploratory stage. Early results for the Wilson line and chiral condensate forfour light flavours showed again a strong variation of both quantities at the samecritical temperature (Polonyi et al., 1984). Since then many simulations, usingdifferent algorithms, quark masses, and larger lattices have been performed. InFig. (20-18) we show the data for the Wilson line and chiral condensate obtained byKarsch, Kogut, Sinclair and Wyld (1987) for four degenerate light quarks, usingthe hybrid algorithm. The results suggest a first order chiral phase transition.Notice that the Wilson line, although it is not an order parameter in full QCD,


still rises steeply across the transition region. This suggests that the mechanismfor deconfinement in the presence of dynamical fermions is that responsible for therestauration of chiral symmetry.

Fig. 20-18 The chiral order parameter (triangles) and the expectation

value of the Wilson line (dots) calculated by Karsch et al. (1987) on an

83 X 4 lattice for QCD with staggered fermions.

Most of the simulations have been carried out with two or four flavours ofmass-degenerate quarks. For some earlier simulations of the more realistic case oftwo equal light quarks and a heavier strange quark see e.g., Kogut and Sinclair(1988), Brown et al. (1990), and more recently Karsch and Laermann (1994),where again chiral symmetry restauration was found to take place at the samecritical temperature where deconfinement sets in. This critical temperature wasdetermined to be Tc « !73MeV.

In the following we will take a look at some Monte Carlo studies of the hightemperature phase, emphasizing, as always, the earlier pioneering work. We shallsay nothing about the full phase diagram of QCD in the temperature-chemicalpotential plane, which is the subject of intensive current research. Monte Carlosimulations with a non-vanishing chemical potential are problematic, since thefermion determinant associated with the lattice action (19.90) is complex. Recall


that the logarithm of this determinant contributes to the effective action. Hence a

direct implementation of Monte Carlo techniques with a Boltzmann factor e~Seif

is not possible. In principle one could write detK — \detK\elT', and only include

Ide^l into the effective action. Denote this action by 5 e / / . The ensemble average

of an observable can then be trivially written in the form

< Oeir ><O>= = ,

where the expectation values on the rhs are now calculated with the (phase

quenched) action Seff. The trouble with this formula is, that in actual numerical

simulations the fluctuation of the phase elT makes this procedure impracticable.

These fluctuations grow with the lattice volume leading to large cancellations

among contributions to the numerator, as well as to the denominator. As a con-

sequence an enormous number of configurations are required to achieve any rea-

sonable accuracy. A number of proposals have however been made to circumvent

this problem. For references to earlier attempts the reader can confer the article

by Fodor and Katz (Fodor, 2002). *

20.8 Some Monte Carlo Results on the High Temperature Phase

of QCD

The confirmation that quarks are confined in hadronic matter at low tempera-

tures, and the (strongly suggested) existence of a phase transition at temperatures

of the order of 1012 Kelvin, are the two most spectacular predictions of lattice

QCD. Such a transition would probably have occured about 10~6 seconds after

the big bang. It has been speculated for some time that the high temperature

phase is that of a quark-gluon plasma (QGP), where hadronic matter is disolved

into its constituents. The possibility of implementing the QCD phase transition

in the laboratory has become feasable within the past decade through heavy ion

collisions carried out at ultrarelativistic energies. In these collisions the initial

kinetic energy is deposited in a very short time interval in a small spatial region,

creating matter with densities 10 to 100 times that of ordinary nuclear matter. Ion

beams of 197Ag, with 11.4 GeV per nucleon have been produced in 1992 by the

AGS accelerator at Brookhaven National Laboratory. At the CERN SPS acceler-

ator beams of ions as heavy as 3 25 have been obtained with energies up to 160

* For more recent proposals see (Fodor, 2002; Allton, 2002; de Forcrand, 2002;

Anagnostopoulos, 2002).


GeV per nucleon. The verification that a quark-gluon plasma is actually formedin such collisions is a very non trivial problem since it requires an unambiguoussignature for its formation. For this a detailed understanding of the dynamics of aplasma is necessary. Lattice calculations can only provide us at present with bulkquantities like the mean energy density, entropy and pressure, but not with theexpected particle yields, momentum spectra, etc., at freeze out.

If the high temperature phase is indeed that of a quark-gluon plasma consist-ing of a weakly interacting gas of quarks and gluons, as is suggested by renormal-ization group arguments (Collins and Perry, 1975), then one would expect thatthe energy density and pressure are approximately that of an ideal gas of quarksand gluons, and that the interquark potential is Debye screened. But because ofthe severe infrared divergencies encountered in perturbation theory, the behaviourof these observables in the deconfined phase could very well turn out to be nonperturbative even at very high temperatures. The only way to estimate the nonperturbative effects is to calculate the above quantities numerically. In the fol-lowing we will take a look at some early Monte Carlo simulations of the hightemperature phase of QCD. We will be very brief, and present here only a few re-sults without going into details. Most of the material in this section has been takenfrom the review article by Karsch published by World Scientific (1990), where thereader can also find an extensive list of references to the work published on thissubject.

Energy density and pressure above Tc

Early Monte Carlo calculations performed in the pure SU(3) gauge theoryshowed that above the critical temperature the energy density approached veryquickly the Stefan-Boltzmann limit for a free gluon gas. An example is shown infig. (20-11). This behaviour of the energy density has also been confirmed in latercalculations performed on larger lattices.

On the other hand, it has been found that the pressure approaches the Stefan-Boltzman limit only slowly. Fig. (20-19) taken from the above mentioned reviewby Karsch, shows the Monte Carlo data for the energy density and pressure ob-tained by Attig et al. (unpublished), (dots and triangles), and Brown et al. (1988),(squares), on a 123 x 4 lattice and 243 x 4 lattice, respectively. The numerical re-sults are compared with the perturbative prediction (dashed horizontal lines) ofHeller and Karsch (1985) for a 123 x 4 lattice. As seen from this figure, the energydensity agrees rather well with (lattice) perturbation theory already at temper-atures slightly above Tc , while the pressure rises only slowly above the phase


transition and approaches the perturbative prediction at best for temperaturesabove 3TC. As we will see below, however, the approach of the energy density tothe Stefan-Boltzmann limit is probably much slower than it appears in the figure.

Fig. 20-19 Energy density and pressure in units of the ideal gas valuesas a function of T/Tc. The dots and triangles are data of Attig et al.(unpublished), and the squares are data of Brown et al. (1988). The linesare drawn to guide the eye. The figure is taken from Karsch (1990)

Note that the pressure seems to be negative slightly above Tc. In a high statis-tics simulation performed on a lattice with spatial volume 163, Deng (1989) hasstudied, in particular, the behaviour of the pressure just above the phase transi-tion. The results are shown in fig. (20-20), where we also display, for comparison,the data obtained by this author for the energy density.

As seen from fig. (20-20b), the pressure is not only negative slightly above Tc

but also appears to be discontinuous at the phase transition. Hence the extractionof the latent heat from a measurement of the discontinuity in (e + p) , or (e —3p), which assumes that the pressure is continuous at the phase transition, willlead to wrong conclusions. Engels et al. (1990) have subsequently discussed theorigin of the problem. As we have seen in section 1, the energy density andpressure can be obtained from the expectation value of plaquette variables, if oneknows the derivatives of the bare gauge coupling with respect to the spatial andtemporal lattice spacings, as and aT , evaluated at as = aT. In most simulations,the thermodynamical quantities have been extracted by assuming that the abovementioned derivatives can be approximated by their leading order weak coupling


Fig. 20-20 MC data of Deng (1989) for (a) the energy density and (b)

the pressure, measured in units of T , as a function of 6/<7Q . The calculation

was performed on a 163 X 4 lattice.

expressions. But for the couplings at which the MC calculations are performed,these derivatives actually deviate substantially from the low order perturbativeresults (Burgers et al., 1988).

The above mentioned authors have therefore used an alternative, non per-turbative, procedure for calculating the pressure. The basic idea is to calcu-late this quantity from the relation p = — / , where / is the free energy density:/ = —T(lnZ)/V.* Of course the free energy cannot be computed directly in aMonte Carlo simulation (which only allows one to calculate expectation values).But its derivative with respect to the coupling A = 6/<?Q can be calculated since—XdlnZ/dX =< SQ >, where SQ is the gauge action.**

By integrating this equation one then obtains, apart from an additive con-stant, the free energy, and hence the pressure, as a function of the coupling (andtherefore also of the temperature). The free energy was normalized by subtractingthe vacuum contribution at zero temperature (actually, the temperature is neverreally zero on a finite periodic lattice). By proceeding in this way one circumventsthe problem of having to compute derivatives of the bare coupling constant. Fig.(20-21) shows the pressure as a function of 6/gg obtained by these authors (solidline) together with the MC data of of Deng (1989) and Brown et al. (1988). Thepressure is seen to be positive everywhere and continuous at the phase transition.

* This relation assumes that the system is homogenous. The authors discussthe validity of this assumption in their paper.

** Recall that we are still discussing the case of a pure SU(3) gauge theory. Inorder to avoid any confusion with the inverse temperature, (3 — 1/T, we havedenoted the coupling 6/<?Q by A and not by /? as is usually done in the literature.


Fig. 20-21 The pressure in units of T 4 obtained by Engelsetal. (1990)

in the pure SU(3) gauge theory (solid line) together with the MC data of

Deng (1989) and Brown et al. (1988) based on the perturbative expressions

for the /3-function.

The above authors have also calculated the energy density, which - as seenfrom eq. (20.8) - requires the knowledge of the /^-function. For couplings 6/g$ <6.1 this ^-function (which can be obtained from a Monte Carlo renormalizationgroup analysis), deviates considerably from that given by the perturbative ex-pression (9.21a,b). By using the non-perturbative /^-function, and their resultsfor the pressure, the authors find that the energy density approaches the Stefan-Boltzmann limit much slower that originally believed, and that the " discontinuity"at the phase transition is much weaker than that suggested by the data displayedin fig. (20-19). This is shown in fig. (20-22).

So far we have only considered the pure SU(3) gauge theory. For full QCDthere exists no such detailed analysis. Simulations with dynamical fermion indi-cated that the behaviour of the energy density and pressure (obtained by usingthe leading order perturbative expressions for the derivatives of the bare couplingconstant) is similar to that encountered in the pure gauge theory. Again the en-ergy density was found to rise steeply in the transition region, approaching rapidlythe ideal gas value (from above), while the pressure was found to rise only slowlyabove the critical temperature.

As an example we show in fig. (20-23), taken from the above mentionedreview by Karsch, the energy density and pressure as a function of the coupling6/g^ for QCD with two light quarks. The data shown is that of Gottlieb et al.,


Fig. 20-22 The energy density in units of T 4 obtained by Engels et

al. (1990) as a function of 6/<7Q (solid line). The calculation is based on

the results for the pressure shown in fig. (20.22) and on a non-perturbative

expression for the /^-function. The MC data are from Brown et al. (1988),

and Deng (1989).

Fig. 20-23 Energy density and pressure in units of T 4 as a function

of 6/<?Q for QCD with two flavours. The data is from Gottlieb et al. (1987).

(1987). In contradistinction to the pure SU(3) gauge theory, the pressure, due to

the quarks and gluons, appears to be positive everywhere and continuous at the

phase transition. The same behaviour was also observed in a simulation carried


out by Kogut and Sinclair (1990) with four dynamical light quarks. The partialpressure due to the gluons was however found to be negative in the vicinity of thephase transition, just as in the pure SU(3) gauge case.

This concludes our discussion of the energy density and pressure in the hightemperature phase of QCD. Let us now take a brief look at the static quark-antiquark potential above the critical temperature.

The qq-potential above Tc

In the presence of a quark gluon plasma the static gg-potential is expected tobe Debye screened. The screening mass is determined, in a way analogous to thatfor an ordinary plasma, from the zero momentum limit of the time componentof the vacuum polarization tensor evaluated at zero frequency. In one loop orderthe electric screening mass is given by (19.88). In two loop order one is facedwith infrared divergencies, and an infinite number of graphs need to be summedto yield an infrared finite result (Toimela, 1985). Hence one can only trust thelowest order perturbative result. The lattice formulation of QCD provides us withthe possibility to study the gg-potential non perturbatively. The results of theMonte Carlo simulations can then be compared with the low order perturbativecalculations.

There are several gg-potentials that can be studied, since the quark-antiquarkpair can be in a singlet or an octet state. We denote these potentials by Vi(R, T)and Vs(R, T), and identify them with the free energy of the system:

{Tre-H'T){l)=e-v'^^'T.

Here the trace is taken over all states of the system with a heavy qq pair in thesinglet (I = 1) or octet (I = 8) state, separated by a distance R. We now define thethermal average of the colour singlet and octet quark-antiquark potential V(R, T)by taking the average of the above expression over the two possible states weightedwith their degeneracy:

e-V(R,T)/T _ I(e-Vi(il,r)/T + 8e-V8(R,T)/T^

The colour averaged potential can be extracted from the following correlationfunction of two Polyakov loops (McLerran and Svetitsky, 1981),

-V{R,T)/T = (TrL(d)Trtf(R))(\L\)2


where

is the Polyakov loop averaged over the N^ spatial lattice sites. By normalizing thecorrelation function in this way, one eliminates divergent self-energy contributionsto the potential. For a detailed discussion of the singlet and octet potentials werefer the reader to the paper by Nadkarni (1986), and to the review article byKarsch cited before. Here we only make some general remarks.

Perturbation theory predicts that the singlet and octet potentials are related

V${R,1)

where

Vl(R,T) = -9-^ê-m^R

in leading order. Hence the singlet potential is attractive, while the octet potentialis repulsive. Their relative strength is such that the colour averaged potentialbehaves like [exp(-2mE(T)R)]/R2.

The determination of the screening mass from Monte Carlo simulations is verydifficult since one needs to study the behaviour of the correlation functions over alarge range of separations of the quark and antiquark. But for large separationsthe signal tends to get drowned in the statistical noise. And this situation wors-ens with increasing temperature, since the screening mass is expected to increasewith temperature. For this reason the region slightly above the phase transition,which is more accessible to numerical computations, has been studied in greatestdetail. Knowledge of the potential in this region is in fact of great importance,since for temperatures just above Tc a quark gluon plasma may have been formedin the experiments performed at CERN, involving the high energy collisions ofheavy nuclei (Abreu et al., 1988). A strongly screened potential would inhibit theformation of bound states with a large radius. The suppression of such boundstates could (possibly) be used as a signal for plasma formation. What numericalsimulations tell us is that close to the critical temperature the colour averagedpotential can actually be well approximated by a simple screened Coulomb formwith an effective screening mass. It is therefore non-perturbative in this region.There exists however some evidence that for temperatures well above the criticaltemperature the colour averaged potential, as well as the potentials in the singletand octet channels, aquire the Debye screened form predicted by perturbation the-ory. This is supported, in particular, by a high statistics calculation of Gao (1990)


performed on a 243 x NT lattice, with NT = 4, 6, and 16 . This author found goodindications that perturbative behaviour sets in at temperatures T > 3.5TC.

20.9 Some Possible Signatures for Plasma Formation

In the previous sections we have seen that there are strong indications fromlattice calculations that at high temperatures QCD undergoes a phase transitionto a new state of matter, the quark gluon plasma. Such a plasma is expected tobe formed in very high energy collisions of heavy nuclei. It is therefore importantto look for a unambiguous experimental signal for plasma formation. Our follow-ing discussion of a possible signature will be more than incomplete and is mainlyintended to stimulate the reader to confer the extensive literature on this fascinat-ing subject. For earlier comprehensive reviews the reader may consult the book" Quark-Gluon Plasma 2F published by World Scientific (1995), and the reviewarticle by Meyer-Ortmanns (1996).

Of the signatures that have been considered in the literature we have chosento discuss in greatest detail the " J / $ suppression", proposed by Matsui and Satz(1986), because of its simplicity, and because it has been the trigger for manysubsequent investigations. This signature has been the subject of much dispute,since it is believed not to provide an unambiguous signal for plasma formation.Other signatures have been intensively discussed since then, and we shall onlymention them briefly at the end.

Many years ago Matsui and Satz (1986) made an interesting proposal fora possible signal of plasma formation. These authors had emphasized that, be-cause the (/g-potential is Debye screened in a deconfining medium, the formationof <j></-bound states will be strongly influenced by the presence of a quark-gluonplasma if the screening length (which can be estimated from lattice calculations)is sufficiently small. This led them to predict that the formation of the J/ip (a351 cc-bound state with a mass of 3.1 GeV) should be strongly suppressed in highenergy nucleus-nucleus collisions if a hot quark-gluon plasma is formed. Since theH+H~ decay of the J/I/J provides a very clear experimental signal for its formation,and since the mechanism for background muon pair production is fairly well under-stood, this resonance appears to be a good candidate for studying the propertiesof the deconfining medium.

Experiments involving high energy collisions of heavy nuclei performed atCERN (Abreu et al. (1988); Grossiord (1989); Baglin (1989)) have shown thatJ/ip formation is substantially suppressed for small transverse momenta of theJ/ip, and for large values of the total transverse energy released in the collisions.


The plasma hypothesis is able to account for the observed suppression pattern.But if J/ip suppression is to be an unambiguous signal for plasma formation, thenone must rule out the possibility that other more conventional mechanisms canalso explain the observed effects. In fact, following the pioneering work of Matsuiand Satz, several authors have pointed out that inelastic scattering of the J/ip ina dense nuclear medium can also lead to substantial suppression (see e.g., Gavin,Gyulassy and Jackson, 1988; Gerschel and Hufner, 1988). Nuclear absorptionalone, however, does not suffice to reproduce the experimental data. But byincluding also initial state interactions, one is able to obtain results compatiblewith the observed J/tp suppression pattern. (Gavin and Gyulassy, 1988; Hufner,Kurihara and Pirner, 1988; Blaizot and Ollitraut, 1989).

In the following we shall take the point of view that a plasma has been formedin the heavy ion experiments performed at CERN, and that the observed J/tpsuppression pattern is due to Debye screening in a plasma. For a review of otherpossible suppression mechanisms see Blaizot and Ollitraut (1990).

What is so attractive about the plasma hypothesis is that it gives a simplequalitative explanation of the effects observed in the NA38 experiments. That aplasma could have been formed is not out of this world. The above mentionedcollaboration has studied, in particular, the J/tp production in collisions of oxygenand sulfur at 200 GeV/nucleon incident on a uranium target. A rough (may betoo optimistic) estimate of the energy density deposited in the collisions (Satz,1990) yields the value e = 2.8 GeV/fm3. Prom lattice calculations one estimatesthe energy density required for deconfinement to be 2-2.5 GeV/fm3. From theseestimates one can at least conclude that the formation of the plasma in the abovecollision process cannot be excluded. But, given the uncertainties in the estimates,one must accept the possibility that a plasma has not been formed.

In the plasma picture, J/tp suppression is not a consequence of individualcollisions between nucleons in the projectile and target nuclei, but is the resultof a collective property of the medium, i.e. the existence of a Debye-screened qqpotential. The details of the suppression pattern will, however, depend on thecharacteristics of the plasma formed in the collision, such as its spatial extension,temperature, hydrodynamical expansion etc.. But independent of such details onecan make the following general statement: since the screening length decreaseswith increasing temperature, the influence of the plasma on bound state formationshould become more pronounced with increasing temperature. Hence there shouldexist a Debye temperature TD above which the potential is so short-ranged thatit can no longer bind a given quark-antiquark pair. This temperature will depend


on the particular pair considered.

For temperatures below the transition to the quark-gluon plasma the static

<?<?-potential is of the form

V(r) = - - + ar, (20.55)r

where a is the temperature dependent string tension, and a the temperature de-

pendent coupling constant. Prom renormalization group arguments a is expected

to decrease with increasing temperature. At the critical temperatur Tc, the string

tension vanishes, and above Tc the colour singlet potential is expected to be re-

placed by a Debye screened Coulomb potential of the form

V(r) = -*e-rlrD(T) ( 2 a 5 6 )r

where ro(T) is the Debye screening length, which can be estimated from lattice

calculations. In the WKB approximation one can determine the minimum screen-

ing radius r^*"' for which the potential (20.6) can still accommodate a bound

state of a quark and antiquark with zero angular momentum and radial excitation

number n (Blaizot and Ollitraut, 1987):

r (min) ( T ) = n2^/2ma, (20.57)

Here m is the mass of the quarks. The J/if> resonance is a 35i bound state of a c and

c quark with n = 1. Inserting in (20.57) some typical values for the charmed quark

mass, mc, and coupling constant a(mc « 1.37 GeV, a « 0.5) determined from

spectrum calculations (Quigg and Rosner, 1979; Eichten et al., 1980), one obtains

that r^71' « 0.45/m. The actual value is however expected to be larger, since

the effective coupling constant decreases with increasing temperature. Making

use of the estimates for ro{T) obtained from lattice calculations, one is led to

the expectation that the J/ip cannot be formed already at temperatures slightly

above the deconfinement phase transition. Other resonances like the ip', which also

decays into a /i+/i~ pair, but have a larger binding radius than the J/tp should be

suppressed already at a lower temperature.

Let us now follow the fate of a cc pair, which is expected to be produced within

a very small space-time volume in the early state of the collision process before the

plasma had the time to form. Because of the large mass of the charmed quarks,

it is unlikely that the cc-pair is produced within the plasma. Thus the creation of

charmed quarks at temperature T should be suppressed by the Boltzmann factor


exp(—mcT).* In the absence of a quark-gluon plasma the c and c quarks wouldbegin to separate and combine to form a J/ip once their separation has reachedthe binding radius of the J/ip (which is about 0.2 - 0.5/m). The time required forthis binding process to take place in the rest frame of the quark pairs is called theformation time TQ. It can be estimated from the radius of the J/ip and the averageradial momentum of the charmed quarks in the J/ip bound state. In addition tothe formation time there are two other important time scales which are relevantfor discussing the fate of the cc-system: the time required for a plasma to beformed in the collision process, and the time required for the plasma temperatureto drop below To, where Debye-screening is no longer effective in inhibiting J/ipformation. This cooling process is associated with an expansion of the plasmawhereby the central hot region shrinks with time. Hence to study the fate of acc-pair we must follow their motion through the dense nuclear medium taking intoaccount the hydrodynamical expansion of the system.** This is a complicatedproblem and one must resort to simple model calculations*** But the generalqualitative features of J/ip suppression can be deduced without performing anyexplicit calculation. The general scenario is the following.

Let us assume that the cc-pair is produced before a plasma has been formed.If the c and c quarks find themselves in a deconfining medium by the time theycould form a J/ip, then the J/ip will not be formed. Hence the c and c quarkswill eventually leave the hot plasma region and combine with other non-charmedquarks (of which there are many around) to form charmed particles (—>• opencharm). The number of cc-pairs which are still trapped within the plasma by thetime their separation is of the order of the bound state radius of the J/ip willdepend on various factors. First of all, if the cc-system carries sufficiently largetransverse momentum pT, then it will be able to escape the hot plasma regionbefore having reached the bound state radius of the J/ip- Hence normal J/ipformation will be possible. These J/ip's decay subsequently into a (J,+(J,~ pair,which can be easily detected in the experiment. Hence for sufficiently large px,J/ip formation should not be suppressed at all. But just how large PT must befor this to be the case depends on the plasma life time and on the region occupiedby the hot plasma in the course of its expansion. Thus if the plasma lifetime

* See e.g. the review of Satz in "Quark Gluon Plasma", edited by R.C. Hwa(World Scientific, 1990).** Unless they have already escaped the critical region by the time the plasma

has been formed. In this case the plasma will not affect J/ip formation.*** See the review by Karsch in the reference given in the previous footnote.


is sufficiently short, then even Jftp's with low transverse momentum should notbe suppressed. But the plasma lifetime is a function of the initial temperature.And the initial temperature depends on the energy deposited in the collision. Thehigher the energy density e, the higher the initial temperature, and hence theplasma life time. If the initial temperature of the plasma is high enough, it willtake a longer time to cool below the critical temperature To where Debye screeningno longer is effective. Now a measure for the energy deposited in a collision is thetotal transverse energy Ex released in the event. This energy tells us how violentthe collision was. Hence for a given initial spatial extension of the hot plasma, weexpect that the screening mechanism will become more effective as ET is increased.

Fig. 20-24 Dilepton spectrum in oxygen-uranium collisions for two

transverse energy cuts observed by the NA38 collaboration at CERN (Abreu

et al., 1988). The figure is taken from the review by Satz (1990).

Let us now look at the experimental situation. Formation of the J/tp shouldshow up as a peak in the invariant mass M of the /i+/i~ pairs at the mass ofthe J/ip- In fig. (20-24) we show the dilepton spectrum in oxygen-uranium colli-sions obtained by the NA38 collaboration at CERN (Abreu et al., 1988) for twotransverse energy cuts. This figure is taken from the review by Satz cited ear-lier. To exhibit the suppression of the J/tp at larger transverse energies, the fitted


H+/j,~ continuum* in the ET < 33 GeV and ET >81 GeV data (arising from otherprocesses than the decay of the J/rp, I(J' etc.) have been matched.

Figure (20-25) shows the dependence of the ratio of the number of J/tp eventsto the number of continuum events N^/Nc as a function of the transverse energy.The figure is taken from the review of Kluberg (1988). The ratio is seen to decreaseby about a factor of 2 between the lowest and highest transverse energy binsconsidered.

Fig. 20-25 Ratio of the number of J/ip events to the number of con-

tinuum events, in oxygen-uranium collisions, as a function of the transverse

energy. The figure is taken from the review by Kluberg (1988).

As we have pointed out earlier, the formation of a plasma should also lead tostronger J/ip suppression for smaller transverse momenta of the J/?/J'S. This effecthas been clearly seen by the NA38 collaboration.

In fig. (20-26) we show the pr dependence of the ratio of J/ip events inthe highest transverse energy bin, to the number of events in the lowest E? bin.The figure is taken from Abreu et al., (1988), and shows the expected decrease

* There are several sources for background /u+/i~ production which need tobe considered in detail in order to determine the actual strength of the observedeffect. In particular one requires detailed information about the dependence of theproduction rate on the transverse energy and momentum of the background fi+n~pairs before one can acertain that the observed effect is not due to an enhancementof fi+fj,~ production in the continuum. For a discussion of this problem see thereviews of Satz and of Blaizot and Ollitrault (1990).


Fig . 20-26 Ratio of J/tp events in the highest transverse energy bin

to the number of events in the lowest Ex bin as a function of the transverse

momentum. The data is from Abreu et al. (1988).

in the J/ip suppression for increasing transverse momenta of the J/tp. Model

calculations based on the plasma hypothesis show that one can get reasonable

quantitative agreement with the experimental data.

F ig . 20-27 Ratio R{PT) of the number of events in the highest trans-

verse energy bin to the number of events in the lowest transverse energy

bin as a function of the transverse momentum of the J/ip- The solid lines

are the results obtained by Hiifner, Kurihara and Pirner (1988). The px

dependence is due to initial state interactions. The data is from Abreu et al.

(1988).

But, as we have already mentioned, a large amount of J/ip suppression can

also be obtained assuming that the J/ip disintegrates due to inelastic scattering


processes in a dense nuclear medium. By including also the effects arising frominitial state interactions, one can obtain a py-dependence of the J/ip suppressioncompatible with the experimental data (Gavin and Gyulassy, 1988; Hiifner, Kuri-hara and Pirner, 1988; Blaizot and Ollitraut, 1989). As an example we show infig. (20-27) the results obtained by Hiifner, Kurihara and Pirner (1988) for theratio of the number of high ET to low ET events as a function of the transversemomentum in oxygen-uranium and sulfur-uranium collisions. The data are fromAbreu et al. (1988). In their analysis gluon multiple scattering was the dominantmechanism for the PT distribution of the J/ip in nuclear collisions. As seen fromthe figure, the agreement with the experimental data is quite good. Hence one doesnot know which of the scenarios for J/ip suppression (if any) is correct. Probably,J/ip suppression is a result of the interplay of several different mechanisms.

We close this chapter with a very brief discussion of two other proposals madein the literature for a possible signature of the formation of a quark-gluon plasma.For details the reader should consult the book mentioned at the beginning of thissection, and the review article by Meyer-Ortmanns (1996). The main objectiveis, as always, to look for features of particle spectra which are sensitive to theirproduction in a deconfmed medium.

Dilepton Production

It has been argued already many years ago that dilepton production couldprovide a clean signal for the formation of a QGP.* Since leptons only interactelectromagnetically or weakly they can escape from the dense nuclear matter with-out further rescattering, and therefore carry the information about their produc-tion at all stages of the collision process, from the initial hadronic phase, throughthe evolution of the plasma until freeze-out. There are several mechanisms fordilepton production which are operative in different kinematical regions. At highinvariant masses (larger than 2-2.5 GeV) we have Drell-Yan production due toparton-antiparton annihilation, and lepton pairs originating from the decay of dif-ferent hadrons. These processes are well understood. Low mass dileptons (withan invariant mass less than 1 GeV) are mainly produced from the decay of neu-tral mesons. If they originate at the late stage of the collision process, where thesystem has cooled down, then they carry no information about the hot and densematter in a plasma. On the other hand suppression lepton pairs in the low massregion could be interpreted as being due to the melting of the p and cp within the

* For a review see Ruuskanen (1992).


plasma. Candidates for a signal of plasma formation are the thermal leptons withan invariant mass in the intermediate mass region. They originate from partoncollisions in a hot and dense medium and their spectrum is sensitive to the tem-perature of the plasma at the time of their formation, as well as to the nature ofthe phase transition, close to Tc (Cleymans et al. (1987)). A typical observable isthe differential multiplicity per invariant mass-squared, transverse momentum andrapidity interval. The prediction of the dilepton spectrum is carried out within thehydrodynamical framework, and involves several assumptions and approximations(see e.g. the review by Meyer-Ortmanns (1996)).

Strangeness Production

It was predicted already some time ago that the production of strange hadronsshould be enhanced in heavy ion collisions [Rafelsky (1981)]. Such an enhance-ment was observed for example for the ratio K+/TT+ [Abbott et al. (1990)]. Themechanism for strangeness production is quite different in a quark-gluon plasma(where strange quark pairs are produced from a very dense medium of quarksand gluons) and in a gas of hadrons (where hadrons with opposite strangeness areproduced in inelastic hadron-hadron collisions). Although the observed enhance-ment of strange particles is consistent with the idea that a QGP has been formed,there are numerous assumptions that go into predicting this enhancement, andthe involved sources of possible systematic errors may be difficult to control [seee.g. the review by Meyer-Ortmanns (1996)]. The problem is that because strangeparticles interact strongly, the accumulation of strangeness proceeds throughoutall stages of the collision process, from the very beginning to the very end. In factthe observed K+/TT+ enhancement can also be be explained in a more conventionalway. That strangeness could play a crucial role in the search for a signature ofQGP formation had been advocated by many physicists. It has also been proposedthat the formation of a quark-gluon plasma in heavy ion collisions could lead tothe production of exotic droplets of stable or metastable strange quark matter,consisting of approximately equal numbers of strange, "up", and "down" quarks[Greiner et al. (1987)]. For a review we refer the reader to the book "Quark-GluonPlasma 2" mentioned earlier.

But what does experiment tell us? The Relativistic Heavy Ion Collider(RHIC) at Brookhaven National Laboratory began its operation in 2000. Twoion beams are brought to collision with a center of mass energy of 200 GeV perNucleon. Whatever the precise implications of the results may turn out to be, itappears that one statement can be made rather safely: the data strongly suggest


that the phenomena observed in the nuclear collisions reflect collective behaviour.The energy density deposited in the early stage of the collision has been estimatedto be 20 GeV/fm3, which exceeds the theoretical estimate given earlier by far. Maybe we are in for many more surprises once sufficient precise data is available fromRHIC.

Much can be said about the very important problem of finding an unambigoussignal for the formation of quark-gluon plasma. We have only taken here a glimpseat a few of the proposals made in the literature so far. Several other signals, likedirect photons, jet production, hadron mass modifications in hot dense media havealso been discussed. But a definite signature for plasma formation is still waiting tobe found. Clearly, the experimental verification of the existence of a quark-gluonplasma would be the most spectacular prediction of lattice QCD.

A P P E N D I X A

The energy sum rule (10.40) has been checked in lattice perturbation theoryby Feuerbacher (2003a,b) up to 0(#o). In this appendix we give a few technicaldetails which are useful for carrying out the very extensive computations. In thefollowing all quantities are understood to be measured in lattice units, i.e., wesuppress the "hat" on the dimensioned quantities.

The first step in verifying the energy sum rule consists in making use ofthe exact action sum rule (10.24a) to cast the energy sum rule in a form whichminimizes the computational effort.

With the definition (10.39b) the energy sum rule (10.30) takes the form

V(R, f) = lim 4 L < -VT + Ps >9?-o + ^ ^ / 3 <VT+VS >,ff_ol ,(A.I)

where < O >qq-o has been defined in (10.24b), and where, in accordance with(10.25a), we have made the replacement

< V'a >gg--o= lim - < Va >g ?_0 (A2)1 —J-OO X

Next we make the decomposition < -VT + Vs > = - < VT + Vs > +2 < Vs >.Then (A.I) takes the following form for SU(N),

V(R,f) = 4 [~§jV- < S >qq-o +§V- < Ss >g?-o +^~ < S >„_„](A3)

where S is the action (10.22) on an isotropic lattice, and Ss is the contribution tothe action arising from the spacial plaquettes only. This form of the energy sumrule is particularily convenient, since one can make use of the exact action sumrule (10.24a)* to express limr-»oo(< S >qq--o /T) in terms of the potential andderivatives thereof. Thus the action sum rule can be rewritten in the form

1 OTV

lim ^ < 5 >99-_o= -3o;q-2 • iAA)

The energy sum rule (for SU(N)) then becomes equivalent to the following state-ment

V - „_ J | | * + bMfrt = lim lVjl < Ss >«_0 . (A5)2Ndg% 2g0 dgl f-+ooT N

* Recall that the action sum rule follows directly from the definition of thepotential via the Wilson loop.

542

Appendix A 543

Since the leading contribution to the potential is of O(g%), and since the same is

true for < Sa >qq-o (a = T,S), as we shall see below, we only need to know ?7_

up to 0(ffo)- N o w uP t o °{9o) V- i s g i v e n by (Karsch 1982),

INr)- = î-+cN (A6)

9owhere c has been determined by this author as a function of the anisotropy (whichhere is set equal to 1).

The potential has been calculated in lattice perturbation theory up to 0{g^)by Kovacs (1982), and by Heller and Karsch (1985), and is given by (9.8). Tocompute the connected correlator

<? <? W ~>< Ss >gg-_0= < ' w > ~ <SS> (A.7)

up to this order, we must expand the action and the Wilson loop in powers of thecoupling. Consider first the expansion of the Wilson loop, normalized for SU(N)conveniently as follows,

W[U] = ±TrHUt, (A.8)tec

where C is a rectangular loop, and Ue. denotes the matrix valued link variableassociated with the link labeled by L Written in terms of the gauge potentials wehave

Ue = ei9oAt . (A.9)

The expansion of W[U] has the form

W = l - gfaW - ff0V3> - 5oV4) + O{g*) . (A10)

Note that because of the trace in (A.8) there is no O(go) term. * Since (A.10)starts with the unit element, the leading term in W which contributes to (A.7) isof C?(<7o)- We therefore only need to expand Ss up to this order:

Ss = S.W + goSP + g*sW + O{gl) . (All)

* Expanding the product of the link variables in (A.8) in powers of the coupling,the leading term is just given by the sum of gauge potentials associated with thelinks along the Wilson loop. For SU(N) (N > 2) these are elements of a Liealgebra with vanishing trace.


The perturbative computation of the (gauge invariant) rhs of (A.5) is done mostconveniently in the Feynman gauge. Since we are computing expectation values ofoperators which are at least of 0(<7Q), the "Boltzmann" factor in the path integralexpressions can replaced by e~Seff, where

Seff = S + g*S™P + glS%\as + O(gl) . (A12)

Here SFp and Smlas are the contributions arising from the Faddeev-Popov deter-minant (associated with the gauge fixing), and from the integration measure (seechapter 15). Up to O(g$) the Boltzmann factor es'tf can be expanded as follows

es^*esi0) (i+goSW+gSlllsWr + SV + sV+SSL]} .

Then up to C(<?o) < $s >qq-Q is given by (Feuerbacher, 2003)

< Ss >99-_o = -gl < S ^ >cm +g* < S^Sû™ >con

- \ < Si°\S^)2^ >con Wo < SJ°>4V2) >con+ gt < S^S£lasJV >cm +ff4 < 5(i)5(i)w(2) >cm {A I 3 Q )

- gt < S^JV >con +g* < S^SU^ >con

- gt < SMu,^ >cm -gl < S(DW(3) >cm

-gt<S^^>con<^>+O{gt),

where, generically

< Ou<"! > c o n =< Ow^ >-<O>< wf") > (A.13.6)

and 5^°) is the free action. The subsript "con" stands for "connected". Note thatall expectation values are now calculated with the Boltzmann factor exp(—5^).Note also that goS^ and g^S1^ are the contributions to the action of the 3 and4-gluon vertex (see chap. 15). For a perturbative calculation one must expressthe above expectation values in terms of expectation values of products of gaugepotentials living on the links of the lattice. Consider e.g. the Wilson loop whoseexpansion determines the a/n)'s in (A.10),

Wc = ^zTr Y[ ei9oA' = jjTreBc (A.14)etc

where Be is again an element of the Lie-algebra of SU(N), and where the productof the exponentials is ordered along the contour C in the counterclockwise sense.

Appendix A 545

The index £ = 1,2,3 ••• labels the successive links along the contour. Makingrepeated use of the Campbell-Baker-Hausdorff formula, the exponent Be will begiven by the sum of X^fgc-^' an<^ higher order commutators in the Bi's, whereBe = igoA(. Hence every term in this expansion will be an element of the Lie-algebra and therefore traceless. For this reason we only need to know Be up toorder g$ in order to calculate the contribution to the Wilson loop up to order g$.We had already made use of this when expanding the plaquette contributions tothe action in section (15.3). In particular we had made use of (15.31) to calculatethe argument of the exponential associated with an elementary plaquette. In fact,this formula remains valid for an arbitrary product of link variables. Thus onecan easily convince oneself that if (15.31) holds for the product eBleB2 • • • eBn =eMn, it also holds for eMneB"+1. An alternative form for (15.31) has be used byFeuerbacher (2003). To keep in the spirit of this author we will make use of itbelow. Up to O{ffo) the exponent Be in (A.14), expressed in terms of the gaugepotentials, is given by

i • (A15)

(li,Ja)<I3 li<h

Expanding the exponential (A.14) one finds that the coefficients u>t in (A.10) aregiven as follows for SU(N) (Heller, 1985),

«"•-iff (!>")'•

where B = 1, • • -N, and

-(3) = JfiTr fe*y + ^ r r fa* ^ [Ah,Ah]) ,\ i / \ i h<h )


\ I J \h<l2 /

- ^ E ^ E uh,Ah},Ah])\ i h<h<h )

-JMTr\Âl E [ i.t . s]]) (A16)\ ' (h,h)<h J

V i h<i2 I

-^( (E^VEIA^V

In perturbation theory the expectation values in (A. 13a) involve propagators con-necting sites located on the Wilson loop and on the boundary of spacial plaquettes,modified by 3 and 4-gluon interactions arising from Seff. In fig. (A-l) we showrelevant diagrams contributing to (A. 13a). The figure is taken from Feuerbacher(2003).

The evaluation of the terms in (A.13a) is evidently quite non trivial, and re-quires substancial gymnastics. When computing the expectation values in (A. 13a)one is confronted with various types of partial sums involving gauge potentials liv-ing on the square contour of the Wilson loop. In the following we give two simpleexamples demonstrating the technicalities involved.

Example 1

Consider e.g. the sum over potentials along the square contour of a Wilsonloop. Let No denote the base point of the Wilson loop with spacial and temporalextent R and T (measured in lattice units). The sum of interest has the form

fi-i

J2 M = ]T {A^No + np.) - A^NQ +TP + np)

(A.17)T-l v '

+ Y2 (AîNo + Rp + nv) - AV(NO + nv))n=0

Appendix A 547

Fig. A - l Connected diagrams contributing to < Ss >qq-o, with

< Ss >qq-o given by (A.13a), in next to leading order (Feuerbacher,

2003b).

Fourier decomposing the potentials as follows*

MN)=L 0r*A{p)eip{NH)' (A18)

and making use of the identity

n=0 Slt l I 2 J

one readily finds that the above sum can be written in the form

J2 At = -2i [dtp Aa{p)eip<no+R^+T^sin(TPv/2)sin{RPli/2)eecw {A }

AQ Msin(pM/2) ^ s i n ( P l , / 2 ) ; '

* Recall that the potentials are evaluated at the midpoints of the links (seechapter 14).

(A19)


where fj, / v. Note that the argument of the exponential involves the lattice sitelocated at the center of the loop.Example 2

As another example consider the restricted sum J2il<e2[Ai1, Ai2], where £\and £2 label the potentials along the Wilson loop in an ordered way. A convenientway for evaluating this sum is the following: a) choose, in turn, £\ to correspondto points on the four sides of the loop; b) for every such choice sum over all £2associated with the remaining sides (thus satisfying £2 > £1)', c) finally considerthe contributions where £\ and £2 correspond, in turn, to points lying on the sameline element of the Wilson loop C. One then readily sees that

4

£[î.^] = £r* (A21)£i<£ 2 fc=l

where

R-l T-l

Ti = []T A^No + np), 52 (AM + RH + n'v) - AU(NO + n'v))]n=0 n'=0R-l R-l

- [Y, A^no + nfi), Y MNo + Tv + n1 ji)]n=0 n'=0R-2 R-l

+ [YJAll{N0 + njJ,), Y MNo + n' A)]n=0 n'=n+lT- l R-l

T2 = (Y AV(NQ + Rp, + ni>), - ^ A^No + Ti> + n'fi)]n=0 n'=0T-l T- l

+ (%2 AV(NO + R(i + nP), - Y, A»(No + n'v)] (A.22)n=0 n'=0T-2 T- l

+ (52 Au(N0 + Rfi + ni>), 52 Al/(N0 + Rp, + n'i>)}n=0 n'=n+l

R-l T-l

T3 = [~52 MNo + Tv + np), - 52 MNo + n'v)]n=0 n'=0.R-l n - 1

+ [- 52 A»(No + Ti> + np), - 52 An(No + T0 + n'p)]n=l ri=0

T-l n-1

Tt = [-52 AU(NO + nu), - Y AU(NO + n'v)] .n=l n'=0

Appendix A 549

The corresponding Fourier integral expressions for Tfc can now be obtained makinguse of (A. 17). Thus for example one readily finds that

J2 £ [A^No + nA), AV(NO + R(i + n'u)} = f (dp)(dp')[^(P). ^ ( P ' ) 1n=On'=O JBZ

;;ci(P+P')^+Rfe+r#)ciP:.#c-^xsin(pMf)sin(p^)

sin(^)sin(4)

Clearly, restricted multiple sums of double commutators of the gauge poten-tials, lead to expressions which are far more complicated. Major simplificationshowever result when one considers the limit T —> oo, which is of interest whenchecking the energy sum rule. Furthermore, a large number of lattice integrals canbe calculated almost analytically. In evaluating the integrals extensive use is madeof the cubic lattice symmetry, and of partial integration. This is demonstrated bythe following examples taken from Feuerbacher (2003). All relations hold in ddimensions.

Consider the following integral which vanishes by construction.

r ddp d sin(pM)

iBzWdp* P2

where

pM = 2sin(-pM)

and

f=£%/ x = l

By performing the differentiation in the integrand one is readily led to

f ddP (l-ffi o f l - ^ - nJBZ i^Y \ P2 (P2)2 ) ~

Because of the cubic symmetry we can replace p^ by \i>2 • One therefore finds that

JBZ (^Y (P2)2 ~ ~ d ~ A ° + d ' ( j 4 - 2 3 a )

where

(A.236)


This integral can be reduced to a one-dimensional integral as follows: We firstwrite (A.22b) in the form

= rdtfr êxp(-4isin2(p/2)))Jo \J-n 2?r J (A.23C)

= -j dte-dtI$(t),

where

Jo(«) = •£- [" dx e~tcosx {A.23d)

is the Bessel function. This integral can be easily evaluated numerically.

Another trivial integral is given e.g. by

f ddp (p2)2

JBZ (27r)d (p2)2 •

Using the cubic symmetry of the lattice we can also write it in the form

JBZ (2*)" (P2)2 + d{d 1} JBZ ( 2 ^ (p2)2 '

where \x / v. Making use of (A.21a) one therefore has that

f ddp PIPI 2d-4

There are many further lattice integrals that can be evaluated with similar tech-niques. For the readers convenience we present some further useful integrals taken

(4.24)

Appendix A 551

from Feuerbacher (2003a):

JBz(^)d(P2)3 d x

f _^P_(^)! = 1 A _ I AJBZ W (P2)3 ™° d*1

f ddp PIPI _ 1 A , 1 A ,

JBZ (2«)d (P2)3 " M ^ I ) ° + d ( d ^ i ) A l ; ^ ^ ^

7 B Z (27T)d (P2)3 d A ° + d d ^

/" ddp P2pg 1 A , 4 A

/• ddp PlPlPl _ 2rf-6 8yBZ(27r^ (p2)3 d ( d - l ) ( d - 2 ) A o d ( d - l ) ( d - 2 ) A l ' / X ^ Z / A '

(yl.25a)where Ai is given by

A i - ( " - 4 ) L l ? w - {A3'-b)

In the limit ( J ^ 4 w e have that Ai ->• 2f^)^- There are many other type of latticeintegrals which need to be evaluated in order to check the energy sum rule (A.I).The complete set of calculations, which go beyond checking the energy sum rule,can be found in (Feuerbacher, 2003).

A P P E N D I X B

Up to O(e2) the vertices in momentum space associated with the axial vector

and pseudoscalar currents (14.41) and (14.42), and with the operator A denned

in (14.43), are easily obtained by Fourier transforming the field as follows,

i/>(x) = [ (dp) fo)?*-*JBZ

4>{x) = f (dp) | (p)e- i"-x

JBZ

A^x) = [ (dk) A»(k)eikx ,JBZ

where, generically,

<*>•<&•

If O(x) stands for any of the operators J5M(X), j<j{x) or A(a;), defined in (14.41),

(14.42), and (14.43), then we define the Fourier transform O by

O(x) = f (dq) e-Ôfa) , (B.la)JBZ

where

O(q) - J(dp')(dp) • • • V(q;p',p, • • •) . (B.lb)

Here the dots stand for possible photon momenta ki (see below), and

V ( q ; p ' , p , • • • ) = ( 2 n ) i 6 ^ ( p ' - p - q - - - - ) V ( ( q ; i / , p , • • •) , (B.lc)

with the vertices V(q;p',p, • • •) denned as follows

v/"HN. ~* e~^ C O S ( ( ^ ) M « ) 7 M 7 5 , (B.2a)

-t-M<QA-M - > -eaSê'^ s in ( (^ ) M a) 7 M 75 , (5.26)

552

Appendix B 553

-a2e26in/SuXe-iSJ^ cos ( (^ -y^- ) M a)7 M 7 5 , {B.2c)

[Mr(p,a) + Mr(p',a)}l5, (B.2d)

( k k 1e r | sin(p + -)Ma + sin(p' - -)Ma J75 , (B.2e)

are2<5MI/1 cos(pH —)Mo + cos(p' —)MaJ75 . (B.2f)

Taking the left derivative of J5M(a;) amounts to contracting the vertices (B.2a-c)with — 2i sm(qfJ,a/2) exp(igô/2). Hence the exponentials appearing in the vertices(B.2a-c) are eliminated by this contraction.

APPENDIX C

Consider a general group element of SU(3) in the fundamental representation:

*=±*T*. (ai)

A=l

We want to compute U~l5U, where SU = U(4> + 8<j>) — U{4>). To this end weintroduce the following matrices (Boulware, 1970)

W(X) = U(\4),

SW(X) = U{$4> + 6(/>)) - U(X(f>),

where A is a real parameter, and where for simplicity, we have suppressed thedependence of W on </>. Next, we derive a differential equation for

5X($ = -iW-l(X)SW(X), (C.2)

and obtain its solution, subject to the condition that Sx(0) = 0. Then U~15U is

given by i#x(l)- We now give the details.

From (C.2) we obtain

^SX(X) = -taWr~A1(AW(A) - iW-1(X)^5W(X). (C.3)

Inserting the expressions

•^-SW(X) = iSW(X)4> + i(W(X) + SW(X))5</>,UA

into (C.3), one finds that

—8X(X) = i[6x(\), 4>} + W + itxWW-

Since 5x(X) is itself of order 5<j>, we have that up to 0(5<j>),

^5X(X) = i[5x(X),} + 5<j>. (C.4)

554

Appendix C 555

Furthermore, up to this order, 5\ is an element of the Lie algebra of SU(3).* Hencewe may write Sx(X) in the form

6X(\) = J2TA6XAW.

A

An analogous decomposition holds of course for 5<j>:

6<t> = Y,TAs<t>A-A

Making use of the commutation relations (15.2a), one finds that (C.4) implies thefollowing differential equation for the components SxA'

^XA(X) = - £ fABc5X

BW<t>C + 54>A. (C.5)B,C

But according to (15.8)

C B

where tc are the generators of SU(3) in the adjoint representation. Hence we maywrite (C.5) in the form

J^X(A) - -iMx(X) + 6$, (C.6)

where 8x and 5<fi are vectors with components 5xA and 5(f>A (A = 1 , . . . , 8), re-spectively, and where

A

is an element of the Lie algebra of SU(3) in the adjoint representation. Thecorresponding generators tA are normalized according to (15.9). The solution to(C.6), subject to the requirement that SxA(0) — 0, is now immediately obtained:

SxW = {—^) **•

* This can be easily seen by applying the Campbell-Baker-Hausdorff formulato i5X{\) = [e-iêiX^+s^ - 1].


Setting A = 1, we therefore find that

U~15U = i^TAMAB{<f>)5(l>B, (C.7a)B

where the matrix M{4>) is given by

MW=m~- {a7b)

This is the result we have been looking for. A similar expression to (C.7a) can bederived for 5UU~1 by solving the differential equation for

5x(X) = -i5W(\)W-l(\),

with W(X) as defined above. One readily finds that

SUitftU-1^) = iJ2TAMAB(-4>)5cf>B. (C.8)A,B

This expression will be useful in the following, where we study how U((j>) transformsunder infinitesimal gauge transformations.

A P P E N D I X D

In this appendix we give a proof of formula (15.19).

Let U{4>) be the group element defined in (C.I). Consider the following in-

finitesimal transformation

eu"U{<P)e-Uu' = U{4> + 6<j>), (D.I)

where 5UJ and 5u>' are infinitesimal elements of the Lie algebra of SU(3) in the

fundamental representation. They can be decomposed as follows

A

6UJ' = J2TA^'A-A

We want to calculate 5<f> up to terms linear in 5u> and 8w'. This can be easilyaccomplished by making use of the results obtained in appendix C. Consider firstthe product

Uê-^' = U(4> + 5K>I)$), (D.2)

where the superscript R on 5f^,~. is to remind us that we are interested in the

change of cf> arising from the group multiplication of U(4>) with e~iSu> from the

right. Clearly the leading contribution to S^,^4> is of order 5ui'. To calculate this

change we write the right-hand side of (D.2) in the form

U(4> + 8?ul)) = U(4>){1 + u~\4>)sum,

where U~l5U is given by (C.7) with <5</> replaced by 6P,-.. But to this order wemay replace exp(—iSui') in (D.2) by 1 — i5w'. Hence we conclude that

5U}'A = -J2MABW5^B, (£>.3a)

B

or

Next we calculate

ei6uU{<p)e-iSu>' = ei5uU((j)'),

where $' = cj> + S^^tj). Let us denote the change in </>' arising from the leftmultiplication of U(<j>') with exp(iSw) by <5f ^ ' ; then

ei'uU(<l>l) = U^' + 6fi)n (DA)

557

(£>.3&)


The right-hand side can be written in the form

utf + sfw)ft) = {1 + 5U{<t>')u-l{4>')}u{4>'),

where 5U{<j>') = U(</>' + S^}(</>')) - U((j>'). On the other hand the left-hand sideof (D.4) can be approximated by (1 + iSuj)U(</>'). Now 5U(<t>')U~l(<f>') is given by(C.8) with 4> replaced by (ft). Hence we are led to the following relation between6IMA and bfu\4>B m leading order

5u>A =YJMAB(-<j>)5fu)4>B.B

Inversion of this equation gives

6U<t>A = Y,MA1B(-<l>)t"B- (D.5)

The total change in 0 induced by the infinitesimal transformation (D.I) is thereforegiven by the sum of (D.3b) and (D,5):

54>A = £ [M^(-0)feB - M2lB{4>WB] . (D.6)

B

Consider now an infinitesimal gauge transformation of the link variables

Let 5(u)4>A(n) denote the change in the group-parameters, defined in (15.1a,b),induced by this transformation. By making the appropriate substitutions in (D.6),one obtains

< W > ) = E [MAB (-4>M)S^B(n) - M^B (^(n))SujB(n + A)] . (D.I)B

This expression may also be written in the form (15.19). To this effect we set

5ujB(n + A) = 5ujB(n) + d*8ujB(n),

where dj} is the right lattice derivative. Then (D.7) becomes

<WM («) = ^ { [ M - H - ^ ) ) - M-l(^(n))]AB5uB(n)B (D.8)

-M^»)5?&;fl(n)}.

Appendix D 559

Making use of the explicit form for M(<f>) given in (C.7b), one can show after somesimple algebra, that

M - 1 (-4v(n)) - M - 1 (<^(n)) = - t fc^n) , (D.9a)

where$» = J»)iA (D.9b)

A

Substituting (D.9) into expression (D.8) we finally obtain

< W » = - E (i$M(«) + M-1 (^(n)) a") ^B(n), (0.10)B

which is the result we wanted to prove.

APPENDIX E

In this appendix we derive a formula which allows us to carry out sums overfermionic Matsubara frequencies in the continuum formulation, where these fre-quencies are not restricted to a finite interval.

Consider the function

h ^ = eifl(tt-l) + i '

where w is a complex variable. It has simple poles with unit residue located atui = uj + ifi, where UJ = UJ~[ are the fermionic Matsubara frequencies (19.57). Letf(u) be a function of the complex variable w which is non-singular for Ira w €[fj, — e, fj, + e], with e infmitessimal. Then

1 °° \ f- Y, /K" + V) = Y~R / d"f(")h(") - OK-1)P fc-cx) 1™P J°

where, for /j, > 0, C is the closed contour depicted in Fig. E-l.

" Imw^

• • • • p

Reu

Fig. E- l Contour of integration C in eq. (E.I)

By making use of the relation

—-— = 1 (E.2)ex + 1 e~x + 1

on the upper branch of the contour we can write (E.I) in the form

- V f(u>7 +in) = — dw f(to) - — / du) . J . \

27T 7-oo+iM-ie ^ e^(—*") + 1

560

Appendix E 561

Consider the case where f(ui) is of the form f(uj) = p(uj)/q(u)), where p(w) andq(ui) are polynomials in w. Then we can close the integration contours in the lasttwo integrals in the lower and upper planes, respectively, and obtain

1 °° 1 /-oo+iu+ie D /•,-, \

Z ^ e-i/3(<Di-i^) + 1 '

where Rf(ujj) stand for the residues at the poles of f(uj) whose location we havedenoted by WJ.

Next consider the integral in (E.3). If f(w) vanishes faster than l^l"1 for\u>\ —> oo, then the residue theorem tells us that

1 r-co+ip+ie 1 /.oo

— / duj f(w) = — du; f(u) - i Yl Rf&)-

Making use of the identity (E.2) we have

0</raiii<M+« 0</mi5;<M+e V J

so that

1 V^ // - , • x r duj ,/ ^ V- Rfa)

_ . y- R(Ql) ^ ' ^Z-/ e-i0(Qi-iiJ,) _i_ 1

7nnSi>0

where we have now set e = 0, since by assumption f(w) is non-singular in the stripImw G [/x — e, fi + e].

(E.3)

APPENDIX F

In this Appendix we derive expressions which are useful for performing sumsover fermionic and bosonic Matsubara frequencies on the lattice. We first considerthe fermionic case.

i) Fermionic Frequency Sums

As we have seen in section (19.11) the following type of sums are of interest:

i{{Pi}) = \ £ ff(e<(<Jr+iA);{Pi}).p I— £

2

where CJJ — -—ij- are the fermionic Matsubara frequencies, with J3 the in-verse temperature measured in units of the lattice spacing. The dependence ofg{elî +*£); {p^}) on the momentum variables {pt} will be suppressed from nowon.

Consider the following function of the complex variable ui:

h(u>) = , ~ ' ^ . (F.I)

It has simple poles located at u> = CJ~[ + ifi, with unit residue. Hence if g(elu) hasno singularities for Imw =G [p, — e, p, + e], then

I V g(e^r^)) = -J- / du,-J^— , (F.2)P ^ 27r/ c eW<o-<A) + i ' V ;

where C is the closed contour depicted in fig. (F-l).

, Ira u

[——F*—]1 1 ^

Fig. F - l Contour of integration C.

562

Appendix F 563

Introducing the variable z = elw, the poles of h(uj) are located on a circle withradius e~^ in the complex z-plane. The upper and lower branches of the contourin fig. (F-l) are mapped onto two circles of radius \z\ = exp(-/i + e) and \z\ =exp(—p, — e), respectively, traversed in a counterclockwise sense. Hence (F.2) takesthe form

a ^ 2™ /ixi=e-A+. * \e^zP +1] ( F 3 )

+ _L/ dz 9(z)2ni /jz|=e-A-e z [efczP + 1] '

If \z\~Pg(z) —> 0 for |z| —> oo, then we can distort the contour in the first integralto infinity, taking proper account of the singularities. For the case where g(z) isa meromorphic function of z, the combined contributions of the two integrals in(F.3) yields

4 y g(e«*r+W)) = y (Res-Z 9-^-) 1 , (FA)

l ~ 2

where ReSzis^- are the residues of g(z)/z at the poles, whose position we have

denoted by z».

ii) Bosonic Frequency Sums

Bosonic frequency sums on the lattice of the type required in section 11 ofchapter 19, i.e.

K(0A) = J E /(Î+;&})»

with w+ = 2£TT//3 can also be readily performed. By considering instead of h(u)in (F.I) the function

-/ x 09^ = ^ r r r •which has simple poles with unit residue located at the bosonic Matsubara fre-quencies o) ~, and following the same line of arguments as in the fermionic case,one readily derives the following summation formula:

i b + ^ Res» (*£)(F.5)

APPENDIX G

The Electric Screening Mass for Naive Fermionsin Lattice QED to One-Loop Order

The electric screening mass squared has been defined in section 6 of chapter19 as the infrared limit of the 44-component of the vacuum polarization tensorevaluated for vanishing photon frequency (cf. eq. (19.71)). In this Appendixwe compute this screening mass for naive fermions (i.e., for vanishing Wilsonparameter), following closely the work of R. Pietig (1994).

In the following the vacuum polarization tensor is defined as the negative ofthe one-particle irreducible diagrams with two external photon lines. The Feyn-man diagrams contribution in O(g2) are shown in fig. (19-6). Using the finitetemperature lattice Feynman rules discussed in chapter 19, one finds, after carry-ing out the traces in Dirac space that

TI{M(M+ h-AP*1 V r ^ sin2(££-+z£)

. 1 U r d3p cos2(i£+-t-a)<r + ifx)[G2 - sin(w++ w£" + ifi)sin(w^ + ifi)}e3^V-*(^ [sin2 (£7 + t£) + £2][sin2(a>+ + wj + i£) + F*} '

(G.la)where

E2 = ^ s i n 2 p i + m2,

i

i

G2 = J s inp j sin(p+ k)i + rh2.i

and where w^ and G)J are the Matsubara frequencies for bosons and fermionsdefined in (19.92b) and (19.93b). All quantities are measured in lattice units. Thefirst integral is the contribution of diagram (b) in fig. (19-6), which has no analogin the continuum. As always, the fermionic Matsubara frequencies appear in thecombination wj + ifi. The frequency sum can be performed by making use of thesummation formula (F.4).

564

(G.lb)

Appendix G 565

Expression (G.la) can be written in the form (recall that we always take /?to be even)

n }K+,*) = \ £ £ (0/(^0' â)

where

fM = )L1 tlV ; (2Ee-*z)2 - (z2e-2» - I)2

( z 2 e - 2 A e i ^ + e - ^ ) 2 (2ze-»G)2 + (z2e-2* - l){eiaie'^z2 - e " ^ )

{2Ee-P-z)2 - {z2e-2fi- - I)2 (2Fe~'i^)2 - (z2e^« e-2A - e-i<:>" )2

(G.26)is a meromorphic function of z. We have suppressed for simplicity the dependenceof f(z) on the momentum variables. Let us rewrite this expression as follows,

_ e4A(22e-2A _ 1 ) 2

ni=i(2-zi)- |"e8A-2^(z2e-2A+^ + e - ^ ) 2 ] 4z2(?2

e-2A + (Z2e-2A - 1 ) (z2e-2Me^ _ e - ^ )

L J n-=i(^-^)(^-^)(G.3a)

wherezx = -z3 = - e * - * ,

22 = - « 4 = e A + * ,

Z l " ^ 3 " e 6 ' (G.36)

z^ = - ^ = e ^ + * e - ^ ,

4> = ar sinh E,

i> = ar sinh F .

Now for |z| —> oo, /(z) approaches a constant. Hence we can make direct use ofthe summation formula (P.4) to calculate the frequency sum (G.2a). Thus

P e=_i i Y4 + i zP + i]

where

(G.4)


are the residues of f(z)/z at z = Zi. A similar statement holds for R{z'i). Noticethat only the poles at z = Zj, z = z[ contribute, since the residue of the aparentpole at z = 0 vanishes. The computation of the residues is straightforward, al-though tedious. One finds that

R(Zl) = R(z3) = -R(z2y = -R(Zi)* = ~tanhj> + H(4>J,G)£ (G.oa)

R{z[) = R{z'3) = -R(z'2y = -R(z'4y = H*$,4>,6) ,

where*H(l *, G) = c o s h ^ - # ) ^ + Sinh0Sinh(^-^)] _ ( G 5 & )

sinh(2()!>) sinh(</> — ip — iut) sinh(<^> + ip — iuin)

Hence (G.4) is given by

I E n«) = - [eHfi-\ +1 - e w + U i ]

r « . . i « . . i i

where we have made used of the fact that e1^* = 1, since $ is even. Now fromthe definitions of 4> and ip given in (G.3b), and the definition (G.lb) we see thatfor any function K (</>, "ip, G),

K(li;,G) -+^K$MG).p->-p-k

Hence if K(4>, ip,G) is a periodic function of p with period 2ir, then

f d3p K0, h G) = r d3p K($, IG) . (G.7)

Making use of this relation, with K given by the rhs of (G.6), we can write (G.la)in the form

* We have supressed in H the dependence on the photon frequency

(G.6)

Appendix G 567

This expression can be decomposed as follows by making use of the identity (ex +

l)-1 + (e-' + l)~1 = l,

n f / t o k) = ni"4ac)(w+,*) + n44(£+, k)f.T., (G.8)

where

(G.9a)

flâc\u,+ ,k) = -4e 2 / * - ^ - [ t a n h ^ - 4Re H(^^,G)} . (G.96)

As we shall see below (G.9a) is the finite temperature (—¥ f.T.) contribution arisingfrom the presence of the heat bath, while f j^" 0 (£>+, k) is the lattice expressionwhich one obtains by using the zero temperature, zero chemical potential latticeFeynman rules, with the fourth component of the photon momentum evaluatedat uf. Indeed, for /t = 0 and $ -> oo, the expression corresponding to (G.la)is obtained by setting p, = 0, replacing LOJ by the continuous variable p±, andthe frequency sum by an integral according to /3~1 J2e —> f*n dp4/2?r. Uponintroducing z = elp4 as a new integration variables, we therefore have that

where f(z) is given by (G.3a) with p, = 0, and where the contour integration iscarried out over a circle in the complex z-plane with unit radius. Since for fi, = 0,the singularities of f(z) inside the circle are located at ±e~* and ±e~^~tu>" , oneis then led to the result (G.9b). As we now show, (G.9) does not contribute to thescreening mass.

Consider (G.9b) for vanishing frequency. One readily verifies that

ni7c)(0, k) = -4e2 f 44f[tanh<£ - 4A(0, j>, G)] , (G.lOa)J-n (. 7rJ

where. - - . G2 coth i+l sinh 20

/l(</>, 1p, G) = . , 2 TT-^T- . (G.10&)

2 sinh(</> — xp) sinh(0 + if))

To compute the corresponding contribution to the screening mass we must take

the limit k -» 0. In this limit •$ -> 0, so that ft(0, i/i, G) becomes singular. This


singularity is however integrable. To see this we make use of (G.7) to rewrite(G.10) in the form

il[v4ac\0,k) = -4e2 f yf^tanh^ - 2/(«^,G)] , (G.ll)

where

f{j>, j>, G) = h(4>, </>, G) + h(j,, j>, G) .

Making use of the relations for hyperbolic functions, /(</>, xjj, G) can be written asfollows

f($, 4>,G) = -, ^ [ 2 r + cosh(0 + </))] .2 sinh((/) + ip) L s i n h ^ s i n h ^ -1

Now for k -> 0, G2 —>• sinh2 0, and V» —> < . Hence

M^GJîtanh^. (G.12)

Inserting this expression into (G.ll) we therefore find that

limf[i7c)(0,I) = 0. (G.13)fc->0

Hence there is no contribution to the screening mass arising from n^â;"1", k).Consider now the second term on the rhs of (G.8), i.e., (G.9a). Let us replace

the second term in the integrand by an expression symmetrized in <j> and ip. Onethen verifies that (G.9a) can be written in the form

n ^ O , * ) = 4e2 ^ •^{t^4-2{f(l$,G)[T)FD(4)+iiFD]+h{4>,i,,G)AT)FD}

(G.14)where

&VFD = [VFDW ~ VFD(IP)} + [VFDW ~ VFDW] .

and (p = 4>, ip)

*FD® = efa-l) + 1 '

*"V) = J^^ , (G.15)

Appendix G 569

are the lattice Fermi-Dirac distribution functions for particles and antiparticles.While h^^jG) is singular in the limit k —> 0, the product h(<f>,'4),G)Ar)FD isfinite, as can be seen by setting ip = 4> + e and taking the limit t —>• 0. Theremaining terms in (G.14) are finite in this limit, as follows from (G.12). One thenfinds that

n - r d3v f eM-fi) eW+£) 1lim n44(0, k) = ml = 4e2/? / ~ ^ { —r-.— + — ^ } .S_>o • / - ( 2 7 r ) 3 \ [ e W - « + l ] 2 [eW+A) + l ] 2 j

(G.16)It is now evident tha t this expression possesses a finite continuum limit, since forP —» oo, /3/i = (3ji fixed, only momenta pj contribute for which (3<f> is finite. Thisimplies tha t E = sinh$ is of 0 (1 / / ? ) . Nevertheless, the above expression is notthe lattice analog of (19.78). First of all there is an extra factor two multiplyingthe integral. This factor arises from excitation in frequency space near the cornerof the Brillouin zone. Such a factor was of course expected since we have carriedout our computat ions with naive fermions. Of course we also expect to see theeffects of the 2 3 douplers arising from excitations in 3-momenta at the edges of theBrillouin zone. This can be easily seen. To this effect we notice the momentumdependence of <j> appears in the form s in 2 ^ - , which also vanishes at the edges of theBrillouin zone. Hence the integral (G.16) is just 2 3 times the integral extendingover only half the Brillouin zone. This integral is now dominated for (3 -+ oo bymomenta pj of the order of 1//3 (for which J3<}> takes finite values). We are thereforenow allowed to replace 4> = arsinhE by \Jp2 + rh2. Introducing the dimensionfulvariables pj,m, [i and /3 by pj = pjd, rh = ma, ji — \xa and 0 = /3/a, we thereforefind tha t the physical screening mass, mei = mei/a, is given by

mll(P,n,m) = lim —m2el(-,iia,ma)

a—*0 Q, Q,

2 f°° d3p j e/3(V'p2+m2-M) ePWP+mt+n) I

7-00 (2TT)3 I [g/JO/^+ro2-^ -)- 1]2 [e/3(V/P2+™2+^) + I ] 2 /

(G.17)Peforming the angular integrations, and an additional partial integration in \p\ wefinally obtain

2 1 R e 2 f™ 2p2 + m2 f 1 1 1mel = 16-T / dp = 1 ,K2 Jo y/p2 + m2 [e/3(vV+m2-/i) + I e/3(Vp2+m2+M) + 1 J

(G.18)which is just sixteen times the result we obtained in continuum perturbation the-ory.


We want to call the readers attention to the fact that the first term in (G.la),arising from the diagram (b) in fig. (19-6), which has no continuum analog, playedan essential role in obtaining (except for the factor 16, of course) the correctcontinuum limit.

REFERENCES

Abbot T. et al., E802 Collaboration, Phys. Rev. Lett. 64. 847 (1990);Phys. Rev. Lett. 66, 1567 (1991)

Abers, E.S. and B.W. Lee, Phys. Rep. 9C, 1 (1973)Abreu M.C. et al. (NA 38 collab.), Z. Phys. C38 117 (1988)Abrikosov A. A., Zh. Eksp. Teor. Fiz. 32, 1442 (1057); Sov. Phys.

JETP 5 1174 (1957)Actor A., Rev. Mod. Phys. 51, 461 (1979)Adler S.L., Phys. Rev. 177, 2426 (1969)Adler S.L., Nucl. Phys. B217. 381 (1983)Albanese M. et al. (APE collab.), Phys. Lett. 192B, 163 (1987);

197B, 400 (1987);Alexandrou C , M. D'Elia and P. de Forcrand, Nucl. Phys. (Proc. Suppl) 83,

437 (2000)Allton C.R., S. Ejiri, S.J. Hands, O. Kaczmarek, F. Karsch, E. Laermann,

Ch. Schmidt and L. Scorzato, Phys. Rev. D66, 074507 (2002)Ambjorn J. and P. Oleson, Nucl. Phys. B170 [FS1], 265 (1980)Anagnostopoulos K.N. and J. Nishimura, Phys. Rev. D66, 106008 (2002)Atiyah M.F. and I.M. Singer, Ann. Math. 87, 596 (1968); 93, 139 (1971)Atiyah M.F., N.J. Hitchin, V. Drinfeld and Yu. I. Manin, Phys. Lett. 65A

185 (1978)Attig N., F. Karsch, B. Petersson, H. Satz and U. Wolff, unpublishedBacilieri P. et al., Phys. Lett. 214B, 115 (1988)Bacilieri P. et. al. (APE collab.), Nucl. Phys. B (Proc. Suppl.) 9, 315

(1989). See also, Phys. Rev. Lett. 61, 1545 (1988)Baglin C. et al. (NA38 collab.), Phys. Lett. 220B, 471 (1989)Baker M. and S.P. Li, Phys. Lett 124B, 397 (1983)Baker M., J.S. Ball and F. Zachariasen, Phys. Rev. D41, 2612 (1990)Bali G.S., K. Schilling and Ch. Schlichter, Phys. Rev. D51. 5165 (1995)Bali G.S. et al., Phys. Rev. D54, 2863 (1996)Banks T., D.R.T. Jones, J. Kogut, S. Raby , P.N. Scharbach, D.K. Sinclair,

and L. Susskind D.K. Sinclair, Phys. Rev. D15, 1111 (1977).Barcielli A., E. Montaldi and G.M. Prosperi, Nucl Phys. B296, 625 (1988);

Erratum Nucl Phys. B303, 752 (1988).Barkai D., K.J.M. Moriarty and C. Rebbi, Phys. Lett. 156B, 385 (1985)Batrouni G.G., G.R. Katz, A.S. Kronfeld, G.P. Lepage, B. Svetitsky and

571


K.G. Wilson, Phys. Rev. D32, 2736 (1985)Belavin A.A. and A.A. Migdal, JETP Lett. 19, 181 (1974)Belavin A., A. Polyakov, A. Schwarz and Yu. Tyupkin, Phys. Lett. 59B

85 (1975)Bell J.S. and R. Jackiw, Nuovo Cimento 60A, 47 (1969)Bender I., D. Gromes, K.D. Rothe and H.J. Rothe, Nucl. Phys. B136,

259 (1978)Bender I., H.J. Rothe, I.O. Stamatescu and W. Wetzel, Z. Physik C58,

333 (1993)Bender CM. and A. Rebhan, Phys. Rev. D41, 3269 (1990)Berezin F.A., The Method of Second Quantization, Academic Press (1966)Berg B., Phys. Lett. 97B, 401 (1980)Berg B., Phys. Lett. B104, 475 (1981)Berg B and A. Billoire, Phys. Lett. 113B, 65 (1982)Berg B., Progress in Gauge Field Theory, Cargese (1983)Berg B. and A. Billoire, Phys. Lett. 166B, 203 (1986)Berker A.N. and M.E. Fisher, Phys. Rev. B26, 2507 (1982)Bernard C.W., Phys. Rev. D9, 3312 (1974)Bernreuther W. and W. Wetzel, Phys. Lett. 132B, 382 (1983)Bhanot G. and C. Rebbi, Nucl. Phys. B180 [FS2], 469 (1981)Bhanot G., U. Heller, and I.O. Stamatescu, Phys. Lett. 129B, 440 (1983)Binder K. and D.P. Landau, Phys. Rev. B30, 1477 (1984)Bitar K.M. and S.-J. Chang, Phys. Rev. D17, 486 (1978)Bitar K.M. et. al., Phys. Rev. D42, 3794 (1990)Blaizot J.P. and J.Y. Ollitrault, Phys. Lett. 199B, 199 (1987)Blaizot J.P.and J.Y. Ollitrault, Phys. Lett. 217B, 392 (1989)Blaizot J.P and J.Y. Ollitrault, in Quark Gluon Plasma, ed. by R.C. Hwa

(World Scientific, 1990)Born K.D., E. Learmann, N. Pirch, T.F. Walsh and P.M. Zerwas,

Phys. Rev. D40, 1653 (1989); see also Nucl. Phys. B (Proc. Suppl.) 9,269 (1989)

Boulware G., Ann. of Phys. 56 (N.Y.), 140 (1970)Bowler K.C., C.B. Chalmers, A. Kenway, R.D. Kenway, G.S. Pawley

and D.J. Wallace, Phys. Lett. 162B, 354 (1985)Bowler K.C., C.B. Chalmers, R.D. Kenway, D. Roweth and D. Stephenson,

Nucl Phys. B296, 732 (1988)Boyko P. Yu, M. Polikarpov and V.I. Zakharov, Nucl Phys. Proc. Suppl.

References 573

119, 724 (2003)Brambilla N., G.M. Prosperi and A. Vario, Phys. Lett. B362, 113 (1995)Brown L.S., and W.I. Weisberger, Phys. Rev. D20, 3239 (1979)Brown F.R., N.H. Christ, Y. Deng, M.Gao and T.J. Woch,

Phys. Rev Lett. 61, 2058 (1988);Brown F.R., F.P. Butler, H. Chen, N.H. Christ, Z. Dong, W. Schaffer,

L.I. Unger and A. Vaccarino, Phys. Rev. Lett. 65, 2491 (1990).Bruckmann F., D. Nogradi abd P. van Baal, Acta Phys. Polon. B34,

5717 (2003), hep-th/0309008Burgers G., F. Karsch, A. Nakamura and I.O. Stamatescu, Nucl. Phys.

B304, 587 (1988)Burkitt A., K.H. Mutter and P. Uberholz; talk presented by K.H. Mutter

at the International Symposium Field Theory on the Lattice, Seillac,France (1987)

Callan C.G., Phys. Rev. D2, 1541 (1970)Callan C , R. Dashen and D. Gross, Phys. Rev. D17, 2717 (1978)Callaway D.J.E. and A. Rahman, Phys. Rev. Lett. 49, 613 (1982);Callaway D.J.E. and A. Rahman, Phys. Rev. D28, 1506 (1983)Campostrini et al, Phys. Rev. D42, 203 (1990)

Caracciolo S., P. Menotti and A. Pelisetto, Nuc. Phys. B375, 195 (1992)Caswell W.E., Phys. Rev. Lett. 33, 244 (1974)Celic T., J. Engels and H. Satz, Phys. Lett. 125B, 411 (1983); 129B,

323 (1983)Chiu T.W., Phys. Lett. B445, 371 (1999)Clarke A.B. and R.L. Disney, Probability and Random Processes: A first

Course with Applications, John Wiley & Sons (1985)Cleymans J, R.V. Gavai and E. Suhonen, Phys. Rep. 130, 217 (1986)Cleymans J., J. Fingberg and K. Redlich, Phys. Rev. D35, 2153 (1987)Close F.E., An Introduction to Quarks and Partons, Academic Press (1979)Coleman S., Uses of Instantons in Aspects of Symmetry, selected

Erice Lectures, Cambridge University press, (1985)Collins J.C. and M.J. Perry, Phys. Rev. Lett. 34, 1353 (1975)Collins J .C, A. Duncan and S.D. Joglekar, Phys. Rev. D16, 438 (1977)Creutz M., Phys. Rev. D15, 1128 (1977)Creutz M., J. Math. Phys. 19, 2043 (1978)Creutz M., Phys. Rev. D21, 2308 (1980)Creutz M. and K.J.M. Moriarty, Phys. Rev. D26, 2166 (1982)


Creutz M., Quarks, Gluons and Lattices, Cambridge University Press (1983a).Creutz M., Phys. Rev. Lett. 50, 1411 (1983b)de Forcrand P. and I.O. Stamatescu, Proc. of the XXIII Int. Conf. in High

Energy Physics, Berkeley 1986; see also P. de Forcrand,V. Linke and I.O. Stamatescu, Nucl. Phys. B304, 645 (1988);P. de Forcrand and I.O. Stamatescu, Nucl. Phys. B (proc. Suppl.) 9,276 (1989)

de Forcrand P., J. Stat. Phys. 43, 1077 (1986)de Forcrand P., R. Gupta, S. Giisken, K.H. Mutter, A. Patel, K. Schilling

and R. Sommer, Phys. Lett. 200B, 143 (1988)de Forcrand P., M. Garcia Perez and I.O. Stamatescu, Nucl. Phys. B499

409 (1997)de Forcrand P. and M. D'Elia, Phys. Rev. Lett. 82, 4582 (1999)de Forcrand P. and M. Pepe, Nucl. Phys. B598, 557 (2001)de Forcrand P. and O. Philipsen, Nucl. Phys. B642, 290

(2002); M. D'Elia and M.P. Lombardo, Phys. Rev. D67, 014505 (2003)Del Debbio L. et a l , Phys. Rev. D55, 2298 (1997)Del Debbio L., M. Faber, J. Giedt, J. Greensite and S. Olejnik, Phys. Rev.

D58, 094501 (1998)DeGrand T.A. and T. Toussaint, Phys. Rev. D22, 2478 (1980)Deng Y., Nucl. Phys. B (Proc. Suppl.) 9, 334 (1989)Di Giacomo A., B. Lucini, L. Montesi and G. Paffuti, Phys. Rev. D61

034503 (2000); 034504 (2004)Di Vecchia P., K. Fabricius, G.C. Rossi and G. Veneziano, Nucl. Phys.

B192, 392 (1981)Ding H.-Q, C.F. Baillie and G.C. Fox, Phys. Rev. D41, 2912 (1990)Ding H.-Q., "Heavy quark potential in Lattice QCD. A review

of recent progress", CALTECH preprint no. 32681 (1990).See also Phys. Rev. D42, 2350 (1990)

Dolan L. and R. Jackiw, Phys. Rev. D9, 3320 (1974)Dosch H.G., O. Nachtmann and M. Rueter, hep-ph 9503386Drouffe J.M. and J.B. Zuber, Phys. Rep. 102, 1 (1983)Duane S., Nucl. Phys. B257 [FSU], 652 (1985)Duane S. and J.B. Kogut, Nucl. Phys. B275 [FS17], 398 (1986)Duane S., A.D. Kennedy, B.J. Pendleton and D. Roweth, Phys. Lett. 195B,

216 (1987)Duru I.R. and H. Kleinert, Phys. Lett. 84B, 185 (1979)

References 575

Eichten E., K. Gottfried, T. Kinoshita, K.D. Lane and T.M. YanPhys. Rev. D21, 203 (1980)

Eichten E. and F. Feinberg, Phys. Rev. D23, 2724 (1981)Engelhardt M. and H. Reinhard, Nucl. Phys. B567, 249 (2000)Engels J., F. Karsch, I. Montvay and H. Satz, Phys. Lett. 102B, 332 (1981a)Engels J., F. Karsch, I. Montvay and H. Satz, Phys. Lett. 1O1B, 89 (1981b)Engels J., F. Karsch, I. Montvay and H. Satz, Nucl. Phys. B205 [FS5,], 545

(1982)Engels J., J. Fingberg, F. Karsch, D. Miller and M. Weber, Phys. Lett.

252B, 625 (1990)Feuerbacher J., (2003a) hep-lat/0309016Feuerbacher J., Nucl. Phys. B674, 484 (2003b)Feynman R.P., Rev. Mod. Phys. 20, 367 (1948)Feynman R.P., and A.R. Hibbs, Quantum Mechanics and Path Integrals,

McGraw Hill (1965).Flower J.W. and S.W. Otto, Phys. Lett. 160B, 128 (1985)Flower J., Caltech preprints, CALT-68-1369 (1986); CALT-68-1377 (1987);

CALT-68-1378 (1987)Fodor Z. and S.D. Katz, JHEP 0203, 014 (2002); JHEP 0404

050 (2004)Fradkin E.S., Proc. Lebedev Physics Institute 29, 6 (1965)Frewer M. and H.J. Rothe, Phys. Rev. D63, 054506 (2001)Fritzsch H. and M. Gell-Mann, in Proc. XVI'th Intern. Conf. on High Energy

Physics, Chicago-Batavia, 1972Fritzsch H., M. Gell-Mann, and H. Leutwyler, Phys. Lett. 47B, 365 (1973)Fucito F., E. Marinari, G. Parisi and C. Rebbi, Nucl Phys. B18O [FS2], 369

(1981)Fujikawa K., Phys. Rev. Lett. 42, 1195 (1979); Phys. Rev. D21,

2848 (1980); ibid D22. 1499(E) (1980); D29, 285 (1984)Fukugita M. and I. Niuya, Phys. Lett. 132B, 374 (1983)Fukugita M. and A. Ukawa, Phys. Rev. Lett. 55, 1854 (1985)Fukugita M. and A. Ukawa, Phys. Rev. Lett. 57, 503 (1986)Fukugita M., Y. Oyanagi and A. Ukawa, Phys. Lett. 203B, 145 (1988)Fukugita M., M. Okawa and A. Ukawa, Phys. Rev. Lett. 63, 1768 (1989);

see also Nucl. Phys. B (proc. Suppl.) 17, 204 (1990)Gao M., Phys. Rev. D41, 626 (1990)Garcia Perez M., O. Phillipsen, I.O. Stamatescu, Nucl. Phys. B551,


293 (1999)Gavai R.V., J. Potvin and S. Sanielevici, Phys. Rev. Lett. 58, 2519 (1987)Gavai R.V., J. Potvin and S. Sanielevici, Phys. Rev. D37, 1343 (1988)Gavin S., M. Gyulassy and A. Jackson, Phys. Lett. 207B, 257 (1988)Gavin S. and M. Gyulassy, Phys. Lett. 214B, 241 (1988)Gerschel C. and J. Hiifner, Phys. Lett., 207B, 253 (1988)Ginsparg P.H. and K.G. Wilson, Phys. Rev. D25, 2649 (1982)Goddard P., J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B56

109 (1973)Golterman M.F.L. and J. Smit, Nucl. Phys. B245, 61 (1984)Golterman M.F.L.and J.Smit, Nucl. Phys. B255, 328 (1985)Golterman M.F.L., Nucl. Phys. B278, 417 (1986a)Golterman M.F.L., Nucl. Phys. B273, 663 (1986b)Gottlieb S.A., J. Kuti, A.D. Kennedy, S. Meyer, B.J. Pendleton, R.L. Sugar

and D. Toussaint, Phys. Rev. Lett. 55, 1958 (1985)Gottlieb S., W. Liu, D. Toussaint, R.L.Renken and R.L. Sugar, Phys.

Rev. D35, 2531 (1987a)Gottlieb S., W. Liu, D. Toussaint, R.L.Renken and R.L. Sugar, Phys.

Rev. D35, 3972 (1987b)Gottlieb, S., W. Liu, R.L. Renken, R.L. Sugar and D. Toussaint, Phys.

Rev. D38, 2245 (1988)Grattinger C , Nucl. Phys. B(Proc. Suppl.) 129 & 130, 653 (2004)Greensite J., Prog. Part. Nucl. Phys. 51, 1 (2003); hep-lat/0301023Greiner C , P. Koch and H. Stocker, Phys. Rev. Lett. 58, 1825 (1987);

C. Greiner, D.-H. Rischke, H. Stocker and P. Koch, Z. Physik C38, 283(1988); Phys. Rev. D38, 2797 (1988); C. Greiner and H. Stocker,Phys. Rev. D44 3517 (1991)

Griffiths L.A., C. Michael and P.E.L. Rakow, Phys. Lett. 129B,351 (1983)

Gromes D., Zeitschrift f. Physik C22, 265 (1984)Gromes D., see the review by this author in Phys. Rep. 200 (Part II),

186 (1991)Gross, D.J. and F. Wilczek, Phys. Rev. D8, 3633 (1973); Phys. Rev.

Lett. 30, 1343 (1973)Gross D.J., R.D. Pisarski and L.G.Yaffe, Rev. Mod. Phys. 53, 43 (1981)Grossiord J.Y. (NA38 coll.), Nucl. Phys. A498, 249c (1989)Gruber C. and H. Kunz, Commun. Math. Phys. 22, 133 (1971)

References 577

Gupta R., G. Guralnik, G. Kilcup, A. Patel, S.R. Sharpe andT. Warnock, Phys. Rev. D36, 2813 (1987)

Gupta R., Nucl. Phys. B (Proc. Suppl.) 9, 473 (1989)Gupta, R., A. Patel, C.F. Baillie, G. Guralnik, G.W. Kilcup and S.R Sharpe,

Phys. Rev. D40, 2072 (1989).Gupta R., Nucl. Phys. B (Proc. Suppl.) 17, 70 (1990)Guth A.H., Phys. Rev. 21, 2291 (1980)Hahn O., E. Schnepf, E. Learmann, K.H. Mutter, K. Schilling and

R. Sommer, Phys. Lett. 190B, 147 (1987)Hamber H.W. and G. Parisi, Phys. Rev. Lett. 47, 1792 (1981)Hamber H.W., Phys. Rev. D39, 896 (1989)Hammersley J.M. and D.C. Handscomb, Monte Carlo Methods, Methuen's

Monographs, London (1975)Hart A. and M. Teper, Phys. Rev. D58, 014504 (1998)Harrington B.J. and H.K. Shepard, Phys. Rev. D17, 2122 (1978);

Phys. Rev. D18, 2990 (1978)Hasenfratz A. and Hasenfratz P., Phys. Lett. 93B, 165 (1980);Hasenfratz A. and Hasenfratz P., Phys. Lett. 104B, 489 (1981a)Hasenfratz A. and Hasenfratz P., Nucl. Phys. B193, 210 (1981b)Hasenfratz P., F. Karsch and I.O. Stamatescu, Phys. Lett. 133B, 221 (1983)Hasenfratz P. and F. Karsch, Phys. Lett. 125B, 308 (1983)Hasenfratz P., Proc. of the XXIII Int. Conf. on High Energy Physics Berkeley

(1986)Hasenfratz P., V. Laliena and F. Niedermayer, Phys. Lett. B427

125 (1998); P. Hasenfratz, Nucl. Phys. B 525, 401 (1988)Haymaker R.W. and J. Wosiek, Phys. Rev. Rapid Comm. D36, 3297 (1987);

Acta Physica Polonica B21, 403 (1990); R.W. Haymaker, Y. Peng,V. Singh and J. Wosiek, Nucl. Phys. B (proc. Suppl) 17, 558 (1990);R.W. Haymaker; V. Singh and Wosiek, Nucl. Phys. B (Proc. Suppl) 20,207 (1991)

Haymaker R. and J. Wosiek, Phys. Rev. 43, 2676 (1991); R. Haymaker,V. Singh, D. Browne and J. Wosiek, Paris Conference on the QCDVacuum (1992)

Haymaker R., V. Singh and Y. Peng, Phys. Rev. D53, 389 (1996)Heller U. and F. Karsch, Nucl. Phys. B251 [F513], 254 (1985)

Nucl. Phys. B258, 29 (1985)Heller U.M., F. Karsch and J. Rank, Phys. Lett. B355, 511 (1995)


Hernandez P., K. Jansen and M. Liischer, Nucl. Phys. B552, 363 (1999)Hiifner J., Y. Kurihara and H.J. Pirner, Phys. Lett. 215B, 218 (1988)Ichie H., V. Bornyakov, T. Streuer and G. Schierholz, Nucl. Phys. A721,

899c (2003)Imry Y., Phys. Rev. B21, 2042 (1980)Ingelfritz E.M., M.L. Laursen, M. Miiller-Preufiker, G. Schierholz and

H. Schiller, Nucl. Phys. B268, 693 (1986)Irback A., P. Lacock, D. Miller and T. Reisz, Nucl. Phys.

B363, 34 (1991)Ishikawa K., A. Sato, G. Schierholz and M. Teper, Z. Physik C21, 167 (1983)Itzykson C. and J.-M. Drouffe, Statistical Field Theory,

Cambridge Monographs on Mathematical Physics, Vol 1 (1989)Iwasaki Y. and T. Yoshie, Phys. Lett. 216B, 387 (1989); see also

Y. Iwasaki, Nucl. Phys. B (Proc. Suppl.) 9, 254 (1989)Ji X., Phys. Rev. D52, 271 (1995)Jolicoeur T., A. Morel and B. Petersson, Nucl. Phys. B274, 225 (1986).Jones D.R.T., Nucl. Phys. B75, 531 (1974)Joos H. and I. Montvay, Nucl. Phys. 225, 565 (1983)Kajantie K., C. Montonen and E. Pietarinen, Z. Physik C9, 253 (1981)Kajantie K. and J. Kapusta, Phys. Lett. HOB, 299 (1982)Kapusta J.I., Nucl. Phys. B148, 461 (1979)Kapusta J., Finite-Temperature Field Theory, Cambridge Monographs

on Mathematical Physics, Cambridge University Press (1989)Kapusta J., Phys. Rev. D46, 4749 (1992)Karsch F. , Nucl. Phys. B205 [FS5], 285 (1982)Karsch F., J.B. Kogut, D.K. Sinclair and H.W. Wyld, Phys. Lett. 188B,

353 (1987)Karsch F., in Quark Gluon Plasma (World Scientific, 1990), ed. by R.C. HwaKarsten L.H. and J. Smit, Nucl. Phys. B183, 103 (1981)Kaste P. and Rothe H.J., Phys. Rev. D56, 6804 (1997)Kawai H., R. Nakayama, and K. Seo, Nucl. Phys. B189, 40 (1981)Kennedy A.D., J. Kuti, S. Meyer and B.J. Pendleton, Phys. Rev. Lett. 54,

87 (1985)Kilcup G.W. and Sharpe, Nucl. Phys. B283, 497 (1987)Kislinger M.B. and P.D. Morley, Phys. Rev. D13, 2771 (1976); Phys. Rep.

51, 63 (1979)Kluberg-Stern H., A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B220

References 579

[FS8], 447 (1983)Kluberg L., Nucl. Phys. A488, 613c (1988)Kogut J.B. and L. Susskind, Phys. Rev. D9, 3501 (1974)Kogut J.B. and L. Susskind, Phys. Rev, D l l , 395 (1975)Kogut J.B., Rev. Mod. Phys. 51, 659 (1979)Kogut J.B., R.P. Pearson, and J Shigemitsu, Phys. Lett. 98B, 63 (1981);

ibid, Phys. Rev. Lett. 43, 484 (1979).Kogut J.B., in Recent Advances in Field Theory and Statistical

Mechanics, Les Houches (1982)Kogut J.B., Rev. Mod. Phys. 55, 775 (1983)Kogut J.B., M. Stone, H.W. Wyld, W.R. Gibbs, J. Shigemitsu, S.H. Shenker

and D.K. Sinclair, Phys. Rev. Lett. 50, 393 (1983a)Kogut J.B., H. Matsuoka, S. Shenker, J. Shigemitsu, D.K. Sinclair, M. Stone

and H.W. Wyld, Phys. Rev. Lett. 51, 869 (1983b)Kogut J.B., H. Matsuoka, S. Shenker, J. Shigemitsu, D.K. Sinclair, M. Stone

and H.W. Wyld, Nucl. Phys. B225 [FS9] 93 (1983c)Kogut J.B., Proceedings of the International School of Physics

Enrico Fermi (1984).Kogut J.B., J. Polonyi, H.W. Wyld, J. Shigemitsu, and D.K. Sinclair,

Nucl. Phys. B251 [FS13], 311 (1985)Kogut J.B. and D.K. Sinclair, Phys. Rev. Lett. 60, 1250 (1988)Kogut J.B. and D.K. Sinclair, Nucl. Phys. B344, 238 (1990)Kovacs E., Phys. Rev. D25, 871 (1982)Kovacs T.G. and E.T. Tomboulis, Phys. Rev. D57, 4054 (1998)Kovalenko A.V., M.I. Polikarpov, S.N. Syritsyn and V.I. Zakharov,

Nucl. Phys. (Proc. Suppl.) 129 & 130, 665 (2004)Kraan T.C. and P. van Baal, Phys. Lett. B435, 389 (1998);

Phys. Lett. B428, 268 (1998); Nucl. Phys. 533, 627 (1998)Kronfeld A.S., M.L. Laursen, G. Schierholz and U.-J. Wiese, Phys. Lett.

B198, 516 (1987); A.S. Kronfeld, G. Schierholz and U.-J. Wiese,Nucl. Phys. 293, 461 (1987)

Kronfeld A.S., J.K.M. Moriarty and G. Schierholz, Comp. Phys. Comm.52, 1 (1988)

Kuti J., J. Polonyi and K. Szlachanyi, Phys. Lett. 98B, 199 (1981)Landsman, N.P., and Ch. G. van Weert, Phys. Rep. 145, 141 (1987)Langfeld K., Phys. Rev. D69, 014503 (2004)Lautrup B., and M. Nauenberg, Phys. Lett. 95B, 63 (1980)


Learmann E., R. Altmeyer, K.D. Born, W. Ibes, R. Sommer, T.F. Walshand P.M. Zerwas, Nucl. Phys. B (Proc. Suppl.) 17, 436 (1990)

Lee K. and P. Yi, Phys. Rev. D56, 3711 (1997)Lee K. and C. Lu, Phys. Rev. D58, 025011 (1998)Linde A.D., Phys. Lett. 96B, 289 (1980)Liischer M., G. Minister and P. Weiss, Nucl. Phys. B180 [FS2] 1 (1981)Liischer M., Lectures given at the Summer School on Critical Phenomena,

Random Systems and Gauge Theories, Les Houches (1984)Liischer M., Lectures given at the Summer School on Fields, Strings

and Critical Phenomena, Les Houches (1988)Liischer M., Phys.Lett. B428, 342 (1999a)Liischer M., Nucl. Phys. B538, 515 (1999b)Mack G., in "Recent Developments in Gauge Theories", edited by 't Hooft

et al. (Plenum, New York, 1980)Mandelstam S., Phys. Rep. 23C, 245 (1976)Matsubara T., Progr. Theor. Phys. 14, 351 (1955)Matsubara Y., S. Ejiri and T. Suzuki, Nucl. Phys. B34 (Proc. Suppl.)

176 (1994)Matsui T. and H. Satz, Phys. Lett. 178B, 416 (1986). For a review

see H. Satz, Z. Physik C62, 683 (1994)McLerran L.D. and B. Svetitsky, Phys. Rev. D24, 450 (1981)Metropolis N., A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller,

J. Chem. Phys. 21, 1087 (1953)Meyer B. and C. Smith, Phys. Lett. B123, 62 (1883)Meyer-Ortmanns H., Rev. Mod. Phys. 68, 473 (1996)Michael C , Nucl. Phys. B280 [FS18], 13 (1987)Michael C. and M. Teper, Nucl. Phys. B314, 347 (1989).Michael C , Nucl. Phys. (Proc. Suppl.) 17, 59 (1990)Michael C, Phys. Rev. D53, 4102 (1996)Migdal A.A., ZETF 69, 810 (1975)Montvay I. and G. Miinster, Quantum Fields on a Lattice (Cambridge

University Press, 1994)Morel A. and J.P. Rodrigues, Nucl. Phys. B247, 44 (1984).Miinster G., Nucl. Phys. B180 [FS2], 23 (1981)Nahm W., v Self dual Monopoles and Calorons", Lecture Notes in Physics

201, 189 (1984)Nadkarni S., Phys. Rev. D33, 3738 (1986); D34, 3904 (1986)

References 581

Negele J.W., Monte Carlo Methods for Hadronic Physics; lectures presentedat the International Spring School on Medium and High Energy,

Nuclear Physics, Taiwan (1988)Neuberger H., Phys. Lett. B417, 141 (1998); ibit B427, 353 (1998)Nielsen H.B. and P. Olesen, Nucl. Phys. B160, 45 (1973)Nielsen H.B. and P. Olesen, Nucl. Phys. B61, 380 (1979)Nielsen H.B. and M. Ninomiya, Nucl. Phys. 185, 20 (1981)Niemi A.J. and G.W. Semenov, Ann. Phys. 152, 105 (1984)Niemi A.J. and G.W. Semenov, Nucl. Phys. B230 [FS10], 181 (1984)Norton R.E. and J.M. Cornwall, Ann. Phys. 91, 106 (1975)Ono S., Phys. Rev. D17, 888 (1978)Osterwalder K. and R. Schrader, Comm. Math. Phys. 31, 83 (1973); 42,

281 (1975); See also K. Osterwalder in Constructive Quantum FieldTheory, ed. by G. Velo and A.S. Wightman, Lecture Notes in Physics 25(Springer, Berlin)

Parisi G. and Y.-S. Wu, Sci. Sin.24, 483 (1981)Patel A., Nucl. Phys. B243, 411 (1984)Peskin M.A., SLAC-PUB-3273 (1983), l l ' th SLAC Summer InstitutePeskin M., Cornell University preprint CLNS 395 (1978), thesisPetcher D.N. and D.H. Weingarten, Phys. Lett. 99B, 333 (1981)Pietig R., Master thesis (1994) unpublishedPisarski R.D. and F. Wilczek, Phys. Rev. D29, 338 (1984)Politzer H.D., Phys. Rev. Lett. 30, 1346 (1973)Politzer H.D., Phys. Rep. 14C, 129 (1974)Polonyi J., J.B. Kogut, J. Shigemitsu, D.K. Sinclair, and H.W. Wyld,

Phys. Rev. Lett. 53, 644 (1984)Polyakov A.M., Phys. Lett. 72B, 477 (1978)Polyakov AM," Gauge Fields and Strings", Harwood Academic Publishers,

(1987)Prasad M.K. and CM. Sommerfield, Phys. Rev. Lett. 35. 760 (1975);

E.B. Bogomol'nyi, Sov. J. Nucl. Phys. 24, 449 (1976)Quigg C. and J.L. Rosner, Phys. Rep. 56, 167 (1979)Rafelski J., in " Workshop on Future Relativistic Heavy Ion Experiments,

GSI-Report 81-6 (1981); see also Phys. Rep. 88, 331 (1982)Reinhardt H., Phys. Lett. B557, 317 (2003)Reisz T., Commun. Math. Phys. 117, 79 (1988a); 117, 639 (1988b); 116,

81 (1988c)


Reisz T. and H.J. Rothe, Phys. Lett. B455, 246 (1999)Reisz T. and H.J. Rothe, Phys. Rev. D62, 014504 (2000)Roepstorff G., Path Integral Approach to Quantum Physics. An introduction

Springer Verlag (1994)Rossi P. and U. Wolff, Nucl. Phys. B248, 105 (1984)Rothe H.J., An introduction to lattice Gauge Theories, Proceedings of the

Autumn College on Techniques in Many-Body Problems, ed. by.A. Shaukat (World Scientific, 1989)

Rothe H.J., Phys. Lett. B355, 260 (1995a)Rothe H.J., Phys. Lett. B364, 227 (1995b)Rothe H.J., Phys. Rev. D52, 332 (1995)Rothe H.J. and N. Sadooghi, Phys. Rev. D58, 074502 (1998).Rusakov B. Ye, Mod. Phys. Lett. A5, 693 (1990)Ruuskanen P.V., in Proc. of Quark Matter '91, Gatlinburg, edited by

V. Plasil et al., Nucl. Phys. A544 (1992)Satz H., in "Quark Gluon Plasma", ed. by R.C. Hwa (World Scientific, 1990)Satz H., in Proc. of Int. Lepton-Photon Symp., Geneve (1991)Scalapino D.J. and R.L. Sugar, Phys. Rev. Lett. 46, 519 (1981)Scalettar R.T., D.J. Scalapino and R.L. Sugar, Phys. Rev. B34, 7911 (1986).Schafer T. and E.V. Shuryak, Rev. Mod. Phys. 70, 323 (1998)Schierholz G., Nucl. Phys. B (Proc. Suppl.) 4, 11 (1988)Schierholz G., Nucl. Phys. B (proc. Suppl.) 9, 244 (1989)Seiler E., in Gauge Theories as a Problem of Constructive Quantum Field

Theory and Statistical Mechanics, Springer Verlag, Berlin, Heidelberg,New York (1982), and references quoted there

Sharatchandra H.S., H.J. Thun, and P.Weisz, Nucl. Phys. B192, 205 (1981)Shiba H. and T. Suzuki, Phys. Lett. B 333, 461 (1994)Shifman M.A., Particle Physics and Field Theory, World Scientific

Vol. 62, 1999H. Shiba, T. Suzuki, PL B333, 461 (1994)Singh V., R.W. Haymaker and D.A. Browne, Phys. Rev. D47, 1715 (1993a)Singh V., D.A. Browne and R.W. Haymaker, Phys. Lett. B306, 115 (1993b)Sommer R. and J. Wosiek, Phys. Lett. B149, 497 (1984);

Nucl. Phys. B267, 531 (1986)Sommer R., Nucl. Phys. B291, 673 (1987); B306, 180 (1988).Soper D.E., Phys. Rev. D18, 4590 (1978)Stack J.D., Phys. Rev. D29, 1213 (1984)

References 583

Stack, J.D., S.D. Neiman and R.J. Wensley, Phys. Rev. D50,3399 (1994)

Stamatescu I.O., Phys. Rev. D25, 1130 (1982)Susskind L., in Weak and Electromagnetic Interactions at High

Energy, Les Houches (1976)Susskind L., Phys. Rev. D16, 3031 (1977)Susskind L., Phys. Rev. D20, 2610 (1979)Suzuki T. and I. Yotsuyanagi,Phys. Rev. D 42, 4257 (1990)Svetitsky B. and L.G. Yaffe, Nucl. Phys. B210 [FS6], 423 (1982);Svetitsky B. and F. Fucito, Phys. Lett. 131B, 165 (1983)Svetitsky B., Phys. Rep. 132, 1 (1986)Symanzik K., Commun. Math. Phys. 18, 227 (1970)Symanzik K., in Mathematical Problems in Theoretical Physics,

eds. R. Schrader et al. (Springer, Berlin, 1982)Symanzik K., Nucl. Phys. B226, 187 and 205 (1983)Takahashi T.T., H. Matsufuru, Y. Nemoto and H. Suganuma, Phys.

Rev. Lett. 86, 18 (2001)Takahashi T.T., H. Suganuma, Y. Nemoto and H. Matsufuru, Phys.

Rev. D65, 114509 (2002)Takahashi T.T., H. Suganuma, H. Ichie, H. Matsufuru and Y. Nemoto, Nucl.

Phys. A721, 926c (2003)Takahashi T.T., H. Suganuma and H. Ichie, Conf. Proc, Color

Confinement and Hadrons in Quantum Chromodynamics, WorldScientific (2004)

Teper M., Phys. Lett. B171, 81 (1986); ibit 183B, 345 (1986)Teper M., Phys. Lett. 183B, 345 (1986a)'t Hooft G., unpublished comments at the conference in Marseille (1972)'t Hooft G., in "High Energy Physics", Ed. A. Zichichi (Editrice Compositori;

Bologna (1976)'t Hooft G., Nucl. Phys. B138, 1 (1978)'t Hooft G., Nucl. Phys. B190 [FS3], 455 (1981)'t Hooft G., Comm. Math. Phys. 81, 267 (1981a)Thacker H.B., E. Eichten and J.C. Sexton, Nucl. Phys. (Proc. Suppl.)

B4, 234 (1988)Tinkham M., Introduction to Superconductivity (McGraw-Hill (1996))Toimela T., Z. Physik C27, 289 (1985)Toussaint D., Introduction to Algorithms for Monte Carlo Simulations


and their Application to QCD; lectures presented at the Symposiumon New Developments in Hardware and Software for Computa-tional Physics, Buenos Aires (1988); Comp. Phys. Commun. 56, 69(1989)

van Baal P. and A.S. Kronfeld, Nucl. Phys. B (Proc. Suppl.) 9, 227 (1989)van Baal P., Nucl. Phys. B (Proc. Suppl.) 63, 126 (1998)van Baal P., hep-th/9912035 (1999)van den Doel and J. Smit, Nucl. Phys. B228, 122 (1983)Wantz O., Diploma thesis, University of Heidelberg (2003)Wegner F.J., J. Math. Phys. 12, 2259 (1971).Weinberg S., Phys. Rev. D9, 3357 (1974)Wetzel W., Nucl. Phys. B255, 659 (1985)Wilson K.G., Phys. Rev. D10, 2445 (1974)Wilson K.G., "New Phenomena in Subnuclear Physics" (Erice, 1975),

ed. A. Zichichi (New York, Plenum, 1975)Wolff U., Phys. Lett. 153B, 92 (1985)Wosiek J. and R. Haymaker, Phys. Rev. D36, 3297 (1987); J. Wosiek

Nucl. Phys. B(Proc. Suppl.) 4, 52 (1988)Yang C.N. and R. Mills, Phys. Rev. 96, 191 (1954)Yoshie, T., Y. Iwasaki and S. Sakai, Nucl. Phys. B (proc. Suppl.) 17,

413 (1990)

INDEX

Action sum rule, 140Abrikosov flux tubes, 365Algorithm:

heat bath, 285Metropolis, 291Langevin, 293molecular dynamics, 295hybrid, 301hybrid Monte Carlo, 304pseudofermion, 307

Anomaly, Adler-Bell-Jackiw, 234, 240, 275, 283Area law, 153Asymptotic scaling, 128

/3-function, 123, 124

Caiorons, 358Center gauge, maximal, 377Center projection, 377, 378Center symmetry, 495Center Vortex, 379Character expansion, 156, 499-505Charge, topological, 347, 349, 350, 354-357Chemical potential, 404, 411, 419, 523Chiral condensate, 328, 522Chiral phase transition, 520Chiral symmetry breaking, 326, 328Cooling procedure, 352, 353Covariant lattice derivative, 249Creutz ratio, 318Critical temperature, 497, 514

Debye screening, 530, 534Deconfinement phase transition, 512Detailed balance, 289Doubling problem, 43

Edinburgh plot, 340Electric screening mass, 448, 452Energy density, lattice expression for, 489Energy sum rule, 142, 145

Faddeev-Popov trick, 247-248

585


Faddeev-Popov ghosts, 250Faddeev-Popov determinant, 250Feynman rules for

Lattice QED, 216Lattice QCD, 268-270

Flux tube, 359, 360, 362Fuzzy link, 334

Gauge field on the latticeabelian, 77non-abelian, 87

Gauge transformations, 78, 82, 83, 88Ginsparg-Wilson fermions, 73, 237Glueballs, 148, 149, 330-336Gluon propagator, 268Grassmann:

algebra, 24integration rules, 25differentiation rules, 28

Haar measure, 244Heat bath algorithm, 285Hopping parameter, 172Hybrid algorithm, 301Hybrid Monte Carlo algorithm, 304

Importance sampling, 284Infrared problem, 482Instanton, 346, 350, 351

J/ip suppression, 532-539

Kogut-Susskind fermions, 57

Langevin algorithm, 293Latent heat, 490, 518Lattice degree of divergence, 206Lattice Hamiltonian, 163Lattice AL- parameter, 272-274Leapfrog integration, 302

Magnetic screening mass, 482Markov chain, 286-288Maximal abelian gauge, 370-372Maximal center gauge, 377Metropolis method, 291

Index 587

Molecular dynamics method, 295Monopoles, 366, 367Monopole current, 369

Nielsen-Ninomiya theorem, 55

Phase transition,deconfining, 512-520chiral, 520-524

Polyakov loop (Wilson line), 490Power counting theorem, 203Pressure, lattice expression for, 489Propagator:

scalar, 40Wilson fermions, 56photon, 219gluon, 268ghost, 268

Pseudofermions, 310Pseudofermion method, 307

Quenched approximation, 109

Reflection positivity, 86Renorrnalization group equation, 122Renormalization of axial vector current, 222Running coupling constant, 126

Scaling, asymptotic, 128Scaling window, 128Screening (of qq potential), 324-325, 530, 534Screening mass, electric, 448, 452

magnetic, 482Staggered fermions:

in configuration space, 57in momentum space, 69

Staggered fermion action, 59String tension, 153, 318, 319

Thermodynamical potential in the 04-theory, 434-437't Hooft loop operator, 382Transfer matrix, 22Topological charge, 347, 354-356

Vertices for QCD, 269, 270Vortex, thin and ideal, 376


Vortices, 373, 375

Wilson fermions, 56

Wilson line (Polyakov loop), 490Wilson loop, 97

Wilson parameter, 56Winding number, 350

Documents

Rothe H.J.; Lattice Gauge Theories an Introduction