MASTER THESIS - NTNU · MASTER THESIS STUD. TECHN. EIRIK EIK SVANES DISCIPLINE: THEORETICAL PHYSICS Norsk tittel: \Den ikke-perturbative renormaliseringsgruppen med anvendelser" English

$Page 1: MASTER THESIS - NTNU · MASTER THESIS STUD. TECHN. EIRIK EIK SVANES DISCIPLINE: THEORETICAL PHYSICS Norsk tittel: \Den ikke-perturbative renormaliseringsgruppen med anvendelser" English$
MASTER THESIS

STUD. TECHN.EIRIK EIK SVANES

DISCIPLINE: THEORETICAL PHYSICS

Norsk tittel: “Den ikke-perturbative renormaliseringsgruppenmed anvendelser”

English title: “The Non-Perturbative Renormalization Groupwith Applications”

This work has been carried out at Department of Physics at the Norwegian University ofScience and Technology, under the supervision of Professor Jens Oluf Andersen.


ABSTRACT

We discuss the exact renormalization group equation for the effective average action. Weproceed to derive the renormalization group equation for the effective potential U(φ2) in thelocal potential approximation, and solve this numerically using a third order Runge-Kuttamethod. We use the O(4)-symmetric linear sigma model as our boundary condition at thecutoff scale Λ. This is a good model for two-flavor QCD, and so we tune the initial param-eters at the cutoff so that it reproduces vacuum physics. Solving the renormalization-groupequation with these couplings at finite temperature, we find a second order phase transitionat a critical temperature Tc = 152 MeV. We compare this with results obtained by e.g. latticesimulations and the Nambu-Jona-Lasinio model.

We then add an isospin chemical potential µI to the theory in order to study the phasediagram in the T − µI plane in both the chiral limit and at the physical point. At the physi-cal point, we add an explicitly symmetry breaking term. We also study the chiral crossoverfor various pion masses, and the competition between the two condensates at the physicalpoint. We find a non-existing charged pion condensate for low isospin chemical potentials,that becomes nonzero and even dominates the chiral condensate for high µI .

We proceed to derive the exact renormalization group equation for the case of an O(2)×O(2)-symmetric theory with two chemical potentials included, one for each O(2). This can be usedas a model for kaon condensation in high density three-flavor QCD, where we thus associatea chemical potential to the neutral kaon µ0, and one to the charged kaon µ+. It is an ef-fective theory describing mesonic degrees of freedom below the color-superconducting energygap ∆ = 30 MeV. We derive phase diagrams in the (µ0, µ+)-plane for different values ofthe temperature, using parameters at the cutoff Λ as effective parameters derived from thehigh-energy theory. We discuss the dependence of the theory on the cutoff. We then imposea charge neutrality condition by adding an electron background field. This gives a relationbetween the chemical potentials, leaving only one of them free to vary. We discuss the effectsof charge neutrality and plot a phase diagram in the chemical potential-temperature plane.


ACKNOWLEDGMENTS

First I would like to thank my supervisor, Professor Jens Oluf Andersen, for giving me thisinteresting and challenging project to work on, and for always helping me if I got stuck orneeded support.

I would also like to thank Tomas Brauner for useful discussions along the way, and MichaelStrickland for giving me hints and help with the numerics.


PREFACE

This master’s thesis discusses the non-perturbative renormalization group with applicationsin high-energy physics. It represents twenty weeks of work at the Norwegian University ofScience and Technology and it corresponds to thirty credit points, or one semester of work.The work is a continuation of the fifteen credit point specialization project that I did in thefall of 2009. In the specialization project, I truncated the RG equations obtained for theeffective potential, and solved the resulting coupled set of differential equations. In this mas-ter’s thesis, I have attacked the full RG equations for the potential, and also introduced morecomplicated models including chemical potentials.

Before I started working on this thesis, I had taken two courses on quantum field theoryat zero temperature, and was therefore quite familiar with concepts such as gauge theo-ries, and methods such as the Feynman path integral. Using the Matsubara formalism, thetransition to thermal field theory where the temperature is nonzero therefore went relativelysmoothly. Simply put, one only needs to go to Euclidean space and integrate over fields thatare periodic (bosons) or antiperiodic (fermions) in the Euclidean time.

It should be mentioned that most of my previous experience with zero temperature quan-tum field theory was through calculating Feynman diagrams for different S-matrix elementsof various theories. This is strictly perturbative work. As most of this thesis has been onthe non-perturbative formalism, I had to familiarize myself more with functional calculus, inparticular when deriving the non-perturbative renormalization group equations. This workwas both challenging and fun. I like this branch of mathematics, though it should be saidthat it is still in its infancy, due to problems concerning a well-defined path-integral measure.

I have also come across problems during my work. The numerical problems have for themost part been overcome, but some theoretical problems still remain, discussed in the conclu-sion. These problems are now subjects of future work. They could perhaps have been tackledhad I had more time on my master’s thesis. However, all in all I think it has gone reasonablywell. I have learned a lot, and had a lot of fun in doing so. I again thank my supervisor JensO. Andersen for inspiration and support during the way. It has been great.


CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV

List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

Part I Introduction

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Path-integral QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Path integrals in QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Symmetries in the path-integral formalism . . . . . . . . . . . . . . . . 91.1.4 Gauge symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.5 TFT and the O(N) Lagrangian . . . . . . . . . . . . . . . . . . . . . . 14

1.2 Perturbative renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.1 Renormalizing φ4-theory . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Renormalizable vs non-renormalizable . . . . . . . . . . . . . . . . . . 21

1.3 Non-perturbative renormalization . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.2 RG equations for the couplings . . . . . . . . . . . . . . . . . . . . . . 231.3.3 Critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2. Quantities of TFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1 Thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.1 Thermal averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.2 Helmholtz-free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1.3 Thermodynamic potential . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Gibbs free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 1PI generating functional . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.3 Computing Γ[M ] in the perturbative framework . . . . . . . . . . . . 32

2.3 Grand canonical partition function . . . . . . . . . . . . . . . . . . . . . . . . 33

3. The Non-perturbative Renormalization Group . . . . . . . . . . . . . . . . . . . . . 353.1 The WP formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Starting point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.2 Block-spin mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 EAA formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 IR regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.2 The effective action Γk[M ] . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 RG equation for Γk[M ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 Approximating Γk[M] . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.2 The RG equation for Uk(M) . . . . . . . . . . . . . . . . . . . . . . . 423.3.3 Spontaneous symmetry breaking and phase transitions . . . . . . . . . 43

Part II The O(N) case

4. Generalizing to O(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1 QCD and O(4) effective theory . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.1 Strong interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2 Chiral limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Finding the equation at zero temperature . . . . . . . . . . . . . . . . . . . . 534.2.1 The O(N) equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2.2 Truncation and coupled equations . . . . . . . . . . . . . . . . . . . . 54

4.3 Turning on the temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3.1 The RG equation for Uk(ρ) . . . . . . . . . . . . . . . . . . . . . . . . 554.3.2 High-temperature flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 The large-N limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5. Tuning the parameters of the O(N) model . . . . . . . . . . . . . . . . . . . . . . . 595.1 Physical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Renormalized couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.1 Critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.2 The quartic coupling g2

k . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Part III Pion condensation

6. Including a chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.1 Deriving the Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2 Including a pion mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.3 Renormalizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7. Full RG-equation for pion condensation . . . . . . . . . . . . . . . . . . . . . . . . 737.1 The full equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.2 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2.1 Charged pion condensate . . . . . . . . . . . . . . . . . . . . . . . . . 757.2.2 BE condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.2.3 Large-N limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

II

8. Numerical results; charged pion condensation . . . . . . . . . . . . . . . . . . . . . 818.1 Leaving it to the numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.2 Analyzing the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.2.1 Including a pion mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.2.2 Interpreting results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.2.3 The chiral limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.2.4 The physical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.3 O(4)-approximation and the large-N limit . . . . . . . . . . . . . . . . . . . . 90

Part IV Kaon condensation

9. Fermions in QFT and TFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959.1 Fermion Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959.2 Fermion Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.3 Fermion partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.4 Fermion free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

10. Kaon condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9910.1 The CFL-phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

10.1.1 SU(3)c+L+R-broken symmetry . . . . . . . . . . . . . . . . . . . . . . 9910.1.2 Inside a neutron star . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.1.3 Charge neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

10.2 Effective Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.2.1 Octet Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.2.2 Expanding the Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 10310.2.3 Kaon effective Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 104

10.3 Charge neutrality condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.3.1 Electron-positron background . . . . . . . . . . . . . . . . . . . . . . . 10510.3.2 Renormalizing the theory . . . . . . . . . . . . . . . . . . . . . . . . . 105

11. The kaon RG equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10711.1 Zero temperature Gibbs free energy . . . . . . . . . . . . . . . . . . . . . . . 10711.2 RG equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10811.3 Equation for T > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10811.4 High-temperature limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.5 Charge neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

11.5.1 Some approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.5.2 One-loop result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

12. Numerical results; kaon condensation . . . . . . . . . . . . . . . . . . . . . . . . . . 11312.1 Numerical results without charge neutrality . . . . . . . . . . . . . . . . . . . 113

12.1.1 The bare potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11312.1.2 Critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11412.1.3 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

12.2 Numerical results with charge neutrality . . . . . . . . . . . . . . . . . . . . . 117

III

Part V Summary, outlook, and conclusion

13. Summary, outlook, and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 12313.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12313.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12413.3 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Appendices

A. Conventions and calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.1 Definitions, conventions and identities . . . . . . . . . . . . . . . . . . . . . . 129A.2 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A.2.1 Matsubara sums and Cauchy integrations . . . . . . . . . . . . . . . . 129A.2.2 The RG equation in the EAA . . . . . . . . . . . . . . . . . . . . . . . 131

A.3 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133A.3.1 RK methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133A.3.2 Finite distance methods . . . . . . . . . . . . . . . . . . . . . . . . . . 134

B. Matlab code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B.1 Potential solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B.2 The iterator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

IV

LIST OF FIGURES

1.1 Low energy scattering→ non-renormalizable coupling . . . . . . . . . . . . . 221.2 Renormalization procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Smoothed out Heaviside function . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 IR regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 SSB and convex effective potentials . . . . . . . . . . . . . . . . . . . . . . . . 443.4 First and second order phase transitions . . . . . . . . . . . . . . . . . . . . . 453.5 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 QCD interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Renormalizing the O(4)-mass parameter . . . . . . . . . . . . . . . . . . . . . 615.2 Effective coupling as function of temperature . . . . . . . . . . . . . . . . . . 635.3 O(4) effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.1 Phase diagram; Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . 78

8.1 Pion-effective potential in the chiral limit . . . . . . . . . . . . . . . . . . . . 848.2 Pion-effective potential as a function of ρ2 in the chiral limit . . . . . . . . . . 858.3 Chiral crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878.4 Phase diagram; charged pion condensation . . . . . . . . . . . . . . . . . . . . 888.5 Charged pion condensate as a function of temperature . . . . . . . . . . . . . 898.6 Condensate competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.7 Phase diagram; Charged pion condensate, O(4)-approximation and the large-N

limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

12.1 Kaon-effective potential as a function of ρ1 at different cutoff . . . . . . . . . 11412.2 Kaon-effective potential as a function of ρ1 at different temperatures . . . . . 11512.3 Neutral kaon condensate at T = 0 . . . . . . . . . . . . . . . . . . . . . . . . 11612.4 Neutral kaon condensate at T = 50 . . . . . . . . . . . . . . . . . . . . . . . . 11712.5 Phase diagram; Kaon condensation . . . . . . . . . . . . . . . . . . . . . . . . 11812.6 m2

+,k as a function of the RG time . . . . . . . . . . . . . . . . . . . . . . . . 11912.7 µ+ as a function of temperature wit neutrality condition . . . . . . . . . . . . 12012.8 Neutral kaon condensate with neutrality condition . . . . . . . . . . . . . . . 120

13.1 Phase diagram; µB-T phase diagram of QCD . . . . . . . . . . . . . . . . . . 124

A.1 Contour integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

V

VI

LIST OF ABBREVIATIONS

RG renormalization groupNPRG non-perturbative renormalization groupQFT quantum field theoryTFT thermal field theoryQCD quantum chromodynamicsQM quantum mechanicsQED quantum electrodynamicsLHS left-hand sideRHS right-had sideBRST Becchi, Rouet, Stora and Tyutin1PI one-particle irreducibleUV ultravioletIR infraredWF Wilson-FisherWP Wilson-PolchinskiEAA effective average actionERGE exact renormalization-group equationLPA local potential approximationRK Runge-KuttaBE Bose-EinsteinNJL Nambu-Jona-LasinioCFL color-flavor lockedPDE partial differential equation

VII


Part I

INTRODUCTION


1. INTRODUCTION

Renormalization is the theory of viewing systems at different scales. What are the long rangestructures resulting from the smaller microscopic physics? Or conversely, what are the lowenergy effects of the high-energy theory? Switching from viewing a system at one scale toanother is known as a renormalization-group (RG) transformation. We could ask: Whathappens when we do this? Does the physics remain the same, as is the case for e.g. a fractal,or does it differ, and if so, how does it change? What happens when we change the scale ofthe system, that is, what happens when we renormalize?

In order to answer these questions, we need a framework in which to tackle them. Thereare two main schools of thought. The first one is the perturbative formalism which is themost widely used. It goes as follows. If we do perturbation theory using Feynman diagrams,we will sooner or later encounter divergent loop diagrams. How do we deal with such di-agrams? Perturbative renormalization has the answer. The trick is to combine them withthe high-energy ”bare” parameters of the theory in order to produce the parameters at thelower-energy scale for which experiment is done, and in this way including the high-energyeffects into the lower energy theory. If this can be done to all orders of perturbation, thetheory is known as renormalizable.

The second formalism is the non-perturbative one and it is this formalism that the mainfocus of this thesis is on. We shall thus investigate the non-perturbative renormalizationgroup (NPRG) and look at some of its applications. The procedure here is to derive a renor-malization group scaling equation for the theory, whereby the higher-energy bare Lagrangian,consisting of bare parameters at some cutoff Λ, gets renormalized as one integrates this equa-tion down to lower energy scales. Exactly how this is done will of course be explained in moredetail later, but before this, we review some preliminaries.

This thesis is organized as follows. In this chapter, we briefly review quantum and thermalfield theory (QFT and TFT) before reviewing the perturbative renormalization procedure. Inthe next chapter, we introduce some useful thermal quantities, and in particular the effectiveaction Γ[M ] of the mean-field 〈φ〉 = M that we shall be concerned with for the most part ofthis thesis. In Chapter 3, we derive the RG-equation for the effective action and for the effec-tive potential in the local-potential approximation. In chapters 4 and 5, we apply the theoryto the O(N) model, explaining its relevance for QCD, while in chapters 6-8 we introduce anisospin chemical potential in order to study charged pion condensation. In chapters 10-12, westudy the general O(2) × O(2)-symmetric Lagrangian relating to kaon condensation at highquark densities. We finally conclude in Chapter 13. We have also included some conventionsand calculations plus a review of the numerics used in Appendix A. Some Matlab source codeused for the numerics is also included in Appendix B.

1.1. PRELIMINARIES 2

1.1 Preliminaries

In this section, we review how QFT and TFT are done in the Feynman path-integral for-malism. The concepts presented here are assumed to be familiar to the reader, but we recallthem for completeness. We will not use the canonical formalism in this thesis. Interestedreaders are referred to e.g. [1, 2, 3] for more details on this. We start by showing how QFT isdone in the path-integral formalism. The transition from a QFT to a TFT (basically going toEuclidean spacetime and turning on the temperature) is shown by means of an example, theO(N)-symmetric Lagrangian. This example is chosen since this Lagrangian is an importantquantity for the remainder of the text.

1.1.1 Path-integral QFT

Before introducing the path-integral formalism for QFT, let us see how it is done in ordinaryquantum mechanics (QM).

Path integrals in QM

We consider, for the moment, ordinary one-particle QM. Let, for the moment, x denote thespace position of the space-time vector (t, x). Say that we wish to calculate the transitionamplitude

mi,f = 〈xf |ei(tf−ti)H(x,p)|xi〉, (1.1)

where x and p are the quantum operators

x|x〉 = x|x〉, (1.2)

p|p〉 = p|p〉, (1.3)

and i and f denote the initial and final times respectively. We work in natural units where~ = c = 1. We divide the time interval from ti to tf into N smaller intervals by introducing

ε =tf − tiN

, (1.4)

where N is a large integer. Then the transition amplitude can be written as

mi,f = 〈xf |N∏j=1

eiεjH(x,p)|xi〉, (1.5)

where each εj is given by Eq. (1.4). We recall that the Hamiltonian for a one-particle systemin regular non-relativistic QM is

H(x, p) =p2

2m+ V (x). (1.6)

We shall use the identity, known as the Trotter formula [4],

limN→∞

(eAN e

BN )N = eA+B (1.7)


for operators A and B. If we take the limit N →∞, we can write Eq. (1.5) as

mi,f = 〈xf |N∏j=1

eiεjp2

2m eiεjV (x)|xi〉. (1.8)

We now introduce N factors of unity of the form

1 =

∫dpk|pk〉〈pk|, (1.9)

where k ∈ 1, ..N, and we have used the completeness relation in momentum space. Weplace these between the exponential factors of Eq. (1.8). We can then write

eiεkp2

2m =

∫dpk|pk〉〈pk|eiεk

p2k2m , (1.10)

where we have let p act to the left in the last equality. We similarly introduce N − 1 com-pleteness relations of the form

1 =

∫dxk|xk〉〈xk|. (1.11)

Using this, we may write

eiεk+1V (x) =

∫dxk|xk〉〈xk|eiεk+1V (xk), (1.12)

where again we have let x act to the left. Note here that the k’th factor of unity correspondsto the (k + 1)’th exponential. With this, the transition amplitude may be written as

mi,f =

∫dpN 〈xN |pN 〉〈pN |xN−1〉e−iεH(xN−1,pN )

∫ N−1∏j=1

dpjdxj〈xj |pj〉〈pj |xj−1〉e−iεH(xj−1,pj)

=

∫ N−1∏j=1

dpjdxj〈xj |pj〉〈pj |xj−1〉e−iεH(xj−1,pj), (1.13)

where we have let x0 = xi and xN = xf and noted that, as N is very large and thus xN−1 ≈ xNand ε ≈ 0, the first exponential may be discarded. From ordinary QM we recall that [5]

〈x|p〉 =1√2πeipx. (1.14)

Using this, we find∫dpj〈xj |pj〉〈pj |xj−1〉e−iεH(xj−1,pj) =

√m

2πiεeiε[m2ε2

(xj−xj−1)2−V (xj−1)], (1.15)

where we have performed the integral over pj . We thus finally arrive at

mi,f = limN→∞

∫ N−1∏i=1

dxi

√m

2πiεeiε

∑N−1i=1

[m2ε2

(xi−xi−1)2−V (xi−1)]

=

∫ xf ,tf

xi,ti

Dxei∫ tfti

dt

(m2

(dxdt

)2−V (x)

)

=

∫ xf ,tf

xi,ti

DxeiS[x], (1.16)


where we have defined the action

S[x] =

∫ tf

ti

dtL(x, x), (1.17)

where L(x, x) = 12mx

2 − V (x) is the Lagrangian. We have also defined the measure

Dx = limN→∞

N−1∏i=1

dxi

√m

2πiε. (1.18)

Note the appearance of the Lagrangian in the exponent. This shows that the importantfundamental quantity of the path integral formalism of QFT is the Lagrangian and not theHamiltonian (though there is a Legendre transformation between the two). In the followingwe abbreviate the above path integral as∫ xf ,tf

xi,ti

Dx =

∫Dx. (1.19)

Operators of QM in the path-integral formalism

We wish to see if the path-integral formalism may be used to calculate expectation valuesof operators, that is, can we write the expectation value 〈O〉 of an operator O using a pathintegral? Indeed we can. To see this, start by considering

P [x(t1), x(t2)] =

∫Dxx(t1)x(t2)eiS[x]

=

∫DxDx1Dx2δ(x1 − x(t1))δ(x2 − x(t2))x1x2e

iS[x], (1.20)

where we have introduced two functional Dirac delta-functions. If we assume t2 > t1, thiscan the be written as

P [x(t1), x(t2)] =

∫Dxx(t1)x(t2)eiS[x]

=

∫Dx1Dx2Dxx(t1)=x1,x(t2)=x2x1x2e

iS[x]

=

∫Dx1Dx2〈xf |ei(tf−t2)Hx2|x2〉〈x2|ei(t2−t1)Hx1|x1〉〈x1|ei(t1−ti)H |x1〉. (1.21)

We now introduce the functional spectral resolution∫Dxx|x〉〈x| = x, (1.22)

in order to get rid of the integrals over x1 and x2. Using this, we obtain

P [x(t1), x(t2)] = 〈xf |ei(tf−t2)H xei(t2−t1)H xei(t1−ti)H |x1〉. (1.23)

Recall thate−itiH xeitiH = xH(ti), (1.24)


where the H denotes the fact that we have gone from Schrodinger to Heisenberg operators.If we also let tf →∞+ iε and ti → −∞− iε, we may write

e−itiH |xi〉 =∑n

e−itiEn |n〉〈n|xi〉 = |0〉〈0|xi〉e−itiE0 , (1.25)

where we have introduced a partition of unity, and |0〉 is the state of least energy E0, i.e. thevacuum. A similar formula holds for |xf 〉. In fact, using Eq. (1.25) it is now easy to see that

mi,f = 〈xf |0〉〈0|xi〉ei(tf−ti)E0 . (1.26)

Thus, we finally arrive at

P [x(t1), x(t2)] = mi,f 〈0|xH(t1)xH(t2)|0〉, (1.27)

i.e.

〈0|xH(t1)xH(t2)|0〉 = 〈xH(t1)xH(t2)〉 =1

mi,f

∫Dxx(t1)x(t2)eiS[x]. (1.28)

Note that in the above we assumed t2 > t1. For general times we must let 〈xH(t1)xH(t2)〉 →〈T xH(t1)xH(t2)〉 where the T denotes time ordering. Also note that we are dealing withHeisenberg operators now.

Eq. (1.28) can be generalized to

〈TO(x)〉 =1

mi,f

∫DxO(x)eiS[x] (1.29)

for some operator O(x), where we have dropped the hats and the Heisenberg index. Thisgives the expectation value for general operators.

Now that we know how the path integral formalism works in QM, we proceed to QFT andsee how it works there.

1.1.2 Path integrals in QFT

We proceed to find the path-integral formalism in QFT. Return now to denoting the spacetimevector by x = (t,x). As usual one writes

L[φ] =

∫d3xL(φx), (1.30)

where L[φ] is the Lagrangian, L(φx) is the Lagrangian density, from here on referred to asthe Lagrangian, and φx = φ(x) = φ(x, t). The action may then be written as

S[φ] =

∫ tf

ti

dt

∫xL(φx) =

∫xL(φ), (1.31)

where we have taken the limits ti → −∞ and tf →∞. We have also introduced the notation∫d3x =

∫x, (1.32)∫

dt

∫x

=

∫x, (1.33)


as explained in Appendix A. If we consider the Lorentz invariant Lagrangian for a one-component scalar field, it reads

L(φ) =1

2∂µφ∂

µφ− U(φ). (1.34)

This has a kinetic term 12∂µφ∂

µφ and a potential term U(φ) known as the potential density.The transition amplitude may now be generalized to

〈φf |e−i(tf−ti)H |φi〉 =

∫ φf ,tf

φi,ti

DφeiS[φ], (1.35)

where φi = φi(x, ti) and φf = φf (x, tf ). The transition is now between configurations of thefield φ, and the path integral is over all possible field configurations between the initial andfinal time.

Correlation functions in QFT

Now that we know how to calculate transition amplitudes in path-integral QFT, we considerhow correlation functions are dealt with. Similarly to the above calculation for QM, it canbe shown that [1]

〈TO(φ)〉 = 〈0|TO(φ)|0〉 =1

Z

∫DφO(φ)eiS[φ], (1.36)

where T denotes time ordering, |0〉 is the vacuum state, and Z is known as the partitionfunction,

Z =

∫DφeiS[φ]. (1.37)

We have also neglected the subscript H for Heisenberg operators, as QFT is usually donein the Heisenberg picture. The path integral is now over all field configurations over all ofspacetime, similar to the limit ti → −∞ and tf → ∞ above. As an example, we considerthe two-point correlation function −iD(x − y) = 〈Tφxφy〉, also known as the propagator.If we add a term

∫x Jxφx to the action, i.e. we couple the field to an external source J , the

partition function may be written as

Z[J ] =

∫Dφ exp

[i

∫x

(L(φx) + Jxφx

)]. (1.38)

The propagator is then defined as

−iD(x− y) = 〈Tφxφy〉 =1

Z[J = 0](−i) δ

δJx(−i) δ

δJyZ[J ]

∣∣∣J=0

. (1.39)

Z[J ] may also be used to generate other correlation functions of various kinds, and is thereforecalled a generating functional.

The Gaussian partition function and the free propagator

The question is now: are there any theories in which Z[J ] is exactly calculable? The answer isyes, and a well known example is the referred to as the Gaussian model due to the quadratic


”free” form of the potential U(φ) = 12m

2φ2. Herem denotes the mass of the particles describedby the field φ. We consider the integral

I =

∫ ∏i

dxie−xiAijxj+Jixi , (1.40)

where A is some invertible matrix. We also assume A to be positive definite, otherwise theintegral will be divergent. Note also that we have assumed the Einstein summation rule, as weshall do for the rest of the thesis unless otherwise stated. When we know what I is, it will beeasy to generalize it to functional integrals. Note that xiAijxj = xixj

12(Aij +Aji) = xiA

Sijxj .

Hence, only the symmetric part AS of the matrix A survives, and we may as well assume Asymmetric in the first place. We perform the coordinate transformation

xi = yi +A−1ij Jj (1.41)

in order to obtain

I = e12JiA−1ij Jj

∫ ∏i

dyie− 1

2yiAijyj . (1.42)

Performing an orthogonal transformation on A so that yiAijyj = yiS−1ik DklSljyj , where D is

a diagonal matrix and detS = 1. we thus get (xi = Sijyj)

I = e12JiA−1ij Jj

∫ ∏i

dxie− 1

2λjx

2j = e

12JiA−1ij Jj

∏k

1√λk

=e

12JiA−1ij Jj

√detA

= exp(1

2JiA

−1ij Jj −

1

2Tr logA

), (1.43)

where we have used the identity log detA = Tr logA, valid for any invertible matrix A, in thelast line. Eq. (1.43) is now easily extended to functional integrals, and we obtain

Z[J ] = exp(− 1

2Tr log

[− i(∂2 +m2)

]+i

2

∫xJx(−∂2 −m2)−1Jx

)(1.44)

for the Gaussian model. We have performed an integration by parts on the action

S[φ] =

∫x(1

2∂µφ∂

µφ− 1

2m2φ2) = −1

2

∫xφ(∂2 +m2)φ. (1.45)

From Eq. (1.44), we see that the propagator of the Gaussian theory becomes D0(x − y) =(−∂2 −m2)−1, where the zero denotes that the theory is free. We then have the identity

(−∂2 −m2)D0(x− y) = δ(x− y). (1.46)

This equation may be Fourier transformed, and is easily solved for D0(x− y) to give

D0(x− y) =

∫p

eip(x−y)

p2 −m2 + iε, (1.47)

where a small term iε has been added in order to prevent the integral from hitting poles inthe p0 plane. With this, we find, in the case of the Gaussian model, that

i〈Tφxφy〉 = D0(x− y) =

∫p

eip(x−y)

p2 −m2 + iε. (1.48)


The perturbative expansion

The generating functional is used to derive the Feynman rules of the theory [1, 2]. These arerules for how to calculate Feynman diagrams. Using Feynman diagrams is a convenient wayof describing and calculating various correlation functions to some perturbative order in thecouplings of the potential U(φ). The couplings of U(φ) are the expansion parameters in frontof the various terms of the Taylor expansion of U(φ), i.e.

U(φ) =1

2m2φ2 + g1φ

3 + g2φ4 + ... , (1.49)

where the gi’s are the couplings. They are used as expansion parameters when one perturbsthe theory from the exactly solvable Gaussian model described above. The Feynman rulesdiffer of course depending on the orders of the coupling terms of the potential U(φ).

As an example, we briefly consider φ4-theory. Here the Lagrangian reads

L =1

2∂µφ∂

µφ− 1

2!m2φ2 − 1

4!λφ4. (1.50)

The Generating functional, expanded in the coupling λ, is

Z[J ] =∞∑n=0

∫Dφei(S0[φ]+

∫x Jφ) 1

n!

( i4!λ

∫xφ4)n, (1.51)

where S0[φ] denotes the free action and we have neglected the spacetime dependence ofthe fields in order to save space. This should then be truncated at the desired order ofperturbation l. Say we wish to calculate some correlation function 〈φn〉, again neglectingspacetime dependence. This reads

〈φn〉 =1

Z[J = 0]

l∑k=0

∫Dφ(φn)eiS0[φ] 1

k!

( i4!

∫xλφ4

)k. (1.52)

The Feynman rules thus require us to write down all the topologically distinct diagrams ofn external legs with up to l internal four vertices (as the coupling is of the form λφ4). Forinstance, if we want to calculate 〈φ4〉 to order two, this reads

〈φ4〉 = + . (1.53)

The different parts of the diagrams are associated to different mathematical expressions givenby the Feynman rules. For instance, internal lines of the diagram (that is, lines not associatedto a leg of the diagram) are given by the free propagator

p p =1

p2 −m2(1.54)

in momentum space, as we saw above. Also, a symmetry factor of 1S is associated to each

diagram, where S is the number of transformations one can do on a diagram without chang-ing it’s topology. The second diagram above for instance has a symmetry factor of 1

2 . Adelta-function is also included per vertex in order to have momentum conservation, and one


should integrate over remaining momenta.

As the main focus of this thesis is on the non-perturbative aspects of field theory, we shallnot go further into detail on how the Feynman rules are derived or used. The interestedreader is referred to e.g. [1, 2, 3] for details. It should also be noted that we have notconsidered fermionic fields yet (the scalar φ-Lagrangian was bosonic in nature), however thecorresponding theory is quite similar and will be introduced when needed.

1.1.3 Symmetries in the path-integral formalism

We now consider how symmetries are treated in the path-integral formalism. We recallNoethers theorem that to every symmetry corresponds a conservation law, usually in theform of an equation. In the path integral formalism it is the partition function that shouldbe invariant under symmetry transformations. We now turn to some examples in order toillustrate the power of using symmetries when deriving certain quantities.

The Euler-Lagrange equations

We first look at the partition function Z = Z[J = 0]. A translation of the field should leave Zinvariant. We thus let φ→ φ+δφ where δφ is an infinitesimal translation (a finite translationcan always be obtained by an infinite number of such). Note that this leaves the measure Dφinvariant. If we have a multicomponent field, i.e. φ = (φ1, .., φN ) we may write (assumingL = L(φ, ∂µφ))

Z =

∫DφeiS[φ] =

∫DφeiS[φ+δφ]

=

∫Dφei

∫x L(φ+δφ,∂µφ+∂µδφ)

=

∫Dφ exp

[i

∫x

(L+ δφi

∂L∂φi

+ (∂µδφi)∂L

∂(∂µφi)

)]=

∫DφeiS[φ]

(1 +

∫xδφi

( ∂L∂φi− ∂µ

∂L∂(∂µφi)

)), (1.55)

where we have performed a partial integration in the last line. Note then that in order forthis to be satisfied, as δφ can be arbitrarily chosen, we must have

∂L∂φi

= ∂µ∂L

∂(∂µφi), (1.56)

i.e. the Euler-Lagrange equations.

Symmetries as a tool in calculations

Symmetries can also be important when calculating correlation functions. Assume for instancethat S = S[Φ,Φ∗], i.e. the action is a functional of a field and its complex conjugate. Then,the partition function reads

Z =

∫DΦDΦ∗eiS[Φ,Φ∗]. (1.57)


We perform the symmetry transformation Φ = eiαΦ and Φ∗ = e−iαΦ∗ for some arbitrary α.Note that the integration measure is invariant under this transformation. We also assume theaction to be invariant under the transformation, and hence the partition function becomesinvariant as well. We calculate the correlation function 〈ΦxΦy〉

〈ΦxΦy〉 =1

Z

∫DΦDΦ∗ΦxΦye

iS[Φ,Φ∗]

=e2iα

Z

∫DΦDΦ∗ΦxΦye

iS[Φ,Φ∗]

= e2iα〈ΦxΦy〉.This implies that 〈ΦxΦy〉 = 0. We shall later see that complex conjugated fields correspondsto particle and anti-particle fields (see below). We have thus shown that a particle field isuncorrelated with itself at different spacetime points. The same holds for the anti-particlefield. The only nonzero correlation functions are thus those that have an equal number ofparticle fields and anti-particle fields. Basically, this is the law of particle conservation, herederived from a symmetry principle.

Note that if a symmetry leaves the action invariant, it will be a symmetry on the classi-cal level where it, according to Noether’s theorem, corresponds to a classical conservationlaw. If, however, the path-integral measure Dφ is not invariant, this may lead to so-calledquantum anomalies. Here, symmetries of the action are not true symmetries when all quan-tum corrections have been included. An example is the so-called axial anomaly of quantumelectrodynamics (QED) [6]. Here, the global chiral symmetry of classical electromagnetismgets broken when quantizing, i.e. when one performs the path integral.

Classical symmetries

Now, we briefly turn to classical symmetries where only the action needs to be invariant underthe symmetry at hand (more generally the classical equations of motion must be invariant, butwe shall assume an invariant action for simplicity). We assume that, under the infinitesimalsymmetry transformation, L → L+ ∂µJ µ, i.e. δL = ∂µJ µ is a total divergence. We thus get

δS[φ] =

∫Od4x∂µJ µ =

∫∂Od3µxJ µ = 0, (1.58)

where O denotes the volume we integrate over, d3µx is an infinitesimal three-surface vector,

and we have used Stokes’ theorem. When O is all of spacetime, ∂O = 0 and the resultfollows (∂O denotes the boundary of O). Hence, if the Lagrangian is transformed by a totaldivergence, the action is invariant and we have a classical symmetry. For the Lagrangian, wefind

L(φ+ δφ, ∂µ(φ+ δφ)

)= L+ δφi

∂L∂φi

+ ∂µδφi∂L

∂(∂µφi)

= L+ ∂µ

(∂L

∂(∂µφi)δφi

), (1.59)

where we have used the Euler-Lagrange equations in the last line. This may be set equal toL+ ∂µJ µ in order to arrive at

∂µ

( ∂L∂(∂µφi)

δφi − J µ)

= ∂µjµi = 0, (1.60)


where we have defined the Noether current

jµi =∂L

∂(∂µφi)δφi − J µ. (1.61)

Eq. (1.60) is the famous Noether theorem for classical symmetries.

1.1.4 Gauge symmetries

There is one kind of symmetries that deserves a bit broader treatment. Namely that of gaugesymmetries. Gauge symmetries and gauge theories are used to describe the fundamentalforces of nature. To each gauge symmetry, there corresponds a gauge group and relatedgauge fields. If the gauge group is of SU(N)-type, the gauge theory is referred to as a Yang-Mills theory [7, 8]. The strong force has the gauge group SU(3), and a corresponding gaugefield describing gluons. The electroweak force has the gauge group SU(2) × U(1) where thecorresponding gauge fields describe the massive W± and Z-particles, and the photon as amassless particle. The W± and Z-particles acquire their mass by spontaneous symmetrybreaking, SU(2)×U(1)→ U(1) (see Section 3.3.3) and the so-called Higgs mechanism [9, 10].This leaves the massless photon associated to the remaining U(1)-symmetry group. Thestrong and electroweak forces may be combined to the gauge group SU(3) × SU(2) × U(1)commonly known as the Standard Model [11, 12, 13]. Even gravity may be written as a gaugetheory [14, 15], though the quantization of this has been less successful.

We shall not touch widely upon gauge symmetries in this thesis, but still, it would be nice toknow a bit about them. We first consider a small example.

U(1) gauge

Consider the Lagrangian

L =1

2∂µφi∂

µφi − U(φ2), (1.62)

where U(φ2) is some potential and φ = (φ1, φ2) is a two-dimensional field vector. We wish towrite this in a slightly different way. We do this by introducing new complex field variables

Φ =1√2

(φ1 + iφ2), (1.63)

Φ∗ =1√2

(φ1 − iφ2). (1.64)

We can thus write the Lagrangian as

L = ∂µΦ∗∂µΦ− U(Φ∗Φ). (1.65)

We now make the symmetry transformation Φ → eiα(x)Φ and Φ∗ → e−iα(x)Φ∗, where we letthe symmetry parameter α be dependent on position and time. L is clearly not invariantunder this transformation. In order to make it invariant, we introduce a so-called covariantderivative

Dµ = ∂µ + igAµ, (1.66)


where Aµ is known as the gauge field, and g is the gauge coupling. We write the newLagrangian as

L = (DµΦ)∗(DµΦ)− U(Φ∗Φ). (1.67)

If we now let the gauge field transform by the gauge transformation

gAµ → gAµ − ∂µα (1.68)

as we transform Φ and Φ∗, we see that this new Lagrangian is invariant under the symmetrytransformation. As the Lagrangian is invariant, the physics will not change under this gaugetransformation.

We have now introduced a new field Aµ. If we want this to be a dynamic field of natureit requires a Lagrangian to describe its dynamics. This ”gauge” Lagrangian should also begauge invariant and is usually taken to be

Lg = −1

4FµνF

µν (1.69)

where the −14 is conventional, and

Fµν = ∂µAν − ∂νAµ (1.70)

is known as the field strength. The new Lagrangian then becomes L → L+ Lg.

We now return to the global symmetry transformation Φ → eiαΦ. As the Lagrangian isinvariant under this transformation, the corresponding Noether current is

jµ =∂L

∂(∂φi)δφi = i(Φ∂µΦ∗ − Φ∗∂µΦ) + 2gAµΦ∗Φ. (1.71)

We see that this is exactly the same as ∂L∂Aµ

, the left hand side (LHS) of the Euler-Lagrange

equations, Eq. (1.56), for Aµ. Calculating the right hand side (RHS) of Eq. (1.56) yields

jµ = ∂νFνµ, (1.72)

which also is the covariant form of Maxwell’s equations assuming jµ is the electromagneticfour-vector current jµem. Thus, if we also associate the field Aµ with the electromagnetic field,Φ and Φ∗ with a charged matter fields, Φ∗ is a matter field with the opposite charge of Φ, i.e.the antiparticle field of Φ, and g with the charge of Φ, we arrive at classical electromagnetismdescribed as a gauge field.

The above was an example of a U(1) gauge symmetry. It is a so-called abelian gauge theoryas the gauge group U(1) is commutative. In general this is not the case, and this leads tomore complicated situations.

General Gauge group

In the more general setting we return to φ a field vector of some kind. We denote the gaugegroup by H. Let G = G(x) ∈ H. Then it can be shown that

L =1

2DµφD

µφ− U(φ2)− 1

4Tr(FµνF

µν) (1.73)


is gauge invariant. Here, the covariant derivative reads

Dµ = ∂µ + gAµ, (1.74)

the gauge transformations are

φ→ Gφ, (1.75)

Aµ → GAµG−1 − 1

g[∂µ, G]G−1, (1.76)

(note then that the gauge field Aµ is in general a matrix), and the field tensor is

Fµν =1

ig[Dµ, Dν ]. (1.77)

The gauge field Aµ is in fact an element of the Lie algebra of H and can thus be written as

Aµ = AiµTi (1.78)

where the T i’s are generators of the Lie group H.

An example of such a non-abelian gauge theory is the theory of quantum chromodynam-ics (QCD), also known as the strong interaction. Here the field comes in a triplet

ψ =

ψ1

ψ2

ψ3

, (1.79)

acted on by the non-abelian gauge group SU(3). The Lagrangian also takes a somewhatdifferent form than (1.73), as we here deal with fermions and not bosons.

Quantizing Gauge theories

Quantizing gauge theories in the path-integral formalism is a bit tricky. The fact that a gaugetransformed gauge field A → Ag, where g ∈ G, the gauge group, corresponds to the samephysics needs to be incorporated in the quantization procedure. One should somehow factorout the gauge group from the path integral as not doing so would result in overcountingphysical states. This is usually done by introducing a so-called Faddeev-Popov determinant∆ [16], and writing the partition function for the gauge field as

Z =

∫DA∆[A]δ

(f [A]

)eiS[A], (1.80)

where f [A] = 0 is called the gauge condition, and is used to eliminate redundant variables.Note that we must have f [Ag] = 0 for a general g ∈ G in order to get rid of all the gaugecopies. The Faddeev-Popov determinant is implicitly determined by

∆[A]

∫Dgδ

(f [Ag]

)= 1, (1.81)


where the integral is over the gauge group. It is easy to see that ∆[A] = ∆[Ag], and hencewe arrive at ∫

DAeiS[A] =

∫DgDA∆[A]δ

(f [Ag]

)eiS[A]

=

∫DgDAg∆[Ag]δ

(f [Ag]

)eiS[Ag ]

=

∫DgZ, (1.82)

giving (1.80) as the partition function with the gauge-group volume∫Dg divided out. The

Faddeev-Popov determinant and the delta-function of Z may further be pulled into the ex-ponential by a process known as Becchi-Rouet-Stora-Tyutin (BRST) quantization [17]. Thisproduces some strange interaction terms in the Feynman diagrams known as Faddeev-Popovghosts. The Faddeev-Popov ghosts are present in order to not overcount different physicalstates as explained above. They do not correspond to real physical fields. We will not go intofurther detail on this, as this thesis is not concerned much with gauge theories. Interestedreaders are referred to e.g. [1, 2, 3] for a more elaborate treatment.

Now that we have gotten a short introduction to the path-integral formalism in QFT, wemove on to statistical mechanics and turn on the temperature, i.e. we consider TFT.

1.1.5 TFT and the O(N) Lagrangian

The transition from a QFT to a TFT is simpler than what one might expect. We will see howit is done for O(N)-symmetric theories, and then postulate that the procedure generalizes toother models. As the rest of this thesis are mostly concerned with theories that are O(N) ornearly O(N) symmetric, this seems like a good place to start.

The general scalar O(N) Lagrangian reads

L =1

2∂µφi∂

µφi − U(φ2). (1.83)

Here φ = (φ1, .., φN ) is a vector of fields, and U(φ2) is some potential. We see that this is aLorentz-invariant Lagrangian. However, in TFT it is often convenient to go to the so-calledimaginary time formalism letting t = −iτ where now τ is the first coordinate of a Euclideanspacetime vector. This changes the metric from a Minkowski metric into a Euclidean metric,and is known as a Wick rotation. The new Euclidean Lagrangian then reads

LE = −L(t = −iτ) =1

2(∇φ)2 + U(φ2), (1.84)

where the ∇ also includes time differentiating. In the future, we will take ∇ to include thetemporal derivative unless otherwise stated. Incidently, LE given by Eq. (1.84) is preciselythe Hamiltonian density of the system. Using this Lagrangian, we may define a Euclideanaction

SE [φ] =

∫xLE(φ) =

∫x

(1

2(∇φ)2 + U(φ2)

), (1.85)


where the integral now is over Euclidean spacetime. If we recall the QFT partition function

Z =

∫Dφei

∫x L[φ], (1.86)

we see that in going to imaginary time we arrive at the partition function

Z =

∫Dφe−

∫x LE(φ). (1.87)

This is the partition function we shall use when we are working in the zero temperature regimein the rest of this thesis.

We now consider a nonzero temperature. Recall that the canonical partition function reads

Z = Tre−βH =∑n

〈n|e−βH |n〉, (1.88)

where β = 1T , and we have set kB = 1 as we work in natural units. We first consider a simple

one-particle example. Here, we recall, the partition function reads

Z = Tre−βH =

∫x〈x|e−βH |x〉, (1.89)

where we have returned to let x denote the position of the particle, i.e. the position part ofthe spacetime vector (t, x). Recall from Eq. (1.16) that the transition amplitude is

〈xf |e−i(tf−ti)|xi〉 =

∫ xf ,tf

xi,ti

DxeiS[x], (1.90)

where the i and the f denote the initial and final times respectively. Using this and going toimaginary time, t = −iτ , we may write the partition function as

Z =

∫x

∫x(0)=x(β)=x

Dxei∫ β0 (−idτ)

[m2

(dx−idτ

)2−V (x)

]

=

∫x(0)=x(β)

Dxe−

∫ β0 (dτ)

[m2

(dxdτ

)2+V (x)

]

=

∫x(0)=x(β)

Dxe−SE [x], (1.91)

where again SE [x] = −S[x(t = −iτ)] denotes the Euclidean action. Note the periodicity inthe imaginary time, x(0) = x(β). This is due to the fact that e−βH is sandwiched betweenequal states when we take the trace in Eq. (1.89).

Eq. (1.91) can easily be generalized to field theory. For the scalar O(N) field we obtain

Z =

∫Dφp exp

[−∫ β

0dτ

∫xLE], (1.92)

where the subscript p denotes the periodicity condition φ(0,x) = φ(β,x) and x = (τ,x) nowis the Euclidean spacetime point. So, if we are in the imaginary time formalism, turning on

1.2. PERTURBATIVE RENORMALIZATION 16

the temperature is easy. Simply let the integration over the imaginary time τ in the Euclideanaction go to an integral from 0 to β. The path integration over the fields is also made into aintegration over periodic fields φ(0,x) = φ(β,x), due to the fact that the ”final” and ”initial”states of the transition amplitudes 〈φ|e−βH |φ〉 integrated over are the same. We shall, in thefuture, often write ∫ β

0dτ

∫x

=

∫x

(1.93)

in order to save space. This should not cause much confusion as it should be clear from thecontext if the temperature is zero or not.

Now that we have had a short introduction to QFT in the path-integral formalism, andan even shorter introduction to TFT, we move on to consider what is the main focus ofthis thesis, namely that of renormalization. We first consider the more familiar perturbativerenormalization procedure, before moving on to the non-perturbative one.

1.2 Perturbative renormalization

In perturbative field theory, one considers an exactly solvable model and adds terms presentin the model under study by means of a perturbative expansion as explained above. Thesimplest O(N) case is the φ4 model. Here, the partition function reads

Z[J ] =

∫Dφp exp(−SE [φ]−

∫xJφ), (1.94)

where J is an external source, and

SE =

∫x

(1

2(∇φ)2 +

1

2m2φ2 +

1

4!λφ4

)=

∫x

(L0(φ) +

1

4!λφ4

)(1.95)

is the Euclidean action. Here, L0(φ) is the Euclidean Lagrangian of the exactly solubleGaussian model Z0[J ]. Expanding the partition function around Z0, we obtain

Z[J ] =

∫Dφp

(1− λ

4!

∫xφ4x +

1

2

(λ

4!

)2 ∫x,yφ4xφ

4y + ..

)exp

[−∫x

(L0 + Jφ

)]. (1.96)

From this expansion one finds the Feynman diagrams that contribute to the correlation func-tions of the theory as explained above. Such an expansion, however, requires that the quarticcoupling λ is small.

Problems arise when trying to calculate transition amplitudes that involve loops in the Feyn-man diagrams. One encounters divergent integrals that completely invalidate the perturbativeexpansion. The problem is however less serious than first anticipated. Since the couplings inthe Lagrangian are so-called ”bare” quantities, that is the quantities when all interactions areabsent, we do not really know what these couplings are in the first place. The bare charge ormass of the electron may very well be infinite. But the coupling we measure in experimentsare not bare, as the experiments are done in a universe with interactions turned on. Theinclusion of this subtlety in the theory is known as renormalization.


In perturbative renormalization we add counterterms to the divergent Feynman diagramsin order to compute physical quantities. The physical quantities may be compared with thosefound by experiment. The counterterms take care of the divergences, and we end up withfinite physical quantities. This can be done order by order in perturbation theory, and hencetransition amplitudes may still be calculated to any order. In a sense, we hide the divergentloop diagrams behind the physical quantities.

Although this thesis is concerned with non-perturbative renormalization, we consider, forcompleteness, how perturbative renormalization works. We look at the simple case of theone-component scalar φ4-theory as an example.

1.2.1 Renormalizing φ4-theory

Consider again the Lagrangian

L =1

2∂µφ∂

µφ− 1

2m2φ2 − λ

4!φ4 (1.97)

of a scalar field. We work in the regime of T = 0, where we have returned to QFT andMinkowksi space for the moment in order not to complicate things unnecessarily. The renor-malization procedure is similar for TFT. We see how the procedure works up to second orderin the coupling λ, which we assume is small in order for the perturbative procedure to work.We then argue from this how to proceed in order to have a consistent theory at all orders. Weshall not derive the Feynman rules for φ4-theory as this is usually the first example encoun-tered in a introductory course on QFT, and interested readers are referred to e.g. [1, 2, 3]for more details and the derivation of the Feynman rules. We follow the renormalizationprocedure of [2] in the following.

Scattering amplitude

We first consider the scattering amplitude of two particles. To second order, we have fourcontributing diagrams. The scattering amplitude M reads

M = + + + . (1.98)

We formally write down the diagrams that contribute, not caring about any factors of i andsuch. The LHS of a diagram denotes before, while the RHS denotes after an interaction. Thisis why the three different loop diagrams are included in M since they constitute differentinteractions, even though they are topologically the same.

In four spacetime dimensions the last three diagrams go as∫d4kk4

and are thus logarith-mically divergent (depends logarithmically on the cutoff Λ). This divergence may howeverbe absorbed into the physical coupling λP , so that the Λ-dependence disappears from thescattering amplitude which is then only dependent of the physical coupling λP . The physicalcoupling can in turn be determined by experiment. We thus write

λP = λ+Kλ2 log

(Λ2

µ2

)(1.99)


where K is some constant that can be calculated from the Feynman rules and µ is sometypical energy scale, usually determined by experiment. The divergence associated with Λ iscanceled by the infinities associated with the bare coupling λ to give the finite quantity λP .This is the first order of perturbative renormalization of the coupling. Can we be sure thisworks to all orders? If it does, the theory will be so-called renormalizable.

Mass and field renormalization

We proceed to calculate the inverse propagator to order λ2. To this order it reads

D−1(k2) =(k k

)−1+ k k

+ k k + k k. (1.100)

Eq. (1.100) is accurate up to order λ2.

Note that only one-particle irreducible (1PI) diagrams contribute to the inverse of the prop-agator. That is, diagrams that cannot be split into two distinct parts with a bare propagatorbetween the parts. This can be seen as follows. We write the full propagator as (neglectingthe propagator momentum for compactness)

= + 1PI + 1PI 1PI + ... . (1.101)

If we define the 1PI diagrams as

1PI = M(k2), (1.102)

where k is the propagator momentum (M must be a function of k2 because of Lorentz invari-ance), we see that we may write the full propagator as

D(k2) = =1

k2 −m2 −M(k2). (1.103)

Hence we arrive at the inverse propagator

D−1(k2) =(k k

)−1− M(k2) (1.104)

which shows that only 1PI diagrams contribute.

The so-called pure loop diagrams, diagrams only containing loops that start and end at


the same place, are all quadratically divergent in the cutoff Λ, and also do not depend on theexternal momentum k. We have, for instance,

= − i2λ

∫|q|<Λ

i

q2 −m2, (1.105)

where |q| is the absolute value of the potential, and the Λ denotes that one should performthe integration up to momentum less than the cutoff Λ. This integral is clearly quadraticallydivergent in Λ and also independent of the external momentum k. It can be shown that thisholds for all such pure loop diagrams.

We now turn to the last diagram of Eq. (1.104) known as the setting sun diagram. Ifwe write it out, it reads

k k =

∫|q|<Λ

∫|p|<Λ

i

p2 −m2

i

q2 −m2

i

(p+ q + k)2 −m2. (1.106)

Due to Lorentz invariance, the diagram must be a function of k2. We can thus write

k k = A+Bk2 + Ck4 + ... . (1.107)

A is found by setting k = 0 in Eq. (1.106) and performing the resulting integral. If we takethe limit of the integrand to high momenta, we find that this integral goes as Λ2, i.e. it isquadratic in the cutoff. B can be found by differentiating the integral of (1.106) with respectto k2 and then setting k = 0. This gives an integral logarithmically dependent on Λ. C andthe higher-order terms are found in a similar fashion. The integrals obtained for these higherorder constants are convergent and thus become cutoff independent as Λ → ∞. They causeno trouble and we don’t need to worry about them.

Thus, to order λ2 we have to change the propagator as follows

1

k2 −m2→ 1

(1 + b)k2 − (m2 − a2), (1.108)

where a and b are quadratically and logarithmically divergent in the cutoff Λ, respectively.We see that the usual pole in k2 is shifted to m2

P = (m2 − a2)(1 + b2)−1. This defines thephysical mass, and the shift is known as mass renormalization. Also note that the residueof the pole is shifted from 1 to (1 + b)−1. This shift is known as field renormalization, itrenormalizes the factor in front of the kinetic term 1

2(∂φ)2 in the Lagrangian. We thus seethat all the terms of our Lagrangian have been renormalized from bare quantities to physicalquantities that do not depend on the cutoff.

The physical Lagrangian

We now wish to write our Lagrangian in terms of physical parameters. The procedure is asfollows. We write our renormalized Lagrangian as

L =1

2(∂φ)2 − 1

2m2Pφ

2 − ΛP4!φ4 +K1(∂φ)2 +K2φ

2 +K3φ4. (1.109)


The terms associated with the Ki’s are known as counterterms. They have their own Feynmanrules associated with them. The Feynman rules for regular φ4-theory hold with the exceptionthat m → mP and λ → λP . When calculating a quantity one should write down all theFeynman diagrams to the given order in λP one wants to calculate. The loops in the diagramsshould be integrated to the cutoff Λ. In addition, one should add so-called crossed counterdiagrams to counter the dependence on the cutoff. These have the Feynman rules [2]

2i(K1k2 +K2) 4!iK3. (1.110)

The constants Ki are determined iteratively as follows. Assume they have been determinedto order λNP . The Ki’s are then determined to order N + 1 by requiring that the propagatorcalculated to this order has a pole at mp with residue 1, and the meson-meson scatteringamplitude discussed above is calculated to be −iλP . This gives three conditions, enough todetermine the three constants.

Degree of divergence

We see that, in our theory, there is enough physical parameters to determine three countert-erms. Is this enough to sweep away all the dependence of our theory on Λ? What about adiagram with six external legs? There is no φ6 counterterm to take care of any cutoff depen-dence should the diagram depend on Λ. This worry is unwarranted as we shall see.

We return now, for the moment, to the unrenormalized theory. Assume that we have adiagram with BE external legs, and let D denote it’s degree of divergence, i.e. it divergesas ΛD. The number of loops L is then equal to the number of internal lines BI minus thenumber of vertices of the diagram minus one, i.e.

L = BI − (V − 1). (1.111)

Each vertex has 4 lines connecting to it, and each internal line is connected to two vertices,while each external line to one vertex. We thus obtain

4V = BE + 2BI . (1.112)

Finally, each loop of the diagram is associated with a 4-dimensional momentum integral, whileeach internal line is associated to a propagator. Translating this to the degree of divergencegives

D = 4L− 2BI . (1.113)

Eliminating BI , L and V yieldsD = 4−BE . (1.114)

We thus see the integrals with six external legs or more are cutoff independent and henceneed no renormalization as they remain finite when we take the limit Λ → ∞. Recall thatthe whole idea of perturbative renormalization was to get rid of any unwanted divergences.If there are no such divergences, there is no need to renormalize.


1.2.2 Renormalizable vs non-renormalizable

The above calculation for superficial divergence can be carried out for a more general La-grangian in d-dimensional spacetime

L =1

2∂µφ∂

µφ− 1

2m2φ2 − g

n!φn. (1.115)

This has been done in e.g. [1] and the result is the equation

d−BE(d− 2

2

)= V

[d− n

(d− 2

2

)]+D. (1.116)

The quantity multiplying V is just the mass/energy dimension of the coupling g. If this quan-tity is negative one only has to add enough vertices in order to make a diagram superficiallydivergent, no matter how many external legs it has. This is the property of a so-called non-renormalizable theory. If a coupling has positive mass dimension, the corresponding operatorφn is known as relevant, while operators associated with couplings of zero mass dimensionare known as marginal. Operators associated with couplings of negative mass dimension areknown as irrelevant. Hence, for four-dimensions φ4-theory, only m2 and λ are renormalizable.The mass term m2 is associated to a relevant operator φ2, while λ is associated to a marginalone φ4. This is also true for the more general O(N) case. Adding a term of order φ6 willmake the theory non-renormalizable. The terms relevant, marginal, and irrelevant will betalked about loosely in the following. For instance, when we say that a coupling is relevant,what we really mean is that the operator associated to the coupling is relevant.

If, on the other hand, a Lagrangian includes irrelevant couplings, these can be thought tobe effective couplings of a higher-energy theory that is renormalizable. An example is theφ4-theory considered above. Say we wish to consider low energy scattering of three particlesin this theory. A contributing diagram will be the diagram depicted on the left in Fig. 1.1.If, however, the momenta of the scattered particles are far smaller than the mass m, we mayapproximate this diagram by the one given on the right in Fig. 1.1. This is a vertex with sixlegs, and hence it will give rise to a non-renormalizable φ6-term in the low-energy effectiveLagrangian. Another example of this is Fermi’s theory for beta-decay. This is now known tobe a non-renormalizable low-energy effective theory of the full electroweak theory developedby Glashow, Salam and Weinberg in the 1960’s, for which they got the 1979 Nobel Prize inphysics [13, 18, 19].

Hence, even though the high-energy theory is renormalizable, we get terms in the low-energyeffective Lagrangian that are non-renormalizable. This is also a trait of the NRRG as weshall see. We may think of the process as putting on blurry ”renormalization” glasses, ef-fectively only seeing the lower energy effects of the high-energy fluctuations. This is what ismetaphorically pictured by the right lens of the glasses depicted the front page of this thesis.The IR lens as we have called it. Thus, there is really no reason why effective Lagrangiansshould not include non-renormalizable couplings, and hence, we would like a framework thatcan describe these situations as well. This is where the NPRG comes in.

1.3. NON-PERTURBATIVE RENORMALIZATION 22

Fig. 1.1: An example of how low-energy three-particle scattering can give rise to a order six couplingof energy dimension −2, in the effective Lagrangian at low energies of φ4-theory.

1.3 Non-perturbative renormalization

We now turn to see how the non-perturbative approach works. We only briefly consider theidea here, as the full theory will come in greater detail in Chapter 3.

1.3.1 The idea

In the non-perturbative framework, instead of talking about the high-energy full theory la-beled by the Lagrangian Lf , also known as the ultraviolet (UV) theory, and the low-energyeffective theory labeled by the Lagrangian Leff , also known as the infrared (IR) theory, oneinstead talks about a family of Lagrangians Lk labeled by a parameter k that denotes theenergy, or renormalization group scale of the system. The name of the game is then to finda differential RG equation

∂kLk = F (Lk, k) (1.117)

so that, if we have the renormalizable full theory at high energies k = Λ, where Λ is theenergy scale of the high-energy theory, commonly known as the cutoff, we can integrate thisdown to obtain effective theories for lower scales k. Integrating this down, we are effectivelydoing RG transformations.

One could ask how we know what the parameters of the cutoff Lagrangian LΛ should be,given that it is the bare theory at the cutoff? This theory is difficult to probe, since we areusually working at energies that are small compared to the cutoff. The answer to the questionis that we do not know what these parameters should be, but we can make educated guesses.

One common way of tuning the bare parameters is to start by assuming a perturbatively renor-malizable theory at the cutoff. When we integrate this down to k = 0, non-renormalizableterms will be generated, and the couplings corresponding to these terms will be functions ofthe high-energy renormalizable bare couplings at the cutoff. The renormalizable terms willalso change, resulting in the full effective theory when k = 0. This theory is then used tocalculate physical quantities, which are compared with the known values for these quantitiesthat have been determined by experiment. In this way, we can tune the bare parameters ofLΛ. This is usually done in the vacuum, where external parameters, such as the temperature


or chemical potentials, are set to zero. We have used this method when tuning the bareparameters in Chapter 5 and Chapter 8.

Another way of setting the bare parameters, is to use a cutoff Lagrangian that is an ef-fective theory of an even higher-energy full theory. The bare parameters are then calculatedfrom the parameters of the higher-energy theory, when approximating this theory by theeffective Lagrangian. This procedure has been used in Chapter 12.

1.3.2 RG equations for the couplings

We can also expand the potential part of Lk and truncate this at some order. It is possible toderive a coupled set of differential RG equations for the couplings involved in the expansion.We also mention that it is possible to find RG equations for the couplings in the perturbativerenormalization scheme as well. This is done by cutting off the loop integrals at some scaleΛ and in this way finding their dependence on Λ which now denotes the RG scale. This canbe done to any order of perturbation theory. However, the RG equation we arrive at withthe non-perturbative approach is exact. This means that all orders have been included! Itshould however be noted that it is impossible to solve analytically. Hence, one has to tend tonumerical methods.

Having found RG equations for the couplings, using either the perturbative or the non-perturbative approach, we may write these as a set

dgidt

= βi(g1, .., gN , t) (1.118)

where t = log( kΛ) is known as the renormalization time, and N is the number of cou-plings (or the order of expansion if one uses the non-perturbative approach). If we thinkof g = (g1, .., gN ) as the coordinates of a particle in N -dimensional space, t as time, andβi(g1, .., gN ) as a position dependent velocity field. The RG equations then describe how theparticle flows through the space of coupling constants.

If βi(g∗) = 0 ∀i, then g∗ is known as a fixed point. If g∗ is such that the ”particle” moves

towards that point as we integrate down towards k = 0, the fixed point is known as stableor attractive. Thus, in order to find the asymptotic behavior of a QFT, one basically has tofind the attractive fixed points of the renormalization group flow.

There is one particular fixed point worth mentioning, namely that of g∗ = 0. This is knownas a free or Gaussian fixed point since here the field theory is free. As the couplings are smallin the vicinity of such a fixed point, we can do perturbation theory there. Thus, fixed pointsin the vicinity of the Gaussian one are in this aspect easier to deal with, and it is usuallyfixed points of this kind that are considered when treating the renormalization procedure ina perturbative manner, see e.g. [20, 21] for examples.

The Gaussian fixed point is often called an ultraviolet (UV) fixed point because one usu-ally starts close to it when integrating down from the cutoff Λ to lower energies (particularlyin the perturbative formalism). As explained above, the higher order non-renormalizable cou-plings can be thought of as effective couplings derived from some fundamental high-energy


renormalizable theory. This gives relations between the renormalizable couplings and the non-renormalizable ones giving a hyperplane in the infinite-dimensional coupling constant spaceof dimension equal to the number of renormalizable couplings (two in the case of φ4-theoryin four spacetime dimensions). Starting with a renormalizable theory where all irrelevantcouplings are set to zero at the cutoff Λ, we may integrate down from Λ to t → −∞. The”particle” will then approach a position on this plane depending on the initial values of therenormalizable bare couplings. We arrive at a so-called infrared (IR) fixed point, a point onthis ”fixed plane” if you will (this is explained in greater detail in [22]). We shall use thisprocedure later in the thesis.

1.3.3 Critical points

In addition to being coupling dependent, the RG equations also depend on external parameterssuch as the temperature T or chemical potentials µ. If we also impose a criticality conditionon the couplings, for instance that they should give the potential at a phase transition, thiswill outline a ”critical submanifold” in the hyperplane of one less dimension. The problem isthen to tune the external parameters so that when we integrate down from the starting point,we arrive at this lower dimensional critical submanifold of the hyperplane. See Fig. 1.2 fora graphical description. If the model discussed is the O(N)-model in four dimensions, and

Fig. 1.2: Starting at k = Λ and integrating down to the hyperplane of dimension equal to the numberof renormalizable couplings. Three cases are shown. One where the temperature is belowcritical, T < Tc, above critical, T > Tc, and critical Tc where we end up at the criticalsubmanifold.

the critical condition is a phase transition from a symmetric to a non-symmetric phase (seeSection 3.3.3), then the point we arrive at on this now one-dimensional submanifold is known


as the Wilson-Fisher (WF) IR fixed point [23] when the temperature is critical. See Chapter5 for a more detailed discussion of this.

We proceed in the next chapters to study the NPRG in greater detail, deriving the RGequation for both zero and finite temperature, and solving these numerically. Before we dothis, however, we need to introduce some more thermodynamical quantities. We will laterderive RG equations for some of these quantities.


2. QUANTITIES OF TFT

In this chapter, we expand upon the concept of TFT introduced in the last chapter, andwe define different quantities appearing therein. We consider only bosonic theories in thischapter. The treatment of fermions is quite similar, and will be introduced when needed, seeChapter 9. We shall introduce the main thermodynamic quantity of this thesis, namely theeffective action Γ[M ] (M = 〈φ〉) and briefly consider how this is calculated perturbativelybefore moving on to the more general concept, that is the non-perturbative renormalizationgroup in the next chapter.

2.1 Thermodynamic quantities

We recall the canonical partition function of TFT

Z[J ] =

∫Dφp exp

[−∫ β

0dτ

∫x(L − Jφ)

], (2.1)

where the subscript p again denotes the periodicity of the fields, φ(x, 0) = φ(x, β), L = L(φ)denotes the Euclidean Lagrangian, and J is an external source field. From now on we assumethat all quantities are given in Euclidean spacetime unless otherwise is stated.

From Z[J ], we can calculate all equilibrium thermal quantities of the theory. The parti-tion function is therefore the most fundamental quantity of TFT. We turn to see how some ofthese quantities are calculated, working for the most part with a scalar one-component fieldφ.

2.1.1 Thermal averages

We are often interested in thermal averages of different quantities, that is, given an operatorO, we would like to calculate its time-ordered thermal average 〈TO〉. As in QFT, it can beshown that this is given by [24, 25]

〈TO(φ)〉 =1

Z[J ]

∫DφpO(φ)e−

∫ β0 dτ

∫x(L(φ)−Jφ). (2.2)

Note that we here have generalized the average to the case of J 6= 0. In the following weassume time ordering and neglect T in our correlation functions. We will also neglect thep denoting the periodicity condition remembering that, for nonzero temperature, the pathintegral includes this periodicity condition.

One important average is the thermal correlation function (also known as the thermal prop-agator) G(x, y) = 〈φxφy〉. This is used to measure correlations of the fields for various

2.1. THERMODYNAMIC QUANTITIES 28

space-time positions x and y. If we assume that our theory is translational invariant, thismeans that G(x, y) can only depend on the difference x − y, i.e. G(x, y) = G(x − y). Notethat if there is no correlation between the fields we would get G(x − y) = 〈φx〉〈φy〉, whichis different from zero. One usually wants uncorrelated fields to have a correlation functionequal to zero. This is dealt with by defining the connected thermal correlation functionGc(x − y) = G(x − y) − 〈φx〉〈φy〉 which measures only the correlation between the points xand y.

2.1.2 Helmholtz-free energy

An important quantity of TFT is the Helmholtz free energy F [J ]. This is defined by addingan external field J to the theory, coupled to the field φ. The Helmholtz free energy is thendefined as

Z[J ] = e−F [J ] =

∫Dφ exp

(−∫x(L(φ)− Jφ)

), (2.3)

where we again have abbreviated∫ β

0 dτ∫x simply as

∫x. F [J ] is often called the generating

functional of the connected correlation functions as these are obtained by acting on F [J ]by various factors of δ

δJ . The connected thermal propagator Gc(x − y) defined above is forinstance given by

Gc(x− y) = − δ

δJy

δ

δJxF [J ]

=1

Z[J ]

∫Dφ(φxφy)e

−∫x(L−Jφ) +

1

Z[J ]2

∫Dφ(φx)e−

∫x(L−Jφ)

∫Dφ(φy)e

−∫x(L−Jφ)

= 〈φxφy〉 − 〈φx〉〈φy〉 (2.4)

which is how we defined the connected correlation function above.

The reason why Gc(x − y) is called connected is because of the fact that only connecteddiagrams contribute to it. There are two contributions to G(x− y)

G(x− y) = x y + x y, (2.5)

where the gray circles denote sums of connected Feynman diagrams. Subtracting 〈φx〉〈φy〉simply means subtracting the last contribution of (2.5). This leaves only the first connectedpart.

In general, it can be shown that the n’th connected correlation function is [1]

Gnc (x1, .., xn) = − δnF [J ]

δJx1 · · · δJxn= 〈φx1 · · ·φxn〉c, (2.6)

where the c again means that this is made up of connected diagrams.

2.1.3 Thermodynamic potential

One is also often interested in a quantity known as the thermodynamic potential Ω. This isdefined as

Ω =1

V βF [0], (2.7)

2.1. THERMODYNAMIC QUANTITIES 29

where V is the spatial volume. Ω is often taken as the quantity from which other quantitiesare calculated. For instance, the average energy density E

E =〈U〉V

= − 1

V

∂

∂βlogZ = Ω + TS (2.8)

where S = −∂Ω∂T is known as the entropy density.

We calculate Ω in the case of scalar free field theory. We recall that the action then reads

S[φ] =

∫ β

0dτ

∫x

[1

2(∇φ)2 +

1

2m2φ2

]. (2.9)

We assume a finite volume, and introduce periodic boundary conditions on V . The Fouriertransformed field in the finite temperature regime then reads

φn,p =1√V β

∫ β

0dτ

∫xφ(x, τ)e−i(ωnτ+p·x), (2.10)

where ωn = 2πnT are the so-called Matsubara frequencies. With this, the fields become

φ(τ,x) =1√V β

∑n,p

φn,pei(ωnτ+p·x), (2.11)

where we get a sum over momenta due to the finite volume V . Using this, the action may bewritten as

S =1

2

∑n,p

φn,p(ω2n + p2 +m2)φ−n,−p. (2.12)

In order for the field φ to be real, one must choose φn,p = φ−n,−p. We find the partitionfunction to be

Z =

∫Dφ exp

(− 1

2

∑n,p

φn,p(ω2n + p2 +m2)φn,p

). (2.13)

If we denote ω2n +p2 +m2 by An,p, Z is can be easily calculated using Eq. (1.43) and we get

Z =1√A

= exp(− 1

2Tr logA

). (2.14)

As the matrix A is diagonal (see Eq. (2.12)), we get

Tr logA =∑n,p

log (ω2n + p2 +m2). (2.15)

We now write∑

p = V∫p, i.e. we take the limit V →∞ in order to obtain

Ω =T

2

∑n

∫p

log (ω2n + p2 +m2). (2.16)

We use Eq. (A.7) which states how the Matsubara sum is done for the differentiated integrandof Eq. (2.16) (differentiated with respect to ω2 = p2 + m2), i.e. we simply need to integratethe RHS of (A.7) in order to calculate the Matsubara sum over n in Eq. (2.16). Doing this,we finally arrive at

Ω =

∫p

(1

2ω + T log (1− e−βω)

). (2.17)

Note the contribution∫p

12ω to the integral. This is the vacuum energy density.

2.2. GIBBS FREE ENERGY 30

2.2 Gibbs free energy

There is another important thermodynamical quantity known as the effective average action,or Gibbs free energy Γ[M ], where

Mx = 〈φx〉 = − δ

δJxF [J ] (2.18)

is the average of the fields. This is the quantity we shall mainly be concerned with in thisthesis. The reason we are more interested in this is because of the easier treatment of Γ[M ]in the NPRG. See the next chapter. As Γ[M ] is the main thermodynamical object of thisthesis, we devote an entire section to it.

2.2.1 Definition

The Gibbs free energy is defined by the Legendre transformation

Γ[M ]− F [J ] =

∫xJM (2.19)

(for both T = 0 and T > 0, as∫x have different definitions for the different cases, as explained

earlier). We compute

δ

δMxΓ[M ] = Jx +

∫y

δJyδMx

My +

∫y

δF [J ]

δJy

δJyδMx

= Jx (2.20)

where we have used the definition of Mx, Eq. (2.18). This shows that, for zero external fieldJ = 0, the effective action satisfies

δ

δMxΓ[M ] = 0. (2.21)

This is the equation that gives the stable mean field solutions Mx of the theory, i.e. it saysthat with zero external field the possible mean fields are at the extrema of the effective action.It is analogous to the Euler-Lagrange equations derived from the classical Lagrangian L(φ)except that it is an equation for the mean field M = 〈φ〉, and thus all quantum (and thermalfluctuations if T > 0) have been included. This explains why Γ[M ] is called the effectiveaction, as it resembles the classical Euler-Lagrange equations

δS[φ]

δφx= 0. (2.22)

If we assumes the field Mx to be invariant under spatial and temporal translations, we maywrite the effective action as Γ[M ] = OU(M) or Γ[M ] = V βU(M) depending on whether wehave set T > 0 or T = 0. Here O denotes the spacetime volume, and we have defined theeffective potential U(M). Eq. (2.21) then becomes

∂

∂MU(M) = 0, (2.23)

giving the translational invariant stable states for zero external field. The bulk of this thesiswill be concerned with deriving the effective potential U(M) for various models, and findingthe physical states Mp that satisfy Eq. (2.23) using a non-perturbative approach.


2.2.2 1PI generating functional

Recall that F [J ] is the generating functional for the connected correlation functions. Onemay ask if Γ[M ] is a generating functional for any such quantities? The answer is yes. Tofind which ones, consider Eq. (2.20). This gives

δ(x− y) =δ2Γ[M ]

δJyδMx=

∫z

δMz

δJy

δ2Γ[M ]

δMzδMx= −

∫z

δ2F [J ]

δJyδJz

δ2Γ[M ]

δMzδMx, (2.24)

where we in the second equality have used the chain rule, and the third equality is therealization that Mx = 〈φx〉 = 〈φx〉c, since only connected diagrams contribute to 〈φx〉. Hence,we find

δ2Γ[M ]

δMyδMx= −

(δ2F [J ]

δJyδJx

)−1

= G−1c (x− y) (2.25)

in the operator sense.

In a similar fashion we look at

G3c(x, y, z) = − δ3F [J ]

δJzδJyδJx=

δ

δJz

(δ2Γ[M ]

δMyδMx

)−1

=

∫wGc(z − w)

∂

∂Mw

(δ2Γ[M ]

δMyδMx

)−1

=

∫u,v,w

Gc(x− u)Gc(y − v)Gc(z − w)∂3Γ[M ]

∂Mu∂Mv∂Mw. (2.26)

In terms of Feynman diagrams this reads

= . (2.27)

The LHS of this equation represents the full three-legged correlation function. The graycircles of the RHS represent full connected propagators while the white circle represents thecontribution from the Gibbs free energy, i.e.

Γ(3)(u, v, w) =∂3Γ[M ]

∂Mu∂Mv∂Mw. (2.28)

But this is precisely the contribution from the one-particle irreducible diagrams. We thusobtain

Γ(3)(x, y, z) = 〈φxφyφz〉1PI . (2.29)

This can be shown to hold more generally as well, see e.g. [1], i.e. for n ≥ 3 we have

Γ(n)(x1, .., xn) = 〈φx1 · · ·φxn〉1PI , (2.30)

where Γ(n)(x1, .., xn) is defined in the obvious way. Thus Γ[M ] is the generating functionalfor the 1PI-correlation functions. Note that also Γ2(x, y) = G−1

c (x, y) is also made up of1PI diagrams, since the inverse propagator consists only of 1PI diagrams as shown in theintroduction.


2.2.3 Computing Γ[M ] in the perturbative framework

Before moving on to see how one computes Γ[M ] in the NPRG formalism, we first considerhow this is done perturbatively. We will only go so far as to consider the first perturbativecorrection to Γ[M ] from the classical action evaluated at mean field, S[M ]. At the lowestorder of perturbation, the so-called one-loop level, we have

∂L∂φx

∣∣∣φx=Mx

− Jx ≈ 0, (2.31)

giving a relation between Jx and Mx. We now write L = L(M)+δL. With this, the partitionfunction may be written as

Z[J ] = e−F [J ] =

∫Dφe−

∫x(L(φ)−Jφ) =

∫Dφe−

∫x(L(M)−JM)e−

∫x δL. (2.32)

We want to find δL to first order of perturbation. We expand the field around its mean valueso that φx = Mx+ηx. The exponential of the first integral in (2.32) may be expanded to firstnon-vanishing order in order to obtain∫

x(L − Jφ) ≈

∫x

(L(M)− JM

)+

1

2

∫x

∫yηx

δ2Lδφxδφy

|φ=Mηy. (2.33)

This then gives

δL =1

2

∫x

∫yηx

δ2Lδφxδφy

|φ=Mηy. (2.34)

Eq. (2.32) can then be evaluated to give

e−F [J ] =

∫Dη exp

(−∫x

(L(M)− JM

)− 1

2

∫x

∫yηxδ2L(M)

δMxδMyηy

)= exp

(−∫x

(L(M)− JM

))exp

[− 1

2Tr log

(δ2L(M)

δM2

)], (2.35)

where we have used Eq. (2.14) again, writing the result more formally this time. Thus, weobtain the one-loop effective action

Γ[M ] = S[M ] +1

2

∫q

log( δ2L(M)

δMqδM−q

), (2.36)

where∫q = T

∑n

∫q when T > 0 as explained in Appendix A. If M is a multicomponent field,

i.e. M = (M1, ..,MN ), the result becomes

Γ[M ] = S[M ] +1

2Tr

∫q

log( δ2L(M)

δMi,qδMj,−q

), (2.37)

where the trace now is over the matrix components i, j.

2.3. GRAND CANONICAL PARTITION FUNCTION 33

2.3 Grand canonical partition function

In the above, we have been concerned with systems in equilibrium that were unable to ex-change particles or charge with the surroundings. If we wish to allow for this, we must usethe grand canonical partition function instead of the canonical one. We investigate this a bitfurther.

We return for the moment to Minkowski space. Recall that the Noether current jµi satis-fies

∂µjµi = ∂0j0 −∇ · j = 0. (2.38)

If we define the i’th charge as

Qi =

∫d3xji,0 =

∫xji,0, (2.39)

we thus getdQidt

=

∫x∇ · ji = 0, (2.40)

where the last equality follows from Stokes’ theorem. Hence Qi is conserved in time.

There is one charge in particular worth mentioning. Recall that, for the Lagrangian (inMinkowski space)

L = ∂µΦ∗∂µΦ− U(Φ∗Φ), (2.41)

the symmetry transformation Φ → eiαΦ (and Φ∗ → e−iαΦ∗) gave rise to a Noether currentwhich was the electromagnetic current jµem. Thus, the conserved charge for this U(1) symme-try is exactly the electromagnetic charge Qem.

Whenever such conserved charges exist, and one wants to allow the system to exchangethese charges with the surroundings, statistical mechanics [26] says that one should use thegrand canonical partition function

Z = Tr(e−β(H−µiQi)

), (2.42)

where we associate a chemical potential to each conserved charge. The easiest way to incor-porate such charges in the theory is to let H → H−µiρi, where ρi denotes the charge densityassociated with the charge Qi and H is the Hamiltonian density (from here on referred to asthe Hamiltonian). We may then use this to derive the Lagrangian, now in Minkowski space,by the Legendre transformation

H(π, φ) + L(φ, ∂µφ) = πi∂0φi, (2.43)

where φ = (φ1, .., φN ), and where

πi =∂L

∂(∂0φi)(2.44)

are the canonical conjugate momenta of the fields φi.

It should be noted that the inclusion of a chemical potential will break Lorentz invariance

2.3. GRAND CANONICAL PARTITION FUNCTION 34

of the theory. This is as it should be since the presence of a charge density ρ of some sortwill always break the Lorentz invariance. The charge density is different in different referenceframes, i.e. Lorentz transforming will change ρ.

Including chemical potentials also allows us to study Bose-Einstein condensation. This is thetheory behind both charged pion condensation and kaon condensation that we shall study indetail later.

Now that we know how the most important quantities of TFT are defined, we turn to seehow they are renormalized.

3. THE NON-PERTURBATIVE RENORMALIZATION GROUP

In this chapter, we study in detail the theory behind the NPRG. There are two main formula-tions. The first is the Wilson-Polchinski (WP) formulation [27, 28]. This is the most famousone, and perhaps the easiest one to grasp conceptually. In return it is difficult to implementinto models, as the objects one deals with are of a very abstract nature. The second one isthe effective average action (EAA) formalism [29]. Though this is the formalism mainly usedin this thesis, we shall, for completeness, consider both approaches. We start with the WPformalism.

We work in the context of thermal field theory when deriving our results. We mostly follow[22] in the derivation of our equations. We also note that we work in Euclidean spacetime forthe remainder of the thesis. Should we return to Minkowski space at some point, this will beexplicitly stated. We also set T = 0 for the moment.

3.1 The WP formalism

We start with the NPRG in the WP formalism. This was the first formalism for derivingexact renormalization group equations (ERGE) and, as stated above, might be the easiestone to grasp conceptually. We move on to see how it works.

3.1.1 Starting point

We mainly follow [30] in the derivation of the WP equation. Start by considering the partitionfunction

Z =

∫Dφ exp

(− S0[φ]− 1

2

∫x,yφxC

−1Λ (x− y)φy +

∫xJxφx

), (3.1)

where C−1Λ (x − y) denotes the inverse propagator, and where we have introduced a scale Λ

as a relevant cutoff. This cutoff corresponds to the microscopic physics of which Z is aneffective field theory. Λ is usually taken to be of the order of the inverse of the defining lengthparameter of the microscopic theory, such as the lattice spacing in an Ising model, or theCompton wavelength of the particles in the case of particle physics. The energy scales of theeffective model should thus be less than Λ in order for the theory to have validity as an effec-tive description of the microscopic theory. This approach is used in both high-energy physics,like elementary particle physics, and low energy physics, like condensed matter physics.

Note that we have pulled the propagator term out of the action, leaving only interactionterms in S[φ]. We have also made the propagator dependent on the cutoff Λ. We write theeffective propagator as

CΛ(p) = (1− θε(|p| − Λ))C(p), (3.2)

3.1. THE WP FORMALISM 36

Fig. 3.1: The smoothed out Heaviside function, smoothed out over an interval of length 2ε

where C(p) is the usual propagator

C(p) =1

p2 +m2=

1

p20 + p2 +m2

(3.3)

in momentum space. Here θε is a Heaviside function smoothed out over an interval of size 2ε:

θε(x) =

0 if x < −ε,12 + x

2ε + 12π sin(πxε ) if −ε < x < ε,

1 if ε < x,

(3.4)

see Fig. 3.1. If we choose ε = 0 we see that θε(x)→ θ(x), the regular Heaviside function, andthe quadratic propagator term becomes sharply cut off at the scale UV-scale Λ, i.e.

1

2

∫|p|<Λ

φ(−p)(p2 +m2)φ(p). (3.5)

This is the kind of cutoff we discussed in the introduction, where the loop integrals were cutoff at a scale Λ.

3.1.2 Block-spin mechanism

We now wish to implement the so-called block-spin mechanism into our field theory. The ideabehind this is to separate the field into ”rapid” and ”slow” modes. This is done with respectto an energy scale k, a one-parameter family used to describe the scale of the renormalizationgroup. That is, k denotes the energy scale we consider the system at, a sort of focus of our”renormalization glasses” if you will. We define

φp = φp<k + φp>k, (3.6)

where φp = φ(p) as explained in Appendix A. Here φp<k denote the modes of momentumless than k and φp>k denote the modes of momentum larger than k. Associate these to

3.1. THE WP FORMALISM 37

propagators

φp → CΛ(p)

φp<k → Ck(p) = C<(p)

φp>k → CΛ(p)− Ck(p) = C>(p). (3.7)

Note that C>(p) is nonzero only for momenta larger than k and less than Λ. This is thuscalled a UV regulator.

There is a mathematical relation∫ ∞−∞

dxe−x

2

2γ−V (x) ∼

∫ ∞−∞

dy

∫ ∞−∞

dze− y

2

2α− z

2

2β−V (y+z)

, (3.8)

where γ = α+β. This relation can easily be generalized to functional integrals by consideringan N -dimensional Gaussian integral and taking the limit N →∞. Doing this yields

Z[J ] =

∫Dφ exp

(− 1

2

∫x,yφ(x)C−1

Λ (x− y)φ(y)− S[φ])

∼∫Dφ<Dφ> exp

(− 1

2

∫x,yφ<(x)C−1

> (x− y)φ<(y)

− 1

2

∫x,yφ>(x)C−1

> (x− y)φ>(y)− S[φ< + φ>] +

∫xJ(φ< + φ>)

), (3.9)

where we now have let φp<k = φ< and similarly φp>k = φ>. Since we do not care aboutmultiplicative constants, we may use this as our new partition function.

In order to simplify the writing, we now introduce the so-called DeWitt notation, see e.g.[31] page 361. In this notation, integrations become dot products and we neglect dependenceon spacetime coordinates. Thus, for example, the above kinetic term becomes

1

2

∫x,yφxC

−1Λ (x− y)φy =

1

2φ · C−1

Λ · φ, (3.10)

a kind of Einstein summation convention for spacetime integrals if you like. With this nota-tion, the new partition function becomes

Z[J ] =

∫Dφ<Dφ> exp

(− 1

2φ< · C−1

< · φ< −1

2φ> · C−1

> · φ>

− S[φ< + φ>] + J · (φ< + φ>)). (3.11)

This can now be written as

Z[J ] =

∫Dφ< exp

(− 1

2φ< · C−1

< · φ<)Zk[J ], (3.12)

where we have defined

Zk[J ] =

∫Dφ> exp

(− 1

2φ> · C−1

> φ> − S[φ> + φ<] + J · (φ> + φ<))

= exp

(− S

[ δδJ

]+ J · φ<

)∫Dφ> exp

(− 1

2φ> · C−1

> · φ> + J · φ>), (3.13)

3.2. EAA FORMALISM 38

as an effective partition function where the rapid modes have been integrated out. We nowperform the integral over φ>, using Eq. (1.43), in order to get

Zk[J ] = exp

(1

2Tr logC> − S

[ δδJ

]+ J · φ< +

1

2J · C> · J

)= exp

(1

2Tr logC> + J · φ< +

1

2J · C> · J

)× exp

(− 1

2Φ · C−1

> · Φ)

exp

(− S

[C> ·

δ

δΦ

])exp

(1

2Φ · C−1

> · Φ), (3.14)

where we have defined Φ = J ·C> +φ<. We see that we can write the last three exponentialsin the above expression as exp

(− Sk[Φ]

), for some Sk[Φ]. We thus end up with

Zk[J ] = exp(1

2Tr logC> +

1

2J · C> · J + J · φ< − Sk[C> · J + φ<]

). (3.15)

Sk[Φ] can be thought of as a kind of effective action, formally incorporating the effects ofhigher energies above k into the theory at scale k. Returning to the metaphor of renormal-ization glasses, we see that this is best described by the right IR lens of the renormalizationglasses depicted on the frontpage, where the higher energy fluctuations have been blurred outinto a wider but smoother curve. We now turn to find an RG equation for Sk[Φ].

From the definition of Zk[J ], Eq. (3.13), we see that

∂kZk[J ] = −1

2

( δ

δJ− φ<

)· ∂kC−1

> ·( δ

δJ− φ<

)Zk[J ]. (3.16)

We now substitute Eq. (3.15) into this and, after a bit of algebra, we finally arrive at

∂kSk[Φ] = −1

2

∫x,y∂kC>(x− y)

[δSk[Φ]

δΦ(x)

δSk[Φ]

δΦ(y)− δ2Sk[Φ]

δΦ(x)δΦ(y)

], (3.17)

where we have returned to the usual notation. This is the WP equation. Note the appearanceof the UV regulator C>(x− y) in the equation. This is an equation for the abstract quantitySk[Φ], the interpretation of which is not very well known, and it is thus hard to deal with.This is the reason for the little practical use of the WP formalism. We shall, in the following,turn to find an ERGE for a less abstract object, namely that of the effective action. Theequation in this formalism is also a bit simpler to derive, and is done in Appendix A.

3.2 EAA formalism

At the end of the 1980s Wetterich proposed an alternative, but equivalent formalism to that ofWilson and Polchinski known as the EAA formalism [29]. Instead of integrating out the rapidmodes to obtain an effective action for the slower modes as is done in the Wilson formalism,one rather computes the effective action Γk[M ] (M = 〈φ〉) of the rapid modes φp>k that havealready been integrated out. This is best described by the left UV lens of the renormaliza-tion glasses depicted on the frontpage, where we have removed all low-energy fluctuations,considering only the high-energy UV modes.

We consider two special cases for k:


• When k = Λ, no modes have been integrated out. Γk[M ] should then be equal tothe classical (Euclidean) action used to describe the microscopic physics of the modelevaluated at mean field, i.e.

ΓΛ[M ] = S[φ = M ]. (3.18)

The microscopic model is usually assumed to be a renormalizable theory.

• When k is lowered to k = 0 and all modes are integrated out, Γk=0[M ] should be thefull effective action of the original model

Γ0[M ] = Γ[M ], (3.19)

i.e. all quantum and thermal fluctuations have been included.

3.2.1 IR regulator

We move on to the problem of finding a one-parameter family of effective actions Γk[M ] la-beled by the renormalization scale k. We wish to decouple the slow modes of the partitionfunction. This can be done by giving them a large mass. A large mass corresponds to a smallCompton wavelength, and thus a small ”interaction length” for the quantum fluctuations.The particles decouple in a sense at low energy, since they contribute only to highly virtualprocesses. These are suppressed by inverse powers of the masses of the heavy particles. Theparticles propagate a distance 1

m , where m is the particle mass.

We therefore use the partition function

Zk[J ] =

∫Dφe−S[φ]−∆Sk[Φ]+

∫x Jφ, (3.20)

where J is an external source, S[φ] is the Euclidean action of the model, and

∆Sk[Φ] =1

2

∫x,yφxRk(x− y)φy =

1

2

∫qRk,qφqφ−q. (3.21)

The function Rk(q) must be defined such that R0(q) = 0 ∀q. We thus recover the full partitionfunction Zk=0[J ] = Z[J ] containing all quantum and thermal fluctuations, when all the modeshave been integrated out. On the other hand, when k = Λ, no modes have been integratedout and all the modes are given a large mass of the order of magnitude of Λ, in order to freezethem out (decouple them). It is therefore instructive to define Rk(q) such that RΛ(q) = ∞,∀q. When 0 < k < Λ Rk(q) must be large for the modes of momentum less than k, and smallfor the modes greater then k. This is a so-called IR regulator, or cutoff-function, opposite tothe UV regulator used in the WP formalism. Fig. 3.2 shows the typical behavior of such acutoff-function. Note also that Rk(q) = Rk(q

2), i.e. the regulator is rotation invariant.

3.2.2 The effective action Γk[M ]

We now proceed to find Γk[M ] in the EAA formalism. We first recall that Helmholtz freeenergy is defined as

Fk[J ] = − logZk[J ], (3.22)


Fig. 3.2: An example of a cutoff-function which may be used in the EAA approach.

see the last chapter or e.g. [25, 24], and the usual Gibbs free energy Γk[M ] is defined by theLegendre transform

Γk[M ]− Fk[J ] =

∫xJxMx, (3.23)

where

Mx = −δFkδJx

= 〈φx〉. (3.24)

In the limit where k → 0 we see that, since R0 = 0, the Gibbs free energy approaches the fullGibbs free energy of the system, where all fluctuations have been included. With the definitiongiven as in Eq. (3.23) however, the Gibbs free energy is not equal to the microscopic actionS[φ] in the limit k → Λ. We must modify the definition in order to achieve this. We thusredefine the Gibbs free energy by the Legendre transform

Γk[M ] + Fk[J ] =

∫xJM − 1

2

∫qRk,qMqM−q. (3.25)

With this, we see that∫Dφe−S[φ]−∆Sk[Φ]+

∫Jφ = e−Γk[M ]−∆Sk[M ]+

∫JM . (3.26)

This gives∫Dφe−S[φ]+

∫Jφ exp

[− 1

2

∫q(φq −Mq)Rk,q(φ−q −M−q)

]= e−Γk[M ]+

∫x JM . (3.27)

When we take the limit k → Λ, the exponential will behave as a functional Dirac delta-function δ(φ−M) aside from maybe an irrelevant constant. We thus obtain∫

Dφe−S[φ]+∫Jφδ(φ−M) = e−Γk[M ]+

∫JM . (3.28)

In this limit, using the definition (3.25) of Γk[M ], we therefore get ΓΛ[M ] = S[M ] as weshould.

3.3. RG EQUATION FOR ΓK [M ] 41

3.3 RG equation for Γk[M ]

We want to find the RG equation for Γk[M ]. This is sometimes called the Wetterich equation[29], and is an equation of the form ∂kΓk[M ] = f(Γk[M ]). The equation is derived in AppendixA, Eq. (A.21) and reads

∂kΓk[M ] =1

2

∫x

∫y∂kRk(x− y)

[Γ(2)[M ](x, y) +Rk(x− y)

]−1, (3.29)

where the −1 means the operator inverse, and

Γ(2)k [M ](x, y) =

δ2Γk[M ]

δMxδMy. (3.30)

Of course, it is impossible to find a general solution to Eq. (3.29), and we therefore need tomake approximations.

3.3.1 Approximating Γk[M]

One common approximation procedure is called the local potential approximation (LPA). Werecall that when integrating down from k = Λ, non-renormalizable terms will be generated,thus the effective action Γk[M ] will in general contain all powers of M and derivatives ofM compatible with the symmetries of the model. We can however expand it in a derivativeexpansion, and truncate this expansion at some order. For the derivative expansion in theLPA, a slowly varying field Mx is assumed, and we thus write the ansatz

Γk[M ] =

∫x

(Uk(Mx) +

1

2Zk(∇Mx)2

), (3.31)

where Zk is the wave-function renormalization (not to be confused with the partition func-tion). Note that we have set all higher order kinetic terms to zero in this expansion (terms thatinvolve derivatives of the fields). Zk is also set to 1 in the LPA, i.e. we neglect wavefunctionrenormalization. The action in the LPA thus reads

ΓLPAk [M ] =

∫x

(Uk(M) +

1

2(∇M)2

). (3.32)

If we set UΛ(M) equal to the classical potential, it is clear that this reduces to the classicalEuclidean action for a one-component scalar field at the cutoff as it should. We neglect thesuperscript LPA in the following.

We could also expand the effective potential Uk(M) to some order. If we assume the po-tential to be Z2 symmetric and expand it to order four, we obtain

Γk[M ] =

∫x

(g2,kM

2x + g4,kM

4x +

1

2(∇Mx)2

). (3.33)

The RG equation then becomes a set of coupled differential equations for the coupling con-stants g2,k and g4,k. We obtain RG equations for the couplings. These are similar to the RGequations obtained using perturbative renormalization as discussed in the introduction.


3.3.2 The RG equation for Uk(M)

We now derive the RG equation for the effective potential Uk(M). This is most simplydone by defining it as Γk[M ], computed for uniform field configurations as described in theintroduction. That is

Uk(Muni) =1

OΓk[Muni] (3.34)

where O is the spacetime volume. With this definition, we obtain

∂kUk(Muni) =1

2O

∫q∂kRk,q

(Γ

(2)k|Muni

+Rk)−1

q,−q, (3.35)

where we have gone to momentum space. We calculate Γ(2)k,x,y[M ] for uniform field configura-

tions in the LPA. In general, we have

Γ(2)k,x,y[M ] =

δ2

δMxδMy

∫z

[Uk(Mz) +

1

2(∇Mz)

2

]

=

∫z

[∂2Uk∂M2

δ(z − y)δ(z − x) +∇δ(z − y)∇δ(z − x)

]. (3.36)

We Fourier transform the integrand in order to obtain

Γ(2)k,x,y[M ] =

∫z

∫q,q′

[∂2Uk∂M2

− qq′]eiq(z−y)+iq′(z−x)

= F

[(2π)dδ(q + q′)

(∂2Uk∂M2

+ q2)], (3.37)

where F denotes the double Fourier transform. We thus find

Γ(2)k,q,q′ [M ]

∣∣∣Muni

= (2π)dδ(q + q′)

[∂2Uk∂M2

(Muni) + q2

]. (3.38)

If we also doubly Fourier transform Rk(x− y), we get

R(q, q′) =

∫x,yRk(x− y)e−i(qx+q′y)

=

∫x,yRk(x− y)e−i(q(x−y)+y(q′+q))

= Rk(q)(2π)dδ(q + q′), (3.39)

where Rk(q) is the single Fourier transform of Rk(x− y).

We recall the general operator identity∫zF (x, z)F−1(y, z) = δ(x− y). (3.40)

The corresponding identity in momentum space reads∫qF (p, q)F−1(q, p′) = (2π)dδ(p− p′). (3.41)


where F (q, q′) now is the Fourier transform of F (x, y). Letting F−1 =[Γk + Rk

]−1, this

implies∫q

[(2π)dδ(p+ q)

(∂2Uk∂M2

+ p2

)+ (2π)dδ(p+ q)Rk(p)

]F−1(q, p′) = (2π)dδ(p− p′), (3.42)

where we have dropped the subscript uni. Solving for F−1(p, p′) we find

F−1(p, p′) =δ(p+ p′)(2π)d(

R(p) + p2 + ∂2Uk∂M2

) . (3.43)

Noting that δ(0) = (2π)−dO, we finally arrive at

∂kUk(ρ) =1

2

∫q

∂kRk(q)

Rk(q) + q2 + ∂2Uk∂M2

. (3.44)

If we assume the potential to be Z2 symmetric, i.e. Uk(M) = Uk(M2) (we shall later let this

Z2 symmetry go to a more general O(N) symmetry when we consider multicomponent fields),we may define ρ = 1

2M2. We thus get an RG equation for Uk(ρ)

∂kUk(ρ) =1

2

∫q

∂kRk(q)

Rk(q) + q2 + U ′k(ρ) + 2ρU ′′k (ρ). (3.45)

It is instructive to use a cutoff-function that makes it possible to calculate the integral inEq. (3.45) analytically. One such function is the sharp cutoff-function [22]

Rk(q2) = (k2 − q2)θ(k2 − q2), (3.46)

where θ denotes the Heaviside step function. With this choice of regulator, the calculation ofthe integral in Eq. (3.45) is easily performed and the result is

∂tUk(ρ) =4vdd

kd+2

k2 + U ′k(ρ) + 2ρU ′′k (ρ), (3.47)

where we again have introduced the renormalization time t = log( kΛ), and where vd is aconstant defined in Appendix A. All that can be learned about this Z2-invariant model atthis level of simplification is contained in the solution of this equation.

3.3.3 Spontaneous symmetry breaking and phase transitions

The potential Uk(ρ) describes the ”field landscape” at the renormalization scale k. Uk=0(ρ)gives the full effective potential. One is usually interested in the minimum (or minima) ofUk=0(ρ), as this gives the physical solution of the mean field M for which the field φ fluctuatesaround. Recall from the last chapter that the physical mean field solves

∂U(M)

∂M= 0, (3.48)

where U(M) = Uk=0(M). Thus φ→ 〈φ〉0 + φ = M0 + φ where 〈φ〉0 = M0 denotes this solu-tion, and φ is the fluctuating field. The solution of Eq. (3.48) is most often at a minimum.


If this minimum is different from zero, we have what is known as spontaneous symmetrybreaking (SSB), i.e. the Z2 symmetry is spontaneously broken and ρ0 > 0, where ρ0 = 1

2M20 ,

see Fig. 3.3(a). The system is then in a different phase than if ρ0 = 0. The phase of SSBis usually called the broken phase, while the phase of ρ0 = 0 is known as the symmetric phase.

More generally, whenever the potential is invariant under transformations of a symmetry-group G, but the minimum of the potential is not invariant under this group, G is said to bespontaneously broken. Acting on a minimum state M0 of the system by an element g ∈ Gproduces a new state gM0. This state is also a minimum state of the system, and is thusequivalent to, but different from, M0. For instance, we see that both M0 and −M0 both solveEq. (3.48) in the above Z2-symmetric case.

It should be noted that the spontaneously broken phase is usually a metastable phase and notthe true statistical minimum of the theory. In order to show what we mean by this, considerFig. 3.3(a). We see that there are two minimum states. Given enough time, large enoughquantum fluctuations may occur causing the system to cross from one minima to the other.When enough such crossings have occurred, the system will have spent an equal amount oftime in both minimum states. Thus, the true effective potential when t → ∞ should reallybe the one where no such crossings can occur, i.e. the one depicted in Fig. 3.3(b). More

(a) (b)

Fig. 3.3: To the left we have the effective potential for metastable states, allowing for SSB. To theright we have the true convex effective potential when t→∞.

generally, it is known as an exact result from statistical mechanics that the true effectivepotential is a convex function [32, 33, 34]. This means that the potential has a non-negativedouble derivative, i.e. U ′′(M) > 0 always. More generally, for multicomponent fields, it meansthat the Jacobian is non-negative, i.e. all the eigenvalues of the Jacobian are non-negative.However, as we for the most part of this thesis shall be concerned with SSB, we will neglectthis trait of the effective potential and only consider the effective potential for the metastablestates where the effective potential is not necessarily convex.

Turning on the temperature, that is including thermal fluctuations as well as quantum fluc-tuations in the theory, usually makes the minimum of U(ρ) tend towards zero. The momentwhen ρ0 = 0, we have a phase transition, and we may write down the critical temperature Tc.

Depending on how the minimum proceeds from ρ > 0 to ρ = 0 as one increases the tem-perature determines what kind of phase transition we are dealing with. If the transition is


discontinuous, the phase transition is of first order, see Fig. 3.4(a). If however the transitionis continuous, we have what is known as a second order, or continuous phase transition, seeFig. 3.4(b). It may also happen that the minimum of the potential approaches zero butnever quite reaches it. It may look like a second order phase transition, but it is not since thephase ρ0 = 0 is never reached. Such a case is known as a crossover, and is depicted in Fig. 3.5.

If the potential is a function of more than one field, more interesting things can happen.The minimum of the potential is free to move both continuously and discontinuously over thespace spanned by the fields giving rise to first- and second-order phase transitions of variousforms. We shall see examples of this later in the thesis.

(a) (b)

Fig. 3.4: We plot the potential at three different temperatures T < Tc, T = Tc and T > Tc, whereTc is the critical temperature, for both a first order (left) and a second order (right) phasetransition. Note in the case of the first order phase transition that the minimum of thepotential jumps discontinuously from ρ = 0 to ρ > 0, while for the second order transitionthis happens continuously.

Fig. 3.5: We plot ρ0 as a function of temperature in the case of a crossover. Note that as the temper-ature goes to infinity, ρ0 becomes smaller and smaller but never zero.


Part II

THE O(N) CASE


4. GENERALIZING TO O(N)

In this chapter, we generalize our theory from the Z2-symmetric one-parameter theory dis-cussed in the last chapter, to an O(N)-symmetric theory with a N -dimensional field vectorφ = (φ1, .., φN ). We derive the RG equation for the effective potential Uk(φ) in the LPA, andsolve this numerically. But before we do this, we discuss why we are interested in O(N) the-ories in the first place. It turns out that they neatly describe QCD under certain conditions.

4.1 QCD and O(4) effective theory

We briefly discuss the importance of O(N) models, and in particular the O(4) model, and howit can be applied as a low energy effective theory for chiral QCD describing massless quarks.In later chapters, we will also include a isospin chemical potential and leave the chiral limit.It is therefore important to have some background knowledge of what we are doing first.

4.1.1 Strong interaction

QCD is the theory of the strong interaction [35, 36]. These are the interactions that mainlygovern what happens in the nucleus of atoms at sub-atomic scales. Like QED, it is a gaugetheory and the fundamental particles are fermions and bosons. The massive interacting par-ticles of the theory are the fermions. These are known as quarks. Quarks are the constituentsof nucleons such as protons and neutrons. These particles are known as baryons (or anti-baryons) and they consist of bound states of three quarks (or anti-quarks) of various types.Quarks can also form mesons (such as the π-meson). A meson is a bound state of one quarkand one anti-quark.

In the old picture of the strong interaction, the mesons where thought of as the mediatorsof the force as described by Hideki Yukawa. In fact, simply from knowing the range of theinteraction, ∼ 10−15 m (radius of the nucleus of an atom), he could predict from the formula

l ≈ 1

m, (4.1)

where l is the interaction length and m is the mass of the mediator, that the mediating par-ticle of the strong interaction should have a mass of about 100 MeV. Later, when mesonswhere discovered, Yukawa got the Nobel Prize of 1949 [37].

The modern picture of the strong interaction is, as mentioned, that it is a gauge theorydescribed by QCD. We recall from the introduction that in a gauge theory the fermionicfields transform under a gauge-symmetry group, and in the case of QCD the gauge groupis SU(3). Remember that the number of generators of the gauge group is in a one-to-onecorrespondence with the number of massless mediators for the interaction. The number of

4.1. QCD AND O(4) EFFECTIVE THEORY 50

generators of SU(3) is eight, and as a result there are eight bosonic mediating fields. Thesefields are known as gluons. We may think of the gluons as ”gluing” quarks together to formhadrons, i.e. baryons and mesons. This is similar to QED, where the gauge group is U(1),and there is thus only one generator corresponding to one mediator, the photon.

There is a subtlety here. Since the electric and magnetic fields are invariant under globaltransformations of the gauge group U(1), see Eq. (1.68), the photon must be have zero elec-tric charge (recall that the charged matter fields came about from global transformations ofU(1)). This is not the case for the eight gluons. They transform under global transformationsof the gauge group SU(3), see Eq. (1.76), and as a result they carry the charge of the stronginteraction known as color. This is also the reason for some strange effects in QCD such asconfinement and asymptotic freedom [38, 39].

The quarks transform under the three dimensional fundamental representation of SU(3),and thus they can carry three different colors (red, green, blue). The gluons transform un-der the eight-dimensional adjoint representation of the group, and carry color differences(as they should, since they propagate between quarks, see Fig. 4.1). In fact, the gluons arein a one-to-one correspondence with the generators of SU(3) as explained in the introduction.

One could argue that there should be nine gluons instead of eight, as there are nine dif-ferent color differences when we have 3 colors. This would be right if the gauge group wasU(3), not SU(3). U(3) has nine generators, not eight. It is the extra condition of SU(3), i.e.

det(U) = 1, (4.2)

that gives a relation between the generators which in turn leaves only eight linear independentgenerators, or color differences if you will.

4.1.2 Chiral limit

If we assume the quarks to be massless, we are in the so-called chiral limit. It is called thechiral limit because massless particles have a well-defined chirality. A photon is either lefthanded or right handed. It is known that quarks come in different types known as flavors.There are six flavors: the up-quark u, the down-quark d, the strange-quark s, the charm-quarkc, the bottom-quark b and the top-quark t. These can be distinguished by mass, charge, mag-netic momentum, etc. However, if one neglects electromagnetic effects and go to the chirallimit, there is really nothing that distinguishes the flavors, and so symmetry transformationsexist between them.

QCD has the chiral flavor-symmetry group SUL(Nf ) × SUR(Nf ) where the L and the Rstand for left-handed and right handed chirality respectively, and Nf is the number of quarkflavors. The number of flavors is, as mentioned Nf = 6. However, if the energy scale is farlower than the mass of the four heaviest quarks (the s-, the c-, the b- and the t-quark), wemay disregard these flavors. This is because they then only show up in virtual processes, andFeynman-diagrams including their interactions are heavily suppressed by various factors oflarge inverse masses squared. In technical terms, we say that the heavy quarks decouple.


Fig. 4.1: An example of an interaction of QCD written with Feynman diagrams. The arrowed linesdenote fermions, in this case quarks and anti-quarks. The bar denotes the anti-quantity ofthe quantity that has the bar. Thus a fermion-line pointing in the opposite time directiondenotes an anti fermion. This also shows that the anti-quarks are in the anti-fundamentalrepresentation of SU(3). The curly line denotes the gluon propagator that carries a colordifference.

The energy scale we are interested in is that of the critical temperature Tc of the phasetransition from the hadronic phase, where the quarks are bound in baryons and mesons, tothe quark-gluon plasma phase. In the chiral limit, this has been calculated to be somewherebetween 150MeV and 200MeV, about the same order as the mass of the lightest heavy quark,the s-quark, which has a bare mass of about 70−130MeV. The s-quark thus has a mass belowthe energy scale we are interested in. However, since the bare masses of the u- and d-quarksare so much lighter, about 3 − 5MeV, these will be far more dominant in the interactions.We thus effectively set Nf = 2 as an approximation at these ”low” energies (we say low inquotation marks as Tc ≈ 1010K which is far above the temperature in the center of the sun forexample). The case Nf > 2 has also been studied. See for example [40] for an example of this.

The flavor symmetry group thus becomes SUL(2)×SUR(2). What is nice about this group isthat it is locally isomorphic to O(4) (in technical terms, they have the same Lie-algebra, andare then locally the same). As the terms in the Lagrangian are of local nature (some forms ofdensities) we may use an O(4) symmetric effective Lagrangian to study the system instead.This is what we have done in this thesis.

In the vacuum, the chiral symmetry gets spontaneously broken and a chiral condensate where〈ψψ〉 6= 0 is formed in the hadronic phase. We here use ψ to describe the quarks and ψ todescribe anti-quarks, and hence ψψ describe mesons. ψψ transforms under SUL(2)×SUR(2)in the same way φ transforms under O(4) due to the isomorphic structure of the groups.Hence this is what we model in O(4) theory by looking at a SSB O(4) → O(3) where


〈φ〉 = 〈0, 0, 0, φ4,0, 〉 6= 0, where we have used the O(4) symmetry to set φ1,0 = φ2,0 = φ3,0 = 0.

We will derive a potential for φ and find its minimum. If we have 〈φ〉 6= 0 at the min-imum, we are in the broken hadronic phase (〈〉 describing the minimum of the potential)while 〈φ〉 = 0 describes the symmetric phase (see Chapter 3). It is worth mentioning that,as discussed above, there are also baryon states, i.e. states consisting of three quarks in thehadronic phase, but we shall ignore these in this thesis.

We recall from the Goldstone theorem [41, 42] that whenever SSB occurs, we should geta massless Goldstone mode from every broken symmetry, i.e. one mode for every generator ofthe group. This theorem is not hard to prove. Recall from Section 2.3 that to every symmetrycorresponds a conserved charge Q. This charge is in a one-to-one correspondence with thegenerator of the symmetry, denoted by the same letter Q. We recall from regular quantummechanics that, if a quantity is conserved in time, we must have

[H,Q] = 0, (4.3)

where H is the Hamiltonian. Let |0〉 denote the vacuum state. We can always tune theHamiltonian (by adding an appropriate constant) to satisfy

H|0〉 = 0, (4.4)

i.e. the vacuum has zero energy. Now, if the symmetry generated by Q is spontaneouslybroken, we have

Q|0〉 6= 0. (4.5)

That is, the vacuum is no longer invariant under the symmetry. We now turn to the stateQ|0〉. This has the energy

HQ|0〉 = [H,Q]|0〉 = 0. (4.6)

So, if |0〉 is the vacuum, Q will generate other states eiαQ|0〉 with the same energy. Thesestates must be massless, see [2] page 198-199. In the case of O(N) we recall that there areN(N−1)

2 generators. Thus, for the SSB O(N)→ O(N −1), there are N −1 broken generators.For O(4) chiral QCD we associate these with the pions π+, π−, and π0. The fact that thevacuum is no longer symmetric will also give rise to a massive mode, as is shown in the nextsection. This mode is associated with the σ-resonance for the case of O(4) chiral QCD. Thisresonance has a mass of about 500 MeV.

At zero isospin chemical potential, there still remains an O(3) symmetry, and there is nopreferred direction with respect to this symmetry. If one introduces an isospin chemical po-tential µI , we shall see that the remaining O(3) symmetry gets further broken, and a preferreddirection is pointed out for the condensate. This is the direction that in QCD correspondsto the direction of the charged pions that carry isospin chemical potential, and we thus get acharged pion condensate. We shall study this in detail later.

If one heats up the chiral condensate of QCD, one will arrive at a critical temperature wherethe SSB disappears (〈ψψ〉 → 0), and the phase becomes symmetric. In QCD, this is knownas the quark-gluon plasma. For two quark flavors, this is known to be a second order phasetransition [43, 44, 45, 46] and this is also what we find. This phase transition can also be

4.2. FINDING THE EQUATION AT ZERO TEMPERATURE 53

modeled by the O(4) theory described above, and this is what we will do. We may also men-tion that ifNf = 3 one finds a first order phase transition [46, 40] instead of a second order one.

In this thesis, we will thus analyze and discuss the O(4) model for different condensatesof QCD (chiral condensate, charged pion condensate etc.), deriving critical temperatures andphase diagrams where the isospin chemical potential has been included. It is also possibleto leave the chiral limit by introducing an explicit symmetry breaking term into the theory.This gives rise to massive pions as we shall see.

We have thus seen how O(N) theories and in particular O(4) theory is relevant to high-energyQCD and in particular pion condensation. We now turn to find the O(N)-RG equation.

4.2 Finding the equation at zero temperature

So far in the derivation of the RG equations, we have only considered a theory consisting of onefluctuating field parameter φ. We want to generalize this to a vector having N components.We denote this vector by φ and its components by φi.

4.2.1 The O(N) equation

The derivation of the RG equation for Γk[M ] is similar as before, although a bit more general.The generalization of the RG equation is in that we must take a trace over internal indicesof the integrand. The result is thus

∂kΓk[M ] =1

2Tr

∫x,y∂kRk,x−y

[Γ

(2)k,x,y +Rk,x−y

]−1. (4.7)

as derived in Appendix A, Eq. (A.24). We work in the LPA again, and write

ΓLPAk [M ] =

∫x

(Uk(Mx) +

1

2(∇Mx)2

). (4.8)

We will neglect the LPA superscript in the following.

We again define the effective potential as U(Muni) = 1OΓk[Muni]. That is, the action Γk[M ]

evaluated at a uniform field, and divided by the spacetime volume O. We can then calculatethe RG equation for Uk(M) in a similar fashion as before, and obtain

∂kUk(M) =1

2Tr

∫q∂kRk(q)

[∂2Uk(M)

∂Mi∂Mj+ (q2 +Rk(q))δij

]−1

, (4.9)

where we again have neglected the subscript uni. Since the model is O(N) symmetric, wemay choose any direction of M we like. We may for instance choose the last componentMN = M and set the remaining components to zero. With this choice, the RG equation forthe potential Uk(M) can again be calculated. Due to the O(N) symmetry, if we let ρ = 1

2M2

we may as well write the effective potential as a function of ρ. If we use the mathematicalidentity (valid for Uk(φ) = Uk(ρ))

∂2Uk∂Mi∂Mj

=∂Uk∂ρ

δij +MiMj∂2Uk∂ρ2

, (4.10)

4.2. FINDING THE EQUATION AT ZERO TEMPERATURE 54

we find that

∂2Uk∂Mi∂Mj

+(q2+Rk(q))δij = diag(q2+Rk(q)+U ′k, . . . , q2+Rk(q)+U ′k, q

2+Rk(q)+U ′k+2ρU ′′k ),

(4.11)where we differentiate Uk(M) with respect to ρ. We may then write down the RG equationfor Uk(ρ),

∂kUk(ρ) =1

2

∫q∂kRk(q)

[N − 1

q2 +Rk(q) + U ′k+

1

q2 +Rk(q) + U ′k + 2ρU ′′k

]. (4.12)

Note the predictions of this equation. We see from Eq. (A.24) that the RHS of this RGequation includes a trace over the full effective propagator. Assuming a potential that has SSB(the minimum of the potential is not at the origin), we see that if one goes to the minimum ρ0

of this potential, the right-hand side of (4.12) will include one ”massive propagator” with massm2 = 2ρU ′′k , and N − 1 ”massless propagators” as U ′k(ρ0) = 0. This is precisely Goldstonestheorem. We shall return to this point later.

4.2.2 Truncation and coupled equations

Using the cutoff-function (3.46), we may evaluate the integral in (4.12) to obtain

∂kUk =4vdk

d+1

d

[1

k2 + U ′k + 2ρU ′′k+

N − 1

k2 + U ′k

]. (4.13)

This must be solved numerically. In order to proceed from Eq. (4.13) without going to nu-merics yet, we can make some more simplifications. This is done by truncating the expansionof the potential Uk(ρ) at some finite order of expansion in ρn. It is known that, even if thepotential has a polynomial form at the microscopic level of k = Λ, all other orders of couplingconstants will get generated when one starts to integrate out fluctuations, and the potentialis therefore in general not a polynomial (at least not a polynomial of finite order). However,as a simplification we can truncate the potential at order ρ2, i.e.

Uk(ρ) = Vk +1

2m2kM

2 + g2kM

4 = Vk +m2kρ+ 4g2

kρ2, (4.14)

where we have defined Vk = Uk(0), m2k = U ′k(0), and g2

k = 18u′′k(0), and neglected higher order

terms. We thus obtain the following equations for the coupling constants:

∂kVk =4vdk

d−1N

d, (4.15)

∂k(m2k) = −

32vdkd+1g2

k(N + 2)

d(k2 +m2k)

2, (4.16)

∂k(g2k) =

256vdkd+1g4

k(N + 8)

d(k2 +m2k)

3. (4.17)

These equations can easily be solved numerically (the first one even analytically) by use ofnumerical methods.

4.3. TURNING ON THE TEMPERATURE 55

The Eqs. (4.15) to (4.17) are slightly different than the ones presented in e.g. [47] at zerotemperature. This is due to the fact that a different cutoff-function is used there, namely

Rk(q) = (k2 − q2)θ(k2 − q2), (4.18)

where q now relates to the d− 1 spatial dimensions of the problem, neglecting the temporaldimension. When the temperature is nonzero this will be a much simpler regulator to workwith, and we shall use this as well when considering T > 0. Note however that the energydimension is neglected in this regulator. Hopefully this will not cause to many problems.

4.3 Turning on the temperature

So far, we have only seen RG equations for T = 0. We now proceed to turn on the temperature.The temperature is an outside parameter, i.e. we consider a heat bath, and it is included inthe theory as follows.

4.3.1 The RG equation for Uk(ρ)

We recall that at finite temperature the T > 0 partition function takes on the form

Z =

∫Dφe−

∫ β0 dτ

∫dd−1xLE , (4.19)

where we recall that β = 1T , T being the temperature of the system, and LE is the Euclidean

Lagrangian density. Remember that going to T > 0 and TFT is the same as going to aperiodic time variable in the fields, and switching to imaginary time. From Eq. (4.19), theGibbs free energy may be calculated as in Chapter 2. We then postulate the Gibbs free energyin the LPA for finite temperature simply to be

Γk[φ] =

∫ β

0dτ

∫dd−1x

[1

2(∇φ)2 + Uk(ρ)

], (4.20)

where ρ = 12φ

2, and we have started to denote Mx = 〈φx〉 simply by φx. We see that thisagain reduces to the classical action at the cutoff Λ.

We recall that with the temperature turned on, the fields become periodic in the imagi-nary time variable. Hence, so does any function of the fields. We recall that the Fouriertransform of a periodic function is a sum over exponentials with different weights. Thus, iff(τ,x) is a periodic function in τ with period β = 1

T it may be written as

f(τ,x) = T∞∑

n=−∞

∫qf(ωn,q)ei(ωnτ+x·q), (4.21)

where ωn = 2πnT are the Matsubara frequencies, and where the weights f(ωn,q) now corre-sponds to the new ”Fourier transform”. Hence, going to momentum space with T > 0 simplymeans that one should sum over Matsubara frequencies in the first variable, the ”Euclideanenergy variable” ωn corresponding to the imaginary time variable τ , i.e.∫

qf(q)→ T

∞∑n=−∞

∫qf(ωn,q). (4.22)

4.3. TURNING ON THE TEMPERATURE 56

In the case of Eq. (4.20), this change of the Fourier transform may be carried out throughthe derivation of the RG equation for the potential with no difficulty, and we thus get theRG equation for the potential at finite temperature

∂kUk(ρ) =T

2

∞∑n=−∞

∫q

[(N − 1)∂kRk(q)

q2 +Rk(q) + U ′k+

∂kRk(q)

q2 +Rk(q) + U ′k + 2ρU ′′k

], (4.23)

where q = (ωn,q). We now see why the choice of cutoff-function Eq. (4.18) is a clever choicewhen T > 0. It means that the integral in Eq. (4.23) can easily be calculated. Doing this weobtain

∂kUk(ρ) =4kdTvd−1

d− 1

∞∑n=−∞

[N − 1

ω2n + k2 + U ′k

+1

ω2n + k2 + U ′k + 2ρU ′′k

]. (4.24)

We use Eq. (A.7) derived in Appendix A to do the Matsubara sum. After this has beenperformed, we finally arrive at the equation

∂kUk(ρ) =2kdvd−1

d− 1

[(N − 1)

ω1,k(1 + 2n(ω1,k)) +

1

ω2,k(1 + 2n(ω2,k))

], (4.25)

where

n(x) =1

exp(βx)− 1(4.26)

is the Bose distribution-function and

ω1,k =√k2 + U ′k(ρ), (4.27)

ω2,k =√k2 + U ′k(ρ) + 2ρU ′′k (ρ). (4.28)

We could Taylor expand the potential around zero again and truncate this expansion at somefinite order. Such an expansion is however only valid for small values of the field parameterρ. If we wish for the potential to allow for SSB, it is better to expand it around its minimumρ0 instead, i.e.

Uk(ρ) = 4g2k(ρ− ρ0,k)

2. (4.29)

We then find the following set of coupled differential equations for the the k-dependent ”run-ning” coupling g2

k and the running minimum ρk,0:

∂kρ0,k =vd−1k

d(N + 2)

(d− 1)ε5k

[(ε2k + 3g2

kρ0,k)(

1 + 2n(εk) + 2εkn′(εk)

)+ 8g2

kρ0,kε2kn′′(εk)

], (4.30)

∂kg2k =

4vd−1kdg4k(N + 8)

(d− 1)ε5k

[3 + 6n(εk)− 6εkn

′(εk) + 2ε2kn′′(εk)

], (4.31)

where εk =√k2 − 8g2

kρ0,k. These equations can be solved to obtain an approximate RG flow

of the coupling g2k and the running minimum ρ0,k. The critical temperature Tc may be found

by requiring that ρ0,k=0 = 0 at Tc. We shall not solve these equations, but rather attack thefull RG Eq. (4.25) in the next chapter.

4.4. THE LARGE-N LIMIT 57

4.3.2 High-temperature flow

We return to expanding the potential around ρ = 0 and see what happens if we increase theorder of expansion. We write the potential as

Uk(ρ) = Uk +m2kρ+ 4g2

kρ2 + 8h2

kρ3. (4.32)

Note that there is something special in expanding to this order when we have four spacetimedimensions. We see that in order for the mass dimension of the potential to become d, makingthe action dimensionless, we must set the mass dimension of the field to d−2

2 . This meansthat when d = 4, the dimension of h2

k must be 6 − 2d = −2, i.e. the theory has becomenon-renormalizable as an old fashion perturbative field theory. This is however not a problemfor the non-perturbative approach, and is, we recall, one of the reasons why this approach isso useful. We should, however, set h2

Λ = 0 if we want the boundary theory at the cutoff to berenormalizable in the perturbative sense.

We recall that, when postulating a potential at scale k = Λ and starting to integrate outmodes, one will always experience that all terms compatible with the symmetries of the the-ory will be generated in the effective Lagrangian even though they are not renormalizable inthe perturbative sense. In the LPA, however, one assumes a slowly varying field so that wecan set all of the higher order ”kinetic terms” to zero, Though they are in general generatedas we integrate out modes. In addition, if the temperature is high enough, we get what isknown as a dimension reduction, i.e. we effectively loose one dimension. We shall see howthis goes.

We recall that in thermal field theory we calculate integrals of the form

T∑n

∫kF[(2πT )2n2 + k2

], (4.33)

where the combination (2π)2T 2n2 +k2 usually occurs in the denominator. Hence, in the high-temperature limit only the n = 0 term has a significant contribution. Thus high temperaturethermal field theory is equivalent to Euclidean QFT in one less dimension. What this meansfor our theory, Eq. (4.32), is that since the field and potential obtain the effective dimensionsd−3

2 = 12 and d− 1 = 3, respectively, it is no longer g2

k that is the marginal coupling constant(coupling constant of dimension 0), but rather h2

k! Thus, in the high-temperature limit, theboundary theory given by Eq. (4.32) is renormalizable.

4.4 The large-N limit

There is something peculiar about considering N > 1 and the large-N limit. Namely the factthat in the limit N → ∞ the LPA becomes exact, at least for constant mean fields which iswhat we are working with in this thesis. We consider a formal proof of this, similar to thatof [48]. We work at T = 0 for simplicity, although the derivation also holds for T > 0. Weneglect spacetime dependence for convenience.

Start by recalling the RG equation, Eq. (A.24),

∂kΓk[φ] =1

2Tr[ δ2Γ[φ]

δφiδφj+ δijRk

]−1, (4.34)

4.4. THE LARGE-N LIMIT 58

where φ again denotes the mean field, Rk is an IR regulator, and the trace is also overspacetime (or momenta if the argument of the trace is in momentum space). We redefine theeffective action Γk[φ] by

Γk[φ]→ 1

2φi ·D−1

0 · φi + Γk[φ], (4.35)

i.e. we have pulled the bare propagator D0 out of the effective action. We now define the fullpropagator as

∆−1k = D−1

0 +Rk. (4.36)

The RG then becomes

∂kΓk[φ] =1

2Tr∂k∆k

[ δ2Γ[φ]

δφiδφj+ δij∆

−1k

]−1. (4.37)

If now Γk[φ] = Γk[φ2] at some k (still allowing for kinetic terms in Γk[φ

2]), we may write

δ2Γk[φ]

δφiδφj= 2δij

δΓk[φ2]

δφ2+ 4φiφj

δ2Γk[φ2]

(δφ2)2. (4.38)

We now use the generalized geometric series

(Aδij −Bij)−1 = A−1δij +A−1BijA−1 +A−1BikA

−1BkjA−1 + ... . (4.39)

It is clear that the first of the above terms will dominate as we let N →∞ and take the trace.In the large-N limit, Eq. (4.37) therefore simplifies to

∂kΓk[φ2] = −N

2Tr

1

∆k∂k∆k

[1 + ∆k

δΓ[φ2]

δφ2

]−1. (4.40)

If we now assume that

Γk[φ2] =

∫xUk(φ

2) (4.41)

at some point, i.e. Γ[φ] is a functional of only φ2 where no kinetic terms are included. Thiswill be the case if we consider constant mean fields, i.e fields independent of spacetime, or ifthere are no higher order derivative terms in Γk[φ

2]. This is the case at k = Λ, at least forthe models we consider. Then the RHS of Eq. (4.40) becomes a functional of only φ2 as well,and so this form of Γ[φ] does not change. Thus the corrections to the effective action do notinclude any derivative terms, and truncating such terms in the LPA becomes exact. This alsoensures that no wave-function renormalization is present. We thus see, in the large-N limit,that the LPA becomes exact, given that the potential takes the assumed form of Eq. (4.41).

Assuming constant mean fields, φ2 = z, a constant, and rescaling the field φ →√Nφ and

potential Uk(z)→ NUk(z), reduces Eq. (4.40) to an RG equation for Uk(z) [48]

∂kUk(z) = −Ω

2

∫ ∞0

dqqD−1

(2π)D1

∆k(q)∂k(q)

1

1 + 2∆k(q)U′k(z)

, (4.42)

where Ω is the solid angle of a (D− 1)-sphere. We recall that we obtained a similar equationin the LPA for the effective potential evaluated at mean field. Hence the LPA becomes exactin the large-N limit, whenever we consider constant mean fields.

The result that the large-N limit allows for no wave-function renormalization is also ob-tained by other methods, such as the ε-expansion of φ4-theory, see the next chapter and[49].

5. TUNING THE PARAMETERS OF THE O(N) MODEL

In this chapter, we start producing some numerical results for the O(N) case. We considerthe case O(4) in particular, as this is a good description for pion condensation in the chirallimit and at zero chemical potential, as described in Chapter 4. We consider Eq. (4.25) again(the RG equation we found for the effective potential at T > 0), and set T = 0 in orderto tune the parameters of Uk=Λ(ρ) so as to agree with vacuum physics as described in theintroduction. Letting T = 0 is the same as setting the Bose distribution-function to zero inthe expression, at least at and to the right of the minimum of the potential. We considera O(N) linear sigma model as our boundary condition at the cutoff k = Λ. When the bareparameters of UΛ(φ) have been tuned, they can be used in the model for T > 0 as well. Weturn to see how this is done.

We use a third order Runge-Kutta (RK) method with finite-distance methods to approxi-mate the derivatives when solving (4.25) numerically, see Appendix A and Appendix B.

5.1 Physical parameters

We consider the classical φ4-O(N) potential

UΛ(ρ) = m2Λρ+ 4g2

Λρ2 (5.1)

as our boundary condition for Uk(φ) at the cutoff k = Λ. Here we have defined ρ = 12φ

2.Adding kinetic terms, this becomes a so-called linear sigma model. Note that we have onlyincluded renormalizable terms to the model at T = 0, i.e. The φ4-term is the highest termincluded. This gives us two parameters to decide, namely m2

Λ and g2Λ.

If we set N = 4, the model is a good approximation to two-flavor QCD in the chiral limit asdiscussed earlier. In chiral QCD, it is known that at T = 0 this symmetry is spontaneouslybroken. This means we must choose m2

Λ < 0 so that O(4) → O(3) spontaneously. We mustalso choose it negative enough, so that it remains negative when k = 0 for temperatures belowthe critical temperature Tc (more precisely m2

k ≡ U ′k(0) < 0 for all k).

With SSB of the O(4) model, Goldstones theorem predicts a massive mode (related to themassive σ particle) and three massless modes (related to the three pions). If we use the O(4)symmetry to rotate the minimum so that it is given by the mean field φ0 = (0, 0, 0, φ4,0), wemay write the fluctuating quantum field as φ = (φ1, φ2, φ3, φ4,0 + φ4). If we Taylor expandthe potential around its minimum, it reads

Uk(φ) = Uk(φ2 + 2φ4,0φ4 + φ2

4,0) = Uk(φ24,0) +

1

2(φ2 + 2φ4,0φ4)2U ′′k (φ2

4,0) + .. (5.2)

5.2. RENORMALIZED COUPLINGS 60

where we have redefined φ2 = φ21 + ..+φ2

4 as the fluctuating field. Note that there is no linearterm as U ′k(φ

20) = 0. The masses of the particles are given by their respective quadratic terms

in the potential. We thus get three massless particles, related to the pions, and one massivemode of mass

m2σ,k = 2ρ0U

′′k (ρ0), (5.3)

where we have renamed the fourth mode as the σ mode. This is exactly the same predictionas in the last chapter. It is also known that, at k = 0, we have [50, 51]

〈φ4〉 = φ4,0 = fπ, (5.4)

i.e. ρ0 = 12f

2π where fπ is the pion decay constant.

We now set mσ = 400MeV and fπ = 93MeV. Using the above equations, we can tunethe parameters at k = Λ so as to obtain these values for the sigma mass and pion decayconstant at k = 0. We choose Λ such that Λ2 = 5m2

σ. Recall that Λ should be chosen to beat least at the highest energy scale of the system, which in this case is mσ.

The choice of Λ2 = 5m2σ might however be a bit high, since if one includes quark and gluon

effects in the model, the chiral symmetry will be restored again at high enough k [51, 52].This scale has been determined to be around 1GeV, about the same scale as we have chosenΛ to be. However, since our predictions should not depend to much on the choice of Λ, wewill stick with this choice.

Using the above, we find that we should set m2Λ = −0.96Λ2 and g2 = 9.39 in order to

obtain the correct values for mσ and fπ at k = 0 and T = 0. We define the renormalizedcouplings at scale k to be

m2k = U ′k(ρ = 0), (5.5)

g2k =

1

8U ′′k (ρ = 0) (5.6)

Note that m2Λ ≈ −5m2

σ, and as m2k=0 is of the order of the σ-mass or smaller, we see that this

parameter gets renormalized a lot! We investigate these couplings a bit further.

5.2 Renormalized couplings

Still letting T = 0, we plot the real part of m2k as a function of the renormalization time

t = log kΛ in Fig. 5.1. Note how the coupling levels off as k → 0. This is in fact characteristic

for all the couplings of the potential and is related to the fact that the couplings arrive atfixed point values of the RG flow as discussed in the introduction.

It should be noted that, to the left of the minimum of the effective potential Uk(ρ) andwhen integrating down from the cutoff, imaginary parts will be added as a result of thesquare roots of Eq. (4.25) becoming imaginary and the fact that the effective mass squaredm2k has leveled off to a negative value. This can be seen in Fig. 5.1, where we have plotted

the real part of m2k and we see that levels off at a small but negative value as the RG time

t→ −∞. As a result, an inflection point is generated to the left of the minimum of the realpart of the effective potential. We do not worry too much about this, since we are primarilyinterested in what happens at the minimum of Uk(ρ), where the potential remains real.


5.2.1 Critical temperature

We now let T > 0 in order to find the critical temperature, i.e. the temperature for whichthe chiral condensate 〈φ4,0〉 vanishes at k = 0. This is equivalent to having the m2

k stabilizingat m2

k=0 = 0. We recall that as k approaches zero, or t → −∞, the couplings of the modellevel off and reach fixed point values. We thus want to tune the temperature so that the fixedpoint value of m2

k=0 is zero. At this temperature, the symmetry gets restored, and the systemgoes to the symmetric phase.

Tuning the temperature in order to achieve this, we end up with a critical temperatureof 0.17Λ = 152MeV. This prediction is within 20 percent of the value other approachespredict, for example that of lattice QCD [53] that predicts a critical temperature of about170MeV, or the Nambu-Jona-Lasinio (NJL) model [54] that predicts a critical temperatureof about 220MeV. In most models, the critical temperature, at zero chemical potential forthe two-flavor chiral condensate, is usually predicted to be from about 150 to around 250 MeV.

In Fig. 5.3, we show the real part of the dimensionless effective potential, that is Re(Uk=0(ρ)

Λ4

),

at different temperatures as a function of ρΛ2 . Note that the transition from a phase of spon-

taneously broken symmetry to the phase where the symmetry is restored is continuous. Thisshows that the phase transition from the chiral condensate to the quark-gluon plasma is asecond order phase transition as discussed in Chapter 4.

Fig. 5.1: The real part ofm2

k

Λ2 as a function of the RG time t = log kΛ at T = 0, where the physical

parameters mσ = 400 MeV and fπ = 93 MeV have been used to tune the couplings of UΛ(ρ),and where Λ2 = 5m2

σ. As the potential Uk(ρ) includes a complex part when k gets lowenough, this means that m2

k will also become complex. Also note the leveling off of m2k at a

slightly negative value as k → 0.


5.2.2 The quartic coupling g2k

We turn to T > Tc and consider the quartic coupling g2k. We recall the definition of the

critical temperature Tc, namely that mk=0 → 0 as T → Tc or equivalently that the min-imum of Uk=0(ρ) vanishes. Thus, by definition, the effective mass parameter, defined bym2k=0 = U ′k=0(ρ = 0), must approach zero when we approach the critical temperature Tc. We

check if this is the case for the effective coupling g2k=0 as well. The results of [55], where an

O(2)-theory of Bose-Einstein condensation is considered very similar to the O(4) theory weconsider, tells us that this is what we expect to happen, i.e. we should have g2

k=0 → 0 asT → Tc. We expect our system to behave similarly due to the fact that the O(2) and O(4)systems belong to the same universality class of O(N) theories, and Fig. 5.2 shows that thisis indeed the case.

The fixed point approached at the critical temperature when t→ −∞ is the WF fixed pointdiscussed in the introduction [23]. It is an IR fixed point, i.e. it is reached when k → 0, orequivalently t→ −∞. We recall that as the temperature grows without bound, we expect toeffectively lose one dimension. If we write the ”effective dimension” as d = 4− ε, this meansthat ε → 1 as T → ∞. This may not have happened yet at T = Tc. However, a nonzerotemperature can be said to reduce the ”effective dimension” slightly from d = 4.

Writing the dimension as d = 4− ε is known as the ε-expansion [20, 49]. We then have

Γk[φ] =

∫ β

0dτ

∫d3−εx

(1

2(∇φ)2 +

1

2m2kφ

2 + 4g2k(φ

2)2 + ...). (5.7)

We work in dimensions of energy. Denote the field dimension by x. In order for the action tobe dimensionless, we see that we must have

x =2− ε

2. (5.8)

This means that the dimension y of g2k must satisfy

y + 2(2− ε) = 4− ε. (5.9)

Solving this equation gives y = ε.

If one defines new ”dimensionless” couplings m2k = k−2m2

k and g2k = k−εg2

k, [20] finds, us-ing perturbative renormalization, that the WF fixed point has m2∗ ∼ ε and g2∗ ∼ ε. As wehave ε > 0, this means we should have m2

k = g2k = 0 which is indeed what we find.


Fig. 5.2: The effective coupling g2k at k = 0 as a function of T−Tc

Λ , where we have used the physicalparameters mσ = 400 MeV and fπ = 93 MeV to tune the couplings of UΛ(ρ) at T = 0, withΛ2 = 5m2

σ. We clearly see that it approaches zero as we approach the critical temperature.

Fig. 5.3: The real part of the dimensionless effective potential Re(Uk(ρ)Λ4 ) at the scale k = 0 for different

temperatures, where we have subtracted the constant added to the potential by renormal-ization. We have used the physical parameters mσ = 400 MeV and fπ = 93 MeV to tunethe couplings of UΛ(ρ) at T = 0, and set Λ2 = 5m2

σ. Note the continuous transition from thebroken to the symmetric phase. This is thus a second order phase transition.


Part III

PION CONDENSATION


6. INCLUDING A CHEMICAL POTENTIAL

We now want to study a charged pion condensate. The basic principle for how this occurs isknown as Bose-Einstein (BE) condensation. The theory of BE condensation has been knownfor quite a while [56, 57], and it can be used to describe many systems, such as e.g. heliumsuperfluidity [58] or even as models for gravity [59]. Though the theory has been known a longtime, it was only as resent as in 1995 that someone was able to produce a pure BE condensatein the laboratory [60]. This was done by Eric Cornell, Carl Wieman, and Wolfgang Ketterle,and it secured them the 2001 Nobel Prize [61].

To study BE condensation, one usually includes a chemical potential into the theory. Inthe case of a charged pion condensate, we must include an isospin chemical potential. Werecall, for N = 4, that at zero chemical potential the system is in an spontaneously brokenphase = O(4)→ O(3). There is, however, still no preferred direction for the remaining O(3)symmetry. With an isospin chemical potential included in the model, a preferred directionis pointed out, namely the direction of the charged pions. Including an isospin chemicalpotential thus turns the chiral condensate at T < Tc into a charged pion condensate, and aBE condensate of charged pions is formed. With such a chemical potential present, it willbe possible to distinguish between the massless Goldstone modes as they now can also carryisospin ”charge”. For N = 4, we thus obtain three pions of different isospin.

Note also that the transition from the O(4) chiral condensate to the O(2) charged pion BEcondensate happens the moment the isospin chemical potential becomes nonzero, at least inthe chiral limit. When we turn on a chemical potential, the original O(4) symmetry is brokento a O(2)×O(2) symmetry. We see how this goes later.

It should be noted that at zero isospin chemical potential, the chiral condensate formedby SSB can in principle be turned in the charged pion condensate direction due to the O(4)symmetry of the problem. In 1984 however, Vafa and Witten proved that, in vacuum QCD(and for ”QCD like theories”), global vector like symmetries (such as isospin) cannot bespontaneously broken [62]. This is known as the Vafa-Witten Theorem. Hence the chiralcondensate at zero isospin chemical potential cannot be a charged pion condensate, thoughthis is not clear from our O(4) simplification. It could also be noted that the chiral conden-sate is parity invariant, while the charged pion condensate is not. Again, Vafa and Wittenproved that in vacuum QCD, parity cannot be spontaneously broken [63], also known as theVafa-Witten Theorem. This also shows that charged pion condensation does not occur in thevacuum.

We return, for the moment, to Minkowski spacetime and let

L =1

2∂µφi∂

µφi − U(φ2). (6.1)

6.1. DERIVING THE LAGRANGIAN 68

6.1 Deriving the Lagrangian

We recall from Chapter 2 and classical field theory [1, 2, 64] that to each conserved current,that is to each symmetry, there is a conserved charge. From thermal field theory we knowthat to each conserved charge we may associate a chemical potential (see Chapter 2). If thetheory involves chemical potentials, we must use the grand canonical partition function [26].This takes the form

Z = Tr[e−β(H−µiQi)

](6.2)

where the Qi’s are the operators corresponding to the conserved charges. Thus includinga chemical potential simply means we let H → H − µiρi where the ρi’s are charge densityoperators.

In the case of O(N) theory, there are N(N−1)2 conserved currents (the same as the num-

ber of generators of the group O(N)). The conserved currents are derived from Noether’stheorem, see Chapter 1, namely

jµj =∂L

∂(∂µφa)δφa (6.3)

where δφa denotes a small symmetry transformation. Now, write an element R ∈ O(N) asR = exp (εjGj) where Gj ∈ Lie

(O(N)

)are generators contained in the Lie algebra of O(N).

If we write the field as φ = (φ1, .., φN ), we see that the potential is a function of φ2. In thisnotation, the field transforms as φ → Rφ, where R is in the fundamental representation ofO(N). We note that the potential U(φ2) is independent of such a transformation as it shouldbe.

The general O(N)-symmetric Lagrangian then reads

L =1

2∂µφ∂

µφ− U(φ2). (6.4)

If we assume a small transformation of the field, it can be written as

φ→ (1 + εjGj)φ. (6.5)

We choose the basis for O(N) where Gij has a 1 in position ij and a −1 in position ji and zeroelsewhere. Note that GTi = −Gi as RRT = RTR = 1. We thus obtain the transformation law

φj → φj + (εjk − εkj)φk, (6.6)

where εjk corresponds to Gjk. This gives the current

jµik = φi∂µφk − φk∂µφi, (6.7)

and there are thus N(N−1)2 of them as stated above. Each current corresponds to a conserved

charge. However, only N2 of these currents may be diagonalized simultaneously, that is, only

N2 currents can coexist at one time. This is due to the fact that each conserved currentcorresponds to a generator of the O(N) group, and only N

2 generators of this group commute.Hence, we can only include a maximum of N

2 chemical potentials. We are, as noted above,only going to include one chemical potential, namely that of the isospin conserved ”charge” I

6.1. DERIVING THE LAGRANGIAN 69

which we associate with the current jµ12. We thus let H → H−µIj012. We recall the definition

of the Hamiltonian density

H(φ, π) = πi∂0φi − L(φ, ∂µφ), (6.8)

where the πi’s denote the canonical conjugate momenta, which in this case are

πi =∂L

∂(∂0φi)= ∂0φi. (6.9)

With a non-zero chemical potential, the Hamiltonian then becomes

H =1

2πiπi −

1

2∂µφi∂

µφi + U(φ2)− µI(φ1π2 − φ2π1). (6.10)

We find the new canonical momenta by using Hamilton’s equation

∂0φi =H(φ, π)

∂πi. (6.11)

From this we find

π1 = ∂0φ1 − µIφ2

π2 = ∂0φ2 − µIφ1

πi = ∂0φi i > 2. (6.12)

By a Legendre transform, we thus obtain the Lagrangian

L = πi∂0φi −H

=1

2∂µφi∂

µφi + µI(φ1∂0φ2 − φ2∂0φ1) +1

2µ2I(φ

21 + φ2

2)− U(φ2). (6.13)

If N is even, we may write N2 complex fields of the form Φa = 1√

2(φ2a−1 + iφ2a) for

a = (1, .., N2 ). Note that these fields are generally charged, as they are complex, see theintroduction. We may then write the Lagrangian (6.1) as

L = ∂µΦ∗a∂µaΦa − U(ΦaΦ

∗a). (6.14)

We see that the inclusion of a chemical potential µa (indexed by a as there are N2 conserved

charges) can be done simply by letting ∂0Φa → (∂0 +iµa)Φa and likewise ∂0Φ∗a → (∂0−iµa)Φ∗awhere no sum over a is implied. Note that the chemical potential occurs as an extra term inthe zeroth component of the gradient of the charged fields. This tells us that it is the chargedfields that carry the chemical potential as we have constructed it, i.e. the chemical potentialcharge is diagonalized with respect to the fields Φa. We also note that Φa and Φ∗a carryopposite chemical potential, i.e. opposite charge means opposite chemical potential. This isin fact just stating that particles and anti-particles have opposite chemical potential. This iswhy π+ and π− carry opposite isospin chemical potential, while π0 is both electrically neutraland carries no isospin charge. The charged pions (π+, π−) are thus related to the complexfields (Φ,Φ∗).

6.2. INCLUDING A PION MASS 70

If we return to Euclidean spacetime, LE = −L(t→ −iτ), we obtain the Euclidean Lagrangian

LE =1

2(∇φ)2 + iµI(φ1∂0φ2 − φ2∂0φ1)− 1

2µ2I(φ

21 + φ2

2) + U(φ2) (6.15)

Note here that the O(N) symmetry has been explicitly broken to an O(2)×O(N−2) symme-try by the inclusion of the chemical potential µI as stated above. This is the Lagrangian weshall use in the next chapters as our boundary condition at the cutoff Λ, when we derive thefull effective RG-equation for the O(4)-model where a chemical potential has been included.

We also note from Eq. (6.15) that the inclusion of an isospin chemical potential lowersthe potential in the (φ1, φ2) direction when we include the isospin term in the potential. Thisis why the chiral condensate goes to a BE condensate of charged pions when we turn on thechemical potential. We investigate this further in Chapter 8.

6.2 Including a pion mass

From now on, we assume that the potential U(φ2) allows for a spontaneously broken symme-try. Up to this point, we have been working in the chiral limit where the pions are massless.We now wish to include a pion mass in our calculation. We recall that in the vacuum, that iswhen T = µI = 0, the three pions are identified with the three massless Goldstone bosons inthe broken phase of the symmetric O(4) theory. That the pions are massive indicates that theO(4)-symmetry is not an exact symmetry, but has been explicitly broken. We thus rewritethe potential as

U(φ2)→ Us(φ2)−Hφ4, (6.16)

where we have added an explicitly symmetry-breaking term Hφ4 and s denotes the symmetricpart of the potential. When H = 0, there are infinitely many ground states resulting in onemassive mode and three Goldstone modes. For H > 0, the situation is different. The potentialis lowered in the direction of φ4 and we thus we expect the ground state to be in this direction.Hence, we expect a vacuum expectation value (vev) 〈φ〉 = (0, 0, 0, σ). We then expand φ4

around its vev and write φ4 → φ4 + σ. We then expand the potential around its minimum,so that

U(φ) = Us(φ2 + 2φ4σ + σ2)−H(φ4 + σ)

=

[Us(σ

2) + (φ2 + 2φ4σ)U ′s(σ2) +

1

2(φ2 + 2φ4σ)U ′′s (σ2) + ..

]−H(φ4 + σ). (6.17)

We write the Lagrangian as

L =1

2∂µφi∂

µφi − U(φ) =1

2φiD

−1ij φj − V (φ), (6.18)

where we have included all the quadratic terms in the first term φiD−1ij φj , where D−1

ij denotesthe inverse propagator, and where V (φ) denotes the remainder. In our case, this propagatoris diagonal, Di = Dii no sum, and we may write

D−1i<4 = −∂2 −m2

π, (6.19)

D−14 = −∂2 −m2

σ, (6.20)

6.3. RENORMALIZING 71

where we have associated the pions with the φk<4 and the sigma particle with φ4. The pionand sigma masses are thus given by

m2π = 2U ′s(σ

2), (6.21)

m2σ = 2U ′s(σ

2) + 4σ2U ′′s (σ2) = m2π + 4σ2U ′′s (σ2). (6.22)

We shall be using these formulas when we tune the parameters of our model later on. An-other important parameter we can use is the so-called pion decay constant fπ. This constantis related to the decay time of the pions. It can be shown that fπ = σ, see e.g. [50, 51, 65].

One could argue that in reality the three pions have distinctly different masses. That istrue, but the differences are not great, and are due to electromagnetic and weak processesthat are severely suppressed by the strong interaction. They will thus be neglected in thisthesis, where we will set the pion masses equal.

We move on to the explicitly symmetry breaking parameter H. This can be found by

∂

∂φ4

(Us(φ

2 + 2σφ4 + ρ2)−H(φ4 + σ))∣∣∣∣φ=0

= 0 (6.23)

If we solve this equation with respect to H, we obtain

H = m2πfπ. (6.24)

6.3 Renormalizing

The above calculations where performed for an O(4)-symmetric theory in vacuum where anexplicit symmetry breaking term had been added. One may wonder if it has any relevance forthe theory we consider? Well, setting µI = 0 makes the theory of Eq. (6.15) O(N) symmetricagain. What about when we include the symmetry breaking term −Hφ4? We will find in thenext chapters that this term retains its bare value as one renormalizes. One can thus add itto the potential Uk=0(φ2).

Also, we recall in the LPA, that one disregards all higher order derivatives of the fieldsthat are generated when one integrates down from an UV scale Λ, and one also neglects anyfield renormalization that occurs. Therefore, in our setting, only the parameters of the po-tential get renormalized, and the calculation above thus holds for any renormalization scaleas long as the potential takes the assumed form (6.16) at that scale, which it will when µI = 0.

Now that we know how to include a chemical potential in the theory, we turn to find theRG equation for the potential in the LPA as we have done before, only now with a chemicalpotential included.

6.3. RENORMALIZING 72

7. FULL RG-EQUATION FOR PION CONDENSATION

In this chapter, we derive the full RG equation for a O(4) scalar theory in the LPA where achemical potential has been included, reducing the symmetry group to O(2)×O(2).

7.1 The full equation

We operate in Euclidean spacetime and let T = 0 for now. It is easy to go to T > 0 when weneed it. We make the following ansatz for the form of the Gibbs free energy Γk[φ] in the LPA

Γk[φ] =

∫x

[1

2(∇φ)2 + iµI(φ1∂0φ2 − φ2∂0φ1) + Uk(φ)

], (7.1)

where the φi’s again denote the mean value of the fields and we have included the chemicalpotential part −1

2µ2I(φ

21 + φ2

2) into the potential Uk(φ). Note that we have again omitted thefield renormalization of the kinetic term.

We recall the exact RG equation, Eq. (A.24)

∂kΓk[φ] =1

2Tr[∂kRk,q

(Γ

(2)k +Rk

)−1

q,−q

], (7.2)

where the trace is over the spacetime momenta q, and indices of the inverse matrix. Weagain define the effective potential by evaluating Γk[φ] in a uniform field configuration (fieldconfiguration independent of spacetime)

Uk(φuni) =1

OΓ[φuni], (7.3)

where O denotes the spacetime volume. If we differentiate this with respect to k, we find theRG equation for the effective potential,

∂kUk(φ) =1

2OTr

[∂kRk,q

(Γ

(2)k |uni +Rk

)−1

q,−q

]. (7.4)

With Γk[φ] given as in Eq. (7.1), we find the Fourier-transformed of Γ(2)k |uni (where uni

means evaluated at uniform field configurations) to be

Γ(2)k,i,j,q,q′ = (2π)dδ(q + q′)

[∂2Uk∂φi∂φj

+ δijq2 − 2µIq0(δj1δi2 − δi1δj2)

](7.5)

where d denotes the spacetime dimension. Note that by the presence of the chemical poten-tial, this is no longer a diagonal matrix. Also note that we can not rotate the field vector

7.1. THE FULL EQUATION 74

φ = (φ1, .., φN ) to point in one specific direction as we no longer have O(N) symmetry. Thishas been explicitly broken to a O(2) × O(N − 2) by the isospin chemical potential µI . Thebest we can do is φ = (φ1, 0, φ3, 0, .., 0) as we now only have a O(2) × O(N − 2) symmetry.This will also give rise to off-diagonal terms in the matrix.

We now set N = 4 in order to consider pion condensation. We may assume, because of theO(2)×O(2) symmetry, that Uk(φ) = Uk(ρ1, ρ2) where ρ1 = 1

2(φ21 + φ2

2) and ρ2 = 12(φ2

3 + φ24).

If we denote the fields by double indices φ = (φ11, φ12, φ21, φ22), we can derive the identity

∂2Uk∂φai∂φbj

=∂Uk∂ρb

δabδij + φaiφbj∂2Uk∂ρa∂ρb

. (7.6)

We now use the O(2)×O(2) symmetry to turn the fields to φ = (φ1, 0, φ3, 0), and recall thatthe Fourier transform of Rk(x − y) is (2π)dδ(q + q′)Rk(q) = Rk,q,q′ . If we add δijRk,q,q′ to

Γ(2)k,i,j,q,q′ , we obtain the matrix[

Γ(2)k +Rk

]q,q′

= (2π)dδ(q + q′)×∂1Uk + 2ρ1∂

21Uk + Fk(q) −2µIq0 2

√ρ1ρ2∂1∂2Uk 0

2µIq0 ∂1Uk + Fk(q) 0 02√ρ1ρ2∂1∂2Uk 0 ∂2Uk + 2ρ2∂

22Uk + Fk(q) 0

0 0 0 ∂2Uk + Fk(q)

,

(7.7)

where Fk(q) = q2 +Rk(q). This is very similar to the inverse of the tree level propagator fora O(4) scalar theory with chemical potential included, see e.g. [50]. It takes the form(

contribution from charged pions coupling termscoupling terms contribution from neutral particles

), (7.8)

where by the neutral particles we mean the π0 and the σ. If we recall the inverse in Fourierspace ∫

q′F (q1, q

′)ikF−1(q2, q

′)kj = (2π)dδ(q1 − q2)δij , (7.9)

we may invert[Γ

(2)k + Rk

]q,q′

to obtain the full k-dependent propagator. This may then be

inserted into Eq. (7.4) in order to obtain the RG equation for the effective potential. If wealso go to T > 0 we should replace the integral over q0 by a Matsubara sum T

∑n. Doing

this, we find

∂kUk =T

2

∑n

∫q∂kRk

[(2(q2 +Rk + ∂2Uk + 2ρ2∂

22Uk)(q

2 +Rk + ∂1Uk + ρ1∂21Uk)

− 4ρ1ρ2(∂1∂2Uk)2 + (q2 +Rk + ∂1Uk)(q

2 +Rk + 2ρ1∂21Uk) + 4µ2

Iω2n

)/(

(q2 +Rk + ∂2Uk + 2ρ2∂22Uk)

[(q2 +Rk + ∂1Uk + 2ρ1∂

21Uk)(q

2 +Rk + ∂1Uk) + 4µ2Iω

2n

]− 4ρ1ρ2(∂1∂2Uk)

2(q2 +Rk + ∂1Uk)

)+

1

q2 +Rk + ∂2Uk

], (7.10)

7.2. SIMPLIFICATIONS 75

where now q2 = ω2n + q2. Using the cutoff-function Rk(q) = (k2 − q2)θ(k2 − q2) as before,

the integral over spatial momenta can easily be performed. The expression obtained is almostidentical to that of (7.10). Simply let q2 + Rk → k2, remove the integral over q and letT2 →

4Tkdvd−1

d−1 .

7.2 Simplifications

It is clear that the Matsubara sum in (7.10) is difficult to perform. To do it in the usualmanner, where it is replaced by a contour integral in the complex plane, one has to factorthe denominator of the first term, which coincidently is also the determinant of the upper left3× 3 matrix in Eq. (7.7). This is a third degree equation and is not easily factorized. Beforewe get our hands dirty doing this, we consider some simplifications.

7.2.1 Charged pion condensate

We wish to study a charged pion condensate, that is ρ1 > 0. In the chiral limit, this rotatesthe fields so that ρ2 = 0 as explained before, see also [50, 65]. Setting ρ2 = 0 in (7.10) weobtain the much simpler RG equation

∂kUk =8Tkdvd−1

d− 1

∑n

[ω2n + k2 + ∂1Uk + ρ1∂

21Uk

(ω2n + k2 + ∂1Uk + 2ρ1∂2

1Uk)(ω2n + k2 + ∂1Uk) + 4µ2

Iω2n

+1

ω2n + k2 + ∂2Uk

]

=8Tkdvd−1

d− 1

∑n

[ 12 +

µ2Iω2k

ω2n + k2 + ∂1Uk + ρ1∂2

1Uk + 2µ2I + ω2

k

+

12 −

µ2Iω2k

ω2n + k2 + ∂1Uk + ρ1∂2

1Uk + 2µ2I − ω2

k

+1

ω2n + k2 + ∂2Uk

], (7.11)

where

ω2k =

√(ρ1∂2

1Uk)2 + 4µ2

I(k2 + µ2

I + ∂1Uk + ρ1∂21Uk). (7.12)

The Matsubara sum is now easily performed. Using Eq. (A.7), we obtain

∂kUk =8kdvd−1

d− 1

[(1

2+µ2I

ω2k

)1 + 2n(ω1)

2ω1+(1

2−µ2I

ω2k

)1 + 2n(ω2)

2ω2+

1 + 2n(ω3)

2ω3

], (7.13)

where

ω1 =√k2 + ∂1Uk + ρ1∂2

1Uk + 2µ2I + ω2

k, (7.14)

ω2 =√k2 + ∂1Uk + ρ1∂2

1Uk + 2µ2I − ω2

k, (7.15)

ω3 =√k2 + ∂2Uk. (7.16)

Note that if µI = 0, we again have O(4) symmetry. Hence ∂2Uk = ∂1Uk. Since ω2k → ρ1∂

21Uk

in this case, we see that we recover the usual O(4) symmetric RG-equation as we should.

Eq. (7.13) is difficult to solve numerically. This is due to the fact that the RHS involvesthe derivative of Uk in the ρ2-direction. As this is calculated numerically by finite-difference


methods, see Appendix A, it is necessary to know Uk at points where ρ2 > 0 too. So, althoughthe equation became simpler, this simplification did not help us much. It will also becomenecessary to know Uk at points where ρ2 > 0 when we leave the chiral limit and go to thephysical point as well.

We could try to expand the potential and find RG equations for the couplings of the po-tential as we did for the O(N) symmetric case. Although the Matsubara sums become nicerto work with (we end up with powers of second degree polynomials in the denominators in-stead of a third order polynomial in ω2

n), there are many more terms to deal with per equation,and the number of equations also increases drastically. We thus postpone solving the exactEq. (7.10) until later, and instead consider the simpler example of an O(2)-model with achemical potential, i.e. a pure BE condensate. As we shall see this can also be viewed as asimplification of our model.

7.2.2 BE condensate

If one, instead of assuming ρ2 = 0, rather assumes that the chiral condensate, connectedto ρ2, and the charged pion condensate, connected to ρ1, are two completely noninteractingsystems, so that Uk(ρ1, ρ2) = U1,k(ρ1) + U2,k(ρ2), we can split our RG-equation into anequation for the charged pion-condensate and one for the chiral condensate. Since we haveset ∂1∂2Uk(ρ1, ρ2) = 0 here, the coupling terms in Eq. (7.8) now vanish. We can then findtwo equations for U1,k and U2,k. They are, after the Matsubara sums have been performed,

∂kU1,k =8kdvd−1

d− 1

[(1

2+µ2I

ω2k

)1 + 2n(ω1)

2ω1+(1

2−µ2I

ω2k

)1 + 2n(ω2)

2ω2

], (7.17)

∂kU2,k =4kdvd−1

d− 1

[1 + 2n(ω3)

2ω3+

1 + 2n(ω4)

2ω4

], (7.18)

respectively. Here we have defined

ω4 =√k2 + ∂2Uk + 2ρ2∂2

2Uk, (7.19)

and ω1-ω3 are as defined above. Note that ∂iUk(ρ1, ρ2) = U ′i,k(ρi). The first of these equa-tions now describes a pure BE condensate, that is, there is no longer any interference fromthe neutral particles π0 and σ, as the off diagonal coupling terms have been set to zero.

The two equations can now be integrated numerically.

Pion Condensate

We are particularly interested in the charged pion condensate. This is described by theexpectation value of ρ1. We thus tackle Eq. (7.17), and let U1,k(ρ1) = Uk(ρ) for ρ1 = ρ. Weassume a boundary potential at scale k = Λ of the form

UΛ(ρ) = (m2Λ − µ2

I)ρ+ 4g2Λρ

2 + 8h2Λρ

3. (7.20)

The last term has been included as the cubic coupling is marginal for high temperatureswhere we have dimensional reduction as discussed in Chapter 4 (for the case of four spacetime


dimensions). However, for T = 0 which is where we start when we want to determine theparameters, the cubic term is non-renormalizable and should be discarded. As usual, we usea third order RK method with finite difference methods to approximate the derivatives whenintegrating the equation. Details are shown in Appendix B.

Determining the parameters

We determine the parameters of the model in Eq. (7.20) by first considering vacuum (i.e.T = µI = 0). If this is to be a model for the charged pion condensate, we should requirem2k=0 = U ′k=0(0) = m2

π, where we set mπ = 140 MeV, thus leaving the chiral limit. We tunethe parameters at scale Λ in order to achieve this. This takes care of one parameter, e.g. m2

Λ.What about the coupling g2

Λ? We will use the same coupling that we found when consideringthe O(4) model at zero chemical potential. At this level of approximation, this seems like agood choice. We recall that the parameters where determined so as to obtain the pion decayconstant fπ = 93 MeV, and mσ = 400 MeV. With these conditions, we found g2

Λ ≈ 10.

We thus set g2Λ = 10, and use the previously mentioned condition to determine the pa-

rameter m2Λ. If we set e.g. Λ = 600 MeV, we find that we must set m2

Λ = −0.58Λ2 in order toarrive at mπ = 140 MeV. Note that this is again heavily renormalized when integrating downto k = 0. We use these bare parameters for nonzero temperature and chemical potential aswell, and plot a phase diagram in the µI − T plane, see Fig. 7.1.

Since we have done a great deal of simplifications in order to get to this point (consider-ing only 2-flavor QCD, working in the LPA, BE condensate and so on), we do not expectthe phase diagram to exactly predict the behavior of the charged pion condensate. We dohowever qualitatively expect to see a behavior comparable with other approaches and models.Comparing with e.g. [50, 43, 66] we see that this is what we get. It is also similar to Fig. 8.4in the next chapter, where we attack the full RG Eq. (7.10).

7.2.3 Large-N limit

We recall that when we let N →∞, the truncation of higher order kinetic terms in the LPA isno longer an approximation. This was shown for the case of an O(N) symmetric theory, butthe inclusion of a chemical potential does not alter the large-N RG equation. The inclusionof a chemical potential gives rise to a 3 × 3 non-diagonal matrix in the propagator, and a(N − 3)× (N − 3) diagonal matrix. Taking the trace and the limit N →∞, it is easy to seethat the diagonal terms dominate. Hence, the equation is not altered in the large-N limit.

Hence, the result that the PLA becomes exact holds for µI 6= 0 as well. Thus, for largeN , we expect our model to be closer to the exact solution than for smaller N . It would there-fore be interesting to see what happens as one takes N →∞, i.e. what is the RG equation inthis limit.


Fig. 7.1: A Phase diagram for a BE condensate when considered as a model for the charged pioncondensation. We have used parameters mπ = 140 MeV to tune the mass parameter at thecutoff scale Λ = 600 MeV in this model. The bare coupling was set to g2

Λ = 10.

Large-N RG equation

We assume that we are dealing with N complex fields as in [50], i.e. we let O(2) × O(2) →O(2)×O(2N − 2). In the large-N limit, the inverse propagator appearing in Eq. (7.7) thengoes from a 4× 4 matrix to a 2N × 2N matrix. if we generalize ρ2 to ρ2 = 1

2(φ23 + ..+ φ2

2N ),then instead of one term of the form ∂2Uk + q2 +Rk, we get 2N − 3 such terms continuing onalong the diagonal. To first order in N , these terms will be dominant as we invert the matrixand take the trace. Thus the RG equation becomes, after performing the Matsubara sum,

∂kUk ≈N4kdvd−1

d− 1· 1 + 2n(ω3)

ω3. (7.21)

If we pull out the −µ2Iρ1 term from the potential so that it is again O(N) symmetric at k = Λ,

we see that ∂2UΛ = m2Λ + 8g2

Λ(ρ1 + ρ2) = m2Λ + 8g2

Λρ where ρ = ρ1 + ρ2. Hence the RHSof Eq. (7.21) becomes O(N) symmetric. Thus the potential remains O(N) symmetric as weintegrate down to k = 0. This simplifies the numerics quite a lot.

By rescaling the fieldsφi →

√Nφi, (7.22)

and the various couplings of the potential with a factor of 1√N

for every negative mass dimen-

sion away from the mass dimension 1, i.e. m2k remains the same, g2

k →1N g

2k and so on, we get

the potential to rescaleUk(φ)→ NUk(φ). (7.23)


This takes care of the N at the RHS of Eq. (7.21). We thus arrive at the RG equation

∂kUk ≈4kdvd−1

d− 1· 1 + 2n(ω3)

ω3. (7.24)

in the large-N limit. Doing the rescaling in this way also ensures that the formulas for mσ,mπ and fπ remain the same after rescaling. That is, the formulas are the same when usingthe rescaled potential. We investigate the large-N limit a bit further in the next chapter.


8. NUMERICAL RESULTS; CHARGED PION CONDENSATION

Since solving the simpler O(2) symmetric BE condensation went so well, we now try tosolve the full RG equation, Eq. (7.10) found in the LPA. Writing the full equation after theMatsubara sum has been performed is simply too cumbersome and would take up severalpages, but there might be some tricks we can apply, and maybe use numerical methods tosolve the problem.

8.1 Leaving it to the numerics

Writing out the roots of the third degree polynomial in ω2n that occurs in the numerator of the

first term on the RHS of Eq. (7.10) would take several pages. However, they can in principlebe found so we may for the sake of argument assume that we have found them. We recall theRG equation

∂kUk =4Tkdvd−1

(d− 1)

∑n

[N(ω2

n)

D(ω2n)

+1

ω2n + k2 + ∂2Uk

](8.1)

where we have defined

N(ω2n) = 2(k2 + ω2

n + ∂2Uk + 2ρ2∂22Uk)(k

2 + ω2n + ∂1Uk + ρ1∂

21Uk)− 4ρ1ρ2(∂1∂2Uk)

2

+ (k2 + ω2n + ∂1Uk)(k

2 + ω2n + ∂1Uk + 2ρ1∂

21Uk) + 4µ2

Iω2n, (8.2)

D(ω2n) = (ω2

n + k2 + ∂2Uk + 2ρ2∂22Uk)

[(k2 + ω2

n + ∂1Uk + 2ρ1∂21Uk)(k

2 + ω2n + ∂1Uk)

+ 4µ2Iω

2n

]+ 4ρ1ρ2(∂1∂2Uk)

2(k2 + ω2n + ∂1Uk). (8.3)

Note that setting µI = 0 will reduce Eq. (8.1) to the fully O(4)-symmetric Eq. (4.24) thatwe solved in Chapter 5. One then has Uk(ρ1, ρ2) = Uk(ρ1 + ρ2). This results in only tworoots of the denominator D(ω2

n). That is both s2 = k2 + U ′(ρ1 + ρ2) and s2 = k2 + U ′(ρ1 +ρ2) + 2(ρ1 + ρ2)U ′′k (ρ1 + ρ2) are roots of the denominator, and they give rise to second orderand first order poles respectively. Hence we assume that µI > 0 from now on, so that weobtain three distinct poles (roots of the denominator) ω2

n = −P1,−P2,−P3, each giving riseto residues R1, R2, R3. We can thus write

N(ω2n)

D(ω2n)

=

3∑i=1

Riω2n + Pi

. (8.4)

If we use Eq. (A.7) in Appendix A, we find

∂kUk =4kdvd−1

(d− 1)

[ 3∑i=1

(1 + 2n(√Pi))Ri

2√Pi

+1 + 2n(ω3)

2ω3

]. (8.5)

8.2. ANALYZING THE EQUATION 82

The only problem now is finding the poles Pi and the corresponding residues Ri. We shallleave this to the numerics. Note that in doing so, we will undoubtedly make our code run abit slower. This is a price we pay for solving the full RG equation.

8.2 Analyzing the equation

We now turn to analyze Eq. (8.5) and derive some interesting results. We compare ourresults with those obtained using other methods to analyze the same problem. Note that Eq.(8.5) was derived in the chiral limit where we still have O(2)×O(2) symmetry. What aboutthe physical point where the pions are massive? We will see that adding a pion mass, andthus explicitly breaking the symmetry of the Lagrangian does not make it much harder. Weproceed to show why this is the case.

8.2.1 Including a pion mass

If we wish to go to the physical point, we remember that we need to add the term −Hφ4 tothe potential. Now comes the crucial question. When should we add the term? Can it beadded after the integration down to k = 0 has been performed? To answer this question weturn to analyze the RG equation a bit further. We recall that the general RG equation in theLPA for the potential, Eq. (4.9), reads

∂kUk(φ) =1

2Tr

∫q

[∂kRk(q)

( ∂2Uk∂φi∂φj

+ (q2 +Rk(q))δij

)−1], (8.6)

where the integral over momenta includes the Matsubara sum if T > 0. Note that we haveset µI = 0 in this equation. The following result does not however depend on µI . We see thatthe RHS of this equation depends on Uk(φ) only through its second derivative with respectto the fields. Hence, any term that is linear in the fields will not contribute to the RHS ofthis RG equation. As a result, we can neglect them in the potential as we integrate down tok = 0. They can be added at whatever scale k we like. This is also true for the term −Hφ4.It does not matter when we add it to the potential. We may just as well add it when we haveintegrated the O(2)×O(2)-symmetric chiral potential down to k = 0 using Eq. (8.5).

This observation is in agreement with what one finds if one renormalizes perturbatively aswell. Here one finds that the factor H retains its ”tree-level”, or bare, value as one renormal-izes. See e.g. [66]. In fact, the parameter H can be considered on the same footing as T andµI , as an external field imposed on the system in order to break the symmetry explicitly. Itis then obvious that H should not depend on the scale. This is what is done in e.g. [67, 68].

8.2.2 Interpreting results

As explained in Chapter 3, the minimum of the potential Uk=0(φ) gives the phase structure ofthe system as this describes the potential for the mean field φ. The field will tend to fluctuatearound the potential minimum φ0. The different phases of the system are connected to theminimum values the fields, i.e. what φi,0, where φi,0 = 〈φi〉 denotes the value of the field φiat the minimum of the potential. φi can be zero or different from zero describing two distinctphases. Thus, for an N -dimensional vector-field φ there are 2N different phases a priori. Sym-metries often tend to reduce this number of possible phases, as is also the case for our example.


In the chiral limit at nonzero isospin chemical potential we have an O(2) × O(2) symme-try. Hence there are four possibilities for the minimum (ρ1,0, ρ2,0). Either both are zero, inwhich case no condensate is formed, or ρ1,0 > 0 and ρ2,0 = 0 in which case we have a chargedpion condensate, or the opposite case, in which we have a chiral condensate, or both ρi,0 > 0,in which case both condensates form. As no chemical potential is introduced for the neutral σand π0, we will see later that it is the second case that takes place for moderate temperatures,i.e. a charged pion condensate is formed.

If we go to the physical point we note that a preferred direction is chosen for the chiralcondensate ρ2,0 = 〈ρ2〉 as well, namely that of φ3,0 = 0 and φ4,0 > 0. We shall also seethat a nonzero H turns the chiral phase transition (the phase transition at µI = 0) into acrossover. This is so since the derivative of Uk=0(φ) in the direction of φ4 is always negativeat φ4 = 0, it is −H, and thus φ4,0 will remain nonzero no matter how high the temperature is.

A multidimensional potential also makes it possible to have more exotic phase transitionsthan those described in Chapter 3 as transitions are possible between all the potentially 2N

different phases.

8.2.3 The chiral limit

We turn to study the phase diagram for pion condensation in the chiral limit in order to seeif any interesting things happen here. We solve Eq. (8.5) numerically, using a third orderRK method as before, and using finite difference methods to approximate the derivatives asexplained in Appendix A. The Matlab code is given in Appendix B.

We consider the boundary potential

UΛ(ρ1, ρ2) = m2Λ(ρ1 + ρ2)− µ2

Iρ1 + 4g2Λ(ρ1 + ρ2)2, (8.7)

i.e. the same model we considered in the O(4) symmetric case, except now we have includedan isospin chemical potential. We have defined ρ1 = 1

2(φ21 + φ2

2) and ρ2 = 12(φ2

3 + φ24). Note

again that we have only included relevant and marginal couplings for the model at T = 0, sothat the theory at the cutoff is renormalizable in the old perturbative fashion.

Vanishing chiral condensate

We determine the parameters m2Λ and g2

Λ in the vacuum by integrating down to k = 0 and us-ing Uk=0(φ) to calculate the parameters mπ, mσ, and fπ as before. Note that as the chemicalpotential is set to zero, we are back to the O(4)-symmetric case. If we use the same physicalparameters as before, fπ = 93 MeV, mσ = 400 MeV and Λ2 = 5m2

σ, we will obtain the samevalues for the couplings as before, i.e. m2

Λ = −0.96Λ2 and g2Λ = 9.39. Using these, we see

what happens when the chemical potential is turned on. We choose e.g. µI = 100 MeV andsee what happens. We plot the real part of Uk=0(ρ1, ρ2) in Fig. 8.1 (recall that an imaginarypart is generated to the potential after some RG time).

We see from Fig. 8.1 that adding an isospin chemical potential rotates the minimum of thepotential Uk=0(ρ1, ρ2) to the ρ1 axis. This is exactly what we expect. As previously stated,


the moment the isospin chemical potential becomes nonzero, a favored direction appears inthe (ρ1, ρ2)-plane, i.e. the direction of the charged pion condensate. This is in agreementwith the results of e.g. [50, 65, 66]. Recall that the phase is determined by the minimumof the potential Uk=0(ρ1, ρ2), and the isospin chemical potential lowers the potential in theρ1 direction, forming a charged pion condensate. But, when the chemical potential is intro-duced, the RG equation also changes, possibly and unexpectedly in such a way as to have aminimum for Uk=0(ρ1, ρ2) where ρ2 > 0 as well. We see from Fig. 8.1 that this is not the case.Hence, as expected, we get ρ1,0 > 0 and ρ2,0 = 0 in the chiral limit with a nonzero isospinchemical potential, where the zero again denotes the minimum of the potential. Thus forµI > 0, we have a charged pion condensate in the chiral limit. It would be interesting to see

Fig. 8.1: The real part of Uk=0(ρ1,ρ2)Λ4 using mσ = 400 MeV, fπ = 93 MeV, to tune the bare parameters

in the chiral limit, and setting Λ =√

5mσ. We have neglected the constant added to thepotential by renormalization and set µI = 100 MeV. Note that Uk=0(ρ1, ρ2) has its minimumon the ρ1 axis as expected (marked by a black dot), i.e. there is only a charged pion condensatepresent.

how much the inclusion of the chemical potential influences the renormalization of the poten-tial Uk(ρ1, ρ2). We consider the potential in the ρ2 direction, i.e. the direction where chiralcondensates may form, setting ρ1 = 0. Doing this, the explicit chemical potential dependentterm in the potential −µIρ1 is set to zero. Hence we consider only the renormalization effectof µI on the potential. We use the same parameters at the cutoff k = Λ =

√5mσ as above,

and compare the case µI = 0 to the case of e.g. µI = 30 MeV. We work at T = 0.

We have chosen to plot the real part of Uk(0,ρ2)Λ4 as a function of ρ2

Λ2 for k = Λ and k = 0in the case of zero and finite chemical potential as an example in Fig. 8.2. We find by con-sidering the graphs of Fig. 8.2 that the inclusion of a finite chemical potential has the effectof increasing the ρ2 value for which Uk(0, ρ2) takes its minimum. We also see from this thatthe renormalization is large when integrating from k = Λ to k = 0, just like in the O(4) caseconsidered in Chapter 4. It should be said that even though U0(0, ρ2) has a minimum for


ρ2 > 0, this does not imply a chiral condensate. One needs to look at U0(ρ1, ρ2) in the entire(ρ1, ρ2) plane and locate its minimum there in order to find the phase structure of the system.If this minimum takes place where ρ2 > 0, chiral condensation will occur.

8.2.4 The physical point

We now add a term −Hφ4 to the potential in order to consider the physical point. The bareparameters H, m2

Λ and g2Λ can then be determined using the equations derived in Chapter 6

in the vacuum. The vacuum potential reads

Uk(φ) = Uk(ρ)−Hφ4, (8.8)

where Uk(ρ) is the totally symmetric part of the potential, and ρ = ρ1 + ρ2. The equationsof Chapter 6 can thus be written as

m2π,k = U ′k

(ρ =

σ2k

2

), (8.9)

m2σ2,k = U ′k

(ρ =

σ2k

2

)+ σ2

kU′′(ρ =

σ2k

2

), (8.10)

fπ,k = σk, (8.11)

where σk = 〈φ4〉k = φ4,k,0 is the vev of φ4 when the symmetry-breaking term −Hφ4 has beenadded. Recall that adding such a term forces φ4 to get a vev, while the other fields fluctuate

Fig. 8.2: The real part of Uk(0,ρ2)Λ4 as a function of ρ2

Λ2 at T = 0 in the chiral limit. The potential isplotted for k = Λ, and for k = 0 in the case of µI = 0 MeV and µI = 30 MeV, where we haveneglected the constant added to the potential by renormalization. We have used mσ = 400MeV and fπ = 93 MeV in order to tune the bare parameters of the potential at the cutoffΛ =

√5mσ. The minima of the renormalized potentials have been marked with transparent

circles making them easier to see.


around zero. We can then tune H, m2Λ and g2

Λ so as to obtain physical values for the pionand sigma masses, and the pion decay constant at k = 0.

There is another way of tuning the bare parameters that reduces the number of parame-ters we need to determine from three to two. Working at k = 0, we know from Chapter 6that H = fπm

2π. Knowing both the pion mass and the pion decay constant means that H

need not be determined as we know what it must be already. We may then use the last twoof the three above equations to tune the two remaining parameters m2

Λ and g2Λ. This reduces

the time needed to tune these parameters considerably. The first of the above equations canthen be used to check our results. That is, see if our numerics works ok since we know thatthis equation should give the pion mass squared. We will use this procedure when tuning thebare parameters of UΛ(φ) at the physical point.

Chiral crossover

As we add the symmetry-breaking term −Hφ4 to the potential, we see that such a term givesrise to a non-vanishing chiral condensate. Note that due to the nature of this term, the chiralcondensate 〈φ4〉 will never vanish. The derivative in the φ4-direction of the potential at φ4 = 0is always −H, no matter how high the temperature is. Thus, at the physical point there is acrossover for the chiral condensate, not a phase transition. We plot this crossover at µI = 0for the chiral limit, where we have the usual phase transition for the O(4)-symmetric case,and where we use the O(4) symmetry to rotate the condensate in the direction of the φ4-axis.We also plot the crossover for mπ = 40 Mev and mπ = 140 MeV (the physical point). Wetune the bare parameters so as to obtain mσ = 400 MeV, mπ = 140 MeV, and fπ = 93 MeVat T = 0 for the physical point as described above. We use the parameters thus obtained forthe other two cases as well. We set Λ =

√5mσ ≈ 900 MeV again and plot the result in Fig.

8.3. We see that the behavior is similar to the results of [51].

Phase diagram

We now turn to plot a phase diagram in the T -µI plane for both the physical point and thechiral limit. Up to this point we have used the physical parameters mσ = 400 MeV, mπ = 140MeV, and fπ = 93 MeV to tune the bare parameters, and we have set Λ =

√5mσ ≈ 900

MeV. mπ = 140 MeV and fπ = 93 MeV have both been determined to a great accuracy byexperiment. This is however not the case for mσ. This is merely a resonance, the mass ofwhich is not well defined, but is known to be in the range of 400−800 MeV. We are thereforefree to let mσ be larger if we want. As for the cutoff Λ, this should be chosen to be greateror equal to the largest physical energy scale of the theory, which in this case is the σ-mass.We are therefore free to vary this as well.

We change the cutoff to Λ = mσ = 600 MeV (the lowest value possible) in order to alsotest the numerics at a different cutoff value as well, and to see how much the theory dependson the cutoff. It could also be interesting to choose new values for the physical parameters,for example mσ = 600 MeV, mπ = 140 MeV, and fπ = 93 MeV, the same values that areused in [50]. At the physical point, we set H = m2

πfπ = 0.00844Λ3. We also use Eq. (8.9)to calculate the pion mass in order to check the accuracy of our numerics. We calculatemπ = 142 MeV, which is pretty close to the physical value of 140 MeV that we used when


calculating H. This gives us confidence in our numerics. We now proceed to make a phasediagram for the charged pion condensate.

In the chiral limit we recall that the charged pion condensate is the only condensate presentat nonzero chemical potential when T = 0. If we go to the physical point however, we seethat a chiral condensate appears in the direction of φ4. For small chemical potentials, wefind that the chiral condensate is the only existing condensate, that is, the minimum of Uk(φ)lies on the φ4-axis and no charged pion condensation takes place. As we increase the chemi-cal potential however, the minimum will eventually move away from the φ4-axis and we getρ1,0 = 〈ρ1〉 > 0, i.e. a charged pion condensate is formed. At T = 0 we find that this happenswhen the chemical potential is about the vacuum pion mass. This is in agreement with theresults of e.g. [65, 66, 69]. We plot our results in Fig. 8.4.

In fact, it can be shown that the critical chemical potential should be exactly the pion massat T = 0. This is also often taken as the definition of the critical isospin chemical potentialµI,c [70]. Using the bare parameters we found at T = µI = 0, we see from Fig. 8.4 that wearrive at a slightly different value for the critical chemical potential, namely µI,c = 135 MeV.The reason we do not get the pion mass is probably that we are working in the LPA, wherewave-function renormalization is neglected. Recall that to include an isospin chemical poten-tial we should let ∂0Φ→ (∂0 + iµI)Φ, where we view the Lagrangian as a function of complex

Fig. 8.3: Chiral transitions 〈φ4〉〈φ4〉0 at µI = 0 for mπ = 0 MeV (dotted line), 40 MeV (dashed line)

and 140 MeV (full line), where 〈φ4〉0 denotes the chiral condensate at T = µI = 0, i.e.〈φ4〉0 = fπ = 93 MeV. We have used mσ = 400 Mev, mπ = 140 MeV and fπ = 93 MeV totune the bare parameters at T = 0 for the physical point. We have set Λ =

√5mσ MeV, and

used the same bare parameters obtained at the physical point for mπ = 40 MeV and in thechiral limit as well.


field variables. This term remains unrenormalized in the LPA, but in reality wave-functionrenormalization would induce changes in this term as we integrate down to k = 0. Hence, ifwe were to include wave-function renormalization, this would hopefully counter the effects ofthe chemical potential in the RG equation in such a way that we arrive at a critical chemicalpotential µI,c = mπ = 140 MeV. Wave-function renormalization has not been included in thisthesis. We have rather ignored such inconsistent effects, since the critical chemical potentialwe arrived at, µI,c = 135 MeV, is quite close to the real value of 140 MeV.

We also note that Fig. 8.4 is quite similar to Figure 5 of e.g. [43]. In this article theso-called Nambu-Jona-Lasinio (NJL) model has been used to study two-flavor pion conden-sation and, although the models are quite different, they arrive at a similar phase diagram aswe do. The phase diagram is also similar to those of e.g. [69, 71, 72, 73, 74] to mention a few,and it has the same structure as the phase diagram of BE condensation that we found earlier,Fig. 7.1. Also note that when µI → ∞, the curves for the chiral condensate and physicalpoint approach each other.

Fig. 8.4: A plot showing the phase diagram for the charged pion condensate in the chiral limit andfor the physical point. We have chosen physical values fπ = 93 MeV, mσ = 600 MeV andmπ = 140 MeV (mπ = 0 in the chiral limit) in order to tune the bare parameters at T = 0.We have also set Λ = 600 MeV. The symmetric phase is above the lines, while the brokenphase is below the lines.

Second-order transition

We recall that we dealt with a second order transition in the O(4)-symmetric case, i.e. in thechiral limit when µI = 0. We check that this is also the case for the charged pion condensate


at µI > 0 and for the physical point. We set µI = 150 MeV using the values for the physicalparameters and the cutoff as above. We plot 〈φ1〉 =

√2ρ1,0 as a function of temperature

for both the chiral limit and the physical point, where we have used the remaining O(2)symmetry to turn the condensate in the φ1-direction. We see from Fig. 8.5 that the phasetransitions in both the chiral limit and at the physical point are of second order.

Fig. 8.5: The relative charged pion condensate, normalized to its zero-temperature value 〈φ1〉0, as afunction of T for both the chiral limit and the physical point, where we have chosen physicalvalues fπ = 93 MeV, mσ = 600 MeV, and mπ = 140 MeV (mπ = 0 in the chiral limit) to tunethe bare parameters at T = 0. We have also set Λ = 600 MeV. We have used the remainingO(2) symmetry to rotate the condensate in the φ1-direction. Note the second order natureof both the transitions.

Competition between condensates

In addition to plotting the chiral crossover as a function of T for different values of the pionmass, we also plot the condensates at T = 0 at the physical point as a function of the chemicalpotential. We use the same physical parameters as above, and set Λ = 600 MeV as before. InFig. 8.6, we plot the condensates 〈φ1〉 and 〈φ4〉 as a function of µI , where we have used theremaining O(2)-symmetry to rotate the charged pion condensate in the φ1-direction again.Note the competition between the condensates. If we go to the chiral limit, the chiral con-densate would disappear completely. It is only because of a finite pion mass that they cancoexist. This is also in agreement with the results of e.g. [43, 69].

From Fig. 8.6 we also see that the phase transition for the charged pion condensate is ofsecond order, also when µI is varied keeping T constant, i.e. for all µI and T .

8.3. O(4)-APPROXIMATION AND THE LARGE-N LIMIT 90

8.3 O(4)-approximation and the large-N limit

As discussed above, the inclusion of an isospin chemical potential in the LPA will give rise tosome inconsistencies. These were ignored in the numerical calculations above. Another wayto handle this problem is to neglect the isospin chemical potential in our RG equation alltogether. This turns the RG equation into the O(4)-symmetric equation again. The chemicalpotential term −µ2

Iρ1 may then be added to the potential once this has been integrated downto k = 0. The potential at scale k = 0 may then be written as

Uk(φ) = Uk(ρ)− µ2Iρ1 −Hφ4, (8.12)

where ρ = ρ1 + ρ2 and Uk(ρ) denotes the totally symmetric potential as before. In order toget a charged pion condensate at k = 0, one needs a negative derivative in the ρ1 direction atthe minimum of Uk(φ) along the φ4 axis, i.e. at φ4,0. When µI is just large enough for thisto happen, we get

∂1Uk=0(φi<4 = 0, φ4,0) = 0 = ∂Uk(φ2

4,0

2)− µ2

I , (8.13)

which exactly states thatmπ = µI . (8.14)

Hence in this approximation we obtain the exact result µI,c = mπ at T = 0. Another nicething about this approximation is that the RG equation becomes O(4)-symmetric again. This

Fig. 8.6: The chiral condensate and the charged pion condensate as a function of isospin chemicalpotential at the physical point at T = 0. The remaining O(2)-symmetry has been used toturn the charged pion condensate in the φ1-direction. We have used physical parametersmσ = 600 MeV, mπ = 140 MeV, and fπ = 93 MeV, and set Λ = 600 MeV to tune the bareparameters.


simplifies the numerics, and reduce simulation times significantly.

Another RG equation that has this property is the large-N RG equation, i.e. Eq. (7.24).We recall that in the large-N limit, the LPA becomes exact. That is, no higher order deriva-tive terms are added to the effective action as we integrate down to k = 0 and wave functionrenormalization disappears. The isospin chemical potential also disappears from the RG equa-tion, rendering it O(N) symmetric again. Hence, in this case as well, we will get the criticalchemical potential µI,c = mπ at T = 0.

We turn to some numerical calculations using the above approximation and the large-Nlimit. We tune the bare parameters in order to obtain mσ = 600 MeV, mπ = 140 MeV, andfπ = 93 MeV as before. We now use a cutoff at the other end of the scale, i.e. we set Λ = 1500MeV in order to check if the numerics works for larger values of the cutoff as well, and alsocheck how cutoff dependent our theory is. We plot the resulting phase diagrams in Fig. 8.7.The first thing to note is how similar the phase diagrams are. We see for instance that the

Fig. 8.7: The phase diagram for the O(4)-approximation (solid line) and the large-N limit (dashedline), where we have set mπ = 140 MeV, fπ = 93 MeV and mσ = 600 MeV to determine theparameters at the cutoff Λ = 1500 MeV in the vacuum. We have also plotted the previousobtained phase diagram for comparison (dotted line).

diagrams both have a critical chemical potential of µI,c = mπ for T = 0 like we predictedabove. Note also that the phase diagram for the O(4)-symmetric approximation reaches thechiral limit line relatively fast as we increase the chemical potential. This takes a somewhatlonger time in the large-N case. The large-N phase diagram is also very close to that of Fig.6 in [50]. This is as it should be, since this figure is also calculated in the large-N limit usingthe same physical parameters as we have done here. We also compare the phase diagramswith that of Fig. 8.4. We see that the diagrams are close to each other, despite the fact that


two completely different cutoffs were used. This tells us that the model is relatively cutoffindependent.

Part IV

KAON CONDENSATION


9. FERMIONS IN QFT AND TFT

We will now turn to study kaon condensation, but before we do this, we need to have a briefreview of fermions in QFT and TFT.

Fermions, and in particular electrons and positrons will be needed when considering kaoncondensation with imposed charge neutrality. We therefore pause a bit to consider fermionsin TFT. We shall not go into great detail on how this is done, but consider some generalconcepts of fermions in the path-integral formalism. The reader is referred to e.g. [1, 2, 3, 24]or any introductory book on QFT or TFT for further details.

9.1 Fermion Algebra

Fermions in QFT (and TFT) are in the path-integral formalism described by so-called Grass-mann numbers (abbreviated Grassmanns). These obey the algebra

η, ρ = 0, (9.1)∫dη = 0, (9.2)∫dηη = 1, (9.3)

where , denotes the anti-commutator. It follows from the above that η2 = 0. This isjust a statement of the Pauli exclusion principle in disguise. We can complex conjugate aGrassmann number η to obtain a distinctly different Grassmann number η∗. More generallywe have

(cn∏i=1

ηi)∗ = c∗

n∏i=1

η∗i , (9.4)

where c is some complex constant. We want η∗η to be ”real”, that is (η∗η)∗ = η∗η. We seethat in order to achieve this, we must choose (η∗)∗ = −η.

Lets have a little fun with Grassmanns. Let A be a hermitian N ×N matrix for simplicity,and consider

I =

∫ ∏i

dηidη∗i e−η∗i Aijηj . (9.5)

We diagonalize A by the transformation Λ = U †AU = diag(λ1, .., λN ) where U is unitary.With this transformation we get η = Uρ and η∗ = U †ρ so that

I =

∫ ∏i

dρidρ∗i e

∑j ρ∗jρjλj =

∏i

∫dρidρ

∗i ρ∗i ρiλi = detA. (9.6)

9.2. FERMION LAGRANGIAN 96

We assumed A hermitian for simplicity, but this holds, in fact, for a general invertible matrixA.

9.2 Fermion Lagrangian

The Lorentz invariant Lagrangian describing a fermionic field reads

L = ψ(iγµ∂µ −m)ψ, (9.7)

where ψ is a Dirac spinor of Grassmanns, ψ = ψ†γ0, and the γµs are the so-called Diracγ-matrices, see e.g. [1, 2, 3] or any introductory book on QFT. The γ-matrices satisfy thealgebra

γµ, γν = 2ηµν . (9.8)

Applying the Euler-Lagrange equations to this gives

iγµ∂µψ = mψ. (9.9)

This is exactly the Dirac equation discovered by Paul Dirac in 1928 [75]. We will not gointo further detail on Dirac spinors or the γ-matrices here. For more details on this, see anyintroductory book on QFT or advanced QM.

The Lagrangian of Eq. (9.7) has a symmetry group and hence a conserved current. Thesymmetry group is in this case U(1), i.e. the transformation of the fields ψ → eiαψ andψ → e−iαψ leaves the Lagrangian invariant, similar to the symmetry transformations ofthe complex bosonic fields considered in the introduction. By Noether’s theorem there is aconserved current

jµ =∂L

∂(∂µψa)δψa, (9.10)

where δψa denotes a small variation of the fields. In the case of the fermion Lagrangian ofEq. (9.7), we have δψ = iαψ. The conserved current is thus

jµ = ψγµψ. (9.11)

This current may be related to the electromagnetic current by a proportionality constant e,i.e. jµem = ejµ, similar to the case of the bosonic theory we studied in the introduction. Infact, e is the coupling of the fermionic field to the gauge field Aµ, i.e. its the charge of theparticles described by ψ. Hence, the charge associated with the current given by Eq. (9.11)becomes

N =

∫xj0, (9.12)

i.e. the number operator.

In order to study fermions in statistical field theory, where we allow the system to exchangecharge densities, given by the zeroth component of this current, with the surroundings, weneed to find their grand canonical partition function. We recall from Chapter 2 that this isobtained from the usual partition function Z = Tre−βH by the substitution H → H − µρwhere ρ is the charge-density operator, and µ is the chemical potential associated with thischarge. We recall that ρ is the zeroth component of jµ, and as jµ contains no derivatives,

9.3. FERMION PARTITION FUNCTION 97

there is no need to go via the Legendre definition of the Hamiltonian in order to derive theLagrangian. We may simply add the term ψµγ0ψ to the Lagrangian of Eq. (9.7) in order toobtain

L = ψ(iγµ∂µ + µγ0 −m)ψ. (9.13)

9.3 Fermion partition function

Fermions in a thermal field theory have a quite similar partition function to that of bosons,only now the fields must be chosen antisymmetric in the imaginary time variable, that is

Z =

∫DψDψψ(x,0)=−ψ(x,β)e

−S[ψ,ψ], (9.14)

where S[ψ, ψ] is now the Euclidean action of the fermions, see e.g. [24] for further details.Note that the fields ψ and ψ are now considered as separate variables. The Euclidean actionreads

S[ψ] =

∫ β

0dτ

∫xψ(γ0(

∂

∂τ− µe)− iγ · ∇+me

)ψ, (9.15)

where the ∇ now only includes spatial derivatives. Note also that the chemical potentialhas been included in S[ψ], hence Eq. (9.14) gives the grand canonical partition function forfermions.

9.4 Fermion free energy

We are now ready to find the free energy in the fermionic case. We first Fourier transformthe fields

ψn,p =1√V β

∫ β

0dτ

∫xψ(x, τ)e−i(ωnτ+x·p), (9.16)

where we have gone to a finite volume denoted by V . Note that we also must use the fermionicMatsubara frequencies ωn = (2n+ 1)πT so that the fermion fields become anti-periodic in τ .The Euclidean action may then be written as

S =∑n,p

ψn,p

(γ0(iωn − µ) + γ · p +m

)ψn,p, (9.17)

where again there is a sum over p due to the finite volume. Using Eq. (9.6), the partitionfunction may then be calculated to give

Z =

∫D[ψ, ψ]e−S[ψ,ψ] = exp

∑n,p

Tr log(γ0(iωn − µ) + γ · p +m

), (9.18)

where we have used the formula

log(detA) = Tr(logA) (9.19)

for any invertible operator A, and the trace is now over the indices of the γ-matrices. Wethus obtain the free energy F = − logZ,

F = −∑n,p

Tr log(γ0(iωn − µ) + γ · p +m

)= −

∑n,p

log (ω2n + ε2−)(ω2

n + ε2+), (9.20)

9.4. FERMION FREE ENERGY 98

where we have defined ε± =√p2 +m2±µ, and used the formula Eq. (9.19) once again, this

time on the γ-matrices. We thus have a thermodynamic potential Ω = 1βV F given by

Ω = −T∑n

∫p

(log (ω2

n + ε2−) + log (ω2n + ε2+)

), (9.21)

where we have taken the limit V → ∞. This can be calculated by a similar procedure asfor the bosonic thermodynamic potential by use of Eq. (A.7) in Appendix A. Doing this, weobtain

Ω = −2

∫p

(Ep + T log

[(1− e−β(Ep−µ))(1− e−β(Ep+µ))

]), (9.22)

where Ep =√p2 +m2, the same as obtained in e.g. [76].

From this potential various thermodynamical quantities may be calculated. Eq. (9.22) willalso be of importance when we consider kaon condensation with imposed electric charge neu-trality in the next chapters.

10. KAON CONDENSATION

Now that we have studied the case of a O(2) × O(2)-symmetric theory with one chemicalpotential included, we could go further and ask what happens if we include two? Does sucha model describe any systems of physical relevance? The answer is yes, in fact it does. Inhigh energy particle physics it applies to e.g. kaons and kaon condensation in dense baryonicmatter. We shall turn to study this now.

10.1 The CFL-phase

We study the case of three flavors, i.e. we neglect the charm, bottom and top quark (dueto their heavy masses they can only interact virtually as described before). Consider thesymmetry group G = SU(3)c × SU(3)L × SU(3)R × U(1)B where the c denotes color, i.e.SU(3)c is the QCD gauge group, SU(3)L × SU(3)R is the chiral symmetry group and U(1)Bconstitutes baryon-number conservation. Recall from the last chapter that fermion-numberconservation is described by a U(1)-symmetry group. It is known that kaons are mesons, i.e.made of one quark and one anti-quark as described in the introduction. The kaons have thefollowing constituents, K0 = 〈ds〉 and K0 = 〈us〉 where the bar denotes anti quarks. It isusual to associate the overall quark density, also known as the baryon density, to a chemicalpotential µB. For large enough chemical potentials, the strange quark mass can be neglectedand the full symmetry group of the system G remains unbroken. It can then be shown that theso-called CFL phase is the ground state of all the states that allow for color superconductivity[76, 77, 78, 79].

10.1.1 SU(3)c+L+R-broken symmetry

As stated above, for sufficiently high densities and low temperatures, it is known that QCDcan be found in the so-called CFL phase [77, 78]. In this state the quarks combine to makeCooper pairs formed by the gluon-gluon interaction making it a color-superconducting phase[77, 78, 80] (it is color superconducting, as there is no way two quarks can be combined intoa color singlet). This is similar to electrons bound in Cooper pairs by a phonon interactionin a regular superconductor. In the CFL phase the group G is spontaneously broken to thegroup SU(3)c+L+R generated by linear combinations of the generators of G. These linearcombinations further lock rotations in color space together with rotations in flavor space,hence the name CFL-phase.

The Goldstone modes arising from the SSB in the CFL phase consists of an octet and asinglet. The octet is due to the broken chiral symmetry, while the singlet is due to the brokenbaryon number symmetry U(1)B. This is just like in vacuum, except that the mesonic statesnow are made up of an antiquark doublet combined with a quark doublet, (qq)(qq), instead ofa quark-antiquark doublet (qq) as in vacuum. This will become apparent later. The broken

10.2. EFFECTIVE LAGRANGIANS 100

baryon number symmetry U(1)B is also responsible for the superfluidity of the phase, makingcolor superconductivity possible.

At sufficiently high densities µB, the Goldstone modes can be thought of as massless andmay be described by an effective field theory similar to the effective theory of pion condensa-tion described in the previous chapters. In the CFL-phase, all excitations of fermionic degreesof freedom are suppressed by an energy gap ∆ ≈ 30 MeV known as the superconducting en-ergy gap. Thus, if we stick to energies below this gap, the Goldstone modes of the CFL phasemay be sufficiently described by an effective theory of bosonic degrees of freedom [77, 78, 79].This is similar to the bosonic description of the previously described pion condensate. Wetherefore start by setting the cutoff scale to Λ = ∆. We shall postulate an effective boundarypotential UΛ(φ), φ denoting the bosonic degrees of freedom, where the parameters of UΛ(φ)are determined as effective parameters from the UV theory as discussed in the introduction.

10.1.2 Inside a neutron star

The mass of the strange quark can be neglected for sufficiently high chemical potentials µBand the Goldstone modes remain massless. For moderate µB this is no longer the case, andchiral symmetry will be explicitly broken rendering only the superfluid mode massless [76, 77].This is the case for densities inside e.g. a neutron star where µB ≈ 500 MeV, approximatelythe constituent mass of the strange quark.

The Goldstone mesons of the octet therefore acquire a mass, and contrary to ordinary vacuumQCD, the lightest mesons are the charged and neutral kaons [76, 77, 78, 79]. We will thereforeintroduce an effective theory for these modes, assuming that the more massive modes decou-ple. We shall also introduce chemical potentials for these modes, allowing for the formationof a BE condensate for large enough chemical potentials.

10.1.3 Charge neutrality

The inside of a neutron star is overall electrically neutral due to the otherwise huge energycost required [76, 81]. This imposes another constraint on the system. The charge neutralityis usually dealt with by introducing an electron-positron background to the Lagrangian andthen a chemical potential µQ for the electric charge. One thus requires charge neutrality as thecalculations are performed, i.e. the total electric charge from the kaons and electron-positronbackground combined is zero. We shall see how this is done in detail later.

It could also be mentioned that when studying the high density phase using other models, e.g.the NJL model, one often has to enforce color neutrality as well as this is not incorporated inthe model a priori. This is because the NJL model has only global symmetries, and no gaugesymmetries that would give rise to a color neutrality condition. See e.g. [82, 83] for examplesof this.

10.2 Effective Lagrangians

We now turn to find the effective Lagrangian for the mesonic degrees of freedom and in par-ticular the kaons. We follow the derivation of [77], [78], and [79]. As the derivation of this


Lagrangian is less important to the main subject of this thesis, the following subsection maybe skipped for the impatient reader. The important result of this derivation is Eq. (10.20),describing the Lagrangian for the kaon fields.

The following derivation also assumes some basic familiarity with particle physics and grouptheory.

10.2.1 Octet Lagrangian

Recall that quarks are fermions, hence a di-quark wave function must be antisymmetric. Itis known that the formation of Cooper pairs by di-quark condensation takes place in theantisymmetric, anti-fundamental (anti-triplet) representation [3]c of the gauge group SU(3)c[77, 78, 79]. If one assumes that the condensation happens in the antisymmetric spin singletchannel as well, a reasonable assumption due to energy considerations as can be seen from anybook on elementary particles, e.g. [84], we must choose the anti-symmetric representation inflavor space in order to have a totally anti-symmetric wave function. This means that we musthave the total wave function Ψ contained in the antisymmetric, anti-triplet representation[3]L ⊕ [3R] of the chiral group SU(3)L × SU(3)R, i.e.

Ψ ∈ [3]c ⊗ ([3]L ⊕ [3R]). (10.1)

Denote the basis of [3]L, [3]R and [3]c by ILi , IRi and Ji respectively, where (ILi )jk = (IRi )jk =(Ji)jk = −iεijk. We may then write the order parameter for color superconductivity as

Ψ = (∆LijJi ⊗ ILj )⊕ (∆R

klJk ⊗ IRl ), (10.2)

where the ∆’s are complex 3× 3 matrices. They transform as

∆L → G†c∆LGL, (10.3)

∆R → G†c∆RGR, (10.4)

under the symmetry group G. Here Gc ∈ SU(3)c, GL ∈ SU(3)L and GR ∈ SU(3)R respec-tively. If we let ∆L

ij = ∆Rij ∼ δij this breaks the symmetry group SU(3)c × SU(3)L × SU(3)R

down to the group SU(3)c+L+R. Hence Ψ becomes invariant under simultaneous rotationsof color, left and right flavor space. This means that chiral symmetry is broken. This is theso-called CFL phase, and it is the ground state in the color-superconductivity phase space[76, 77, 78, 79].

The low-energy mesonic excitations are associated with rotations of the phases of ∆l and∆R [85, 86]. As we are interested in finding an effective theory for the mesonic degrees offreedom, we neglect the norm of the ∆’s, an thus assume them to be unitary.

We define the mesonic field asΣ = (∆L)†∆R. (10.5)

As the ∆’s are unitary, so is Σ. Thus Σ ∈ U(3) and has nine degrees of freedom. Oneof these is the η′ connected to the breaking of the U(1)-axial symmetry. This symmetry ishowever violated at the quantum level for moderate densities due to the axial anomaly, i.e.instanton interactions. In the introduction we briefly discussed how such quantum anomalies


may occur. For more information regarding anomalies, see e.g. [1] Chapter 19, or [87]. As theU(1)-axial symmetry is broken at the quantum mechanical level, we will ignore it in the fol-lowing and assume Σ ∈ SU(3). It should, however, be mentioned that the violation becomesnegligible for asymptotically high µB due to a screening factor 1

µBsuppressing the instanton

density. Therefore, at asymptotically high densities, the U(1)-axial symmetry is restored [71].

The mesonic field Σ is easily seen to be a color singlet and it transforms under the chiralgroup as Σ → V †LΣVR where (VL, VR) ∈ SU(3)L × SU(3)R. Note that Σ is a product of aleft-handed anti-diquark condensate and a right-handed diquark condensate and thus carriesthe quantum numbers of two flavors and two antiflavors. Writing the flavors as (u, d, s), wefind

Σ =

(ds)L(ds)R (ds)L(us)R (ds)L(ud)R(us)L(ds)R (us)L(us)R (us)L(ud)R(ud)L(ds)R (ud)L(us)R (ud)L(ud)R

. (10.6)

Note that the new high density mesonic states are composed by states of the form qqqq asopposed to the usual chiral condensates qq in the vacuum. Nevertheless, similar quantumnumbers are found.

The effective Lagrangian for the meson fields may now be written as [76, 77, 78, 79, 85, 86]

L =1

4f2πTr[(∂0Σ+i[A,Σ])(∂0Σ†−i[A,Σ†])−v2

π∂iΣ∂iΣ]+af2

π

2detMTr

[M−1(Σ+Σ†)

](10.7)

where we have defined the ”gauge field”

A = µQQ−M2

2µ. (10.8)

Here, Q = diag(23 ,−

13 ,−

13) and M = diag(mu,md,ms) give the electric charges and masses

of the various flavors respectively. Note the factor v2π in front of the spatial kinetic term

signaling the fact that we are no longer in vacuum, but rather in a dense quark medium. Inthe vacuum, Lorentz invariance would force vπ = 1. In a high density state, like the one weare considering here, this is no longer the case. As discussed in Chapter 2, the inclusion of achemical potential will break Lorentz invariance and hence we can no longer expect this tobe a viable symmetry of our Lagrangian.

For asymptotically high densities we may treat the particles of the system as ultra rela-tivistic massless particles. In the rest frame of the dense medium we have the equation ofstate

p =1

3E , (10.9)

where E denotes the energy density and p the pressure of this rest frame. This formula canbe found in most introductory books to general or even special relativity, e.g. [88, 89, 90].The speed of sound v for a relativistic fluid in this reference frame is given by [89, 91]

v2 =∂p

∂ε. (10.10)

Thus the speed of sound in a high density medium is v = 1√3, a bit more than half of the

light speed. In [71] they show that the meson associated with the phase of the baryonic


symmetry group U(1)B acquires a velocity vB = 1√3

at asymptotically high densities. This

is not surprising as the U(1)B phase is the variable conjugate to baryonic density variations,which are responsible for sound waves. They also show that this holds for the other mesonsas well, i.e. vπ = vB = 1√

3.

By extrapolating to high densities and using perturbative QCD, one may calculate the otherconstants, fπ and a, as well. They are found to be [71, 92, 93]

f2π =

21− 8 log 2

18π2µ2B, (10.11)

a =3∆2

π2f2π

, (10.12)

where ∆ is the previously mentioned color-superconducting gap. Note that these are onlyvalid for extremely high densities. For more moderate densities, such as those found in aneutron star, they may differ.

10.2.2 Expanding the Lagrangian

We now write Σ = ei θfπ . If we subtract the CFL ”ground state” part Σ = 1 from the

Lagrangian, we may write the new Lagrangian as L = L0 + L1 where [77]

L1 =f2π

2Tr

[A2 −

(A cos θ′

)2−(A sin θ′

)2− 2a(detM)M−1(1− cos θ′)

], (10.13)

L1 =f2π

4Tr

[(∂0 cos θ′)2 + (∂0 sin θ′)2 − v2

π

((∇ cos θ′)2 + (∇ sin θ′)2

)]+ i

f2π

2Tr

[(∂0 cos θ′[A, cos θ′]) + (∂0 sin θ′[A, sin θ′])

], (10.14)

where θ′ = θfπ

, and where the ∇ again denotes the spatial derivatives. We expand these tofourth order in the mesonic field θ in order to obtain

L0 =1

2Tr[Xθ2 − (Aθ)2

]+

1

2f2π

Tr

[1

3(Aθ)2θ2 − 1

12Xθ4 − 1

4(Aθ2)2

], (10.15)

L1 =1

4Tr

[(∂0θ)

2 − v2π(∇θ)2 + 2i(∂0θ)[A, θ]

], (10.16)

where we have definedX = A2 − adet(M)M−1 (10.17)

in order to simplify the writing. We wish to find a potential for the mean field 〈θ〉, i.e. apotential we can use as our boundary condition at the scale Λ = ∆. We thus proceed to meanfield theory and write θ → θ + Φ where Φ = 〈θ〉 is the mean field and θ denotes fluctuationsaround the mean. Using these definitions, we obtain the potential

U(Φ) =1

2Tr[XΦ2 − (AΦ)2

]+

1

2f2π

Tr

[1

3(AΦ)2Φ2 − 1

12XΦ4 − 1

4(AΦ2)2

]. (10.18)

10.3. CHARGE NEUTRALITY CONDITION 104

10.2.3 Kaon effective Lagrangian

We proceed to define new fields φa by

Φ = 〈θ〉 = φaTa, (10.19)

where the Ta’s are the eight Gell-Mann matrices [94]. As the kaons are expected to be thelightest modes [76, 77, 85, 86], we neglect the other modes and consider only the kaons, i.e.a = (4, 5, 6, 7). We then obtain the O(2)×O(2)-symmetric effective Euclidean Lagrangian

L =[(∂0 + µ0)Φ∗1

][(∂0 − µ0)Φ1

]+ v2

π(∂iΦ∗1)(∂iΦ1) +

[(∂0 + µ+)Φ∗2

][(∂0 − µ+)Φ2

]+ v2

π(∂iΦ∗2)(∂iΦ2) +m2

0Φ∗1Φ1 +m2+Φ∗2Φ2 +

λ0

2

(Φ∗1Φ1

)2

+λ+

2

(Φ∗2Φ2

)2+ λH

(Φ∗1Φ1

)(Φ∗2Φ2

), (10.20)

where we have defined the new complex fields K0 = Φ1 = 1√2(φ6 + iφ7) and K+ = Φ2 =

1√2(φ4 + iφ5) and where we have defined new chemical potentials, masses, and couplings

[76, 77]:

µ+ = µQ +m2s −m2

u

2µB, (10.21)

µ0 =m2s −m2

d

2µB, (10.22)

m2+ = amd(ms +mu), (10.23)

m20 = amu(ms +mu), (10.24)

λ0 =1

3f2π

(4µ20 −m2

0), (10.25)

λ+ =1

3f2π

(4µ2+ −m2

+), (10.26)

λH =1

2(λ+ + λ0)− 2

(µ0 − µ+

2fπ

)2

. (10.27)

This is the effective microscopic Lagrangian for the kaon degrees of freedom below the color-superconducting gap ∆. It is the Lagrangian we shall use as the boundary condition at thecutoff when we derive the non-perturbative RG-equation for the kaon effective potential inthe next chapter.

It should be noted that the above expansion of the potential in the fields is only valid for smallmean field values φ. This means that the value φ0 at the potential minimum cannot becometoo large in order for the model to be valid. We are concerned with φ0 as this describes thekaon condensate. Recall that the Lagrangian of Eq. (10.20) should describe the microscopicphysics at renormalization scale k = Λ. This in turn means that we should have m2

i − µ2i

small, i.e. mi ≈ µi where i denotes 0 or +.

10.3 Charge neutrality condition

We now turn to consider the effects of an electric charge neutrality condition imposed onthe system. As discussed above, one often requires charge neutrality for the kaon condensate


when considering physical systems, e.g. neutron stars. It could therefore be interesting toinclude a neutrality condition on the model.

10.3.1 Electron-positron background

The neutrality condition is usually achieved by adding an electron background Le to theLagrangian of Eq. (10.20) of the (Euclidean) form

Le = ψ(iγµ∂µ + γ0µee−me)ψ, (10.28)

where ψ is the electron field, µe is the electron chemical potential, e is the electric charge andthe γ’s are the Dirac gamma matrices. This is the fermion Lagrangian from Eq. (9.13) de-scribing a background of electrons and positrons, where µe is the chemical potential associatedwith the electric charge density . The electron thermodynamic potential is

Ωe = −2

∫p

Ep + T log

[(1− e−β(Ep−µQ))(1− e−β(Ep+µQ))

], (10.29)

where Ep =√p2 +m2

e and µQ = eµe. If we neglect the electron mass in the thermody-namic electron-potential and disregard the vacuum-energy contribution from the Ep term,the temperature dependent integrals may be done to give [76]

Ωe = − 1

12π2µ4Q −

1

6µ2QT

2 − 7π2

180T 4. (10.30)

In general, we recall there is a Legendre transformation between the Helmholtz free energyF [J ] and the Gibbs free energy Γ[Φ], Eq. (2.19), where Φ again denotes the mean field. Asthe external field is here set to zero, and Ωe is independent on the fields, Ωe may be simplyadded the effective potential as a constant term. Note also that Ωe is only dependent onexternal parameters T and µe, and thus will not change as we renormalize the potential.

10.3.2 Renormalizing the theory

We shall use the same procedure as before when renormalizing, deriving an RG equation forthe potential Uk(Φ), where the boundary condition at k = Λ is given by Eq. (10.20) combinedwith Eq. (10.30). Integrating down, one obtains the full effective potential Uk=0(Φ). At theminimum of the potential Φ0 we have [95]

Uk=0(Φ0) = E − TS − µini, (10.31)

where E again denotes the energy density, S the entropy density, µi the various chemicalpotentials (µ0, µ+ and µQ), and ni the associated particle densities. The electric chargedensity from the electron-positron background, measured in units of e = 1, is thus given by

ne = −∂Uk=0

∂µQ=

1

3π2µ3Q +

1

3µQT

2. (10.32)

This does not vary with renormalization scale, i.e. choosing a different k would produce thesame result.


Recall that the kaons are basically given as K0 = 〈ds〉 and K+ = 〈us〉. This gives

µ0 = µd − µs, (10.33)

µ+ = µu − µs, (10.34)

as particles and antiparticles have opposite chemical potentials as we recall from Chapter 6.Thus we have µ+−µ0 = µu−µd. If we assume equilibrium of the weak process d+ν u+e−

and that the neutrinos leave the system so that µν = 0, this implies µQ = µ+ − µ0.

The contribution to the scale dependent charge density from the kaons then becomes

nk,+ = −∂Uk(Φk,0)

∂µ+= 2µ+Φ∗2,k,0Φ2,k,0 −

∫ k

Λdk

∂

∂µ+∂kUk|Φ=Φk,0 , (10.35)

where Φ2,k,0 denotes the minimum of Uk(Φ) at the scale k. The full contribution to the chargedensity from the kaons is given at k = 0, where Uk=0(Φ) is the potential of the full effectiveaction including all thermal and quantum fluctuations.

Requiring charge neutrality means that we should set the total charge density to zero, i.e.

ne + nk=0,+ = 0. (10.36)

This gives another relation between µ+, µ0, and µQ leaving only one of them free to vary.We shall investigate the effects of this charge neutrality condition in Chapter 12.

11. THE KAON RG EQUATION

In this chapter, we derive the full RG equation for the case of kaon condensation in the LPA.It is similar to that of the pion condensation case, except that one now includes two chemicalpotentials, and the masses differ for the different fields. Also, more couplings are introducedat the scale k = Λ. Thus the equation is bound to be more complicated. In return, as weshall see, the parameters at scale Λ will be easier to determine, as they are given by Eqs.(10.21) to (10.27) obtained as effective parameters from the higher-energy theory.

11.1 Zero temperature Gibbs free energy

We recall the effective kaon Lagrangian given by Eq. (10.20) of the last chapter. This is thenon-renormalized (but perturbatively renormalizable) bare Lagrangian density at the scalek = Λ, and thus provides us with our initial condition when we integrate down to k = 0 fromk = Λ. Working in the LPA and setting T = 0 for simplicity, we may, from Eq. (10.20),postulate the form of the kaon Gibbs free energy in a dense medium

Γk[φ] =

∫x

[1

2(∂0φ)2 +

v2π

2(∇φ)2 + iµ0(φ1∂0φ2 − φ2∂0φ1) + iµ+(φ3∂0φ4 − φ4∂0φ3) + Uk(φ)

],

(11.1)where φ again denotes the mean field, v2

π = 13 , ∇ denotes the spatial derivatives, and the

integral is over Euclidean spacetime. Recall that the value vπ = 1√3

was found by extrapolating

to asymptotically high densities, and may thus differ from this value at moderate densities.We shall, nonetheless, use this value in the rest of the thesis. If we write

Φ1 =1√2

(φ1 + iφ2), (11.2)

Φ2 =1√2

(φ3 + iφ4), (11.3)

we see that this reduces to the classical Gibbs free energy at the cutoff Λ as before.

As this is an O(2) × O(2)-symmetric model, we know that we can assume the potentialUk(φ) to be of the form Uk(ρ1, ρ2) where ρ1 = 1

2(φ21 +φ2

2) is associated with the K0 field, andρ2 = 1

2(φ23 + φ2

4) is associated with the K+ field. We can read off the boundary condition atk = Λ from Eq. (10.20) as

UΛ(ρ1, ρ2) = (m20 − µ2

0)ρ1 + (m2+ − µ2

+)ρ2 +λ0

2ρ2

1 +λ+

2ρ2

2 + λHρ1ρ2, (11.4)

where the parameters of the potential are given by Eqs. (10.21) to (10.27). This is thebare potential we wish to integrate down from. In order to do so, we need to derive an RGequation.

11.2. RG EQUATION 108

11.2 RG equation

The procedure to derive the RG equation is the same as that of Chapter 7. One startsfrom Eq. (A.24) in Appendix A for the Gibbs free energy and derives an RG equation forthe potential Uk(ρ1, ρ2) where the full k-dependent propagator appears on the RHS of theequation. As the derivations are so similar, the only difference is that we associate a chemicalpotential with the fields φ3 and φ4 as well, we omit this part of the derivation and write downthe result

∂kUk =1

2Tr

∫q∂kRk×

∂1Uk + 2ρ1∂21Uk + Fk(q) −2µ0q0 2

√ρ1ρ2∂1∂2Uk 0

2µ0q0 ∂1Uk + Fk(q) 0 02√ρ1ρ2∂1∂2Uk 0 ∂2Uk + 2ρ2∂

22Uk + Fk(q) −2µ+q0

0 0 2µ+q0 ∂2Uk + Fk(q)

−1

,

(11.5)

where Fk(q) = v2πq

2 + Rk(q) and the trace is over the matrix. Note the similarity betweenthe propagator on the RHS of this equation and the inverse propagator of [76]. Turning onthe temperature is now easy. Simply replace the integral over q0 by a sum over Matsubarafrequencies, i.e.

∫dq0 → T

∑n, while q0 → ωn.

11.3 Equation for T > 0

We wish to do the integral in Eq. (11.5) analytically and we thus use the previous usedregulator Rk(q) = (k2 − v2

πq2)θ(k2 − v2

πq2), rescaled to account for the fact that we are in a

dense medium. With this regulator, the RG equation for the potential becomes

∂kUk =4Tvd−1k

d

v3π(d− 1)

∞∑n=−∞

Tr×

∂1Uk + 2ρ1∂

21Uk + ω2

k −2µ0ωn 2√ρ1ρ2∂1∂2Uk 0

2µ0ωn ∂1Uk + ω2k 0 0

2√ρ1ρ2∂1∂2Uk 0 ∂2Uk + 2ρ2∂

22Uk + ω2

k −2µ+ωn0 0 2µ+ωn ∂2Uk + ω2

k

−1

. (11.6)

where ω2k = k2 +ω2

n. Writing the trace of the RHS of this equation specifically is unnecessarycompared to the small amount of new insight it will bring. Instead we analyze the currentform of the equation.

We see that the trace of the inverse matrix on the RHS of the equation has the form ofa third degree equation in ω2

n divided by a fourth degree equation in ω2n. We shall assume

that the fourth degree polynomial has four different roots (a reasonable assumption, giventhe complexity of the above expression). A check that this is the case will be implemented inthe numerics. The equation may then be written as

∂kUk =4Tvd−1k

d

v3π(d− 1)

∑n,i

Riω2n + Pi

(11.7)

11.4. HIGH-TEMPERATURE LIMIT 109

where Ri denotes the residue of the i’th pole −Pi. We have thus split the fraction that is thetrace of the inverse matrix into four more manageable terms. Using Eq. (A.7) in AppendixA we may write this as

∂kUk =4vd−1k

d

v3π(d− 1)

∑i

(1 + 2n(√Pi))Ri

2√Pi

. (11.8)

This is a much shorter form of the equation. We shall leave it to the numerics to find theRi’s and Pi’s, and from now on focus on the results of this equation. In the remainder of thethesis, we set d = 4.

11.4 High-temperature limit

We will later find that the mass parameters and chemical potentials in the bare potentialUΛ(ρ1, ρ2) are of order a few MeV which is smaller than the cutoff scale Λ, while the couplingsof the potential are also quite small. This will result in a high critical temperature comparedto, say, the mass parameters of UΛ(ρ1, ρ2) or even Λ. It is therefore instructive to find an RGin this limit. Expanding the RHS of Eq. (11.6) in 1

ω2n

, it can be shown that

∂kUk =Tk4

v3π6π2

[(6ρ1∂

21Uk

(4

3ρ2∂

22Uk(

1

2(∂2Uk + ∂1Uk) + k2)

+ (∂2Uk + k2)(k2 +1

3∂2Uk +

2

3∂1Uk)

)+ 6(∂1Uk + k2)(k2 +

1

3∂1Uk +

2

3∂2Uk)ρ2∂

22Uk

+ 4(

(k2 + ∂2Uk)(k2 + ∂1Uk − 2ρ1ρ2(∂1∂2Uk)

2))

(k2 +1

2∂1Uk +

1

2∂2Uk)

)×(

(k2 + ∂1Uk)(k2 + ∂2Uk)

(2ρ1∂

21Uk(k

2 + ∂2Uk + 2ρ2∂22Uk) + 2ρ2∂

22Uk(k

2 + ∂1Uk)

− 4ρ1ρ2(∂1∂2Uk)2 + (k2 + ∂1Uk)(k

2 + ∂2Uk)))−1

+∑n6=0

(4

ω2n

−(

2ρ1∂21Uk + 2ρ2∂

22Uk + 2∂2Uk + 2∂1Uk + 4j2 + 8µ2

0 + 8µ2+

) 1

ω4n

+O(1

ω6n

)

)]. (11.9)

We see that the so-called dimensional reduction discussed in Chapter 4 takes place. Forexactly what temperature this becomes a good approximation is debatable and will dependon the various parameters of UΛ(ρ1, ρ2). We shall see in the next chapter that for T ≈ 20MeV and above, Eq. (11.9), with higher order terms in 1

ω2n

neglected, can be used to good

accuracy, i.e. we only consider the n = 0 term. This can at least be done for the parametersof UΛ(ρ1, ρ2) that we will consider.

The RG equation in the high-temperature limit is easier in a couple of ways. First of all, noMatsubara sum has to be performed, so no splitting of the fraction needs to be done in orderto perform the residue integrals. As this splitting is a slow process numerically (simulationtimes are increased about tenfold), using Eq. (11.9) instead saves us a lot of time. This isthus very fortunate for our calculations.

11.5. CHARGE NEUTRALITY 110

11.5 Charge neutrality

We recall that imposing a charge neutrality condition gives a relation between the chemicalpotentials µ0 and µ+, leaving only one of them free to vary. If we require charge neutralityand let one of the chemical potentials vary freely, we wish to tune the other chemical potentialso that the charge neutrality condition is satisfied. We choose µ0 to be the chemical potentialthat we vary freely, and tune µ+ so the charge neutrality condition is satisfied.

We recall the contribution to the charge density from the kaons at the RG scale k

nk,+ = − ∂

∂µ+Uk(φk,0) = 2µ+ρ2,k,0 +

∫ Λ

kdk

∂

∂µ+∂kUk(φk,0), (11.10)

where φ0 denotes the minimum of Uk(φ). We wish to allow for a neutral kaon condensate,i.e. ρ1,0 > 0, where the zero denote the minimum of the potential. In the next chapter, wewill choose parameters of UΛ(ρ1, ρ2) such that ρ2,Λ,0 = 0. This implies that ∂2Uk,0 > 0 for allk, and thus ρ2,k,0 = 0 for all k. Hence, we have

nk,+ =

∫ Λ

kdk

∂

∂µ+∂kUk(φk,0). (11.11)

We recall that Uk(ρ1, ρ2) has a part explicitly depending on µ+, −µ+ρ2 (this part is notrenormalized), and a part implicitly depending on µ+ through renormalization. Partial dif-ferentiating with respect to µ+ leaves only the explicitly dependent term. The same is truefor the µ+ dependence of ∂kUk(φk,0). We thus find (after a bit of algebra)

∂

∂µ+∂kUk =

2k4T

3v3ππ

2

∑n

(∂2Uk)2 + 2(k2 − ω2

n)∂2Uk − 2ω2n(2µ2

+ + k2) + k4 − 3ω4n(

(∂2Uk)2 + 2(k2 + ω2n)∂2Uk + 2ω2

n(2µ2+ + k2)2 + k4 + ω4

n

)2 , (11.12)

where we have set ρ2,0 = 0 and performed the integral over momenta using the same regulatorRk(q) = (k2 − v2

πq2)θ(k2 − v2

πq2) as before. Note that the denominator is a factor of two

second order polynomials in ω2k. This Matsubara sum is then much easier to perform than if

there were higher order polynomials in ω2n in the denominator as we have had before. It can

be done analytically without too much trouble. We thus perform the Matsubara sum usingthe methods derived in Appendix A. After some algebra we find

∂

∂µ+∂kUk =

k4 sinh (βµ+) sinh (β√k2 + µ2

+ + ∂2Uk)

24v3ππ

2T√k2 + µ2

+ + ∂2Uk

(cosh2

(12βµ+

)− cosh2

(12β√k2 + µ2

+ + ∂2Uk))2

(11.13)where β = 1

T as usual. At scale k = 0 we thus have

n+ =

∫ Λ

0dk

k4 sinh (βµ+) sinh (β√k2 + µ2

+ + ∂2Uk,0)

24v3ππ

2T√k2 + µ2

+ + ∂2Uk,0

(cosh2

(12βµ+

)− cosh2

(12β√k2 + µ2

+ + ∂2Uk,0))2

(11.14)where Uk,0 denotes the potential evaluated at (ρ1,0, ρ2,0). As ∂2Uk,0 > 0 for all k, this impliesthat the integrand of the above expression goes to zero as we take T → 0. This gives µ+ = µ0,see Eqs. (10.32) and (10.36). It can also be shown that this expression grows as T for largeT , T ≈ Λ, due to the finite integration limit Λ.


11.5.1 Some approximations

There are two approximations we can make for the above expression. Firstly we set ∂2Uk(ρ1, 0) ≈m2

+ − µ2+, i.e. we assume a small renormalization of ∂2Uk(ρ1, 0). This is true for low temper-

atures in particular, though we shall see later that it holds quite well for higher T as well.This makes the integrand independent of the fields φ. Secondly, if the temperature is lowenough, T << Λ, we can approximate the upper limit by ∞. After a partial integration anda suitable variable change we thus obtain

n+ ≈1

8π2

∫ ∞0

dx

√x sinh (βµ+)

cosh2(

12β√v2πx+m2

+

)− cosh2

(12βµ+)

. (11.15)

11.5.2 One-loop result

It would be interesting to compare the above result for the kaon-charge density with the resultobtained at one loop. We recall the one loop result for the potential, see Eq. (A.26),

Uk(φ) = UΛ(φ) +1

2Tr log

(D−1

0,ij + δijRk

)−1

q,−q, (11.16)

where the trace is over momenta as well as the matrix indices and where

D−10,ij(q,−q) =

δ2L0(φ)

δφi,qδφj,−q(11.17)

denotes the original inverse propagator at the cutoff. Thus L0(φ) is the bare lagrangian, andis given by Eq. (10.20). From this, we arrive at the charged kaon density

nk,+ = −∂Uk(φ0,k)

∂µ+= 2µ+ρ2 −

1

2Tr

(∂D−1

0

∂µ+Dk

), (11.18)

where φ0,k denotes the minimum of Uk(φ), and the matrix Dk has components Dk,ij =(D−1

0,ij + δijRk)−1 and is the full k-dependent propagator at one loop. If we again assume

ρ2,0 = 0, this can be written as

nk,+ = −T∑n

∫p

4µ+

(v2πp

2 +Rk(p) +m2+ − µ2

+ − ω2n

)×(Rk(p)2 + 2(v2

πp2 +m2

+ + ω2n − µ2

+)Rk(p) + µ4+ + 2µ2

+(ω2n −m2

+ − v2πp

2)

+ (ω2n + v2

πp2 +m2

+)2)−1

. (11.19)

Note that this is no longer dependent on the background fields φ. We perform the Matsubarasum, obtaining

nk,+ =

∫p

sinh (βµ+)

cosh2(

12β√v2πp

2 +Rk(p) +m2+

)− cosh2

(12βµ+)

. (11.20)


We recall that at k = 0 we have Rk=0(q) = 0 for all q. We thus obtain

n+ =

∫p

sinh (βµ+)

cosh2(

12β√v2πp

2 +m2+

)− cosh2

(12βµ+)

=1

8π2

∫ ∞0

dx

√x sinh (βµ+)

cosh2(

12β√v2πx+m2

+

)− cosh2

(12βµ+)

. (11.21)

This is precisely the low temperature approximation we found above. Also note that thisintegral is not cut off at Λ. This results in a T 2-dependence for large T instead of a T depen-dence as we got in the exact RG flow case.

The fact that we get exactly the result of Eq. (11.15) in the one-loop case should not come asa surprise. The above replacement ∂2Uk(ρ1, o) → m2

+ − µ2+ is equivalent to the replacement

of the action Γk[φ] by the one-loop result in Eq. (A.24). One might worry that we still onlyintegrate up to the cutoff Λ in Eq. (11.14). However, this should not matter if T << Λ,and if T ≈ Λ the theory does not work well anyway, as the cutoff should be larger than anyrelevant energy scales of the model.

12. NUMERICAL RESULTS; KAON CONDENSATION

In this chapter, we analyze the RG equation derived in the last chapter for kaon condensationin a dense medium, derived in the last chapter. We first consider the system without theelectric charge neutrality condition imposed. We then turn to see what the effects of imposingthe charge neutrality are. We again use the third order RK method described in Appendix Ato solve the RG equation. We also use finite distance methods to approximate the derivativesas in previous chapters.

12.1 Numerical results without charge neutrality

In this section, we derive some numerical results for the model using the RG equation obtainedin the last chapter, Eq. (11.8). We first see what the parameters of the cutoff potentialUΛ(ρ1, ρ2) should be. We then move on to derive some interesting numerical results for themodel.

12.1.1 The bare potential

We look at T = 0 for the moment, and consider the parameters of UΛ(ρ1, ρ2). We rememberthe expansion of UΛ(ρ1, ρ2),

UΛ(ρ1, ρ2) = (m20 − µ2

0)ρ1 + (m20 − µ2

0)ρ2 +λ0

2ρ4

1 +λ+

2ρ4

2 + λHρ1ρ2. (12.1)

This expression is assumed to be valid for Λ ≤ ∆ = 30 MeV, i.e. below the fermionic-degrees-of-freedom-suppressing superconducting gap ∆. We also recall that this expansion in the fieldsis only valid when the chemical potentials are close to the respective masses. We shall choosem0 = 4 MeV and m+ = 5 MeV as is done in [76, 77], if not otherwise stated. These valuesare found by inserting reasonable values for the quark masses and baryonic chemical potential(µB ≈ 500 MeV) into Eqs. (10.23) and (10.24), where we use fπ and a calculated from Eqs.(10.11) and (10.12), respectively. We should then choose values of the chemical potentialsclose to their respective masses in order for the expansion to be physically valid. We chooseµ0 = µ+ = 4.5 MeV, as is done in [76, 77], if not otherwise stated. Using Eqs. (10.25)-(10.27)with these values, we obtain λ0 = 1.25 · 10−3, λ+ = 1.08 · 10−3, and λH = 1.16 · 10−3, wherewe use fπ = 93 MeV.

ERRATUM: We used the above values for the couplings when performing thenumerical calculations of this chapter. These values are unfortunately half ofwhat they should be: λ0 = 2.5 · 10−3 MeV, λ+ = 2.16 · 10−3 MeV, and λH = 2.32 · 10−3

MeV.

In [77], they use the above values without renormalizing the theory. It could therefore be

12.1. NUMERICAL RESULTS WITHOUT CHARGE NEUTRALITY 114

Fig. 12.1: The real part of UΛ(ρ1, 0), and Uk=0(ρ1, 0) for Λ = 30 MeV and Λ = 100 MeV, all asfunctions of ρ1

Λ2 at T = 0 where the constant added to the potential by renormalization hasbeen discarded. Note how little the potential is renormalized.

argued that we should choose our parameters at scale k = Λ so as to obtain these values inthe potential expansion after we have renormalized to k = 0, as we have done before. If wehowever plot the potential at k = 0 and k = Λ as a function of e.g. ρ1 at e.g. ρ2 = 0 usingthe above values, we see that it does not change much, i.e. the renormalization is quite smallat T = 0, see Fig 12.1. Hence, it does not really matter if we use these values at scale k = Λ,or if we tune the parameters at k = Λ in order to arrive at the above values for the expan-sion parameters of Uk=0(ρ1, ρ2). The results are essentially the same. We shall thus use theabove values for the parameters of the model at scale Λ, as this is by far the simplest approach.

Note also that no charged kaon condensate will result when using the above parameters.This is clear as the renormalization procedure usually tends to reduce the value of the fieldφ0 for which the potential takes its minimum value. As ρ2,0 = 0 already at scale k = Λ, thisis not likely to change as we renormalize. Hence, there is no charged kaon condensate. See

also Fig. 12.6, showing an increase in m+,k =√∂2Uk(0, 0) + µ2

+ as we renormalize (though

this plot is for Λ = 100 MeV with charge neutrality imposed).

12.1.2 Critical temperature

We now turn to find the critical temperature when using the parameters given above. Notefrom Fig. 12.1 that the value for which the potential takes its minimum, ρ1,0 > 0, clearlydoes not vanish as one integrates down to k = 0 at T = 0. We wish to find the critical tem-


perature for which this minimum does disappear. Turning up the temperature to criticality,we arrive at a critical temperature of 215 MeV. This is quite a lot higher than that obtainedin [76] (Tc = 118.5 MeV) and almost five times that of [77] (Tc ≈ 45 MeV). The critical

Fig. 12.2: The real part of Uk=0(ρ1, 0) as a function of ρ1 for four different temperatures, T = 0,T = 20, T = Tc = 61 and T = 100 MeV. We have neglected the constant added to thepotential by renormalization, and set Λ = 100 MeV.

temperature we found is also much higher than the cutoff value Λ = 30 MeV. This poses aproblem. Remember that the cutoff scale should be larger than the largest relevant energyscale of the model in order for the model to describe the physics at energy scales from k = Λto k = 0. If we calculate physical quantities of the theory that fall outside this range, thevalues obtained cannot really be trusted as good predictions as they fall outside the energyrange for which the model has predictive power. This is the case for the calculated criticaltemperature above. This has not been a problem in any of the previous models consideredso far, e.g. charged pion condensation, as all these gave critical temperatures well below Λ.

We therefore increase the cutoff scale until we find it to be well above the critical temperature.If we set Λ = 100 MeV, we arrive at a critical temperature of Tc = 61 MeV, well below thecutoff scale. We shall therefore use Λ = 100 MeV in the following. Note that increasing thecutoff Λ will increase the amount of renormalizing that takes place both for T = 0 and T > 0.This will in turn increase the difference between the parameters of the potential at k = Λ andk = 0, see Fig. 12.1. The difference is still not large however, and we shall continue to usethe couplings, masses, and chemical potentials given above as the parameters of UΛ(ρ1, ρ2) ifnot otherwise stated. We also note that the critical temperature varies quite drastically withthe cutoff. This is also bad as we usually want our physical quantities to be as independentof the cutoff as possible. The behavior is however not unwarranted due to the large criticaltemperature for Λ = 30 MeV.


One could object that choosing such a high cutoff might invalidate the theory of Eq. (12.1) asa good description for the kaons at the microscopic level. Remember that Eq. (12.1) is validfor Λ ≈ ∆ = 30 MeV and below, where other degrees of freedom are suppressed. Assumingthat this holds for energies as high as 100 MeV as well might be stretching it a bit. It shouldthough be mentioned that superconducting energy gaps as high as ∆ = 100 MeV have beencalculated [78], although be it for very high baryonic densities. Hence, setting Λ = 100 MeVmight not be such a stretch after all.

Using Λ = 100 MeV, we turn to plot the real part of the renormalized potential Uk=0(ρ1, 0) asa function of ρ1 for different values of the temperature T . This is done in Fig. 12.2 using thesame parameters at the cutoff as before. We also check that we get the dimensional reductiondiscussed in the last chapter when T ≈ 20 MeV. That is, calculating Uk=0(ρ1, ρ2) using Eqs.(11.8) and (11.9) give approximately the same result at this temperature.

Note that the critical temperature Tc = 61 MeV is still above the color-superconducting gap∆ = 30 MeV. This is somewhat unpleasant, as temperatures above ∆ may induce fermionicexcitations (recall from Chapter 10 that these where suppressed by ∆). The fermionic ex-citations may again invalidate our model, Eq. (12.1), as a good description of the system.In fact, they make the kaons unstable [85, 86]. This is also discussed in [76, 77] where theyobtain critical temperatures above the superconducting gap as well. Also note that that thenew critical temperature is much closer to those found in [76, 77].

Fig. 12.3: The neutral kaon condensate φ1,0 as a function of the chemical potentials µ0 and µ+ atT = 0, and Λ = 100 MeV. Note the change of phase transition from first order to secondorder at the critical point (µ0, µ+) = (4.18, 5.13) marked by a black dot.

12.2. NUMERICAL RESULTS WITH CHARGE NEUTRALITY 117

12.1.3 Phase diagrams

We now turn to plot some phase diagrams for the model. We plot the neutral kaon condensateφ1,0 =

√2ρ1,o as a function of the chemical potentials µ0 and µ+, where we have used the

O(2)-symmetry to rotate the condensate in the φ1-direction. We do this for T = 0 MeVand T = 50 MeV, and plot the results in Figs. 12.3 and 12.4 respectively. Note the

Fig. 12.4: The neutral kaon condensate φ1,0 as a function of the chemical potentials µ0 and µ+ atT = 50 MeV, and Λ = 100 MeV. We have used the O(2)-symmetry to rotate the condensatein the φ1-direction. Note that the second order phase transition takes place at a slightlyhigher µ0, i.e. the critical point is now (µ0, µ+) = (4.44, 5.33) marked by a black dot.

transition from a second order to a first order phase transition which happens at the criticalpoint (µ0, µ+) = (4.18, 5.13) MeV for T = 0, and at (µ0, µ+) = (4.44, 5.33) MeV for T = 50MeV. These critical points have been marked in the figures by black dots. The first-ordertransition takes place when the minimum of the potential Uk=0(ρ1, ρ2) jumps discontinuouslyfrom (ρ1,0 > 0, ρ2,0 = 0)0 to (ρ1,0 = 0, ρ2,0 > 0), hence going from a neutral kaon condensateto a charged kaon condensate. Note the vertical wall of the figures, describing the first ordertransition from the neutral kaon condensate to the charged kaon condensate. We also plot aphase diagram at different temperatures, showing the different phase structures in Fig. 12.5.

12.2 Numerical results with charge neutrality

Turning on the charge-neutrality constraint, we recall that this means we can only vary one ofthe chemical potentials µ0 and µ+ freely. We have chosen µ0 to be this potential, and hencewe must tune µ+ in order to achieve charge neutrality. Remember from the last chapterthat we expect the n+ result to be quite similar to the one-loop result for low temperatures,


T << Λ. Tuning µ+ in order to satisfy Eq. (10.36), we also find that µ+ becomes similar tothe one-loop result for low T . We use the above parameters at k = Λ and set µ0 = 4.5 MeVin order to plot the µ+ as a function of temperature for different values of the cutoff Λ. Wedo this in Fig. 12.7.

Note from Fig. 12.6 that m+,k does not get renormalized very much. This holds not onlyfor low temperatures, but also for higher ones to some degree. Thus, setting ∂kUk(ρ1, 0) =m2

+ − µ2+ is not such a huge simplification. We thus expect µ+ to follow the one-loop result

quite well, only to bend away when T approaches the cutoff.

From Fig. 12.7 we indeed see that, for relatively low temperatures, the one-loop resultand the RG-equation results agree quite well (holds better for larger Λ), but as the tempera-ture increases the RG-flow results bend upwards. This bend upwards is due to the fact thatn+ ∼ T for large T , as explained in the last chapter. Hence, as T grows, the T 2 part of ne(see Eq. (10.32)) will dominate the total charge density n = n+ + ne, and in order to havethe total charge density equal to zero, we must set the term in front of T 2 equal to zero, i.e.µ+ = µ0 in the limit T → ∞. This is thus the behavior of the model in the temperatureregime for which the model loses its validity as discussed above. Nonetheless, we note thatµ+ stays below µ0 for all temperatures. Hence, if we require charge neutrality, there is nopossibility for a charged kaon phase, as µ+ must at least be greater than m+,k=0, which isgreater than 5 MeV as we see from Fig 12.6. This is also what is found in [76].

Fig. 12.5: The phase structure of the system at Λ = 100 MeV for the temperatures T = 0 MeV (dashedlines), T = 50 MeV (dotted lines) and T = 150 MeV (dash-dotted lines). The connectedline denotes the first order phase transition between the neutral and charged kaon phase.


Fig. 12.6: We plot m2+,k =

√∂2Uk(0, 0) + µ2

+ as a function of the RG time t = log10kΛ at different

temperatures. We have used Λ = 100 MeV. The dotted line is at T = 100 MeV, the dashedline is at T = 50 MeV, and the full line is at T = 0.

We also note that µ+ < µ0 in the case of the one-loop result. Here we recall that n+ ∼ T 2 asT → ∞, and hence µ+ will stabilize on a value below µ0 in this case. This can also be seenfrom Fig. 12.7. The behavior here is similar to the one obtained in [76, 77].

We turn to plot the neutral kaon condensate as a function of temperature T and µ0 inFig. 12.8. Note here that the transition φ1,0 = 0 → φ1,0 > 0 for different temperatures isof second order as expected, that is, no first order transition to a charged kaon condensatetakes place. This is similar to Fig. 10 of [76]. We also find that the critical temperature forwhich the transition takes place is slightly higher than what we obtained before. That is,without the charge neutrality condition, we obtained a critical temperature of Tc = 61 MeVat µ0 = µ+ = 4.5 MeV. With the charge neutrality condition, this now becomes Tc = 64 MeVat µ0 = 4.5 MeV (and µ+ = 1.82 MeV). This is similar to the results of [76, 77] where theyalso obtain an increase of the critical temperature when imposing the neutrality condition.


Fig. 12.7: µ+ as a function of T for three different values of the cutoff Λ. I.e. Λ = 30 MeV, Λ = 100MeV and Λ = 300 MeV. We also plot the one-loop result for comparison.

Fig. 12.8: φ1,0 as a function of T and µ0 with for an electrically neutral system. We have used Λ = 100MeV.

Part V

SUMMARY, OUTLOOK, AND CONCLUSION


13. SUMMARY, OUTLOOK, AND CONCLUSION

13.1 Summary

In this thesis, we have introduced the concept on non-perturbative renormalization. Thisis a way of handling the renormalization procedure in both QFT and TFT (with a nonzerotemperature), different from the more conventional perturbative formalism. We derived RGequations in both the WP and the EAA formalism, and noted differences and advantages/-drawbacks between the formalisms. We continued with the EAA formalism in the LPA,where we derived an RG equation for the effective potential evaluated at a constant meanfield. Having done this, we considered some examples and applications of the theory.

The first example considered was that of O(N) theories, and in particular O(4) theory asan effective theory for two-flavor low-energy (chiral) QCD. Here, we used a linear sigmamodel as the boundary condition at the cutoff scale Λ. The parameters of the boundarypotential were then tuned to reproduce vacuum physics. Having done this, we proceeded toderive some results, such as a second-order phase transition at a critical temperature, thevanishing of the effective coupling at this temperature and more.

We continued by expanding on the O(4) model by including an isospin chemical potentialµI , reducing the symmetry to O(2) × O(2). We also added an explicit symmetry breakingterm to the model, going to the physical point where the pions have a finite mass. We thenplotted a phase diagram for BE condensation of charged pions at both the physical pointand the chiral limit. A second-order phase transition was found for µI > 0 as well. We alsofound that the charged pion phases at the chiral limit and the physical point became similarfor large chemical potentials as expected. We studied the chiral phase transition, noting thatthis became a crossover at the physical point. We also studied the condensate competition asµI →∞. We proceeded to study the large-N limit where the LPA becomes exact, deriving asimilar phase diagram as in the O(4) case, but using a much larger cutoff. We compared thediagrams, finding them to be quite similar. This, in turn implies a small cutoff dependenceof the theory.

We moved on to consider kaon condensation at high baryon densities. We derived the bareLagrangian as an effective theory from the physics at high energy. This Lagrangian is validfor energies below the color-superconducting gap. We therefore started by setting the cutoffto this gap. Using this Lagrangian and reasonable parameters at the cutoff, we derived acritical temperature far above the cutoff. We therefore changed the cutoff in order to arriveat a critical temperature below it. Doing this, we found that the critical temperature was verydependent on the cutoff. We allowed for BE condensation for both the neutral and chargedkaons by introducing a chemical potential for both K0 and K+. Plotting phase diagrams, wefound a second-order phase transition from the symmetric phase to the neutral kaon phase

13.2. OUTLOOK 124

or to the charged kaon phase, while the phase transition between the kaon phases was of firstorder. We then proceeded to impose a charge-neutrality condition on the system, using abackground of electrons and positrons. With the given parameters of the bare Lagrangian,we then found that a charged kaon phase became impossible with this condition. We alsofound a slight increase of the critical temperature.

13.2 Outlook

Next, we could apply the theory to new problems. We could for instance introduce a baryonchemical potential µB to the theory, and thus derive a phase diagram in the µB-T -plane.Doing this, we should find a first-order phase transition for low T as we increase µB. This hasbeen done before, see e.g. [95]. It turns into a second order transition for high enough T , andit is thus possible to find the critical point when this happens, see Fig. 13.1. Using the NPRGhere tuns out to have advantages over other approaches and lattice simulations in particular,as the NPRG approach has no sign problem [96, 97]. Other examples where the NPRG maybe applied also exist, not only in high-energy physics. Non-perturbative renormalization hasmany applications in e.g. low energy solid state physics and superconductivity theory in par-ticular. Another thing that could be done is to address the problem arising within the LPA

Fig. 13.1: A plot of the µB-T phase diagram of QCD. Note the critical point where the transition fromthe hadronic phase goes from a first order (solid line) to a second order (dashed line) phasetransition. This figure is taken from [98].

when considering a nonzero chemical potential. We recall that the RG equation is changedwhen we include an isospin chemical potential. In the LPA, there is no wave-function renor-malization to compensate for this, resulting in inconsistencies, e.g. µI,c 6= mπ at T = 0. Oneway to deal with this problem could be to leave the LPA, and consider the full RG equationfor the effective action, though this will probably lead to greater challenges, both analyticallyand numerically. Another solution could be to change the regulator to account for the ef-fects of the chemical potential, though it is not clear to this writer how this would be done.However, we recall that the problem disappears in the large-N limit, as the LPA is exact there.

13.3. DISCUSSION AND CONCLUSION 125

We could also make some improvements in our numerics. We have throughout this thesisused a third-order RK method, combined with third-order finite distance methods to approx-imate the derivatives when solving our RG equations. One improvement is to increase theorder, thus making our solutions more exact, but in return increasing the simulation timesneeded to find them. In order to counter the increase of simulation time, ”faster” program-ming languages could be used. The alert reader may have noticed that the current code iswritten in Matlab, a fine language when it comes to solving linear problems, but quite slowwhen it comes to non-linear ones. As a result, simulation times can become very long. Fig.12.4, for example, took more than a day to simulate.

13.3 Discussion and conclusion

We have seen that the non-perturbative formalism is flexible enough to deal with manyproblems, and it has some advantages over the perturbative formalism. In the perturbativeformalism, one must add more and more Feynman diagrams in order to increase the level ofaccuracy, making it increasingly difficult. In the non-perturbative formalism, we derive anexact renormalization group equation, and if we manage to solve this one, the problem issolved. The problem is that this equation is often very involved, and is usually (if not always)unsolvable as it stands. Thus, simplifications are needed in order to get anywhere analytically(LPA, sharp cutoff-function, expanding the potential, etc). Some of these simplifications alsolead to inconsistencies with known physics, for example that we got a critical isospin chemicalpotential different from the pion mass at T = 0 when we considered charged pion condensa-tion. This is however not a problem in the large-N limit, as the LPA becomes exact here,and the chemical potential disappears from the RG equation completely, rendering it O(N)symmetric again. As the equations are so involved, one also needs a great deal of numericsin order to solve them. This in turn makes the simulations run slower.

Nonetheless, it is in this writers opinion that non-perturbative renormalization is here tostay, and progress concerning it may be made in the future. It is also in in this writers opin-ion that the non-perturbative formalism gives a better understanding of what renormalizationreally is, and that is an important point.

13.3. DISCUSSION AND CONCLUSION 126

APPENDICES


Appendix A

CONVENTIONS AND CALCULATIONS

A.1 Definitions, conventions and identities

• We use of course the Einstein summation convention, where repeated indices are summedover unless otherwise stated.

• We work in natural units where ~ = c = kB = 1.

• For T = 0, we define the integral operators, in coordinate and momentum space respec-tively, as ∫

x=

∫ddx ,

∫q

=

∫ddq

(2π)d. (A.1)

• Similarly for T > 0, we define the same integral operators to mean∫x

=

∫ β

0dτdd−1x =

∫ β

0dτ

∫x

,

∫q

= T∑n

∫dd−1q

(2π)d= T

∑n

∫q. (A.2)

where β = 1T .

• We often abbreviate φ(x) = φx, φ(p) = φp and so on in our equations. Also, if insideintegrals, φ might be understood to depend on the variable that is integrated over, orit may be neglected altogether. For example

L =

∫xL(φ) =

∫xL(φx) =

∫xL. (A.3)

• A useful integral identity:∫qf(q2) =

1

2dπd/2Γ(d2)

∫ ∞0

dxxd/2−1f(x) = 2vd

∫ ∞0

dxxd/2−1f(x),

where we have defined the constant vd.

A.2 Derivations

A.2.1 Matsubara sums and Cauchy integrations

We derive some useful formulas used in the text for summation of Matsubara frequencies. Weconsider

T

∞∑n=−∞

f(ωn),

A.2. DERIVATIONS 130

where ωn = 2πnT . If the function f(ωn) does not have any poles on the imaginary axis, wehave the following useful identity [99]

T∞∑

n=−∞f(ωn) =

1

2πi

∮C1

n(z)f(−iz)dz =1

2πi

∮C1

1

2

(coth

βz

2− 1)f(−iz)dz, (A.4)

where the contour C1 is the first contour shown in Fig. A.1, and

n(ω) =1

eβω − 1(A.5)

is the Bose distribution-function. The formula holds since the cotangent has poles at βz2 = iπn

where the residue is 2β . C1 is closed at infinity, so all the poles are included. We may shrink

the semicircles at infinity to a noncontributing size. If f(iz) is such that the integral alongcircular semicircles at infinity vanishes, the integral may be evaluated along the contours C2

instead (with an additional minus sign, due to the switch in contour orientation, see Fig.A.1). We use the formula (A.4) to calculate the following simple Matsubara sum

−K K

↑ C1

(a)

−K K

↑ C2

C2 ↓

(b)

Fig. A.1: The contours we are integrating over. To the left is the contour C1, and to the right thecontour C2.

T∞∑

n=−∞

1

ω2n +K2

= − 1

2πi

∮C1

n(z)1

z2 −K2dz (A.6)

as an example. Here the integrand does shrink fast enough so that the integrals over semi-circles at infinity do not contribute. The sum is then

T∞∑

n=−∞

1

ω2n +K2

= − 1

2πi

∮C1

n(z)

z2 −K2=

=1

2πi

∮C2

n(z)

z2 −K2=n(K)− n(−K)

2K=

1 + 2n(K)

2K,


where again n(K) is the Bose distribution-function. Note that we lose the minus sign in thelast line due to the opposite orientation of C2. We have thus derived

T∞∑

n=−∞

1

ω2n +K2

=1

2K

[1 + 2n(K)

]. (A.7)

This formula holds for K real. We may, however, analytically continue the formula andassume it to be true for complex K 6= 0 as well, i.e. we use the RHS of the above equation asthe analytic continuation of

H(K) = T

∞∑n=−∞

1

ω2n +K2

(A.8)

into the complex plane. The formula then holds for all complex K 6= 0 (the RHS of the aboveequation diverges at K = 0). Eq. (A.7) has been used repeatedly throughout this thesis.

A.2.2 The RG equation in the EAA

In this section, we derive the RG equation described and used in the text. We recall thepartition function of the EAA

Zk[J ] =

∫Dφe−S[φ]−∆Sk[φ]+

∫x Jφ, (A.9)

where

∆Sk[φ] =1

2

∫x,yφxRk(x− y)φy =

1

2

∫qRk(q)φqφ−q, (A.10)

and Rk(x − y) is an IR regulator. We want to find the RG equation for the modified Gibbsfree energy Γk[M ] defined by the Legendre transform

Γk[M ]− Fk[J ] =

∫xBM − 1

2

∫x,yMxRk(x− y)My (A.11)

where Fk[J ] is the dimensionless Helmholtz free energy,

Fk[J ] = − logZk[J ]. (A.12)

We start by differentiating Zk[J ],

∂kZk[J ] = −1

2

∫Dφ∫x,yφxRk(x− y)φye

−S[φ]−∆Sk[φ]+∫x Jφ

= −1

2

∫x,yRk(x− y)

δ

δJx

δ

δJyZk[J ]. (A.13)

As Zk[J ] = e−F [J ], it is easy to see that

∂kFk[J ] =1

2

∫x,y∂kRk(x− y)

[δFk[J ]

δJ(x)

δFk[J ]

δJ(y)− δ2Fk[J ]

δJ(x)δJ(y)

]. (A.14)


Note the similarity between this equation and the WP Eq. (3.17).

We see, by differentiating Eq. (A.11), that

δΓk[M ]

δMx= Jx −

∫yRk(x− y)My. (A.15)

We now act on Eq. (A.11) with ∂k|J , the derivative with respect to k and with J held fixed,in order to obtain

∂kΓk[M ]|J − ∂kFk[J ]|J =

∫xJ∂kM |J

− 1

2

∫x,yMx∂kRk(x− y)My −

∫x,y∂kMx|JRk(x− y)My. (A.16)

We transform the derivative with respect to k with J kept constant into the derivative withrespect to k with M kept constant by the transformation law

∂k|J = ∂k|M +

∫x∂kMx|J

∂

∂Mx. (A.17)

If we substitute the Eqs. (A.14), (A.15), and (A.17) into (A.16), we obtain, after a bit ofalgebra,

∂kΓk[M ] =1

2


δ2FkδJxδJy

. (A.18)

We recall that the mean field is defined as

Mx = −δFkδJx

= 〈φx〉. (A.19)

If we differentiate this with respect to Mz, we obtain

δ(x− z) = − δ2FkδJxδMz

= −∫y

δ2FkδJxδJy

δJyδMz

= −∫y

δ2FkδJxδJy

( δ2ΓkδMyδMz

+Rk(x− y))

= −∫yF

(2)k,x,y

(Γ

(2)k +Rk

)y,z, (A.20)

where we have used Eq. (A.15), and defined F(2)k and Γ

(2)k . Considered as an operator, we

see that −F (2)k is the inverse of what is inside the parenthesis. We finally arrive at the RG

equation for Γk[M ]

∂kΓk[M ] =1

2


(Γ

(2)k +Rk

)−1

x,y. (A.21)

If the effective action Γk[M ] is a functional of more fields, i.e. M = (M1, ..,MN ), this equationmay easily be generalized. The generalization is in that we must take a trace over internalindices of the integrand. The result is thus

∂kΓk[M ] =1

2Tr

∫x,y∂kRk(x− y)F

(2)k,x,y, (A.22)

A.3. NUMERICS 133

where now F(2)k,x,y is an N ×N matrix with components

F(2)k,ij(x, y) ≡ δ2Fk

δJi,xδJj,y. (A.23)

With this definition, noting that F(2)k is the inverse of (Γ

(2)k +Rk) (again defined as a matrix),

we find the RG equation

∂kΓk[M ] =1

2Tr

(∂kRk(x− y)

[Γ

(2)k,x,y +Rk(x− y)

]−1), (A.24)

where the trace also includes integration over x and y. This is sometimes called the Wetterichequation [29], and it has a simple diagrammatic form

∂kΓk =1

2, (A.25)

where the full k-dependent propagator(Γ

(2)k,ab +Rk

)−1is associated with the propagator line,

and where the dot denotes the insertion of the regulator part ∂kRk(q). This closely resembles

a one-loop RG equation. In fact, if we replace Γ(2)k,ab in the equation by S(2)ab where S denotes

the classical action as a first approximation, we obtain

Γk[M ] = S[M ] +1

2Tr log

[S

(2)ab +Rk

](A.26)

where the trace is over spacetime and indices of the matrix. We immediately notice the log-arithm dependence that is characteristic of the one-loop free energy. Note also the similaritybetween Eq. (A.26) and Eq. (2.37). They become equal when k = 0, as they should.

As an aside, we may also mention that the form of Eq. (A.24) for the RG equation alsoholds if the action includes spinor fields and gauge fields of different kinds as well, with minordifferences. The trace is then also over whatever indices accompanying such fields.

A.3 Numerics

In this section, we discuss the numerical method used in this paper when integrating Eq.(A.24). Obviously this equation cannot be integrated analytically and so numerical methodsare needed. We thus turn to see how Eq. (A.24) may be integrated numerically

A.3.1 RK methods

We consider a first order partial differential equation (PDE) of the form

∂kFk(q) = H[Fk(q)] (A.27)

where q is some vector of parameters and H is some functional of Fk(q) that may in principleinvolve all kinds of differentiations and integrations of Fk(q) with respect to the parametersq. If we also know Fk(q) at a given point k0, we have a initial value problem.

A.3. NUMERICS 134

The Euler Method

The crudest way of integrating this is by the Euler method. Say we wish to integrate Eq.(A.27) from k0 to k. We first discretize the interval into n smaller intervals of equal lengthε = 1

n(k − k0). We then have

Fki+1(q) = Fki(q) + εH[Fki(q)] +O(ε2). (A.28)

That the error is of order ε2 is seen from a simple Taylor expansion of Fk(q) around Fki(q).If we iterate this process, we eventually get from k0 to kn. Summing up the errors give usn error terms of order ε2. As ε is of order 1

n , we therefore get a total error of order ε. Ofcourse, as we let n → ∞ this method becomes exact, but the larger we choose n, the longerthe numerical calculation is going to take. We would therefore like to use some method thathas an error of order smaller than ε.

RK method

In this paper we use a third order RK method, i.e. the accumulated error of order ε3, hencethe first order RK method is just the regular Euler method. This method is a combinationof two second order methods.

In the first method we consider an interval of length ε. We calculate the value Fk+ 12ε(q)

as the midpoint value of the interval using the first order Euler method described above. Wethen calculate K 1

2= H[Fk+ 1

2ε(q)] and write Fk+ε(q) = Fk(q) + εH[Fk+ 1

2ε(q)]. The error of

the Euler method is of second order. Combining this with the ε in the last expression givesan error of third order, and thus an accumulated error of order two.

In the second approach we write K0 = H[Fk(q)] and K1 = H[Fk+ε(q)]. We take the av-erage K = 1

2(K1 −K0), and write

Fk+ε(q) = Fk(q) + εK. (A.29)

The error here is also clearly of second order.

If we combine these two methods, we end up with an iteration scheme of the form of Eq.(A.29) where now K = 1

6(K0 + 4K 12

+K1). The method thus obtained has an accumulated

error of third order, much better than the simple first order Euler method. This is the methodwe use to integrate Eq. (A.24) in this thesis.

A.3.2 Finite distance methods

Note that H[Fk] is a functional of Fk(q) and may involve various orders of derivatives orintegrations of Fk(q) with respect to the parameters q. In this thesis we consider equationswhere q is two dimensional, and H[Fk] involves both first and second order and even mixedderivatives with respect to the parameters. As we have no hope of performing these deriva-tions analytically, we need numerical methods to approximate their values.

The best (if not the only) way of doing this is by finite difference methods, and this isthe method also used in this text. It goes like this. Assume we have a function F (x, y). We

A.3. NUMERICS 135

wish to find an approximation for ∂x∂yF (x, y) at x = y = 0 for simplicity. Discretize anarea around 0 (for simplicity we use the same discretization length h in both the x and ydirections). Then decide to what order in the discretization length h we would like to knowthe derivative. Recall that we used a third-order RK method when integrating Eq. (A.24).We may as well use the same order of accuracy when doing the finite distance approximations.We write

F (x, y) = F (0, 0) + ∂xF (0, 0)x+ ∂yF (0, 0)y +1

2∂2xF (0, 0)x2 + ∂x∂yF (0, 0)xy +

1

2∂2yF (0, 0)y2

+ ∂x∂2yF (0, 0)xy2 +

1

2∂y∂

2xF (0, 0)yx2 +

1

4∂2x∂

2yF (0, 0)x2y2 +O(x3) +O(y3)

= A+Bx+ Cy +Dx2 + Exy + Fy2 +Gxy2 +Hyx2 + Ix2y2 +O(x3) +O(y3),(A.30)

where we have defined A-I. We then choose what values of x and y we want to insert intothis equation, depending on where we are on the (x, y) grid. Say we only know F for positivex and y. Assume that F is given as F (xi, yj) = F (h · i, h · j) = Fij , and set, for the moment,the grid resolution to h = 1 (we can rescale back later). Eq. (A.30) then gives us the nineequations

F (0, 0) = A, (A.31)

F (1, 0) = A+B +D, (A.32)

F (2, 0) = A+ 2B + 4D, (A.33)

F (0, 1) = A+ C + F, (A.34)

F (1, 1) = A+B + C +D + E + F +G+H + I, (A.35)

F (2, 1) = A+ 2B + C + 4D + 2E + F + 2G+ 4H + 4I, (A.36)

F (0, 2) = A+ 2C + 4F, (A.37)

F (1, 2) = A+B + 2C +D + 2E + 4F + 4G+ 2H + 4I, (A.38)

F (2, 2) = A+ 2B + 2C + 4D + 4E + 4F + 8G+ 8H + 16I. (A.39)

This set of linear equations is easily inverted and we may read of the solution for e.g. E =∂x∂yF (0, 0). We find

∂x∂yF (0, 0) =1

4h2

[9F00− 12F10 + 3F20− 12F01 + 16F11− 4F21 + 3F02− 4F12 +F22

]+O(h3),

(A.40)where h has been scaled back . The mixed derivatives in other places of the grid are calculatedin a similar manner, as are the ”pure” derivatives in the x and y directions.

A.3. NUMERICS 136

Appendix B

MATLAB CODE

In this appendix, we include some of the Matlab code used to solve the RG equations ofthis thesis. The code used does not change much from case to case, and we therefore onlyinclude the code used to solve the pion-condensation case as an example. We also only includethe most important code, that is, the code used to solve the RG equation for the effectivepotential. Code used to make phase diagrams and such has been neglected to save space.

B.1 Potential solver

This is the main function used to calculate the effective potential. Various quantities are alsocalculated, including the pion and sigma masses. A step function is used for every iterationstep. This one is shown in the next section. The potential solver reads

function [U, p1 , p2 , k , msigma ,mpi , msigma0 , f0 , fp i ,H, vp , vc , n ,m,U1 ] = . . .U12 integra te ( U zero , msq lamb , gsq lamb , hsq lamb , fp i , mpi ,T, u ,Lam , . . . .p end , incr , d iv )

%Calcu la t ed q u a n t i t i e s :

%U; the p o t e n t i a l g i ven as U=U(k , p2 , p1 )%(measured in un i t s o f Lamˆ4)

%p1 ; the f i r s t f i e l d parameter (measured in un i t s o f Lamˆ2)%p2 ; the second f i e l d parameter (measured in un i t s o f Lamˆ2)%k ; the renorma l i za t i on s c a l e (measured in un i t s o f Lam)%msigma ; the sigma mass at the p h y s i c a l po in t%mpi ; the pion mass at the p h y s i c a l po in t%msigma0 ; the sigma mass in the c h i r a l l im i t%f0 ; the pion decay cons tant in the c h i r a l l im i t%f p i ; the pion decay cons tant in the c h i r a l l im i t%H; the symmetry−break ing parameter%vp ; the pion condensate%vc ; the c h i r a l condensate%U1; the f i n a l p o t e n t i a l wi th the symmetry−break ing term added

%(measured in un i t s o f Lamˆ4)

% The func t i on tak e s in the f o l l ow i n g v a r i a b l e s :

%U zero ; the i n i t i a l cons tant term of the p o t e n t i a l a t the UV s ca l e%( u s u a l l y s e t to zero )

%msq lamb ; the mass parameter o f the UV s c a l e (measured in un i t s o f Lamˆ2)%gsq lamb ; the qua r t i c coup l ing o f the UV s ca l e ( d imens ion l e s s )%hsq lamb ; the s e c t i c coup l ing o f the UV s c a l e

%( u s u a l l y s e t to zero and measured in un i t s o f Lamˆ−2)%f p i ; the known phy s i c a l pion decay cons tant

B.1. POTENTIAL SOLVER 138

%mpi ; the known phy s i c a l pion mass%T; the temperature%u ; the chemica l p o t e n t i a l%Lam; the c u t o f f s c a l e%p end ; the endpoint o f the p o t e n t i a l (measured in un i t s o f Lamˆ2)%incr ; the incrementat ion l en g t h per k s t ep%div ; the r e s o l u t i o n in the (p1 , p2 ) p lane

%I n i t i a t e v a r i a b l e s :

i =1;j =1;i n t =1;k ( i n t )=1;p1 ( j )=0;p2 ( i )=0;

%I n i t i a t e p o t e n t i a l :

U( int , i , j )=U zero+(msq lamb )∗p1 ( j )+msq lamb∗p2 ( i )+4∗gsq lamb ∗( p1 ( j ) . . .+p2 ( i ))ˆ2+8∗ hsq lamb ∗( p1 ( j )+p2 ( i ) ) ˆ 3 ;

while p2 ( i )<(p end−p end /500)p2 ( i+1)=p2 ( i )+p end/div ;U( int , i +1, j )=U zero+(msq lamb )∗p1 ( j )+msq lamb∗p2 ( i +1)+. . .

4∗ gsq lamb ∗( p1 ( j )+p2 ( i +1))ˆ2+8∗hsq lamb ∗( p1 ( j )+p2 ( i +1))ˆ3;i=i +1;

end

i =1;

while p1 ( j )<(p end−p end /500)

p1 ( j+1)=p1 ( j )+p end/div ;U( int , i , j+1)=U zero+(msq lamb )∗p1 ( j+1)+msq lamb∗p2 ( i )+ . . .

4∗ gsq lamb ∗( p1 ( j+1)+p2 ( i ))ˆ2+8∗ hsq lamb ∗( p1 ( j+1)+p2 ( i ) ) ˆ 3 ;

while p2 ( i )<(p end−p end /500)p2 ( i+1)=p2 ( i )+p end/div ;U( int , i +1, j+1)=U zero+(msq lamb )∗p1 ( j+1)+msq lamb∗p2 ( i +1)+. . .

4∗ gsq lamb ∗( p1 ( j+1)+p2 ( i +1))ˆ2+8∗hsq lamb ∗( p1 ( j+1)+p2 ( i +1))ˆ3;i=i +1;

end

i =1;j=j +1;

end

%Sta r t i t e r a t i o n :

while k ( i n t)>− i n c r

i n ti n t=in t +1;

Ua( : , : )=U( int − 1 , : , : ) ;

B.1. POTENTIAL SOLVER 139

[ k ( i n t ) , p1 , p2 ,U1 ] = Step U2D (k ( int −1) ,Ua , p end ,T, u , incr , d iv ) ;U( int , : , : )=U1( : , : ) −U1( 1 , 1 ) ;

end

%inc l ude the i s o s p i n chemica l p o t e n t i a l in the p o t e n t i a l :

i n t =1;while int <=(1/ i n c r+1)

l =1;while l<=numel (U( 1 , 1 , : ) )

h=1;while h<=numel (U( 1 , 1 , : ) )

U( int , l , h)=U( int , l , h)−uˆ2∗p1 (h ) ;h=h+1;

endl=l +1;

endi n t=in t +1;

end

%Ca l cu l a t e msigma ,mpi , msigma0 , f0 , f p i ,H, vp , vc , and U1:

j=numel (U(1 , : , 1 ) ) −1 ;

H=mpiˆ2∗ f p i /Lamˆ3 ;

%lo c a t e the p o t e n t i a l minimum in the p2 d i r e c t i o n in the c h i r a l l im i t :

while U(1/ incr , j ,1)<U(1/ incr , j +1 ,1)j=j −1;

endj=j +1;

%ca l c u l a t e f0 and msigma0 :

f 0=sqrt (2∗p1 ( j ) )∗Lam;msigma0=sqrt (2∗p1 ( j )∗DD2(U, j , 1 ) ) ∗Lam;

%ca l c u l a t e U1 :

U1( : , : )= real (U(1/ incr , : , : ) ) ;

s=1;while s<(div+1)U1 ( : , s )=U1 ( : , s)−H∗sqrt (2∗p2 ( : ) ) ;s=s+1;end

%f ind the minimum of U1 :

m=1;n=1;

r=1;s=1;

B.2. THE ITERATOR 140

a=0;

while s<(div+1)while r<(div+1)

i f U1( r , s)<aa=U1( r , s ) ;n=r ;m=s ;

endr=r+1;

endr=1;s=s+1;

end

%ca l c u l a t e vc and vp :

vc=sqrt (2∗p2 (n ) )∗Lam;vp=sqrt (2∗p1 (m))∗Lam;

%ca l c u l a t e mpi , msigma , and f p i ;

mpi=sqrt (D1(U, n ,m)+uˆ2)∗Lam;x=D2(U, n ,m) ;msigma=sqrt ( x+2∗p2 (n)∗DD2(U, n ,m))∗Lam;f p i=sqrt (2∗p2 (n ) )∗Lam;

function [D1 ] = D1(U, n ,m)max=1/ i n c r ;D1=real ( (1/(60∗ p end/div ))∗(−147∗U(max, n ,m)+360∗U(max, n ,m+1 ) . . .

−450∗U(max, n ,m+2)+400∗U(max, n ,m+3)−225∗U(max, n ,m+4 ) . . .+72∗U(max, n ,m+5)−10∗U(max, n ,m+6)) ) ;

end

function [D2 ] = D2(U, n ,m)max=1/ i n c r ;D2=real ( (1/(60∗ p end/div ))∗(−147∗U(max, n ,m)+360∗U(max, n+1,m) . . .

−450∗U(max, n+2,m)+400∗U(max, n+3,m)−225∗U(max, n+4,m) . . .+72∗U(max, n+5,m)−10∗U(max, n+6,m) ) ) ;

end

function [DD2] = DD2(U, n ,m)max = 1/ in c r ;DD2 = (1/(180∗ ( p end/div )ˆ2 ) )∗ real ( (938∗U(max, n ,m) . . .

−4014∗U(max, n+1,m)+7911∗U(max, n+2,m)−9490∗U(max, n+3,m) . . .+7380∗U(max, n+4,m)−3618∗U(max, n+5,m)+1019∗U(max, n+6,m) . . .−126∗U(max, n+7,m) ) ) ;

endend

B.2 The iterator

We here write down the step function used to iterate down from k = Λ to k = 0. It uses athird order Runge-Kutta method as shown is Appendix A. It goes as follows

function [ k , p1 , p2 ,U] = Step U2D ( k old , U old , p end ,T, u , h , div )


%This i s the incrementat ion func t i on . I t t a k e s in the o ld po t en t i a l ,%U old , the o ld va lue o f the sca l e , k o ld , the width o f the area on%which the p o t e n t i a l i s de f i end , p end , the temperature , T, the chemica l%po t en t i a l , u , and the r e s o l u t i on , d i v .

%I n i t i a t e v a r i a b l e s :

i = 1 ;j = 1 ;p1 ( j )=0;p2 ( i )=0;k=k old−h ;

%Perform the i t e r a t i o n in the f i r s t boundary po in t ( 1 , 1 ) :

U 0 = F( k old , p1 ( j ) , p2 ( i ) , U old ( i , j ) , U old ( i , j +1) , U old ( i , j + 2 ) , . . .U old ( i +1, j ) , U old ( i +1, j +1) , U old ( i +1, j +2) , U old ( i +2, j ) , . . .U old ( i +2, j +1) , U old ( i +2, j +2) , U old ( i , j +3) , U old ( i +3, j ) . . .,T, p end , u , 1 , 1 ) ;

U 1 = F( k old−h , p1 ( j ) , p2 ( i ) , U old ( i , j )−h∗U old ( i , j ) , U old ( i , j +1 ) . . .−h∗U old ( i , j +1) , U old ( i , j+2)−h∗U old ( i , j +2) , U old ( i +1, j ) . . .−h∗U old ( i +1, j ) , U old ( i +1, j+1)−h∗U old ( i +1, j +1) , U old ( i +1, j +2 ) . . .−h∗U old ( i +1, j +2) , U old ( i +2, j )−h∗U old ( i +2, j ) , U old ( i +2, j +1 ) . . .−h∗U old ( i +2, j +1) , U old ( i +2, j+2)−h∗U old ( i +2, j +2) , U old ( i , j +3 ) . . .−h∗U old ( i , j +3) , U old ( i +3, j )−h∗U old ( i +3, j ) ,T, p end , u , 1 , 1 ) ;

U ha l f = F( k old−(h/2) , p1 ( j ) , p2 ( i ) , U old ( i , j )−(h/2)∗U old ( i , j ) , . . .U old ( i , j +1)−(h/2)∗U old ( i , j +1) , U old ( i , j +2)−(h/2)∗U old ( i , j + 2 ) , . . .U old ( i +1, j )−(h/2)∗U old ( i +1, j ) , U old ( i +1, j +1)−(h/2)∗U old ( i +1, j +1 ) . . ., U old ( i +1, j +2)−(h/2)∗U old ( i +1, j +2) , U old ( i +2, j )−(h/2)∗U old ( i +2, j ) . . ., U old ( i +2, j +1)−(h/2)∗U old ( i +2, j +1) , U old ( i +2, j +2 ) . . .−(h/2)∗U old ( i +2, j +2) , U old ( i , j +3)−(h/2)∗U old ( i , j + 3 ) , . . .U old ( i +3, j )−(h/2)∗U old ( i +3, j ) ,T, p end , u , 1 , 1 ) ;

kU=1/6∗U 0+4/6∗U hal f+1/6∗U 1 ;U( i , j )=U old ( i , j )−h∗kU;i=i +1;

%Perform the i t e r a t i o n f o r the boundary po in t s (2 ,1)−( div −1 ,1):

while i<(numel ( U old ( : , j ) ) )p2 ( i ) = p2 ( i−1)+p end/div ;

U 0 = F( k old , p1 ( j ) , p2 ( i ) , U old ( i −1, j ) , U old ( i −1, j +1) , U old ( i −1, j +2 ) . . ., U old ( i , j ) , U old ( i , j +1) , U old ( i , j +2) , U old ( i +1, j ) , . . .U old ( i +1, j +1) , U old ( i +1, j +2) , U old ( i , j +3) ,0 ,T, p end , u , 1 , 2 ) ;

U 1 = F( k old−h , p1 ( j ) , p2 ( i ) , U old ( i −1, j )−h∗U old ( i −1, j ) , . . .U old ( i −1, j+1)−h∗U old ( i −1, j +1) , U old ( i −1, j +2 ) . . .−h∗U old ( i −1, j +2) , U old ( i , j )−h∗U old ( i , j ) , U old ( i , j +1 ) . . .−h∗U old ( i , j +1) , U old ( i , j+2)−h∗U old ( i , j +2) , U old ( i +1, j ) . . .−h∗U old ( i +1, j ) , U old ( i +1, j+1)−h∗U old ( i +1, j +1) , U old ( i +1, j +2 ) . . .−h∗U old ( i +1, j +2) , U old ( i , j+3)−h∗U old ( i , j +3) ,0 ,T, p end , u , 1 , 2 ) ;

U ha l f = F( k old−h/2 , p1 ( j ) , p2 ( i ) , U old ( i −1, j )−(h/2)∗U old ( i −1, j ) , . . .


U old ( i −1, j +1)−(h/2)∗U old ( i −1, j +1) , U old ( i −1, j +2 ) . . .−(h/2)∗U old ( i −1, j +2) , U old ( i , j )−(h/2)∗U old ( i , j ) , U old ( i , j +1 ) . . .−(h/2)∗U old ( i , j +1) , U old ( i , j +2)−(h/2)∗U old ( i , j +2) , U old ( i +1, j ) . . .−(h/2)∗U old ( i +1, j ) , U old ( i +1, j +1)−(h/2)∗U old ( i +1, j +1 ) , . . .U old ( i +1, j +2)−(h/2)∗U old ( i +1, j +2) , U old ( i , j +3 ) . . .−(h/2)∗U old ( i , j +3) ,0 ,T, p end , u , 1 , 2 ) ;

kU=1/6∗U 0+4/6∗U hal f+1/6∗U 1 ;U( i , j )=U old ( i , j )−h∗kU;i = i +1;

end

%Perform the i t e r a t i o n f o r the boundary po in t ( div , 1 ) :

p2 ( i ) = p2 ( i−1)+p end/div ;

U 0 = F( k old , p1 ( j ) , p2 ( i ) , U old ( i −2, j ) , U old ( i −2, j +1) , U old ( i −2, j +2 ) , . . .U old ( i −1, j ) , U old ( i −1, j +1) , U old ( i −1, j +2) , U old ( i , j ) , U old ( i , j + 1 ) , . . .U old ( i , j +2) , U old ( i , j +3) , U old ( i −3, j ) ,T, p end , u , 1 , 3 ) ;

U 1 = F( k old−h , p1 ( j ) , p2 ( i ) , U old ( i −2, j )−h∗U old ( i −2, j ) , U old ( i −2, j +1 ) . . .−h∗U old ( i −2, j +1) , U old ( i −2, j+2)−h∗U old ( i −2, j +2) , U old ( i −1, j ) . . .−h∗U old ( i −1, j ) , U old ( i −1, j+1)−h∗U old ( i −1, j +1) , U old ( i −1, j +2 ) . . .−h∗U old ( i −1, j +2) , U old ( i , j )−h∗U old ( i , j ) , U old ( i , j +1 ) . . .−h∗U old ( i , j +1) , U old ( i , j+2)−h∗U old ( i , j +2) , U old ( i , j +3 ) . . .−h∗U old ( i , j +3) , U old ( i −3, j )−h∗U old ( i −3, j ) ,T, p end , u , 1 , 3 ) ;

U ha l f = F( k old−h/2 , p1 ( j ) , p2 ( i ) , U old ( i −2, j )−(h/2)∗U old ( i −2, j ) , . . .U old ( i −2, j +1)−(h/2)∗U old ( i −2, j +1) , U old ( i −2, j +2 ) . . .−(h/2)∗U old ( i −2, j +2) , U old ( i −1, j )−(h/2)∗U old ( i −1, j ) , . . .U old ( i −1, j +1)−(h/2)∗U old ( i −1, j +1) , U old ( i −1, j +2 ) . . .−(h/2)∗U old ( i −1, j +2) , U old ( i , j )−(h/2)∗U old ( i , j ) , U old ( i , j +1 ) . . .−(h/2)∗U old ( i , j +1) , U old ( i , j +2)−(h/2)∗U old ( i , j +2) , U old ( i , j +3 ) . . .−(h/2)∗U old ( i , j +3) , U old ( i −3, j )−(h/2)∗U old ( i −3, j ) ,T, p end , u , 1 , 3 ) ;

kU=1/6∗U 0+4/6∗U hal f+1/6∗U 1 ;U( i , j )=U old ( i , j )−h∗kU;

j=j +1;

%Perform the i t e r a t i o n f o r the boundary po in t s (1 ,2)− (1 , div −1):

while j<(numel ( U old ( i , : ) ) )p1 ( j ) = p1 ( j−1)+p end/div ;

U 0 = F( k old , p1 ( j ) , p2 ( i ) , U old ( i −2, j −1) , U old ( i −2, j ) , U old ( i −2, j +1 ) . . ., U old ( i −1, j −1) , U old ( i −1, j ) , U old ( i −1, j +1) , U old ( i , j − 1 ) , . . .U old ( i , j ) , U old ( i , j +1) ,0 , U old ( i −3, j ) ,T, p end , u , 2 , 3 ) ;

U 1 = F( k old−h , p1 ( j ) , p2 ( i ) , U old ( i −2, j−1)−h∗U old ( i −2, j − 1 ) , . . .U old ( i −2, j )−h∗U old ( i −2, j ) , U old ( i −2, j +1 ) . . .−h∗U old ( i −2, j +1) , U old ( i −1, j−1)−h∗U old ( i −1, j −1) , U old ( i −1, j ) . . .−h∗U old ( i −1, j ) , U old ( i −1, j+1)−h∗U old ( i −1, j +1) , U old ( i , j − 1 ) . . .−h∗U old ( i , j −1) , U old ( i , j )−h∗U old ( i , j ) , U old ( i , j +1 ) . . .−h∗U old ( i , j +1) ,0 , U old ( i −3, j )−h∗U old ( i −3, j ) ,T, p end , u , 2 , 3 ) ;


U hal f = F( k old−h/2 , p1 ( j ) , p2 ( i ) , U old ( i −2, j − 1 ) . . .−(h/2)∗U old ( i −2, j −1) , U old ( i −2, j )−(h/2)∗U old ( i −2, j ) , . . .U old ( i −2, j +1)−(h/2)∗U old ( i −2, j +1) , U old ( i −1, j − 1 ) . . .−(h/2)∗U old ( i −1, j −1) , U old ( i −1, j )−(h/2)∗U old ( i −1, j ) , . . .U old ( i −1, j +1)−(h/2)∗U old ( i −1, j +1) , U old ( i , j − 1 ) . . .−(h/2)∗U old ( i , j −1) , U old ( i , j )−(h/2)∗U old ( i , j ) , U old ( i , j +1 ) . . .−(h/2)∗U old ( i , j +1) ,0 , U old ( i −3, j ) . . .−(h/2)∗U old ( i −3, j ) ,T, p end , u , 2 , 3 ) ;

kU=1/6∗U 0+4/6∗U hal f+1/6∗U 1 ;U( i , j )=U old ( i , j )−h∗kU;j = j +1;

end

%Perform the i t e r a t i o n f o r the boundary po in t (1 , d i v ) :

p1 ( j ) = p1 ( j−1)+p end/div ;

U 0 = F( k old , p1 ( j ) , p2 ( i ) , U old ( i −2, j −2) , U old ( i −2, j −1) , U old ( i −2, j ) , . . .U old ( i −1, j −2) , U old ( i −1, j −1) , U old ( i −1, j ) , U old ( i , j − 2 ) , . . .U old ( i , j −1) , U old ( i , j ) , U old ( i , j −3) , U old ( i −3, j ) ,T, p end , u , 3 , 3 ) ;

U 1 = F( k old−h , p1 ( j ) , p2 ( i ) , U old ( i −2, j−2)−h∗U old ( i −2, j − 2 ) , . . .U old ( i −2, j−1)−h∗U old ( i −2, j −1) , U old ( i −2, j )−h∗U old ( i −2, j ) , . . .U old ( i −1, j−2)−h∗U old ( i −1, j −2) , U old ( i −1, j−1)−h∗U old ( i −1, j − 1 ) , . . .U old ( i −1, j )−h∗U old ( i −1, j ) , U old ( i , j−2)−h∗U old ( i , j −2) , U old ( i , j − 1 ) . . .−h∗U old ( i , j −1) , U old ( i , j )−h∗U old ( i , j ) , U old ( i , j − 3 ) . . .−h∗U old ( i , j −3) , U old ( i −3, j )−h∗U old ( i −3, j ) ,T, p end , u , 3 , 3 ) ;

U ha l f = F( k old−h/2 , p1 ( j ) , p2 ( i ) , U old ( i −2, j−2)−(h/2)∗U old ( i −2, j − 2 ) , . . .U old ( i −2, j−1)−(h/2)∗U old ( i −2, j −1) , U old ( i −2, j ) . . .−(h/2)∗U old ( i −2, j ) , U old ( i −1, j−2)−(h/2)∗U old ( i −1, j − 2 ) , . . .U old ( i −1, j−1)−(h/2)∗U old ( i −1, j −1) , U old ( i −1, j ) . . .−(h/2)∗U old ( i −1, j ) , U old ( i , j−2)−(h/2)∗U old ( i , j −2) , U old ( i , j − 1 ) . . .−(h/2)∗U old ( i , j −1) , U old ( i , j )−(h/2)∗U old ( i , j ) , U old ( i , j − 3 ) . . .−(h/2)∗U old ( i , j −3) , U old ( i −3, j )−(h/2)∗U old ( i −3, j ) ,T, p end , u , 3 , 3 ) ;

kU=1/6∗U 0+4/6∗U hal f+1/6∗U 1 ;U( i , j )=U old ( i , j )−h∗kU;

i =1;j =2;

%Perform the i t e r a t i o n f o r the boundary po in t s ( div ,2)−( div , div −1):

while j<(numel ( U old ( i , : ) ) )

U 0 = F( k old , p1 ( j ) , p2 ( i ) , U old ( i , j −1) , U old ( i , j ) , U old ( i , j +1 ) . . ., U old ( i +1, j −1) , U old ( i +1, j ) , U old ( i +1, j +1) , U old ( i +2, j − 1 ) , . . .U old ( i +2, j ) , U old ( i +2, j +1) ,0 , U old ( i +3, j ) ,T, p end , u , 2 , 1 ) ;

U 1 = F( k old−h , p1 ( j ) , p2 ( i ) , U old ( i , j−1)−h∗U old ( i , j − 1 ) , . . .U old ( i , j )−h∗U old ( i , j ) , U old ( i , j +1 ) . . .−h∗U old ( i , j +1) , U old ( i +1, j−1)−h∗U old ( i +1, j −1) , U old ( i +1, j ) . . .−h∗U old ( i +1, j ) , U old ( i +1, j+1)−h∗U old ( i +1, j +1) , U old ( i +2, j − 1 ) . . .


−h∗U old ( i +2, j −1) , U old ( i +2, j )−h∗U old ( i +2, j ) , U old ( i +2, j +1 ) . . .−h∗U old ( i +2, j +1) ,0 , U old ( i +3, j )−h∗U old ( i +3, j ) ,T, p end , u , 2 , 1 ) ;

U ha l f = F( k old−h/2 , p1 ( j ) , p2 ( i ) , U old ( i , j − 1 ) . . .−(h/2)∗U old ( i , j −1) , U old ( i , j )−(h/2)∗U old ( i , j ) , . . .U old ( i , j +1)−(h/2)∗U old ( i , j +1) , U old ( i +1, j − 1 ) . . .−(h/2)∗U old ( i +1, j −1) , U old ( i +1, j )−(h/2)∗U old ( i +1, j ) , . . .U old ( i +1, j +1)−(h/2)∗U old ( i +1, j +1) , U old ( i +2, j − 1 ) . . .−(h/2)∗U old ( i +2, j −1) , U old ( i +2, j )−(h/2)∗U old ( i +2, j ) , . . .U old ( i +2, j +1)−(h/2)∗U old ( i +2, j +1) ,0 , U old ( i +3, j ) . . .−(h/2)∗U old ( i +3, j ) ,T, p end , u , 2 , 1 ) ;

kU=1/6∗U 0+4/6∗U hal f+1/6∗U 1 ;U( i , j )=U old ( i , j )−h∗kU;j = j +1;

end

%Perform the i t e r a t i o n f o r the boundary po in t ( div , d i v ) :

U 0 = F( k old , p1 ( j ) , p2 ( i ) , U old ( i , j −2) , U old ( i , j −1) , U old ( i , j ) , . . .U old ( i +1, j −2) , U old ( i +1, j −1) , U old ( i +1, j ) , U old ( i +2, j − 2 ) , . . .U old ( i +2, j −1) , U old ( i +2, j ) , U old ( i , j −3) , U old ( i +3, j ) . . .,T, p end , u , 3 , 1 ) ;

U 1 = F( k old−h , p1 ( j ) , p2 ( i ) , U old ( i , j−2)−h∗U old ( i , j −2) , U old ( i , j − 1 ) . . .−h∗U old ( i , j −1) , U old ( i , j )−h∗U old ( i , j ) , U old ( i +1, j − 2 ) . . .−h∗U old ( i +1, j −2) , U old ( i +1, j−1)−h∗U old ( i +1, j −1) , U old ( i +1, j ) . . .−h∗U old ( i +1, j ) , U old ( i +2, j−2)−h∗U old ( i +2, j −2) , U old ( i +2, j − 1 ) . . .−h∗U old ( i +2, j −1) , U old ( i +2, j )−h∗U old ( i +2, j ) , U old ( i , j − 3 ) . . .−h∗U old ( i , j −3) , U old ( i +3, j )−h∗U old ( i +3, j ) ,T, p end , u , 3 , 1 ) ;

U ha l f = F( k old−h/2 , p1 ( j ) , p2 ( i ) , U old ( i , j−2)−(h/2)∗U old ( i , j − 2 ) , . . .U old ( i , j−1)−(h/2)∗U old ( i , j −1) , U old ( i , j )−(h/2)∗U old ( i , j ) , . . .U old ( i +1, j−2)−(h/2)∗U old ( i +1, j −2) , U old ( i +1, j − 1 ) . . .−(h/2)∗U old ( i +1, j −1) , U old ( i +1, j )−(h/2)∗U old ( i +1, j ) , . . .U old ( i +2, j−2)−(h/2)∗U old ( i +2, j −2) , U old ( i +2, j − 1 ) . . .−(h/2)∗U old ( i +2, j −1) , U old ( i +2, j )−(h/2)∗U old ( i +2, j ) , . . .U old ( i , j−3)−(h/2)∗U old ( i , j −3) , U old ( i +3, j ) . . .−(h/2)∗U old ( i +3, j ) ,T, p end , u , 3 , 1 ) ;

kU=1/6∗U 0+4/6∗U hal f+1/6∗U 1 ;U( i , j )=U old ( i , j )−h∗kU;i=i +1;

%Perform the i t e r a t i o n f o r the boundary po in t s (2 , d i v )−( div −1, d i v ) :

while i<(numel ( U old ( : , j ) ) )

U 0 = F( k old , p1 ( j ) , p2 ( i ) , U old ( i −1, j −2) , U old ( i −1, j −1) , U old ( i −1, j ) . . ., U old ( i , j −2) , U old ( i , j −1) , U old ( i , j ) , U old ( i +1, j − 2 ) , . . .U old ( i +1, j −1) , U old ( i +1, j ) , U old ( i , j −3) ,0 ,T, p end , u , 3 , 2 ) ;

U 1 = F( k old−h , p1 ( j ) , p2 ( i ) , U old ( i −1, j−2)−h∗U old ( i −1, j − 2 ) , . . .U old ( i −1, j−1)−h∗U old ( i −1, j −1) , U old ( i −1, j ) . . .−h∗U old ( i −1, j ) , U old ( i , j−2)−h∗U old ( i , j −2) , U old ( i , j − 1 ) . . .−h∗U old ( i , j −1) , U old ( i , j )−h∗U old ( i , j ) , U old ( i +1, j − 2 ) . . .


−h∗U old ( i +1, j −2) , U old ( i +1, j−1)−h∗U old ( i +1, j −1) , U old ( i +1, j ) . . .−h∗U old ( i +1, j ) , U old ( i , j−3)−h∗U old ( i , j −3) ,0 ,T, p end , u , 3 , 2 ) ;

U ha l f = F( k old−h/2 , p1 ( j ) , p2 ( i ) , U old ( i −1, j − 2 ) . . .−(h/2)∗U old ( i −1, j −2) , U old ( i −1, j−1)−(h/2)∗U old ( i −1, j − 1 ) , . . .U old ( i −1, j )−(h/2)∗U old ( i −1, j ) , U old ( i , j − 2 ) . . .−(h/2)∗U old ( i , j −2) , U old ( i , j−1)−(h/2)∗U old ( i , j −1) , U old ( i , j ) . . .−(h/2)∗U old ( i , j ) , U old ( i +1, j−2)−(h/2)∗U old ( i +1, j − 2 ) , . . .U old ( i +1, j−1)−(h/2)∗U old ( i +1, j −1) , U old ( i +1, j ) . . .−(h/2)∗U old ( i +1, j ) , U old ( i , j − 3 ) . . .−(h/2)∗U old ( i , j −3) ,0 ,T, p end , u , 3 , 2 ) ;

kU=1/6∗U 0+4/6∗U hal f+1/6∗U 1 ;U( i , j )=U old ( i , j )−h∗kU;i = i +1;

end

i =2;j =2;

%Perform the i t e r a t i o n in the r e s t o f the square :

while j<(numel ( U old ( i , : ) ) )while i<(numel ( U old ( : , j ) ) )U 0 = F( k old , p1 ( j ) , p2 ( i ) , U old ( i −1, j −1) , U old ( i −1, j ) , U old ( i −1, j +1 ) . . .

, U old ( i , j −1) , U old ( i , j ) , U old ( i , j +1) , U old ( i +1, j − 1 ) , . . .U old ( i +1, j ) , U old ( i +1, j +1) ,0 ,0 ,T, p end , u , 2 , 2 ) ;

U 1 = F( k old−h , p1 ( j ) , p2 ( i ) , U old ( i −1, j−1)−h∗U old ( i −1, j − 1 ) , . . .U old ( i −1, j )−h∗U old ( i −1, j ) , U old ( i −1, j +1 ) . . .−h∗U old ( i −1, j +1) , U old ( i , j−1)−h∗U old ( i , j −1) , U old ( i , j ) . . .−h∗U old ( i , j ) , U old ( i , j+1)−h∗U old ( i , j +1) , U old ( i +1, j − 1 ) . . .−h∗U old ( i +1, j −1) , U old ( i +1, j )−h∗U old ( i +1, j ) , U old ( i +1, j +1 ) . . .−h∗U old ( i +1, j +1) ,0 ,0 ,T, p end , u , 2 , 2 ) ;

U ha l f = F( k old−h/2 , p1 ( j ) , p2 ( i ) , U old ( i −1, j − 1 ) . . .−(h/2)∗U old ( i −1, j −1) , U old ( i −1, j )−(h/2)∗U old ( i −1, j ) , . . .U old ( i −1, j +1)−(h/2)∗U old ( i −1, j +1) , U old ( i , j − 1 ) . . .−(h/2)∗U old ( i , j −1) , U old ( i , j )−(h/2)∗U old ( i , j ) , U old ( i , j +1 ) . . .−(h/2)∗U old ( i , j +1) , U old ( i +1, j−1)−(h/2)∗U old ( i +1, j − 1 ) , . . .U old ( i +1, j )−(h/2)∗U old ( i +1, j ) , U old ( i +1, j +1 ) . . .−(h/2)∗U old ( i +1, j +1) ,0 ,0 ,T, p end , u , 2 , 2 ) ;

kU=1/6∗U 0+4/6∗U hal f+1/6∗U 1 ;U( i , j )=U old ( i , j )−h∗kU;i = i +1;endi =2;j=j +1;

end

%Define the d e r i v a t i v e used in the i t e r a t i o n above by a t h i r d order%method . The s denotes the type o f po in t .%s=1: S t a r t po in t%s=2: Middle po in t%s=3: End po in t


function [D] = D(U1 ,U2 ,U3 , p end , s )

i f s==1D = (1/(2∗ ( p end/div )))∗(−3∗U1+4∗U2−U3 ) ;

end

i f s==2D = (1/(2∗ ( p end/div ) ) ) ∗ (U3−U1 ) ;

end

i f s==3D = (1/(2∗ ( p end/div ) ) ) ∗ (U1−4∗U2+3∗U3 ) ;

end

end

%Define the second d e r i v a t i v e used in the i t e r a t i o n above by a t h i r d%order method :

function [DD] = DD(U1 ,U2 ,U3 ,U4 , p end , s )

i f s==1DD = (1/ ( ( p end/div )ˆ2 ) )∗ ( 2∗U1−5∗U2+4∗U3−U4 ) ;

end

i f s==2DD = (1/ ( ( p end/div ) ˆ2 ) )∗ (U1−2∗U2+U3 ) ;

end

i f s==3DD = (1/ ( ( p end/div )ˆ2))∗(−U4+4∗U1−5∗U2+2∗U3 ) ;

end

end

%Define the mixed d e r i v a t i v e used in the i t e r a t i o n above by a t h i r d%order method . Here a and l denote the p o s i t i o n s on the p lane :%a=1 corresponds to a po in t ( x , 1 ) ( x random)%a=2 corresonds to a po in t between ( x , 1 ) and ( x , d i v )%a=3 corresponds to a po in t ( x , d i v )%l=1 corresponds to a po in t (1 , y ) ( y random)%l=2 corresonds to a po in t between (1 , y ) and ( div , y )%l=3 corresponds to a po in t ( div , y )

function [ D12 ] = D12(U11 ,U12 ,U13 ,U21 ,U22 ,U23 ,U31 ,U32 ,U33 , p end , a , l )

i f l==1i f a==1

D12 = (1/(4∗ ( p end/div )ˆ2 ) )∗ ( 9∗U11−12∗U21+3∗U31−12∗U12 . . .+16∗U22−4∗U32+3∗U13−4∗U23+U33 ) ;

end

i f a==2D12 = (1/(4∗ ( p end/div )ˆ2 ) )∗ ( 3∗U11−3∗U13−4∗U21+4∗U23+U31−U33 ) ;

end


i f a==3D12 = (1/(4∗ ( p end/div )ˆ2))∗(−9∗U13+12∗U12−3∗U11+12∗U23 . . .

−16∗U22+4∗U21−3∗U33+4∗U32−U31 ) ;end

end

i f l==2i f a==1

D12 = (1/(4∗ ( p end/div )ˆ2 ) )∗ ( 3∗U11−3∗U31−4∗U12+4∗U32+U13−U33 ) ;end

i f a==2D12 = (1/(4∗ ( p end/div ) ˆ2 ) )∗ (U11−U13−U31+U33 ) ;

end

i f a==3D12 = (1/(4∗ ( p end/div )ˆ2))∗(−3∗U13+3∗U33+4∗U12−4∗U32−U11+U31 ) ;

endend

i f l==3i f a==1

D12 = (1/(4∗ ( p end/div )ˆ2))∗(−9∗U31+12∗U21−3∗U11+12∗U32 . . .−16∗U22+4∗U12−3∗U33+4∗U23−U13 ) ;

end

i f a==2D12 = (1/(4∗ ( p end/div )ˆ2))∗(−3∗U31+3∗U33+4∗U21−4∗U23−U11+U13 ) ;

end

i f a==3D12 = (1/(4∗ ( p end/div )ˆ2 ) )∗ ( 9∗U33−12∗U23+3∗U13−12∗U32 . . .

+16∗U22−4∗U12+3∗U31−4∗U21+U11 ) ;end

end

end

%Define the Boltzmann func t i on :

function [ n ] = n(v ,T)n = 1/(exp( v/T)−1);end

%Define the main i t e r a t i o n func t i on F:

function [F]=F(k , p1 , p2 , U11 ,U12 ,U13 ,U21 ,U22 ,U23 ,U31 ,U32 , . . .U33 ,U1 ,U2 ,T, p end , u , a , l )

%Define d e r i v a t i v e s :

D12 = D12(U11 ,U12 ,U13 ,U21 ,U22 ,U23 ,U31 ,U32 ,U33 , p end , a , l ) ;

i f l==1D1 = D(U11 ,U12 ,U13 , p end , a ) ;

end


i f l==2D1 = D(U21 ,U22 ,U23 , p end , a ) ;

endi f l==3

D1 = D(U31 ,U32 ,U33 , p end , a ) ;end

i f a==1D2 = D(U11 ,U21 ,U31 , p end , l ) ;

endi f a==2

D2 = D(U12 ,U22 ,U32 , p end , l ) ;endi f a==3

D2 = D(U13 ,U23 ,U33 , p end , l ) ;end

i f l==1D11 = DD(U11 ,U12 ,U13 ,U1 , p end , a ) ;

endi f l==2

D11 = DD(U21 ,U22 ,U23 ,U1 , p end , a ) ;endi f l==3

D11 = DD(U31 ,U32 ,U33 ,U1 , p end , a ) ;end

i f a==1D22 = DD(U11 ,U21 ,U31 ,U2 , p end , l ) ;

endi f a==2

D22 = DD(U12 ,U22 ,U32 ,U2 , p end , l ) ;endi f a==3

D22 = DD(U13 ,U23 ,U33 ,U2 , p end , l ) ;end

omega = sqrt ( kˆ2+D1 ) ;omega2 = sqrt ( kˆ2+(1/2)∗(D1+D2)+2∗(p1+p2 )∗D12 ) ;

%Return func t i on f o r the chemica l p o t e n t i a l u=0:

i f u==0i f T==0

F = ((4∗ kˆ4)/(3∗8∗ pi ˆ2) )∗ (1/(2∗ omega2)+3/(2∗omega ) ) ;end

i f T>0F = ((4∗ kˆ4)/(3∗8∗ pi ˆ2))∗((1+2∗n( omega2 ,T) )/ (2∗ omega2 ) . . .

+3∗(1+2∗n(omega ,T) )/ (2∗ omega ) ) ;end

end

%Return func t i on f o r chemica l p o t e n t i a l s u>0:

i f u>0


%Define f r a c t i on , ( b1∗xˆ2+b2∗x+b1 )/( a1∗xˆ3+a2∗xˆ2+a3∗x+a4 ) :

b1=3;b2=6∗kˆ2+2∗D2+4∗p2∗D22+4∗(D1−uˆ2)+4∗p1∗D11+4∗uˆ2 ;b3=2∗(kˆ2+D2+2∗p2∗D22)∗ ( kˆ2+D1−uˆ2+p1∗D11)+(kˆ2+D1−u ˆ 2 ) ∗ . . .

( kˆ2+D1−uˆ2+2∗p1∗D11)−4∗p1∗p2∗D12ˆ2 ;

a1=1;a2=−3∗kˆ2−D2−2∗p2∗D22−2∗(D1−uˆ2)−2∗p1∗D11−4∗uˆ2 ;a3=(kˆ2+D2+2∗p2∗D22)∗ (2∗kˆ2+2∗(D1−uˆ2)+2∗p1∗D11+4∗u ˆ 2 ) . . .

+(kˆ2+D1−uˆ2+2∗p2∗D22)∗ ( kˆ2+D1−uˆ2)−4∗p1∗p2∗D12ˆ2 ;a4=−(kˆ2+D2+2∗p2∗D22)∗ ( kˆ2+D1−uˆ2+2∗p1∗D11)∗ ( kˆ2+D1−u ˆ 2 ) . . .

+4∗p1∗p2∗D12ˆ2∗( kˆ2+D1−u ˆ2 ) ;

%Use re s i due theorem to s p l i t ac t i on in t o re s idues , r ,%and po les , p :

[ r , p , s ] = residue ( [ b1 , b2 , b3 ] , [ a1 , a2 , a3 , a4 ] ) ;

%Return func t i on F:

i f T==0F = ((4∗ kˆ4)/(3∗8∗ pi ˆ2) )∗ ( r (1 )/ (2∗ sqrt (p ( 1 ) ) )+ . . .r (2 )/ (2∗ sqrt (p (2)))+ r (3 )/ (2∗ sqrt (p (3)))+1/(2∗ omega ) ) ;

end

i f T>0F=((4∗kˆ4)/(3∗8∗ pi ˆ 2 ) ) ∗ ( . . .

(1+2∗n( sqrt (p ( 1 ) ) ,T) )∗ r (1 )/ (2∗ sqrt (p ( 1 ) ) )+ . . .(1+2∗n( sqrt (p ( 2 ) ) ,T) )∗ r (2 )/ (2∗ sqrt (p ( 2 ) ) )+ . . .(1+2∗n( sqrt (p ( 3 ) ) ,T) )∗ r (3 )/ (2∗ sqrt (p ( 3 ) ) )+ . . .(1+2∗n(omega ,T) )/ (2∗ omega ) ) ;

endend

endend


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