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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
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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.
33
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
34
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,