Neutrino Mass, Dark Matter, Gravitational Waves, Monopole Condensation, and Light Cone Quantization
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Edited by
Stephan L. Mintz Florida International University Miami,
Florida
and
Springer Science+Business Media, LLC
Library of Congress Catalog1ng-1n-Publtcat lon Data
N e u t r i n o m a s s , dark n a t t e r , g r a v i t a t i o n
a l w a v e s , monopo le
c o n d e n s a t i o n , and l i g h t cone q u a n t i z a t i o
n / e d i t e d by Behram N.
K u r s u n o g l u . , S t e p h a n L. M t n t z and A r n o l d
P e r l m u t t e r .
p . cm.
" P r o c e e d i n g s of t h e I n t e r n a t i o n a l C o n f
e r e n c e on O r b l s S c i e n t l a e
1996 f o c u s i n g on n e u t r i n o n a s s , d a r k m a t t e
r , g r a v l a t l o n a l w a v e s ,
c o n d e n s a t i o n o f atoms and m o n o p o l e s , l i g h t
cone q u a n t i z a t i o n , h e l d
J a n u a r y 2 5 - 2 8 , 1996, 1n Miami B e a c h , F l o r i d a
. " — T . p . v e r s o .
I n c l u d e s b i b l i o g r a p h i c a l r e f e r e n c e s
and i n d e x .
ISBN 978-1-4899-1566-5 1. P h y s i c s — C o n g r e s s e s . 2 .
A s t r o p h y s i c s — C o n g r e s s e s .
3 . P a r t i c l e s ( N u c l e a r p h y s i c s ) — C o n g r e
s s e s . 4 . Dark m a t t e r
( A s t r o n o m y ) — C o n g r e s s e s . 5 . N e u t r i n o s
— M a s s — C o n g r e s s e s .
I . K u r s u n o g l u , B e h r a m , 1922 - . I I . M 1 n t z ,
S t e p h a n L. I I I . P e r l m u t t e r , A r n o l d , 1928 -
. IV . I n t e r n a t i o n a l C o n f e r e n c e on
O r b i s S c i e n t l a e (1996 : M iami B e a c h . F l a .
)
QC1 .N487 1996 5 3 0 — d c 2 0 9 6 - 4 3 7 1 3
C I P
Proceedings of the International Conference on Orbis Scientiae
1996, focusing on Neutrino Mass, Dark Matter, Gravitational Waves,
Condensation of Atoms and Monopoles, Light Cone Quantization, held
January 2 5 - 2 8 , 1996, in Miami Beach, Florida
This volume was taken from a series of conferences sponsored by
Global Foundation, Inc., Coral Gables, Florida
ISBN 978-1-4899-1566-5 ISBN 978-1-4899-1564-1 (eBook) DOI
10.1007/978-1-4899-1564-1
© 1996 Springer Science+Business Media New York Originally
published by Plenum Press, New York in 1996 Softcover reprint of
the hardcover 1st edition 1996
All rights reserved
10 9 8 7 6 5 4 3 2 1
No part of this book may be reproduced, stored in a retrieval
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without written permission from the Publisher
PREFACE
The International Conference, Orbis Scientiae 1996, focused on the
topics: The Neutrino Mass, Light Cone Quantization, Monopole
Condensation, Dark Matter, and Gravitational Waves which we have
adopted as the title of these proceedings. Was there any exciting
news at the conference? Maybe, it depends on who answers the
question. There was an almost unanimous agreement on the overall
success of the conference as was evidenced by the fact that in the
after-dinner remarks by one of us (BNK) the suggestion of
organizing the conference on a biannual basis was presented but not
accepted: the participants wanted the continuation of the tradition
to convene annually. We shall, of course, comply.
The expected observation of gravitational waves will constitute the
most exciting vindication of Einstein's general relativity. This
subject is attracting the attention of the experimentalists and
theorists alike. We hope that by the first decade of the third
millennium or earlier, gravitational waves will be detected,
opening the way for a search for gravitons somewhere in the
universe, presumably through the observations in the CMBR. The
theoretical basis of the graviton search will take us to quantum
gravity and eventually to the modification of general relativity to
include the Planck scale behavior of gravity - at energies of the
order of 1019Ge V.
We were very pleased to welcome the 1995 Nobel Laureate Frederick
Reines to the Orbis Scientiae 1996, who moderated the conference
session on neutrino masses. Professor Reines has been an
enthusiastic participant of the Coral Gables Conferences, and in
1980 was awarded the J. Robert Oppenheimer Memorial Prize. We
preceded the Nobel Committee!
The Trustees and Chairman of the Global Foundation wish to extend
special thanks to Edward Bacinich of Alpha Omega Research
Foundation for his generous support of the 1996 Orbis
Scientiae.
Behram Kursunoglu Stephan L. Mintz Arnold Perlmutter
v
The Global Foundation, Inc., utilizes the world's most important
resource ... people. The Foundation consists of distinguished men
and women of science and learning, and of outstanding achievers and
entrepreneurs from industry, governments, and international
organizations, along with promising and enthusiastic young people.
These people convene to form a unique and distinguished
interdisciplinary entity to address global issues requiring global
solutions and to work on the frontier problems of science.
Global Foundation Board of Trustees
Behram N. Kursunoglu, Global Foundation, Inc., Chairman of the
Board, Coral Gables M. Jean Couture, Former Secretary of Energy of
France, Paris Manfred Eigen*, Max-Planck-Institut, Gottingen Robert
Herman, University of Texas at Austin Willis E. Lamb*, Jr.,
University of Arizona
Walter Charles Marshall, Lord Marshall of Goring, London Louis
Neel*, Universite de Gronoble, France Frederick Reines *,
University of California at Irvine Abdus Salam*, International
Centre for Theoretical Physics, Trieste Glenn T. Seaborg*, Lawrence
Berkeley Laboratory Henry King Stanford, President Emeritus,
Universities of Miami and Georgia
*Nobel Laureate
Global Foundation's Recent Conference Proceedings
Making the Market Right for the Efficient Use of Energy Edited by:
Behram N. Kursunoglu Nova Science Publishers, Inc., New York,
1992
Unified Symmetry in the Small and in the Large Edited by: Behram N.
Kursunoglu and Arnold Perlmutter Nova Science Publishers, Inc., New
York, 1993
Unified Symmetry in the Small and in the Large· 1 Edited by: Behram
N. Kursunoglu, Stephen Mintz, and Arnold Perlmutter Plenum Press,
1994
Unified Symmetry in the Small and in the Large· 2 Edited by: Behram
N. Kursunoglu, Stephen Mintz, and Arnold Perlmutter Plenum Press,
1995
Global Energy Demand in Transition: The New Role of Electricity
Edited by: Behram N. Kursunoglu, Stephen Mintz, and Arnold
Perlmutter Plenum Press, 1996
Economics and Politics of Energy Edited by: Behram N. Kursunoglu,
Stephen Mintz, and Arnold Perlmutter Plenum Press, 1996
Neutrino Mass, Dark Matter, Gravitational Waves, Condensation Of
Atoms And Monopoles, Light Cone Quantization Edited by: Behram N.
Kursunoglu, Stephen Mintz, and Arnold Perlmutter Plenum Press,
1996
Contributing Co-Sponsors of the Global Foundation Conferences
Gas Research Institute, Washington, DC General Electric Company,
San Jose, California Electric Power Research Institute, Palo Alto,
California Northrop Grumman Aerospace Company, Bethpage, New York
Martin Marietta Astronautics Group, Denver, Colorado Black and
Veatch Company, Kansas City, Missouri Bechtel Power Corporation,
Gaithersburg, Maryland ABB Combustion Engineering, Windsor,
Connecticut BellSouth Corporation, Atlanta, Georgia National
Science Foundation United States Department of Energy
vii
(NEUTRINO MASS, DARK MATTER, GRAVITATIONAL WAVES, CONDENSATION OF
ATOMS AND MONOPOLES, LIGHT CONE QUANTIZATION •••• )
(24TH IN A SERIES OF CORAL GABLES CONFERENCES ON ELEMENTARY
PARTICLE PHYSICS AND COSMOLOGY SINCE 1964)
JANUARY 25.28, 1996
PROGRAM
THURSDAY •. January 25.1996 (Cotillion Ballroom) 8:00 AM· Noon
REGISTRATION at the entrance of the Cotillion Ballroom
1:30 PM SESSION I:
BEHRAM N. KURSUNOGLU "Creation of Matter via Condensation at
Absolute Zero and Planck-Scale Temperatures" C.W. KIM, Johns
Hopkins University "Scale Dependent Cosmology for an Inhomogeneous
Universe" KAZUHIKO NISHUIMA, Chuo University, Tokyo "Unbroken
Non-Abelian Gauge Symmetry and Confinement"
Annotators: ALAN KRISCH, University of Michigan JOSEPH LANNUTTI,
Florida State University LARRY RATNER, University of Michigan
Session Organizer: BEHRAM N. KURSUNOGLU
3:30 PM Coffee Break
ix
3:45 PM SESSION II: INSPIRATIONS FROM COSMOLOGY AND ELEMENTARY
PARTICLE PHYSICS
Moderators: KATHERINE FREESE, University of Michigan ROBERT HERMAN,
University of Texas
Dissertators: KATHERINE FREESE "Inflationary Cosmology: From Theory
to Observation and Back" HARRISON PROSPER, Florida State University
"Bayesian Analysis of Solar Neutrino Data" EDWIN L. TURNER,
Princeton University Observatory "Do the Cosmological Parameters
have Natural Values?"
Annotators: ANDREW HECKLER, Fermilab
Session Organizer: KATHERINE FREESE
5:00 PM SESSION III: PROGRESS ON SOME NEW AND OLD IDEAS· I
Moderator: KAZUHIKO NISHUIMA, Chuo University, Tokyo GEORGE
SUDARSHAN, Center for Particle Physics, University of Texas
Dissertators: VERNON BARGER, University of Wisconsin "Fixed Points
in Supersymmetry: R-Parity-Violating Yukawa Couplings" GERALD B.
CLEAVER, Ohio State University, Columbus "Grand Unified Theories
from Superstrings" V ASKEN HAGOPIAN, Florida State University
"Capability of Future CMS Detector at the LHC Searching for Dark
Matter" FREYDOON MANSOURI, University of Cincinnati "Supersymmetric
Wilson Loops and their Stringy Extensions" KA TSUMI TANAKA, Ohio
State University, Columbus "Comments on the Symmetry Breaking Terms
in the Quark Mass Matrix" YUN WANG, Fermilab, Batavia,illinois
"Statistics of Extreme Gravitational Lensing Events"
Annotators: RICHARD ARNOWITT, Texas A & M University
Session Organizer: Dissertators
7:30 PM Orbis Scientiae adjourns for the day
FRIDAY, .January 26 1995 (Mona Lisa) 8:30 AM SESSION IV:
GRAVITATIONAL WAVES
Moderator: SYDNEY MESHKOV, Cal. Tech.
Dissertators: BARRY BARISH, Cal. Tech. "Status of LIGO"
x
Annotators: RICHARD P. WOODARD, University of Florida EDWARD KOLB,
Fermilab
Session Organizer: SYDNEY MESHKOV, CALTECH
10:15 AM Coffee Break
Moderator: BARRY BARISH, Cal. Tech.
Round Table Dissertators: SAMUEL FINN, PETER FRITSCHEL, PETER
SAULSON
Moderator: FREDERICK REINES, University of California, Irvine
Dissertators: MAURY GOODMAN, Argonne National Laboratory, Argonne
"Oscillation Searches Using Atmospheric Neutrinos and Long Baseline
Neutrino Experiments" C. W. KIM, Johns Hopkins Univ., Baltimore
"The Role of the Third Generation in the Analysis of Oscillation
Experiments" WILLIAM LOUIS, Los Alamos National Laboratory "Ongoing
Neutrino Oscillation Searches at Accelerators" RABIADREA MOHAPATRA,
Univ. of Maryland, College Park "Neutrino Mass Textures and New
Physics Implied by Present Neutrino Data" NEVILLE REAY, Kansas
State Univ., Manhattan "Fermilab MUNI Project" JOHN WILKERSON,
Univ. of Washington, Seattle "Solar Neutrino Measurements: Current
Status and Future Experiments" LINCOLN WOLFENSTEIN, Carnegie-Mellon
Univ., Pittsburgh "Theoretical Ideas About Neutrino Masses"
Annotators: JEREMY MARGULIES, Los Alamos National Laboratory
HARRISON PROSPER
Session Organizer: STEPHAN MINTZ, Florida International
University
3:30 PM Coffee Break
xi
3:45 PM ROUND TABLE DISCUSSION of Neutrino Masses by the Above
Dissertators
Moderator: WILLIAM LOUIS, Los Alamos National Laboratory
5:30 PM SESSION VI: STRINGS AND FIELD THEORY
Moderator: LOUISE DOLAN, University of North Carolina
Dissertators: LOUISE DOLAN, Department of Physics, University of
North Carolina "BPS States and Type II Superstrings" BRIAN GREENE,
Department of Physics, Cornell University "Changing the Topology of
the Universe" RENAT A KALLOSH, Department of Physics, Stanford
University "F and H Monopoles" JEFFREY MANDULA, DOE, Washington
D.C.
Annotators: GERALD B. CLEAVER, Ohio State University,
Columbus
Session Organizer: LOUISE DOLAN
7:00 PM Orbis Scientiae adjourns for the day
SATURDAY, January 27,1996 (Mona Lisa) 8:30 AM SESSION VII: DIRAC'S
LEGACY: LIGHT- CONE QUANTIZATION
Moderator: STANLEY BRODSKY, SLAC
STAN BRODSKY, SLAC "Applications of Light-Cone Quantization"
HANS-CHRISTIAN PAULI, Max Plank Institute, Heidelberg "Discrete
Light-Cone Quantization"
10:00 AM Coffee Break
xii
Dissertators: ALEX KALLONIA TIS, University of Erlangen-Nurnberg,
Erlangen "2-D Non-Perturbative Light-Cone Results" DAVE ROBERTSON,
Ohio State University "The Light-Cone Gauge and Zero Modes" BRETT
VAN DE SANDE, Max-Planck Institute, Heidelberg "Tube Model
Solutions of QCD"
Annotators: ZACHARY GURALNIK, Princeton University
Session Organizer: STEPHEN PINSKY
12:00 PM Lunch Break
Moderator: EDW ARD KOLB, FNAL, Chicago
Dissertators: RICHARD ARNOWITT, Texas A&M University "SUSY Dark
Matter with Non-Universal Soft Breaking Masses" SHARON HAGOPIAN,
Florida State University "Search for SUSY in the DO Collider
Experiment" ANDREW HECKLER, Fennilab "On the Fonnation of a
Hawking-Radiation Photoshpere: The Cloak Around Microscopic Black
Holes" EDW ARD KOLB, Fennilab "Light Photinos as Dark Matter" IGOR
TKACHEV, Ohio State Univ. "Primordial Axions Appearing as Dark
Matter and Other Astrophysical Objects"
Annotators: V ASKEN HAGOPIAN
3:30PM Coffee Break
3:45 PM SESSION IX: PROGRESS ON SOME NEW AND OLD IDEAS II
Moderators: FRED ZACHARIASEN, CALTECH
Dissertators: PRAN NATH, Institute for Theoretical Physics, Santa
Barbara, CA "Superunification and Planck Scale Interactions" MARK
SAMUEL, Oklahoma State University "Going to Higher Order - The Hard
Way and the Easy Way: The Agony and the Ecstasy" INA SARCEVIC,
University of Arizona "Domain Structure of a Disoriented Chiral
Condensate from a Wavelet Perspective" RICHARD P. WOODARD,
University of Florida "Quantum Gravity Slows Inflation"
Annotators: GERALD GURALNIK, Brown University
Session Organizer: SESSION DISSERTATORS
7:30 PM Conference Banquet - MONA LISA ROOM
SUNDAY. January 28. 1996 (Key Biscayne Room) 8:30 AM SESSION X:
PROGRESS ON SOME NEW AND OLD IDEAS - III
Moderator: DON B. LICHTENBERG, Indiana University
xiii
Dissertators: DON B. LICHTENBERG, Indiana University "Superflavor
Symmetry and Relations Between Meson and Baryon Masses" PAUL H.
FRAMPTON, University of North Carolina at Chapel Hill "Constraining
a(Ma) From the Hidden Sector" LUCA MEZINCESCU & RAFAEL
NEPOMECHIE, University of Miami "Integrable Systems with
Boundaries" GREGORY TARLE, University of Michigan "Cosmic Ray
Signatures for Neutralinos: New Measurements and their
Implications"
Annotators: G. BHAMA THI, University of Texas at Austin
Session Organizer: DISSERTATORS
10:30 AM SESSION XI: EXACTLY SOL V ABLE QUANTUM MODELS
Moderator: ANDRE LeCLAIR, Newman Laboratory, Ithaca, New York
Dissertators: PAUL FENDLEY, University of Southern California, Los
Angeles "Two-Dimensional Field Theory Meets Experiment" SERGEI LUKY
ANOV, Newman Laboratory, Cornell University "Yang-Baxter Equation
and Baxter's Q-Operators in CFT" GIUSEPPE MUSSARDO, Scuola
Internationale Superiore di Studi Avanzati, Trieste, Italy "Form
Factor Approach to Integrable Quantum Field Theory: The spin-spin
correlation function of 2-d Ising Model in a magnetic field" LUC
VINET, Laboratorie de Physique Nuc1eaire et Centre de Recherches
Mathematiques, Montreal, Canada "Exact Operator Solution of the
COLOGERO SUTHERLAND Model"
Annotators: ZACHARY GURALNIK, Princeton University
Session Organizer: ANDRE LeCLAIR, Newman Laboratory, Ithaca, New
York
12:30 AM Lunch Break
xiv
Dissertators: ALAN CHODOS, Yale University "Sonoluminescence and
the Heimlich Effect" ZACHARY GURALNIK, Princeton University
"Critical Phenomena and the Boundary Conditions for Schwinger-Dyson
Equations" GERALD GURALNIK, Brown University
"Using Symmetry to Numerically Solve Quantum Field Theory" GEOFFREY
WEST "Glueballs, the Essence of Non-Perturbative QeD" DONALD
WEINGARTEN, IBM, New York "Evidence for the Observation of a
Glueball"
Annotators: SESSION MODERATORS
xv
CONTENTS
Unbroken Non-Abelian Gauge Symmetry and Confmement
....................... 13 K. Nishijima
SECTION II - PROGRESS ON NEW AND OLD IDEAS - A
R-parity-violating Yukawa Couplings
........................................................ 19 V.
Barger, M.S. Berger, RJN. Phillips, and T. Wohrmann
Grand Unified Theories from Superstrings
................................................. 31 Gerald B.
Cleaver
Searching for Dark Matter with the Future LHC Accelerator at CERN
Using the CMS Detector
............................................................
43
Vasken Hagopian and Howard Baer
A Scale Invariant Superstring Theory with Dimensionless Coupling to
Supersymmetric Gauge Theories
............................................................
49
M. Awada and F. Mansouri
Superstring Solitons and Conformal Field Theory
...................................... 57 L. Dolan
Comments on Symmetry Breaking Tenns in Quark Mass Matrices
............ 65 K. Tanaka
SECTION III - GRA VIT A TIONAL WAVES
LIGO: An Overview
................................................................................
73 Barry C. Barish
Cosmology and LIGO
................................................................................
79 Lee Samuel Finn
Interferometry for Gravity Wave Detection
................................................ 95 Peter
Fritschel
xvii
Theoretical Ideas about Neutrino Mass
.................................................... 111 Lincoln W
olJenstein
A Bayesian Analysis of Solar Neutrino Data
............................................ 115 Harrison B.
Prosper
Ultrahigh-Energy Neutrino Interactions and Neutrino Telescope Event
Rates
.............................................................................................
121
Raj Gandhi, Chris Quigg, MH. Reno, and Ina Sarcevic
SECTION V - DIRAC'S LEGACY: LIGHT-CONE QUANTIZATION
Dirac's Legacy: Light-Cone Quantization
............................................... 133 Stephen S.
Pinsky
Light-Cone Quantization and Hadron Structure
........................................ 153 Stanley 1.
Brodsky
Discretized Light-Cone Quantization
....................................................... 183
Hans-Christian Pauli
Possible Mechanism for Vacuum Degeneracy in YM2 In DLCQ
.............. 205 Alex C. Kalloniatis
The Vacuum in Light-Cone Field Theory
................................................. 223 David G.
Robertson
The Transverse Lattice in 2+ 1 Dimensions
.............................................. 241 Brett van de
Sande and Simon Dalley
SECTION VI - THE MATTER OF DARK MATTER
SUSY Dark Matter with Universal and Non-Universal Soft Breaking
Masses
.....................................................................................
253
R. Arnowitt and Pran Nath
Search for SUSY in the D0 Experiment..
................................................ 265 Sharon
Hagopian
Formulation of a Photosphere around Microscopic Black Holes
............... 273 Andrew F. Heckler
A Supersymmetric Model for Mixed Dark Matter.
................................... 283 Antonio Riotto
Light Photinos and Supersymmetric Dark Matter
..................................... 287
Edward W. Kolb
Non-Universality and Post-GUT Physics in Supergravity Unification
...... 301 Pran Nath and R. Arnowitt
Pade Approximants, Borel Transform and Renormalons: The Bjorken Sum
Rule as a Case Study
................................................... 309
John Ellis, Einan Gardi, Marck Kanliner, and Mark A. Samuel
Hadron Supersymmetry and Relations between Meson and Baryon Masses
........................................................................................
319
DB. Lichtenberg
Constraining the QCD Coupling from the Superstring Hidden Sector
....... 323 Paul H. Frampton
Weak Interactions with Electron Machines: A Survey of Possible
Processes ......................... ;
......................................................... 331
SL Mintz, M.A. Barnett, G.M. Gerstner, and M. Pourkaviani
SECTION VIII - EXACTLY SOLUBLE QUANTUM MODELS
Matrix Elements of Local Fields in Integrable QFr
.................................. 349 G. Delfino and G.
Mussardo
Boundary S Matrix for the Boundary Sine-Gordon Model from
Fractional-Spin Integrals of Motion
................................................. 359
Luca Mezincescu and Rafaell. Nepomechie
SECTION IX - EPILOGUE
Boundary Conditions for Schwinger-Dyson Equations and Vacuum
Selection
..................................................................................................
377 Zachary Guralnik
Numerical Quantum Field Theory Using the Source Galerkin Method
..... 385 G. S. Guralnik
Index
.......................................................................................................
395
COLD VERSUS HOT CONDENSATION TO CREATE MATTER
The process of condensation of the monopoles carrying magnetic
charges gn with n ranging from zero to infinity, to create an
orbiton was first obtained twenty years ago in my paper in Physical
Review D Vol. 13, Number 6,15 March 1976, (see especially the pages
1539 and 1551). In contrast to the Bose-Einstein condensation of a
dense gas of atoms near absolute zero termperature in the recent
experiments (July 1995) by Eric A. Cornell and his colleague Carl
Wieman to create a condensate as a "large atom", condensation of
the magnetic charges at the dawn of the universe was taking place
in an inferno at Planck-scale temperatures (- 1030 degrees Kelvin)
to create an orbiton (a quark with structure). In both instances of
the resulting condensates the distribution of atoms and monopoles,
respectively, range from packed to sparse. The color picture above
for orbiton represents a layered structure of magnetic charges with
alternating signs and decreasing magnitudes. The painting was
commissioned in 1980 to Ms. Sheila Rose of Miami. It appeared in
black and white on page 1539 of the referred Physical Review paper.
The confined magnetic charges, resulting from the condensation
prior to the Big-Bang creation of the universe, do also confine the
electric charges. For more scientific discussions, see my paper
"After Einstein and SchrOdinger: A New Unified Field Theory ,"
Journal of Physics Essays, Vol. 4, No.4, pp 439- 518, 1991 and the
references there in pages 517 and 518. See also the 1994, 1995 and
1996 proceedings of the Coral Gables Conferences sponsored by the
Global Foundation and published by Plenum Publishing Company, New
York.
The similarities of the Bose-Einstein condensation of a dense gas
of atoms at a temperature of absolute zero and that of magnetic
charges at Planck -scale temperatures are most striking. Do recent
experiments pertaining to Bose-Einstein cold condensation also
vindicate the hot condensation pertaining to magnetic charges to
create matter?
xx
INTRODUCTION
I would like to present some new results regarding the ongm of mass
and distributions of electric and magnetic charges in an elementary
particle. The discreteness of the electric charge distribution and
its confinement results from the layered distribution, with
alternating signs, of the magnetic charge. It is found that
magnetic charge layers thin out towards the surface of the particle
while the electric charge increases so that most of it resides on
the particle' s surface. The fact of the electric and
magnetic charges lying on a circle [i.e., ro2 = (2G/c4) (e2+g2)]
implies that the
discreteness of the magnetic charge distribution will result in the
discreteness of electric charge distribution.
In my January 1995 Coral Gables conference presentation (II "Exact
Solutions for Confinement of Electric Charges via Condensation of a
Spectrum of Magnetic Charges" it was pointed out that layered
magnetic charge constituency with alternating signs corresponds to
the structure ot: for example. a quark of spin angular momentum K
It was clear that a point like quark could not possibly carry a
spin and have mass. On this occasion I shall provide, based on
recent experiments. more theoretical evidence on the. however
small. ultimate structure of matter. In what follows I would like
to discuss some remarkable similarities of the condensation of gas
of atoms near absolute zero temperature with the condensation of a
gas of monopoles at a temperature of the order of 1032 degrees
kelvin that may have prevailed during the Big Bang creation of the
universe.
The July 14, 1995 issue of the New York Times contained a spread
announcing the experimental results on the theoretical prediction
of the Bose-Einstein condensation phenomenon. The Eric A. Cornell
et al experiment with rubidium gas cooled near absolute zero
revealed the creation of a Bose-Einstein condensate. The same type
of condensate was. after a month. obtained in an experiment with
lithium gas at Rice University. The details of the first experiment
appearcd also in the July 14, 1995 issue of the AAAS Science
magazine in full color to illustrate the distribution of the atoms
in the condensate. The color picture was also included in page 19
of the August 1995 issue of Scientific American. The recent 1996
calendar received from the American Physical Society is graced by
the same color picture of the condensate. Here I would like to
discuss this and another kind of condensation during the early
universe referring to monopoles and the corresponding condensate,
the elementary particle. The latter type of condensation was for
the first time introduced in my paper in Physical Review D, Volume
13, Number 6, 15 March 1976.
3
4
BOSE-EINSTEIN CONDENSATION AT T ~O
The most remarkable aspect of the condensate' s picture was the
distribution of atoms at 35 nanokelvin -- 35 billionths of a degree
above zero -- across 100 microns from packed (red rimmed portion in
the picture) to .Iparse (yellow rimmed portion in the picture) i.e.
decreasing density qjatoms with the distance ji-om the origin. The
Colorado group saw the condensate formed at around 20 nanokelvins,
the lowest temperature ever achieved and included around 2000
atoms. The Rice University group achieved the condensation of some
100.000 atoms at a temperature between 100 and 400
nanokelvins.
The Bose-Einstein condensation, compared to other phase transitions
governed by the forces between atoms and molecules, is driven by
the quantum mechanical concepts. In accordance with the uncertainty
principle the position of the atoms are spread
proportional to their wave-lengths A related to their momentum p by
the relation Ap=h. When coiled near the zero temperature. atoms are
barely moving. their positions become uncertain. Correspondingly
the wave function of the atoms spreads out and merge leading to a
quantum state occupied by a large number of atoms. As the
temperature dropped so did the size of the condensate atoms.
CorneIrs group while scanning the cloud of rubidium atoms with a
laser found a sharp increase in density toward the middle. The
properties oj the condensate ineludes a survival lime of one minute
bej(n'e .freezing into rubidium-R7 ice.
In all these. bosons lose their individual identities. condensing
into a part of a superboson or a superatom. It is this loss of
identity ncar absolute zero temperature that the quantum mechanical
wave function of neighboring atoms overlap and lead to the
formation of a condensate. Thus. for a condensate to emerge the
experiment must overcome the fact that the long-lived atoms are
composite products and can stick together not allowing the
formation of a condensate. However, with the lowering of the
temperature. the atoms' wave lengths become longer and they can be
packed close enough together to merge to create a condensate.
The Rice University group used Iithium-7 gas which. unlike
rubidium-87 atoms repelling each other weakly (residual forces
arising from their orbiting electrons). consists of atoms that
at/ract each other. This meant that they would form a liquid and
drain away long before the formation of a condensate. However, the
Rice group seems to have achieved the condensation of 100.000 atoms
of lithium. The rather brief discussions of the experimental
findings on the Bose-Einstein condensation occurring at absolute
zero temperature will now be compared with the monopole
condensation during the early universe and creation of matter or
quarks as condensates of monopoles at Planck-scale temperatures (~I
01' degrees kelvin).
A BRIEF OUTLINE OF THE GENERALIZED THEORY OF GRA VITA nON
The idea of monopole condensation was inferred from the spherically
symmetric form of the generalized theory of gravitation (121). The
theory was originated from the nonsymmetric structure of general
relativity in the presence of an electromagnetic field where
electric charges were not present. The basic nonsymmetric field
variables in general relativity expressed in their contravariant
form are given by
(1)
which can be obtained. to order qo-l from the inverse of the
covariant nonsymmetic tensor(2)
(2)
where the constant qo has the dimensions of an electric field and
the tensors 9J..!v and
<l>J..!v represent the generalized gravitational and
generalized electromagnetic fields.
respectively. The addition of the anti symmetric tensor
<l>J..!v to' 9J..!v as in (2) is
equivalent to turning on the electric and magnetic charges.
The Largrangian of general relativity can be expressed in terms of
the
nonsymmetric contravariant tensor (I) provided the constant qo is
restricted by the
fundamental relation (3)
r 2q 2= c4/2G o 0 , (3)
where both real r 0 (fundamental length). qo and their purely
imaginary forms ir 0 • iqo are allowed since in both cases the
field equations of general relativity are unchanged. This kind of
invariance is referred to here as a super,l)lfl1mefry degeneracy of
general relativity where electric and magnetic charges are not
included and the concept of spin angular momentum does not come in.
The use of the word supersymmetry here is not related to its use in
the conventional elementary particle physics. General relativity
predicts the existence of gravitational waves which carry energy
and momentum but. because of supersymmetry degeneracy or because of
the symmetric field variables. they do not can')' mass. However.
for nonsymmetricfield variables arising from turning on electric
and magnetic charges the resulting theory, besides massless
gravitational waves, predicts the existence of massive waves
carrying spin O. 1. and 2. Thus, in place of Higgs bosons. as in
the conventional theory. as the origin of mass. we find that the
nomymmetTy
of the field variahles g~lV defined by definition (2) is the
fill1dwnental basis for the genera/ion ofll1([ss. It must also be
understood that the nonhermitian and the hermitian
field variables 9 J..!V are SO(2) and lJ( I) gauge invariant.
respectively. The two
supersymmetric real and complex field variables describe fermi-like
and bose-like paliicles. respectively.
However. if the nonsymmetric tensor g~lV as given by the definition
(2) is used as
the basis of the generalized theory of gravitation then the
supersymmetry degeneracy is removed (2) and we obtain a theory
which includes electric and magnetic charges along with particles
of half integral and integral spin angular momenta. In this case
the
.., fundamental relation (3) between qo- (energy density). and the
fundamental length ro can be interpreted as an eqllation o!sliJte
and is most versatile in its cosmological and
elementary particle physics implications. For example. if ro is
taken as large as the size
"'1 .., of the universe and qo ~ C- represents the average mass
density in the universe then we
find that the equation (3) yields the results in the ballpark. The
equation of state (3) can also be written as
(4)
5
where Eo = q02 = energy density and Po = C2/2Gr02 = mass density.
The r o-2
can be interpreted as the average curvature of space. If r 0 is of
the order of the size of
the universe then the curvature of space is very small and the
field equations of the generalized theory of gravitation yield flat
space-time solutions and therefore the universe is approximately
flat where the mass density is, as the universe keeps expanding,
constantly decreasing. The mass of a particle or the universe
itself can be defined by
(5)
which can also be obtained by integrating the equation (4) over the
r 0 -space. Here
again, by substituting the value of ro as the size of the universe
we obtain the ballpark
value for the mass of the universe (~ 1 022Mo ' Mo = total solar
mass).
The mass relation (5) \vhen written in the f0l111 '
(6)
is reminiscent of the Schwarzschild singularity in general
relativity but "0 not refening
to coordinates, it is not related to that singularity. However. the
relation (6) is
reminiscent of gravitational col/apse yielding a particle where ro
is its gravitational size
lying inside the particle and M is its corresponding mass. The
relation (6) is. of course.
independent of the coordinate system. From (5) we can write the
relation
(7)
Hence, if we consider the special case of Planck particle we
obtain
ropo = Yzft , (8)
where we choose
(10)
CREATION OF MATIER VIA MONOPOLE CONDENSATION
At the instant of creation of the universe from a region of the
vacuum of size r 0
at the prevailing Planck-scale temperatures (~l 0" kelvin) the
monopoles of positive and negative magnetic charges lost their
individual identity and with a mass small compared
to their energy Cp began to condense. The monopoles' wave-lengths
were of the order
of Planck length and they were closely packed to merge and to form
a monopole condensate. The monopole condensate could have lasted
only a Planck-time duration (10·,]1 sec) to change phase and could
have '"frozen" into an orbiton (quark with structure)
6
or an antiorbiton (antiquark with structure) as illustrated in the
figures 1,2. arid 3. If an orbiton represents a quark with
sfructure then it can constitute with additional quarks (or
antiquarks) elementary particles like. for example, protons,
neutrons and the variety of bosons (figures 3. 4). In the
approximate solutions of the spherically symmetric field
equations where an angular (or hyperbolic) function <l> is a
constant then the
fundamental length r 0 is obtained as
(11 )
where e and 9 represent fundamental units of electric and magnetic
charges,
respectively. The numbers N± and :Jvl± can be expressed as
(12)
where nand n' range over (0. I. 2 ....... 60 ....... ). The minus
signs in the definitions
(12) refer to elementary particles while the plus signs are related
to the size and the expansion of the universe (creation of electric
and magnetic charges from the vacuum)
where, for example. for n = n' = 60 one obtains ro ~ 1028 cm .. the
size of the universe.
For example, to obtain the proton mass from (5) we must choose the
gravitational size of
proton relative to gravitational size of the universe (i.e. r 0 ~
10'8 cm.) to be of the order
of 10.52 cm. so as to yield the ratio of the mass of the universe
to proton mass to be of the order of 10so, the number of particles
in the universe. Thus. in accordance with Mach' s principle the
inertia of a mass is due to the distribution of the rest of the
mass in the ulllverse.
Figure I. Figure 2.
Figure l. Confinement of the magnetic charge. Layered distribution
of the magnetic charges with alternating signs and decreasing
amounts generates short-range forces to confine all the
layers.
Figure 2. Confinement of the electric charge. Layered distribution
of the electric charges of the same signs, within the magnetic
charge layers, with increasing amounts leads to the confinement of
the electric chatges residing mostly in the outer magnetic charge
layers or on the "surface" of the elementary particles.
7
Figure 3. Magnetic charge dipole. Represents orbiton-antiorbiton
synthesis to create spin zero or spin one particles. It could also
correspond to quark-antiquark combinations. The arrows represent
directions of spins. Addition of two spin angular momenta yields 0
or 1, -1 units represented by antiparallel and parallel spin
directions, respectively.
For the special case where n = n' = 0 we have the simple
relation
(13)
where the discrete values gl' (l' = I. 2. 3, .... ) of the magnetic
charge implies discrete
distribution of the electric charge itself which is also related to
magnetic charge by
(14)
whereJll2 [= g2/(e2 + g2] is the eigen-value which appears in the
field equations
for regions of zero magnetic charge density (i.e. the interface
between positive and negative magnetic charges in an orbiton.) We
note that for a "free" monopole, as shown by P.A.M. Dirac a long
time ago. the electric and magnetic charges are related by
eg = (Yl)nhc , (15)
where n is an integer. However. in our theory. as seen from
relation (11) or relation (13).
there exists no free monopoles since both the electric and magnetic
charges are confined to constitute the elementary particles i.e ..
the monopoles are hidden. Such a distribution
8
I , Figure 4. Proton's constituents consist of three orbitons (or
quarks with structure). Addition of the three spin angular momenta
of 1/2 units where the latter refers to a particle different from
the proton.
of magnetic charge. where Ign = 0 represents the vanishing of the
infinite sum of
magnetic charges which generate short-range force. We also observe
that if all matter is made of confined positive and negative
magnetic charges then the ratio of dark matter to
luminous matter can be represented by g/e = eg/e2 = (Y2)nfzc/e2 ~
68n. Hence.
depending on the choice of the integer n(= L 2. ... ) the universe
may consist predominately of dark matter.
The solutions of the tield equations where thc angle (or
hyperbolic) function <D is
not a constant may have linear dependence on the positive and
negative electric and magnetic changes. Such solutions are expected
to remain unchanged under the interchange of the positive and
negative electric charges and should lead to the existence of
electric and magnetic dipole moments. In fact the theory yields
four sets of generalized Dirac wave equations ,·1 with the mass
defined by (5). and with diicrete space-time symmetries. In the
meantime. aside from various electric and magnetic
moments. the discussion of the relation (13). for a given r 0-
constitutes a definite proof
for the confinement of the electric charge. because of the discrete
nature of the magnetic charge distribution. most of it resides on
the surface of the elementary particle and as seen in figures 1 and
2. it remains stable.
The distribution of electric and magnetic charges in an elementary
particle as predicted by the generalized theory of gravitation
yields a dual running coupling of the fields. Thus. at very high
energy (like. for example. in the canceled sse accelerator of
9
the order of 40 Tev). scattering of charged particles (or
preferably proton-antiproton collisions) while the electromagnetic
coupling decreases with increasing energy the strong coupling.
experienced during interparticle penetration. increases. The
running coupling constants which appear in the field equations for
the spherically symmetric fields are given by
where the electric charge e increases towards the surface of the
particle whilst the
magnetic charge 9 increases towards the origin of the particle. In
the same way e and 9 decrease towards the origin and towards the
surface. respectively. Both e and 9 assume
zero values at the origin. Experimentally very high energy
proton-antiproton scattering may provide, hopefully. some clues
with respect to the nature of the above mentioned structural
properties of the elementary particles.
DISCRETENESS OF CONFINED ELECTRIC CHARGE DISTRIBUTION
If in the fundamental relation (13) we represent the fundamental
length r 0 in
discrete units of the Planck length by substituting
(17)
N ~ 10±n , n = 0, 1, 2, .... , (19)
and where e and 9 represent electric and magnetic charges.
respectively. as restricted by
the relation (18). and they lie on a circle of radius
(N/(",J2))(-V(hc)). The
discreteness of the magnetic charge 9 (= 9" 92, 93, ..... 9n, .....
) implies.
because of (18). a discrete spectrum or the quantization of the
electric charge e (=e 1,
e2, e3, ..... en, ..... ) where
• ~9n =0, 9n=(-lt 19n1, 19n1 > 19n11, Lim. 9n =0, (20)
(21)
The distribution of the confined magnetic charges as quantified by
the relation (18) and portrayed in the figure 1 determines the
confined electric charge distribution as described by the relations
(21) and portrayed in figure 2. It is clear that most of the
electric charge. as confined by the magnetic charge distribution,
resides on the surface of the elementary patticle.
10
REFERENCES
1. Behram N. Kursunoglu. Unified Symmeli)' In the Small and In the
Large, 1995, Volumes 1 and 2, Plenum Press, New York. edited by
Behram N. Kursunoglu et al.
2. Behram N. Kursunoglu, .Journal of Physics Essays. Vol. 1. No.4.
pp. 439-518, 1991, University of Toronto Press.
3. Behram N. Kursunoglu, Physical Review, 88. 1369 (\ 952).
4. Behram N. Kursunoglu. Physical Review D. Volume 12. Number 6, 15
March 1976.
5. See the July 14. 1995 issue of the New York Times. AAAS Science
Magazine. and the August 1995 issue of Scientific American (page
19).
11
K. Nishijima Department of Physics, Chuo University
Bunkyo-ku, Tokyo 112, Japan
It is shown that color confinement is an inevitable consequence of
unbroken color symmetry and asymptotic freedom of QCD.
1. Interpretation of Color Confinement
The quark model of hadrons has been so successful that we can no
longer think of any other substitute for it. All the experimental
evidences for this model have been indirect, however, since no
isolated quarks have been observed to date. Thus the hypothesis of
quark confinement emerged implying that isolated quarks are in
principle unobservable. Later, this was promoted to the hypothesis
of color confinement that implies the unobservability of all the
isolated colored particles including gluons.
Then a natural question is raised of whether we can account for
this hypothesis within the framework of the conventional QCD or we
need a new additional principle. It is the purpose of the present
paper to stress that color confinement is an inevitable consequence
of the conventional QCD provided that color symmetry is not
spontaneously broken and that asymptotic freedom is valid. Since
the mathematical details of its proof have been published
elsewhere,03>, we shall give here the basic ideas underlying
this approach.
The solution of this problem is decomposed into two steps. First,
we have to find a proper interpretation or definition of color
confinement, and then we have to prove it. In fact, there is a
variety of interpretations of confinement. To quote a few, Wilson's
area law4)
in the lattice gauge theory leads to the linear potential between a
pair of a quark and an antiquark that holds the system to be always
in bound states. Another example is the recent supersymmetric
theory of Seiberg and Witten5>,6l in which the duality between
electric and magnetic fields holds, and confinement is then a
consequence of the condensation of magnetic monopolies. Therefore,
speaking of confinement we have to specify what it means.
We start looking for a known example of confinement within the
framework of known field theories. Then it occurs to us that we
have a prototype example of confinement in QED. let us quantize the
electromagnetic field in a covariant gauge, say, in the Fermi
gauge, and we recognize that there are three types of photons,
namely, transverse, longitudinal and scalar photons. Of these three
types only the transverse photons are subject to observation, and
the latter two escape detection. This is indeed a typical example
of confinement, and we shall recapitulate the underlying
implication.))
Quantization of the electromagnetic field introduces the indefinite
metric that was inherited from the Minkowski metric. In order to
adopt the probabilistic interpretation of quantum mechanics to QED,
it is necessary to confine ourselves to physical states which
13
are free of negative probability. 'In fact, the Lorentz condition
selects such states, and in particular those states that involve
only transverse photons and changed particles belong to the
physical subspace of the whole state vector space. The S matrix,
then, transforms a physical state into another physical
state.
Let us consider the unitarity condition of the S matrix between two
transverse photon states, then the intermediate states are
saturated by physical states. In fact, both longitudinal and scalar
photons show up in the intermediate states, but their contributions
cancel themselves leaving only those of the transverse photon
states. As a result, longitudinal and scalar photons are not
observable, implying confinement of these unphysical photons. This
mechanism of confinement may be referred to as metric cancellation
since it is due to the indefinite metric.
We are now concerned with how we should extend this interpretation
of continement to QCD which is a typical non-abelian gauge theory.
In QCD we introduce a pair of so-called Faddeev-Popov ghost fields
in order to keep the S matrix unitary. They are anticommuting
hermitian scalar fields denoted by c and c respectively. Since they
violate Pauli's theorem on the connection between spin and
statistics we are obliged to introduce indefinite metric
again.
For the gauge fields as well as quark fields we can introduce local
gauge transformations. Let us consider an infinitesimal local gauge
transformation and replace the intinitesimal gauge function by
either c or c, and we obtain the BRS or anti-BRS transformation of
the respective fields. For the ghost fields local gauge
transformations cannot be defined, but their BRS or anti-BRS
transformations can be defined so as to keep the total Lagrangian
density invariant. Then Noether's theorem leads to the conserved
charges corresponding to the BRS and anti-BRS invariances,
respectively. They are called BRS charges.
In QCD the subsidiary condition corresponding to the Lorentz
condition is the Kugo-Ojima condition.7), 8) Namely, physical
states are defined as those states that are annihilated by applying
the BRS charge. The collection of physical states forms the
physical subspace of the whole state vector space, and it is an
invariant subspace of the S matrix in QCD just as in QED.
Therefore, when color multiplet states, such as isolated quark or
gluon states, do not belong to the physical subspace, they escape
detection in the same sense as the longitudinal and scalar photons
do in. QED. We may interpret this as color confinement, and we now
know what we should prove for color confinement.
2. A Sufficient Condition for Color Confinement
Now that we have introduced an interpretation of confinement in the
preceding section we have to investigate the condition under which
confinement is realized.
Because of the presence of the gauge-fixing term in the Lagrangian
density the equation for the gauge field deviates from the standard
Maxwell equation. The current corresponding to this deviation can
be obtained can be obtained by applying the BRS and anti-BRS
transformations successively to the gauge field. Then let us
introduce three-point Green functions involving this current and a
pair of quark fields or gluon fields. By taking the four-divergence
of these Green functions with respect to the space-time coordinates
of this current we tind that they can be equated to two-point
functions of the quark fields or gauge fields, respectively, by
making use of the modified Maxwell equation. They are the so-called
Ward-Takahashi identities and we can write them down for any pair
of colored fields provided that their transformation properties
under the color SU(3) group are known.
When isolated quark states as well as gluon states are not
annihilated by applying the BRS charge, then they escape detection
since they are not physical states. In order to relate Green
functions to quark or gluon states for the purpose of checking the
above
14
condition we have to refer to the LSZ reduction formula. 8)
Therefore, we shall assume it~ validity in what follows. Then we
can prove that the asymptotic gluon field, either incoming or
outgoing, consists of two terms, one corresponding to the gluon and
the other to the gradient of a massless spin zero (ghost) particle.
By combining the Ward-Takahashi identities with the LSZ reduction
formula we can show that color confinement is realized provided
that successive applications of the BRS and anti-BRS
transformations annihilate the second term in the asymptotic gluon
field. This is indeed a sufficient condition to be referred to as
the condition A. Since the asymptotic field is a rather complicated
object, we shall express this condition in terms of Heisenberg
operators.
For this purpose let us consider a two-point function defined as
the vacuum expectation value (VEV) of the time-ordered product of
the gauge field and the current corresponding to the deviation from
the Maxwell equation. When the residue of the massless spin zero
pole of this two-point function vanishes, the condition A is
satisfied.
Let us denote this residue as C, then it cannot vanish when color
symmetry is spontaneously broken. Indeed, the non-vanishing residue
is a signature of the emergence of the Nambu-Golstone boson.
Therefore, color symmetry should not be broken for the realization
of color confinement.
In general this constant C is gauge-dependent and satisfies a
simple renormalization group (RG) equation. This equation alone
cannot determine C, however, unless a proper boundary condition or
a normalization condition is given. Thus we shall stand on a
slightly different point of view. We realize that this constant C
can be expressed as the VEV of an equal-time commutator (ETC)
between two Heisenberg operators in the two-point function
mentioned above. This ETC is given as a sum of two terms. The first
term denoted by a is equal to the inverse of the renormalization
constant of the gluon field Z3' and the second term is the
so-called Goto-Imamura-Schwinger (GIS) term 10), II) that was first
discovered in the evaluation of the ETC between the space and time
components of the charge current density in QED. For the GIS term
both the RG equation and the boundary condition are known, and it
can be expressed uniquely in terms of a vanishes, so that the
evaluation of a is now the central issue.
The renormalization constant Z3 is gauge-dependent in QCD, but the
concept of confinement extends to other gauges in which a does not
vanish.
Fortunately it is possible to evaluate a exactly with the help of
RG, and we can prove that we can always find gauges in which a
vanishes exactly provided that asymptotic freedom is valid.
Thus we conclude that confinement is an inevitable consequence of
an unbroken nonabelian gauge symmetry and asymptotic freedom. The
electro weak interactions are not related to confinement since the
original non-abelian gauge symmetry SU(2) x U(l) is spontaneously
broken and reduces to the abelian gauge symmetry U(1).
Reference
1) K. Nishijima, Int. 1. Mod. Phys. A9 (1994) 3799. 2) K.
Nishijima, Int. 1. Mod. Phys. AI0 (1995) 3155. 3) K. Nishijima,
Czech. J. Phys. 46 (1996) 1. 4) K. Wilson, Phys. Rev. D14 (1974)
2455. 5) N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19. 6)
N. Seiberg and E. Witten, Nucl. Phys. B431 (1994) 484. 7) T. Kugo
and L Ojima, Phys. Lett B73 (1953) 255. 8) T. Kugo and L Ojima,
Prog. Theor. Phys. SuppL No.66 (1979) 1. 9) H. Lehmann, K. Symanzik
and W. Zimmermann, Nuovo Cim. 1 (1955)205. lO) T. Goto and T.
Imamura, Prog. Theor. Phys. 14 (1955) 396. 11) 1. Schwinga, Phys.
Rev. Lett. 3 (1959) 296.
15
IDEAS- A
(a) Physics Department, University of Wisconsin, Madison, WI 53706,
USA
(b)Physics Department, Indiana University, Bloomington, IN 47405,
USA
(C) Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OXIl OQX,
UK
ABSTRACT
We discuss the evolution of R-parity-violating (RPV) couplings in
the minimum super symmetric standard model, assuming a hierarchy
for coupling strengths and empha sising solutions where
R-conserving and R-violating top quark Yukawa couplings both
approach infrared fixed points. We show that fixed points offer a
new source of bounds on RPV couplings at the electroweak scale, and
that lower limits on the top quark mass lead to RPV constraints at
the GUT scale. We show how the evolution of CKM matrix elements is
affected. Fixed-point behaviour is compatible with present
constraints, but for top-quark couplings would require
participating sleptons or squarks to have masses 2:: mt to avoid
unacceptable top decays to sparticles.
1. INTRODUCTION
Supersymmetry is a very attractive extension of the Standard Model
(SM), so its low energy implications are being vigorously pursued.
1,2 In the minimal supersymmetric standard model (MSSM), with
minimum new particle content, a discrete symmetry (R-parity) is
assumed to forbid rapid proton decay. The R-parity of a particle is
R == (_1)3B+L+2s, where B, Land S are baryon number, lepton number
and spin; thus R = + 1 for particles and R = -1 for sparticles. An
advantage of R-conservation is that the lightest sparticle is
stable and hence provides a candidate for cold dark matter.
However, since R-conservation is motivated empirically and not by
any known
·Talk presented by V. Barger
19
(1)
there are two classes of R-violating couplings in the MSSM
superpotential, allowed by supersymmetryand
renormalizability.3
The first class of superpotential terms violates L,
W = ~AabcLLLtER + A~bcLLQtlJR + JliH2Li'
while the second class violates B,
W I,,, D-aD-bU-c = 2"abc R R R·
(2)
(3)
Here L, Q, E, lJ, {j denote the doublet lepton, doublet quark,
singlet antilepton, singlet d-type antiquark, singlet u-type
antiquark superfields, respectively, and a, b, c are gen eration
indices. (V)ab, (D)ab and (E)ab in Eq. (1) are the Yukawa coupling
matrices. In our notation, the superfields above are the weak
interaction eigenstates, which might be expected as the natural
choice at the grand unified scale, rather than the mass
eigenstates.
The term JliLiH2 in the superpotential can be rotated away into the
R-parity con serving term JlHi H2 via a SU( 4) rotation between
the superfields Hi and Li. However this operation must be performed
at some energy scale, and the mixing is regenerated at other scales
through the renormalization group equations.
To forbid fast proton decay, it is sufficient to forbid either
L-violating couplings or B-violating couplings, while retaining the
other class of RPV interaction. We follow this course.
The Yukawa couplings Aabc and A~bc are antisymmetric in their first
two indices because of superfield anti symmetry, so there are 9
independent couplings of each kind. There are also 27 independent
A~bc couplings, making 45 altogether. These superpo tential terms
lead to the interaction lagrangians
£' A:bc {vaLdcRdbL + dbLdcRVaL + (dcR)*(iJaL)CdbL -eaLdcRubL -
ubLdcReaL - (dcR)*(eaL)CUbd + h.c. ,
£" = 1 A" { cdcd* + cd*dc + -*dcdC} + h 2 abc Uc a b Uc a b Uc a b
.c.
(4)
(5)
(6)
There are phenomenological upper limits on the various couplings
Aabc, A:bc, A~bc from colliders and low-energy data,3-8 from proton
decay9 and from cosmology,lO but con siderable latitude remains
for RPV. These limits are generally stronger for couplings with
lower generation indices.
There are far too many RPV parameters for comfort. However, we know
that the dominant Higgs couplings are the third generation, At, Ab,
An and there may plausibly exist a similar generational hierarchy
among the RPV couplings. We shall therefore retain only A233, A;33,
A~33' which have the maximum of third-generation indices and are
also the least constrained phenomenologically.
The renormalization group evolution equations (RGE), relating
couplings at the electroweak scale to their values at the grand
unification (GUT) scale, have given new
20
insights and constraints on the observable low-energy parameters in
the R-conserving scenario. Let us see what can be learned from RGE
in RPV scenarios. An initial study of A~33 and A~33 evolution8 was
later extended to all baryon-violating couplings A;jkY Our present
work is a somewhat more general study of the RGE for RPV
interactions, emphasising solutions where R-conserving and
R-violating top Yukawa couplings both simultaneously approach
infrared fixed points. 12 Such fixed-point behaviour requires a
coupling A, )..', or A" to be of order unity at the electroweak
scale. We implicitly assume that RPV couplings do not have
unification constraints at the GUT scale,9 which would forbid this
behaviour. After our study was completed, two related works on RGE
for RPV couplings appeared,!3,14 which however have a different
focus and are largely complementary to the present work. In Ref.
14, de Carlos and White have studied the evolution of the soft
supersymmetry-breaking terms and find strong limits can be placed
on R-parity violating couplings by imposing neutrino mass limits
and bounds on lepton flavor violation.
2. RENORMALIZATION GROUP EQUATIONS AND FIXED POINTS
The evolution of the couplings d abc with the scale p, for any
trilinear term in the superpotential dabcq>aq>bq>c, is
given by the RGE
p ~ d abe = ,: debe + Ib d aec + I~ d abe , (7)
where the I~ are elements of the anomalous dimension matrix. With
the simplifying assumption that only third-generation Higgs and our
selected
RPV couplings contribute in the Yukawa sector, the one-loop RGE
become
dQi
L bi Q 7 , bi = {33/5, 1, -3}
~ It (6It + y" + Y' + 2Y" - !QQ3 - 3Q2 - liQI) 27r 3 15
2~ Yb (It + 6Yb + Y,. + 6Y' + 2Y" - .1fQ3 - 3Q2 - tsQI)
1 ( '9 ) 27r Y,. 3y" + 4Y,. + 4Y + 3Y - 3Q2 - 5QI
1 ( '9 ) 27r Y 4Y,. + 4Y + 3Y - 3Q2 - 5QI
~Y' (It + 6y" + Y; + Y + 6Y' - !QQ3 - 3Q2 - lQI) 27r ,. 3 15
2~ Y" (2It + 2Yb + 6Y" - 8Q3 - ~QI) .
(8)
(9)
(10)
(11)
(12)
(13)
(14)
Here Qi = 4~g; , the variable is t = In(p/ Ma) where p is the
running mass scale and Ma is the GUT unification mass, and we
define
Ii = ~A7 (i = t,b,r), 47r
It is understood that we take either Y = Y' = 0 or Y" = o.
2.1. At fixed point in the MSSM
1 2 Y = -A233. 47l'
An extremely interesting possibility is that It is large at the GUT
scale and conse quently driven toward a fixed point at the
electroweak scale. 15,16 In the pure MSSM
21
(RPV neglected), the fixed-point condition dY,./dt ~ 0 at J-t ~ mt
gives
(15)
Now At and Ab at J-t = mt are related to running masses
(16)
( In )-1/2 where v = v2 GF = 246 GeV and tan/1 = V2/VI is the ratio
of the Higgs vevs. Here 1Jb gives the QeD/QED running of mb(J-t)
between J-t = mb and J-t = mt; T/b ~ 1.5 for o:.(mt) ~ 0.10.16
Then
taking mb(mb) = 4.25 GeV, mt(mt) = 167 GeV, and hence
1/,(mt) ~ 3 X 10-4 tan2,B yt(mt).
(17)
(18)
For moderate values tan /1 ~ 20, we can neglect 1/" and the the
approximate values 0:3 = 1/10, 0:2 = 1/30, 0:1 = 1/58 at J-t = mt
then give
(19)
A more precise numerical analysis shows that At --+ 1.06 as J-t --+
mt. Since At(mt) = V2mt( mt)/( v sin /1), this leads to the
relation16
mt(pole) = (200 GeV) sin/1, (20)
where mt(pole) is the mass at the t-propagator pole. It is
interesting to examine the impact of RPV couplings on this
result.
2.2. A", At simultaneous fixed points
In the B-violating scenario with Y = Y' = 0 and Y" non-zero, the
possibility that both Yt and Y" approach fixed-point limits was
found numerically in Ref. 8 (note that these authors use a
different definition of A~bc). The corresponding conditions dYt/dt
~ 0 and dY"/dt ~ 0 at J-t ~ mt give
6yt + 1/, + 2Y" - !f0:3 - 30:2 - R0:1
2Yt + 21/, + 6Y" - 80:3 - ~O:l
Solving for Yt and Y" we obtain (if 1/, « Yt)
At ~ 0.94, A~33 ~ 1.18,
with At displaced downward due to A~33.
0,
(23)
This large fixed-point value of A~33 would give strong t --+ bs, sb
decay, if kinemati cally allowed.
With both At and A~33 at fixed points as above, the predicted top
quark mass becomes
mt(pole) ~ (150 GeV) sin/1. (24)
22
Even for moderate values of tan,8 (tan,8 > 5) one has sin,8 ~ 1
(sin,8 > 0.98). This prediction is therefore at the lower end of
the present data: 17,18
mt = 176 ± 8 ± 10 GeV (CDF) , mt = 199!~~ ± 22 GeV (DO). (25)
More precise data could eventually exclude the fixed-point
possibility for A~33 . In the case of large tan,8, the coupling Yi,
is non-negligible and may even be near
its own fixed point given by dYi,jdt ~ 0; then
(26)
Here Y,. can be related to Yi, since A'T (mt) = V2m'T (mt) j ('rJ'T
V cos ,8), and hence
(27)
by arguments similar to those above relating Ab(mt) to At(mt). Then
we have three simultaneous equations in three unknowns, with the
solutions
(28)
2.3. N or A, At simultaneous fixed points
If instead fixed points should occur simultaneously for yt and Y'
(with Y" = 0), the conditions dytj dt ~ 0 and dY' j dt ~ 0 at J1 ~
mt give
yt
(29)
(30)
If Y is small and we also neglect Yi, and Y'T (assuming small
tan,8), then yt and Y' approach almost the same fixed-point
value
(31)
In this case At(mt) is only slightly displaced below the MSSM
value, while A;33 has quite a large value. The latter would imply
substantial t -+ bT, fb decays, if kinematically allowed; the t -+
bT mode is more likely, since T is usually expected to be lighter
than b, and we discuss its implications later. Alternatively, if Y'
is negligible, yt and Y can approach fixed points simultaneously;
in this case the two conditions dytjdt ~ 0 and dYjdt ~ 0
essentially decouple, giving the MSSM result for yt. Neglecting Yi,
and Y,., the solution is
(32)
but if Yi, too is large and approaches its fixed point, the three
corresponding conditions give
(33)
while the A233 fixed point is very small and never truly reached in
numerical studies. It is also not possible for Y, Y' and yt to have
simultaneous fixed points; the conditions dY j dt = dY' j dt = dytj
dt. = 0 cannot be satisfied with all three couplings
positive.
23
2.4. CKM evolution
The presence of non-zero RPV couplings can also change the
evolution of CKM mixing angles. Assuming, as we do, that only the
RPV couplings A233, A;33 or A~33 are non zero, it turns out12 that
the one-loop RGE for mixing angles and the C P-violation parameter
J = Im(v"d v". v,,*. v,,'d) have the same forms as in the MSSM,
namely19
dW W (2 2) dt = - 811"2 At + Ab , (34)
where W = lv"bI2, lv"bI2, IVtdI 2 , IVt.12 or J. Nevertheless the
evolution of CKM angles differs from the MSSM because the evolution
of the Yukawa couplings on the right hand side is altered by the
RPV couplings.
3. NUMERICAL RGE STUDIES
It is instructive to supplement our algebraic arguments above with
explicit numerical solutions of the RGE. Figure 1 shows the
fixed-point behaviour of the three RPV couplings considered in this
paper, (A~33' A;33, A233) along with the corresponding fixed point
behaviour for At, assuming that tan,8 is small so that Ab and AT
are negligible. We see that for all A ;::: 1 at the GUT scale, the
respective Yukawa coupling approaches its fixed point at the
electroweak scale. These infrared fixed points provide theoretical
upper limits for the RPV-Yukawa couplings at the electroweak scale,
summarized in Table 1. The numerical evolution of the fixed points
approaches but does not exactly reproduce the approximate
analytical values Eqs. (28), (31) and (32).
Table 1: Fixed points for the different Yukawa couplings A in
different models for i) tan,8 ,$ 30 and ii) tan,8 '" mt/mb. In the
case oflarge tan,8, Ab also reaches a fixed point.
Model At Ab A233 A;33 A~33 i) MSSM 1.06
Lepton # Violation (A » N) 1.09 0.90 Lepton # Violation (N » A)
1.03 1.01
Baryon # Violation 0.90 1.02 ii) MSSM 1.00 0.92
Lepton # Violation (N » A) 1.01 0.72 0.71 Baryon # Violation 0.87
0.85 0.92
We remark in passing that RPV couplings must be well above their
fixed-point values to explain5 the apparent discrepancy between
theory and experiment for Rb = r(Z -+ bb)/r(Z -+ hadrons).
We obtain additional limits on the RPV couplings from the
experimental lower bound on mt (that we take to be mt > 150
GeV17,18). These are shown in Fig. 2; the dark shaded region is
excluded in all types of models only by assuming this lower bound
on the top mass.
Finally we examine RPV effects on the evolution of off-diagonal
terms in the CKM matrix. When the CKM masses and mixings satisfy a
hierarchy, the evolution from electroweak to GUT scales is given
by
W(GUT) = W(p)S(p),
5 nI, M(GUT) nI, M(GUT)
a) Baryon # violation b) Baryon # violation 4 -- A,(t) A,(G T)= 4.0
4 -- A'233(t) A'2J3(GUT) = 4.0
for A'~m(GUT) = 2.0 for A,(GUT) = 2.
3.0 3.0
1.0
-30 -20 -10 0 -30 -20 -10 0 5 5
c) Lepton # violation d) Lepton # violation 4 -- A,(t) A,(GUT) =
4.0 4 -- Am(t)
for Am(GUT) = 2.0. for A,(G T) = 2.8, A:I33(GUT) = 0.2 A~\J3(G
T)=0.2
3 3.0 3 3.0
1.0 1.0 -- 0.2 0.2 0 0
-30 -20 -10 0 -30 -20 -10 0
c) Lepton # violation I) Lepton # violation
4 - - A, (t) A,(G T) = 4.0 4
for Am(GUT) = 0.2. ~ r A,(G T)=2.8, A~l33(GUT) = 2.0 Am(G T) =
0.2
3.0 3.0
-30 -20 -10 o -30 -20 -10 o
Fig. 1. Couplings>. as a function of the energy scale t for
>'t in (a) baryon number RPV, (c) lepton number RPV with
>'233 » >';33 and (e) lepton number RPV with >';33 »
>'233 for different starting points at the GUT scale (t = 0).
Panels (b), (d) and (f) show the same for >'~33' >'233
(>'233 » >'~33) and >'b3 (>'~33 » >'233)
respectively. Here t ~ -33 represents the electroweak scale, where
these couplings reach their fixed points.
25
A,(GUT) A,(GUT)
Fig. 2. Excluded regions in the (a) At(GUT), A~33(GUT) plane and
(b) At(GUT), A233(GUT) (A233(GUT) = A;33(GUT)) plane obtained from
mt > 150 GeV.
where W is a CKM matrix element connecting the third generation to
a lighter gen eration and S is a scaling factor19 found by
integrating Eq. (34) with the other RGE. The remaining CKM elements
do not evolve to leading order in the hierarchy. Figure 3 shows how
S depends on the GUT-scale RPV couplings '\233, '\;33 and
.\~33'
P ;:J Q
,",(GUT)
4
Fig. 3. Contours of constant Sl/Z for different values of (a)
A~33(GUT) and At{GUT) (baryon number violation) and (b) AZ33(GUT) =
A~33(GUT) and At(GUT) (lepton number violation).
4. RPV DECAYS OF THE TOP QUARK
The RPV couplings .\~33 and .\~33 would give rise to new decay
modes of the top quark,20 if the final-state squark or slepton
masses are small enough. L-violating .\~33 leads to tR -t bRTR,
bRfR decays, with partial widths20
26
r(t-tbT)
r(t-tbf)
(35)
(.\;33)2 m (1 _ m~/m2)2 321T t b t , (36)
neglecting mb and mT' The former mode is more likely to be
accessible, since sleptons are expected to be lighter than squarks.
Since the SM top decay has partial width
(37)
the ratio of RPV to SM decays would be typically
It is natural to assume that T would decay mostly to T plus the
lightest neutralino X~ followed by the RPV decay X~ --t bbvT(VT),
with a short lifetime21
giving altogether (40)
This mode could in principle be identified experimentally, e.g. via
the many taggable b-jets and the presence of a tau. However, it
would not be mistaken as the SM decay modes t --t bW+ --t
bqij',bfv, (f = e,p), that form the basis of the presently detected
pp --t tEX signals in the (W --t fv) + 4jet and dileptonchannels
(neglecting leptons from T --t fvv that suffer from a small
branching fraction and a soft spectrum). On the contrary, the RPV
mode would deplete the SM signals by competition. With m T rv Mw ,
fixed-point values >'~33 ~ 0.9 (Fig.l) would suppress the SM
signal rate by a factor (1 + o. 70( >'~33)2t2 ~ 0.4, in
contradiction to experiment where pp --t {EX --t bbWW X signals
tend if anything to exceed SM expectations.17,18 We conclude that
either the fixed-point value is not approached or the T mass is
higher and reduces the RPV effect (e.g. m T = 150 GeV with >'~33
= 0.9 would suppress the SM signal rate by 0.88 instead). Note that
our discussion hinges on the fact that the RPV decays of present
interest would not contribute to SM top signals; it is quite
different from the approach of Ref. 7, which considers RPV
couplings that would give hard electrons or muons and contribute in
conventional top searches. _
Similarly, the B-violating coupling >'~33 leads to tR --t bRsR,
bRsR decays, with partial widths
r(t --t bs) = r(t --t bs) = (~~~2 mt (1 - m~/m;)2 , ( 41)
neglecting mb and m. and assuming a common squark mass mb = m. = m
q. If the squarks were no heavier than 150 GeV, say, the ratio of
RPV to SM decays would be
r(t --t bs,bs)/r(t --t bW+) ~ 0.16 (>'~33? (for mq = 150 GeV) .
(42)
These RPV decays would plausibly be followed by ij --t qX~ and X~
--t cbs, cbs (via the same >'~33 coupling with a short lifetime
analogous to Eq.(39)), giving altogether
t --t (bs,sb) --t bsX~ --t (cbbbs,cbbbs). ( 43)
This all-hadronic mode could in principle be identified
experimentally, through the multiple b-jets plus the t --t 5-jet
and X~ --t 3-jet invariant mass constraints. However, it would not
be readily mistaken for the SM hadronic mode t --t bW --t 3-jet,
and would simply reduce all the SM top signal rates. If the
coupling approached the fixed-point value >'~33 ~ 1.0, while mq
~ 150 GeVas assumed in Eq.( 42), the SM top signals would
27
be suppressed by a factor (1 + 0.I6(A~3J2)-2 '::::' 0.75, which is
strongly disfavored by the present datal7,18 but perhaps not yet
firmly excluded.
If indeed the s- and b-squarks were lighter than t to allow the
B-violating modes above, it is quite likely that the R-conserving
decay t --t ix~ would also be allowed, followed by i --t ex~ (via a
loop) and B-violating decays for both neutralinos, with net
effect
- 0 0 0 -- ---- t --t tx 1 --t ex IX 1 --t (eecbbbb, ccbbchb,
cccbbbb). ( 44)
This seven-quark mode would look quite unlike the usual SM modes
and would further suppress the SM signal rates. Depending on
details of the sparticle spectrum, however, other decays such as i
--t bW X~ might take part too, leading to different final states;
no general statement can be made except that they too would dilute
the SM signals and therefore cannot be very important.
5. CONCLUSIONS
• We have shown how the RGE for SM Yukawa couplings and CKM
elements would be affected by RPV, assuming hierarchical
couplings.
• We have identified fixed points in the RPV couplings and At
simultaneously.
• These give upper bounds on RPV couplings at the electroweak scale
[Fig.I].
• There are large tan f3 scenarios where Ab too has a fixed
point.
• The fixed point values [Table 1] are compatible with present
constraints.
• However, fixed-point values of A~33 or A~33 would require the
corresponding slep tons or squarks to have mass 2: mt, to avoid
strong top decays to sparticles.
• The fixed points give constraints, correlating the RPV couplings
with At at the GUT scale, from lower bounds on mt [Fig.2).
• RPV couplings affect the evolution of CKM mixing angles
[Fig.3].
Acknowledgements
VB thanks Herbi Dreiner for a discussion and the Institute for
Theoretical Physics at the University of California, Santa Barbara
for hospitality during part of this work. RJNP thanks the
University of Wisconsin for hospitality at the start of this study.
We thank B. de Carlos and P. White for pointing out our omission of
the Higgs-lepton mixing anomalous dimension in an earlier version
of our RGE. This research was supported in part by the U.S.
Department of Energy under Grant Nos. DE-FG02-95ER40896 and
DE-FG02-9IER4066I, in part by the National Science Foundation under
Grant No. PHY94-07194, and in part by the University of Wisconsin
Research Committee with funds granted by the Wisconsin Alumni
Research Foundation and support by NSF. TW is supported by the
Deutsche Forschungsgemeinschaft (DFG).
28
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29
ABSTRACT
Department of Physics The Ohio State University Columbus, OH
43210
I discuss how traditional grand unified theories, which require
adjoint (or higher representation) Higgs fields for breaking to the
standard model, can be contained within string theory. The status
of stringy free fermionic three generation SO(10) SUSY -GUT models
is reviewed. Progress in classification of both SO(lOh charged and
uncharged embed dings and in N = 1 spacetime solutions is
discussed.
SUSY-GUTs and Strings
Elementary particle physics has achieved phenomenal success in
recent decades, resulting in the Standard Model (SM), SU(3)c
XSU(2)L xU(l)y, and verification to high precision of many SM
predictions. However, many aspects of the SM point to a more
fUlldamental, underlying theory:
• the SM is very complicated, requiring measurement of some 19 free
parameters,
• the SM has a complicated gauge structure,
• there is a naturalness problem regarding the scale of electroweak
breaking,
• fine-tuning is required for the strong CP problem, and
• the expected cosmological constant resulting from electroweak
breaking is many, many orders of magnitude higher than the
experimental limit.
Since the early 1980's, these issues have motivated investigation
of Grand Unified Theories (GUTs) that would unite SM physics
through a single force at higher tem peratures. Superstring
research has attempted to proceed one step further and even merge
SM physics with gravity into a "Theory of Everything."
Perhaps the most striking evidence for a symmetry beyond the SM is
the predicted coupling unification not for the SM, but for the
minimal supersymmetric standard
31
model (MSSM) containing two Higgs doublets. [1] Renormalization
group equations applied to the SM couplings measured around the Mzo
scale predict MSSM unification at Munif ~ 2.5 X 1016 GeV. However,
this naively poses a problem for string theory, since the string
unification scale has been computed, at tree level, to be one order
of magnitude higher. That is, Mstring ~ gs X 5.5 X 1017 GeV, where
the string coupling gs ~ 0.7.[2] In recent years, three classes of
solutions have been proposed to resolve the potential inconsistency
between Munif and Mstring:
• The unification of the MSSM couplings at 2.5 x 1016 GeV should be
regarded as a coincidence. Munif could actually be higher as a
result of
1. SUSY ~breaking thresholds,
4. non-perturbative effects.
• Mstring could be lowered by string threshold effects, or
• Munif and Mstring remain distinct: there is an effective GUT
theory between the two scales. MSSM couplings unify around 1016 GeV
and run with a common value to the string scale.
I have been investigating this third possibility. The rationale for
this research has been further strengthened recently by findings
suggesting that stringy GUTs and/or non~MSSM states between 1 Te V
and Munif are the only truly feasible solutions on the list (except
perhaps for unknown non~perturbative effects). Shifts upward in
Munif
from SUSY ~breaking and/or non~standard hypercharges appear too
small to resolve the conflict and string threshold effects in
quasi~realistic models consistently increase Mstring rather than
lower it. [3]
The "birth" of string GUTs occurred in 1990, initiated in a paper
by D. Lewellen.[4] wherein Lewellen constructed a four~generation
SO(10) SUSY ~GUT built from the free fermionic[5, 6] string. This
quickly inspired analysis of constraints on and properties of
generic string GUTs.[7, 8] Following this string GUT research laid
dormant until searches for more phenomenologically viable GUTs
commenced in 1993 and 1994. Ini tial results during this "infancy"
stage of string GUTs seemed to suggest that three generation
string~derived GUTs were fairly simple to build and were numerous
in number. [9, 10] However, eventually subtle inconsistencies
became evident in all these models. The methods used to supposedly
yield exactly three chiral generations were inconsistent with
worldsheet supersymmetry (SUSY) and, relatedly, unexpected tachy
onic fermions were found in the models. Understanding how to
produce three gen erations consistent with world sheet SUSY
spurred the current "maturation stage" of string GUT research.[ll,
12, 13]
String GUTs and Kac-Moody Algebras
Besides being the possible answer to the Munir/ Mstring
inconsistency, string GUTs possess several distinct traits not
found in non~string~derived GUTs. First, string~ derived models can
explain the origin of the extra (local) U(I), R, and discrete
symme tries often invoked ad hoc. in non-string GUTs to
significantly restrict superpotential
32
terms.[14J. The extra symmetries in string models tend to suppress
proton decay and provide for a generic natural mass hierarchy, with
usually no more than one generation obtaining mass from cubic terms
in the superpotential. All string GUTs have upper limits to the
dimensions of massless gauge group representations that can appear
in a given model. Further, the number of copies of each allowed
representation is also con strained; there are relationships
between the numbers of varying reps that can appear. These features
suggest the opportunity for much interplay between string and GUT
model builders.
At the heart of string GUTs are Kat-Moody (KM) algebras, the
infinite dimen sional extensions of Lie algebras. [15J (See Table
1.) A KM algebra may be generated from a Lie algebra by the
addition of two new elements to the Lie algebra's Cartan sub
algebra (CSA), {Hi}. These new components are referred to as the
"level" K and the "scaling operator" La. K forms the center of the
algebra, i. e. it commutes with all other members. Therefore, K is
fixed for a given algebra in a given string model and is nor
malized to a carry a positive, integer value when the related Lie
algebra is non-abelian. La appears automatically in a string model
as the zero-mode of the energy-momentum
Table 1. Kac-Moody Algebras -vs- Lie Algebras
LIE ALGEBRA with rank l:
• FINITE dimensional algebra
a(Hi)E"
0,
if a + {3 is a root; if a + {3 = 0; otherwise.
AFFINE KAC-MOODY ALGEBRA with rank l + 2:
(18)
• New elements in CSA are "LEVEL" K (center of group) and
"scaling/energy operator" La
• INFINITE dimensional algebra: m, n E 7L
[H:n,H~] [H~, E~]
a(H~)E~+n
{ E(a,{3)E~1n, ;2 [a . H m+n + K mOm,-n], 0,
[K,E~J = 0
if a + {3 is a root; if a + {3 = 0; otherwise.
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operator. These new elements transform the finite dimensional Lie
algebra of CSA and non-zero roots {Hi, E"'} into an infinite
dimensional algebra, {K, La, H~, E~J, by adding a new indice m E 7L
to the old elements. A KM algebra is essentially an infinite tower
of Lie algebras, each distinguished by its m-value.
These KM algebras conspire with conformal and modular invariance (
i. e. the string self-consistency requirements) to produce tight
constraints on string GUTs. There are three generic string-based
constraints on gauge groups and gauge group reps. The first
specifies the highest allowed level K; for the ith KM algebra in a
consistent string theory. The total internal central charge, c,
from matter in the non-supersymmetric sector of a heterotic string
must be 22. The contribution, Ci, to this from a given KM algebra
is a function of the level Ki of the algebra,
K;dimC i c = L Ci = LT - ~ 22 .
i i Ki + hi (1)
dim Ci and hi are, respectively, the dimension and dual Coxeter of
the associated Lie algebra, Ci . Eq. (1) places upper bounds of 55,
7, and 4, respectively, on permitted levels of SU(5), SO(10), and
E6 KM algebras.[7, 8]
Once an acceptable level K for a given KM algebra has been chosen,
the next constraint specifies what Lie algebra reps could
potentially appear. Unitarity requires that if a rep, R, is to be a
primary field, the dot product between its highest weight, ),R, and
the highest root of the KM algebra, W, must be less than or equal
to K.
(2)
For example only the 1, 10, 16, and 16 reps can appear for SO(10)
at levell. (See table 2.) For this reason adjoint Higgs require K
:2: 2 for SO(10) or any other KM algebra.
Table 2. Potentially Massless Unitary Gauge Group Reps
k=l k=2 k=3 k=4 SU(5) c=4 C = 48/7 c=9 c = 32/3
rep h rep h rep h rep h 5 2/5 5 12/35 5 3/10 5 4/15
10 3/5 10 18/35 10 9/20 10 2/5 15 4/5 15 7/10 15 28/45 24 5/7 24
5/8 24 5/9 40 33/35 40 33/40 40 11/15 45 32/35 45 4/5 45
32/45
75 1 50 14/15 70 14/15 75 8/9
SO(10) c=5 c=9 c = 135/11 c = 15 rep h rep h rep h rep h 10 1/2 10
9/20 10 9/22 10 3/8 16 5/8 16 9/16 16 45/88 16 15/32
45 4/5 45 8/11 45 2/3 54 1 54 10/11 54 5/6
120 21/22 120 7/8 144 85/88 144 85/96
210 1
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Masslessness of a heterotic string state requires that the total
conformal dimension, h, of the non-supersymmetric sector of the
state equal one. Hence the contribution hR coming from rep R of the
KM algebra can be no greater than one. For a fixed level K, hR is a
function of the quadratic Casimir, CR, of the rep,
hR = Cr/iJ!~ . K+h
(3)
Requiring hR ~ 1 presents a stronger constraint than does
unitarity. For instance, although all SO(10) rep primary fields
from the singlet up through the 210 are allowed a.t level 2, only
the singlet up through the 54 can be massless. In particular, the
126 ('annot be massless unless K :::: 5.
Free fermionic string models impose one additional constraint.[12]
Increasing the level K decreases the length-squared, Q;ooo of a
non-zero root of the KM algebra by a factor of K. In free fermionic
strings Q;oot at level 1 is normalized to 2 for the long roots.
Thus,