SERRCHING FOR THE SUPERWORlO
World Scientific Series in 20th Century Physics Published Vol. 20 The Origin of the Third Family edited by 0. Barnabei, L. Maiani, R. A. Ricci and F. R. Monaco Vol. 21 Spectroscopy with Coherent Radiation (with Commentary) edited by N. F. Ramsey
- Selected Papers of Norman F. Ramsey
Vol. 22 A Quest for Symmetry - Selected Works of Bunji Sakita edited by K. Kikkawa, M. Virasoro and S. R. Wadia Vol. 23 Selected Papers of Kun Huang (with Commentary) edited by B.-F. Zhu Vol. 24 Subnuclear Physics - The First 50 Years: Highlights from Erice to ELN by A. Zichichi edited by 0. Barnabei, P. Pupillo and F. Roversi Monaco Vol. 25 The Creation of Quantum Chromodynamics and the Effective Energy by V. N. Gribov, G. ’t Hooft, G. Veneziano and V. F. Weisskopf edited by L. N. Lipatov Vol. 26 A Quantum Legacy - Seminal Papers of Julian Schwinger edited by K. A. Milton Vol. 27 Selected Papers of Richard Feynman (with Commentary) edited by L. M. Brown Vol. 28 The Legacy of Leon Van Hove edited by A. Giovannini Vol. 29 Selected Works of Emil Wolf (with Commentary) edited by E. Wolf Vol. 30 Selected Papers of J. Robert Schrieffer - In Celebration of His 70th Birthday edited by N. E. Bonesteel and L. P. Gor’kov Vol. 31 From the Preshower to the New Technologies for Supercolliders - In Honour of Antonino Zichichi edited by B. H. Wiik, A. Wagner and H. Wenninger Vol. 32 In Conclusion - A Collection of Summary Talks in High Energy Physics edited by J. 0.Bjorken Vol. 33 Formation and Evolution of Black Holes in the Galaxy - Selected Papers with Commentary edited by H. A. Bethe, G. E. Brown and C.-H. Lee Vol. 35 A Career in Theoretical Physics, 2nd Edition by P. W. Anderson Vol. 36 Selected Papers (1945-1 980) with Commentary by Chen Ning Yang Vol. 37 Adventures in Theoretical Physics - Selected Papers with Commentaries by Stephen L. Adler Vol. 38 Matter Particled - Patterns, Structure and Dynamics - Selected Research Papers of Yuval Ne’eman edited by R. Ruffini and Y. Verbin Vol. 39 Searching for the Superworld - A Volume in Honour of Antonino Zichichi on the Occasion of the Sixth Centenary Celebrations of the University of Turin, Italy edited by S. Ferrara and R. M. Mossbauer
For information on Vols. 1-19, please visit http://www.worldscibooks.com/series/wsscp-seres.shtml
VO,I 39
World Scientific Series in 20th Century Physics
SEARCHING FOR THE SUPER#ORlU A Volume in Honor ofAntonino Zichichi on the Occasion of the Sixth Centenary Celebrations of the University of Turin,Italy
editors
Sergio Ferrara CERN Geneva, Switzerland
Rudolf M Mossbauer Echnischen UniversitatMiinchen, Germany
N E W JERSEY
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LONDON
6 World Scientific 1 : SINGAPORE
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BElJlNG
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SHANGHAI
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TAIPEI
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CHENNAI
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SEARCHING FOR THE SUPERWORLD: A Volume in Honour of Antonino Zichichi on the Occasion of the Sixth Centenary Celebrations of the University of Turin, Italy World Scientific Series in 20th Century Physics -Vol. 39 Copyright 0 2007 by “Ettore Majorana” Foundation and Centre for Scientific Culture All rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
ISBN- 13 978-98 1-270-018-6 ISBN-10 981-270-01 8-8
Printed in Singapore by B & JO Enterprise
V
CONTENTS
...
Xlll
PREFACE
E. Pellizzetti
INTRODUCTORYPAPERS THERUNNING OF GAUGINO MASSES,THE GAP AND THE SINGLE-PHOTON FINALSTATES IN THE SEARCH FOR THE SUPERWORLD
R.M. Mossbauer SUPERSYMMETRY, SUPERSPACE AND THE SUPERWORLD
M J . Duffand S. Ferrara SELECTED PAPERS ON SEARCHING FOR THE SUPERWORLD BY A. ZICHICHI AND COLLABORATORS
PARTA THEEFFECTIVE EXPERIMENTAL CONSTRAINTS ON Msusy AND MGUT F. Anselmo, L. Cifarelli, A . Peterman and A. Zichichi I1 Nuovo Cimento 104 A (1991) 1817.
27
THEEVOLUTION OF GAUGINO MASSESAND THE SUSY THRESHOLD F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi I1 Nuovo Cimento 105 A (1992) 581.
47
THECONVERGENCE OF THE GAUGECOUPLINGS AT E G U T AND ABOVE:CONSEQUENCES FOR a3(Mz) AND SUSY BREAKING F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi I1 Nuovo Cimento 105 A (1992) 1025.
THESIMULTANEOUS EVOLUTION OF MASSESAND COUPLINGS: CONSEQUENCES ON SUPERSYMMETRY SPECTRA AND THRESHOLDS F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi I1 Nuovo Cimento 105 A (1992) 1179. ANALYTICSTUDY OF THE SUPERSYMMETRY-BREAKING SCALE AT TWO LOOPS F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi I1 Nuovo Cimento 105 A (1 992) 1201.
63
85
109
vi
A STUDY OF THE VARIOUS APPROACHES TO MGUT AND %UT F. Anselmo, L. Cifarelli and A. Zichichi I1 Nuovo Cimento 105 A (1992) 1335. LEP DATA, A X2-TESTTO STUDY THE al, a2, a3 CONVERGENCE FOR HIGH-PRECISION HAVINGIN MINDTHE SUSY THRESHOLD F. Anselmo, L. Cifarelli and A. Zichichi I1 Nuovo Cimento 105 A (1992) 1357.
117
141
UNDERSTANDING WHERE THE SUPERSYMMETRYTHRESHOLD SHOULD BE A. Zichichi Proceedings of the Workshop on “Ten Years of SUSY Confronting Experiments”, CERN, Geneva, 7-9 September 1992, CERN-PPE/92-149, CERN/LAA/MSL/92-017 (7 September 1992), and CERN-TH.6707/92 - PPE/92-180 (November 1992) 94.
157
WHERE WE STAND WITH THE REAL SUPERWORLD A. Zichichi Proceedings of the XXX Course of the International School of Subnuclear Physics: “From Superstrings to the Real Supenvorld”, Erice, 14-22 July 1992, World Scientific - The Subnuclear Series 30 (1993) 1.
181
AND STRING VACUA: A SUPERSYMMETRIC ON A CLASS OF FINITESIGMA-MODELS EXTENSION A. Peterman and A. Zichichi I1 Nuovo Cimento 106 A (1993) 719.
A SEARCH FOR EXACTSUPERSTRING VACUA A. Peterman and A. Zichichi I1 Nuovo Cimento -Note Brevi 107 A (1994) 333. BETWEEN DOUBLE SCALING LIMITAND PROOF OF TIIE EQUIVALENCE FINITE-SIZE SCALING HYPOTHESIS A. Peterman and A. Zichichi 11 Nuovo Cimento - Note Brevi 107 A (1 994) 507.
OF GRAVITATIONAL WAVES EXPLICIT SUPERSTRING VACUA IN A BACKGROUND AND DILATON A. Peterman and A. Zichichi I1 Nuovo Cimento 108 A (1995) 97.
195
20 1
21 1
217
vii
PARTB TROUBLES WITH THE MINIMAL su(5)SUPERGRAVITY MODEL Jorge L. Lopez, D. V. Nanopoulos and A . Zichichi Physics Letters B 291 (1992) 255.
TESTSFOR MINIMALsu(5) SUPERGRAVITY AT FERMILAB,GRAN SASSO, AND LEP Jorge L. Lopez, D. V. Nanopoulos, H. Pois and A. Zichichi
226
PROPOSED
SUPERKAMIOKANDE
239
Physics Letters B 299 (1993) 262. IMPROVED LEP LOWERBOUNDON THE LIGHTESTSUSY HIGGSMASSFROM
BREAKING AND ITS EXPERIMENTAL CONSEQUENCES RADIATIVE ELECTROWEAK
247
Jorge L. Lopez, D. V. Nanopoulos, H. Pois, Xu Wang and A . Zichichi Physics Letters B 306 (1993) 73. TESTSAT FERMILAB: A PROPOSAL Jorge L. Lopez, D. V. Nanopoulos, Xu Wang and A . Zichichi Physical Review D 48 (1993) 2062. SUPERSYMMETRY
255
SUSY SIGNALS AT DESY H E M IN THE NO-SCALEFLIPPEDsu(5)SUPERGRAVITY MODEL Jorge L. Lopez, D. V. Nanopoulos, Xu Wang and A. Zichichi Physical Review D 48 (1993) 4029.
269
SPARTICLE AND HIGGS-BOSON PRODUCTION AND DETECTION AT CERN LEP 11 IN T W O SUPERGRAVITY
MODELS Jorge L. Lopez, D. V. Nanopoulos, H. Pois, Xu Wang and A. Zichichi Physical Review D 48 (1993) 4062.
279
TOWARDS A UNIFIED STRING SUPERGRAVITY MODEL Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi Physics Letters B 319 (1993) 45 1.
295
SIMPLEST, STRING-DERIVABLE, SUPERGRAVITY MODELAND ITS EXPERIMENTAL
PREDICTIONS Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi Physical Review D 49 (1994) 343.
303
...
Vlll
STRONGEST EXPERIMENTAL CONSTRAINTS ON su(5) X
u(1) SUPERGRAVITY MODELS
317
Jorge L. Lopez, D. V. Nanopoulos, Gye i? Park and A. Zichichi Physical Review D 49 (1994) 355. SCRUTINIZING SUPERGRAVITYMODELS THROUGH NEUTRINO TELESCOPES
33 1
Raj Gandhi, Jorge L. Lopez, D. V. Nanopoulos, Kajia Yuan and A. Zichichi Physical Review D 49 (1994) 3691. NEWPRECISION ELECTROWEAK TESTSOF su(5) X u ( 1) SUPERGRAVITY Jorge L. Lopez, D. V. Nanopoulos, Gye i? Park and A. Zichichi Physical Review D 49 (1994) 4835.
347
THETOP-QUARK MASSIN SU(5) x U( 1) SUPERGRAVITY Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi Physics Letters B 327 (1994) 279.
357
EXPERIMENTAL ASPECTSOF su(5) X u ( 1) SUPERGRAVITY Jorge L. Lopez, D. V. Nanopoulos, Gye T. Park, Xu Wang and A. Zichichi Physical Review D SO ( 1994) 2 164.
367
NEWCONSTRAINTS ON SUPERGRAVITY MODELSFROM b -+ Sy Jorge L. Lopez, D. V. Nanopoulos, Xu Wang and A . Zichichi Physical Review D 51 (1995) 147.
397
EXPERIMENTAL CONSEQUENCES ON ONE-PARAMETER NO-SCALESUPERGRAVITY MODELS Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi International Journal of Modern Physics A 10 (1 995) 424 1. CONSTRAINTS ON NO-SCALESUPERGRAVITY MODELS
41 1
43 7
S. Kelley, Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi Modern Physics Letters A 10 (1995) 1787.
A LIGHTTOP-SQUARK AND ITS CONSEQUENCES AT HIGHENERGY COLLIDERS Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi Modern Physics Letters A 10 (1995) 2289.
447
SUPERSYMMETRY DILEPTONS AND TRILEPTONS AT THE FERMILAB TEVATRON
457
Jorge L. Lopez, D. V. Nanopoulos, Xu Wang and A . Zichichi Physical Review D 52 (1995) 142.
ix STRING NO-SCALESUPERGRAVITY MODELAND ITS EXPERIMENTAL CONSEQUENCES
467
Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi Physical Review D 52 (1995) 4 178.
EXPERIMENTAL CONSTRAINTS ON A STRINGY S U ( ~x) U( 1) MODEL Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi Physical Review D 53 (1996) 5253. SUPERSYMMETRIC PHOTONIC SIGNALS AT THE CERN efe- COLLIDER LEP IN LIGHTGRAVITINO MODELS Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi Physical Review Letters 77 (1996) 5168.
475
483
SINGLE-PHOTON SIGNALS AT CERN LEP IN SUPERSYMMETRIC MODELSWITH A LIGHTGRAVITINO Jorge L. Lopez, D. V. Nanopoulos and A. Zichichi Physical Review D 55 ( 1997) 58 13.
489
LIGHTGRAVITINO PRODUCTION AT HADRONCOLLIDERS Jaewan Kim, Jorge L. Lopez, D. V. Nanopoulos, Raghavan Rangarajan and A. Zichichi Physical Review D 57 (1998) 373.
505
LISTOF PUBLICATIONS ON SEARCHING FOR THE SUPERWORLD BY A. ZICHICHI AND COLLABORATORS
517
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xi
PREFACE
On the occasion of the 6th Centenary of its foundation, the University of Turin has awarded to Professor Antonino Zichichi an honorary degree (Laurea Honoris Causa) for his discovery of nuclear antimatter. In the official ceremony the subject chosen for his magistral lecture has been “The Superworld”, a subject bound to become even more topical owing to the start of LHC, the new accelerator of CERN. Superworld is a formidable new reality to which Zichichi has contributed with a series of important papers of phenomenological and theoretical nature. These papers represent an estimable contribution that will surely remain of great interest, not only for their originality but also for their completeness in terms of phenomenological analysis of what could be expected, taking into account our present knowledge of the physics world. In recognition of the value of these papers, the University of Turin has decided to group them in a unique volume in order to provide the scientific community with a text in which the crucial aspects of the Superworld phenomenology are rigorously and exhaustively treated. Ezio Pelizzetti Rector of the University of Turin
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1
INTRODUCTORY PAPERS
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3
THE RUNNING OF GAUGINO MASSES, THE GAP AND THE SINGLE-PHOTON FINAL STATES IN THE SEARCH FOR THE SUPERWORLD Rudolf M. Mossbauer Technischen Universitat Miinchen, Garching bei Miinchen, Germany
Sometimes ago, in the middle eighties, I received a call from my friend and colleague, Nino Zichichi; he had decided to devote a great attention to the new theoretical set of discoveries called “supersymmetry”. He wanted to have my opinion in order to decide on the forthcoming Courses of the Subnuclear Physics School. Not being my field of direct interest, I was not very enthusiastic for the simple reason that supersymmetry was lacking a direct impact with experiments. Nino sent me his review paper [l] delivered at the EPS Conference in Geneva (1979) where he said: <
>. This statement was based, as he has emphasized in many occasion, on a work with AndrC Peterman. Nino and his friend AndrC, using the renormalization group equations, had realized that the problem of the convergence of the slopes (versus energy)
4
of the three gauge-couplings, al a2a3 , could receive a new degree of freedom from supersymmetry. At the time of the Geneva 1979 EPS Conference, the three gauge couplings, a l a2 a3 , were not converging in a point but in a sort of triangle. The new degree of freedom in the three slopes of the gauge couplings had as experimental impact the energy-threshold where to search for the first example of a superparticle. Lot of problems needed to be clarified and Nino, a few months later, informed me that he wanted to devote the forthcoming Courses of the Erice Subnuclear Physics School to supersymmetry. He later send me the three volumes “Superworld I, 11, III” of the Erice Schools (1986, 1987, 1988) [2]. These were just premises to what he told me in 1991, when, being engaged with is group in searching for the first experimental evidence of the Superworld in the 50 GeV mass range, the best theoretical prediction was giving as lower bound for the supersymmetry threshold the level of 21 TeV
131. This
prediction
was
based
on
the
evolution
of
the
gauge
couplings (a1a2 a3)computed neglecting the effects due to the evolution of the masses. Once this effect is introduced, the energy threshold, for the detection of the first signal from the Superworld, is lowered by nearly three orders of magnitudes, more exactly a factor 700 [4]. The following Figure is taken from this paper [4]and illustrates the value of introducing the running of the gaugino masses in the determination of the supersymmetry breaking threshold.
5
I
Predictions for SUSY-threshold lower bound
loo
lo2
lo4
lo6
lo8
lolo
IO’*
1014
I
1oI6
(GeV) Figure published in I1 Nuovo Cimento 106A, 581 (1992). Predictions for supersymmetry threshold lower bound, with and without corrections for the evolutions of the gaugino masses (EGM effect). The reference numbers are those specified in the original paper quoted above.
This result triggered a new revived interest in the search for the Superworld using the (e+e-) collider at CERN, LEP.
In fact other discouraging “theoretical” papers had been published, the most advertized one being that where the SUSY threshold was predicted to be above the TeV range [5].
6
As illustrated on the last five columns of the following Table, these “predictions” were neglecting, not only the EGM effect, but also many other “details”. These “details” illustrate how many important properties, of the Supenvorld physics to be described, had been neglected by many authors, including those whose claim was to “predict” the energy scale at which supersymmetry was expected to be broken. This Table has been presented by
A. Zichichi in his “Lezione Magistrale” at the University of Turin, 21 May
2004. The numbers of the first column indicate the references quoted in his Lecture.
Authors ACPZ [47, 49-54]
Authors AdBF (551
Msusv ttpldata IErrors EC WA e 2 0 allpossible Yes solutions(24) lnputdata onlyone experiment
1 Errm 1 (J
EC
Msusy
onlyone solution
Yes
CC UC AT/ M x A T H EGM physical Yes Yes Yes Yes Yes
CC IUC A T L M x ATH EGM OmWaI ’ No No No No No
7
Let me say a few words on the five “details” reported in columns 6 to 10 of the Table above: i) The unification of the gauge couplings ( a 1 a 2a3) must continue
; condition is above the energy level where they all converge ( E G ~ )this indicated as UC. ii) The low energy threshold must be described by a spectrum, ATL , and not using a sharp step at a given energy value. iii) The mass scale (M,) where the Grand Unified Theory (GUT) breaks into SU(3) x SU(2) x U( 1) has to be investigated in terms of the experimental results in the low energy range, around the Fermi scale. iv) The high energy threshold were the Grand Unified Theory breaks into the Standard Model (SU(3) x SU(2) x U(1)) cannot be a &function of the energy, but a spectrum. This spectrum, ATH, has in fact consequences not only on the low energy threshold for supersymmetry breaking but also on the possible existence of a Gap between two basic energy scale: one is the energy level where all gauge couplings (ala2 a3)converge, EGUT,and the other is the energy level, Esu, where the string theory predicts the unification of all forces (on the basis of the Newton gravitational coupling). The problem concerning the existence of the Gap [6] is another topic which Nino liked to discuss with his friends.
v) Finally, the evolution of the gaugino masses (EGM) which, as said before, has the effect of lowering the low energy supersymmetry threshold by nearly three orders of magnitude, must be duly taken into account.
I would like to mention another contribution by Nino which has been of remarkable value for the experimental search performed at CERN during more than a decade. It was Zichichi who actually called everybody’s attention on the
8
fact that also the “single-photon” final states had to be searched for as a basic signal for the Superworld [7]. In this note I have restricted my introduction to three effects which have attracted my interest, thanks to Nino’s discussion on several occasions during many years, and in particular in 1979, and then in the years following 1991, when he decided to step in the field in order to avoid the general “theoretical” trend discouraging to search for superparticles below the TeV-mass-range. The analysis of all experimental data and of all quantities having an effect on the experimental search for the Supenvorld is documented in the papers by A. Zichichi and collaborators reproduced in the present volume. This will be a guide not only for the forthcoming experiments in the new energy range opened by the CERN LHC Collider, but also for the future searches of the Supenvorld.
References [l] A. Zichichi, Rivista del Nuovo Cimento 2, n. 14 (1979). [2] The Superworld I, 11, 111, A. Zichichi ed, Plenum Press, New York and London (1987, 1988, 1989). [3] J. Ellis et al., Physics Letters B 260, 131 (1991) 141 F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, I1 Nuovo Cimento 106A, 581 (April 1992)l. [5] U. Amaldi, W. de Boer and H. Furstenau, Physics Letters
m,447 (1991).
[6] F. Anselmo, L. Cifarelli and A. Zichichi, I1 Nuovo Cimento 105A, 1335 (1992). [7] Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi, Physical Review Letters 77, 5168 (1996); and Physical Review D,5813 (1997)l.
9
SUPERSYMMETRY, SUPERSPACE AND THE SUPERWORLD M.J. Duff(*) and S. Ferrara(**) (*) Imperial College London, UK; (**) CERN, Geneva, CH
One of the most fascinating ideas in the quest for unity of the basic forces of nature is certainly the one that states that the physical laws governing elementary particle processes have a higher, yet undiscovered, invariance called supersymmetry [l, 2, 3,4, 51. In the search for experimental evidence for supersymmetry, A. Zichichi with Peterman, Cifarelli and Anselmo played a leading role, especially in the study of gauge coupling evolutions and the determination of the low-energy supersymmetric threshold('). Zichichi further proposed with Lopez and Nanopoulos a particularly appealing string-derived no-scale supergravity model where a realistic and predictable scenario for new physics beyond the standard model is examined('*). In this volume where some of the most relevant contributions of A. Zichichi and collaborators are presented we thought interesting to illustrate the foundations of supersymmetry and of some of the basic theoretical constructions which led to the proposal of the superworld. Supersymmetry is a symmetry that intertwines the basic interactions of the two classes of elementary particles out of which everything else is made, fermions and bosons.
(0) (00)
These papers are reproduced in Part A of the present volume. These papers are reproduced in Part B .
10
Fermions and bosons obey two different kinds of statistics, Fermi-Dirac and Bose-Einstein respectively and, because of the Pauli exclusion principle, their roles in making our Universe as it is seem quite distinct. For example, their relativistic kinematical attributes, such as the spin, in Planck units is half-integer for fermions and integer for bosons, because of the CPT-spin statistics theorem.
Also fermions appear as the basic constituents of nucleons, atoms, and molecules built out of quarks, protons, neutrons and electrons. On the other hand, owing to the quantum nature of the subnuclear world, the forces underlying their mutual interactions are due to the exchange of particle quanta which are bosons, such as photons, gluons and weak vector bosons. Last but not least, the tiniest of these forces, gravity, when extended to the quantum domain, is believed to be due to an exchange of a spin two particle, the graviton, the basic carrier of the gravitational interaction. The great importance of the supersymmetry hypothesis lies in the fact that such proposed invariance would not only predict the existence of new particles, as new symmetries often imply, but would unravel a drastic change of our notion of space and time. In fact, because of its intrinsic mathematical structure, it would imply the existence of extra-coordinates; rather than the usual kind of coordinates, described by complex numbers, these would be defined by anticommuting “numbers”, known to mathematicians as Grassmann variables. Such space, named “superspace” by its discoverers (Salam and Strathdee [6]), carries D bosonic dimensions xp (in Minkowski space it has one time and D
-
1 space dimensions) ( p = 0, ..., D
-
l), but also 2[D/21
anticommuting dimensions 0, ( [ 0 / 2 ] is the integral part of D/2). Indeed, while the bosonic coordinates have a “vector” index (a vector in D dimensions has D components), so that their component number grows linearly in D , the 0, coordinates are spinors, which obey the mathematical theory of spinors mainly
11
developed by Cartan and Weyl in the first half of the last century, and their number thus grows exponentially in D. A particle moving in superspace is called a superparticle. When seen from ordinary space-time, it corresponds to a collection (or multiplet) of ordinary particles with spins differing by a half-unit. For instance, in ordinary four-dimensional space-time, there are four 8’s and the simplest multiplet, carrying two bosons and a (Weyl) chiral fermion, is the so-called Wess-Zumino multiplet [7]. These authors were the first to propose an interacting Lagrangian for this multiplet in 1973 [8], by extending to D = 4 [9] a symmetry discovered two years before, in two-dimensional systems describing the world-sheet of strings, by Neveu, Schwarz, Ramond, Gervais and Sakita [ 10, 1I]. In the same years, quite independently from one another, the Russian physicists Gol’fand, Likhtman [12] and Volkov, Akulov [13] proposed an algebraic structure, extending the PoincarC algebra, called graded Lie algebra by mathematicians (or superalgebra). Superalgebras turn out to be the mathematical machinery underlying supersymmetry, extending the concept of ordinary Lie algebras and Lie groups, the basic tools used to describe ordinary continuous symmetries in physics. As the group of motion of a point-particle in ordinary space-time is realized by the PoincarC group, a group of motion in “superspace” is induced by a supersymmetry transformation (super-Poincar6 group). A Lagrangian in superspace corresponds to an ordinary Lagrangian for an entire “supermultiplet” of ordinary particles in ordinary space-time. This is due to properties of functions of “anticommuting” variables, and the so-called Berezin integration rule, which also underlies the path integral approach in quantum field theory when extended to fermionic variables [4].
As seen in superspace, the original Wess-Zumino Lagrangian appears as
a “cubic” self-interaction of a “chiral” superfield [7], the simplest interaction that could be imagined in superspace. Soon after it was realized that such interactions not only are renormalizable, when extended to the quantum theory, but they have milder “ultraviolet” properties than ordinary renormalizable field theories. This property was not just a mathematical curiosity: it also opened the way to thinking that supersymmetric field theories may solve the problem of quantum gravity and the so-called hierarchy problem of Grand Unified Theories (GUTS) [3, 4, 51. In the latter, the separation of two different physical scales, such as the Fermi scale (250 GeV) and the Grand Unification scale (1016 GeV), is made quite unnatural when quantum corrections are taken into account. This is due to quadratic divergences occurring in the effective scalar potential, which controls the symmetry breaking in these theories. Two pivotal developments were the unification of supersymmetry with ordinary gauge invariance, which led to super-Yang-Mills theories [ 141, and the unification with gravity, which led to supergravity [15]. This unification made it possible to confront supersymmetry with an extension of the standard model of electroweak and strong interactions called the MSSM (Minimal Supersymmetric Standard Model). The discovery of supergravity [15], [23] made it possible to study quantum theories of gravity, where a new fundamental gauge particle of spin 3/2, the supersymmetric partner of the graviton, called the gravitino, plays a major role. An amazing discovery made in studying fundamental interactions in superspace, partly due to W. Nahm [16] (who classified the PoincarC supermultiplets in arbitrary dimensions), is that supersymmetry gives an upper bound to the dimension of bosonic and fermionic coordinates. For example, supersymmetric interactions of only spin 0 and spin 1/2 particles
13
exist only up to D = 6 (and 8 fermionic coordinates), super-Yang-Mills theories exist up to D = 10 dimensions (and 16 fermionic coordinates), supergravity theories exist up to D = 11 dimensions (and 32 fermionic coordinates). The maximal superspaces, with total dimension 26 and 43, play a major role in superstring [lo, 111 and M-theory [17, 181. These theories, which admit maximal supersymmetry, generalize point-like particle quantum field theories to extended objects of different space extension. It is crucial, for the consistency of these theories, that two kinds of supersymmetry are at work, one present on the surface of the world-volume of a given extended object (called p-brane [19] if it has a p space extension on a p
+
1 Lorentzian world-volume), the other present on the “bulk”, i.e. the D-
dimensional ambient space. Such interplay of symmetries on the brane and bulk spaces and their dynamical role has made a major appearance in recent works on extra dimensions [20], realization of the so-called Randall-Sundrum [ 211 scenario, and the possibility of having two kinds of elementary particles, one living on the bulk and the other on the brane, something that makes their interactions quite distinct. It is at present believed, especially by the practitioners of superstring and M-theory, that supersymmetry should play a basic role at the Planck scale
to make these theories quantum-mechanically consistent. However, since supersymmetry is not observed in our physical world, the case of supersymmetry requires an understanding of how this symmetry is broken and at what scale the symmetry breaking occurs [ l , 2, 3 , 4 , 51. After the discovery of supersymmetric Yang-Mills theories, in the mid ~ O ’ S ,P. Fayet made the proposal [22] of a minimal extension of the spectrum of
the standard model, in which all ordinary particles, such as quarks, leptons, as well as the force carriers, are extended to supermultiplets. In doing this, it was
14
noticed that, to realize the Higgs breaking, avoiding anomalies, two Higgs doublets were required, something that already goes beyond the minimal spectrum of the standard model. This implies, in particular, that in any supersymmetric extension of the standard model a new (adimensional) parameter arises, namely the ratio of the VEVs (tan
of the two Higgs doublet
scalars [4, 51. If supersymmetry is spontaneously broken in these theories, the low-energy theorems can be studied and the gravitino acquires a mass through the supersymmetric version of the Higgs mechanism. The mass of the gravitino is closely related to the mass scale of supersymmetry breaking. The compelling reason to invoke a low-energy supersymmetry for nongravitational interactions essentially comes from the so-called hierarchy problem of GUTS [ 3 , 4, 51. A basic tree-level quadratic-mass sum rule implies that the effective potential of a supersymmetric gauge theory does not receive quadratic corrections above some cut-off scale A, at energies at which supersymmetry is effectively unbroken. So if the effective scale of supersymmetry breaking is Eo, quadratic corrections disappear for A >> Eo. In particular, if the Fermi-scale is close to Eo, then it is stable with respect to a Grand Unification scale which occurs close to the string or Planck scale E p , since E p >> Eo. It is of primary importance to disentangle the experimental constraints that make this scenario plausible: if it turns out to be correct, supersymmetry may indeed be discovered at present or future atom smashers such as the Tevatron in the US and the forthcoming LHC at CERN. There are many indirect pieces of evidence for the introduction of supersymmetry in the basic laws of physics, both of theoretical and of “observational” nature. As previously alluded to, supersymmetry seems a basic ingredient for a theory that encompasses the gravitational force. The
15
point-like particle limit, at low energies, of such a theory must be supergravity coupled to matter field multiplets, the latter describing the non-gravitational forces. Supergravity is the gauge theory of supersymmetry as much as standard general relativity is the gauge theory of the PoincarC group, the space-time symmetry of relativistic systems. If gravity is extended at microscopic scales, such as the Planck scale, it is believed that the basic objects become extended objects [lo, 111. The ultimate theory based on such principles lives in D = 11 dimensions: it describes membranes and five-branes and reduces to all known string theories when some dimensions get compactified. This theory, called Mtheory, can reproduce GUTS and supersymmetric extensions of the standard model
when
seven
dimensions
are
compactified
and
32
original
supersymmetries get reduced to 4. It is also hoped that such theories might explain the present smallness of the cosmological constant and the inflationary evolution of our Universe. In all these constructions, there is nothing that fixes the supersymmetrybreaking scale. However, such a scale becomes relevant when supersymmetry is advocated at low energies, in the TeV range, to solve the hierarchy problem and eventually to explain the origin of the Fermi scale. Indirect signals of such low-energy supersymmetry are usually claimed to be the following: 1) The non-observation of proton decay as predicted by the minimal Georgi-Glashow SU(5) GUT; 2 ) The LEP precision measurements; 3 ) The unusually large top-Yukawa coupling;
4) Possible candidates, such as neutralinos for dark matter.
16
Important ingredients in low-energy supersymmetry are the low-energy supersymmetry-breaking parameters, which are usually restricted by embedding supersymmetric gauge interactions in a supergravity low-energy effective theory [24, 251. In the early SO’S, the idea of supergravity, as a messenger of supersymmetry breaking to the observed elementary particles, such as quarks, leptons, Higgs and their superpartners, was elaborated [26]. A fundamental role is played by the gravitino mass M3/2 which has its origin in the super-Higgs mechanism. An appealing class of theories, which recently found place in superstring constructions, are the so-called no-scale supergravities [27]. In such theories the gravitino mass is a sliding scale at the tree level and it is dynamically fixed by radiative corrections, with the possibility of a hierarchical suppression with respect to the Planck or GUT scale [28]. Among the experimental implications of the supersymmetric extension of the standard model and of its GUT extensions, there is a prediction for the gauge-coupling unification and the supersymmetric threshold, namely the scale at which the supersymmetry breaking occurs. Peterman and Zichichi realized in 1979 that supersymmetric particles would imply a better convergence of gauge couplings [29] because of the strong modifications on the
functions due to
superpartners of the particle spectrum of the standard model. The main interest was to see if the energy threshold for the lightest supersymmetric particle could be predicted. Peterman and Zichichi realized that many problems needed to be worked out before any reasonable prediction could be made. Zichichi became a strong supporter of supersymmetry and in order to encourage a fruitful activity, decided to have a sequence of three Subnuclear Physics Erice Schools devoted to supersymmetry [30]. But high precision experimental data were missing.
17
They came with LEP and allowed a detailed analysis of all problems involved [31, 32, 331 in the attempt to make predictions on the lightest supersymmetry particle. This analysis gave a supersymmetric spectrum, showing that the supersymmetric threshold can be made as low as the Fermi scale, for a unification of couplings at 1016 GeV, making supersymmetry detectable in a wide energy range [31, 321. This was possible by exploiting some earlier work of Peterman and Zichichi in the late 70’s [33]. Indeed, in 1974, it was already known that, in pure super-Yang-Mills theory (without matter), the one-loop p function is given by [ 141
where C2 is the quadratic Casimir of G in the adjoint representation and
for N = 0, 1, 2 and 4 supersymmetry respectively. We therefore see that the function is less and less negative as we have more and more supersymmetry, and that it vanished in the maximally extended N = 4 Yang-Mills theory, which is conformal-invariant. In a series of seminal papers, reproduced in Part-A of the Selected Papers, Peterman and Zichichi, in collaboration with Anselmo and Cifarelli, critically made a series of detailed studies [34] on the basic problems connected with gauge-coupling unification. The goal of these studies was to have a deeper understanding on the lowest possible value of the supersymmetry threshold. It is shown in particular that the evolution of the gaugino masses, one of the supersymmetric soft-breaking terms, has the extremely important effect of lowering by orders of magnitude the supersymmetric threshold [35]. Two-loop
18
effects were also included [36]. The consequence of these studies was that the rudimental prediction [37] that the unification of gauge couplings (based on the initial
data
given by
the
LEP precision
measurements),
implies
a
supersymmetry breaking at the TeV scale, is not correct. A further original contribution of Zichichi and his collaborators has been the detailed study for the possible existence of a gap between the GUT scale (2: 1016 GeV) and the string scale (E 1018 GeV) [38].
A plausible and economic theoretical scenario for string unification has been
given
by
Lopez,
Nanopoulos
and
Zichichi
in
a
series
of
Papers (reproduced in Part-B of the Selected Papers) with a string-derived noscale supergravity model based on a single parameter, a universal gaugino mass [39, 401. The observable sector gauge group is SU(5) x U(l) [40]. An important element that deserves explanation and can be accommodated in the SU(5) x U(1) model is the gap [38] between the GUT scale
(2:
1016 GeV) and
the string scale (E 1018 GeV). There are nowadays other possible scenarios to explain a unification scale below the string or Planck scale. One is to work in the strongly coupled heterotic string which is related to weakly coupled type I string or M-theory on a segment [41]. The other possibility is to invoke large extra dimensions [20] and to have a different running depending on the energy scale with respect to the Kaluza-Klein masses [42].
References [ 11
G.R. Farrar, “Supersymmetry in Nature”, S. Ferrara, “Supersymmetric Theories of Fundamental Interactions”, A. Zichichi, “Supersymmetry and s U ( 2 ) ~x U ( ~ ) L + R ”in, “The New Aspects of Subnuclear Physics”, vol. 16 of Subnuclear Series, Erice 1978 (A. Zichichi ed., Plenum Press, New YorkLondon).
19
[2] E. Witten, “Introduction to Supersymmetry”, in “The Unity of the Fundamental Interactions”, vol. 19 of Subnuclear Series, Erice 1981 (A. Zichichi ed., Plenum Press, New York-London).
[3] “Supersymmetry”,vols. I, 11, reprints volumes (S. Ferrara ed., North Holland and World Scientific, 1987). [4] “Supersymmetry and Supergravity”, a reprint volume of Phys. Rep. (M. Jacob ed., North Holland and World Scientific, 1986). [5] S. Weinberg, “The Quantum Theory of Fields”, vol. 3 of “Supersymmetry” (Cambridge University Press, 2000). [6] A. Salam and J. Strathdee, “Supergauge Transformations”, Nucl. Phys. (1 974). [7]
m,477
J. Bagger and J. Wess, “Supersymmetry and Supergravity”, JHU-TIPAC-9009 (Princeton University Press, 1992).
(81 J. Wess and B. Zumino, “A Lagrangian Model Invariant Under Supergauge 52 (1974). Transformations”, Phys. Lett.
m,
[9] J. Wess and B. Zumino, “Supergauge Transformations in Four Dimensions”, Nucl. Phys. B70, 39 (1974). [lo] “Superstrings, the First I5 Years of Superstring Theory”, vols. ed.), reprints volumes (World Scientific, 1985).
1, II (J. Schwarz
[ 1 11 M. Green, J. Schwarz and E. Witten, “Superstring Theory” (Cambridge
University Press, 1987). [12] Y.A. Gol’fand and E.P. Likhtman, “Extension of the Algebra of Poincare‘ Group Generators and Violation of P”, JETP Lett. l3, 323 (1971) [Pisma Zh. Eksp. Teor. Fiz. l3,452 (197 l)]. [ 131 D.V. Volkov and V.P. Akulov, “Is the Neutrino a Goldstone Particle?”, Phys.
Lett. M, 109 (1973). [ 141 S. Ferrara and B. Zumino, “Supergauge Invariant Yang-Mills theories”, Nucl.
m,
Phys. 413 (1974); A. Salam and J. Strathdee, “Supersymmetry and 353 (1974). Nonabelian Gauges”, Phys. Lett.
m,
20
[15] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, “Progress Toward a 3214 (1976); S. Deser and B. Zumino, Theory of Supergravity”, Phys. Rev. “Consistent Supergravity”, Phys. Lett. 335 (1976).
m,
m,
1161 W. Nahm, “Supersymmetries and their Representations”, Nucl. Phys. B 135, 149 (1978). [ 171 M.B. Green, “Superstrings, M-Theory and Quantum Gravity”, in “Highlights of
Subnuclear Physics: 50 Years Later”, vol. 35 of Subnuclear Series, Erice 1995 (A. Zichichi ed., World Scientific). [18] M.J. Duff, “M Theory (The Theory Formerly Known as Strings)”, Int. J. Mod. Phys. All,5623 (1996) [arXiv:hep-th/9608117]. [ 191 M.J. Duff, “Not the Standard Superstring Review” and “From Superspaghetti to
Superravioli”, in “The Superworld II”, vol. 25 of Subnuclear Series, Erice 1987 (A. Zichichi ed., Plenum Press, New York-London) [QCD161:165:19871. [20] I. Antoniadis, “Physics with Large Extra Dimensions”, CERN-TH/2001-3 18, Lecture given at the “2001 European School on HEP”, Beatenberg, Switzerland, 2001. [21] L. Randall and R. Sundrum, “A Large Mass Hierarchy From a Small Extra Dimension”, Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hep-ph/9905221]; L. Randall and R. Sundrum, “An Alternative to Compactifcation”, Phys. Rev. Lett. 83,4690 (1999) [arXiv:hep-th/9906064]. [22] P. Fayet in refs. 141 and 151. 1231 D.Z. Freedman and P. van Niewenhuizen, “Supergravity and the Unification of the Laws of Physics”, Scientific American, No. 238 (1978). [24] R. Arnowitt, “Supergravity Models”, in “From Supersymmetry to the Origin of Space-Time”, vol. 31 of Subnuclear Series, Erice 1993 (A. Zichichi ed., World Scientific). [25] R. Barbieri, “Supersymmetric Particles”, in “The Superworld III”, vol. 26 of Subnuclear Series, Erice 1988 (A. Zichichi ed., Plenum Press, New YorkLondon).
21
1261 R. Barbieri, S. Ferrara and C.A. Savoy, “Gauge Models with Spontaneously Broken Local Supersymmetry”, Phys. Lett. B 119, 343 (1982); A.H. Chamseddine, R. Arnowitt and P. Nath, “Locally Supersymmetric Grand Unification”, Phys. Rev. Lett. 49, 970 (1982); L.J. Hall, J. Lykken and S. Weinberg, “Supergravity as the Messenger of Supersymmetry Breaking”, Phys. Rev. 2359 (1983).
m,
[27] E. Cremmer, S, Ferrara, C. Kounnas and D.V. Nanopoulos, “Naturally 61 Vanishing Cosmological Constant in N = 1 Supergravity”, Phys. Lett. (1983).
m,
[28] J.R. Ellis, A.B. Lahanas, D.V.Nanopoulos and K. Tamvakis, “No-scale Supersymmetric Standard Model”, Phys. Lett. B134, 429 (1984); J. Ellis, C. Kounnas and D.V. Nanopoulos, “No scale Supersymmetric Guts”, Nucl. Phys. B247, 373 (1984). For a review see A.B. Lahanas and D.V.Nanopoulos, “The Road to no Scale Supergravity”, Phys. Rep. 145,1 (1987). [29] A. Zichichi, “New Developments in Elementary Particle Physics”, Rivista del Nuovo Cimento 2.14, 1 (1979), Plenary Lecture given at the Closing Session of the 4th General Conference of the EPS on “Trends in Physics”, York, UK, 25-29 September 1978. [30] “The Superworld I”, “The Superworld Ir’and “The Superworld ItI‘?, vol. 24, vol. 25 and vol. 26 of the Subnuclear Series, Erice 1986, 1987, 1988 (A. Zichichi ed., Plenum Press, New York-London). 1311 F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, “The Effective Experimental Constraints on Ms”;sy and MGuT)’, Nuovo Cimento 104A, 1817 (1991). [32] F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, “The Simultaneous Evolution of Masses and Couplings: Consequences on Supersymmetry Spectra and Thresholds”, Nuovo Cimento 105A, 1 179 (1992). [33] A. Zichichi, “Subnuclear Physics. The First 50 Years: Highlights from Erice to ELN” (0.Barnabei, P. Pupillo, F. Roversi Monaco eds.), a joint publication by University and Academy of Sciences of Bologna, Italy, 1998; 20th Century Physics Series, vol. 24 (0.Barnabei, P. Pupillo, F. Roversi Monaco eds., World Scientific, 2000).
22
[34] A. Zichichi, “Where we stand with the Real Superworld’, in “From Superstrings to the Real Superworld’, vol. 30 of Subnuclear Series, Erice 1992 (A. Zichichi ed., World Scientific) and “Where can SUSY be?”, in “From Supersymmetry to the Origin of Space-Time”, vol. 31of Subnuclear Series, Erice 1993 (A. Zichichi ed., World Scientific). [35] F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, “The Evolution of Gaugino Masses and the SUSY Threshold”, Nuovo Cimento 105A, 581 (1992). [36] F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, “Analytic Study of the Supersymmetry-Breaking Scale at Two Loops”, Nuovo Cimento 105A, 1201 (1992). [37] U. Amaldi, W. de Boer and H. Furstenau, “Comparison of Grand Unified Theories with Electroweak and Strong Coupling Constants Measured at LEP’, Phys. Lett. B260,447 (1991). [38] F. Anselmo, L. Cifarelli and A. Zichichi, “A Study of the Various Approaches to M G ~ and T a~uf’, Nuovo Cimento 105A, 1335 (1992). 1391 J.L. Lopez, D.V. Nanopoulos and A. Zichichi, “A String No-scale Supergravity Model and its Experimental Consequences ”, Physical Review 4178 (1995) [arXiv:hep-ph/9502414].
m,
[40] J.L. Lopez, D.V. Nanopoulos and A. Zichichi, in “From Superstring to the Real Superworld”, vol. 30 of Subnuclear Series, Erice 1992 (A. Zichichi ed., World Scientific). [41] E. Witten, “Strong Coupling Expansion of Calabi- Yau Compactijcation ”, Nucl. Phys. 135 (1996) [arXiv:hep-ph/9602070].
m,
[42J I. Antoniadis, “Experimental Signatures of Strings and Branes”, in “Towards New Milestones In Our Quest To Go Beyond The Standard Model”, to be published in vol. 43 of Subnuclear Series, Erice 2005 (A. Zichichi ed., World Scientific).
23
SELECTED PAPERS ON SEARCHING FOR THE SUPERWORLD BY A. ZICHICHI AND COLLABORATORS
THEPAPERS REPRODUCED CAN BE GROUPED IN TWO CLASSES.
PARTA These papers deal with the problems of the convergence of the three fundamental forces of Nature measured by the gauge couplings, “ 1 , a2, a3, The effect of the convergence on the energy threshold for the production of the lightest supersymmetric particles is studied with great accuracy. In this class there are the theoretical two-loop calculations, the evolution of the gaugino mass in the RGEs which lowers the low-energy supersymmetry threshold by nearly three orders of magnitude and the study of the possible existence of a gap between the energy where the gauge couplings converge and the energy of the Planck scale.
PARTB The papers in this class deal with a search for a theoretical model with the minimum number of parameters, possibly one; the model which best agrees with all conditions found by strings theories, including the extremely small value of the cosmological constant, turns out to be a “one-parameter nomodel whose experimental consequences are scale supergravity investigated for present and jkture facilities aimed at the discovery of the first example of superparticle. ”
This page intentionally left blank
25
SELECTED PAPERS ON SEARCHING FOR THE SUPERWORLD BY A. ZICHICHI AND COLLABORATORS
PARTA
This page intentionally left blank
27
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi
THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON Msusy AND MGm
From I1 Nuovo Cimento 104 A (1991) 1817
This page intentionally left blank
29 IL NUOVO CIMENTO
Dicembre 1991
VOL. 104A, N. 12
The Effective Experimental Constraints on MsusY and MGUT(*).
',
F. ANSELMO L. CIFARELLI172v3, A. PETERMAN 11476 and A. ZICHICHI'
'
C E R N - Geneva, Switzerland Universita di Napoli, Italy INFN, Sezione di Bologna, Italy World Laboratory - Lausanne, Switzerland CPT, CNRS - Luminy, Marseille, France
(ricevuto il 15 Luglio 1991; approvato il 2 Settembre 1991)
Summary. -A comprehensive analysis of the world-data on a6is reported together with its average value at the Zomass. The effective constraints on Msusy and MGUT are given. The care needed to reach any conclusion on Msusy is discussed. For example, taking for the a1 , a2, as coupled equations a numerical solution (it should be the most reliable one) and the two standard deviation limits in the uncertainty of the ag(Mp)world average, the expected MSUSY values range from 10°.6'0.5 GeV to 105*lGeV, i.e. from GeV t o PeV.
PACS 11.30.Pb - Supersymmetry.
1.
- Introduction.
The purpose of this paper is twofold: i) t o report on a comprehensive analysis of the world-data on as (the QCD coupling constant); ii) t o work out which prediction, if any, can be made on MsusY(the ad hoc energy chosen for the introduction of SUperSYmmetry breaking, assuming that all sparticles have the same mass) and on MGUT (the unification mass corresponding t o the energy where the effective electromagnetic (a1),weak (az) and strong (a3) couplings converge). Let us specify these couplings in the SU(3)c0 s U ( 2 )0 ~ U(1) Standard Model: (1)
a1
= (5/3)(g"/LIT) = 5a/3 COS'
,
a2 = g2/4n= a/sin'
em ,
a3 = g:/4n,
where g ' , g and g, are the U(l>, s U ( 2 ) and ~ SU(3)c couplings, respectively; a is the fine-structure constant, and sin't3m is the mixing angle in the model, the so(*) Due to the relevance of its scientific content, this paper has been given priority by the Journal Direction.
1817
30 1818
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
TABLEI. - The { b i } and {bij} matrices for the two cases: without SUSY and with SUSY. Standard model Without SUSY [3] 1/10
i:
{bij} = 0
0
-136/3 0
]
1
19/15 + N F 1/5 - 102 11/30
3/5 4913 3/2
44/15
] [.to130/6 9/50
9/10
0
+NH
76/3
With SUSY [4] 3/10
38/15
6/5
88/15
9/50 ]+NH[3to
11/15
3
68/3
9/10
712
4 0
TABLE11. - Approximate analytical solutions of egs. (2) in the text. Formula 1: ref. [5]
ref. [8] Formula 3: same as Formula 2 but with aj( p ) in the poi expression replaced by the 1-loop equation solution.
31 THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON
Msusy
AND
1819
MGUT
called modified minimal subtraction scheme [l].The normalization factor 513 comes from the SU(5) condition that, at the unification scale, sin' Ow be equal to 3/8[21. I n the 2-loop approximation, the basic <
(2)
+ C(b,/8xz)
aja;
3
where the indexes i and j run from 1to 3, and the { bi } and { b, } matrices are given in table I, both for the non-SUSY [3] and the SUSY [4] cases. As discussed in sect. 3, these coupled equations (2) can be solved [5,6] either via numerical integration [7] or via approximate analytic expressions. These possible solutions are given in table 11. Recently an analysis [8] using one of these solutions has been reported. 2.
- Analysis
of a,.
The experimental determination of as ranges from 1.5GeV up to 91 GeV, i . e . the Zo mass rangef91. As reported in the review by S. Bethke [lo], the values of a8 obtained by LEP and SLC experiments have been grouped according to the various final-state analyses. These data are reproduced in tables 111-V for the convenience of the reader: Jet Rates (table 111), Energy-Energy Correlations (EEC, table IV) and Asymmetry in Energy-Energy Correlations (AEEC, table V). The average value relative to the so-called Event Shapes analysis [ll,121, which is however affected by a larger error, is reported in table VI, together with the other averages[lO]. The world average that can be TABLE111. -
a8(Mz)determination from
Jet Rates. ~~
~~~~
Value
Experimental
Mark I1 ALEPH DELPHI L3 OPAL
0.123 0.115 0.118
k 0.009 k 0.004 -+ 0.005 k 0.005 f 0.003
k 0.011 k 0.010 -+ 0.007
f 0.005 f 0.007 -+ 0.003 f 0.003 f 0.004
Average
0.119
f 0.002
k 0.007
f 0.003
0.121 0.114
Scale
~
Group
zk 0.008
Hadronization
TABLEIV. - a, ( M z ) detemzinution from EEC. Group
Value
Experimental
ALEPH
0.117
k 0.005
Scale
Theoretical
+ 0.006 - 0.009
L3
0.121
rt 0.004
+ 0.009
k 0.006
- 0.006 OPAL
0.124
k 0.006
k 0.007
k 0.007
Average
0.121
k 0.004
rt 0.007
f 0.006
32 1820
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
TABLEV. - uS(Mz) determination from AEEC. Group
Value
Experimental
Scale
Theoretical
DELPHI
0.106
t 0.004
L3
0.115
+ 0.008 - 0.006 + 0.007
+ 0.003 - 0.000 + 0.002
+ 0.003
OPAL
0.117
- 0.000
* 0.000 * 0.001
- 0.009
Average
0.113
t 0.007
- 0.005
+ 0.006
- 0.002 f 0.004
TABLEVI. - Final compilation on a,(Mz). Method
Value
Jet Rates EEC AEEC Event Shapes
0.119 k 0.008 0.121 ? 0.010 0.113 ? 0.008 0.126 t 0.016
TABLEVII. - Summary of
Unweighted a, average Weighted a, average Number(*) of events .lo6
aBresults
at LEP.
DELPHI
ALEPH
OPAL
L3
0.110 t 0.004 0.107t 0.004(**) 0.05
0.119k 0.009 0.119 t 0.009 1.06
0.120k 0.009 0.119 t 0.009 0.70
0.117 t 0.008 0.117 k 0.008 0.83
(*) Maximum number of events used either in Jet Rates or EEC or AEEC analysis by the four LEP experiments. (**) The weighted a, average quoted in ref. 181 is 0.108 k 0.006. This slight difference is likely due to rounding off when using the values of tables 111-V.
derived from LEP and SLC data is:
(Mp)= 0.118 f 0.008.
(3)
as
In addition table VII shows a summary of all LEP results, for each experiment separately. From the results obtained with Jet Rates, EEC and AEEC, the weighted and unweighted averages for each experiment have been worked out for the sake of commrison. As indicated in tables 111-V, there are essentially three possible sources of errors as:
experimental uncertainties (both statistical and systematic), uncertainties due to the scale truncation, theoretical uncertainties (in particular due to the various possible hadronization models).
33 THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON
Msusy
AND
MGUT
1821
Let us attribute the index j to specify each of the above sources of errors. The index i specifies a given process and analysis technique used to extract the value of as. the error will be: Then, for each measured value
(4)
j=l
Notice that when asymmetric errors (0; , u;) appear in tables 111-V, an average value (oij = ( m i + g;)/2) is used. To compute the weighted averages, the weights {wi= = 1/$} are used. We have assumed in table VII the errors to be the same for both kinds of averages, weighted and unweighted. In fact, since the various sources of errors are not purely statistical, moreover since possible correlations might exist among the theoretical errors involved in the different analyses, the resultant error of each experiment is taken to be: (5)
u
0.32
= mb[uili=1,2..... n
9
E
0.28
0.24
0.2 0.16
0.12
0.08
0.04
0 20
E ,, Fig. la. - The
a,
60
40
[
80
100
GeV1
fit (*fit I n in the text) with (1.5+59.5)GeV data (symmetric error).
34 1822
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
where n is the total number of analysis techniques used in the experiment. Notice that the unweighted and weighted aB averages in table VII are essentially the same for all LEP experiments but DELPHI, whose a, values are smaller and in particular the weighted average used in ref. [8] to compute the a l , a 2 , a3 evolution. As an additional piece of information, the number of events analysed by each collaboration[l3-16] is reported in table VII. Since this number is not always the same for different a, estimates by the same collaboration, moreover since possible overlaps of event samples, if any, are not specified, the maximum number of events analysed (N)is quoted in table VII, ie.:
N = max [Nili=1,2,..., n .
(6)
We have also analysed non-LEP data in order to get an extrapolated value of a, at the Zo mass, a,(MZo), from a, measured at lower centre-of-mass energies. This has been done by fitting the 1.5 GeV to 59.5 GeV ag values with the 2-loop QCD formula
0.32
0.28
0.24 0.2
0.16
0.12
0.08 0.04 0
20
60
40
E,,
[
80
100
GeV3
Fig. lb. - The a, fit (<
35 THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON
for
ag running
(7)
Msusy
AND
MGUT
1823
[17]: =
12n (33 - 2nf)In (p2/~2QcD)
6(153 - 19nf) In [In ( P 2 / d & ~ ) l
(33 - 2%)'
In (p2/A&d
nf being the number of flavours with mass smaller than p. The world average,
obviously not been included in the fit. The only free parameter in the fit which turns out to be:
as(MZo),has
is A,,,,
AQcD = (260+40) MeV.
(8)
The results are shown in fig. la. We call this fit, where all data from 1.5 to 59.5 GeV are included, dit 1)). Notice that the value of A Q ~ Dremains practically unaltered if the world average, as(Mzo), is also included in the fit. The +-a error on AQcD is represented in fig. la via the two dashed-line curves.
0.45
.............................................................................
.
0.4 0.35 0.3 0.25
0.2 0.1 5
0.1
0.05
n 20
40
E ,,
60
[
80
100
GeVl
Fig. 2. - The a6 fit (.fit 2, in the text) with (29 + 59.5) GeV data (symmetric and asymmetric errors coincide).
36 1824
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
The same data and fit are shown in fig. l b but with a different error-band. ‘i’his is obtained by randomly excluding one or more data points from the fit and by evaluating the spread of the fit results obtained in this way. As shown in fig. l b , this spread turns out t o be asymmetric with respect to the central fit curve. Therefore, the extrapolated a, value at the Zo mass can be quoted, according to figs. la and b, respectively, as: [a,(Mp)]fit1= 0.115
(9)
k 0.002 (symmetric error),
[a,(MZo)lfit1 = 0.115T::ii
(10)
(asymmetric error).
The same exercise is repeated using the more reliable higher-energy data: i.e. those between 29 and 59.5 GeV, only. This fit is called <
-4 I I I I I
I
World Average
r I I I I I
L3
f. I I I I
L
I
I
I I I I I
dl
+I
0 AL
11
I I I
I
4
I
D LPHl
I l l
I I I I I
I
0.10
0.1 1
ALEPH
te from 29 - 59 GeV data (fit 2) I
I
0.1 2
0.1 3
0.14
Fig. 3. - The allowed a, (Mzo)ranges from low-energy data (fit 1 and fit 2), compared with the a,(MZo) measurements from LEP experiments. The world average for a,(MZo) is also shown.
.fit 2, [AQcD = (390 If: 70) MeVl are such as to indicate possible sources of unquoted <<systematic>> errors. In fact the inclusion of the low-energy data lowers the value of A,,,. (*) Note that the values of A Q ~ D for dit 1, [AQCD= (260 k 40) MeVl and for =
37
THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON
M~usyAND MGUT
1825
Here again, the value of AgcD remains practically unchanged when, in the best fit, the world average, a , ( M z ~ ) is , included. The +o error-band is shown via the dottedline curves in fig. 2. Using the technique of random exclusion of data points (described above for fig. l b ) , this error-band remains almost unchanged. The corresponding extrapolated as(MZo)values are: (12) (13)
[a,(Mzo)]fit2= 0.121 k 0.004 (symmetric error), [a,(Mz0)]fit2= 0.121+8::!
(asymmetric error)
in fair agreement with the previous results. Figure 3 shows a compilation of LEP data (see table VII), where the asymmetric error-bands from figs. l b and 2 are superimposed. Clearly there is no reason to select a particular value of as (as done in ref. [81, with the DELPHI result) as the most significant one. The conclusion is that the world average for a, (Mzo)(which by the way is consistent with the extrapolated [a, (MZo)]fit1or [a,(Mp)lfit2values), should be the basis for all speculations on possible new physics. This world average (3) is indicated in fig. 3. 3.
- Consequences on MsusY and MGUT.
The a1 , a 2 , as evolution us. p can be worked out in the Standard Model framework, using as input the values of as(M2o)(see sect. 2) with given hypotheses on NH (number of Higgs doublets), N F (number of flavour Families) and using, as final input, the value of sin2 Om(Mzo), as indicated in (1). Since 1978 SUperSYmmetry and its importance was emphasized [HI. The convergence of a l , a 2 , a3 using SUperSYmmetry was computed [MI, and published for the first time in 1981[19]. These papers showed that the couplings evolve better towards a common Grand Unification point (MGUT) when SUperSYmmetry is introduced. However, the values of MsusYand MGUTwill obviously depend on the many ingredients adopted. In fact, in addition t o the experimental determination of a,(Mzo) and sin20m(MZo),there are the number of Higgs doublets (NH),the number of Families ( N F )and the choice of the particular analytical expression used as an approximate solution of the coupled 2-loop equations (table 11). Moreover, these differential equations can also be solved numerically, via a Runge-Kutta-Merson method of integration “71. This method should provide the most reliable approximation. In the present study, we have investigated many possibilities, using: i) a set of fixed ingredients, i.e.:
(14)
1
NF=3, NH=2,
sin2 Om (Mp)= 0.2334 k 0.0008 (*) [201 ;
ii) the different values of
a,(MZo) and the various possibilities for the solution of the
(*) This error includes the uncertainty on
T,,and mH
38 1826
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
0.1 08 f 0.005 0.1 15 k 0.008
4
0.1 18 f 0.008 0.120 f 0.009
15.0
15.5
16.0
16.5
17.0
Log M G U T ( G W Fig. 4a. - Results for Msusy and MGUT using Formula 1 (table 11) as solution of eqs. (2) in the text, for different values of a,(MZo), as specified.
A
E CJ)
0
A
0.108 f 0.005
A
0.115 ~t0.008
A
0.1 I 8 ~t0.008
A
0.1 20 0.009
-I
15.0
15.5
16.0
Log
GUT
16.5
17.0
(GeV)
Fig. 4b. - Results for MS,,, and M G U T using Formula 2 (table 11) as solution of eqs. (2) in the text, for different values of a,(Mp), as specified.
39 THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON
Msusy AND MGUT
1827
as (MZO ) o
0.108 f 0.005
0 0.1 15 k 0.008
Fig. 4c. - Results for MsusYand M G U T using Formula 3 (table text, for different values of a,(Mp), as specified.
0
0.118+0.008
0
0.120 f 0.009
as solution of eqs. (2) in the
0.1 15
+ 0.008
0.1 18 f 0.008
rn
15.0
15.5
16.0
16.5
0.120 _+ 0.009
17.0
Log M GUT ( G W Fig. 4d. - Results for M s u s ~and MGUTsolving eqs. (2) in the text by the Runge-Kutta-Merson method, for different values of a,(Mp), as specified.
40 1828
F. ANSELMO, L. CIFARELLI, A. PETERMAN
in
d
and
0 -
A. ZICHICHI
41
THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON
a3
kfsusy
AND
MGUT
1829
t
Fig. 6. - The a1 , aZ , agevolution curve represented as a series of points whose darkness increases with energy. Also shown as a series of smaller points the al = a2 = a3 straight line. The two curves meet at the unification point (&IcuT).
coupled a1 , a2, a3 2-loop equations. Four different values of a6 (Mza) have been considered, in order to scan the region allowed by the present experimental and theoretical uncertainties. The values chosen range from the lowest ( a , = 0.108, see table VII and footnote) to the highest ( a , = 0.120) LEP result, with an intermediate value at a, = = 0.115 k 0.008 and the world average: a, = 0.118 k 0.008, The masses of the lightest and the heaviest Higgs doublet have been assumed t o be MZoand Msusy, respectively. A simple fit procedure, applied to a1, a2, a3 evolution, with MGUT and Msusy as free parameters, has been performed. The results are shown in figs. 4a-d in terms of MsusY us. MGUT,for the various possibilities discussed so far. As can be seen from fig. 5, Msusy can range within one standard deviation from very small values up t o several tens of TeV and, accordingly, MGUT from 1017 down to almost 1015GeV. In order to see clearly the route followed by the three couplings a1 , a 2 , a 3 , towards the common point of convergence at M G u T , we have plotted a three-dimensional graph with a1, a 2 , a3 as orthogonal axes. Figure 6 shows an example worked out for Msusy = 1TeV. The various points are at energies specified by increasing darkness: the higher the energy, the darker the point. The fwst point corresponds t o 0.1 TeV,
42
1830
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
the second to lTeV, the third to 10TeV and so on, up to the unification point (*). We have also studied the convergence of the three running couplings a l ,a 2 , a3 without SUSY and found that this is still possible, provided NH= 6. The detailed analysis of these and other results is reported elsewhere[21]. 4. - Discussion on the care needed to reach conclusions on MsusY.
The use of one standard deviation limit for the study of possible consequences of quantities experimentally measured with their standard errors has become an accepted trend, as if these errors had more precise meanings than those specified by the well-known laws of statistics. The minimum range of safety is the 95% confidence level, i.e. f20. when Therefore we have studied what would happen in terms of MsusYvs. illGUT, applying our fit procedure not only with a , ( M p ) = 0.118+0.008 (the world average given in sect. 2), but also with the central value shifted by two standard deviations up and down: as(Mzo)= 0.134f0.008 and as( M p ) = 0.102+0.008, respectively. The results are shown in figs. 7a-c. The four possible solutions of the coupled 2-loop equations for a],a2, a3 evolution (discussed in sect. 3) are all included. The values of Msusy and M G U T scan a large range of possibilities. For instance, if (a, 2a) is taken at M p
+
Formula 2 Formula 3
urnerical solution
15.0
15.5
16.0
16.5
17.0
Log MGUT (GeW Fig. 7a. - Results for Msusy and MGUTusing for a, (Mp) the central value of the world average. The four possibilities correspond to the four possible solutions of the coupled equations (see table 11). (*) We are aware that the requirement of a unique crossing of a 1 , a 2 , a3 trajectories is not a compulsory condition for GUT.
43 THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON
Msusy
AND
1831
MGUT
0.134+0.008
Formula 1
A
Formula2
0
Formula3 Numerical solution
15.0
15.5
16.0
16.5
17.0
Log MGUT (GeV) Fig. 7b. - Same as fig. 7a using for aR( M p ) the central value of the world average shifted upwards by two standard deviations.
7
6
5
4
Formula 1
3
2
A
Formula 2
0
Formula 3 Numerical solution
1
0
15.0
15.5
16.0
16.5
17.0
Log MGUT (GeV) - Same as fig. 7a using for a , (Mp)the central value of the world average shifted down-
Fig. 7c. wards by two standard deviations. 116 - I1 Nuovo Gimento A
44 1832
F. ANSELMO, L. CIFARELLI, A. PETERMAN
cu +I
+I
0
-
0
0
T
v
and
m
+I
m
-
0
0
A. ZICHICHI
45
THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON
Msusy
AND
MGUT
1833
(fig. 7b), the values Msusy = 10°.6'0.6GeV and M G U T = 10'6.8'o.' GeV are obtained using the Runge-Kutta-Merson method of integration. The corresponding values for (as- 2g) are MsusY = lo6" GeV and MGUT = 10'6.4*0.2GeV, as shown in fig. 7c. We would like to emphasize that if new knowledge on Msusy should seriously be claimed by present experimental data [8], this new knowledge should be based on the 95% confidence level. This is why in fig. 8 we give a summary of the results derived in our analysis whti-] as the basic scenario t o study the evolution of a1 , a2, a3 with SUperSYmmetry introduced at the various energy levels, as specified. From these studies it comes out that care is needed before quoting[8] a value of MsusYwhere the expected SUSY-break would occur. 5. - Conclusions.
As reported in fig. 3, the values of a,(MZo) have a large range of uncertainty. No value of aB can be considered as privileged with respect to any other in the quoted figure. The only correct choice which takes all data with their errors into account is the world average (3). Moreover, the various approximate solutions of the coupled equations for a1 , a:!, a3 evolution must be considered as another source of uncertainty. Consequently, the values of MsusYand MGUT are in the range reported in fig. 8 for 95% confidence level. Claims for better knowledge on Msusy are unjustified.
*** We are grateful to S. Bethke, J. Ellis and W. J. Stirling for a series of interesting discussions. We would also like to acknowledge the collaboration of G. La Commare, M. Marino and G. Xexeo.
REFERENCES [l] W. A. BARDEEN,A. BURAS,D. DUKE and T. MUTA: Phys. Rev. D, 18, 3998 (1978).
[2] H. GEORGIand S. L. GLASHOW:Phys. Rev. Lett., 32, 438 (1974). [3] D. G. UNGERand Y. P. YAO:University of Michigan, Report No. UMHE 81-30 (unpublished). See also D.R.T. JONES: Phys. Rev. D, 25, 581 (1982). [4] M. B. EINHORN and D. R. T. JONES: Nucl. Phys. B , 196, 475 (1982); J. ELLIS,D. V. NANOPOULOS and S. RUDAZ:Nucl. Phys. B , 202,43 (1982). For lectures and references on all these topics see: Superworld I, edited by A. ZICHICHI(Plenum Press, New York-London, 1986); Superworld ZZ, edited by A. ZICHICHI(Plenum Press, New York-London, 1987); Superworld ZZZ, edited by A. ZICHICHI(Plenum Press, New York-London, 1988). [5] W. J. MARCIANO and A. SIRLIN:Invited Talk, Second Workshop on Grand Unification, Ann Arbor, April 1981. [6] J. ELLIS,S. KELLEYand D. V. NANOPOULOS: Phys. Lett. B , 249, 441 (1990) and 260, 131 (1991); P. LANGACKER: University of Pennsylvania, Preprint UPR-0435T (1990). In their analyses of LEP data these authors avoid overoptimistic estimates in the context of Supersymmetry. "71 See G. N. LANCE:Numerical Methodsfor High-speed Computers (Iliffe & Sons, London, 1960). [81 U. AMALDI, W. DE BOER and H. FURSTENAU: Phys. Lett. B , 260, 447 (1991).
46
1834
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
[91 For a review see, for instance, S. BETHKE:Preprint LBL-28112, November 1989; W. J. STIRLING and M. R. WHALLEY:Preprint RAL-87-107 and DPDG/87/01. [lo] S. BETHKE:Preprint CERN-PPE/91-36. [113 ALEPH COLLABORATION: Phys. Lett. B , 255, 623 (1991). [121 N. MAGNOLI, P. NASONand R. RATTAZZI:Phys. Lett. B, 252, 271 (1990). [13] DELPHI COLLABORATION: Phys. Lett. B , 247, 167 (1990) and 252, 149 (1990). 1141 ALEPH COLLABORATION: Phys. Lett. B , 255, 623 (1991) and 257, 479 (1991). [15] OPAL COLLABORATION: Phys. Lett. B, 235, 389 (1990) and 252, 159 (1990). [16] L3 COLLABORATION: Phys. Lett. B, 248, 464 (1990) and L3 Preprint No. 023, December 1990. [171 W. J. MARCIANO:Phys. Rev. D, 29, 580 (1984). [18] A. ZICHICHI:Closing Lecture at the E P S Conference, York, UK, 25-29 September, 1978; Opening Lecture at the E P S Conference, Geneva, C H , 27 June-& July, 1979 (CERN Proceedings) and New Developments in Elementary Particle Physics, Riv. Nuovo Cimento, 2, No. 14 (1979). The statement on p. 2 of this paper, *Unification of all forces needs first a Supersymmetry. This can be broken later, thus generating the sequence of the various forces of nature as we observe them*, was based on a work by A. PETERMAN and A. ZICHICHI where the renormalization group running of the couplings using Supersymmetry was studied with the result that the convergence of the three couplings improved. This work was not published. [19] S. DIMOPOULOS and H. GEORGI:Nucl. Phys. B, 193,150 (1981); L. IBANEZand G. G. ROSS: Phys. Lett. B , 105,439 (1981). See also J. ELLIS,D. V. NANOPOULOS and S. RUDAZ:Nucl. Phys. B , 202, 43 (1982) and the three volumes of the Proceedings of the UEttore Majoranaz International School of Subnuclear Physics (ref. [41).In all these volumes the fact that Supersymmetry allowed a better convergence of the basic running couplings ( a l a2, , a s ) was filly discussed by many authors, using different models. [ZO] P. LANGACKER and M. LUO: Preprint UPR 0466-T. [21] F.ANSELMO,L. CIFARELLI,A. PETERMANand A. ZICHICHI:Preprint in preparation.
47
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi
THE EVOLUTION OF GAUGINO MASSES AND THE SUSY THRESHOLD
From I1 Nuovo Cimento 105 A ( 1992) 581
1992
This page intentionally left blank
49
IL NUOVO CIMENTO
VOL. 105A, N. 4
Aprile 1992
The Evolution of Gaugino Masses and the SUSY Threshold. F. ANSELMO('),L. C I F A R E L L I ( ~ , A. ~ , ~PETER ), MAN(^,^^^) and A. ZICHICHI(') CERN - Geneva, Switzerland Dipartimento di Fisica, Universith di Pisa, Italy (3) INFN - Bologna, Italy (4) World Laboratory - Lausanne, Switzerland ( 5 ) Centre de Physique Thkorique, CNRS - Luminy, Marseille, France (I)
(2)
(ricevuto il 17 Febbraio 1992; approvato il 27 Febbraio 1992)
Summary. - We propose a numerical iterative method to account for the evolutions of the gaugino masses (EGM). The effect of these evolutions on the most exhaustive model of SUSY breaking is presented. From above 21 TeV, the one-o lower bound for SUSY breaking is brought down by more than two orders of magnitude. The model without EGM needed two CT to reach the Zo-massrange. The same model with EGM needs only one o to reach the same level. The conclusion is that the SUSY threshold can be anywhere, including within the mass range where LEP I, HERA, L E P I1 and other colliders are working or planning to work. PACS 11.30.Pb - Supersymmetry.
1. - Introduction.
We first review the status of SUSY in order to focus the main problem of concern in this exciting field of particle physics. Then we choose what appears at present to be the .best model. for the SUSY-threshold predictions. A numerical iterative method to account for the evolution of the gaugino masses is discussed, together with its consequences on the <
One of the central problems in particle physics is to establish if SUSY exists. From the theoretical viewpoint, there are good basic motivations for its existence. 581
50 582
F. ANSELMo, L. CIFARELLI, -4.PETERMAN
and
A. ZICHICHI
Historically, supersymmetry [l-41 has given rise to supergravity [5]; this produced superstring theory, which has no-scale supergravity as infrared solution [6-91. Thus supersymmetry breaking can be produced dynamically, via (> considerations and are far from being <<predictions.. The first work where a model has been built in order to attempt a prediction for the SUSY threshold was published soon after our work [24] by Ellis et al. [26,271. The conclusion of this paper was that, taking into account all known data with their errors and making all possible reasonable estimates of the unknowns, the SUSY threshold is, at the one-a level of confidence, predicted to be:
Furthermore, in order to get this threshold in the ZO-mass range, two standard deviations are needed. This produced a lot of interest, because it was the result of the most exhaustive model ever attempted for SUSY-threshold predictions: the best model. Discouragement affected a large physics community engaged in the search for SUSY particles [28-441. For this reason, we decided to study this model [26,271, trying to contribute to its improvement. In the present work, we show that improvements are possible. In fact, using the
51
THE EVOLUTION O F GAUGINO MASSES AND TH E
SUSY
THRESHOLD
583
same range of experimental and theoretical uncertainties, the above limit (1) can go below the Z o mass, for example
thus giving a good motivation for further searches in the low-energy domain of LEP I. And taking out of despair those physicists (including some of us) who are planning not only to continue the search for SUSY particles at LEP, but also to go on at HERA and LEP 11. Our conclusion is that, on the basis of the present experimental and theoretical knowledge, there is no lower bound for the SUSY threshold, even at the one-a level. Claims for the SUSY threshold based on a partial and restricted use [251 of our experimental and theoretical knowledge are not very significant.
3. - The SUSY threshold: two problems. Predicting where the SUSY threshold could be corresponds to facing at least two problems: the structure of the SUSY spectrum and the evolution of the masses involved. Many examples exist where a unique SUSY-breaking effective mass has been chosen, thus neglecting the effect of the mass spectrum of the various sparticles. To assume that all sparticles have the same mass is unphysical. Moreover, the evolution of these masses cannot be neglected. In the next section we introduce a numerical method in order to deal with these two problems: the spectrum and the evolution of the gaugino (g and W) masses involved in the light threshold for SUSY breaking.
4. - The evolution of the gaugino-mass spectrum and its effect on the lower bound for the SUSY threshold.
We chose a model where the gauge couplings satisfy evolution equations with the strong Grand Unification boundary conditions:
supplemented by the unification condition for the gaugino masses:
The evolution of the gaugino masses, mEand mm, is given by the coupled equations 38 - I1 Nuovo Cimento A
52 584
F. ANSELMO, L. CIFARELLI,
A. PETERMAN
and
A. ZICHICHI
[45]:
For the evolution of
a3
and
a2
we use the one-loop approximation [17,46-481:
The above formulae (5a)-(5c) and (6a)-(6c) are based on an approximate treatment of the threshold. This is justified, since we are computing corrections to genuine threshold parameters, i.e. mEand mw. Note that renormalization group calculations give approximately [27]:
But these proportionality factors cannot stay constant. When the gaugino masses increase, a2 and a3 evolve and the net effect of this evolution is a dumping on the expected increase. The mechanism of this self-dumping works as follows. Consider for example eq. (5a). For a large value of m112,there will be an increase of mg which would scale with ml12if a 3 ( m g ) / a G U T ( m 1 /would 2) stay constant. This is not the case. When mEincreases, a3 (mi1 will decrease, thus dumping the increase of mg. On the other hand, mil2is proportional, as we will see later [eq. @)I, to a positive power of a3(mE), therefore an increase of mil2, inducing a decrease of a 3 ( m p ) , will self-consistently dump the initial increase of mil2. We have elaborated a program which, by iteration, calculates sequentially the set of SUSY masses (mil2,mg ,rnm) until it reaches a set of convergent solutions. We proceed as follows. The starting points are the values of (@GUT )o, a3 (mZ1, sin2o(mZ), a e m ( m Z ) ,F ( X ) , and ( m G U T ) O , mg,mw, milz [see table 1 and formula (ll)]. Using the one-loop approximation (6a)-(6c), we evaluate a3 ( m E )a2 , (mw) and aGUT(m1/2 >.The
53 THE EVOLUTION O F GAUGINO MASSES AND THE
SUSY
THRESHOLD
585
TABLEI. - The inputs of the numerical iterative solution of the coupled equations f o r the evolutions of the gaugino masses. For details, see ref. [24]. ARBITRARY INPUTS 1
1
ml/z
INPUTS FROM DATA (EXPERIMENT AND THEORY) [a3(mZ)]-'= 8.85 [a,,(mz)]-' = 127.9 sin20(mZ)= 0.2331
next step is the use of the coupled equations (5a)-(5c) to work out mz and mq. The last equation to be used is [271:
which allows us to work out the value of m l p These values for mi, m q , m l p are the new inputs for the next iteration. We find that after less than 100 iterations the values for mg, m q , mll2 converge towards a stable solution: this is the result of the evolutions of the gaugino masses (EGM). The details of this work will be published elsewhere [491. Here we will limit ourselves to the main steps which allow the derivation of (8). This will illustrate the exhaustive nature of the model [27] chosen for SUSY-breaking predictions. Let us start with sin2%(mz), which can be written as follows [271: sin'
(m, ) = 0.2 +
7aem (mz ) 15as(mZ)
+ 0.0029 + A , (heavy) + ZSch + A T (light) .
The first two terms in (9) are the well-known one-loop formula. The third matches numerically the two loops [26,271. The sum of these two numbers, 0.2029, is the first term in (8). The term Adheavy) stands for the contribution of the GUT superheavy mass fields [501. Following Ellis et al. [27], it will be neglected: in fact any contribution from A T (heavy) can only increase the lower bound for mIj2. The effects of
54
586
F. ANSELMO,
L. CIFARELLI,
A. PETERMAN
and A. ZICHICHI
the conversion of the renormalization schemes are contained in Ssch: this term depends on the way the threshold matching is treated. At the foreseen level of precision for sin'e, this term can be neglected, even if it is straightforward to account for it numerically. The term AT(light) accounts for the threshold effects and has a lot of physics in it. Different, contradictory formulations exist in the literature. We have worked on a few of them [26,27,51], using the old-fashioned method of determinants, and checking their p-function structure. We agree with only one of the various formulations for AT, namely that given in [27]. This reads:
-3 log($)
+ $log(
2)
-3
log($)
+ 2 log(
2)
-
To extract useful information from (101, it is necessary to include a spectrum for the gauginos [451 and the squarks-sleptons t521 which can be expressed in terms of two universal SUSY-breaking parameters: mo and mlIz. The part concerning the squarks and sleptons, i . e . the last five terms in (lo), is a complete representation of SUSY-SU(5) and therefore enjoys, at one-loop, a scale-invariance property. Owing to this approximate scale invariance, the contribution of the last five terms in (9) is small. It is synthesized byf(y, w)in (8) and a plausible estimate has been worked out [27]: (0.2k0.2). Compared with other uncertainties in the accepted ranges of unknown masses and parameters, this contribution is small. The first two terms in (10) allow us to extract log(ml/z/mz)vs. mE and m%, i . e . the basic eq. (8), which can be rewritten as follows:
Notice that in formula (11) and from now on "GUT = aGUT(ml/2). The function F ( X ) , with its error + A , synthesizes, in this field of physics, all our knowledge (experimental data and theoretical estimates with their limits), except the effects due to the EGM. In order to proceed further, we need the numerical expression for (11). Following Ellis et al. [27], this is: (12)
log
(z)
= [10.96 5
7.3 5 6.9 ic 0.3
-3.3 k 0.2 +. 0.81 + +0.3
55 THE EVOLUTION OF GAUGINO MASSES AND THE
SUSY
THRESHOLD
587
The value 10.96 corresponds to the .central value. of F ( X ) , from experimental data and theoretical estimates of the unknown quantities entering in (8). The first three errors are related to the experimental measurements of sin2B, a3 and aem at the Z o mass. The last three correspond to theoretical uncertainties. The focal point of our work is to compute the last two terms in (12). They represent the contribution to ml12 of the evolutions of the gaugino masses, i.e. the EGM effect. So far, only the evolution of the couplings (a1 , a 2 ,aa) has been accounted for in dealing with SUSY-threshold problems, not the evolution of the masses. In order t o see the effect of this evolution, we proceed to a straightforward comparison with the .best model. [27], so far the most exhaustive. To this end, we take exactly the same input values, errors, treatment of errors, plausible ranges of unknown parameters ratio, the off-diagonal elements in the stop-mass matrix, the such as the m0/m1/2 treatment of the two Higgs doublets and the theoretical ranges of p and mh with their central values: all this in order to parallel the predictions of [27]. This is why we start with formula (8) above and use its numerical expression (12). In order to calculate the evolutions of the gaugino masses, the value of F ( X ) is needed as an input to our iterative process (see table I). As the experimental and theoretical knowledge will improve, +A in (11) will become negligible. At present it is so large that it can shift the <
+ 1.3,
(theoretical) = - 4 . 3 .
The experimental errors are added in quadrature. The result is: 10(experimental) = ? 10.05. The one-5 level is obtained, combining experimental errors and theoretical
TABLE11. - Experimental errors combined with theoretical uncertainties and the values of [ F ( X )+.A] and [ F ( X )fA' 1 corresponding to 10 and 25, respectively. Notice that since the .central value. of F ( X )i s quoted with f o u r significantfigures in ref. [27], we keep the (15 and 25) errors o n F ( X ) with the same significant figures.
1
15=
+ 11.35 - 14.35
+ 21.40 - 24.40
F(X)-tA= F(X)& A '
=
+ 22.31 - 3.39
+ 32.36 - 13.44
56 588
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
uncertainties, as follows: la*
j
la' = 5' (theoretical)
+ a (experimental) = +11.35,
la- = 5- (theoretical) - a (experimental) = - 14.35.
For the two-a level, the theoretical uncertainties are taken to be the same. The results are summarized in table 11. Within la, the .central value. of F ( X ) = +10.96 in (11) and (12) can be as high as +22.31 and as low as -3.39.
a, (m> a GUT 3
2.5
2
1.5
1
0.5
0 -10
-5
0
5
10
15
20
25
30 F(X)
Fig. 1. -The values of a3 ( ~ ~ ) / Q uand T a2(mm)/aCuTvs. F ( X ) . The function F ( X )represents all our knowledge in SUSY physics, i.e. experimental data and theoretical estimates of the unknowns. The range where F ( X ) values lie depends on the experimental errors and the theoretical uncertainties. At present, within one 5 , F ( X ) can be in the range from + 22.31 t o - 3.39. The two-5 limits are from + 32.36 down to - 13.44. However for very negative values of F ( X ) the formula allowing predictions for the lower bound of the SUSY threshold needs to be reconsidered.
57
THE EVOLUTION OF GAUGINO MASSES AND THE
SUSY
589
THRESHOLD
The evolution of the gaugho masses depends on a3(mg) and a2 (mw). The variation of these couplings vs. the input values for F(X)is reported in fig. 1. For example, if we compute, with our iterative method, the values of ag and az, using as fixed input the .central value. of F(X),the results are:
a2(mw) aGUT
= 0.922.
As a consequence, the EGM effect, i.e. the contribution of the EGM to m112,is:
This value (14) should be compared with the result [27]:
It follows that the estimates for ml12 with and without the corrections for the evolutions of gaugino masses (EGM) are in the ratio: (16)
( m 1 / 2 )without EGM
= exp [8.84 - 2.701 = 4.64.
lo2.
( m 1 / 2 )with EGM
The result of Ellis et al. [27] for the one-5 lower bound of SUSY threshold was: (17)
without EGM
-
(mlj2)lower bound - 21 TeV .
Using our correction, the new value is:
The EGM effect, as mentioned above, depends on the experimental and theoretical knowledge, synthesized in the function F(X). The smaller +A, the better we can estimate the EGM effect. At present, f A is quite large, as shown in table 11. We report in fig. 2 the EGM effect vs. the allowed input values for F ( X ) . The higher are the masses of the gauginos, the smaller is the correction for the evolutions of these masses. If these masses are very high, there is not much to evolve before reaching the unification limit. This trend is present in fig. 2, where at high F ( X ) values the EGM effect is minimum. In fact, high F ( X ) corresponds to high energy. Apart from the EGM effect, the result (18) followed the method, adopted by Ellis et al. [27], of using the .central value. for F ( X ) and then subtracting one standard deviation. The introduction of the EGM effect allows different possibilities to reach the -la- level for the m112lower bound. For example, if the input [F(X)- A] is taken to compute the gaugino evolution and no subtraction of lo- is applied to m1l2,the mllz lower bound is found to be at 1.9 TeV. On the other hand, if the evolution is computed taking as input [F(X)+A] and the subtraction of 2a- is applied to m1I2,the m112lower bound goes again below the
58 590
F. ANsELMO,
L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
EGM effect 10
8
6
4
2
0
-2
-10
-5
0
5
10
15
20
25
30
F(X) Fig. 2. - Showing the contribution to log(ml,Z)mz) from the evolutions of the gaugino masses, i.e. the so-called EGM effect, as computed with our numerical iterative method. The EGM effect depends on the input value for F(X), the function which represents all knowledge on SUSY (experimental data and theoretical estimates), as discussed in the text.
ZO-mass range. So if the search is for a lower bound, the result is that within lo, this is below the Z'mass. Without the EGM effect [27], in order to go below 21 TeV and reach the Zo-mass range, two standard deviations are needed. Figure 3 shows the range where the new lower bound of the SUSY threshold is predict.ed to be. 5. - Conclusions.
The evolution of the gaugino masses should be taken into account when dealing with the problem of predicting where the lower bound for SUSY breaking is expected to be. We have introduced a numerical method of iteration to deal with the evolution of the involved masses. We have shown that this correction is not negligible. When
59 THE EVOLUTION O F GAUGINO MASSES AND THE
SUSY
THRESHOLD
Predictions for SUSY-threshold lower bound
loo
lo2
lo4
lo6
lo8
1olo
1014
1o16
(GeV) Fig. 3 - Predictions for SUSY-threshold lower bound, with and without corrections for the evolutions of the gaugino masses (EGM). The grey range indicates the physics troubles when the lower bound for SUSY threshold is too low, as discussed in the text.
applied to the <>for SUSY-breaking predictions, it produces a decrease of more than two orders of magnitude in the lower bound of the SUSY-breaking threshold. Already at the one standard deviation level of confidence, the predicted lower bound for SUSY breaking is below the Z o mass. We emphasize that all experimental and theoretical knowledge in SUSY physics is included in the model adopted. Models based on a partial and restricted use of experimental and theoretical knowledge are not very meaningful. Now a comment. The real problem is the discovery of the first example of a SUSY particle or of a clear SUSY effect. Experimental searches should be encouraged, incorrect claims for a
60 592
F. ANSELMO,L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
high SUSY threshold [231 stopped, and attention concentrated on t h e solution of t h e real problems, which are very numerous: both in t h e technological and in t h e theoretical areas.
REFERENCES [l] Y. GOL'FAND and E. LIKHTMAN: J E T P Lett., 13, 323 (1971). [2] D. VOLKOVand V. AKULOV:Phys. Lett. B, 46, 109 (1973). [3] J. WESS and B. ZUMINO:Nucl. Phys. B, 70, 39 (1974). (North-HollandNorld Scientific, [4] Report Collection:Supersymmetry, edited by S. FERRARA 1987). [5] S.FERRARA, D. FREEDMAN and P. VAN NIEUWENHUIZEN: Phys. Rev. D , 13, 3214 (1976). S.FERRARA, C. KOUNNAS and D. NANOPOULOS: Phys. Lett. B, 133,61(1983). [6] E. CREMMER, [7] J. ELLIS,A. B. LAHANAS,D. NANOPOULOS and K. TAMVAKIS: Phys. Lett. B, 134, 429 (1984). Nucl. Phys. B, 247, 373 (1984). [8] J. ELLIS,C. KOUNNASand D. NANOPOULOS: [9] E. WITTEN:Phys. Lett. B, 155, 151 (1986). [lo] L. E. IBANEZ and G. G. ROSS: Phys. Lett. B, 110, 215 (1982). [ll] L. E. IBANEZ:Nucl. Phys. B, 218, 514 (1983). J. POLCHINSKI and M. B. WISE: Nucl. Phys. B, 221, 495 (1983). [12] L. ALVAREZ-GAUME, and K. TAMVAKIS: Phys. Lett. B, 125, 275 [13] J. ELLIS,J. S.HAGELIN,D. V. NANOPOULOS (1983). [14] L. E. IBANEZand C. LOPEZ:Phys. Lett. B, 126, 54 (1983). [15] C. KOUNNAS,A. B. LAHANAS,D. NANOPOULOS and M. QUIROS:Phys. Lett. B, 132, 95 (1983). [16] M. VELTMAN:Acta Phys. Pol. B , 12, 437 (1981). [17] S. DIMOPOULOS and S. RABY:Nucl. Phys. B, 192, 353 (1981). [l8] L. MAIANI:Proceedings of the S u m m e r School on Particle Physics, Gif-sur-Yvette, France, 1979 (IN2P3, Paris, 1980), p. 1. [19] E. WITTEN:Nucl. Phys. B, 188, 513 (1981). and F. WILCZEK: in K Unity of the Fundamental Interactions., edited by A. [20] S. DIMOPOULOS ZICHICHI(Plenum, New York, N.Y., 1983), p. 237. Closing Lecture at the E P S Conference, York, UK, 25-29 September 1978; also [21] A. ZICHICHI: Opening Lecture at the E P S Conference, Geneva, C H , 27 June-4 J u l y 1979, CERN Proceedings; and New Developments in Elementary Particle Physics: Riv. Nuovo Cimento, 2, No. 14, 1 (1979). The statement on p. 2 of this article, <
61 THE EVOLUTION O F GAUGINO MASSES AND THE
SUSY
THRESHOLD
593
[25] U. AMALDI,W. DE BOER and H. FURSTENAU: preprint CERN-PPE/91-190, November 1991. [26] J. ELLIS,S. KELLEYand D. V. NANOPOULOS: Phys. Lett. B, 260, 131 (1991). [27] J. ELLIS,S. KELLEYand D. V. NANOPOULOS: preprint CERN-TH6140-91. [28] ALEPH COLLABORATION (D. DECAMPet al.): Phys. Lett. B , 236, 86 (1990). [29] ALEPH COLLABORATION (D. DECAMPet al.): Phys. Lett. B , 237, 291 (1990). [30] ALEPH COLLABORATION (D. DECAMPet al.): Phys. Lett. B , 244, 541 (1990). [31] ALEPH COLLABORATION (D. DECAMPet al.): Phys. Lett. B , 265, 475 (1991). [32] DELPHI COLLABORATION (P. ABREUet al.): Phys. Lett. B , 241, 449 (1990). [33] DELPHI COLLABORATION (P. ABREU et al.): Phys. Lett. B , 245, 276 (1990). [34] DELPHI COLLABORATION (P. ABREUet al.): Phys. Lett. B , 247, 148 (1990). [35] DELPHI COLLABORATION (P. ABREUet al.): Phys. Lett. B , 247, 157 (1990). [36] L3 COLLABORATION (B. ADEVAet aZ.1: Phys. Lett. B , 233, 530 (1989). [37] L3 COLLABORATION (B. ADEVAet al.): Phys. Lett. B , 251, 311 (1990). [38] L3 COLLABORATION (B. ADEVAet aZ.1: Phys. Lett. B , 252, 511 (1990). [391 OPAL COLLABORATION (M. Z. AKRAWYet aZ.): Phys. Lett. B , 240, 261 (1990). [401 OPAL COLLABORATION (M.Z. AKRAWYet al.): Phys. Lett. B , 242, 299 (1990). [41] OPAL COLLABORATION (M. Z. AKRAWYet al.): Phys. Lett. B , 248, 211 (1990). [42] OPAL COLLABORATION (P. D. ACTONet al.): Phys. Lett. B , 268, 122 (1991). [43] MARK I1 COLLABORATION (T. BARKLOWet al.): Phys. Rev. Lett., 64, 2984 (1990). [44] M. DAVIER:preprint LAL 91-48, December 1991, presented at the LP-HEP 91 Conference,
Geneva, 1992. [45] J. POLCHINSKI: Phys. Rev. D , 26, 3674 (1982). [46] H. GEORGI,H. R. QUINNand S. WEINBERG:Phys. Rev. Lett., 33, 451 (1974). [47] D. R. T. JONES: Nucl. Phys. B, 87, 127 (1975). [481 M. B. EINHORN and D. R. T. JONES: Nucl. Phys. B, 196, 475 (1982). [49] F. ANSELMO,L. CIFARELLI,A. PETERMAN and A. ZICHICHI:preprint in preparation. [50] R. BARBIERIand L. J. HALL: preprint LBL-31238, September 1991. [51] J. ELLIS: preprint CERN-TH/6193-91. 1521 A. B. LAHANASand D. V. NANOPOULOS: Phys. Rep., 145, 1 (1987).
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63
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi
THE CONVERGENCE OF THE GAUGE COUPLINGS AT EGU AND ABOVE: CONSEQUENCES FOR a,(M,) AND SUSY BREAKING
From I1 Nuovo Cimento 105 A ( I 992) 1025
I992
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65 I L NUOVO CIMENTO
VOL. 105A, N. 7
Luglio 1992
The Convergence of the Gauge Couplings at EGUTand above: Consequences for a3 (M,) and SUSY Breaking. F. ANSELMO (l), L. CIFARELLI (l) (2) (l) (2)
(3) (4)
(5)
(3),
A. PETERMAN (l) (4)
(6)
and A. ZICHICHI(l)
CERN - Geneva, Switzerland Physics Department, University of Pisa - Pisa, Italy I N F N - Bologna, Italy World Laboratory - Lausanne, Switzerland Centre de Physique Thdorique, CNRS-Luminy - Marseille, France
(ricevuto il 13 Dicembre 1991; approvato 1'11 Marzo 1992)
Summary. - We work out an iterative solution of the evolution equations for a l , a2,a3 in the Minimal Supersymmetric SU(5) Grand Unifed model with the condition that the gauge couplings should converge smoothly towards E G U T and should not separate above E G U T . The work is done at two levels of theoretical accuracy: one loop and two loops. Improved accuracy favours high values of a3 (M,) with respect to the world average. i
PACS 11.30.Pb - Supersymmetry.
1. - Introduction.
Many authors have used the evolution of the gauge couplings al, a2,a3,in order to find the energy level of their convergence at a common point E G U T , without considering [l-41 that above E G U T the couplings a1, a2 and a3 separate. This is obviously unphysical. There is another effect which is neglected in spite of the fact that it needs to be there: the high-mass-threshold effect below E G U T . We have investigated what happens in the accessible range of energies when these two basic features are taken into account (*). We proceed as follows. In sect. 2 we study the complete equations for the gauge couplings using one-loop and two-loop approximations, in order to compare the results and see which trend is expected when the approximation is improved (two-loops). High values of a 3 ( M z ) with respect to the present world average are favoured. On the other hand increasing values of a3(Mz)imply increasing values for E G U T and, as already pointed out [6], decreasing values for Msusv (the energy scale for
(*) For a similar study in the context of the Standard Model, see [5].
1025
66 1026
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
'SUSY breaking). Section 3 deals with the convergence of the gauge couplings a l , a 2 , a3 above E G U T and their smooth convergence below EGUT:i.e. the heavy-mass-threshold effect. Finally in sect. 4 we discuss the results and present our conclusions. 2. - The evolution of the gauge couplings from Grand Unification down to M Z . 2'1. One-loop approximation. - The work presented here starts from the strong unification hypothesis that all gauge couplings ( a 1 , a', a3) have the same value a5 at some extreme energy point EMAX. We take for this extreme energy point the arbitrary value: EMAX
= 1020 GeV.
Thus we have as boundary conditions: al(EMAX) = az(EMAX) = a3(EMAX)
= a5(EMAX)
where a5 is the gauge coupling of the Supersymmetric Grand Unified Group. Among all possible SUSY-GUTS we choose the minimal SUSY-SU(5) [7-101 whose coupling a5 evolves above E G U T following a3. The evolutions of a l , a', a3 are worked out starting from E m and going towards lower energies. I n the next subsection we shall deal with the two-loop equations, in order to compare the results. The one-loop equations are:
(3)
where the F"'function (or F"'1ater on) is reported in the appendix, a l , a2 and a3 are the couplings of the gauge groups SU(3), SU(2) and U(1), Q1 is the extreme energy = Q1, and Qo is the low-energy point where input-output quantities can value, EMAX be experimentally measured [61. We take as input the value of a3 measured at the Z o mass: a 3 (Qo )
= a3 ( M z)
.
This is in fact the only input for eqs. (1)-(3). Thus the threshold scale M xis worked out as a function of this single input. We evolve a3 from its value at the Zomass (which, we repeat, is our unique input) up to the fixed scale, E MAX = 10" GeV, where the unification of al , a2 and a3 has been assumed to have already occurred. Then we evolve al , a 2 , a3 down to the Zo mass (or Qo) and get from eqs. (1)-(3) the value of Mx fitted in such a way as to give for aem( M z) and sin' O(M, ) the values which are as close as possible to the experimentally measured quantities. The quantity M x represents the mass of the superheavy gauge bosons which
67 1027
THE CONVERGENCE OF THE GAUGE COUPLINGS ETC.
become massive as a result of the SUSY-SU(5) breaking. Consequently, the unique value of a5 is split into three values: alla z , a 3 , corresponding to SUSY-[SU(3) x x SU(2) x U(l)]. We assume for the superheavy bosons a degenerate spectrum: as if all superheavy particles, which become massive because of SUSY-SU(5) breaking, had the same mass. However we perform an exact treatment of the threshold. If we call EGUT the energy where the three p-functions can be considered as being equal, the mass value of the superheavy bosons M x is about two times smaller than 1. In EGUT. We show in fig. 1 the range of variation of M xvs. the input values of a3 (Mz fig. 2 we report how E G U T varies with the input values of a 3 ( M z ) . Let us now discuss the values of sin20 at M Z . We recall that we are still dealing with the one-loop approximation. Figure 3 shows the variation of sinzB(MZ) us. the input values of a 3 ( M z ) .The results seem to favour values of a 3 ( M Z )below the world average a y A ( M z )= 0.118. Note that the value of aem(Mz) is practically insensitive to a3 ( M z ), as shown in fig. 4. However, here again, the values below the world average for a s ( M Z ) seem to be favoured. We report in table I data where several values of a 3 ( M z ) have been taken as inputs. For the definition and discussion of the appearing in table I, see (10) in subsect. 2'3. We now turn to the two-loop equations.
2
2'2. Two-loop approximation. - The two-loop equations for same approach as above are:
W . 6 log (Qf /&,") - 57.6 F "'(Q: 24~c
- --
a1, a2, a3
using the
, Qf , M i )I + TI + FfZ) (Qf , Q," , M i ) ,
The Ti terms are defined as follows: (7) where the bj and bij coefficients are given [S] by:
68 1028
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
5 4.5
-2
4 3.5
0
-
z
0
5 3
3 2.5
2 1.5 1
0.5
Ooti
I
' ' '
1
0.11 I 8
1
3
8
' I0.12 t ' '
8
8
8
*
3
0.13 I I
8 . 1
8
I
' ' ' 0.14 L
4 M J Fig. 1. - 1-loop results for the heavy mass M x as a function of the input a 3 ( M z )in the range: world average k2u.
Fig. 2. - 1-loop results for the GUT energy scale as a function of the input a 3 ( M z )in the range: world average *20.
69
1029
THE
14
Fig. 3. - 1-loop results for the sin20(MZ)as a function of the input a g ( M z )in the range: world average + 2 ~ .
WA+2u I I I I I I
il -LOO4 1 LSULTS I
128
..
!...
I ....
.....:.
......
...
I...
WA
.
.
I I
I I
I I
4
Fig. 4. - 1-loop results for the l/a,,(MZ) as a function of the input a 3 ( M z ) in the range: world average k2a. 61
- I1 Nuovo
Cimento A
70
1030
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
TABLEI. - One-loop results of thefitting procedure to get the best M , value in order to be as close as possible to the experimental values for sin28and l/ae,,, at M Z , i.e. sin28 = 0.2334 2 0.0008 and l/ae,,, = 127.9 t 0.2.
0.102
0.70
+-
+-
127.9
0.04
0.2358 -c 0.0001
0.008 0.107
1.02
0.2341
127.9
5
+
?
-
0.003
0.05
0.0001
0.2
0.111 -c 0.008
1.3
0.2329
127.9
t 0.1
t
+-
0.0001
0.2
0.113
1.54
0.2323
127.9
t
t
t
?
0.003
0.08
0.0001
0.2
0.119
2.2
0.2307
127.9
?
+
?
?
0.003
0.1
0.0001
0.2
0.123
2.8 2
0.008
0.2
0.2297 -c 0.0001
127.8
?
0.125
3.1 -c 0.2
0.2292
+ 0.008
0.0001
127.8 -c 0.2
0.134
4.8
0.2273
127.8
t
+-
t
t
0.008
0.2
0.0001
0.2
-
I
(9)
+-
+
I
199/25 { b V }= 9/5 11/5
0.8
0.4
1.9
11.7
21.8
?
0.2
2
I
8.7
0.2
I
27/5 25 9
27.7
59.3 I
8815 241. 14
With the two-loop equations (4)-(6) we use an iteration procedure, The results are shown in figs. 5-8. In fig. 5 we see that M x goes from 0 9 x 1016GeV to 3.8 x 1OI6 GeV when a, ( M , ) varies from the minimum (ayA- 2 ~ to) the maximum + 2a) values, respectively. The range of variation for E G U T is shown in fig. 6, while sin2B(MZ)vs. a 3 ( M Z )is reported in fig. 7. It is interesting to remark that, in order to approach the world average value for sin28(MZ),it is necessary to have for a3 (MZ ) a value slightly above the world average, even if the world average a?* = = 0.118 is certainly well fitted by the two-loop approximation. Contrary to the one-loop
(arA
71
1031
THE CONVERGENCE OF THE GAUGE COUPLINGS ETC.
WA+2a .
4.5
.
.
.
.
:
. . . . . . . . . .;
2-coo~.REs"LTs I . . .I . .;..
..
I I
r"
,
I ........ I
: I . . . . . . . ., . . . I . . . . . .
:
/I
:
. . . .
.........................
2 1 :
...;. . . . .
I
; j
......
. . . . . .
I
I1
' '
I
. I . ;
14
Fig. 5. - 2-loop results for the heavy mass M x as a function of the input a 3 ( M z ) in the range: world average 220.
. . . . .I... . . . . . .
..........
I I
. . . . . . . . . . . . . . . . . . . . . . . . I.. .:. . . . . . . . . . . . . . . . . . I I
:
........
I
A
20
'0
0 F
X
v
5
W
4
Fig. 6. - 2-loop results for the GUT energy scale as a function of the input a 3 ( M z ) in the range: world average 520.
72
1032
1?\,
F. ANSELMO, L. CIFARELLI, A. PETERMAN
0,238 0,236
-.
WA
A. ZICHICHI
WA+2u
I
j
1
:
: ;
1-LOOF)iESU.LTS I '
.!. . . . . . . . . . . . . .
and
I I I
;
.... :........ !I
I
:
:
I
I
:
:
I
:
I ............................................................. iWA+2c
I
- _ _ _ - _ _ I_ -
--I--
.............. l
: ., .......................
:
r--I
,,,.I. WA'""'
- - - - _" . . - k
-----I-&
1 : .I_..! ... . . . . . . .-.. . ...
-
--
j
I
:
I
.....I -. . .-. . .-. ,Lwby&
I
0.23
. I --..I..
. I . . . :. . . . . . . . . . . . .
-
0 228
1 I I - I.. . . . . . . - 1
1,;
1 I : I : . . . . . . . . . . I . .:..
,,,,,,
0.2260,
I
:
I I
:
.........I
I
: :
I I .I. I I
;
I
: . . . . .
. . .
0,
Fig. 7. - 2-loop results for the sin20(MZ)as a function of the input as(MZ) in the range: world average 2 2 ~ .
WA
128.1
WA+2u
I I
j
I
I
1 I
I-LOOd I ISULTS I I I
..................
128
I I
<
r"
W
d 127.9
_. _. _.
-. -.
I
:
I
: I ...................................... : I
I
n
I :
:
I
: : :
I I I
;
I I
WA
II I
\ 7
I I I I . . . . . . . . . . . . .I...
127.8
............
j
I
j
l
: I : I . . . . . . . ..I. . . . . . . .
I I
: :
I
:
I
I
: :
I I
I I
u
I I
: I I L I I I I I I , ,
0.14
..
n
..
r.ig. 8. - Z-loop results for the average - 1 2 ~ .
.
I
, . - as \ a runction ,. .. of".. . . .. . . . . . the input a 3 ( M Z ) in the range: world
l/a,,(MZ)
13
1033
THE CONVERGENCE OF THE GAUGE COUPLINGS ETC.
TABLE11. - Two-loop results of the fitting procedure to get the best M x value in order to be as close as possible to the experimental values for sin20 and l/aem at M Z , i.e. sin20= 0.2334 2 t 0.0008 and l/aem= 127.9 2 0.2.
0.2389
0.008
0.59 -e 0.03
0.107 rt
0.102
128.0
rt
f
0.0001
0.2
0.86
0.2372
t
2
0.003
0.04
0.0001
128.0 -e 0.2
0.111
1.11
0.2360
127.9
f
-
rt
f
0.008
0.05
0.0001
0.2
0.113
1.26
0.2355
127.9
f
rt
rt
Itl
0.003
0.06
0.0001
0.2
0.119 -e 0.003
1.80
0.2339
127.9
f
+.
rt
0.09
0.0001
0.2
0.2330
127.9
rt
+
0.008 rt
0.008
rt
f
0.0001
0.2
0.2325
127.9
f
f
0.0001
0.2
0.2306
127.9
46.9
23.2
11.1
6.9
0.4
0.3
1.3
rt
0.0001
case, here the trend is to favour the high values of a3 ( M , 1. As for the one-loop case, a,,(MZ) is practically constant. However the values of a 3 ( M z ) above the world average are favoured again. As for the one-loop case we report in table I1 all two-loop data in detail. 2'3. Comparison between the one-loop and the two-loop results. - We now compare the one-loop and the two-loop results. In order to do this we define a 2 as follows: (10)
=
(
sin2O(Mz) - 02334 0.0008
)' ( +
1/a,, ( M , ) - 127.9 02
This definition is of course valid for the one-loop as well as for the two-loop results.
74 1034
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
I
I I I
50
I\
2-LOOPS
~
I I
I I I
I I
\
lo
-
I I
II
I
\
Fig. 9. ? 20.
I
I
\I
30 li;l I Nx
1-LOOP
- -
2-loop vs. 1-loop
0.10;0.11
2
I
0.115 0.12 0.125 0.13 0.135 0
4
as a function of the input a3(MZ) in the range: world average
The ;i!values are reported in tables I and I1 for the one-loop and the two-loop cases, respectively. The results are shown in fig. 9 where it is evident that the two-loop equations for al , a:!, a 3 do favour values of a 3 ( M Z) above the present value of the world average. Low values of a 3 ( M z ) are clearly disfavoured. A direct comparison between the one-loop and the two-loop results is reported in table I11 for three values of a 3 ( M Z), i.e. the world average and its corresponding ? 2g values.
MX ( x 1016GeV)
sin20(MZ)
llat!m(Mz)
li"
0.70 f 0.04 0.59 t 0.03
0.2358 1- 0.0001 0.2389 t 0.0001
127.9 ? 0.2 127.9 f 0.2
8.7 46.9
= 0.118 one-loop two-loops
2.1 t 0.1 1.7 k 0.1
0.2309 k 0.0001 0.2342 ? 0.0001
127.9 ? 0.2 127.9 ? 0.2
9.6 1.1
+ 20- = 0.134 one-loop two-loops
4.8 ? 0.2 3.8 2 0.2
0.2273 2 0.0001 0.2306 t 0.0001
127.8 t 0.2 127.9 ? 0.2
59.3
a 3 ( M z1
a?" - 20- = 0.102
one-loop two-loops
mrA a?"
12.1
75 1035
THE CONVERGENCE OF THE GAUGE COUPLINGS ETC.
The trend is clear: when the approximation is improved from one to two loops the high values of a 3 ( M z ) are favoured. 2'4. Comparison between our ?-test a n d other claims. - Some authors [3] have claimed to be able to know where the SUSY breaking threshold should be, by constructing a ?-test based on the goodness of the al , a 2 , a3 convergence. These
authors [31 have ignored the heavy threshold, needed to allow the breaking of the supergroup into SUSY [SU(3) x SU(2) x U(l)]. Notice that if the goodness of the convergence is the crucial quantity to construct a 2-test, the heavy threshold is the basic feature for a physically sound convergence of the couplings towards E G U T . We show now what should have been the meaningful $-test. High-precision LEP data produce the best values for a e m ( M z ) ,sin20(MZ), and a 3 ( M z ) . In order to study the best convergence of the couplings we use only one input: a 3 ( M z ) . The evolution of a3 is worked out using the Renormalization Group Equations up to E MAX = 10'' GeV, where we impose the condition: a1( EMAX ) = = a2 ( E MAX ) = a3 ( E MAX). Then we evolve all couplings down to M Z , crossing the heavy threshold somewhere a t M x, below E MAX. We work out the values of sin2O(MZ) and a,,(MZ) and construct the ?-test (10) whose minimum provides Mx.The ?-test shows that there is no need for SUSY threshold anywhere down to the Z-mass. We emphasise that if the basic convergence of the couplings a t E G U T is the guideline to search for the SUSY threshold, the correct $-test is the one introduced now. Unfortunately the authors of [3] have introduced a $-test deprived of physical meaning because the "1, a2, a3 convergence is studied regardless the heavy threshold effects and the fact that above EGUT the couplings should not diverge. The next section is devoted to a detailed study of these physically essential features of the problem. a l , a 2 , a3
3. - The smooth convergence of al , a 2 , a3 towards EGUT and the unified evolution above E G U T .
The a l , a 2 , a3 evolution equations features:
(1)-(3) and
(4)-(6)
do guarantee two
1 - A smooth convergence of a l , a2, a3 up to E G U T . This is due to the treatment of the M x threshold. 2 - A fully unified evolution of a1, a2, a3 above E G U T .
The detailed evolutions for al , a 2 , a3, showing the above-quoted physically expected features, are reported in fig. 10-12. In fig. 10 the evolutions are calculated using as input value the world average for a3 ,
i.e. $*(AdZ) = 0.118.
In order to show the detailed behaviour in the high-energy range, fig. 10a is a
blow up of fig. 10 around EGUT. The other figures, 11 and 12, refer to the values of a r A ( M , ) with k 2 5 . There is a crucial point to emphasise: the results presented herein show that the smooth convergence of the gauge couplings towards EGUT and their unified evolution above EeUTdo not need any SUSY breaking all along the range from EGUT down to E = 10'GeV.
76 1036
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
80
SUPERSYMMETRIC SM a,(M,)=O.l 18; M X = l , 7 x 1 0l6d e V ..................
60
................
._ 40
a
\ v-
/
20
0
.......
:
I
.............
,
4
I
,
/
.i.. . . . . . . . . . ..i. .( . . . . . . . . . . . ..... I
I
,
I
8 lO91oP
,
I
12
,
,
d
,
1
20
(GeV)
Fig. 10. - a l , a2, a3 evolution as a function of the scale for a 3 ( M z ) = 0.118 (world average) and for the M x value resulting from the fitting procedure with that a3(MZ) value.
40
36
32
>ti-
28
24
20
Fig. 10a. - Exploded view of fig. 10.
77 1037
T H E CONVERGENCE OF T H E GAUGE COUPLINGS ETC.
80
SUPERSYMMETRIC SM a,(M,)=O.l02 MX=0,59x1O l 6 GeV . . . . . . . . . . . . . . . .
60
., . . . . . . . . . .
................................
._ 40
a
\ 7
\ --.
w
/ "
20
0
Fig. 11. - a l , a2, a3 evolution as a function of the scale p for a3(MZ) = 0.102 (world average -2g) and for the M x value resulting from the fitting procedure with that a3(MZ) value.
40
-
,"
..,S~P~R~YM~E~RIC . . .. .. .~. .M . . ..i.~ . . . .. . .. . .. . . . . . . .
a,'(M,)=O. M,=0,59x
162 1 GeV
36 -. . . . . . . . .:. . . . . . . . . . . . . . . . .
32
23-
\28 r-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;. . . . . . . . . . . . . . . . . .
-.
_........ ...................
24 -. . . . . . . .j . . . . . . . . . . . . . . . . . i . . . . .
20
- . . . . . . . . :. . . . . . . . . . . . .
/;
I. . . . . . . . . . . .
j
............... ................
I..
. . . . . . . . . . . :. . . . . . . . . . . . . . . . .
I
I
I
I
12
14
16
18
Fig. 11a. - Ex[ploded view of fig. 11.
1
J
78 1038
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
80
SUPERSYMMETRIC SM a,(M,)=O. 134 MX=3,8x1 0l6GeV 60
zj\
40
\ . ---^ -
r--
\ -
20
0
:
-_
: / . j.. . . . . . . .:... . . /&
. . . . . . .
I
,
4
I
,
I
,
8 I^-
1
..
1"YlOP
,
1
,
12
,
,
1
,
,
16
,
1
20
/Om\/\
\bev)
Fig. 12. - a1 , a2, a3 evolution as a function of the scale ,u for a 3 ( M z )= 0.134 (world average and for the M x value resulting from the fitting procedure with that a 3 ( M z ) value.
40
PERJYMMETRI~..SM.~.
................
/
. . . . . . . . . . .
M,)=O. 1 3 4 =3.8x1016 GeV 36
. . . . . . . . . . . .
. . . . :. . . . . . . . . . . .
/
.
.
.
............
32
> t(
28
24
20
0
Fig. 12a. - Exploded view of fig. 12.
+ 20)
79 THE CONVERGENCE OF THE GAUGE COUPLINGS ETC.
1039
40
36
32
24
20
log,oP (GeV)
Fig. 13. - Figures 10a, l l a and 12a superimposed.
This is compatible with the threshold for SUSY breaking being in the lo2 GeV range. In other words the naive belief that SUSY breaking is needed in the TeV range in order to obtain a better convergence [3] for the gauge couplings al , a2, u3 is shown not to be true. What is needed is a set of correct equations which take into account the smooth convergence towards EGUT. In fig. 13 we present, for a direct comparison, the exploded view around EGUT of all evolutions together, when a3 ( M Z) takes the three values a?* ( M Z) and u r A(Mz )& 2 2u.
4. - Discussion and conclusions.
It is interesting to compare the results of our present work with other studies of ours [6,11]. We started [6] to investigate how MsusYand EGUT varied as a function of a 3 ( M z ) and of the different possible solutions (four) of the coupled evolution equations for the gauge couplings. The results are reported in figs. 14-17. The interest of these figures is the clear anticorrelation between MSUSY and E G U T . In fact, for all solutions, increasing the value of a 3 ( M z ) decreases Msusy but pushes E G U T towards higher values. This work was based on the naive trend of ignoring many problems in which our group has been and is presently engaged, such as the effect of the evolution of the gaugino masses (EGM) [lll. The conclusion of our work [6] was that, within -+2u, the SUSY breaking could occur in a very wide range: from GeV to PeV.
80 1040
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
7
Fig. 14. - Fitted Msusv and EGUT as functions of as(MZ) according to the <
A 16.5 t
7
GeV)
Fig. 15. - Fitted Msusv and E,,, of [6].
as functions of a g ( M z ) according to the 4'ormula 1,,
81
1041
THE CONVERGENCE OF THE GAUGE COUPLINGS ETC.
7 FORMULA 2
:I-
I
@ a,(M,)=O.l @ a,(M,)=U I
* T -
I
-
18 (WA)
134 (WA-I-2
\I
3 -
Fig. 16. of [GI.
102 (WA-2
a,(M,)=O.
0
,
Fitted Msusv and EGu, as functions of
,
,
I
a3(MZ)
I
,
,
,
according to the <(Formula2>,
7
FORMULA 3 6 -
0
e% oc,(M,)=U
I 1 8 (WA)
5 -
@ a,(M,)=O
134 (WA+2o)
1 0 2 (WA-2o)
a,(M,)=O
4 3 -
2 -
1 -
'15'
"
"
"
"
I
'
"
'
Fig. 17. - Fitted M s u s ~and EGUTas functions of a 3 ( M z ) according to the <
82 1042
F. ANSELMO, L. CIFARELLI, A. PETERMAN
/
NUMERICAL SOLUTION
0"
'
'
'
'
0.1 1
'
'
'
'
'
0.12
'
'
'
'
'
0.13
a&)
Fig. 18. - Msusy vs. ag(Mz), using the .Numerical Solution. of [6].
7 6
5
4 3
2
1
0
0 11
0 12
%(MJ
Fig. 19. - Msusv vs. ag(MZ), using the <(Formula2. of [6].
0 13
'
1
and
A. ZICHICHI
83
THE CONVERGENCE O F THE GAUGE COUPLINGS
1043
ETC.
In the present paper we follow a completely different approach. Starting from and using a correct treatment of the degenerate heavy Mx threshold, above, at E we study the compatibility of our model with MsusYbeing as low as possible. We find that MsusYcould very well be in the 102GeV range. Let us summarise these three contributions to the same problem: 1 - In [ 6 ] the main point was to work out a correct estimate of the range where MsusY could be by: i) studying four possible solutions of the all a z , a3 coupled evolution equations, including a new ((direct,, numerical solution; ii) using the world average for a 3 ( M z ) and its k 2 0 values. 2 - In [ l l ] the main point will be to introduce the evolution of the gaugino masses. 3 - In the present work the main point is to derive the consequences of the superheavy threshold effects below EGUT , with pure Supersymmetric evolution (*), It is interesting to find out that these different approaches towards an understanding of where Msusy could be, all show that Msusy could be where LEP-I and the Tevatron are working, or in the forthcoming accessible range of energies where HERA and LEP-I1 will soon operate.
A P P E N D I XA The simplest threshold function which we used to perform our calculation is a fitted function which minimises the properties of the genuine threshold functions. We give it as an example of what is indicated by F'" in formulae (1) and (2):
In this simple case we have consequently:
Now, the genuine threshold functions are not the simple logarithm g.:en in (A.2). More generally the F's are of several types: HG , H F, HHaccording to their connection with gauge, fermion, Higgs particles. We have followed a technique sketched by Paschos [12]. As an example we give Paschos' HG and its contribution to F'": (A.3)
HG( Q 2 ,
1= 1 t 2M:/Q2
+v
1 + 2M$/Q2 -
(*)
q
m
See [5] for a similar study of the Standard Model without Supersymmetry.
w
)+ $1.
84 1044
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
Then
P ) ( Q ; , Q;, it421
(A.4)
M:)
- H~(Q;,
M:).
In this way we may accommodate a non-degenerate spectrum of gauge bosons. The other varieties of H can be treated similarly (for the fermionic part see [13]). As a final remark, since we do not know the two-loop mass dependent p-functions which are needed in order to formulate the threshold behaviour a t two loops, the formula (A.2) can be used safely with a F(') given by (A.1). At the usual level of precision the use of (A.l) or more sophisticated functions like (A.4) is in general undistinguishable.
REFERENCES
111 J. ELLIS,S. KELLEYand D. V. NANOPOULOS: Phys. Lett. B, 249, 441 (1990). [23 P. LANCACKER and M. LUO:preprint UPR 0466-T. DE BOERand H. FURSTENAU: Phys. Lett. B, 260, 447 (1991). [3] u. AMALDI, preprint CERN-PPE/91-190, November [4] U. AMALDI, W. DE BOER and H. FURSTENAU: 1991. [5] D. A. Ross: Nucl. Phys. B, 140, 1 (1978). A. PETERMAN and A. ZICHICHI, preprint CERN-PPE/91-123, [6] F.ANSELMO,L. CIFARELLI, 15 July 1991 and Nuovo Cimento A, 104, 1817 (1991). Note that for notations, formulae and other details not discussed in the present paper, we refer the reader to this work. [7] H. GEORCIand S. L. GLASHOW:Phys. Rev. Lett., 32, 438 (1974). [8] M. B. EINHORN and D. R. T. JONES: Nucl. Phys. B , 196, 475 (1982). [9] M. E . MACHACEK and M. T. VAUGHN: Nucl. Phys. B, 236, 131 (1984). [lo] For lectures and references on Supersymmetry, Supersymmetric theories and, in particular (Plenum Press, New York-London, SUSY-SU(5) see: Superworld I , edited by A. ZICHICHI 1986); Superworld ZZ, edited by A. ZICHICHI(Plenum Press, New York-London, 1987); Superworld ZZZ, edited by A. ZICHICHI(Plenum Press, New York-London, 1988). [ I l l F. ANSELMO, L. CIFARELLI, A. PETERMAN and A. ZICHICHI:The Evolution of Gaugino Masses, work in progress. [12] E. A. PASCHOS:preprint DO-TH 80/19. 1131 A. PETERMAN: as( & ' ) Extrapolation with Analytic Thresholds, to appear as CERN-TH preprint in 1992.
w.
85
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi
THE SIMULTANEOUS EVOLUTION OF MASSES AND COUPLINGS: CONSEQUENCES ON SUPERSYMMETRY SPECTRA AND THRESHOLDS
From
I1 Nuovo Cimento 105 A ( 1992) 1179
I992
This page intentionally left blank
87
IL NUOVO CIMENTO
VOL. 1 0 5 4 N. 8
Agosto 1992
The Simultaneous Evolution of Masses and Couplings: Consequences on Supersymmetry Spectra and Thresholds. F. ANSELMO (I), L. CIFARELLI (l) (2) (3), A. PETERMAN (l) (4) (I) CERN - Geneva, Switzerland (') Physics Department, University of Pisa - Pisa, Italy (3) (4)
(5)
P) and
A. ZICHICHI(l)
INFN - Bologna, Italy World Laboratory - Lausanne, Switzerland Centre de Physique The'orique, CNRS-Luminy - Marseille, France
(ricevuto il 2 Aprile 1992; approvato 1'8 Luglio 1992)
Summary. - We use the Renormalization Group Equations to work out, at the one-loop level, the simultaneous evolution of all masses and couplings and show explicitly the self-consistency of the whole scheme. A thorough examination is performed of the light Supersymmetry threshold in the Minimal Supersymmetric extension of the Standard Model (MSSM). All fundamental quantities a G W , M,,,, a3(mZ), az(mZ),a l ( m z )(consequently sin' e(mz)) are given in terms of the detailed spectrum of all particle and sparticle thresholds. Examples of Supersymmetry spectra are given as function of a3(mz)and of the other essential parameters. The results of this study, where the evolution of masses is extended to all possible masses, confirm om previous conclusions on the EGM effect for the Supersymmetry threshold lower bound. Examples of the predictive power of our method are given. PACS 11.30.Pb - Supersymmetry.
1.
- Introduction.
This work is in the line of o w previous study[l] on the light threshold for Supersymmetry breaking[2], where a range for the primordial parameter mtjz was derived by comparing the experimental value sin2 e(mZ)- with sin2 O(mZ Ith, z.e. the two-loop-corrected minimal SUSY-SU(5) theoretical prediction, without threshold effects. The difference between these two quantities was then accounted for by the light threshold contribution, thus establishing, by means of the light spectrum, the range of energy where ml/2 should lie. In our previous work[l] we introduced the EGM effect, i.e. the evolution of the gaugino masses, in the light threshold study for SUSY breaking. Here we extend the light threshold effects to all unification parameters like MGUT and aGUTand obtain entirely analytic solutions for the one-loop evolution equations of the gauge 1179
88 1180
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
couplings. The two-loop analytical formulation of threshold effects is being completed with a single discontinuity in the two-loop p-function coefficients b, at ml12[3]. The structure of this paper is as follows. In sect. 2 we give the analytic formulae for MGuT and aGm with the detailed contribution of the light threshold. Then the evolution equations for a l , a 2 , a3 down to the Z-mass value (mZ) are given, starting from MGm, with due account of the complete light SUSY spectrum. Consistency checks of the whole formalism are given. In sect. 3 we indicate the level of approximation used when dealing with the evolution of the various parameters. We follow the proposal of Ellis et al.[2], for the mass degeneracy of all squarks except stops: &, &. The main points where we depart from[2] are: i) to account for the evolutions of the gaugino masses, as already done in[l], and to extend these evolutions to all masses; ii) to relax the fixed input values for both the higgsino and the Higgs masses, mi; and mH, respectively; these are let to evolve according t o known equations[4]; furthermore we transfer the inputs for mi; and mH into more general primordial parameter ratios. In sect. 4 we summarize the results obtained by our group about the SUSY threshold problem and discuss the trends which emerge from our studies. In sect. 5 we present the conclusions. 2. - Formulae for masses.
aGm,MGm and
the simultaneous evolutions for couplings and
We consider a spectrum where all Standard Model particles are included with the addition of a second Higgs doublet (H) and the Supersymmetric partners of the Standard particles. This spectrum, in spite of its complexity, is somewhat simplified as we treat all squarks in the same way, with an exception for the left stop and the right stop. The sleptons, left and right, are obviously included. The simplification in the squark and slepton spectra follows Ellis et al. [2], who introduced this treatment in computing the threshold contribution to sin2 6(mz). The estimate of this contribution by Ellis et al.[2] is confirmed in our work for the whole range of evolution of the squark and slepton masses. Since the squarks and sleptons contribute t o sin2 6(mz>by less than 5.10-5, to neglect them in our previous study[l] when computing the EGM effect on the SUSY lower bound was obviously justified. Moreover this study[l], based on a single discontinuity threshold, is confirmed by our present exhaustive iterative method involving all evolutions of couplings and masses. Our previous approximation is therefore numerically confirmed. Let us remind the reader that the key point of our work is to derive a range for the primordial SUSY parameters by comparing [2] the experimental value sin26(mZ)exp with the two-loop-theoretical calculation based on the minimal SUSY-SU(5), without threshold effects: sin2 6(mz)n= 0.2
+
7a2
(mZ)
1-54? (mz1
+ 0.0029.
This is the starting point for all predictions on where the SUSY spectrum could be. These predictions, started by Ellis et al. [2], were improved by the introduction of the EGM effect[l], and are now worked out through a numerical method which automatically adjusts the whole light SUSY spectrum in such a way that the light
89 THE SIMULTANEOUS EVOLUTION OF MASSES
ETC.
1181
threshold contributions, ATL, to sin2 e(mz)make the theoretical prediction coincide exactly with the experimental value. Notice that the present method would be straightforwardly modified if a light sparticle were found below the Z mass. The solutions of the evolution equations for aGUT and M G W , modified by the threshold effects, together with all couplings and masses involved, have been worked out at the one-loop level with the appropriate spectrum of thresholds. The detailed formulae grouped into and .masses. are as follows:
1
(2)
1 1 - al(p)
aGUT
MGUT
49
27r
10
(3)
1 -a2(p)
(4)
GUT
[
+log ( M ; m ) * -log( 27r
1 - -+ a3(p)
GUT
2x
[
(
?)* -
-31og M;m)*-2lOg(3*-
90 1182
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
where log (2)*= 8(2 - 1)*log(2),
+ 19 ~ j 2~ l o g ( ~ ) - ~ l o g ( ~ ) + g l o g ( ~ ) -
where:
91
1183
THE SIMULTANEOUS EVOLUTION OF MASSES ETC.
wl = he-structure constant of squarks,
at =
ht2 = top-Yukawa coupling, 4x
-;.-i.
T = l o g ( M8UT
Three are the primordial masses, mlj2, m, and m4, whose role, at follows: ml12 parametrizes the gauginos; m, parametrizes squarks and sleptons; m4 parametrizes the Higgsino; m, and m4 parametrize the Higgs. 77 - I1 Nuovo Cimento A
MGW,
is as
92 1184
F. ANSELMO, L. CIFARELLI, k PETERMAN
and k
ZICHICHI
In the course of the evolution, squarks, sleptons, Higgs and Higgsino get additional contributions parametrized also by ml12. Notice that, in the degenerate case when all mi= mlI2, we find our previous formulae for M G and ~ aGm [l]:
A self-consistency check reads as follows: inserting for instance (1) and ( 5 ) into (3) for p = mZ, we get:
or
with
and
.[
5 2) + 2)+ ; 2) 2)+ ( 2 ) (2;) ( 2 ) 7 (:)-3,2 log(
-log
-
log(
- -410g
log(
- 3 log(
- -310g - + - l o g
-
2 log(
2)
-log
(31
-
- ,
which is our formula (10) in [l]. Another complementary test can be obtained inserting (1) and (5) into (2) for p = = mz. One gets, with obvious notations:
Formulae (2)-(4) and (8)-(26) (or their equivalent) can be found in the literature (in[4] and references therein). The others are our responsibility. 3.
- Primordial
ratios and mass spectra.
To study the full evolution of all Supersymmetric masses is the main task of this paper. As we have already shown[l], the evolution of the gaugino masses (EGM) has
93
1185
THE SIMULTANEOUS EVOLUTION O F MASSES ETC.
important effects. For example it lowers by more than two orders of magnitude the value of the primordial mass, m1/2,predicted without the EGM effect [2]. In contrast with the case of the gauginos, the evolution of squark and slepton families has practically no effect. This was already implicit in[2] and the reason for it was discussed in[l]. These features are by now well known and need not be further discussed. The only relevant members of the squarks to be kept in the evolution equations are the left and right stops: iL,iR. These squark states have their evolution partly dictated by the top-Yukawa coupling, which is expected to be potentially large. This coupling is the one which drives negative the squared mass of the Higgs, thus producing the electroweak symmetry breaking. If this scenario has to be considered seriously, it may happen that for large primordial and/or m4, which determine the Higgs mass at M c W , one needs a large top-Yukawa coupling. It becomes therefore important to check that, in the case of the stop-squarks, the Yukawa-driven part will not win over the gauge contribution to their evolution. If this were the case, colour breaking would occur because the Yukawa part would make negative the mass squared of the squarks. This aspect of OLE study t o predict the mllz mass must always remain under control.
si n%( rn,) =0.2334
aJ(mZ)=O.I I 5
10 ............................. .................................................................................
................... . . . . . . . . . . . . . . . . . . . . . . . . . . 1
...
n
> Q,
...................... ..................... .................... .................... .....................
I0
.................... ..................... + - - - - -.-. ................
c3
W
L .....................................................
10
......................
..................... ......................
..................... ...................... ..................._ ..... ............................... ..................... .................... .......................................................... ..................... .......................................................... ..........., ......................................................
1
i .
.......
r ' . .......
I
I
I
I I Ill
I
I
I
I I I I I
I
I
I 1 1 1 1 1
I
I
I
I I I I J
1 o2 Fig. 1. - The value of the W-in0 mass, mw, vs. the ratio of the primordial SUSY breaking parameters (mo/ml12)2. The other parameters a3(mZ),sin2O(m,) and (m4/m,lz)2 are fixed as and lo2, have been chosen for the latter. indicated. Two extreme values,
94
1186
F. ANSELMO, L. CIFARELLI, k P E T E R W
and
A. ZICHICHI
Having accounted for the evolutions of gaugino masses [ll, the other delicate point is to relax the fixed input values for mi;and mH,and to use new inputs, namely the primordial mass ratios:
1
I
I
I
I
I
I
I
I
I
I
I
I
O*
I
1o-2
I
J
1
0.231 8
I
si ~AY(m),
0.2350
1
Fig. 2. - The value of the W-in0 mass, ma, vs. aj(mz),sin2O(mz)and (m4/m,i2)2. Each curve is the result of our iterative system of seventeen evolution equations (see sect. 2). The curve showing the variation of ma, vs. aS(mZ)), or sin2 f3(mz), or (m,/m,12)2,is computed keeping the other parameters futed at the values indicated by the arrows. For the experimentally measured values, a 3 ( m ~and ) sin2 O(mz),the world averages have been chosen. For the ratio of primordial masses the value (m4/m,lz)2 = 1 has been taken. The top mass is kept at m, = 125 GeV. Please note that our analysis is valid only above mz. The ma values shown below this limit are for illustrative purposes.
95 THE SIMULTANEOUS EVOLUTION OF MASSES ETC.
1187
and
This does not introduce any new additional input, as we know, at any energy, the connection between mi;and m 4 ,and also between mH and m,, m l f z ,m 4 .In fact the evolutions of mi;and mH are given in terms of the primordial mass ratios (27) and (28).
cx,(m,)
= 0.1 18 (WA)
sin*$(m,)
=
0.2354 (WA)
mo/m1,2
= 1
m4/m1,2
= 1
480
n
> Q)
c3 W
E
Fig. 2a. - The value of the W-in0 mass versus m, for given values of the other inputs, as indicated. This is the only case (mw)where the dependence on m,is shown. We do not show the results for m,, mc, mi.
96 1188
F. ANSELMO, L. CIFAFLELLI, A. PETERMAN
and
A. ZICHICHI
The range where mlp varies is very sensitive to m4. If the value of mL is taken fixed at 100GeV, this corresponds to a value of m4 of the same order of magnitude. It seems to us not very consistent - as some authors do - to keep m4 f i e d at low values even if, as a result of the analysis, ml12turns out to be very large. However there are no compelling arguments to prevent mh from being small. Often in the literature high values for mK and mH are limited by arguments based on <
- m; = f(~J(mZ>> - - mc = f(sin*+(m,)) -.10
............
n
> Q)
...........................................................
10
c3
...........
W
10
10
0.1 I I
0.23 18
0.1 3 0.2.33 4
I
I
sin2,$(mJ
Fig. 3. - Same as fig. 2 when the gluino mass, m ~is,computed.
0,2350
97 1189
THE SIMULTANEOUS EVOLUTION OF MASSES ETC.
The first seventeen formulae reported in sect. 2 represent the system of coupled non-linear equations for couplings and masses which describe in great detail the evolution of particles and sparticles interacting via SU(3) x SU(2) x U(1) gauge forces. In the system of coupled equations (1)-(17) we have neglected the heavy threshold, ATH, ie. the spectrum associated with the Supergroup breaking into SUSY-[SU(3) x SU(2) x U(l>] at very high masses. This threshold can be described in terms of three basic masses [5] (Mv, M H, Mx). Our method (*) allows to treat in detail ATH and its influence on the SUSY spectrum[61.
- ma = f(4mz)) 0
0
-.-
mg = f(sinz$(mz)) m; = f((m4/m,,J2)
................. .............................. .................. .............................. .. .................. . . . . . . . . . . . . . . . . . .............................. .............................. .. . . . . . . . . . . . . . . . . . ............................. .. . . . . . . . . . . . . . . . . . . .............................. . . ................ .. . . . . . . . . . . . . . . . . . . .............................. . .
10
_
1.
- ............................. n
> a,
10
..
(m,/m,,J2i= 1 ............................. ~.
.............................
............................. ..............................
............................. ............................. ................
,... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... ..-. .. .. .. .. .. .. .. .. ..
0
-
W
-. . -. .:'
10
........... . . . . . . . . . . . .%*. ....
L. . . . . . . . . . . . . . . . . . .
:.. . . . . . . . . . . . . . . . . . . . . . . . . .
.\.
............
......
-................... .: . . . . . . . . . . . . . . . . . . .
10 2
1
1
I
I
0.1 1
L
0.231 8
I
I
\1/
0.2334
I 1o-2
0.13
I 0.1 2
1
s in2?P(mJ
0.2350
1
Fig. 4. - Same as fig. 2 when the squark mass, m,, is computed. (*) Using a simple treatment of the heavy threshold (only one heavy mass) we have studied the a l , az, a3 evolutions [7] in order to have a smooth convergence towards M,, and a unified
evolution of the couplings above MGm. In this evolution [7] no light threshold effects were included.
98 F. ANSELMO, L. CIFARELLI, A. PETERMAN and A. ZICHICHI
1190
To the system of coupled equations (1)-(17) we apply our iterative procedure[l] in order to work out the quantitative results wanted. The first of these results is that the EGM effect on the SUSY threshold lower bound[l] is confirmed, i.e. the lower bound on m112is in the Z-mass range. The next result refers to the spectrum of SUSY particles. For example let us focus our attention on the value of the SUSY threshold for a typical sparticle, such as the W-ino. In the following we ignore particle mixing. The first remark is that the W-in0 mass is insensitive to the ratio (mo/ml,2)2 as shown in fig. 1. Over four orders of magnitude for this ratio, from to lo2, the W-in0 mass slightly decreases by no more than 20%. On the contrary - as shown in fig. 2 - the W-ino mass is very )2. The range of variation for sensitive to a3 (mz), sin2 e(mZ) and to the ratio (m4/ml12 the measured quantities a3(m2) and sin2Nm,) has been taken in the k 2 0 interval around their world average (WA) values, while for (m4/mIl2)'we have computed the W-in0 mass letting this ratio vary by four orders of magnitude: from t o lo2. In fig. 2, each curve is computed keeping the other variables constant. The fixed values are indicated by the arrows in fig. 2. For example the W-in0 mass us. a3 (mz) is computed keeping sin2 e(m,) at its world average value (0.2334) and the ratios (mo/ml12)2 = (m4/m112)2 = 1. It is interesting t o observe that - as shown in fig. 2 -
a,(m,)=O. 1 1 5
10
.
s in'a( m,) =0.2 334
(m,/mr,2)2 = 10' .......
iiQ-5
...............................................................................
..................
...............
.........
.//
.......
1o2 Fig. 5. - Same as fig. 1,when the slepton case is worked out. Note that the left (upper curve) and right (lower curve) slepton masses are very close.
99
1191
THE SIMULTANEOUS EVOLUTION OF MASSES ETC.
the W-in0 mass decreases with increasing values of a 3 ( m z ) , sin26(m,) and (m4/mlI2)'.All these results are obtained with m, = 125 GeV. We have also studied how the W-in0 mass varies with m, in the range (90 S m, S 180>GeV,once all other parameters are kept fixed as indicated in fig. 2a. We have then computed the detailed behaviour of the gluino, squark and slepton masses. For the gluino mass we find again that it is insensitive to the ratio (mo/ml,z)2 (range studied: from lo-' to 10') while it decreases (as for the W-in0 case) with )'. The results are reported in increasing values of a3 ( M , ), sin2 6(Mz) and (m4/ml12 fig. 3. The squark mass is found to be insensitive to the ratio (mo/m,12)2 in the range to 10'. From 10' to 10' it increases by less than an order of magnitude. The from
-
m L R
= f(al(mZ))
--
mxLR= f(sinz$(mz))
-*-
mILR
=
f((m4/ml/,)2>
10 ................................... .................................
...............................
n
> a,
..................
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ., .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ................................. ................................... ................................... . . . . . . . . . . ,. . . . . . . . .
10
c3
W
a
....................
-.-,
1
: -0
.
-.
............... 15.................... .r( .................... ......
10
....................
.................... ....................
.........
10
0.1 I I
0.23 I a
0.13 0,2334
I
J
sin2$( rn,)
0.2350
Fig. 6. - Same as fig. 2, when the slepton masses are computed. The left (upper curve) and right (lower curve) slepton masses almost coincide.
100
1192
Mass/-
F. ANSELMO, L. CIFARELLI, k PETERMAN and k ZICHICHI
PREDICTED MASS SPECTRA World averages for the experimentally measured quantities: a3(mZ)WA= 0.1 18 f 0.008 sin28(mZ)WA= 0.2334 l/ae,(mZ)WA
f 0.0008
= 127.9 f 0.2 ~~~~
Ratios of primordial parameters:
I
* Fig. 7. - The predicted SUSY mass spectrum for three cases, when the measured quantities (aa(mZ), sin20(mz), aem(mZ))are taken a t their world average values and at k2u. The ratios of the primordial parameters are kept equal to one. Note the large range where the SUSY spectra could be, on the basis of our best experimental and theoretical knowledge.
101 1193
THE SIMULTANEOUS EVOLUTION OF MASSES ETC.
squark mass is very sensitive to a 3 ( M z ) ,sin' e(mz)and (m4/m1/2)2: it decreases with increasing values of a3(mz),sin' e(mz) and (m4/m112)2. The results are shown in fig. 4. The slepton (left and right) masses are more sensitive than the squark one to the /mllz)'for the obvious reason that they carry no colour. The slepton masses ratio (mo increase when the ratio (q,/mll2)'increases: in the range from lo-' t o 102, by one order of magnitude, as shown in fig. 5. On the other hand the trend versus a3 (mZ ), sin' O(mz)and (m4/ml12)' is like the W-ino, gluino and squark cases: the slepton (left and right) masses decrease with increasing values of a 3 ( m z ) , sin' O(mz) and (m4/mllz )', as shown in fig. 6. This is the effect of the coupled evolutions. These detailed studies are the basis for our overall analysis of the SUSY-mass spectra shown in fig. 7, where for the primordial masses the value (mo/mll2)'= = (m4/ml12 )' = 1 has been chosen. The spectra have been worked out, taking for the experimentally measured quantities, a3(mZ ), sin' O(mZ), aem(mz), three values, ie. the world average and the corresponding &2u values [%lo]:
* 0.008, sin2 O(mz)WA = 0.2334 * 0.0008, l/aem(mZ)wA = 127.9 * 0.2. a3 (
m)wA ~ = 0.118
SUSY PARTICLE MASS SPECTRUM
a&)
= 0.123
l/a,,(rnz) = 127.9 sin2O(rnz) = 0.2334
rn,= 125 GeV n
L 100
mH m -
'R
mfima rnp
I1 300
ieV
Fig. 8. - A detailed spectrum of SUSY particles showing that presently existing colliders (LEP-I and Tevatron) and forthcoming ones (HERA and LEP-11) have the same chance as future colliders (SSC and LHC) to discover SUSY.
102 1194
F. ANSELMO, L. CIFARELLI, A. PETERMAN
a3(m,)=0.1 1 5
sin%(m,)=0.2334
and
A. ZICHICHI
m, = 160 GeV
// .
.
.
:..
. ...:
..
.
.. ....
- . . . . . .. . . . I
,
. . _ _. . . . . . . . , . . _
.....___.
2.5
2
'1 .5
'1
Fig. 9a. - Three-dimensional plots of the primordial parameters m1,2, m,, m4for a,(mz) = 0.115. For sin2 O(mz) and a,,(mZ), the world average values are used. The top mass is kept at m, = 160 GeV.
It is interesting t o remark that the results of these detailed studies confirm our first conclusion[8] - obtained using only a single MsusY- namely that within the minimum confidence level of our knowledge, ie. &2u, the SUSY particle masses can be in the very wide range which goes from mZto PeV. A more detailed study of the SUSY spectrum shows that the mass sequence between all members (%, g, 4,i) changes when the ratios (mo/ml~z)', (m4/m1/2)' change. The W-in0 is the lightest member of the spectrum for the example shown in fig. 8. We have investigated the effect on the W-in0 mass when the experimentally measured quantities are allowed to take values within the +2u range. The most important quantity is a 3 ( M z ) . From [a3( M , )wA - 2G] t o [a3 + 2u] the W-in0 mass changes by more than four orders of magnitude. For sin' O(Mz), the W-in0 mass variation is less than two orders of magnitude. For aem( M z ) the corresponding effect is a variation of about 25%. These results remain practically unchanged when we allow the ratios of the primordial
103 ~~
~
THE SIMULTANEOUS EVOLUTION OF MASSES
a3(m,)=0. 1 18
ETC.
1195
m, = 160 GeV
sin*dJ(m,)=0.2334
...__
4.8 4.4
. . . .
4
, . . .
5.6
3.2 2.8 2.4 2 3
2.5
2
1.5
0.5
(y
+
Fig. 9b. - Three-dimensionalplots of the primordial parameters mIl2,motm4 for ag(mZ) = 0.118. For sin2O(m,) and a,,(mz), the world average values are used. The top mass is kept at m, = 160GeV.
masses, (m,-,/ml12)2 and ( ~ ~ 4 / m , / zto ) ~ ,go from l o p 2 to 10' (see table I). The quantitative details concerning the W-in0 mass are as follows: if we allow all parameters to change within ? 2a [for the experimentally measured quantities, a3(mZ), sin2 O(mz),a,,(mz)] and within and lo2 [for the ratios of the primordial masses, ( q , / m l 1 2 ) 2and (m4/m1/2)2],the maximum value for the W-in0 mass is 4.2PeV and the minimum is below mZ (although we should recall that our formulae are only valid above the Zo mass). The search for SUSY particles should therefore be encouraged where right now there are colliders at work, i.e. LEP-I and Tevatron. Moreover, the forthcoming HERA and LEP-I1 are t o be considered excellent tools t o search for SUSY, whose mass spectrum (see fig. 7) scans nearly an order of magnitude, once the lightest member of the sparticles has been identified. For the sake of completeness we show in fig. 9a-d how the primordial parameters m o ,m1/2,m4 are coupled together for different values of aS(mZ): 0.115, 0.118, 0.123 and 0.125. The value chosen for sin2 O(mz)is the world average and for the top mass
104 1196
F. ANSELMO, L. CIFARELLI, A. PETERMAN
a3(m,)=0.1 23
sin%(m,)=0.2334
and
A. ZICHICHI
mt = 160 GeV
3.6 n
2
5.4
5.2
U
n
5 2.8
E 2.6 2.4 0 - 2.2 2
Fig. 9c. - Three-dimensional plots of the primordial parameters mlI2,m,, m4for a3 ( m Z )= 0.123. For sin2 O(mz)and a,(mz), the world average values are used. The top mass is kept at m, = 160 GeV.
m, = 160GeV. These figures 9a-d synthetize the predictive power of our iterative method, in terms of primordial masses, while fig. 8 gives an example of this power in terms of SUSY mass spectrum. Notice that, in the example shown, the W-in0 is the lightest particle. We emphasise the relevance of this spectrum: it has been computed using the best of our experimental and theoretical knowledge. It is as reliable - and in fact by far more reliable - than other claims towards higher SUSY mass values which could be reached only when the new generation of colliders (SSC or LHC) will be available. Let us repeat: the search for SUSY needs not to wait for the new generation of colliders. Those available now (LEP-I, Tevatron) and in the near future (LEP-11, HERA) are in the energy range <<predicted.by the best of OUT present knowledge. In this connection it is interesting to discuss the trend which emerges from ow various approaches to this key problem of finding where the SUSY threshold could be.
105
1197
THE SIMULTANEOUS EVOLUTION OF MASSES ETC.
a3(m,)=0.1 25
/q-
sin%(mZ)=0.2334
m, = 160 GeV
.........................................................................
3
5 a, 2.8 (3 U n
> h(
2.6
E 2.4
c
W
E!
m 0 2.2 -
2
Fig. 9d. - Three-dimensional plots of the primordial parameters m1/2, mo, m4for q ( m z )= 0.125. For sin2@(mz) and aem(mZ), the world average values are used. The top mass is kept at m, = 160 GeV.
4.
- Discussion
on the trends.
Let us start recalling a few definitions: ATH:
ATL:
EGM:
indicates that the heavy threshold, when SUSY-SU(5) is broken into SUSY-[SU(3) x SU(2) x U(l)], has been taken into account (using only one single heavy mass). indicates that we have taken a detailed SUSY spectrum to account for the threshold effects when SUSY-[SU(3) x SU(2) x U(l)] is broken into either SU(3) x SU(2) x U(1) or SU(3) x U(l>,. indicates that the simultaneous evolution of the couplings and of the gaugino masses has been worked out.
106 1198
F. ANSELMO, L. CIFARELLI, A PETERMAN
and
A. ZICHICHI
TABLEI. - The variation of the W-in0 mass corresponding to 2 25 variation of the experimental inputs, a3(mz), sin' e(mz),ae,(mz), with respect to their world average values, and to the variation of ( ~ / m I l 2and ) ' (m4/m112)2 in the indicated mnge. The duta are o!erivedfrom our system of seventeen coupled evolution equations (see sect. 2). The dominant effect i s clearly due to a3(mz). Please note that our e q u a t i m are valid only above the Zo mass. Nevertheless the results in t e r n of mw ratios are given, for illustrative purposes, even when the + 20 limit of the experimental inputs pushes mw below mZ.
mw (a3(mzlWA- 20.) = 2.104 mw (a3(mz)WA + 2u) IWA values for the experimental inputs: sin2e(mz),a m ( m z ) ]
mw (sin' e(mz)WA - 20) mw (sin' e(mZlwA + 26)
= 20
-+ 50
WA values for the experimental inputs: a3(mz), aem(mz)] mw (I/am (mzlWA - 20) = 1.2 + 1.3 mq ( I / a m (mz)wA + 20) IWA values for the experimental inputs:
a3(mz),sin2
e(mz)l
Mowed range of variation for primordial SUSY breaking mass ratios:
EGM*: indicates that the analysis has been done working out the simultaneous evolution of the couplings and of all the masses (gauginos, squarks, Higgsino, etc.)
A:
stands for accuracy. There are in fact cases where we have worked out identical quantities using one-loop and two-loop approximations. The purpose being to find out the trend when the theoretical accuracy is improved. Here are the main results from our group in order to contribute understanding the
SUSY problem: I. In our first paperr81 we have introduced a numerical solution t o the coupled differential equations (not to an approximate analytical solution of these equations) for a l , a2, a3 evolutions. We have used only one step function for SUSY breaking and no A T L , no ATH,no EGM. We have compared one-loop and two-loop results. We find that, when A improves, the value of MGm diminishes. On the other hand, we also find that when the value of a 3 ( m Z ) increases, the value of MGm increases and the value of M s u s ~decreases. 11. In a second step[l] we have introduced an iterative method to take into account the evolution of the gaugino masses: the EGM effect. In the present work we
107 THE SIMULTANEOUS EVOLUTION OF MASSES
ETC.
1199
BASIC TRENDS
Fig. 10. - The trend emerging from our three approaches, as discussed in sect. 4. When the theoretical accuracy increases, A ( '! ), high values of a3(mz) are preferred: a3(mz)( '! 1. When a3 (mz)increases ( ? ), MGbTincreases ( ? ) and M s u s decreases ( ). These trends have been studied allowing the measured quantities to go from the minimum (world average minus 2 ~ to) the maximum (world average plus 2 4 values.
extend the simultaneous evolutions of couplings and masses t o all other masses (EGM*); ATL is in, not AT,. The formulae are at the one-loop level. We find that increasing values of a 3 ( m z ) produce higher values of MGUT and lower values of MsusY *
111. Our third approach[7] is based on a straightforward comparison between one-loop and two-loop calculations, using for ATH only one single heavy mass in the 1016GeV range. The only input is a3(mz). The purpose of this approach was to see what the theoretical accuracy prefers as values of a3(mZ) once a 2 is defined in terms of the measured quantities sin2 O(mz)and n,,(mz>. The results are that increasing accuracy prefers a 3 ( m Z ) values on the high side (ie. above the world average) and MGm values on the low side. On the other hand it is found that higher values of a3(mZ)increase M G m and decrease Msusy. A synthesis of the trends is shown in fig. 10. A remark is in order: the fact 78
- I1 Nuovo Cimento A
108
1200
F. ANSELMO, L. CIFARELLI, A. PETERW
and
A ZICHICHI
that the proton lifetime limit is high ( T > 5.5~ lo3'~ years) ~ indicates that MGm cannot be too low. But our study on the trends indicates that high values of MGm imply low values for Msusy. In fact a possible scenario for the SUSY light threshold has been shown in fig. 8. 5.
- Conclusions.
The exhaustive treatment of the light threshold confirms the results obtained with the EGM on the SUSY-threshold lower bound. Our iterative method has been shown to be very powerful in predicting SUSY spectra as a function of the experimentally measured quantities a3 (mZ1, sin' 8(mZ), aem(mz ) and of the ratios of the primordial parameters mIl2, rq, and m4. From our approaches to the SUSY problem we conclude that the following trends appear: 1. When the theoretical accuracy (from one loop to two loops) improves, the values of a3 ( m Z) are preferred to be on the high side (ie. above the world average), and the values of MGuT on the low side (20% decrease). 2. Higher values of a 3 ( m z ) are correlated with higher values of MGUT (formula (8)). 3. Higher values of MGm are correlated with lower values of Msusy (formulae (5) and (20)). 4. Higher values of a3 (mz ) are correlated with lower values of Msum (formulae (6) and (7)).
At present, increasing theoretical accuracy favours a3 ( m Z) t o be on the high side. Consequently Msum is lowered. The experimental searches for SUSY particles at LEP-I, at the Tevatron and in the forthcoming HERA and LEP-I1 colliders should be firmly encouraged.
REFERENCES [ l ] F. ANSELMO,L. CIFARELLI,A. PETERMAN and A. ZICHICHI:preprint CERN-TH.6429/92; Nuovo Cimento A, 105, 581 (1992). [2] J. ELLIS, S. KELLEYand D. V. NANOPOULOS: Nucl. Phys. B, 373, 55 (1992). [3] F. ANSELMO, L. CIFARELLI,A PETERMAN and A. ZICHICHI: preprint CERN.TH.WB2; Nuovo Cimento A, 105, 1201 (1992). [4] A B. LAHANAS and D. V. NANOPOULOS: Phys. Rep., 145, 1 (1987). [5] R. BARBIERIand L. J. HALL: Phys. Rev. Lett., 68, 752 (1992). [6] work in progress. [7] F. ANSELMO, L. CIFARELLI,A. PETERMAN and A. ZICHICHI:Nuovo Cimento A, 105, 1025 (1992). [8] F. ANSELMO, L. CIFARELLI,A. PETERMAN and A ZICHICHI:preprint CERN-PPE/91-123, 15 July 1991; Nuovo Cimento A, 104, 1817 (1991). Please note that for notations, formulae and other details not discussed in the present paper, we refer the reader to this work. [9] P. LANGACKERand M. Luo: Phys. Rev. D, 44, 817 (1991). [lo] G. BURGERS and F. JEGERLEHNER: 2 Physics at L E P I , editors G. ALTARELLI,R. KLEISS and C. VERZEGNASSI: CERN Report 89-08, Vol. 1 (1989), p. 55.
109
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi
ANALYTIC STUDY OF THE SUPERSYMMETRY -BREAKING SCALE AT TWO LOOPS
From
I1 Nuovo Cimento 105 A (1992) 1201
I992
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111 IL NUOVO CIMENTO
VOL. 105A, N.8
Agosto 1992
Analytic Study of the Supersymmetry-Breaking Scale at Two Loops. F. ANSELMO (I), L. CIFARELLI (l) ( 2 ) (3), A. PETERMAN (l) (*) (5) and A. ZICHICHI(l) CERN - Geneva, Switzerland Physics Department, University of Pisa - Pisa, Italy (3) INFN - Bologna, Italy (*) World Laboratory - Lausanne, Switzerland ( 5 ) Centre de Physique The'orique, CNRS-Luminy - Marseille, France
(I)
(2)
(ricevuto il 15 Maggio 1992; approvato 1'8 Luglio 1992)
Summary. - We study the impact of a threshold parametrized by the primordial gaugino supersymmetry-breaking mass scale m,lz, at the two-loop level. Its contributions to sin2B(mZ),aCm and M,,, are given. It is found that the inclusion of this two-loop threshold increases by - 50% the supersymmetry-breaking scale, with respect to the case where it is ignored. All expressions are given analytically and are therefore exact. The numerical analysis is done by an iterative procedure. Note that numerical corrections, inserted by hand in order to fit the two-loop accuracy, become unnecessary.
PACS 11.30.Pb - Supersymmetry.
1.
- Introduction.
The comparison between the experimental value of sin20q(rnZ) and the theoretical prediction for this quantity, in the framework of the minimal supersymmetric SU(5), has been and still is a powerful tool for the determination of the scale range of the light threshold [l-41. This scale is most conveniently represented by the primordial gaugino supersymmetry-breaking mass mlp. So far, analytic studies have been made in terms of a detailed one-loop light threshold, parametrized by primordial parameters, supposed to have their origin in a hidden sector of N = 1 supergravity. Since the range of ml12 (which parametrizes gauginos and partly squarks and sleptons) is very sensitive to the value of sin2O(mZ), this quantity should be known with the greatest possible accuracy. However in the absence of a full two-loop analytic treatment, one fits the results of the analytic one-loop calculations to a numerical two-loop calculation for central values of the input [ 5 ] . 1201
112
1202
F. ANSELMO,
L. CIFARELLI,
A. PETERMAN
and
A. ZICHICHI
For sin20(mZ),this procedure corresponds to adding the term 0.0029, if the value of a3(mz) is taken in the range a3(mZ) = 0.112 f 0.118 [51. This numerical correction is very important, as we will see later. The purpose of the present paper is to give a full two-loop analytic treatment to the problem of finding a favoured range for mllz. The two-loop approximation being a correction to the one-loop leading p-function coefficients, we use a single discontinuity threshold at ml12(*). Our motivation is to see what happens t o the favoured range of ml12 when the two-loop calculations including threshold are performed. The main conclusions are: 1) the scale of supersymmetry breaking increases by roughly 50%, thus allowing physically plausible spectra to be obtained above the Z scale for values of a3(mz) as high as 0.130; 2) the two-loop analytic formulae for sin2e(mz), dGm and MGm , are given for completeness. They can be incorporated in any numerical analysis, making numerical fits to sin2f3(mz), aGm and MGUT unnecessary, since they are automatically and exactly taken into account for any value of the various inputs. 3) The two-loop analytic corrections with a light threshold at ml12 are equally given and incorporated. Although relatively modest, they cannot be neglected, especially for the case of sin20,which, as already mentioned above, is a crucial and extremely sensitive quantity for ml12. Note that light-threshold spectra will not be discussed systematically here. This will be the subject of a subsequent paper [6]. 2.
- Outline of
the formalism.
The procedures used to deal analytically with the gauge-coupling evolution are well known and straightforwardly taken over from the general coupled RGEs:
C being a geometrical factor (see, for instance,[7]). The solution to (1) can be cast in the following form:
-GUT
+1 2x
[
( MGUT mZ ) * - + l o g ( z ) * -
-log 33
-
glog($t)*-
4
(*) We used the same approximation for computing the EGM effect [2] and a subsequent detailed threshold analysis [3] proved the goodness of such an approximation.
113 1203
ANALYTIC STUDY O F THE SUPERSYMMETRY-BREAKING SCALE ETC.
(2c)
~
1 a3(mZ)
-
-+
L[- 3
UGUT
2x
log( m,)* MGUT - 2 log(
$)*
-
where
The Ci stand for the following combinations of two-loop matrix coefficients:
for i = 1, 2, 3. The b; and bj" are supersymmetric two-loop and one-loop coefficients of the ,B function, respectively. The bi> and b; are the corresponding non-supersymmetric ones. From (Ba), (2b) and (Zc), ff& and 1/2x log (MGW/ m Z )can be deduced, and then sin20th(mZ)predicted, all at the two-loop level with a threshold at m1/2,where the transition from the non-SUSY case t o the full SUSY case takes place. We have discussed in the introduction the justification of such an approximation. We stress once again that the two-loop contribution is by itself a correction to the leading so that an approximate treatment of the threshold in one-loop terms (see eq. (l))(*), this contribution is a fair approximation and will prove t o be satisfactory to examine the trends induced by its presence in the equations. Note that the parametrization of this two-loop threshold by ml12allows it t o adjust itself as a kind of centre of mass of the detailed threshold structure.
O b i ) with respect to the one-loop bi. M(') means two loops; generally: (1)= two loops; (0) = one loop.
(*)
means one loop. More
114 1204
F. ANSELMO, L. CIFARELLI, A. PETERMAN
and
A. ZICHICHI
Our main results read as follows: in terms of the Ci,eq. (3), the two-loop contributions including threshold are:
(8)
1 aGUT
-
&AT
43 27~
+1 6 log(
2)+
$log(
2)+
2)+
$log(
+lo g (
2)+
20
sin2e(mZ)= 0.2
(9)
+
7aKp(mz) + sin20(mZ)(1) 15ayP(mZ)
+ AT^^^^^
with (10)
AT,,,, =
(mz) 20x
a2'
- 3 log(
2)+
2 log(
2) 2) -lo g (
-4
log(
2)
-
115
1205
ANALYTIC STUDY OF THE SUPERSYMMETRY-BREAKING SCALE ETC.
The pure two-loop corrections without threshold can be recovered of course by putting ml12= mZ in (3). In doing this we do not account for the presence of a light threshold in the two-loop expression, but we account automatically and analytically for the numerical fits done up to now to the one-loop analytical results. As announced in the introduction, the effect of a threshold in the two-loop approximation increases the favoured range for mIl2.No systematic studies have been performed up t o now. Nevertheless we can give an example of the trend when accuracy (loop approximation, two-loop threshold) is improved. Let us first define cases I, 11, I11 and IV as follows:
I: 1-loop, no 0.0029 term added. 11: 1-loop, 0.0029 term added. 111: 2-l00ps, no 2-loop threshold. I V 2-loops, 2-loop threshold a t mIlz. The results are shown below for two sets of input values. Notice how ml12 changes.
I11
840
Iv
1290
(2) 2
Inputs:
ag
(mZ= ) 0.118,
(z)
sin2O(m,) = 0.2334,
m, = 100 GeV,
= 0.27,
2
= 0.27.
Inputs:
a3 ( m Z = )
0.122,
( z)2 1. =
sin2e(mZ)= 0.2331,
m, = 100 GeV,
($Y=l,
116
1206
F.
ANSELMO, L. CIFARELLI, A.
PETERMAN and A. ZICHICHI
3. - Conclusions.
As a conclusion, we would like to insist on two main points: 1) The numerical fit to analytic one-loop calculations is in fair agreement with the genuine analytic two-loop corrections. 2) The inclusion of the full two-loop gauge-coupling evolution, with a threshold also at two loops for mIl2,increases the favoured ml12range by an amount of roughly 50% in the cases investigated. This fact means that it is therefore possible t o deal with a3 ( m Z ) values as high as 0.130 with physically acceptable spectra above m Z . Let us give an example:
I I1
Inputs: a3 (mZ> = 0.130,
( $) 0.27.
3.7 85
sin2e(mZ) = 0.2326,
m, = 100 GeV,
($)' 0.27, =
2
=
Note: as additional information, we give the correction t o sin' 8 computed for the cases I11 and IV: 111: 0.00334, IV total two-loop correction: 0.003438. Two-loop correction due to threshold only: 1.23.
REFERENCES [l] J. ELLIS, S. KELLEYand D.V. NANOPOULOS: Nucl. Phys. B, 373, 55 (1992). [2] F. ANSELMO, L. CIFARELLI,A. PETERMAN and A. ZICHICHI:preprint CERN-TH.6429/92; Nuovo Cimento A, 105, 581 (1992). [3] F. ANSELMO, L. CIFARELLI,A. PETERMAN and A. ZICHICHI:The simultaneous evolution of
masses and couplings: consequences on supersymmetric spectra and thresholds, preprint C E R N / L W S L / 9 2 - 0 0 8 , 2 April 1992 and CERN-PPE/92-103, 22 June 1992; Nuovo Cimento A, 105, 1179 (1992). [4] J. ELLIS, S. KELLEY and D. V. NANOPOULOS:preprint CERN-TH.6481/92, CTP-TAMU-42/92, ACT-10/92. Phys. Lett. B, 260, 131 (1991). [5] J. ELLIS, S. KELLEYand D. V. NANOPOULOS: [6] F. ANSELMO, L. CIFARELLI,A. PETERMAN and A. ZICHICHI:work in preparation. [7] M. E. MACHACEKand M. T. VAUGHN: Nucl. Phys. B, 236,221 (1984); M. B. EINHORN and D. R. T. JONES: Nucl. Phys. B, 196, 475 (1982).
117
F. Anselmo, L. Cifarelli and A. Zichichi
A STUDY OF THE VARIOUS APPROACHES TO MG,
AND C ~ G ,
From I1 Nuovo Cimento 105 A ( 1992) I335
I992
This page intentionally left blank
119
IL NUOVO CIMENTO
VOL. 1 0 5 4 N. 9
A Study of the Various Approaches to MGUTand F. (l)
(2) (3)
Settembre 1992
“GUT.
L. CIFARELLI (1>(2)(3> and k ZICHICHI (l) CERN - Geneva, Switzerland Physics Department, University of Pisa - Pisa, Italy INFN - Bologna, Italy h S E L M 0 (l),
(ricevuto il 20 Maggio 1992; approvato il 20 Agosto 1992)
Summary. - An exhaustive study is presented of the different M G m and aGUT values obtained using the renormalization group equations for the evolution of the couplings, with and without the evolution of the masses, with and without light and heavy-threshold effects, using different levels of theoretical accuracy: one and two loops, simple and detailed threshold treatments. A structure of the heavy threshold with mass ratios as large as lo6 and values of a z ( M Z ) as high as the world average which are still an order of plus 25 are needed in order to reach high values of MGUT, A discussion concludes the paper. magnitude below MPlanek.
I
PACS 11.30.Pb - Supersymmetry.
1. - Introduction. One of the most interesting problems in high-energy physics is the energy level where the couplings of the known SU(3) x SU(2) x U(1) gauge forces merge, M G U T , and the value of the unified coupling, a G U T . The results from
- Definitions
a1
, a2, a3 should become equal,
and input values.
In order to work out the value of the energy at which the known couplings
1335
120
1336
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
the basic ingredient is the renormalization group equations (RGEs). There are, however, different approaches which allow all couplings to be equal (1) at a given energy, M G U T . Our group has studied these approaches in order to investigate the SUSY-threshold problem [l-51. Here we focus our attention on MGm and aGm. Let us review the different approaches aimed at deriving the values of these basic quantities. The definitions that we will use are the following: AT,: indicates that the heavy threshold, when SUSY [SU(5)] is broken into SUSY [SU(3) X SU(2) x U(l)], has been taken into account (using either a simple degenerate mass or a structure with three masses). ATL: means that we have taken into account a complex structure for the light threshold effect when SUSY [SU(3) x SU(2) x U(l)] breaks either into SU(3) X x SU(2) x U(1) or into SU(3) x U(l)em.
EGM: indicates that the simultaneous evolution of the couplings and of the gaugino masses has been studied. The various approaches have as experimental inputs the world average values and their limits at 22u. We have taken for the world averages and their ? 1 u errors the following values [11: as (Mz)wA= 0.118 2 0.008,
sin2O(Mz )wA = 0.2334 2 0.0008, l/aem(Mz IwA = 127.9 & 0.2.
Other input values are needed for the heavy and light thresholds calculations. For the case of the heavy threshold [2,6,7], the input is the ratio Mv/M,, which is allowed to vary in a wide range: 10’ + lo6. For the light thresholds the inputs are the two ratios of the primordial masses: (mo/ml/2 )2 and (m4/mllz)2 [41, which are allowed to vary in a range + lo2. All results for MGm and aGm will always be quoted with uncertainties corresponding to 220 of the experimental inputs. For the Standard Model with three families and one Higgs doublet[81, the following matrix coefficients w i l l be used in the text: 41/10 { b j ’ } = -19/6
[
199/50
27/10
11/10
9/2
(3)
, 44/5 121. -26
For the Minimal Supersymmetric Standard Model (MSSM) with three families and
121 A STUDY OF THE VARIOUS APPROACHES TO
M G AND ~ aGm
1337
two Higgs doublets[9], the matrix coefficients will be:
[ 23] 3315
(4)
{b?> =
(5)
I
199125 2715
3.
{b;} =
- The MGUTvalues from
915 1115
25 9
8815 24 14
different approaches.
3a). The value of M G musing the simplest approach: evolution of al,a2,a3 at one loop; only one Msusv (step finetion). - The evolution equations are (if
0.1 4
a-0.12
0.1
Fig. 1. - Correlation between M G Uand ~ a,(MZ) for the approach 3a): a l , a2, a3 evolution at one loop; only one Msusy (step function). 81 - I1 Nuovo Cimento A
122 1338
F. ANSELMO, L. CIFARELLI
and A.
ZICHICHI
(7)
The result for
.
0.1 4
0.1
MGuT
is as follows:
- - - - - -
-....................
-
- - - - - --
-
- -
-
- - - -. -
-
- - -
-
-
,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
............................................................................
14
15
18
Fig. 2. - Correlation between MGw and a3(Mz) for the approach 3b): a l , az, a3 evolution a t two loops; only one Msusy (step function).
123 A STUDY OF T H E VARIOUS APPROACHES TO
MGUTAND
1339
aGm
Figure 1 shows the correlation between the possible input values for a3 ( M z) and the results for M G U T .
3b). The value of MGm using the next-to-simplest approach: evolution of at two loops; only one MsusY (step jknction). - The next-to-simplest approach is based on a new numerical solution applied directly to the two-loop
a ] , u z , a3
-----.................. ......................
0.14
-, - - - - -. - ..,....................
n
r"
W
go.12
0.1
c 14
. . . . . . + ............
. ..f .......
__f
.....
18
Fig. 3. - Correlation between MGm and a3( M z ) for the approach 3c): al, a2, a3 evolution at one loop; ml12 (step function) with EGM (one loop). No AT,.
coupled RGEs for the a1, a 2 , u3 evolution (previous approaches were based on numerical solutions of analytic approximations to the a ] , a2, a3 evolutions):
with coefficients bi = b/ , bij = b i if p < MsusYand SUSY coefficients bi = b f , bij = b$ if p > MsUsy (assuming Msusy > M Z 2 mJ.
124 1340
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
The results are: MGm = (1.6Tf;) . 10l6 GeV,
(Ila)
Note that increasing accuracy lowers the value of MGW. Figure 2 shows the correlation between the possible input values for a3( M , ) and the results for MGW. 3c). The value of MCm with a l , a z , a3 evolution at one loop introducing the evolution of the gaugino masses at one loop. - Keeping the one loop for the evolution of the couplings, we introduce a one-loop treatment of the threshold when SUSY [SU(3)x SU(2) x U(l)l breaks either into SU(3) x SU(2) x U(1) or into
0.1 4
W
30.12
0.1
tI
I
I
14
I
1
I
I
I
I
I
I
15
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
16
I
I
I
17
I
I
I
I
I
I
I
I 1
18
(GeV)
~O910MGUT
Fig. 4. - Correlation between MGm and a s ( M z ) for the approach 3d): al, a 2 , a3 evolution at one loop; mllz (step function) with EGM (one loop) plus AT, (one loop). The ratios M v / M , = = M v / M H = 1are taken. The leftmost curve corresponds to (mo/ml12)2 = (m,/ml/z)2 = lo-', the central one to (mo/m,/2)2= (m4/m1/2)' = 1, and the rightmost one to (mo/m1/2)2 = = (wL~/wz~= / , )102. ~
125 A STUDY OF THE VARIOUS APPROACHES TO
MGm
1341
AND aGUT
SU(3) x U(l)em.First we consider the light threshold in the simplest way, using only one primordial mass, mIl2,which parametrizes the gauginos, and allowing these masses to evolve [3]. The equations are:
(14)
-
- - -
---
............................
0.1 4
- -. .- -. - -
-
a-0.1 2
-
-
--
- -
- - -
-
............................
-
-
-
- - - -
-
- - - - - -
- -. - - - - -
0.1
--
..........................
- - , - - - - - ._ -
15
18
Fig. 5. - Correlation between MGUT and a 3 ( MZ )for the approach 3e): a l , a2, a3 evolution at one loop; mlI2 (step function) with EGM (two loops) plus ATL (one loop).
126 1342
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
0.1 4
n
0.1 2
0 .1
~
1
14
1
1
1
1
I l l i l l l l l l l l l i l l l l l l , l l i l l l , l l l , l 15 16 17 18 1
Fig. 6. - Correlation between MGm and a3 (M,) for the approach 3f): a1 , a2, a3 evolution at one loop; m1lz(step function) with EGM (one loop) plus ATL (one loop). The ratios Mv/M,= lo2 and = ( m 4 / ~ ~ 1 / 2 )=' lo-', the Mv/MH = 1 are taken. The leftmost curve corresponds to (mo/mllz)2 central one to (mo/ml12)2 = (m4/ml/z)2 = 1. The rightmost curve corresponds to (mo/mllz)2 = = (m4/ w L ~ / Z = I@.
127 A STUDY OF THE VARIOUS APPROACHES TO
MGm
1343
AND aGUT
The results, with one loop for the evolution of al, a2,a3,one loop for the evolution of the gaugino masses, are: (20a)
= (l.lO!j:g).
MG,
10l6 GeV,
Notice that the introduction of the evolution of the gaugino masses lowers MGuT (see (9a)). Figure 3 shows the correlation between the possible input values for a3( M z ) and the results for MGW. 3d). The value of MGm with a l ,a2,a3 evolution at one loop introducing the evolution of all masses and a detailed spectrum for the light threshold at one loop. This approach should be compared with the approach 3c), where we have one loop for the evolution of the couplings (a1,a2,a3) and one loop for the evolution of the gaugino masses. Here we extend the evolution[4] to all masses (squarks, sleptons, Higgsino, ...) and consider a detailed light-threshold spectrum.
t ---__
0.14
...............
n
r”
W
$0.1,
- - _ _ _
0.1
...............
tI
14
I
I
I
I
I
I
I
l
i
I
I
I
I
I
I
I
I
I
II
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
18
Fig. 7. - Correlation between MGUT and a 3 ( M Z for ) the approach 3g): a l , a p , a3 evolution at one loop; ATH (one M,, smooth threshold) at one loop.
128 1344
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
The equations for the couplings are:
[: (z:)+ :i (z:)+
+2X
+-log 1 5
-log
-
-log
-
-log 43 240
(z.) -
+
(;:) + z'o (a)+(2 ) 120 -
-log
+
+
83
0.1 4
W
n
80.12
0.1
Fig. 8. - Correlation between MGuTand a3( M z ) for the approach 3h): al,a2, a3 evolution at two loops; ATH (one M x ,smooth threshold) at two loops.
129
MGm
A STUDY OF THE VARIOUS APPROACHES TO
(22)
1 al(p)
1 +I [ GUT
2x
*
*)*
log(
1345
AND aGUT
-
%log(
?)*-
1
30 --
10 (23)
1 - 1 + I[log(!!5E)*-log(~)*az(,u)
GUT
--log 21
(24)
1 = -+ a3(p)
GUT
--log 2 3
log(?)*-
2x (:)*-;log(?)*-
-271.
[
+log(?)*-
-31og (M;uT)* - - 2 log(
(:)*-
where
For the masses the equations axe:
+log(
?)*-
+log(y)*],
?)*-
+log(?)*-
+log(?)*].
130 1346
F. ANSELMO, L. CIFARELLI
where
(36)
(37)
(38)
mH
=
d
m
(at scale m H ) ,
where (39) with h,2 4x
at = - = top-Yukawa
coupling
and
A. ZICHICHI
131
A STUDY O F THE VARIOUS APPROACHES TO
MGUTAND
1347
aGUT
and
T
= log[-;.-i. M&JT
The results, with one loop for the evolution of a l , a2, a3 and one loop for the evolution of all masses with a detailed light-threshold spectrum at one loop, are:
Wa)
MGm = (1.3+f::).1016GeV,
(42b)
l / a G U T = 25.7+;:$.
The result (42a) should be compared with (20a): a detailed light-threshold spectrum slightly pushes M G U T towards higher values. Figure 4 shows the correlation between the possible input values for a3 (Mz) and the results for MGUp
3e). The value of MGm with a ] , a 2 , a3 evolution at two loops introducing the evolution of the gaugino masses at two loops, with a detailed spectrum of the light threshold at one loop. - In this approach we have two loops for the evolution of the couplings ( a l , a 2 , a 3 ) and of the gaugino masses [5]. The equations are: -- 1
al(Mz)
3 - -a, 5
-1
)*-: *):(
1 +1 33 log MGUT ( 1 - sin2@= ~ G U T 2x MZ 5
(
[
-"log(z)*30
&log(a)'-
$log(z)*-
-log
-
-
* lo g (K)miL * -
c 1
1
- sin2e -
az(Mz)
aem
GUT
-
a3(Mz) 1
-
"GUT
+ I[ log( 2x
(
log mt )* MZ
[
+2x
-31og
-lo g (
z)*
4rr'
-
ilog( 3)' ilog( ")' -
-
MZ
MZ
(-
%;)*-2log(z)*-
flog(z)*-
z)* z)* z)*]
-Aog( 3
%)*
+
- +lo g (
-
+lo g (
+ -.(73
4x
132 1348
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
The Ci terms stand for the following combinations of matrix coefficients:
The results, with two loops for the evolution of a l , u z , a3 and of the gaugino masses and with the light threshold at one loop, are:
MGuT= (LO?;::).
(47~) (47b)
10l6 GeV,
l / m G u T = 25.3fi:i.
The result ( 4 7 ~ should ) be compared with (42a). Again, increasing accuracy favours lower M G U T values. Figure 5 shows the correlation between the possible input values for a3( M z ) and the results for MGm. 3f). The value of M G U T introducing the simplest structure in AT* at one loop. We now study the approach 3d), where we allow ATH to have a structure. The simplest one is in terms of three masses: Mv, MH, M s . Two of them need to be very high and quasi-degenerate, M , and M H ,owing to the limits on 5.The one which could be low is M v . Let us consider for ATH the ratio Mv/ M v . We let this ratio be in the range 10' i lo6. The equations are the same as in subsect. 3d), but with the following changes: eqs. (21), (23), (24) should be modified respectively into (48), (49), (50):
65 +-log 48
-
43 + -log 240
6
+ 3 log ( 2 + ) 20 M Z
(49)
1 =-
(zi)+ -
-log
(2;) - t
5 =)+
log( 3) + log( MZ MZ
133
A STUDY OF T H E VARIOUS APPROACHES TO
(50)
1 a3(p)
+ -2x
*GUT
--log( 5 3
[
MGm
1349
A N D CCGUT
-310g ( M;m)*-210g(7)*-
+) m- * - +log(
%L
;log(?)*-
*-6 1 log( T miR ) * - 3 log(
z)].
The right-hand side of eq. (26) should be multiplied by (Mv/M,)3/10. To the right-hand side of eq. (27) a term (6119)log(Mv/M\-) should be added. If Mv /Mx= l o 2 , the results, using the al, a2, a3 evolutions at two loops, ATL at one loop and AT, at two loops, are: MGm = (4.8?i?j9)-1016GeV,
(514
This is the highest possible value for MGm so far. Figure 6 shows the correlation between the possible input values for the results for MGw.
a3 ( M z)
and
Note that:
goes 1) The ratio MGw/MFGF changes by a factor of almost 2 when (ml/m,12)2 from lo-' to lo2. This ratio remains practically unchanged when Mv/Mx goes from 1 to lo6. The MGUT value obtained when Mv - - lo6
and
(z) 2
=
10'
is: MGm = 8.87.10'' GeV . to 2) The ratio MGv/M/$$ changes by 20% when (mo/ml/2)2 goes from lo2. This ratio remains practically unchanged when Mv/M, goes from 1 to lo6. In order to see how high MGuT can be, we have used as inputs the values:
with
a3 (M,) = 0.134 (WA
+ 20). The result is: MGw = 3.14.10'* GeV ,
which represents the highest possible value for the energy level where the 4ow-energy. measurements allow al, a 2 , a3 to merge. Table I summarizes the results of our approach 3f). 3g). The value of MGm introducing the smooth convergence towards MGm and the unified evolution above MGm at one loop. - We now follow a different approach[2]. The evolution of the couplings ( a l , a 2 ,a3) is worked out from above, starting from
134
1350
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
the high side and descending down to lo2 GeV. Going from the high-energy side down, a supergroup, for example SUSY [SU(5)1, will break into SUSY [SU(3)X SU(2) X X U ( l ) ] , crossing a heavy threshold at M x . In this approach we ignore the light-threshold problems and concentrate on the heavy threshold, which is treated in the simplest way: assuming only one heavy mass and taking as unique input the value TABLEI. - The values of MGm for the approach described in subsect. 3f) when as(MZ) ranges f i o m the world average (WA) value to 2 lo, 225, and the ratio MV/M,goesfiom 10' to lo6. The quoted values are for (m,/m1,2)2 = (mo/ml/2)2 = 1.
I
0
0.118 (WA)
I
I
16.12
2
16.68
4
17.24
6
17.80
0
I
16.39
1
I
~~
2
0.126 (WA + lo)
16.96
4 I
0.134 (WA
+ 2o)
17.52 1
I
6
18.09
0
16.61
2
17.19
4
17.76
I
6
I
0
I I
18.33 15.79
2
16.35
4
16.90
6
17.46
1 1
15.41
17.06
of a3 at the Zo mass: a3(MZ). We then allow the program to choose the value for MGuT that best fits the other two experimentally measured quantities: sin28(MZ) and aern (Mz1.
135 A STUDY OF T H E VARIOUS APPROACHES TO
MGW
AND aGm
1351
The equations are:
where p1is a scale above the Mxthreshold where unification is assumed to have taken place, po is a scale below M x and
(55)
I
a
(ATH only one mass - two loops)
136 1352
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
Figure 7 shows the correlation between the possible input values for a3(Mz) and the results for MGm.
3h). The value of MG, introducing the smooth convergence towards MGUT and the unified evolution above M,, at two loops. - Here we repeat the same approach as in 391, improving the theoretical approximation via the two-loop calculation. The equations are:
(ala2a3two loops - EGM two loops)
0 (a,a2a3one loop - EGM & ATL one loop)
20-
A
(ai0+a3 one loop - EGM one loop)
A
(ATH only one mass - two loops) (aiaZa3
0
15
16
17
two loops - one Msusy)
(aia2a3
one loop - one Msusy)
10gioMGUT
(GeV)
Fig. 10. - The aGUT values corresponding to the different values of MGuT resulting from the different approaches described in the text. The uncertainties are computed taking k 2 0 around the world average for a3(Mz).For the two approaches: ( a l , az, a3 one loop; EGM and AT, one loop) and (AT, two masses and two loops; EGM and ATL one loop), the central values are computed with m, = 125GeV; error bands also include uncertainties on m, (95GeV < < m, < 180 GeV). The primordial mass ratios are mo/ml/2 = m4/ml/z = 1.
137 A STUDY OF T H E VARIOUS APPROACHES TO
MGm
AND aGm
1353
where
(60)
0.15 /
0
World Average: WA
@
W A + lo WAf20
A
Fig. 11. - The dependence of M w T on aZ(Mz) and on the ratio of the two crucial heavy-threshold masses, M,/M,, as described in the text. Note that the extreme value for MGUT is above 10l8 GeV.
138 1354
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
and
The results, with AT, at two loops, are: (624
MGm = (3.0?::i).10’6 GeV,
(62b)
l / a G m = 24 2 1 .
A s for the approach 3b), increasing accuracy lowers the value of M G U T . Figure 8 shows the correlation between the possible input values for a the results for M G U T .
3 ( M ~ )and
TABLE11. - The trends are as follows: increasing accuracy (>om one loop to two loops) lowers MGm. The introduction of ATL has no effect on MGm. The introduction of a structure in the heavy threshold pushes MCUT upwards and the introduction of EGM lowers MGm. Higher values of a3(Mz), sin2e(Mz)and ae,,,(Mz), all push MGuTup.
139 A STUDY OF THE VARIOUS APPROACHES TO
4.
- Discussion
MGUTAND aCm
1355
and conclusions.
All MGm values calculated using different approaches and theoretical levels of accuracy are reported in fig. 9. We report in fig. 10 the correlation between these values of MGuTand the associated aGm. Finally in fig. 11 we show how MGUT depends on a3( M z ) and on the ratio Mv/ M v . The lowest value of MGmcorresponds to a value of a3 ( M Z) which is equal to the world average minus 2a and to a ratio MV/ M v = 10'. This value is MGW= 2.5.lOl5 GeV; it is lower than the M x value allowed by the experimental lower limit of T ~ If. we assume a quasi-degenerate structure for the heavy threshold, the ratios of the primordial masses equal to 1 and the central values for the experimentally measured quantities (a3 ( M z), sin2O(Mz 1, aem(Mz I), the value of MCUTis 1.3.1016GeV. However, this estimate depends on the ratio of the primordial masses and is dominated by the ratio Mv/M,. If this ratio is lo6 and a3( M z ) has the world average value + 2a, the result is MGm= 2.1 * 10l8 GeV, a value by almost one order of magnitude. With the same inputs, if we take 102 below MPlanck for the ratios of the primordial masses (m4/ml2I2 and (m, /m1/2)2,the result for MGm is 3.1-10'* GeV. This is the extreme limit where 4ow-energp data can push M G U T . The depends on the knowledge of these key existence of a gap between MGm and MPlanek quantities. Thanks to our studies on MGUT as a function of experimental and theoretical inputs and uncertainties, the trends shown in table I1 can be worked out. The gap between Mc, and MPlanck appears to be on safe grounds. In fact even with extreme input conditions M G u remains ~ safely below Mplanck. The consequences of the existence of this gap in the range of accessible energies should be extensively studied.
REFERENCES [ l ] F. ANSELMO,L. CIFARELLI,A. PETERMAN and A. ZICHICHI:Nuovo Cimento A, 104, 1817 (1991). We refer the reader t o this work for notations, formulae and other details not discussed in the present paper. [2] F. ANSELMO,L. CIFARELLI, A. PETERMAN and A. ZICHICHI:Nuovo Cimento A, 105, 1025 (1992). [3] F. ~ S E L M O L., CIFARELLI,A. PETERMAN and A. ZICHICHI:Nuovo Cimento A, 105, 581 (1992). A. PETERMAN and A. ZICHICHI:Nuovo Cimento A, 105, 1179 [4] F. ANSELMO,L. CIFARELLI, (1992). L., CIFARELLI,A. PETERMAN and A. ZICHICHI:Nuovo Cimento A, 105, 1201 [5] F. ~ S E L M O (1992). NucL Phys. B, 373, 55 (1992). [6] J. ELLIS, S. KELLEYand D. V. NANOPOULOS: [71 R. BARBIERIand L. J. HALL: Phys. Rev. Lett., 68, 752 (1992). [8] D. G. UNCER and Y. P. YAO: University of Michigan, Report No. UMHE 81-30 (unpublished). See also D.R.T. JONES: Phys. Rev. D, 25, 581 (1982). and D. R. T JONES: Nucl. Phys. B, 196, 475 (1982). [91 M. B. EINHORN
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141
F. Anselmo, L. Cifarelli and A. Zichichi
A x*-TEST TO STUDY THE a,, a2,a3CONVERGENCE FOR HIGHPRECISION LEP DATA, HAVING IN MIND THE SUSY THRESHOLD
From I1 Nuovo Cimento 105 A (1992) I357
I992
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143
IL NUOVO CIMENTO
VOL. 1 0 5 4 N. 9
Settembre 1992
A ?-Test to Study the a l , a2, a3 Convergence for High-Precision LEP Data, Having in Mind the SUSY Threshold. F.
ANSELMO (’),
L.
CIFARELLI (1)(2)(3)
and k ZICHICHI ( l )
CERN - Geneva, Switzerland (2) Physics Department, University of Pisa - Italy (3) INFN - Bologna, Italy (l)
(ricevuto il 20 Maggio 1992; accettato il 20 Agosto 1992)
Summary. - A ?-test is proposed in order to establish in a physically sound way the meaning of the convergence of the a l , a2, a3 couplings, taking correctly into account the high-precision L E P data. The naive belief that SUSY breaking must be in the TeV range in order to obtain the < convergence of the couplings is shown not to be correct.
PACS 11.30.Pb - Supersymmetry. I
1. Introduction. L E P allows to measure, with high precision, in the Zo-mass range, the couplings associated with all known gauge forces (excluding gravity). The purpose of this paper is to discuss how the high-precision L E P data can be used to understand what happens with Supersymmetry. The fact that Supersymmetry was needed to unify SU(3) X SU(2) X U(1) is known since 1979 [l]. The difficulty is in solving the many problems arising when the evolution of the couplings needs to be worked out over more than ten orders of magnitude in energy. The Renormalization Group Equations (RGEs) allow a rigorous extension of our knowledge over many orders of magnitude, provided we take into due account the physical effects connected with the evolution of the couplings (EC). These effects are the light threshold (ATL),the heavy threshold (ATH) and the evolution of masses (EGM), as we will see later. Let us suppose that a Supergroup (for example SUSY [SU(5)]) containing the known gauge groups really exists. At some energy level this Supergroup will split into the three known gauge groups SUSY [SU(3)x SU(2) x U(l)], whose couplings a1, a2, a3 have the same value a G W , just before the splitting occurs. Let us call E G U T the energy where (1)
a1 @GUT ) = a2 (EGUT ) = a3 @GUT ) = “GUT @GUT ) .
1357
144 1358
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
Just below EGW, at some energy M,, if we had a collider, we would discover the production of very heavy particles and could measure the <
M Z : where high-precision measurements are performed;
MsusY : where the SUSY threshold is; M,: where the heavy threshold is. The detailed spectrum of the light and heavy thresholds is a further complex problem connected with our main task ( C C ) . The purpose of this paper is t o study if high-precision L E P data can allow to know where a l ,a 2 , a3 unify and where the SUSY threshold could be. The structure of the paper is as follows. In sect. 2 we use the a1 , a2, a3 evolution equations with SUSY coefficients at the two-loop level in order to study the convergence of the couplings at some E G U T , taking into account the heavy-threshold effects. In sect. 3 we define a ?-test in order t o compute the best M x value and the best a3(Mz ). In sect. 4 we study the light-threshold effects as a cross-check of the results obtained in sect. 3. In sect. 5 we compare our ?-test with other claims. The conclusions are in sect. 6. 2. - The evolutions of
a1,
a 2 , a3 using the two-loop approximation.
Among all possible SUSY-GUTS we choose Minimal SUSY [SU(5)][3-61 whose coupling a5 evolves above EGUT following a3 . The evolution of the couplings al , a2, a3 is worked out starting from some very high energy, EMAX, and going towards lower energies. Here are the three evolution equations at the two-loop level:
-
--[
24n
39.6 log
145
A ?-TEST
(3)
1359
TO STUDY THE a,, a2, a3 CONVERGENCE ETC.
1 -1 -
I
a2 (p?
a2
(d)
The Ti terms are defined as follows: (5) where the bj” and b$ SUSY coefficients are given[5] by:
(b!) =
[
3315 ‘3],
199125 2715
8815
(7)
1115
14
9
and the F functions by (see appendix of 171):
for i, j = 1, 2, 3 and k = 1, 2. As mentioned in the introduction, a3, a2 and al are the couplings of the gauge groups SU(3), SU(2) and U(l), while p1is the extreme-energy value, EMAX= pl, and po is the low-energy point where input-output quantities can be experimentally measured[8]. Table I shows the high-precision LEP data adopted. As the key TABLEI. - High-precision LEP data. a,(MZ) = 0.118 -t 0.008
sin20(Mz)= 0.2334
I
k
0.0008
l/a,,(Mz) = 127.9 2 0.2
I
146 1360
F. ANSELMO, L. CIFARELLI
and
A. Z I C H I C H I
0.24 0.238
0.236 0.231 9 N
.-ccn
0.232 0.23
Fig. 1. - 2-loop results for sin2B(MZ)as a function of the input aq(MZ) in the range (World Average ? 20-1.
condition is the convergence of the couplings (CC), expressed in (l),we start with only one input, a3 ( M z): a3 (PO) = a3 ( M z
1.
With the two-loop equations (2)-(4) we use an iteration procedure and evolve a 3 , from its value at the Zo mass (which, we repeat, is our unique input) up to an arbitrary high-energy scale where the unification condition (1) is assumed to have already occurred. Then we evolve a l , a 2 , a3 down to the Zo mass (or po) and get from eqs. (2)-(4) the values for M,, aem( M z ) and sin2 O(MZ). These three results are worked out as a function of the single input, a3( M z ) . In fig. 1 we show the correlation between the values of sin2 6 ( M z ) and a3(M2). It is interesting to remark that, in order to approach the world average value for sin' O(MZ), it is necessary t o have for a3(M2) a value slightly above the world average, even if the world average, a?*= 0.118, is certainly well fitted by the
147
WA-20
128.1
WA
I
-
1 1
-
1 I
_
-
r"
-
I
I I I I I I .....................
;
I I I I ......................................
: : :
I I
: :
I I
;
I I
;\ n
: :
iZ-LOOdRESULTS
I
1 28
WA+2a
1 I
I 1
i
1 I I I . ?,-
: :
I I
1. I . '
:
I
WA
W
\
1 27 8
-
1
I
I 1 I
1
;
-
1
;
1
;
I I
I 1 . . 1.
..................................
I I
-
I
-
1
-
1
I
127.7 ~ , 0.1
I I I I
I
: : .
I
.............................................
I
:
I
:
I I
I I
: :
I I
I I
: :
I I
I 1 ' l l l t l I I t l l I 1 l i I I I I 1 l l l l l I ' I ' l i ' l ' t '
0.1 1
0.1 2
0.1 3
0.1 4
two-loop approximation. Figure 2 shows that l/aem( M , ) varies less with a3 ( M , ). However as for sin2 t9(Mz), slightly higher values than the world average are ) has to be at its world average value (*). preferred for a3 ( M z) if a,, (Mz 3. - The ?-test proposed to use the high-precision LEP data.
Having investigated the correlation between the three experimentally measured quantities, we now proceed to get the best value for Mx.For this we need to define the 2. There are three experimentally measured quantities. The quantity a3 (Mz ) has
(*) We recall that sin' 0 = al /(a1 factor).
+ 5/3. a')
(where the 5/3 coefficient is the SU(5) normalization
148
1362
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
60
50
30
.............
......... . . . . . . . . . . . . . . . . . . ......................
....................
.........
.........
s
........ ...... . _ . ......
N
x 20
.......
10
. ....,
.........................
;
................ '
/
/'
:/'
OO
1
2
3
4
M, (x I 0" GeV) 2
Fig. 3. - The 2 values us. Mx.The depends on the values of sin2e ( M z )and aem( M z) predicted by our coupled evolution equations, as explained in the text. Note that the only input of our model is a3(Mz) and the chosen values are in the range ? 2u around the World Average value.
been chosen as the input. The 2 can only be defined using the other two quantities,
i. e.
149
A
TEST TO STUDY THE a l , az, a3 CONVERGENCE 5
4.5 4
wA-2~ -
I
; 2-
--
I
:
I
-
I 1
-....................
-..,
i
. . . . ........ L . . . . . . . . . I I
1 .........
'.........
...........................
_
1
:
............. .........
-. . - I
I I : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .I. . . . :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .I. . . . . . - I I :: I
0.5 1.................... I
L
-
'-
-
n U
I I
- I - I .................... - 1
-
1
f
.........
.....)
2.5 -
1.5
I
: I - I 1
2
;
I : -. . J . . . . . . . . . . . . . . . . . : - . . . . . . . . . . . . . . . . .I. . . .
3.5 3
I
.cook.REsuLrs
L . . .J. . . . . . . . . . . . . . . . i . 1
WA+2a
WA
I
-
1363
ETC.
I I
0.1
I
I
I
I
I
I
I
I
I
; . . . . . . . . .I. . . . . . . . .
. . . . . . . . . . . . . . . . . I... 1 . . . . . . . . . . . . . . . . . . . . . I : I : I
I
I
I
I
I
I
I
I
1
1
1
1
1
1
1
1
I I 1
1
1
0.13
1
1
I
I
~
I
I
~
0.1 4
Fig. 4. - 2-loop results for the heavy mass M x as a function of the input a3(Mz) in the range (World Average 2 20).
The results are shown in fig. 3 where the <
The quantity M x represents the mass of the superheavy gauge bosons which become massive as a result of the SUSY [SU(5)]breaking. This threshold must exist if it is wanted that the unique value of a5 splits into three values: a1, a:!, a3, corresponding to SUSY [SU(3)x SU(2) x U(l)]. We assume for the superheavy bosons a degenerate spectrum: as if all superheavy particles, which become massive because of SUSY [SU(5)]breaking, had the same mass. However we perform an exact treatment of the threshold. If we call EGUT the energy where the three p-functions can be considered as being equal, the mass value of the superheavy bosons M x is about two times
150 1364
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
-1
ai Only one input
as(Mz) = 0.118
GeV
Fig. 5 . - The smooth convergence of a l , a2, a3 towards EGUT and the unified evolution above following our model. We emphasise that our model has only one input: q ( M z ) .
EGUT,
smaller than E G U T . We show in fig. 4 the range of variation of Mx vus. the input values of a3(Mz ). Figure 5 shows the smooth convergence of al , a2, a 3 towards E G m and the unified evolution of the couplings above E G U T . Notice that smooth convergence towards E G U T and unified evolution above should be the key features for all attempts aimed at giving physical significance to the fact that three gauge couplings a l , a2, a3 unify at some E G m (CC). Let us remark that the convergence of the couplings has as neighbour the heavy threshold, while the SUSY threshold is far away. Both threshold effects should be included in the evolution equations of the couplings. The heavy threshold is certainly closer to E G U T than the SUSY threshold and therefore in no case can it be neglected when dealing with CC problems (1). Now we want to use the information coming from the fact that the heavy threshold has been introduced into the evolution equations, to see what is the best value of a3(Mz). To this purpose we use the same definition (10). The next step is to apply the ?-test to a3(Mz). We thus use different input values of a3(Mz) in our iterative process with the evolution equations (2)-(4). To each a3(MZ)will correspond a minimum (10). The distribution of these minima is shown in fig. 6. It is interesting to note that the value of a 3 ( M z ) which minimizes the distribution of these minimum values of J is
2
2
(12)
a3 (Mz ) = 0.121
,
i.e. a value slightly above the world average (WA) (see table I). This value for
a3 ( M z )
15 1 A ?-TEST
1365
TO STUDY THE a I , a2, a3 CONVERGENCE ETC.
60
_ ................................................
50
. . .........................
...............
. . _.
40
...............
. . . . . . . . . . . . . . . . . . .................
50 N
x .................
20
.:. . . . . ./. / ..
10
:/’i :
/ , ’ : A
u
0.1
I
I
I
I
1
0.1 1
0.1 2
0.13
1
1
,
0.1 4
01304) Fig. 6. - The distribution of the ?-minima vs. a 3 ( M z ) in the range (World Average 2 2 ~ ) .
represents the best fit coming from the RGEs once the heavy threshold effect, to ensure a smooth convergence towards E G U T and a unified evolution above EGW,has been accounted for. The striking feature of our analysis is that the evolution equations for a l , a2,a3 show an excellent smooth convergence at EGUTwithout any necessity to introduce SUSY breaking above M Z .This means that, insofar as the wanted feature is CC (l), the SUSY threshold can be as low as the Zo mass. Therefore the light threshold AT, is expected to be very far away from A T H . As a cross-check to complete our work we need to study the light-threshold effects, using as input parameter the best fit (12) for a3(Mz).
152 1366
4.
L.
F. ANSELMO,
- The
CIFARELLI
and
A. ZICHICHI
light threshold.
In a recent paper we have studied the evolution of the couplings (EC) taking into account a detailed SUSY spectrum (at one loop) and the evolution of the gaugino masses (EGM) at two loops [9]. Here are the basic equations: ~
1 - 3 a,(MZ) 5
-1
(1- sin2,j) = GUT
2x
17
- 3 log 5 1 - sin2e a2(Mz) aem
GUT
( 2)* - $ log ( z)* 10log ( a)*] + 4x C1 ,
[ (%)* 2)'
+log 2~
- log(*):
1 a3(MZ)
1
[
+1 - 3 log 27c
~GUT
1
-
-
;log(
-
-log(
-
2)'
-
( %)* - ( 2 log
flog(
z)*
z)* -
-
log
( 2)*
-
MZ where log (x)* = O(x
-
1).log (2) and O(y)
=
0 if 1 if
yo.
The Citerms stand for the following combinations of matrix coefficients:
for i,j = 1, 2, 3. The b$ and b; are again the supersymmetric two-loop and one-loop coefficients of the p functions, respectively. The b i and b/ are the corresponding non-supersymmetric ones [5]. To eqs. (13)-(15),we apply our iterative procedure [lo] using as input values for a3(Mz) the result of our best fit (12) and for the other inputs, sin28(Mz) and a e m ( M z ) ,their world average values (table I).
153
A ?-TEST
TO STUDY THE a l , a2, a3 CONVERGENCE ETC.
1367
The result is that Msusy can be as low as the Zo mass. This is an excellent cross-check. In fact our study, intended to work out a ?-test for the three experimentally measured quantities sin2O(M, ), a3 ( M z), a,, ( M , ) in terms of the al , a2, a3 convergence (CC) at some EGUT value, indicated that Msusy could be as low as the Zo mass. 5.
- Comparison between
our ?-test and other claims.
Some authors [ l l ] have claimed to be able to know where the SUSY breaking should be, by constructing a ?-test based on the goodness of the a l ,a 2 , a3 convergence.
5'1. The weak points of re$ [ll]. - Although some weak points have already been discussed in the literature by us [7-10,151 and other authors [12-141, we would like to discuss them in more detail, since they appear not to be generally known. Here they are: i) Only one measurement of a3 ( M z) has been taken as input, not the world average. ii) Only one (presumably not the most accurate) of the many possible solutions of the a l , a 2 , a3 coupled evolution equations has been considered. iii) The confidence level chosen to quote the uncertainties has been the lowest possible, i.e. ? lo. iv) A ?-test deprived of any physical meaning to study the goodness of the al , a 2 , a3 convergence at EGUThas been adopted. v) The light threshold has been accounted for assuming mass degeneracy for all SUSY particles and using a step function. vi) The heavy threshold has been ignored, despite the emphasis on the search for al , a 2 , a3 convergence at E G U T . vii) The evolution of masses has been ignored. viii) The divergence of couplings above E G U T has been ignored. The first three weak points quoted above have been discussed in [8], while points iv), vi) and viii) have been briefly mentioned in[7]. The problems raised by the light threshold (point v) above) have been thoroughly considered in[9,10], while the evolution of masses (point vii)) has been the subject of [10,15] at one loop and of [9] at two loops. In the present paper we concentrate our study on points iv), vi) and viii). 5'2. The ?-test of re$ [ll]. - The authors of ref. [ l l ] emphasised that their goal was the goodness of the al , a2, a3 convergence (CC) but ignored the heavy threshold needed to allow the breaking of the Supergroup into SUSY [SU(3)x SU(2) X U(1)3. If the physical requirement is the <
154 1368
F. ANSELMO, L. CIFARELLI
and
A. ZICHICHI
where there is no reference to the three basic effects (heavy threshold, light threshold, evolution of masses) needed to describe the evolution of the couplings and therefore their convergence. Nevertheless these authors claim to study how well the condition (1) can be satisfied using high-precision L E P data. In the high-energy range where the couplings should converge, their 2 ignores the heavy-threshold effects on the convergence of the couplings; their 2 is purely <>. 6. - Conclusions.
The
a1, a2, a 3
evolution equations (2)-(4) do guarantee two features:
1 - A smooth convergence of a l , a z , a 3 up to E G m . This is due to the treatment of the M x threshold. 2 - A fully unified evolution of a l , a2, a3 above E G U T .
The detailed evolution for a l , a2,a3, showing the above-quoted physically expected features, is reported in fig. 5, where the evolutions are calculated using as input value the world average for a3, i.e. a?* (Mz ) = 0.118.
There is a crucial point to emphasise: the results presented herein show that the smooth convergence of the gauge couplings towards E G m and their unified evolution above E G U T do not need any SUSY breaking all along the range from EGm down to E = 10' GeV. This is compatible with the threshold for SUSY breaking being in the 10'GeV range. The $-test from the EGm range on the high-precision LEP data provides the .best, value for a3(Mz). This value is used for the detailed light-thresholds study including EGM effects (at two loops) and confirms that the SUSY threshold can be as low as the Zo-mass range: a cross-check of great interest. The naive belief that SUSY breaking is needed in the TeV range in order to obtain the .best. convergence[ll] of the gauge couplings a l , a 2 ,a3 is shown not to be correct. What is needed is a detailed study of the effects expected to exist below E G U T and a set of correct equations which take into account the smooth convergence towards E G U T . Once all this is done, the convergence of the couplings (CC) is .perfect. and Msusy can be as low as the energy range of L E P I , Tevatron, HERA and LEPII.
REFERENCES [l] k ZICHICHI: Closing Lecture at the E P S Conference (Yo& UK, 25 - 29 September 1978); also Opening Lecture at the E P S Conference (Geneva, CH, 27 June - 4 July 19791, CERN Proceedings; and New Developments in Elementary Particle Physics, Riv. Nuovo Cimento, 2, No. 14 (1979). The statement on p. 2 of this paper, Unification
of all forces needs first a Supersymmetry. This can be broken later, thus generating the sequence of the various forces of nature as we observe them, was based on a work by A. PETERMAN and A. ZICHICHI where the renormalization group running of the couplings using Supersymmetry was studied with the result that the convergence of the three couplings improved. This work was not published, but perhaps known
155
A ?-TEST
TO STUDY THE a l , a2, a3 CONVERGENCE ETC.
1369
to a few. The interest in the Erice Schools d3uperworld I>>,aSuperworld 11, and &uperworld 111, sparked from this work. M. DAVIER: Proceedings of the Joint International Lepton-Photon Symposium and Europhysics Conference o n High Energy Physics, Geneva, Switzerland, 25 July - 1 August 1991, edited by S. HEGARTY, K. POTTERand E. QUERCIGH (World Scientific, Singapore, 1991), Vol. 11, p. 151. H. GEORGIand S. L. GLASHOW:Phys. Rev. Lett., 32, 438 (1974). For lectures and references on Supersymmetry, Supersymmetric theories and, in particular, SUSY [SU(5)] see: Superworld I , edited by A. ZICHICHI(Plenum Press, New York-London, 1986); Superworld ZZ, edited by k ZICHICHI(Plenum Press, New York-London, 1987); Superworld 111, edited by A. ZICHICHI(Plenum Press, New York-London, 1988). M. B. EINHORNand D. R. T. JONES: Nucl. Phys. B, 196, 475 (1982). M. E. MACHACEKand M. T. VAUGHN: Nucl. Phys. B, 236, 131 (1984). F. ANSELMO,L. CIFARELLI, k PETERMAN and A. ZICHICHI:Nuovo Cimento A, 105, 1025 (1992). F. ANSELMO, L. CIFARELLI, A. PETERMAN and k ZICHICHI:Nuovo Cimento A, 104, 1817 (1991). Note that for notations, formulae and other details not discussed in the present paper, we refer the reader to this work. F. ANSELMO,L. CIFARELLI, k PETERMAN and k ZICHICHI:Nuovo Cimento A, 105, 1201 (1992). F. ANSELMO, L. CIFARELLI,A. PETERMAN and A. ZICHICHI:Nuovo Cimento A, 105, 581 (1992). u. AMALDI,W.DE BOERand H. F ~ S T E N APhys. U : Lett. B, 260, 447 (1991). S. GLASHOW:Proceedings of the Joint International Lepton-Photon Symposium and Europhysics Conference o n High Energy Physics, Geneva, Switzerland, 25 July - 1 August 1991, edited by S. HEGARTY, K. POTTERand E. QUERCIGH (World Scientific, Singapore, 1991), Vol. 11, p. 467. J. ELLIS,S. KELLEYand D. V. NANOPOULOS: Nml. Phys. B, 373, 55 (1992) and Phys. Lett. B, 287, 95 (1992). R. BARBIERIand L. J. HALL: Phys. Rev. Lett., 68, 752 (1992). F. ANSELMO, L. CIFARELLI, A. PETERMAN and A. ZICHICHI:Nuovo Cimento A, 105, 1179 .(1992).
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A. Zichichi
UNDERSTANDING WHERE THE SUPERSYMMETRY THRESHOLD SHOULD BE
From Proceedings of the Workshop on “Ten Years of SUSY Confronting Experiment ”, CERN, Geneva, 7-9 September 1992 CERN-PPE/92-149, CERN/LAA/MSL/92-017 (7 September 1992) and CERN-TH,670 7/92 - PPE/92- I80 (November 1992) 94
1992
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159
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
CERN-PPE/92-149 CERN/LAA/MSL/92-017 7 September 1992
UNDERSTANDING WHERE THE SUPERSYMMETRY THRESHOLD SHOULD BE
A. Zichichi
CERN, Geneva, Switzerland
Abstract
A supersynthesis is presented of the most relevant results obtained together with my collaborators during April 1991 - September 1992 on the search to understand where the Supersymmetry threshold should be and how its structure would look
Presented at the Workshop on “Ten Years on SUSY Confronting experiment” CERN,Geneva, 7-9 September 1992
160
1.
Introduction
During April 1991 - September 1992, together with my collaborators (Franc0 Anselmo, Luisa Cifarelli, Jorge L. Lopez, Dimitri V. Nanopoulos and Andre Peterman) we have studied a series of problems in Supersymmetry in order to contribute towards clarifying and understanding this fascinating field of Physics. A supersynthesis of the most relevant results is presented. For details we refer the reader to our original papers, [l-71 and [lo-111, whose abstracts are reported in the Appendix.
2.
Where is EGUT
Let us start with the question: where is EGUT(the energy level where all gauge couplings a1, a2, a3 meet):
The answer [l] is shown in Figure 1, where the three axes show EGUTversus a 3 ( M ~ and ) the heavy threshold mass ratio (Mv/Mz). Mv is the typical gauge boson mass of the Super Grand Unified Gauge Group (SGUGG) and M c represents the mass characterizing the remnants from the SGUGG breaking. There are two extreme points: &m(min) 2 1016 GeV &m(max) 5 1018 GeV The minimum value is obtained taking the World Average (WA) values minus 2 0 for the experimentally measured quantities, and all mass ratios equal to 1. The maximum value is obtained taking WA+20 for the experimentally measured quantities, pushing the heavy threshold mass ratio Mv/Mc up to 106 and the primordial mass ratios (mg/m1/2)2 = (mq/m1/2)* = 102. The value for &m(max) (- 1018 GeV) is consistent with the fact that there is no GAP between the string unification scale Esu (- 1018 GeV) and the gauge couplings unification point &m.
3.
The effect of ATL, ATH, EGM on MSUSYand EGUT
However, if we take the central values for the measured quantities and all mass ratios equal to 1 the gap appears, as shown in Figure 2,
2
161
where the evolutions of the gauge couplings are shown to meet at a point &UT, to continue to evolve above &UT, and to be consistent with the SUSY breaking to be quasi degenerate with the Electroweak breaking. In the graph are indicated the light threshold ATL, the heavy threshold ATH and the evolution of masses. Indeed during the last year we have studied [l-71 how these effects influence Msusy. Of particular relevance are the increased accuracy in a 3 ( M ~ and ) the Evolution of Gaugino Masses (EGM). We have learned that ATH pushes Msusy up and EGM brings Msusy down [4]. On the other hand increasing 0l3(Mz) accuracy corresponds to higher values of a3(Mz) and this means Msusy down. The study of ATL allows to know the mass range where all sparticles are expected to lie [4]. We have also calculated the EGM at two loops [ 6 ] and the result is that Msusy goes up with respect to the one-loop calculation. For example values of a3(Mz) as high as 0.130 are no longer forcing Msusy to be below Mz [ 6 ] . We have also studied the effects of ATL, ATH and of the mass evolution (EGM) on EGUT(or MGUT).All these effects are synthetically reported in Tables 1 and 2. These results bring us to two conclusions: 1) nothing can be said on Msusy and MGUTif the above-quoted detailed studies on ATH,ATL,EGM are not made. 2) The Electroweak and the SUSY breaking can be degenerate. 4.
What is the best candidate for the GAP ?
From our analysis it follows that the Gap between &UT and Esu could really be there. If this was the case new Physics should show up. The best candidate to fill the gap is SUSY [SU(S)xU(l)], i.e. the socalled “flipped SU(5)” [8]. This gauge group has the great advantage of being derivable from “string”. On the other hand SU(5) Supergravity [9] is in trouble [lo] for two reasons: 1) fine tuning is needed for more than two orders of magnitudes (see Figure 4). 2) The neutralino relic density exceeds the allowed cosmic value for most of the parameter space. We have selected a set of points on the boundary of the region which satisfies proton decay constraints for mt=125 GeV. On the other 3
162
hand, points inside the allowed region have larger values of the neutralino relic density RX.h2. The details are shown in Table 3 and reported in Figure 5.
5.
Spectra from Supergravity flipped SU(5)
Having found that SU(5) Supergravity is in trouble, we [ 111 have decided to work out the spectra for SUSY particles using “no-scale” Supergravity flipped SU(5). To comply with the typical “no-scale” requirement we have put two of the five parameters equal to zero: mg E A = 0. Thus our parameter space has only three axes: mg, mt, tanP (plus the sign of p). Two spectra are shown in Figures 6a,b for positive and negative p values, respectively. The top mass has been taken mt=130 GeV and the gluino mass m;g = 300 GeV. The vertical axis indicates various levels: 1, 3, 5, 7, 9. Each level corresponds to a set of particles: level 1 = neutralinos, level 3 = charginos, level 5 = sleptons, level 7 = Higgs, level 9 = squarks and gluino (Cj indicates the average of the first and second family squark members). The values for the masses are computed for (2 5 tanp 532). The masses change according to the type of particle and go from A d m 10-3 (gluino) up to 32% (Higgs). The details are shown in Table 4.
6.
Conclusions
In order to understand where the Supersymmetry threshold should be we have studied the light threshold effects (ATL), the heavy threshold effects (ATH), the convergence of the gauge couplings al, 0 2 , a3 taking into account the evolution of masses. We have worked out spectra under different assumptions with the boundary conditions imposed by nuclear stability and cosmological constraints. Our studies are all pointing towards the “prediction” that the Supersymmetry threshold could be as low as the Zo mass range and, in any case, accessible with existing colliders: LEP I, Tevatron, HEM and LEP 11. The claim that the SUSY threshold can be estimated from a
4
163
parameter like M s u s ~derived from the “geometrical convergence” of the gauge couplings is misleading.
7. Acknowledgements The data reported here are the results of a collaboration with Franco Anselmo, Luisa Cifarelli and Andre Peterman [l-71 and with Jorge L. Lopez and Dimitri V. Nanopoulos [lo-1 13. To all of them I would like to express my gratitude for the excellent and effective collaboration in this exciting field of Physics.
5
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Appendix The abstracts of all papers whose main results have been summarized in the present work. THE EFFECTIVE EXPERIMENTAL CONSTRAINTS ON Msusy AND MGUT F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi A comprehensive analysis of the world-data on a, is reported together with its average value at the ZO mass. The effective constraints on Msusv and MG, are given. The care needed to reach any conclusion on Msusy is discussed. For example, taking for the a 1 , ~ 2 , ~coupled ~ 3 equations a numerical solution (it should be the most reliable one) and the two standard deviation limits in the uncertainty of the a,(MZO) world average, the expected Msusy values range from 100.5fO-5GeV to lO5+_1GeV, i.e. from GeV to PeV. CERN-PPE/91-123; CERN/LAA/MSL/91-015, July 15, 1991; 11 Nuovo Cimento Vol. 104A, N. 12 (1991) 1817.
THE CONVERGENCE OF THE GAUGE COUPLINGS AT EGUTAND ABOVE: CONSEQUENCES FOR a3(Mz) AND SUSY BREAKING F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi We work out an iterative solution of the evolution equations for a1, a2, a3 in the Minimal Supersymmetric SU(5) Grand Unified model with the condition that the gauge couplings should converge smoothly towards E G and ~ should not separate above &UT. The work is done at two levels of theoretical accuracy: one loop and two loops. Improved accuracy favours high values of a3(Mz) with respect to the world average. CERN/LAA/MSL/91-026, December 1991, to appear in “I1Nuovo Cimento”.
6
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THE EVOLUTION OF GAUGINO MASSES AND THE SUSY THRESHOLD F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi We propose a numerical iterative method to account for the evolutions of the gaugino masses (EGM). The effect of these evolutions on the most exhaustive model of SUSY breaking is presented. From above 21 TeV, the one-cs lower bound for SUSY breaking is brought down by more than two orders of magnitude. The model without EGM needed two-c3 to reach the ZO-mass range. The same model with EGM needs only one-c3 to reach the same level. The conclusion is that the SUSY threshold can be anywhere, including within the mass range where LEP I, HERA, LEP II and other colliders are working or planning to work. CERN-TH.6429/92; CERN/LAA/MSL/92-003, February 1992; I1 Nuovo Cimento Vol. 105A, N. 4 (1992) 581.
THE SIMULTANEOUS EVOLUTION OF MASSES AND COUPLINGS: CONSEQUENCES ON SUPERSYMMETRY SPECTRA AND THRESHOLDS F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi
We use the Renormalization Group Equations to work out, at the one-loop level, the simultaneous evolution of all masses and couplings and show explicitly the self-consistency of the whole scheme. A thorough examination is performed of the light Supersymmetry threshold in the Minimal Supersymmetric extension of the Standard Model (MSSM). All fundamental quantities a G U T , MGUT,a3(mz), a2(mz), a l ( m z ) [consequently sin28(mz)] are given in terms of the detailed spectrum of all particle and sparticle thresholds. Examples of Supersymmetry spectra are given as function of ag(mz) and of the other essential parameters. The results of this study, where the evolution of masses is extended to all possible masses, confirm our previous conclusions on the EGM effect for the Supersymmetry threshold lower bound. Examples of the predictive power of our method are given. CERN-PPE/92- 103, 22 June, 1992 and CERN/LAA/MSL/92-008, 2 April, 1992, to appear in “I1 Nuovo Cimento”. 7
166
ANALYTIC STUDY OF THE SUPERSYMMETRY-BREAKING SCALE AT TWO LOOPS F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi We study the impact of a threshold parametrized by the primordial gaugino supersymmetry-breaking mass scale mln, at the two-loop level. Its contributions to sin%(mz), CXGUTand MGUTare given. It is found that the inclusion of this two-loop threshold increases by 50% the supersymmetry-breaking scale, with respect to the case where it is ignored. All expressions are given analytically and are therefore exact. The numerical analysis is done by an iterative procedure. Note that numerical corrections, inserted by hand in order to fit the two-loop accuracy, become unnecessary.
-
CERN-TH.6543/92, CERNLAAMSW92-009, June 1992, to appear in “I1Nuovo Cimento”.
A STUDY OF THE VARIOUS APPROACHES TO MGUTAND GUT F. Anselmo, L. Cifarelli and A. Zichichi An exhaustive study is presented of the different MGUTand CXGUT values obtained using the renormalization group equations for the evolution of the couplings, with and without the evolution of the masses, with and without light and heavy threshold effects, using different levels of theoretical accuracy: one and two loops, simple and detailed threshold effects. A structure of the heavy threshold with mass ratios as large as 106 and values of a3(Mz) as high as the world average plus 2 0 are needed in order to reach high values of MGUT,which are still an order of magnitude below Mplanck. A discussion concludes the paper. CERNLAA/MSL/92-011, July 1992, to appear in “I1 Nuovo Cimento”.
8
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A x2-TEST TO STUDY THE a1012a3 CONVERGENCE FOR HIGH PRECISION LEP DATA, HAVING IN MIND THE SUSY THRESHOLD F. Anselmo, L. Cifarelli and A. Zichichi A x2-test is proposed in order to establish in a physically sound couplings, taking way the meaning of the convergence of the al,a2,~3 correctly into account the high precision LEP data. The ndive belief that SUSY breaking must be in the TeV range in order to obtain the “best” convergence of the couplings is shown not to be correct. CERN-PPE/92-122, CERN/LAA/MSL/92-012, 20 July, 1992
TROUBLES WITH THE MINIMAL SU(5) SUPERGRAVITY MODEL Jorge L. Lopez, D.V. Nanopoulos, and A. Zichichi We show that within the framework of the minimal SU(5) supergravity model, radiatively-induced electroweak symmetry breaking and presently available experimental lower bounds on nucleon decay, impose severe constraints on the available parameter space of the model which correspond to fine-tuning of the model parameters of over two orders of magnitude. Furthermore, a straightforward calculation of the cosmic relic density of neutralinos ( x ) gives RXh2D 1 for most of the allowed parameter space in this model, although small regions may still be cosmologically acceptable. We finally discuss how the no-scale flipped SU(5) supergravity model avoids naturally the above troubles and thus constitutes a good candidate for the low-energy effective supergravity model. CERN-TH.6554/92, CTP-TAMU-49/92, CT- 14/92, June 1992, submitted to “Physics Letters”.
9
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I
Msusy
TRENDS
1
OATH pushes MsUsyUP ATL allows to know the mass range of MSUSY EGM pushes MSUSY DOWN A(a3)pushes MSUSY DOWN
A(EGM) pushes MSUSY UP
Table 1 The summary of all our studies [l-71 on the trend for MSUSY. (A) stands for increasing accuracy.
10
169
From 1loop to 2 loops ATj(0ne mass)
Table 2 The trends are as follows: increasing accuracy (from one loop to two loops) lowers MGUT. The introduction of ATL has no effect on MGUT. The introduction of a structure in the heavy threshold pushes MGUTupwards and the introduction of EGM lowers MGUT. Higher values of a3(MZ), sinW(MZ) and G~(MZ), all push M Gup.~
11
I a
74
8.1
-5.4
205
625
33
I21
15.9
b
122
6.5
-3.2
340
845
53
218
27.9
c
187
5.4
-1.6
520
I110
78
364
5.13
d
267
4.5
-.41
740
I370
I04
530
3.74
e
364
3.8
+.55
IOIO
I645
133
726
3.20
T Table 3 The set of five points (a, b, c, d, e) on the boundary of the parameter space of SU(5)Supergravity (allowed by T,, limits) with the correspondent values for the key quantities. Notice the last column where the neutralino relic density is reported.
171
II-laSS
(GeV) 163.7 290.1 29.2
% variation
211.5
17.0 0.1 18.3 2.1 11.9 12.8
53.1
9.3
210.3 82.0 185.1 179.3 197.5 267.2 289.8 170.4 362.6 99.6 67.7 69.4 56.7 105.3
10.9 19.1 32.0 29.7 24.3 3.2 1.o 7.8 1.3 2.2 4.2 8.5 17.9 6.8
61.6 174.1
Table 4 The mass of the various members of the Supersymmetry set of sparticles and the Higgs (corresponding to Figure 6b). The last column indicates the (%) variation in mass for the range of tanp = 2+32.
13
172
References [l]
F. Anselmo, L. Cifarelli and A. Zichichi, preprint CERN/LAA/MSL/92-011 July 1992, to appear in “Il Nuovo Cimento”.
[2]
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, I1 Nuovo Cimento vo1.104 A, N. 12 (1991) 1817.
131
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, preprint CERN/LAA/MSL/91-026, December 1991, to appear in “I1 Nuovo Cimento”.
[4]
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, I1 Nuovo Cimento vo1.105 A, N. 4 (1992) 581.
[a
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, preprint CERN-PPE/92-103,22 June 1992, to appear in “I1 Nuovo Cimento”.
[6]
F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, preprint CERN-TH.6543/92 June 1992, to appear in “I1 Nuovo Cimento”.
[J F. Anselmo, L. Cifarelli and A. Zichichi, preprint CERN-PPE/92-122, 4 July 1992, to appear in “I1 Nuovo Cimento”.
[8]
S. Barr, Phys. Lett. B112 (1982) 219, Phys. Rev. D40 (1989) 2457; J. Derendinger, J. Kim, and D.V. Nanopoulos, Phys. Lett. B 139 (1984) 170. I. Antoniadis, J. Ellis, J. Hagelin, and D.V. Nanopoulos, Phys. Lett. B 194 (1987) 231. I. Antoniadis, J. Ellis, J. Hagelin, and D.V. Nanopoulos, Phys. Lett. B268 (1991) 359; For a recent review see J.L. Lopez and D.V. Nanopoulos, in Proceedings of the 15th Johns Hopkins Workshop on Current Problems in Particle Theory, August 1991, p.277; ed. by G. Domokos and S. Kovesi-Domokos.
191
R. Arnowitt and P. Nath, Texas A & M University preprint CTPTAMU-24/92.
[lo] J.L. Lopez, D.V. Nanopoulos and A. Zichichi, Texas A & M University preprint CTP-TAMU-49/92 and CERN-TH.6554/92. [ll] J.L. Lopez, D.V. Nanopoulos and A. Zichichi, Texas A & M
University preprint and CERN-TH, September 1992 (in preparation). 14
173
Figure Captions Figure 1 These data show the dependence of ECUT (or MGUT)from a3(Mz) and the ratio of the two crucial heavy threshold masses Mv/Mz. Note that the extreme value for EGUTis above 1018 GeV. Figure 2 The evolution of the gauge couplings (011, 012, a3) versus E. Note the value of &UT at -1016 GeV and the string Unification Energy (Esu) at -1018 GeV. Where the light threshold and the heavy threshold are is also shown. Figure 3 Taking the most probable value for E!&uT, the GAP with Esu appears to be there. In this case SUSY [SU(5)] needs to be replaced. The best candidate is SUSY [SU(S)xU(l)]: a gauge group derivable from strings. Above Esu another big problem opens up: the origin of space-time. Figure 4 Fine tuning coefficient ct as a function of mt for five points (a, b, c, d, e) which satisfy the proton decay constraints in the minimal SU(5) Supergravity model. Figure 5 The lightest neutralino relic density as a function of mt for the five points of the minimal SU(5) Supergravity model. The region with QX.h2 > 0.25 (dotted line) is excluded on cosmological grounds. A restricted range (below the dotted line) remains allowed. Figures 6 Spectra of sparticles (neutralinos, charginos, sleptons, squarks) and Higgs, predicted by flipped SU(5) Supergravity for p > 0 (a) and p < 0 (b).
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WHERE WE STAND WITH THE REAL SUPERWORLD
From Proceedings of the X X X Course of the International School of Subnuclear Physics: “From Superstrings to the Real Superworld”, Erice, 14-22July 1992, World Scientific - The Subnuclear Series 30 (1993) I
I993
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183 FROM SUPERSTRINGS TO THE REAL SUPERWORLD
1
WHERE WE STAND WlTH THE REAL SUPERWORLD
A. Zichichi CERN, Geneva, Switzerland
During the last year or so, a topic of interest to me more than ten years ago [11 came to my attention. The novelty being the high precision LEP data. The problem: can SUSY threshold be “predicted”?. The basis for this new frontier in physics - i.e. the existence of a Superworld - is the Renormalization Group Equations (RGEs) which allow to span an energy range as large as 14-17 orders of magnitude. In fact the high precision LEP data are at 102 GeV and we would like to understand what happens all the way up to &UT (the energy where the three gauge forces SU(3)c, s U ( 2 ) ~and U(1)y - characterized by the three couplings
- unify).
al,a2,a3
Recently a great deal of confusion has been raised by the authors of Ref. [2] who claimed that it was possible to predict the Supersymmetry threshold on the basis of a Xz-test on the convergence of the couplings 011,012,a3at EGW. This paper has many weak points [3-91, the weakest one being a logical inconsistency. In fact if the source of our knowledge about the Supersymmetry threshold is the “convergence” at &UT, the top priority problem is to study what happens at &m. This means the study of the threshold effects in the very high energy limit (1015-1017 GeV) because what happens at &m is supposed to have consequences in the energy range m y orders of magnitude below. Moreover, it is contradictory to work out a ~ 2 test for the geometrical convergence at &m of the three gauge couplings (al,a2,~~3) and then let them diverge again [2], above &m. Physics is not Euclidean geometry. The synthesis of this logical inconsistency is shown in Fig. 1 (which is the key figure of the paper quoted above 121). The reason for my interest in this paper is because it produced a lot of discouragement in the physics community, including my young
-
184
2
collaboratorsengaged in searching for a supersymmetric signal with existing facilities. We have put order in this field [3-91 and the conclusion is that, in addition to all weak points and logical inconsistencies, the quantity Msusy is meaningless. We have worked out the spectra expected [6] and found that the lightest detectable supersymmetric particle could be as light as 50 GeV in mass, with the Msusy parameter more than one order of magnitude fat away. Furthermore we have pointed out that the evolution of masses needs to be included Of special relevance is the evolution of the gaugino masses, the so-called EGM effect [5]. A great development in our way of thinking [lo] is that, not only the gauge couplings evolve with energy, but the masses as well. A careful analysis shows [S] how interesting are the consequences of this conceptual development: the Msusy threshold goes from 21 TeV down to the present LEP energy scale range. This is shown in Fig. 2. Let me emphasize again that Msusy is a parameter. It does not correspond to any physical mass. We keep it for the sake of comparison with other calculations without EGM effect [ll]. The physically interesting results are those concerning the spectra of the lightest supersymmetric particles: charginos (xf12),neutralinos ( ~ o ~ ~ , ~ , ~ ) , gluinos, sleptons and squarks. An example of mass spectra prediction [12] is shown in Fig. 3. Moreover, to account for the light threshold (ATL), the heavy threshold (ATH) and the radiative effects due to the evolution of masses is perfectly possible and it allows the gauge couplings to converge at ~ I J T ,not to diverge above EGUTand have the lightest supersymmetric observable signal in the energy range of present existing facilities (Fig.4). This figure brings me to a very interesting point: i.e. &m and ESU. The first, h, is obtained from "below"; the second, E ~ uthe , String Unification Energy level, is obtained from "above". Taken at their central value these two quantities are two orders of magnitude apart. Nevertheless it should be realised that many uncertainties come into the derivation of J~JJT. We have performed a detailed study of this problem [8] and the results are shown in Fig. 5 . Indeed it is possible to get &UT as high as 1018 GeV when the various parameters are chosen accordingly. For the sake of discussion suppose EGUTwithin the minimal SU(5) model was definitely determined to be at 1016 GeV. Then the best candidate to fill the gap between &UT and Esu would be SU(S)xU(l), the so called "flipped" SU(5) whose most
185 3
interesting feature is that it is derivable from strings. On the other hand Nature, for the electroweak interactions has decided not to choose SU(2) but SU(2)xU(1). The most natural extension of the minimal SU(5) supergravity model is SU(S)xU(l). A detailed comparison of these two supergravity models, SU(5) and SU(S)xU(l), will be presented at the School [13]. Let me close this lecture with a graphical representation of the three coupled differential non linear equations describing the evolution, versus 4 2 , of the three gauge couplings CX~,CQ,CX~. The standard RGEs at two loops are used. The results are shown in Fig. 6 where the correlation among the three gauge couplings is shown in steps - each one being an order of magnitude above the previous one. The first point is at@=O.l TeV; the last point at 1016-2GeV, which is the value of &m obtained with the input values for a 3 and sin28w as indicated. If Nature would have followed the apparently simplest way (the straight line) we could not be here. Nature has followed the road illustrated by the sequence of the big dots in Fig. 6 and these predictions could not be there without the knowledge of the RGEs, the number of families and the values of ~~1,0r2,a3 measured at LEP.
186 4
References
[l]
A. Zichichi, Closing Lecture at the EPS Conference (York, UK, September 25-29, 1978); also Opening Lecture at the EPS Conference (Geneva, CH,27 June - 4 July, 1979) - CERN Proceedings; and "New Developments in Elementary Particle Physics", Rivista Nuovo Cimento, Vol 2, No. 14 (1979) 1. The statement on page 2 of this paper, "Unijcation of allforces needsfirst a Supersymmetry. This can be broken later, thus generating the sequence of the various forces of nature as we observe them was based on a work by A. Petermann and A. Zichichi where the renormalization group running of the couplings using Supersymmetry was studied with the result that at one loop the convergence of the three couplings was theoretically granted. This work was not published, but known to a few. 'I,
[2] U. Amaldi,W. de Boer,H.Fiirstenau, Phys. Lett. B 260 (1991) 447.
[3] F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, I1 Nuovo Cimento 104A (1991) 1817. [4] F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, I1 Nuovo Cimento 105 A (1992) 1025. [S] F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, I1 Nuovo
Cimento 105 A (1992) 581.
[6] F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, I1 Nuovo Cimento 105 A (1992) 1179. [7] F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, I1 Nuovo Cimento 105 A (1992) 1201.
[8] F. Anselmo, L. Cifarelli and A. ZichichiJ Nuovo Cimento 105 A (1992) 1335.
187 5
191 F. Anselmo, L. Cifarelli and A. Zichichi,Il Nuovo Cimento 105 A (1992) 1357. [lo] For a review see A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1. [ll] J. Ellis, S. Kelley, D.V. Nanopoulos, Nuclear Physics B 373 (1992) 55.
[12] J.L. Lopez, D.V. Nanopoulos and A. Zichichi, preprint CERNTH.6667/92, CTP-TAMU-68/92, ACT-20/92 (to be published in Nuclear Physics B). [I31 See "The Superworlds of SU(5) and SU(S)xU(l): A Critical Assessment and Overview" by J.L.Lopez, D.V. Nanopoulos and A. tichichi, in this volume.
188
6
Figure Captions Fig. 1
The wrong approach to understand SUSY [2].
Fig. 2
The Evolution of Gaugino Masses (the EGM effect) on the prediction for SUSY thresholds 151.
Fig. 3
An example of spectra for neutralinos (x01,23,4), charginos (x*l,2), sleptons, squarks, gluinos and Higgs as predicted by a Supergravity model with inputs (mt=130 GeV, ng=300 GeV and cl
Fig. 4
The correct approach to understand SUSY [3-111.
Fig. 5
The dependence of MGUTfrom a g ( M z ) and the ratio of the two crucial heavy threshold masses Mv/Mz, as described in the 183. Note that the extreme value for M Gis ~ above 1018 GeV.
Fig. 6
The evolution of the three gauge couplings (ai,a2,a3)and their mutual correlation as described by a supergravity model. Note that the number of families (N+) and the knowledge of a3 are vital inputs.
189 7
Physics is not Euclidean geometry
p1
x
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
M-
EGeV1
,M Figure 1
[GeVI
190
8
I
Predictions for SUSY threshold lower bound
B > Y ] without EGM corrections
? I TeV
191 9 0 0
1-
I'
I'
I'
>
GRAND UNIFICATION WITH SUSY
[ 1I a
24
I
.
Anselmo .Cllarelll~Pelsrmnnn Zkhkhl (ACPZ)
EVOLUTION OF MASSES INCLUDED
1 /a, 60
40
20
: :1
. .World Averages (at mz) = 127.9f0.2
1
sin 8 = 0.2334f 0.0008 = 0.1 18 f 0.008
GeV
IGRANSASSO I ATH z Heavy Threshold
Figure 4
193
11
0 World Average: WA WAflo WA f20
Figure 5
194 12
Supergravity Model NF 2 3 NH = 2
Msun = l o 2 GeV Em = 10'6'GeV CL,(M~)= 0.1 18 f 0.008
I
sin2e,(MZ)= 0.2334 & 0.0008
Figure 6
195
A. Peterman and A. Zichichi
ON A CLASS OF FINITE SIGMA-MODELS AND STRING VACUA: A SUPERSYMMETRIC EXTENSION
From
I1 Nuovo Cimento 106 A ( I 993) 719
1993
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197
IL NUOVO CIMENTO NOTE BREVI
VOL. 1 0 6 4 N. 5
Maggio 1993
On a Class of Finite Sigma-Models and String Vacua: a Supersymmetric Extension. A. PETER MAN(^)(^) and A. ZICHICHI(~) (l) CERN - Geneva, Switzerland (2) CNRS - Luminy, Marseilles, France (ricevuto il 15 Febbraio 1993; approvato il 20 Marzo 1993)
Summary. - Following a suggestion made by Tseytlin, we investigate the case when one replaces the transverse part of the bosonic action by an n = 2 supersymmetric sigma-model with a symmetric homogeneous Kahlerian target space. We demonstrate what has been conjectured by Tseytlin: i.e. the metric is shown to be exactly known since the exact expression of the beta-function reduces to its one-loop value.
PACS 11.17 - Theories of strings and other extended objects. PACS 04.60 - Quantum theory of gravitation. PACS 11.30 - Symmetry and conservation laws.
To find exact solutions to string vacua is one of the most interesting problems of present-day physics. In fact the hope is that some string vacua will have physical interpretation. An example of this type is the one[l] which exhibits blackhole-type solutions in two-dimensional target space. This example has been discovered within some gauged WZNW coset models [ll. Recently a new class of models has been introduced by Tseytlin[2]. These are a-models with Minkowskian signature. The key idea was t o have them finite. This can be done choosing those with symmetric target space metric and covariantly null Killing vector. In this note we plan to investigate a suggestion made by Tseytlin himselff21 and concerning the case when the so-called <
(*) All along this note, we shall refer to the terminology, to the notations and results, as published in[2]. In particular, formulas (4), (5), (6) and (7) are the formulas ( l l ) , (12), (19) and (20) of ref. [2], respectively.
719
198
720
A. PETERMAN and A. ZICHICHI
Reference [21 considers the line element: (1)
ds'=G,,dX'dx'=
-2d~d~+f(~)yij(x)dx~d~j with p, v = 0, 1, ..., N , N
+ 1; i,j, ..., N ,
and the yzj.corresponding to a symmetric space (constant curvature). It is then shown that, with a specific choice of f(u>,the o-model with (1) as target space metric is ultraviolet finite. The part of (1) proportional tof(u) is referred to as the <
K
=R/N,
said maximally symmetric, is considered in order t o make easier the perturbative expansions of the various quantities necessary to the interpretation of the model. We shall not discuss any further Tseytlin's model except for the parts relevant to our purpose. The finiteness of the model on a flat 2d background (for renormalization techniques in non-flat spaces, see for instance ref.[41) needs the condition D(,M") = 0
Pf" + D,,M",
(3)
=0
with pFV referring t o the full 0-model with target space metric (1)beta function and M , a vector to be determined for each separate case in order to satisfy (3). The analysis of ref.[2] leads, among others, t o the following basic results: (4) with (5)
P(f) = a + ( N - 1)-l u'f-' + N+3 ( N - 1)-' a 3 f P 2+ O ( a 4 f - 3 ) , 4
u
= a'K.
f-' plays a role of coupling of the symmetric space o-model and satisfies
with has (7)
'C
a kind of RG &me,) parameter defined just after (18, ref.[2]). Of course one
f(u)= U ( T
+ (N - I)-' log + 'C
0cT-l))
;
T
=
T(u)
when P(f) is given by (5) above. Now we come t o the transverse part, which, we recall, is N-dimensional in the target space. One knows from previous studies that this transverse part enjoys an n = 2 supersymmetry if its transverse space is Kahlerian. Moreover we know from a particular example[3] that its beta function is exactly given by its first term (1-loop). The example we refer to requires in addition that the transverse space not only be Kahlerian but also homogeneous. Appropriate Kahler manifolds of this type can be found in the literature (ref. [3] and reference therein).
199 72 1
ON A CLASS OF FINITE SIGMA-MODELS ETC.
Suppose our choice of space is such a manifold, in this case ( 5 ) and (7) reduce t o
(8) (9)
p(f)
=
const
= C,
f(u) = c.z(u) .
z is well defined in each case in terms of u, as said before and therefore the metric (1) is given exactly by
(10)
d s 2 = -2dudv+C.zyij(x)dxidx’.
We conclude that the results @)-(lo) confirm, in the specific case considered, the conjecture made by Tseytlin in the footnote 3 of ref. [2]. At this point we have not yet identified any string vacua. However, depending on whether the vector M , in (3) is an exact gradient, the appropriate dilaton field can be exactly determined in order to satisfy Weyl invariance of the model which then leads to exact string vacua represented by the resulting backgrounds.
REFERENCES A. SENGUPTA and S. WADIA:Mod. Phys. Lett. A, 6 , 1685 (1991); R. DIJKGRAAF, [l] G. MANDAL, H. VERLINDE and E. VERLINDE: Nucl, Phys. B, 371, 269 (1991); A. A. TSEYTLIN: Phys. Lett. B, 268, 175 (1991) and Johns-Hopkins Univ. Preprint JHU-TIPAC 91009; E. WITTEN:Phys. Rev. 0, 44, 314 (1991); M. MULLER:Nucl. Phys. B, 337, 37 (1990); S. ELITZUR, A. FORGE and E. RABINOVICI:NucL Phys. B, 359, 581 (1991); M. ROCEK,K. SCHOUTENS and A. SEVRIN:IAS Preprint IASSNS-HEP-91/14. [2] A. A. TSEYTLIN:Cambridge preprint DAMTP-92-26 hepth@xxx/9205058. [31 A. MOROZOV, A. PERELOMOV and M. SHIFMAN: Nucl. Phys. B, 248, 279 (1984); A. MOROZOV and A. PERELOMOV: 2. Eksp. Teor. Fiz. Pis’ma Red., 40, 38 (1984). [4] A. A. TSEYTLIN: Nucl. Phys. B, 294, 383 (1987); I. JACK and H. OSBORN:Nucl. Phys. B, 343, 647 (1990) and references therein.
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A. Peterman and A. Zichichi
A SEARCH FOR EXACT SUPERSTRING VACUA
From I1 Nuovo Cimento - Note Brevi 107 A ( 1994) 333
1994
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203 IL NUOVO CIMENTO
VOL.107A, N. 2
Febbraio 1994
NOTE BREW
A Search for Exact Superstring Vacua. A. PETERMAN and A. ZICHICHI CERN - Geneva, Switzerland (ricevuto il 16 Luglio 1993; approvato il 21 Ottobre 1993)
Summary. - We investigate 2d u-models with a (2 + N)-dimensional Minkowski signature target space metric and Killing symmetry, specifically supersymmetrized, and see under which conditions they might lead to corresponding exact string vacua. It appears that the issue relies heavily on the properties of the vector M,, a reparametrization term, which needs to possess a definite form for the Weyl invariance to be satisfied. We give, in the n = 1 supersymmetric case, two non-renormalization theorems from which we can relate the u component of M , to the pfu function. We work out this (u, u ) component of the pG function and find a non-vanishing contribution at four loops. Therefore, it turns out that at order a there are in general non-vanishing contributions to M u that prevent us from deducing superstring vacua in closed form. 14,
PACS 11.17 - Theories of strings and other extended objects. PACS 11.30.Pb - Supersymmetry. PACS 04.60 - Quantum theory of gravitation.
1. - Introduction. To find exact string vacua is one of the most interesting problems of string theory. In fact, the symmetries of the vacuum are the symmetries of the World[l], the constants of the vacuum are the constants of the World[2] and therefore to know the string vacua is indeed the first crucial step towards a TOE. In the past few months, attention has been brought to 2d finite sigma-models with a (2 N)-dimensional Minkowski signature target space metric with a covariantly constant null Killing vector. They can be considered as describing string tree-level backgrounds consisting in plane gravitational-wave type, supplemented by a dilaton background. Such kinds of solution of Einstein equations have been discussed long ago by Brinkmann [3]. These models have been extensively studiedl41 in the bosonic case and show several important features such as: 1) the W finiteness (on shell) of these models; 2) that, given a non-conformal a-model with Euclidean N-dimensional target space (the so-called transverse space), there exists a conformal invariant Minkowskian a-model in 2 + N dimensions;
+
333
204
334
k PETERMAN
and
A. ZICHICHI
3) that, because of the Killing symmetry, the (2 + N)-dimensional metric does not depend on one of the two extra coordinates u and v;
4) the fact that the 2 + N dimensional metric is expreyed in terms of the running coupling of the transverse theory. Later, Tseytlin [5] and the present authors [6] discussed independently a specific supersymmetric extension of this class of models, which allows one to know exactly the transverse beta-function, thanks to a non-renormalization theorem [7], and therefore to be able to give the line element in closed form. But, having in mind exact vacua identification, we need also to know an appropriate dilaton field such that the Weyl invariance conditions are satisfied. Only then will the resulting models correspond to string vacua. That such a dilaton field exists has been proved order by order in a perturbation expansion in a ' . However, this does not allow in general a formulation in closed form. The crucial point is a piece of information on W,, the u-component of W,, a vector that enters linearly in the reparametrization vector M, (cf. sect. 2); W, is a covariant vector originating through the mixing under renormalization of dimension-2 composite operators [8]. While earlier discussions on this point had been based on conjectures (see [5,6,9, lo]), none of them seems to be satisfactory, being either incomplete [5,10] or only necessary but not sufficient [6] (see also [ll]). It is the aim of this paper to go deeper into this question. In order to get concrete information, we have been looking at the first non-vanishing contribution to W,, if it exists at all. In the bosonic case, it is well known[4] that W, starts to be nonvanishing at a l 3 and behaves like u - 3 (u is a light-cone coordinate) (*). In the n = 1 supersymmetric case under examination in the present paper, W, has been computed to order a in two essentially different ways: first directly; and secondly via the nonrenormalization theorem, which binds 3, W, = W, and the ,BEu component of ,B,"y, the beta function of G,". The two coinciding results show that W,, with a u - behaviour, ~ starts to be non-vanishing at order ar4for general N. This work is organized as follows: in sect. 2 we recall the general features of the n = 1 supersymmetric model, which we shall investigate and the two non-renormalization theorems uncovered in this model, in addition to the well-known theorems on the homogeneous Kiihler transverse space (hence n = 2 supersymmetric). In sect.3, we give the results for the ,:p component as derived from two different sources and for the direct computation of the W, component of W, . In sect. 4, we give and discuss these results for a simple homogeneous manifold as transverse space. Finally, in sect. 5, we present our conclusions. l4
2. - The supersymmetric model and the non-renormalization theorems.
In the class of finite 2d a-models introduced in ref.[4], the N-dimensional transverse space was supposed to be a Euclidean symmetric space. Order by order it was shown that there exists a dilaton which, together with the (2 + N)-dimensional metric background, solves the Weyl invariance conditions. The metric of the (2 N ) -
+
(*) For instance, the %loop contribution to p,, due to the term D,Ra,raD,Rafirs reads -(d3/16)f2 f - 4 RZ . .N Rijkl;i, j, k, 1 are transverse indices, and f = bu.
205
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A SEARCH FOR EXACT SUPERSTRING VACUA
dimensional space turns out to give the line element ds2 = GrYdxp dx' =
(1)
- 2 dudv
+ f (u)yij dxi dxj , p,
!J=o, ..., N + 1; i , j = 1, ..., N .
With a specific choice off (u),the model is shown to be W-finite; f(u) is bound to satisfy a first-order RG equation (2)
pf=
f= a,f;
P(f 1;
p
= const
with p( f ) defined by the transverse p$
p$
(3)
= P ( f ) yij
*
In order to complete the proof of finiteness of the model on a flat 2d background, the beta function of the 2 N a-model with target space metric GPYhas to vanish up to a reparametrization term [S]:
+
pFv + 2D,, M,,
(4)
=0.
M, is not arbitrary and to establish that the c-model based on (1) is Weyl-invariant, one needs to show that a dilaton field exists such that M , in (4)can be represented by
+
M,
(5)
=.'a,+
+ -21 w,,
the origin of W, having been specified in the introduction. The n = 1 supersymmetric extension [5,6] of the model with bosonic action [4]has been done as schematically indicated below. One replaces the bosonic action
I* = ( 4 x u ' ) - '
\ d2zfi[GpY(x)dOlxPdaxXY + u'R(~)+(x)]
by the following superfield action:
I = Ic + I,
+
ey.3, and E-' is the determinant of the n = 1 supervielbein. hen obvious how t o specialize the general metric GrYto a null Killing vector metric as in (l),in terms of the real superfields U,V and Xt
(9)
I=
(4xu')-'
\ d2~d20[-2DUDV+gij(UX).DXiDXj].
The generalization of the bosonic case studied in[4] to the supersymmetric case is
206 336
A. PETERMAN
and
A. ZICHICHI
then straightforward and the finiteness condition for symmetric spaces (2)(*) will also be determined by the beta-function of the transverse part of (9). If this transverse space is chosen to be Kiihlerian, the N-dimensional part is n = 2 supersymmetric. The choice in [5] and [6] was even more restrictive, assuming the transverse space to be a symmetric homogeneous Kiihler manifold. This was dictated by the fact that known examples of these manifolds exist, for which the betafunction reduces to its one-loop expression and is therefore exactly known “7,121. In this case, p ( f ) reduces to a constant a, depending on the manifold chosen and its symmetries (remember that f is the inverse of the generic transverse r-model coupling A); f ( u ) becomes equal to
with b = u p - ’ , and the transverse metric g i j ( u , 2) is b u y i j . Evidently the n = 2 supersymmetry is not shared by the full 2 + N model with Minkowski signature studied here and has only n = 1 by construction (cf. eqs. (71, (9)). However, use can be made of the result [13] that in the n = 2 case, the dilaton coupling does not get renormalized, so that some quantities appearing in the Weyl anomaly coefficients of the transverse part do vanish in the minimal subtraction scheme we use throughout (seeL51 for details). Therefore the 2 N model with n = 1 supersymmetry and with homogeneous symmetrical Kiihler transverse subspace has a simplified structure, as compared with the generic n = 1 r-models. However, the u components of the various key quantities such as W,, pz, and the <
+
for symmetric transverse spaces, do not seem to benefit from the special properties of the transverse part. Nevertheless, one can formulate two non-renormalization theorems concerning the three quantities @(u),W, and pf,: i)
46 + W, = 0 beyond
ii) W,
+ 2pE,
=0
one loop,
beyond one loop
+
(at the one-loop level, however, 4& W, = N/2u and W, + 2pf, = - N 12u 2>. As already emphasized in the introduction, the issue of whether W, is identically zero or starts being non-vanishing at some high order is crucial. For, if zero, then the dilaton field can be integrated to a closed form. If not, this closed form will only be the starting value of a perturbation expansion in a’ and the associated string vacuum, though existing, can only be expressed perturbatively. No string vacuum can be given in closed form if W, f 0, according to the present methods. In a first attempt to clarify the situation, we computed both W, and pf, in the n = 1 supersymmetric case. The results are shortly presented in the next section.
(*) With metric tensor given by
g i j ( u , z) = f(u)yij(z).
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A SEARCH FOR EXACT SUPERSTRING VACUA
3.
- W,
and p:, at four loops.
We have explicitly worked out at four loops the quantity W, and the (u, u ) component of the n = 1 supersymmetric beta-function in the simplest non-trivial case, when the subspace (N-dimensional transverse space) is locally symmetric. The direct calculation of W, comes out with the result
Although we can derive the p:, from (12) and the non-renormalization theorem, we evaluated it, as a cross-check, from two different sources:
a) the direct calculation[14] of the beta-function in n = 1 supersymmetric non-linear generic a-models, with the result
means R i j k l ( Y ) ; b) the simple derivatives of a scalar built out from the sum of two different contractions of the product of three Riemann tensors[ll], confirming eq. (13). Accessorily, it can be verified that
Rijkl
.
1 2
4 = -pG
1 . w,, at four loops 4
= --
uu
so that
(15)
p:,
+ mu+ 28 =p:,
=0,
which is the Weyl invariance condition for the (u,u)component of the gravitational pfv Weyl anomaly coefficient. This shows, if necessary, that the dilation field +, eq. (ll), will receive contributions from higher order, which are due to a-model interactions. 4. - Concrete example.
We want to apply here the general formulas we have found in the previous sections and verify accessorily that the non-renormalization theorems are satisfied. There are general Riemannian manifolds called spaces of constant curvature, i.e. whose curvature is independent both of the surface direction and the position (for details, see, e.g., ref.[3b], sect. IS). These manifolds have a particularly simple (*) The covariant derivatives D, differ from ordinary a,, the connections r;, being non-vanishing for i, j transverse.
208
338
A. PETERMAN
and
A. ZICHICHI
curvature tensor, said maximally symmetric, which reads
R being the constant curvature and N the dimension of space (transverse in our case). Of course these manifolds, for instance the N-dimensional sphere S N embedded in a Euclidean R N + l space, are Riemannian but not Kahlerian in general. Therefore they are not suitable for our transverse space, which we assumed to be homogeneous Kahler. However, it happens that for N = 2 this Riemannian space is also homogeneous Kahler, due to the accidental isomorphism between S2= S0(3)/S0(2) and CP'. So, for N = 2, this model is homogeneous Kiihler and possesses all the properties wanted for our N-dimensional transverse space. Also the metric (16) can be used and it is straightforward to establish that (12) takes the value
and (13)
Equations (17) and (18) obviously verify (14) and the non-renormalization theorem of sect. 2. One notes also that the generic four-loop term contributes, for n = 1 supersymmetry, a quartic expression in terms of the curvature tensor, which is the only one to survive in locally symmetric spaces for general N. However, using the metric (16), it gives a contribution proportional to ( N - 2) and therefore vanishes in our example for which N = 2. This exemplifies the consistency of the approach: the quartic term has a pure Riemannian origin, but must be absent in a Kahler geometry, as it does in our example. 5. - Conclusions.
In spite of the well-known fact that, even in the simplest models with n = 1 supersymmetry, the four-loop beta-function is non-vanishing, witnessing the tree-level string theory graviton scattering modification to Einstein action [16], the model introduced by Tseytlin [4] and conveniently supersymmetrized [5,6] is so specific the hope was not unreasonable to see it escaping tne four-loop contributions component of the beta-function. As a matter of fact, and as a posteriori to the (u,u) justification of our attempt, the genuine Riemannian d4 contribution to P,"", proportional to R 4 , produces zero contribution to ,BEu in the present model(*),
(*I As it does also of course to ,B$ or ,B$ since the N-space has been assumed (homogeneous) Kiihlerian.
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A SEARCH FOR EXACT SUPERSTRING VACUA
while it is fully contributing to the generic n = 1 supersymmetric 0-model (see for instance [ll]).This is an example of its specificity. Summarizing, it turns out that only an n = 4 supersymmetry allows a perfect knowledge of the backgrounds in closed form. With a hyper-Kahler transverse space, the beta-function of this transverse part is identically zero, in all renormalization schemes (RS) and f is constant. The metric is trivially simple. Moreover, if non-renormalization theorems might exist too, they will not necessarily be the same as those of sect. 2. But if W, starts getting non-zero contributions at some high order, even if it is at the four-loop level, there will necessarily be a scheme in which it vanishes identically and for which the beta function is still identically zero, since this property is RG-invariant. This is in contrast with the situation in the present paper. We can indeed find a scheme in which W, is zero at all orders. However, we will lose the exact knowledge of the metric due to p(f) = a , the .vanishing of all loop contributions but the first. being not an RG-invariant property, but specific to particular schemes. We plan t o come back on these general questions in a forthcoming publication [12].
*** We would like to acknowledge several enlightening conversations and correspondence with A. A. Tseytlin. One of us (AP) thanks also the Theoretical Physics Centre of the CNRS at Luminy-Marseilles for the hospitality.
REFERENCES
Secret symmetry: a n introduction to spontaneous symmetry breakdown and [l] S. COLEMAN: gauge fields, in Laws of Hadronic Matter, edited by A. ZICHICHI(Academic Press, New York and London, 1975), p. 139. [2] G. VENEZIANO:Quantum strings and the constants of nature, in The Challenging Questions, edited by A. ZICHICHI(Plenum Press, New York, N.Y., 1990), p. 199. [3] a ) H.-W. BRINKMA": Proc. Natl. Acad. Sci. U.S.A., 9, 1 (1923); Math. Ann., 94, 119 (1925). See also b ) W. PAULI:Theory of Relativity (Pergamon Press, Oxford, 1958). [4] A. A. TSEYTLIN:Phys. Lett. B, 288, 279 (1992) and DAMTP-92-26; Nuch Phys. B, 390, 153 (1993). [5] A. A. TSEYTLIN:Phys. Rev. D, 47, 3421 (1993) and Imperial/TP/92-93/7. [6] A. PETERMA"and A. ZICHICHI:Nuovo Cimento A, 106, 719 (1993), CERN/TH.6828/93; CPT-93/P2879. A. PERELOMOV and M. SHIFMAN:Nucl. Phys. B, 248, 279 (1984) [7] A. MOROZOV, 181 A. A. TSEYTLIN:Nucl. Phys. B, 294, 383 (1987); H. OSBORN:Nucl. Phys. B, 294, 595 (1987). [91 A. A. TSEYTLIN:CERN/TH.6820/93. 1101 A. A. TSEYTLIN:CERN/TH.6783/93. [Ill Q.-HANPARKand D. ZANON:Phys. Rev. D, 35, 4038 (1987). [121 A. PETERMANN and A. ZICHICHI:work in preparation. [13] P. E. HAAGENSEN: Int. J. Mod. Phys. A, 5, 1561 (1990); I. JACK and D. R. T JONES: Phys. Lett. B, 220, 176 (1989); M. T. GRISARUand D. ZANON:Phys. Lett. B, 184, 209 (1987). [141 M. T. GRISARU,A. VAN DE VEN and D. ZANON:Nucl. Phys. B, 277, 409 (1986). [15] M. T. GRISARUand D. ZANON:Phys. Lett. B, 177, 347 (1986): M. FREEMAN,C. N. POPE, M. SOHNIUS and K. STELLE: Phys. Lett. B, 178, 199 (1986). [16] D. J. GROSSand E. WITTEN:Nucl. Phys. B, 277, 1 (1986).
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21 i
A. Peterman and A. Zichichi
PROOF OF THE EQUIVALENCE BETWEEN DOUBLE SCALING LIMIT AND FINITE-SIZE SCALING HYPOTHESIS
From
I1 Nuovo Cimento - Note Brevi 107 A ( I 994) 507
I994
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IL NUOVO CIMENTO
VOL. 1074 N. 3
Marzo 1994
NOTE BREVI
Proof of the Equivalence between Double Scaling Limit and Finite-Size Scaling Hypothesis. A. PETERMANand A. ZICHICHI CERN - 1211 Geneva 23. Switzerland (ricevuto il 10 Settembre 1993; approvato il 3 Febbraio 1994)
Summary. - The computation of the exponents of the string partition function is one of the most crucial problems in string theory. There are very powerful and well-experienced statistical mechanics tools, devised for exponents calculation. If these tools could be used in string theory, a great step forward could be foreseen in the computation of the exponents of the string partition function. For this step to be accomplished, it would be necessary to show the complete equivalence of the double scaling limit in stringy matrix models and the finite-size scaling hypothesis in statistical systems. We start with two-dimensional systems and prove this equivalence. On the other hand, the finite-size scaling can be extended to arbitrary d-dimensional (d 2 1) systems. PACS 11.17 - Theories of strings and other extended objects.
I) In early 1993 a very interesting paper appeared [l] on N-vector models with the aim of computing accurately the critical exponents. Instead of going into the more complex case of the N x N matrix model extensively studied in connection with string theory, the investigation for N-vector models was made with the following idea in mind. Brezin and Zinn-Justin [2] had proposed an RG-based method, together with a practical method, to obtain the relevant beta-function. The straightforward integration over a part of degree of freedom of the matrix in the RG spirit led, to first order, to qualitatively reasonable values of the exponents, but the authors of ref. [l] did not succeed in improving this result when proceeding to higher orders. Whence their interest in a simpler case in order t o see if an RG-based approach had some meaning and validity. However, their approach was exactly in the same spirit as that of ref. [21 (using double scaling limit appropriate to d = 1 and RG equation ii la Callan-Symanzik). In a straightforward calculation of the {pi} function, they found unsatisfactory results and made the observation that, in order t o determine unambiguously the RG flow in the coupling space, it was necessary to perform field redefinitions. The reparametrization identities they found in proceeding this way supplemented the straightforward method. Applying this procedure to the one507
214
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coupling case, they were able to reproduce the exact results for the rn = 2 critical point, namely a fured point of the beta-function at - 1/4 with the susceptibility exponents 3/2 and 1/2 for y1 and y o , respectively (extension to coupling space of higher dimensions was as successful). Reading these conclusions, it appeared to us that the double scaling limit was not unrelated to fmite-size scaling in statistical mechanics and if it could be made concrete, this similarity would allow to utilize the methods used there, in general numerical, but ancient enough to have proved their efficiency in the calculation of critical exponents. This way we might avoid the reparametrization procedure. In fact, there is a complete similarity between the double scaling limit in N-dependent models and finite-size scaling of size L statistical systems. 11) First we recall that:
a) The double scaling limit in string theory asserts that: lim ((A * - A ) G -2/Y1)
A+A*;
with C finite ,
= C,
G+O,
with A the cosmological constant, G the string coupling, y1 - string susceptibility exponent. Starred quantities are their fured point values. Translated in matrix model language, it reads: lim(g - g*)N2/y1= C , g+g*; N + W .
(C finite at disposal) ;
Here g is the matrix model coupling; N, the size of the matrix. b) Now we quote some formulae (taken over from <<paperscollection. publications in finite-size scaling, namely ref. [3]) which trivially show the equivalence of both formulations (double scaling and finite-size scaling). The finite-size scaling hypothesis assumes that the ratio of t;/a and L / u remains finite close to the critical point, the microscopic lattice spacing <> dropping out (t; is the correlation length of the system). More precisely, L / t ( t ) ,with t - T, - T , should tend to a fixed limit C when t - t * + O and L+ a. As t ( t ) - It\-”, L/t;(t)- ( t - t*)L’/’ and
(C f i i t e ) ;
lim(t-t*)L1/’=C,
t+t*;
L4W.
The strict anology with (1) is evident as v , the correlation length exponent, can be shown to satisfy by explicit RG calculations (3)
1
v = -
6;
(for 2nd order transitions);
,f3;
being the value of -
Then, the identification of ref. [2]
p ; = 2/yl, completes the argument.
for d = 2
215
PROOF OF THE EQUNALENCE BETWEEN DOUBLE SCALING LIMIT ETC.
509
The identification of a) and b) cases leads to the conclusion that the double scaling limit in matrix model is the fmite-size scaling hypothesis, at least for such d = 2 statistical systems. The size N of the matrix corresponds to the size L of the statistical system in a given geometry, and the string susceptibility exponent is related by a precise relation to the correlation length exponent. So the RG machinery can be put into action. The onset of a real space RG is (see sect. 1.1 of [3a] and formula (1.4) in it, for instance) the following formula for the free energy: (4)
f( ) = free energy and b is the coarse graining of the lattice spacing; a + ba; b > 1, according to the RG ideology. This leads for the free energy, when we deal with a single coupling {k} = t , to the following form of the singular part off near the critical point: (5)
f(L,t ) - t2’$(tLl/”);
for d = 2
or, in matrix model notations, mutatis mutandis
-
f(N, 9) ( g * - g ) 2 / P m 7 * - g)N@il
since v - l = p : . Or, with the identification y o + y1 = 2; y1 = 2/pL (6)
f(A, G) = (A* - ~ i ) ~ - y o $ [ ( A *- A ) G-2/Yl]
the known singular formula of strings(*). 111) With these precise analogies, we suggest to use very efficient techniques in the determination of g, , p( g ) hence p‘(gz), issued from statistical mechanics. We see different possibilities of investigation, namely:
i) We may treat matrix models with finite-size scaling and apply real space RG which has been extensively studied numerically on a great variety of problems and is by now very powerful. ii) Alternatively, going back to Lee and Yang[4], one may enter the complex temperature domain (or more generally the coupling complex space with flows). The study of Itzykson et al. [5] of the complex zeroes of the partition function and their scaling shows how this can also be a very powerful tool in statistical mechanics. Using it for the string partition function will certainly provide fair numerical estimates of the string y1 exponent.
(*) The case d = 2 treated in this note can be extended to any d 3 1, by retaining d unspecified in the generic formulae. With d = 1,we tried to repeat the one coupling case with c = 0 of ref. [l], which inspired the present note, using two different numerical methods. Fair agreement with the m = 2 exact result obtained. With satisfactory convergence, if we consider the rather crude approximation which we used. Careful investigation of more elaborate cases are planned in order to try to select the most efficient methods at disposal in statistical systems.
216 510
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iii) The more recent proposal of [2], to use conventional RG & la CallanSymanzik might perhaps also lead to solutions, although the techniques are presently not as powerful as those developed in the examples mentioned above. And as said at the beginning of this note, the field reparametrizations done with success in the N-vector model ( d = 1) [l] are quite cumbersome and will most probably be more tedious in the N x N matrices case ( d = 2). Therefore we suggest that, owing to the perfect analogy with finite-size scaling, the techniques developed in this field since almost two decades should be applied with possible success in the computation of string exponents. Though numerical and approximate, they lead in general to an excellent level of precision. Here we have particularly in mind the <>pioneered by Nightingale [6]. Then using this renonnalization interpretation of finite systems, we might exploit it to infer the crucial quantities like P(t),P’(t = t * ) of the bulk from the finite systems, under the form of Roomany-Wyld approximants [7]. Indeed the latter can vanish for finite systems, for which the exact beta-function never does.
IV) In conclusion, although the remarkable equivalence we found of the double scaling limit in matrix models with the finite-size scaling of two-dimensional statistical systems is interesting p e r se, we think it unlikely that it could be of no help when we restrict ourselves to exponents calculations. REFERENCES [l] S. HIGUCHI,C. ITOI and N. SAKAI:preprint TIT/HEP-215, NUP-A/93-4, hepth@xxx/ 9303090, March 1933. [Z] E. BREZIN and J. ZINN-JUSTIN:Phys. Lett. B, 288, 54 (1992). See also J. ALFARO and P. DAMGAARD: Phys. Lett. B, 289, 342 (1992). [3] a ) J. L. CARDY(Editor): Finite-Size Scaling (North-Holland, Elsevier Science Publ., 1988); b ) M. N. BARBER:Finite-size scaling, in Phase Transitions and Critical Phenomena, edited by C. DOMBand J. LEBOWITZ, Vol. 8 (Academic Press, 1983). [4] T. D. LEE and C. N. YANG:Phys. Rev., 87, 410 (1952); M. E. FISHER: Phys. Rev. Lett., 40, 1610 (1978); A. KOCIC:Phys. Lett. B, 281, 309 (1992). [5] a ) R. B. PEARSON: Phys. Rev. B, 26, 6285 (1982); b) C. ITZYKSON, R. B. PEARSON and J. B. ZUBER:Nucl. Phys. B, 220,415 (1983); c) for Monte Carlo computing the partition function for complex temperatures, see, for instance, M. FALCONI, E. MARINARY, M. PACIELLO, G. PARISI and B. TAGLIENTI: Phys. Lett. B, 108, 331 (1982). [6] M. P. NIGHTINGALE: Physica A, 83, 561 (1976); Phys. Lett. A, 59, 486 (1977). [7] H. H. ROOMANYand H. W. WYLD:Phys. Rev. D, 21, 3341 (1980). See also[3b] for a review.
@ by SoeietA Italiana di Fisica Proprieta letteraria riservata Direttore responsabile: RENATO ANGEL0 RICCI Questo fascicolo B stat0 realizzato in fotocomposizione dalla Monograf, Bologna e stampato dalla tipografia Compositori, Bologna su SERENA MATT, earta patinata ecologica non riciclata prodotta dalle Cartiere del Garda S.p.A., Riva del Garda (TN) nel mese di aprile 1994
Questo periodic0 6 iscritto all’Unione Stampa Periodica Italiana
217
A. Peterman and A. Zichichi
EXPLICIT SUPERSTRING VACUA IN A BACKGROUND OF GRAVITATIONAL WAVES AND DILATON
From I1 Nuovo Cimento 108 A ( I 995) 97
I995
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219
IL NUOVO CIMENTO
VOL.lOSA, N. 1
Gennaio 1995
Explicit Superstring Vacua in a Background of Gravitational Waves and Dilaton. A. PETERMAN (I) (2) and k ZICHICHI(2) ( I ) CNRS - L u m i n y - Marseille, France ( 2 ) CERN - Geneva, Switzerland (ricevuto il 15 Luglio 1994; approvato il 10 Ottobre 1994) I
Summary. - We present an explicit solution of superstring effective equations, represented by gravitational waves and dilaton backgrounds. Particular solutions will be examined in a forthcoming note. PACS 11.30.Pb - Supersymmetry. PACS 11.17 - Theories of strings and other extended objects. PACS 04.60 - Quantum theory of gravitation.
1. - Introduction. In the conclusions of a recent paper on supersymmetric non-linear sigma models with Brinkmann metric[l], we announced the possibility to display explicit metric and dilaton backgrounds which satisfy the Weyl conditions. The assumption, for obtaining this result, was that the transverse space of the model be hypergahler, therefore implying for this model an N = 4 supersymmetry. Such supersymmetric model has been worked out explicitly. It provides the backgrounds for both the metric and the dilaton in closed form and is the purpose of this paper. After a brief outline of the hyperKahler manifold chosen for the transverse space of the model, we give the metric in this space, that was taken in the Calabi series [21, i:e. a cotangent bundle over CP1:T*(CP'). The background for this model is notoriously self-dual: of the Eguchi-Hanson type. At this point the crucial feature is that N = 4 supersymmetric sigma-models (requiring a manifold which is hyperKahler), are known [3] to have an identically zero beta-function (*). Accordingly the
(*) Notice that preservation of N = 4 supersymmetry in perturbation theory requires that the metric plus eventual counterterms preserve the Ricci-flatness [4].
97
220 98
A. PETERMAN
and
A. ZICHICHI
Weyl-invariant background for the metric is equal, up to a multiplicative constant (*), to the original metric one started with. Moreover, the generic reparametrization term M,, reduces to M,, = a,,# since W,, can be shown to be zero for such models having vanishing beta-function of their transverse part. Finally, as these hyperKahler spaces are necessarily Ricci-flat [41, the dilaton may be reduced to its linear form in the O-th and (n 1)-th coordinates of the model, i.e. the light cone coordinates u and v. A s conclusions and perspectives, we call the reader’s attention on the following fact: metric on various hyperKahler manifolds are seldom explicitly known, although, for most of them, existence theorems can be proven (for instance the celebrated K3-spaces). Recent studies, however, have considerably enlarged the field [5]. The progresses in this direction should allow one t o formulate a whole series of valuable models with non-trivial space-time. The existence of an N = 4 world sheet superconformal symmetry seems to have a stabilizing effect on the perturbative solution. Actually we refer to the advances mades in[5c] on superstrings in wormhole-like backgrounds. The basic features of the supersymmetric non-linear sigma-model with Brinkmann metric, Minkowski signature and covariantly constant null Killing vector, will of course not be recalled here, as a series of rather exhaustive papers on the subject have appeared these last two yearsE6-91. We recall the general form of the line element for a suitable choice of coordinates
+
(1)
ds2 = g,,” dx“ dx ” = - 2 du d~ + gij(x, U ) dxi dxj ,
+
1, ... , n,n 1; i ,j = 1, 2, ..., n (real indices). The transverse part (with Latin indices) has a metric that we will choose in such a way as it complies with the manifold described above on which transverse strengths take values. We will start to alleviate the computation by assuming that gij(x, u ) can be brought t o the form
p , v = 0,
(2)
) (XI gij (x, U ) = f ( ~yij
as was done in [6-91. This by no means prohibits a possible but more involved treatment of the problem by invoking further additional symmetries which we shall not need to take into account here. The function f(u)in (2) can be shown to be the inverse coupling of the supersymmetric transverse sigma-model and is therefore defined for a family of theories with various values of u. It can be shown that f ( u )is running with u and will satisfy a RG-like equation (3) with p a constant and by the transverse p; (4)
P(f) the beta-function of the transverse sigma-model, defined
P; = P(f> y i j
*
Finiteness of the model on a flat 2d background requires in addition that the n
+2
(*) In principle the sigma-model inverse constant f might be a complicated function of the ( n + l ) - t h coordinate u. Here it is constant.
22 1
99
EXPLICIT SUPERSTRING VACUA IN A BACKGROUND ETC.
sigma-model with target space metric gPv beta-function has to vanish up to a reparametrization term D(,My). M, is not arbitrary, and to establish Weyl invariance of the model, the existence of an adequate dilaton background must be proved, such that M,, is represented by
+
M,, = u'a,+
(5) 2.
- The
+ -21 W,
explicit model.
The origin of W, has been discussed in several papers (see[6-91 and[113 for instance) and we do not repeat here the information we have on it. Similarly we do not repeat either the N = 1 supersymmetric extension [8,9] of the model with bosonic action [6, '71. In previous works on supersymmetric models of the kind studied in this note, the number of supersymmetries considered for the transverse part was N = 2. Accordingly, Kiihler manifolds were considered and especially symmetric and homogeneous Kahler manifolds. These had the properties that the beta-function in (3) reduced t o a constant and, by integration, f(u)was a constant times u, i.e. f ( u >= bu
(Kahler transverse space, homogeneous and symmetric). In the present note on the same class of models, we assume the transverse space to be an hyperKahler manifold. We will suppose the reader somewhat familiar with differential geometry and Lie groups in order to avoid to discuss here the holonomy group of such manifolds, in particular the absence of a U(1) factor in the holonomy group(*) being the signature of a Ricci-flat manifold. Also we suppose the reader acquainted with the Clifford algebra fulfilled by their three complex structures. Anyhow, one of the main features that must be kept in mind is that hyperK5hler manifolds are Kahlerian and quaternionic. For concreteness we specify the metric of our transverse space, as explained at the beginning of this note, to be that of a cotangent bundle over CP': T*(CP'). The local hyperKahler structure of this space is proved by observing three linearly independent covariantly constant complex structures (or equivalently three closed 2-forms on T*(M)) and show their interrelation through the SU(2) = Sp( 1) group transformations. This can be found in the literature [12-141. Another feature to take into account is that, as can be seen by construction, a higher symmetry of the metric on T* ( M ) is guaranteed or at least coinciding with the symmetry G of the initial manifold M , here CP' . The latter is known to be a symmetric space isomorphic to SU( 2 ) / U (1) (or SO( 3)/SO(2)). With our example (T* (CP')) we automatically specialize to a transverse space which is symmetric. Thus we can appreciably simplify the analysis(**). In particular, there are both: 1)
(*) The generator of this U( 1) factor is the Ricci form R&jdzYA dZd (complex indices). (**) Rigorously speaking, we have already implicitly used this symmetry property in writing (3)
and (4),although parent relations can be written down for non-symmetric spaces a t the price of a much more involved analysis (for comments see, for example, ref. [8]). Personally we did not try to examine this case at this moment.
222 100
A. PETERMAN
and
A. ZICHICHI
restrictions on the form of the dilaton potential which must be independent of the transverse space coordinates and can be written as $ = pv
(6)
+ $(u>
and 2) non-renormalization theorems for the dilaton[10] which were already true at the level of N = 2 supersymmetric (ie. Kiihler transverse spaces), studied in [8] and [91 and which can be shown to be a fortiori valid for hyperK2hler spaces like the one appearing in the present note. This property considerably simplifies the Weyl invariance differential equation which was, in the N = 2 case (7)
.
w,
A
$ = -f-'P(f)P
4 +q;
A, p , q constants, f constant, as p( f )= 0 (q = 0 for critical D = 10). We have explicitly shown in [9] that W, was not vanishing on homogeneous Kiihler spaces, due to the role of f ( u )= bu in raising and lowering the indices by gij(x, u ) = = f ( u )yij(x) and its inverse, so that W, is proportional to a,S(u) f 0, S(u) being the well-known globally defined trace of curvature power series (see also[151). In the present case f is a constant and as such cannot introduce u dependence. W,, like the other components of W, is vanishing in the present instance. Furthermore the first term in (7) right-hand side is vanishing as p( f ) does, and finally the answer for the dilaton backgrounds reads (8)
$ = $(u,v) = pv
+ qu + $0
($o constant) as announced at the very beginning of this note.
What remains now is to display the line element in this model. As mentioned we took it as the simplest case (n = 1)in the Calabi series. That means we have to display the metric for a cotangent bundle over S2= CP'. This metric exists in the literature at several places (e.g. [2]). Taking complex coordinates for convenience and short-hand notations, the metric on any T * ( M ) has a block-form
(
A
gT*M =
B
D)
7
with complex n x n matrices A, B, C, D, expressed in particular through the Kiihler metric on M (9)
g 4 = d,d$r(xl, Zl);
( x 1" (cc = 1, ...,n) a n-component complex variable), K(x 1, X l ) being the Kiihler potential. One denotes another 2n real coordinates in cotangent space through 22,(Z2,), ( a = 1, ..., n). In our case, with A4 = CP', n = 1, we have a 4-dimensional cotangent bundle T* (CP') as the transverse space in (1).Out of these variables, the following scalar quantity can be defined
223
101
EXPLICIT SUPERSTRING VACUA IN A BACKGROUND ETC.
and the application (11)
A(t)
= [-1
+ (1 + 4t)1'2]/2t
is a solution of the Ricci-flatness condition
det ( g T * M ) = det
(
A
B D ) = const,
with adequate normalizations and regular behaviour at infinity. One must say that it is remarkable that the determinant does not generally depend on z l and 22 separately, but well on t only. This feature allows one to deduce the solution (11).It is necessary in order to obtain (11).So, a candidate for a Kiihler potential on T*(CP') can be guessed in the form (12)
KT'M(Z1, X l , 22, X2) = K(z1, 51) + U(t).
In doing the calculations of det(gT*M),it appears soon that the candidate (12) is improper for most of the Kahler manifolds M in T* ( M ) .But it appears also that in the Calabi series ( M = CP),(12) is the right form to consider and in particular for T" (CP'),
A(t)
= U'(t)
3
dU(0 . dt
After some algebra, we can write the line element for the transverse space in the following form (depending on gq, its inverse, U(t), and derivatives with respect to t, and on t itself of course): (13)
d$ = Ad dz 1"dXlj + B,Bdz 1"dX2j + Cfd z l j dz2,
+ Dd dz2, dX2j ,
with a, B, C and D given in appendix. Therefore (1) becomes (14)
d s 2 = - 2 d u d v + f d s $ , f const.
Hence, the resulting backgrounds (14), ((15) below) and (8) represent exact explicit solutions of superstring effective equations, the so-called fixed-point direct product solutions. What is given below can be found at several places in the literature. It seemed to the authors convenient for the reader to have this material close at hand.
224 102
A. PETERMAN
and
A. ZICHICHI
3. - Mathematical appendix.
1) A, B, C and D appearing in formula (13) of the text read
gap , g a p, T .i .y, Riy: . . are, respectively, the Hermitian metric, its inverse, the connection curvature tensor on CP". A is given by (11) in the text.
Complex conjugation:
-
-
Aup= A , , ;
BPa = Cta .
Dap =
Conditions fcr the metric on T" ( C P ) to be hyperKahler:
I
D ~ A -- c-"B/'= 6" P
PY
C;DPY
=D
Y
Y '
dC#,
BPA= A= P-BP = Pr r
1
plus three analogous conditions, got from (16) by complex conjugation.
-
4) CP' case. The indices u,p , ..., Cr, p, ... can be dropped, as we have only two complex variables z l , 22 and their complex conjugates X l , X2. Using the explicit values of the metric, the connection and curvature tensor for the M = CP' case, the transverse line element can be cast in the simple form
with the G's given by
225 EXPLICIT SWERSTRING VACUA I N A BACKGROUND ETC.
103
REFERENCES [l] [2] [3] [4] [5]
PETERMAN A. and ZICHICHIA., Nuovo Cimento A, 107 (1994) 333. CALABIE., Ann. Sci. Ecole Norm. Sup., 12 (1979) 266. ALVAREZ-GAUME L. and FREEDMAN D. Z., Commun. Math. Phys., 80 (1981) 443. ALVAREZ-GAUME L. and GINSPARGP., Commun. Math. Phys., 102 (1985) 311. a ) SPINDEL P., SEWNA, TROOSTW. and VAN PROYEN A., Nucl. Phys. B, 308 (1988) 662; 311 (1988) 465. b ) SIEGEL W,, Phys. Rev. Lett., 69 (1992) 1493. c) KOUNNAS C., Phys. Lett. B, 321 (1994) 26; CERN-TH.7169/94 (hep-th 9402080); ANTONIADIS I., FERRARA S. and KOUNNASC., CERN-TH.7148/94 (hep-th 9402073). d ) BERKOVITSN. and VAFA C., HUTP-93/A031; BASTIANELLIF. and OHTA N., NHI-HE 94-10 (hep-th 9402118); with PETERSEN J., NHI-HE 94-08 (hep-th 9402042); see also GOMISJ. and SUZUKIH., Phys. Lett. B, 278(1992) 266; FIGUEROSA-O’FARRIL J., Phys. Lett. B, 321 (1994) 344. [61 TSEYTLINA., Phys. Lett. B, 288 (1992) 279. “71 TSEYTLINA., Nucl. Phys. B, 390 (1993) 153. [81 TSEYTLINA., Phys. Rev. D, 47 (1993) 3421. [91 PETERMAN A. and ZICHICHIA, Nuovo Cimento A, 106 (1993) 719. [lo] HAAGENSEN P., Znt. J. Mod. Phys. A, 5 (1990) 1561; JACKJ. and JONES D., Phys. Lett. B , 220 (1989) 176; GRISARUM. and ZANOND., Phys. Lett. B, 184 (1987) 209. [ l l ] HOROWITZG. and STEIFA., Phys. Rev. D, 42 (1990) 1950; RUDDR., Nucl. Phys. B, 352 (1991) 489. [12] KOBAYASHI S. and NOMIZUK., Foundation of Diflerential Geometry, Vol. 2 (Wiley Interscience, New York, N.Y.) 1963; HELGASON S., Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, New York, N.Y.) 1978. [131 YANOK., Differential Geometry on Complex and Almost Complex Manifolds (McMillan) 1965; LICHNEROWICZ A., The‘orie globule des connections et des groupes d’holonomie (CNR, Rome) 1962. [141 a) ZHELOBENKO D., Compact Lie Groups (AMS, Providence, R.I.) 1973. b) For readers not familiar with complex spaces, we recommend the illuminating lecture by ALVAREZ-GAUM~ L. and FREEDMAN D. Z., A simple introduction to complex man7$olds at the Europhysics Conference on Unification of the Fundamental Particle Interactions, edited by S. FERRARA,J. ELLIS and P. VAN NIEUWENHUIZEN (Plenum Press, New York and London) 1980, p. 41. [I51 FREEMAN M. D., POPEC., SOHNINSM. and STELLE K., Phys. Lett. B, 178 (1986) 199; NAN-PARKQ. and ZANOND., Phys. Rev. D, 35 (1987) 4038.
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SELECTED PAPERS ON SEARCHING FOR THE SUPERWORLD BY A. ZICHICHI AND COLLABORATORS
PARTB
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Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi
TROUBLES WITH THE MINIMAL SU(5) SUPERGRAVITY MODEL
From Physics Letters B 291 ( I 992) 255
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Physics Letters B 291 (1992) 255-262 North-Holland
PHYSICS LETTERS B
Troubles with the minimal SU ( 5 ) supergravity model Jorge L. Lopez a*b, D.V. Nanopoulos a
‘
‘ybsc
and A. Zichichi
Centerfor Theoretical Physics, Department of Physics, Texas A&M University, College Station. TX 77843-4242, USA Astroparticle Physics Group, Houston Advanced Research Center (HARC). The Woodlands, TX 77381, USA Theory Division, CERN, CH-1211 Geneva 23, Switzerland CERN, CH-1211 Geneva 23, Switzerland
Received 9 July 1992
We show that within the framework of the minimal SU( 5 ) supergravity model, radiatively-induced electroweak symmetry breaking and presently available experimental lower bounds on nucleon decay, impose severe constraints on the available parameter space of the model, which correspond to fine-tuningof the model parameters of over two orders of magnitude. Furthermore, 3* 1 for most of the allowed parameter space a straightforward calculation of the cosmic relic density of neutralinos ( x ) gives in this model, although small regions may still be cosmologically acceptable. We finally discuss how the no-scale flipped SU(5 ) supergravify model avoids naturally the above troubles and thus constitutes a good candidate for the low-energy effective supergravity model.
1. Introduction
Mathematical simplicity or economy does not always imply physical simplicity or entails on nature what choices to make. A typical example is electroweak unification, where the simplest possible mathematical choice which encompasses charged ( W’ ) and electromagnetic ( y ) currents is SU( 2 ) [ 1 1. But we all know that nature’s preferred choice is S U ( 2 ) x U ( l ) [2]. One ofthe best reasons we can offer is the availability of non-vector representations in S U ( 2 ) x U ( l ) . In the case of SU(2) there is no reason for the fermion masses to be s O ( M w ) , whereas in SU ( 2 ) XU ( 1 ) at least we understand naturally why this must be so. The point we would like to make here is that one can either follow a big fundamental principle to choose the model and /or follow indicative clues from the available experimental data. String theory may eventually provide us with a unique vacuum (i.e., model) and then all discussions would be hushed. For the time being though, we have to struggle between the two above mentioned ways to select the potentially right model. Daring minds have used the absence of adjoint Higgs representations [ 31 in level-one Kac-Moody
superstring constructions [ 41 as a super-clue to select flipped SU( 5) X U (1 ) [ 5,6] as the “chosen” model [ I ] .For those who feel queasy by the presence of the extra U ( 1 ) factor, we remind them of the opening paragraph (i.e., SU(2):out; SU(2) X U (1 ) : in) and of the fact that in string theories U ( 1) factors proliferate. Here we will follow well established theoretical prejudices (radiative electroweak breaking) and available lower bounds on the nucleon decay lifetime to comer the minimal SU ( 5 ) supergravity model, and then use calculations of the cosmic neutralino relic density (which turns out to be almost always very large sZ,>> 1 ), to cast doubts about the candidacy of this model as the effective low-energy supergravity model. The minimal SU( 5) supergravity model [ 8 ] can be generically described by the following observable sector superpotential:
where C is the 24 of SU ( 5 ) whose scalar components acquire a VEV Zx,,=M[2Sxy-S(S&,+S444,> 1
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which breaks SU(5) down to S U ( ~ ) X S U ( ~ ) X tuning which is not protected by any symmetry of the U ( 1); h, k are 5, 5 Higgs superfields; @ is a singlet theory. This problem is solved by going to no-scale which induces SU(2),xU( l),,+U( l)e. (treesupergravity [ 141 where the vacuum energy is autolevel) breaking although it can be omitted if the elecmatically zero even after supersymmetry breaking. troweak symmetry is broken radiatively; and F , A are Another problem of the hidden sector in the minimal the usual 10,s matter fields. The most pervasive difmodel is that the scale of supersymmetry breaking ficulty encountered in this model is the needed doumust be put in by hand as one of the parameters and blet-triplet mass splitting of the Higgs pentaplets. As there is no justification for the phenomenologically it is, in eq. (1.1 ) the choice M‘=M gives massless acceptable choice. In contrast, no-scale supergravity doublets and O ( M ) triplets [ 91. Even though stable owes its name to the fact that in this class of theories under radiative corrections, this solution is rather ad it may be possible to determine the magnitude of the hoc. A more natural solution is provided by the “slidsupersymmetry breaking parameters dynamically. Another solution to this problem of the hidden sector ing singlet mechanism” [ 101 in which the M’hkterm is dropped but the singlet @ gets a large VEV which of the minimal model is obtained in dynamical again keeps the doublets massless. In either case the models of supersymmetry breaking [ 15 ] where this remaining physical degrees of freedom of the singlet scale is related to the scale where hidden sector gaugare light, and after supersymmetry breaking the Higgs ino condensates form. doublets become massive as follows. The hk@term gives rise to supersymmetry breaking trilinear scalar couplings A@H2f12and A@H3H3, and hk& 2. Proton decay and fine-tuning MuH3Ef, to a soft supersymmetry breaking scalar mass squared M u f i which splits the H3 supermulThe parameter space of unified supergravity models tiplet (which acquires an average mass - M u ) . These with universal soft supersymmetry breaking can be three couplings generate a one-loop H2 self-energy described by just five parameters: universal scalar tadpole diagram mediated by @ and having H3 circu( m o )and gaugino ( m , / 2 )masses, universal cubic lating in the loop, which gives a (Murk)I / ’ mass to scalar couplings ( A ) ,the ratio of the Higgs vacuum the scalar components of H,, H2 [ 1 1 1. Thus the light expectation values (tan p ) , and the top-quark mass ( m , ) .The remaining parameters in the model are desinglet and the large Higgs triplet mixing destroy the termined from the experimental values of M,, mb,m,, doublet-triplet splitting. a3,aem, the renormalization group equations, and the The best known solution to this problem utilizes constraint of radiative electroweak symmetry breakthe “missing partner mechanism” [ 121 wherein a 75 ing. (The sign of the Higgs mixing parameter p reof SU(5) breaks the gauge symmetry and new mains underermined.) A detailed study of this pa5 0 , g representations are introduced: the three hk terms in eq. ( 1.1 ) are replaced by 50-75-h and 3- rameter space using the one-loop Higgs effective potential has been recently given in ref. [ 161. The 75-fi. The doublets now remain massless automatifive-dimensional parameter space is restricted by , cally (since there are no doublets in the 50, %I) several consistency and phenomenological conwhereas the triplets acquire -Mu masses. Unfortustraints. This space is bounded in the tan p and m, nately, the SU( 5 ) gauge symmetry breaking through directions but there is no firm upper bound on the a VEV of the 75 causes several cosomological diffisoft supersymmetry breaking parameters mIl2, culties [ 131. Let us now turn to the hidden sector superpotential to=m o / m l / 2and , &=A/mlI2.A semi-quantitative upper bound on m l 1 2(as a function of &) can be obwhich gives rise to soft supersymmetry breaking tained by demanding a “not-too-much-fine-tuning” masses and couplings upon spontaneous breakdown condition. In ref. [ 161 two fine-tuning coefficients c, of supergravity. In the minimal supergravity model and c, have been defined which have the property that there is a problem with the vacuum energy which is if c,,, < A then the observed value of Mz is derived generically M & after supergravity breaking, unless within the model with cancellations among the releone arranges the parameters in the hidden superpovant parameters of less than log(A)-orders of magtential to obtain zero vacuum energy. This is a fine-
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nitude. The physical meaning of these coefficients can be readily grasped by studying the following tree-level Higgs potential minimization constraint f M 2Z -
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mL1-mL2 tan2@ -p2 tan2@- 1
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where mHI,zare the Higgs soft supersymmetry breaking masses. Since the whole theory (the Higgs potential and the RGEs) scales with m,/,, as mIl2 grows the first term on the right-hand side will tend to grow forcing p to larger values to keep Mz at its observed value. This results in large values of c,= I ( p 2 / M $ ) aM$/ap21=2p2/M$. It could also happen that p is kept artificially small by some cancellations within the first term on the right-hand side of eq. (2.1 ) due to fine-tuned choices of m,. This results in large values of c, = I (m:/M:) aM$/am? I . The main result is that generically c,,, a m:,2 ( a b{$ ), where a and b are some m,- and tan @-dependentfunctions. The latter generally increases with m,. In ref. [ 161 it was found that for m,,, 2 400 GeV, cP,,2 100 for all values of &. The choice A = 100 thus corresponds to the usual "naturalness" bound of mg= 2.77m,,, 5 1 TeV. Smaller values of m,,2 allow some range of values of tosuch that c,,,
+
mental bound. This requirement imposes severe constraints on the parameter space of the model [ 18201. Without getting into the calculational details, one can simply state that this requirement implies an upper bound on a quantity P=P(mII2,(0, 24, m,, tan 8) < ( 103 f 15)MH,/Mu [ 201. The function P i s reduced by small values of mIl2 and large values of and it grows with tan @, P a ( 1 tan2@)/tan @.If one assumes that MH3< 3Mu so that the Yukawa coupling which gives rise to the H3 mass is reasonably small [ 201, then an absolute upper bound on P results. In ref. [ 201 the resulting allowed parameter space is given for values of m,=125 GeV and tan @= 1.73 and a calculable value of &. The values of m,/2and ma are varied over a range which presumably respects the "naturalness" constraints. We have selectedfive ( m I l 2(p", , &) sets (given in table 1 ) which are on the boundary of the allowed region (i.e., on the horizontal line in fig. 2 of ref. [ 201 ). Values of & smaller than these (for fixed mIl2) give values of P inconsistent with the experimental upper bound. For the purposes of this paper the choice & =to"'"will suffice. For tan @= 1.73 we have computed c, and e, for varying values of m, for these five points # I , as shown in fig. 1. (These coefficients do not depend on the sign ofp. ) It is clear that c, exceeds A = 100 for all values of m, and for all the selected points. Since c,,, grow with <$, values of &, > (i.e., inside the allowed region) give even larger values of c ~Furthermore, , ~ if we take MH3=Mu as a reasona-
+
c0,
The values of m, vary over the shown finite range which is completely determined by the choices of the other four parameters [ 161. Also, all calculations in this paper are performed using the one-loop Higgs effective potential.
Table 1 Selected set of points which are on the boundary of the allowed region which satisfiesproton decay constraints for ml= 125 GeV, tan ,9= 1.73, and p > 0, and the corresponding gluino and average squark masses, fine-tuning coefficients, and neutralino relic density. Points inside The "symbol" corresponds to the curves shown in figs. 1 and 2. All the allowed region have &,> (0"'" and larger values of mpand lQZ. masses in GeV. Symbol
w2
(?in
ti
me
m,
cs
C,
QXh2
a b
74 122 187 267 364
8.1 6.5 5.4 4.5 3.8
-5.4 - 3.2 - 1.6 -0.41 +0.55
205 340 520 740 1010
625 845 1110 1370 1645
33 53 78 104 133
121 218 364 530 726
15.9 27.9 5.13 3.74 3.20
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10 100
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these coefficients) which satisfy the proton decay constraints in the minimal SU( 5 ) supergravity model. Values of c=A indicate a finetuning of model parameters of log(d)-orders of magnitude.
ble central value for MH3,then the upper bound on P gets reduced by a factor of 3 and the values of
3. The neutralino relic density The parameter space of the minimal SU ( 5 ) supergravity model is highly constrained by the experimental proton decay bounds as discussed above. We now show that for the allowed region the cosmic relic density of the lightest supersymmetnc particle (LSP), i.e., the lightest neutralino, is generally very large, i.e., Qfi2 >> 1, where 0.5< h < 1 is the Hubble parameter. We have calculated Qfi’ in this model for the five
25 8
points given in table 1 following the methods described in ref. [ 21 1. This is a numerically intensive calculation which differs from the usual analyses in that in the computation of the LSP annihilation cross section the masses and couplings of all particles involved can be determined for any choice of the five model parameters. That is, no ad hoc assumptions are made about the masses of the exchanged and final-state particles. The present calculation includes the one-loop corrected Higgs boson masses as well #2. The results depend on the sign of ,u and are given in fig. 2 as a function of m, for the five points in table 1 and tan p= 1.73. These are actually lower bounds since &,>{p increases Q,h2. Recall also that MH3< 3Mu leads to larger values of {om’” . In all honesty, only the values of Q$’ for p > O and m,= 125 112
A detailed study of the cosmic relic density of neutralinos in the SSM is given in ref. [ 221.
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mt (GeV)
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mt (GeV)
1.o
0.8
nih2
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mt (GeV) Fig. 2. The lightest neutrino relic density as a function of m,for the points in table I (and tan p= 1.73 and both signs o f p ) which satisfy the proton decay constraints in the minimal SU(5 ) supergravity model. ( a ) Values of Q$*> 1 (dashed line) are in conflict with current cosmological observations; (b) small cosmologically allowed regions of parameter space may sill exist for special values of m,.If h=0.5 then l2$'>0.25 (dotted line) is excluded.
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GeV (i.e., those given in table 1) can be used to further constrain the allowed parameter space. This is so because for other values of m, and/or the sign of p, the value of the function P (see section 2) will be different from the one which follows for the points in table 1, i.e., P = 309 [ 201. A detailed analysis of these cosmological bounds on the minimal SU (5) supergravity model will be given elsewhere [ 231. It is clear that generally Qxh2>> 1 (see fig. 2a) unless m, takes “special” values as shown in detail in fig. 2b. The explanation for this phenomenon is simple: for values of m, close to the low end of their allowed range, p is relatively small (see fig. l , keeping in mind that in tree-level approximation c,= 2p2/ M $ ), and the LSP composition is “mixed”, allowing for “normal” levels of annihilation, and therefore small values of 52$ [ 2 1 1. Note though that since the quarks, sleptons, and the three heavier Higgs bosons are rather heavy due to the proton decay constraints [ 201, only the Z and lightest Higgs h remain as efficient annihilation mediators. When m, (and therefore p ) grows, the LSP becomes increasingly more a nearly pure bino state (see e.g. fig. 1 in ref. [ 2 1 ] ) and its couplings to 2 and h tend to zero [ 2 1 3, resulting in a large relic density. Some of the curves (curve a for p > 0 and curves b and c forp < 0 ) exhibit a non-monotonic dependence of Bxh2on m,. This is due to poles and thresholds of the LSP annihilation cross section #3 for special values of m,. For example, in curve b for p> 1 [ 25 1. In fact, most studies indicate that Q< 1. On phenomenological grounds, cold dark matter seeded structure formation models require Q< 1 [25]. On theoretical grounds Q= 1 is the only “timestable” value, in that smaller values must be finetuned to be very close to unity otherwise the Universe would have re-collapsed on a Planck time scale. t13
Our calculational scheme breaks down for points near these special regions; detailed calculations show that the correct result vanes more smoothly than our figures indicate [ 241.
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Inflationary models of course predict this precise value of 52 [ 261. Recent data from the COBE DMR instrument [ 271 showing a non-vanishing quadrupole moment of the cosmic microwave background anisotropy appear to confirm the basic predictions of inflationary models [ 281 #4. Furthermore, the “best fit” to the data seems to be given by a mixture of cold and hot dark matter [30,31], as originally proposed in ref. [32], once more disfavoring values of Q;2>1. Moreover, h =0.5 appears to be strongly favored over h = 1 [ 3 1 1, in which case values of Qxh2>0.25 are disfavored on cosmological grounds. Table 1 then shows that the particular set of representative points chosen which satisfy the proton decay constraints in this model are in gross conflict with cosmological observations. It may be possible to find small regions of parameter space where the value of in, is tuned (as fig. 2b shows) to be within narrow intervals such that Qxh2< 1. Note that there are several obstacles hampering the identifications of these possible cosmologically allowed regions: (i) values of to> increase Qxh2, (ii) values of MH3<Mu inand therefore Q,h2, (iii) h= 0.5 decreases crease tr1” the allowed region considerably since in this case QXh2>0.25 is excluded.
4. Discussion
We have shown that proton decay constraints force the minimal SU( 5) supergravity model into a region of parameter space where the Z-mass is obtained within this model subject to cancellations among the model parameters of at least two orders of magnitude. Furthermore, within this allowed region the relic density of neutralinos is generally in gross conflict with current cosmological observations, although small regions of parameter space may still exist which are cosmologically acceptable. In our opinion, these results cast doubts on the candidacy of the minimal SU( 5) supergravity model as the correct low-energy effective supergravity model. We now present an alternative supergravity model 114
In fact, the needed small density perturbations 6p/p=O( lo-’) are only compatible with supersymmetric inflationary models 1291.
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based on no-scale supergravity with the gauge group flipped SU ( 5 ) which does not suffer from any of the troubles discussed in this paper. The doublet-triplet splitting of the Higgs pentaplets is achieved through the SU( 5 ) invariant couplings HHh and HHk [ 6 1, where H , are 10, 3 Higgs representations which effect the SU ( 5 ) x U ( 1) +SU ( 3) X SU ( 2 ) X U ( 1 ) symmetry breaking when their neutral components acquire non-zero VEVs. This is possible because the distribution of the quarks and leptons in the usual 5 and 10 representations is "flipped" (Le., u-d, W U ) relative to their usual assignments, and therefore H x uh, If= uh #'. The above couplings then give HHh+M&H, and HHfi+MudhH3, making the triplets heavy and leaving the doublets massless. Note that this pattern of symmetry breaking is unique [ 6 1, thus avoiding the cosmological multiple-vacua problem of regular SU ( 5 ), As far as the dimension-five proton decay operators are concerned, note that the Higgs triplet mixing term ~I@+M,H,L~, (cf.hfiZ+MuH3H3) is small, whereas the triplet masses themselves are large, This results in a MWMu suppression (in the amplitude) of these operators relative to the regular SU ( 5 ) case [ 331, thus making them completely negligible. This implies that values of the supersymmetry breaking parameters do not need to be as large and therefore the fine-tuning coefficients can be naturally small. Furthermore, Qxh2 can be within current cosmological bounds for a wide range of model parameters, perhaps even providing interesting amounts of astrophysical dark matter [ 221. The small Higgs triplet mixing term also prevents the light H 2 , H 2 doublets from acquiring large masses through the one-loop tadpole diagram discussed in the introduction. Indeed, in the case of flipped SU( 5 ) the induced H2 scalar mass is ( M w f i ) ' / 2 - MW . Let us finally remark that no-scale supergravity ameliorates considerably the cosmological constant problem, i.e., A - M L . Whereas A - M & in minimal supergravity. Also, superstring models realize the noscale ansatz automatically [ 34 ], and interesting flipped SU ( 5 ) models have been constructed within this framework [ 71. We thus propose the no-scale
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lrs
The fact that symmetry breaking does not require adjoint Higgs representations is crucial to the derivation of flipped S U ( 5 ) models from superstring theory [ 31.
24 September 1992
flipped SU( 5 ) supergravity model as a very good candidate for the low-energy effective supergravity model.
Acknowledgement
We would like to thank H. Pois and K. Yuan for useful discussions. This work has been supported in part by DOE grant DE-FG05-9 1-ER-40633.The work of J.L. has been supported in part by an ICSC-World Laboratory Scholarship. The work of D.V.N. has been supported in part by a grant from Conoco Inc. We would like to thank the HARC Supercomputer Center for the use of their NEC SX-3 supercomputer.
References [ I ] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 28 (1972) 1494. [ 2 ] S.L. Glashow, Nucl. Phys. B 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Proc. 8th Nobel Symp. (Stockholm, 1968), ed. N. Svartholm (Almqvist and Wiksell, StocWlolm, 1968) p. 367. [ 3 ] J. Ellis, J. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font, L. Ibaiiez and F. Quevedo, Nucl. Phys. B 345 (1990) 389. [ 4 ] See e.g. M. Dine, ed., String theory in four dimensions (North Holland, Amsterdam, 1988); A.N. Schellekens, ed., Superstring construction (NorthHolland, Amsterdam, (1989). 51 S . Barr, Phys. Lett. B 112 (1982) 219; Phys. Rev. D 40 (1989) 2457; J. Derendinger, J. Kim and D.V. Nanopoulos, Phys. Lett. B 139 (1984) 170. 61 I. Antoniadis, J. Ellis, I. Hagelin and D.V. NaLlupobL4s, Phys. Lett. B 194 (1987) 231. 71 I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, Phys. Lett. B 231 (1989) 65; J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 268 ( 1991 ) 359; for a recent review see J.L. Lopez and D.V. Nanopoulos, in: Proc. 15th Johns Hopkins Workshop on Current problems in particle theory (August 1991), eds. G. Domokos and S. Kovesi-Dornokos, p. 277. [ 81 A. Chamseddine, R. Arnowitt and P. Nath, Phys. Rev. Lett. 49 (1982) 970. [ 9 ] S. Dimopoulos and H. Georgi, Nucl. Phys. B 193 (1981 ) 150. [ 101 L. Ibaiiez and G. Ross, Phys. Lett. B 110 (1982) 215; D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B I10 (1982) 449.
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[ 1 1 ] H.P. Nilles, M. Srednicki and D. Wyier, Phys. Lett. B 124
[ 2 1 ] J.L. Lopez, D.V. Nanopoulos and K. Yuan, Nucl. Phys. B
(1983) 337; A. Lahanas, Phys. Lett. B 124 (1983) 341. A. Masiero, D.V. Nanopoulos, K. Tamvakis and T. Yanagida, Phys. Lett. B 115 (1 982) 380; B. Grinstein, Nucl. Phys. B 206 (1982) 387. C. Kounnas, D.V. Nanopoulos, M. Quiros and M. Srednicki, Phys. Lett. B 127 (1983) 82; T. Hiibsch, S. Meljanac, S. Pallua and G. Ross, Phys. Lett. B 161 (1985) 122. [ 141 For a review see A. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1. [ 151 J.P. Derendinger, L. Ibafiez and H.P. Nilles, Phys, Lett. B 155 (1985) 65; M. Dine, R. Rohm, N. Seiberg and E. Witten, Phys. Lett. B 156 (1985) 5 5 . S. Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois and K. Yuan, Texas A&M University preprint CTP-TAMU-16/92 and CERN-TH.6498/92. S. Weinberg, Phys. Rev. D 26 (1982) 287; N. Sakai and T. Yanagida, Nucl. Phys. B 197 (1982) 533. [ 181 B. Campbell, J. Ellis and D.V. Nanopoulos, Phys. Lett. B 141 (1984) 229. [ 191 M. Matsumoto, J. Arafune, H. Tanaka and K. Shiraishi, University of Tokyo preprint ICRR-267-92-5 (April 1992). [20] R. Arnowitt and P. Nath, Texas A&M University preprint CTP-TAMU-24/92.
370 (1992) 445. [22] S. Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois and K. Yuan, in preparation. [ 231 J.L. Lopez, C.V. Nanopoulos and H. Pois, in preparation. [24] K. Griest and D. Seckel, Phys. Rev. D 4 3 (1991) 3191; P. Gondolo and G. Gelmini, Nucl. Phys. B 360 (199 1 ) 145. [25] See e.g., E. Kolb and M. Turner, The early universe (Addison-Wesley, Reading, MA, 1990). [ 261 For recent reviews see e.g., K. Olive, Phys. Rep. 190 ( 1990) 307; D. Goldwirth and T. Piran, Phys. Rep. 214 ( 1992) 223. [27] G.F. Smoot, et al., COBE preprint (1992). [28] E.L. Wright, et al., COBE preprint (1992). [29] S. Hawking, Phys. Lett. B 115 (1982) 295; A. Guth and S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 11 10; J. Ellis, D.V. Nanopoulos, K. Olive and K. Tamvakis, Phys. Lett.B 120 (1982) 331. [ 301 R. Schaefer and Q. Shafi, Bartol preprint BA-92-28 (1992); G. Efstathiou, J.R. Bond and S.D.M. White, Oxford University preprint OUAST/92/ 1 1. [ 3 1 ] A.N. Taylor and M. Rowan-Robinson, Queen Mary College preprint (June 1992). [32] D.V. Nanopoulos and K. Olive, Nature 327 ( 1987) 487. [33] J. Ellis, J. Hagelin, S. Kelley and D.V. Nanopoulos, Nucl. Phys.B311 (1988/89) 1. [34] E. Witten, Phys. Lett. B 155 (1985) 151.
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Jorge L. Lopez, D.V. Nanopoulos, H. Pois and A. Zichichi
PROPOSED TESTS FOR MINIMAL SU(5) SUPERGRAVITY AT FERMILAB, GRAN SASSO, SUPERKAMIOKANDE AND LEP
From Physics Letters B 299 ( I 993) 262
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Proposed tests for minimal SU ( 5 ) supergravity at Fermilab, Gran Sasso, SuperKamiokande and LEP Jorge L. Lopez a,b, D.V. Nanopoulos a,b,c, H. Pois a,b and A. Zichichi ‘ Centerfor Theoretical Physics, Department ofphysics, Texas A & M University, College Station, TX 77843-4242, USA Aslroparticle Physrcs Group, Houston Advanced Research Center (HARC), The Woodlands, TX 77381, USA Theory Division, CERN, CH-1211 Geneva 23, Switzerland CERN, CH-1211 Geneva 23, Switzerland Received 1 1 November 1992
A series of predictions are worked out in order to put the minimal SU( 5 ) supergravity model under experimental test. Using the two-loop gauge coupling renormalization group equations, with the inclusion of supersymmetric threshold corrections, we calculate a new value for the proton decay rate in this model and find that SuperKamiokande and Gran Sasso should see the proton decay mode p - t t X + for most of the allowed parameter space. A set of physically sensible assumptions and the cosmological requirement of a not too young Universe give us a rather restrictive set of allowed points in the parameter space, which characterizes this model. This set implies the existence of interesting correlations among various masses: either the lightest chargino and the next-to-lightest neutralino are below % 100 GeV (and therefore .observable at the Tevatron) or the lightest Higgs boson is below 5 50 GeV (and therefore observable at LEP 1-11). These tests are crucial steps towards selecting the correct lowenergy effective supergravity model. We also comment on the compatibility of the model with sin2Bw(MZ)measurements as a function ofa,(M,).
One of the more interesting problems in high-energy physics is to disentangle the right model for the description of all particles and all interactions. Recently several rather restrictive constraints on the minimal SU ( 5 ) supergravity model have been pointed out [ 1-31, Here we continue the study of this model and determine a set of predictions which could be experimentally verified with existing colliders and detectors. These predictions mainly concern the lightest chargino and the next-to-lightest neutralino, the lightest Higgs boson, and the proton lifetime. Unification of the standard model particle interactions at very high energies into larger models requires the presence of low-energy supersymmetry to avoid the notorious gauge hierarchy problem. Moreover, supergravity models allow one to explicitly calculate the phenomenologically necessary soft supersymmetry breaking terms which split the ordinary particles from their supersymmetric partners. The recent LEP measurements of the low-energy gauge couplings and their use to study gauge coupling unification [ 4-61 constitutes a nice example of the validity 262
of this scenario. In this paper we restrict ourselves to the minimal SU ( 5 ) supergravity model. Here the introduction of the new light supersymmetric degrees of freedom raises the unification scale M a and makes the dimension-six-operator mediated proton “partial” lifetime (p+e’ao) much longer than experimentally required. However, dimension-five proton decay operators [ 71 arise due to the exchange of a heavy colored Higgs triplet supermultiplet H3 and can easily give unacceptable proton lifetimes [ 8 1, unless MH3?MU and the supersymmetric spectrum is not too light [ 9, l o ] . In fact, demanding MH3< 3MU so that the Yukawa coupling generating the H3 mass remains perturbative [ 9,1] and the naturalness criterion m4,6< 1 TeV, it has been shown [ 1 ] that the proton decay mode p+VKc is sufficiently suppressed only if the squarks and sleptons are heavy, and the two lightest neutralinos and the lightest chargino are much lighter. It has also been shown [ 21 that for representative points in the proton-decay allowed five-dimensional parameter space of the model, the relic abundance of Elsevier Science Publishers B.V.
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the lightest neutralino - a stable particle in the minimal model with R-parity conservation - is in gross conflict with current cosmological observations mainly due to a lack of efficient pair-annihilation channels. The cosmologically allowed set of points was subsequently determined in an extensive search of the parameter space [ 31. This set of points shows an experimentally interesting correlation: if the lightest Higgs boson mass is above -80 GeV, then the lightest chargino and the second-to-lightest neutralino masses are below = 100 GeV, thus making the observability of at least one of these particles quite likely at LEPII. In this letter we present a refinement of one of the most important elements in the proton lifetime calculation, by determining the unification mass Mu using [ 11 ] the two-loop renormalization group equations (RGEs) for the gauge couplings including the threshold effects of the supersymmetric particles. (In ref. [ 31, Mu was calculated using the one-loop gauge coupling RGEs with a common supersymmetric threshold at M,.) The main effect of these corrections is a well known systematic reduction [ 4,12 ] of the value of Mu which in turn reduces the upper bound on the proton lifetime (since T, x M L 3 and we take MH3< 3Mu) rendering most of the originally allowed points unacceptable. We also explore some previously neglected regions of parameter space (where tan/3<2) which maximize the proton lifetime and increase the number of allowed points. The final allowed set of points entails a limit on the lightest chargino and second-to-lightest neutralino masses of m,f,xS ,< 150 GeV and on the lightest Higgs boson mass of mh,<100 GeV. Since neutralino-chargino hadro-production at Fermilab should be able to probe m,;,,; 5 100 GeV through the tri-lepton signal [ 1315 1, most of the parameter space of this model can be probed in the near future. Furthermore, if m,;,,; 2 100 GeV, then mh,<50 GeV, making observability of the latter at LEP a sure bet. Let us first sketch the calculational procedure. The low-energy sector of the minimal SU ( 5 ) supergravity model can be described in terms of five parameters: the universal soft-supersymmetry breaking scalar (ma) and gaugino ( inll2)masses and the trilinear coupling ( A ) ,the ratio of vacuum expectation values (tan p ) , and the top-quark mass ( m , ). This reduced set of parameters results from the use of the RGEs to
28 January 1993
relate high- and low-energy parameters and the requirement of radiative breaking of the electroweak symmetry (enforced by using the one-loop effective potential) [ 16 I. For quick reference, ml12determines the gluino mass mg= (a,/au)ml,2,while mo and m,,, determine the squark and slepton masses mi,prn:+cim:/,, with ci-6 (0.5) for squarks (sleptons). The Higgs mixing parameter ,u is calculable, it roughly scales with mo or ml12(whichever is dominant) and grows with increasing m,; its sign remains undetermined. In ref. [ 31 three of us conducted a discrete scan of this five-dimensional parameter space and determined those points which satisfied all experimental bounds on the sparticle and one-loop corrected Higgs boson masses. We used one-loop gauge coupling RGEs and a common supersymmetric threshold at M,, to determine Mu, au, and sin%,, once a3(Mz)=o.113, 0.120 [17] and a;l(MZ)=127.9 were given. We then computed the proton lifetime into the dominant decay mode P - ’ F ~ , ~ , ~ assumK+, ing MH3< 3Mu, and obtained an upper bound on T,. About 10% of the otherwise allowed points satisfied sP > = 1 x 1032yr [ 18 1. We also computed the relic abundance of the lightest neutralino Q,h:, where Qx=p,/pa is the present neutralino density relative to the critical density and 0.5
zrXp
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Table 1 The value of the one-loop unification mass M g ' , the two-loop and supersymmetric threshold corrected unification mass range M C ) ,the ratio of the two, and the range of the calculated sin2&, for the indicated values of a3 (the superscript ( - ) denotes p > O (c0))and a;' = 127.9. The sin2& values should be compared with the current experimental + l a range sin2&,= 0.2324k 0.0006 [ 191. Lower values of a3drive sin2& to values even higher than for a3= 0.1 18. All masses in units of 1 0l6GeV.
+
013
Mg'
M$j'
M $ j ) / M g ) sin2&
0.126+ 0.1260.1 18+ 0.1 18-
3.33 3.33 2.12 2.12
1.60-2.13 1.60-2.05 1.02-1.35 1.02-1.30
0.48-0.64 0.48-0.61 0.48-0.64 0.48-0.61
0.2315-0.2332 0.2313-0.2326 0.2335-0.2351 0.2332-0.2345
28 January I993
dicates that this upper bound is reduced down to tanPS3.5. Here we consider also tan/?= 1.5, 1.75 since low tan /? maximizes T ~ C Csin22/?.These add new allowed points (i.e., T ~ O > ) rFP) to our previous set, although most of them (>,75%) do not survive the stricter proton decay constraint ( T:') > 57"") imposed here. In fig. 1 we show the re-scaled values of T, versus the lightest chargino mass m x ; . All points satisfy ~ 0 = m o / m l , , ~ 6 a n d m x5; 150GeV,whicharetobe contrasted with COX 3 and mx; 5 225 GeV derived in ref. [ 31 using the weaker proton decay constraint. The upper bound on m,,t derives from its near proportionality to mg, m,.. -0.3m, [ 1,3], and the result m,S 500 GeV. The latter follows from the proton decay constraint to>,6 and the naturalness requirement m4xJ= f m , W < 1 TeV. Within our naturalness and H3 mass assumptions, we then obtain #'
lated range of sin%,. These values are obtained after all constraints have been satisfied, the proton decay being the most important one. Note that for a3=0.1 18 (and lower), sin20, is outside the experimental t l a range (sin20w=0.2324+0.0006 [ 19]), ~ ~ d 3 (.31. 4 ) ~ l O ~ ~for y r ,u>O ( p < O ) . (1) whereas a3= 0.126 gives quite acceptable values. We do not specify the details of the GUT threshThe p + P K + mode should then be readily observable olds and in practice take the usual three GUT mass at SuperKamiokande and Gran Sasso since these ex) be degenerate with Mu parameters (Mv,Mz, M H 3 to periments should be sensitive up to T,= 2 X 1033yr. Since we then allow MH3< 3MU,table 1 indicates that Note that if MH3is relaxed up to its largest possible in our calculations M H 3< 6.4X 10l6GeV. In ref. [20] value consistent with low-energy physics, M,, = it is argued that a more proper upper bound is 2.3x10" GeV [20], then in eq. ( 1 ) T ~ - ~ T ~ MH3c: 2Mv,but M , cannot be calculated directly, 4.0(4.8) X yr, and only part of the parameter only (MGM,) ' I 3 < 3.3 x 10I6 GeV is known from '' Note that in general, TpaMk,l,[m:/m,:]2aMk,[m1(6+ low-energy data [ 201. If we take MZ=Mv, this would ti) 1' and thus T~ can be made as large as desired by increasing give M,, < 2Mv < 6.6 x 10l6 GeV, which agrees with sufficiently either the supersymmetric spectrum or M H , , our present requirement. Below we comment on the case M,< M , With the new value of M u we simply rescale our previously calculated 7, values which satisfied 4 ~ " ' " " ' " ' " 7 f ) > ~ 7and ~ , find that T ~ ' ) = T ~ , ~ ) [ M ~ ) I M ~ ~ ) I ~ >rFP for only 525% of the previously allowed points. The value of a3has a significant influence on the results since (see table 1) larger (smaller) values of a3increase (decrease) Mu, although the effect is more pronounced for low values of a3.To quote the most conservative values of the observables, in what follows we take a3at its 1a value ( a3= 0.126). As discussed above, this choice of a3 also gives sin20, Fig. 1. The calculated values of the proton lifetime into p-CKK+ values consistent with the k 1a experimental range. versus the lightest chargino (or second-to-lightest neutralino) Finally, in our previous search [ 31 of the parameter mass for both signs ofp. Note that we have taken a,+ l o in order space we consider only tan /?=2,4,6, 8, 10 and found to maximize T~ Note also that future proton decay experiments that tan P 6 was required. Our present analysis inshould be sensitive up to T,- 20 x yr.
4m
+
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"3
P <
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space of the model would be experimentally accessible. However, to make this choice of M H , consistent with high-energy physics (i.e., MH,< 2Mv) one must have Mv/Mz> 42. In this model the only light particles are the lightest Higgs boson ( m h 5 100 GeV), the two lightest neutralinos ( mxy = f m,! 5 75 GeV), and the lightest chargino ( m,; = m,? 5 150 GeV). The gluino and the lightest stop can be light ( m g = 160-460 GeV, mi; = 170-825 GeV), but for most of the parameter space are not within the reach of Fermilab. In fig. 2 we present the results for the experimentally interesting correlation between mh and m,: =rnm,q, which shows #‘that rnh580(95)GeVfor,u>O (p 106(92) GeV (for p > o ( p < O ) ) , we obtain mh550 GeV and Higgs detection at LEP should be immediate #3. This updated
” 41)
In fig. 2 only tan p= 1.5, 1.75, 2 are shown. For the maximum allowedtanBvalue (=3.5), mhSlOOGeV. In fig. 2 forpz 0, m h c50 GeV, and m,. 2 100 GeV, there is a sparsely populated area with highly fine-tuned points in parameter space ( m ,c 100 GeV, tan p.: 1.5, t;R = A / m l l z c&,= 6 ).
40
60
00
LOO
m,: (CeV)
120
140
40
60
60
100
120
140
m,; @V)
Fig. 2 . The correlation between the lightest chargino (or secondto-lightest neutralino) and the lightest Higgs boson masses for both signs of p. The bands for low values of mh correspond to ta np=l. 5, 1.75. The plus signs indicate points where the branching ratio into three charged leptons for neutralino-chargino hadro-production becomes negligible due to the opening of the channel x;-,yyh.
28 January 1993
prediction ( mh2 50 GeV 3 m,:,q 5 100 GeV) is much sharper that the previous one ( m h 2 80 GeV 3 m x f , x5; 100 GeV) in ref. [ 31. Interestingly enough, it has been recently pointed out [ I41 that Fermilab has the potential of exploring most of the LEPII parameter space, before LEPII turns on. This would occur through the process pp+ ~ 4 x 1 ’which has a cross section 2 1 pb for m,;,,q 5 100 GeV. The further decay into three charged leptons has very little background [ 13,2I , 141 and possibly sizeable branching ratios which, with an integrated luminosity of = 100 pb-’, should yield a significant number of candidate events. A detailed calculation of this process in this model is in progress [ 15 1. One concern which is usually brushed aside is whether the decay channel ,y; + ~ y his open, since in this case the branching ratio into three charged leptons is expected to be negligible. We have checked that in this model this channel is indeed open, although mostly for m,;,,q 2 100 GeV, and thus mostly inconsequential for Fermilab searches. These points are represented by plusses in fig. 2 and, given the approximate mass relations in this model, roughly speaking correspond to mh,< fmx;,xq. In sum, within our physically reasonable assumptions, the minimal SU ( 5 ) supergravity model should be fully testable at a combination of present and near future experimental facilities. At Fermilab, charginoneutralino hadro-production and decay into three charged leptons should probe rn,;,,; ,< 100 GeV. If this process is not observed at Fermilab, then LEP should see the lightest Higgs below = 50 GeV. If Fermilab does see the chargino-neutralino, then LEPII should confirm the model by observing the Higgs at the appropriate mass. Independently, SuperKamiokande and Gran Sasso should see the p+PK+ decay mode for most of the allowed parameter space. Of course, if any of the above predictions fails to be confirmed, then under our physically reasonable assumptions the minimal SU ( 5 ) supergravity model will be excluded. It is worth pointing out three basic features of this model: ( i ) the gauge group SU( 5 ) , (ii) the unification scale M U - 10l6 GeV, and (iii) the exclusion of the “no-scale” type supergravity boundary conditions [ 1 ] (i.e., &, = 0; here <, 2 6 is required). The first and second features are not found in any known string model. On the other hand, noscale supergravity appears as the low-energy limit of
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superstring theory. Therefore, the possible exclusion of this model could be taken as evidence against nonstring-derived models. This work has been supported in part by DOE grant DE-FG05-91-ER-40633. The work of J.L. has been supported by an SSC Fellowship. The work of D.V.N. has been supported in part by a grant from Conoco Inc. We would like to thank the HARC Supercomputer Center for the use of their NEC SX-3 supercomputer.
References [ I ] R. Arnowitt and P. Nath, Phys. Rev. Lett. 69 (1992) 725; P. Nath and R. Arnowitt, Phys. Lett. B 287 (1992) 89; B 289 (1992) 368. [ 21 J.L. Lopez, D.V. Nanopoulos and A. Zichichi, Phys. Lett. B 291 (1992) 255. [ 3] J.L. Lopez, D.V. Nanopoulos and H. Pois, Texas A & M University preprint CTP-TAMU-6 1/92 and CERNTH.6628192. [ 4 ] J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B 249 (1990) 441. [ 51 J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B 260 (1991) 131; P. Langacker and M.-X. Luo, Phys. Rev. D 44 ( 199 1 ) 8 17; F. Anselrno, L. Cifarelli, A. Peterman and A. Zichichi, NuovoCimento 104A (1991) 1817. [ 6 ] J. Ellis, S. Kelley and D.V. Nanopoulos, Nucl. Phys. B 373 (1992) 55; Phys. Lett. B 287 (1992) 95; R. Barbieri and L. Hall, Phys. Rev. Lett. 68 ( 1992) 752;
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F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, NuovoCimento 105A (1992) 581; J. Hisano, H. Murayama and T. Yanagida, Phys. Rev. Lett. 69 (1992) 1014. [ 7 ] S. Weinberg, Phys. Rev. D 26 (1982) 287; N. Sakai and T. Yanagida, Nucl. Phys. B 197 (1982) 533. [ 81 J. Ellis, D.V. Nanopoulos and S. Rudaz, Nucl. Phys. B 202 (1982) 43; B. Campbell, J. Ellis and D.V. Nanopoulos, Phys. Lett. B 141 (1984) 229. [ 9 ] K. Enqvist, A. Masiero and D.V. Nanopoulos, Phys. Lett. B I56 (1985) 209. [ 101 P. Nath, A. Charnseddine and R. Arnowitt, Phys. Rev. D 32 (1985) 2348; P. Nath and R. Arnowitt, Phys. Rev. D 38 (1988) 1479. [ 1 1 ] F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi, Nuovo Cimento 105A (1992) 1201, and references therein. [ 121 F. Anselrno, L. Cifarelli and A. Zichichi, CERN-PPE/92145 and CERN/LAA/MSL/92-OII (July 1992), Nuovo Cimento, to appear. [ 131 P.NathandR.Arnowitt,Mod. Phys.Lett.A2 (1987) 331. [ 141 H. Baer and X. Tata, Florida preprint FSU-HEP-920907. [ 151 J.L. Lopez, D.V. Nanopoulos and X. Wang, in preparation. [ 161 For a detailed description of this general procedure see, e.g., S. Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois and K. Yuan, Texas A & M University preprint CTP-TAMU-161 92 and CERN-TH.6498/92. [ 171 S. Bethke, Talk given at XXVI Intern. Conf. on High energy physics (Dallas, TX, August 1992). [ I S ] Particle Data Group, K. Hikasa et al., Review of particle properties, Phys. Rev. D 45 (1992) SI. [ 191 P. Langacker and N. Polonsky, University of Pennsylvania preprint UPR-05 13T (October 1992). [ 201 J. Hisano, H. Murayama and T. Yanagida, Phys. Rev. Lett. 69 (1992) 1014; Tohoku University preprint TU-400 (1992). [ 2 I ] R. Barbieri, F. Caravaglios, M. Frigeni and M. Mangano, Nucl. Phys. B 367 (1991) 28.
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247
Jorge L. Lopez, D. V. Nanopoulos, H, Pois, Xu Wang and A . Zichichi
IMPROVED LEP LOWER BOUND ON THE LIGHTEST SUSY HIGGS MASS FROM RADIATIVE ELECTROWEAK BREAKING AND ITS EXPERIMENTAL CONSEQUENCES
From Physics Letters B 306 (1993) 73
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249
PhysicsLetters B 306 (1993) 73-78 North-Holland
P H Y S I C S LETTERS B
Improved LEP lower bound on the lightest SUSY Higgs mass from radiative electroweak breaking and its experimental consequences Jorge L. Lopez a,b, D.V. Nanopoulos a
a,b,
H. Pois a,b, Xu Wang a,b and A. Zichichi
Centerfor Theoretical Physics, Department of Physics, Texm A&M University, College Station, TX 77843-4242, USA Astroparticle Physics Group. Houston Advanced Research Center (HARC). The Woodlands, TX 77381, USA CERN, CH-1211 Geneva 23, Switzerland
Received 1 I February 1993 Editor: R. Gatto
We show that the present LEPI lower bound on the Standard Model Higgs boson mass (M&60 GeV) applies as well to the lightest Higgs boson ( h ) of the minimal SU( 5) and no-scale flipped SU( 5 ) supergravity models. This result would persist even for the ultimate LEPI lower bound (MH>70 GeV). We show that this situation is a consequence of a decoupling phenomenon in the Higgs sector driven by radiative electroweak breaking for increasingly larger sparticle masses, and thus it should be common to a large class of supergravity models. A consequence of mh>60 GeV in the minimal SU(5 ) supergravity model is the exclusion from the allowed space of “spoiler modes” (xs-xph) which would make the otherwise very promising trilepton signal in p p - x f x q X unobservable at Fermilab. Within this model we also obtain stronger upper bounds on the lighter neutralino and chargino masses, i.e., mxt 5 50 GeV, mx;,x:5 100 GeV. This should encourage experimental searches with existing facilities.
1. Introduction
The current renewed interest on supersymmetry and its phenomenological consequences can, within the context of the MSSM, only go so far. This limitation is due to the large size (at least 2 1-dimensional) of the parameter space that should be explored. In practice people routinely impose certain “grand unification-” and “supergravity-inspired” relations among the model parameters, although usually not in a completely consistent way and omitting several equally well motivated constraints. This hodgepodge approach to minimizing the number of assumptions in order to get the most “model-independent’’ results can be misleading #’. Supergravity models with radiative breaking of the electroweak symmetry [ 2 ] have a much reduced parameter space (three super#’
A relevant example of this occurs in the Higgs sector where
the decay mode h-AA can be a spectacular signature in the MSSM but has been shown to be forbidden in supergravity models [ 1 1. Elsevier Science Publishers B.V.
symmetry breaking parameters ( m,,2, mo,A ) , tan j3, and the top-quark mass) and are therefore highly predictive and falsifiable. We have recently studied the experimental signatures [3,4] for two such models: ( i ) the minimal SU ( 5 ) supergravity model including the stringent constraints of proton stability [ 5-7 ] and cosmology [ 6-9 1, and (ii ) the no-scale flipped SU ( 5 ) supergravity model [ 10 1. In this note we focus on the constraints from current LEPI data on the lightest Higgs boson mass in these two models. We show that due to the nature of the Higgs masses, couplings, and branching fractions, at LEPI for mh5 70 GeV the h particle should be basically indistinguishable from the Standard Model Higgs boson, and therefore the experimental lower bound on the latter should also apply to the former. We then explore the generality of this result and give arguments, based on the built-in radiative electroweak breaking mechanism, for its validity in a more general class of supergravity models. The pre-existing severe limits on the minimal SU( 5 ) supergravity model (based on mh> 43 GeV) are shown to be even stronger, imply73
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ing that LEPII and Fermilab should be able to explore a yet larger portion (if not all) of the parameter space of this model.
27 May 1993
From eq. (2.1 ) one can deduce the integrated luminosity achieved, I u d t = 3 / [ U s ~ ( 6 1 . 6 ) B R , ~ ] .In analogy with eq. (2.1 ), we can write # eventssusy = ~ S U S (Ymh ) BRSUSY 9dt
2. Improved lower bounds on mh
= [email protected] (mh) /%M ( 6 1.6)
The current LEPI lower bound on the Standard Model (SM) Higgs boson mass (mH>61.6 GeV [ 111) is obtained by studying the process e+e--+Z*H with subsequent Higgs decay into two jets. The MSSM analog of this production process leads to a cross section differing just by a factor of sin2(a - p ) , where a is the SUSY Higgs mixing angle and tan p= u2/ul is the ratio of the Higgs vacuum expectation values [ 121. The published LEPJ lower bound on the lightest SUSY Higgs boson mass (mh>43 GeV) is the result of allowing sin2(a-p) to vary throughout the MSSM parameter space and by considering the e+e-+Z*h, hA cross sections. It is therefore possible that in specific models (which embed the MSSM), where sin2(a-p ) is naturally restricted to be near unity (as for example discussed in the next section), the lower bound on mh could rise, and even reach the SM lower bound if BR (h+ 2 jets) is SM-like as well. This we will show is the case for the two supergravity models in hand. Non-observation of a SM Higgs signal puts the following upper bound in the number of expected 2-jet events: -# eventssM
=a(e+e-+Z*H)sM BR(H+2 jets)sM
[Ydt (2.1 1
(3.
The SM value for BR(H-+2 jets)sMzBR(H+ b b + c ~ + g g ) ~ ~ ~ 0[ 121 . 9 2corresponds to an upper bound on o(e+e-+Z*H),,. Since this is a monotonically decreasing function of mH, a lower bound on mH follows, i.e., mH> 6 1.6 GeV as noted above. We denote by usM(6 1.6) the corresponding value for a(e+e--+Z'H),,. For the MSSM the following relations hold
a(e + e-
+
Z*h)susy
=sin2(a- P)o(e +e- +Z*H)sM ,
BR(h-+2jets),,,,=fBR(H+2 74
jets)sM.
(3.
This immediately implies the following condition for allowed points in parameter space: foSUSY (mh)
< oSM(61 .6)
.
(2.4)
The cross section asusy ( mh) is shown in fig. 1 for both models. The values shown for the minimal SU ( 5 ) model also correspond to the SM result since once can verify that sin2(a -8) > 0.9999 in this case. For the flipped model there is a hard-to-see (sin2(a - p) > 0.95 ) drop relative to the SM result (as shown on the top row plots) for some points. The ratiofversus mh is shown in fig. 2 #2. It is interesting to remark that the two models differ little from the SM and in fact the proper lower bound on mh (which follows from the use of eq. (2.4) ) is marked by the set of arrows near 60 GeV in fig. 1. Note that the bound is lowest (given by the left-most arrow of the four near 60 GeV in fig. 1 ) for the minimal SU ( 5 ) ( p < 0) since the fratio is smallest in this case. A similar analysis shows that ifMH> 70 GeV is established at LEPI, then m h z70 GeV would also follow (note the second set of arrows in fig. 1 around 70 GeV). The present bound on MH has been obtained with N 3 x 1O6 hadronic Z-decays, which is nearing the ultimate number achievable at LEPI. Therefore if Higgs events are at all seen, LEPI should not be able to distinguish these two models from each other or from the SM. Such a differentiation would require a detailed study of the branching fractions [ 41.
3. Radiative breaking and decoupling The results in the previous section may signal a general feature of supergravity models. In fact, the two examples considered above can be taken as extreme cases of generic supergravity models with radiative
(2.2a) #'
(2.2b)
(2.3)
In the calculation of BR(h-2 jets)susu we have included all contributing modes, in particular the invisible h-+xyxpdecays.
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Fig. 1. The cross section o(e+e-+Z'h+uUh) as a function of the lightest Higgs boson mass mh at the 2-pole for both supergravity models. The arrows near 60 GeV indicate the improved lower bounds on mh (relative to the MSSM lower bound of 43 GeV), while the arrows near 70 GeV indicate possible future lower bounds if LEPI were to establish a lower bound on the SM Higgs boson mass of 70 GeV. The gaps on the curves are to be understood as tilled by intermediate points in the discrete parameter space explored.
electroweak breaking, in that the no-scale flipped model has &=molm,/,=O, whereas the minimal model has &k3. The Higgs sector of the MSSM is generally specified at tree-level by the arbitrary choice of two parameters, e.g., tan Band mA.At one-loop the whole spectrum enters, but for the present purposes it will suffice to describe it in terms rn, and m4 [ 131. It is well known that in the limit of mA>>Mz one recovers a SM-like theory for the h the Higgs couplings to fermions and vector bosons, with mh at its maximum value (mi= cos2pM$+ A m i ) and sin2(a! - B ) , Isincu/cos/.?l, Icosa!/sinpl = 1. Also, the H , A , H + decouple: they become increasingly heavy, degenerate, and their couplings to fermions and vector bosons are driven to zero. Since in the MSSM mA is a physical input parameter, there is no a priori preferred value; experimentally, mA>23 GeV. What mechanism may enforce mA>> Mz is a question beyond the MSSM. On the other hand, radiative electroweak breaking in supergravity models determines
m i in terms of other sectors of the theory,
=m$l+m$2+2p2+Amf,,
(3.1)
where p, B, m b l , m$2 are parameters in the Higgs potential (see e.g., ref. [ 141 ), and Am: represents the one-loop correction. Since the renormalization group equations (RGEs) (which determine m H I , mH2a n d p ) scale with m,/2ccm2[ 141, increasing mg will drive mAto larger values and the Higgs sector to the SM-like limit. Furthermore, if the initial conditions for the m$1,2RGEs at the unification scale, i.e., mo= tom,,, are increased, the stated behavior should be accelerated: decoupling should be approached for lower values of mB.To verify these qualitative statements we have studied a class of minimal supergravity models with cO=O, 1, 2 and determined the sin2(a!- p ) = 0.99 contours in the (mi, tan p )
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Fig. 2. The ratiof=BR(h-2
PHYSICS LETTERS B
jets),,,,/BR(h+2
plane #3. These contours are shown in fig. 3 (points to the right (left) of a given contour have larger (smaller) values of sin2( a- /3) ) and help to quantify our previous qualitative remarks, and to explain the behavior observed in the two sample models considered in section 2. In fig. 3 contours for 2 < lo< 2.8 occur to the left of the shown & = 2 contour, but still have some points with theoretically and experimentally allowed values of the gluino mass. For & > 3 contours have no points for allowed values of me Therefore, any minimal supergravity model with lo> 3, implies sin2(a-p) >0.99. Precise statements about lower bounds on mhdepend on BR(h+2Jets) which is quite model dependent (see for example fig. 2). However, the only deviation from SM rates will arise from loop-induced decay modes (e.g., h-rgg) and non-SM final states (e.g., ~ F + X : X ~ ) . The point to be stressed is that if the supersymmetric Higgs sector is found to be SM-like, this could be are not sensitive to the choice of C, m,,or the sign ofp, except for the portion of the contours which may become phenomenologically excluded.
113 The results
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jets)sM for both models as a function of mh.
taken as indirect evidence for an underlying radiative electroweak breaking mechanism #4, since no insight could be garnered from the MSSM itself.
4. Experimental consequences for the minimal SU(5) supergravity model
The improved bound m,,? 60 GeV mostly restricts low values of tan D and therefore the minimal SU ( 5 ) supergravity model where t a n p s 3 . 5 [ 7 1. (The noscale flipped SU ( 5 ) supergravity model is also constrained for small tan 8, but since in this model tan p can be as large as 32, only a small region of parameter space is affected.) In ref. [ 71 we obtained upper bounds on the light particle masses in the minimal SU ( 5 ) supergravity model ( f ,h, x : , ~ x, I' ) for mh > 43 GeV. In particular, it was found that m,: 2 100 GeV was only possible for mhs50 GeV. The improved bound on mh immediately implies the following considerably stronger upper bounds: A4
Or as a "cosmic conspiracy", whichever one likes better.
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5. Conclusions
P
m
i
mc(GeV)
Fig. 3. The contours of sin2(a - /3) ~ 0 . 9 in 9 generic supergravity models in the ( mp,tan /3) plane for increasing values of &,. For &,>3 all contours fall below the experimental lower bound on mp The no-scale flipped model has &,=O, while the minimal model requires &2 3. Since areas to the right of the contours have yet larger values of sinz(a-/3), the plot shows the approach to the SM-like limit of the Higgs sector (i.e., sin2(a-/3)-+l) for increasingly larger gluino masses. This trend is accelerated by increasing &, values. (For the &,=0 case consistency conditions impose an upper bound on tan /3( 5 IS) [ 141.)
We have shown that the current experimental lower bound on the SM Higgs boson mass can be used to impose constraints on the Higgs sector of supergravity models, which are stronger than those possible in the generic MSSM. This is the direct result of the underlying radiative electroweak symmetry breaking mechanism which links the Higgs sector to the sparticle sector of the theory. In fact, such a link leads to the SM-like limit of the Higgs sector in a natural way for c0> 3 in any minimal supergravity model, since independent gluino searches have obtained a lower limit to the scale involved in radiative electroweak = O ) , the possibility breaking. In the no-scale case (to for significant deviations from the SM-like limit exists [ 151 but requires m h k 80 GeV and large tan p. This region is obviously beyond the reach of LEPI, but is accessible at LEPII [4]. We have pursued the consequence of these ideas explicitly in two realistic supergravity models, and have obtained rather stringent indirect constraints on the lighter neutralino and chargino masses of the minimal SU ( 5 ) supergravity model with the new h mass limit mh2.0 GeV. We conclude that well motivated theoretical assumptions open the way to observe experimentally a SMlike Higgs boson.
m,:,552(50) G e V ,
(4.la)
mxq5 103(94) G e V ,
(4.lb)
mxF 5 104(92) G e V ,
(4.1~)
Acknowledgement
mg5320(405) G e V ,
(4.ld)
J.L. would like to thank S. Katsanevas and J. Hilgart for very helpful discussions. This work has been supported in part by DOE grant DE-FG05-91ER-40633. The work of J.L. has been supported by an SSC Fellowship. The work of D.V.N. has been supported in part by a grant from Conoco Inc. The work of X.W. has been supported by a T-1 WorldLaboratory Scholarship. We would like to thank the HARC Supercomputer Center for the use of their NEC SX-3 supercomputer and the Texas A&M Supercomputer Center for the use of their CRAY-YMP supercomputer.
for p > O ( p < O ) . Imposing mh>70 GeV does not change these results. A related consequence is that the mass relation mxq > mx:, mh is not satisfied for any of the remaining points in parameter space and therefore the xp -+ xyh decay mode is not kinematically allowed. Points where such a mode was previously allowed (see the symbols in fig. 2 in ref. [ 71 ) led to a vanishing trilepton signal in the reaction pp-xfx: at Fermilab (thus the name “spoiler mode”) [ 3 1. The improved situation now implies at least one event per 100 pb- I for all remaining points in parameter space.
+
“+”
References [ 1 ] M. Drees and M.M. Nojiri, Phys. Rev. D 45 (1992) 2482.
77
254 Volume 306, number 1,2
PHYSICS LETTERS B
27 May 1993
[ 21 For a review see, A.B. Lahanas and D.V. Nanopoulos, Phys.
[ 10J J.L. Lopez, D.V. Nanopoulos and A. Zichichi, Texas A&M
Reu. 145 (1987) 1. [ 31 J.L. Lopez, D.V. Nanopoulos, X. Wang and A. Zichichi, Texas A&M University preprint CTP-TAMU-76/92, CERN/LAA/92-023, and CERN-PPEl92-194. [4] J.L. Lopez, D.V. Nanopoulos, H. Pois, X. Wang and A. Zichichi, Texas A&M University preprint CTP-TAMU-891 92, CERN-TH.6773193, and CERN/LAA/93-01. [ 51 R. Arnowitt and P. Nath, Phys. Rev. Lett. 69 (1992) 725; P. Nath and R. Arnowitt, Phys. Lett. B 287 (1992) 89; B 289 (1992) 368. [ 61 J.L. Lopez, D.V. Nanopoulos and H. Pois, Phys. Rev. D 47 (1993) 2468. [ 7 ] J.L. Lopez, D.V. Nanopoulos, H. Pois and A. Zichichi, Phys. Lett. B 299 (1993) 262. [ 81 J.L. Lopez, D.V. Nanopoulos and A. Zichichi, Phys. Lett. B 291 (1992) 255. [ 9 J R. Arnowitt and P. Nath, Phys. Lett. B 299 (1993) 5 8 , and Erratum; Texas A&M University preprint CTP-TAMU-661 92, NUB-TH-3066192 (revised).
University preprint CTP-TAMU-68/92, CERN-TH.66671 92, and CERN-PPE/92-I 88. J. Hilgart, Talk presented at the 1993 Aspen Winter Conf. See e.g., J. Gunion, H. Haber, G. Kane and S. Dawson, The Higgs hunter’s guide (Addisson-Wesley, Redwood City, CA, 1990). Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys.85 (1991) l;Phys.Lett.B262 (1991) 54; J. Ellis, G.Ridolfi and F. Zwirner, Phys. Lett. B 257 (1991 ) 83; H. Haber and R. Hempfling, Phys. Rev. Lett. 66 (1991) 1815. S. Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois and K. Yuan, Texas A&M University preprint CTP-TAMU-I 6/92, Nucl. Phys. B 398 (1993) 3. S. Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois and K. Yuan, Phys. Lett. B 285 (1992) 61.
78
[ 11 [12
[ 13
[ 14
[15
25 5
Jorge L. Lopez, D.V. Nanopoulos, Xu Wang and A. Zichichi
SUPERSYMMETRY TESTS AT FERMILAB: A PROPOSAL
From Physical Review D 48 ( 1993) 2062
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257 PHYSICAL REVIEW D
VOLUME 48, NUMBER 5
I SEPTEMBER 1993
Supersymmetry tests at Fermilab: A proposal Jorge L. Lopez, D. V. Nanopoulos, and Xu Wang Center for Theoretical Physics, Department of Physics, Texas A&M Uniuersity, College Station, Texas 77843-4242 and Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, Texas 77381
A. Zichichi CERN, Geneva, Switzerland
(Received23 November 1992; revised manuscript received 28 May 1993) We compute the number of trilepton events to be expected at Fermilab as a result of the reaction pp-+,y$,&, where ,y$ is the lightest chargino and y ,! is the next-to-lightest neutralino. This signal is expected to have very little background and is the best prospect for supersymmetry detection at Fermilab if the gluino and squarks are beyond reach. We evaluate our expressions for all points in the allowed parameter space of two basic supergravity models: (i) the minimal SUM supergravity model including the severe constraints from proton decay and a not too young universe and (ii) a recently proposed no-scale flipped SU(5) supergravity model. We study the plausible experimental scenarios and conclude that a large portion of the parameter spaces of these models could be explored with 100 pb-’ of integrated luminosity. In the minimal SU(5) supergravity model chargino masses as high as 100 GeV could be probed. In the no-scale flipped model it should be possible to probe some regions of parameter space with m 5 175 GeV, therefore, possibly exceeding the reach of the CERN LEP I1 for chargino and x:
neutralino (since rn = m
8
5 ) masses.
In both models such probes would indirectly explore gluino
XI
masses much beyond the reach of Fermilab. PACS numbeds): 14.80.Ly, 12.10.Gq, 13.85.Qk, 13.85.Rm
It is becoming ever more apparent that supersymmetry is “the way to go” beyond the standard model. In addition to its numerous theoretical niceties, such as its role in solving the hierarchy problem, in explaining the lightness of the Higgs boson, in the unification of the gauge couplings, in the unification with gravity, and in superstrings, supersymmetry entails a rather predictive and experimentally appealing framework. On the most general grounds, however, all one can say is that we expect the set of superpartners of the ordinary particles to appear somewhere below 1 TeV. Moreover, the number of parameters needed to describe the new particles is rather large (at least 21), and therefore a full exploration of the parameter space of a generic low-energy supersymmetric model is impractical. On the other hand, things become much simpler if one studies specific models which embody a minimal set of well-motivated theoretical assumptions, including spontaneously broken supergravity with universal softsupersymmetry breaking, and radiative electroweak symmetry breaking [l]. In fact, the dimension of the parameter space of these models is quite minimal: three softsupersymmetry-breaking masses (rn 1,2,mo,A ), the ratio of Higgs vacuum expectation values (tanp), and the topquark mass ( m , ) . Among these models there are two which stand out because of their high predictive power: (i) the minimal SU(5) supergravity model including the severe constraints from proton decay [2,3] and a not too young universe [4,5], and (ii) a recently proposed no-scale
flipped SU(5) supergravity model [ 6 ] . In fact, these two models can be viewed as prototypes of (i) traditional supergravity grand unified theory (GUT) models and (ii) string-inspired supergravity models, respectively. In this paper we begin our study of supersymmetric signals from these two models at the Fermilab Tevatron collider by considering the trilepton signal which occurs in the decay products of the reaction p$j-+x:x!$, where is the lightest chargino and X; is the next-to-lightest neutralino. This hadronically quiet signal [7]has been shown to have very little background [8-101 and is expected to be the best one for exploring the neutralinochargino sector of supersymmetric models at Fermilab [lo]. In fact, even if the squark and gluino masses are well beyond the reach of Fermilab, the two models under consideration still predict trilepton signals that can be directly observable. Conversely, the potential exclusion of certain regions of the neutralino-chargino parameter space would entail indirect exploration of a large range of squark and gluino masses. We first compare the features of both models and their predicted supersymmetric spectra. We then compute in succession the p$j-xFx:X cross section, the branching ratio of Y,: to one charged lepton, and the branching ratio of X! to two charged leptons. We find that the branching ratios in the no-scale flipped SU(5) supergravity model depend crucially on the slepton mass spectrum since it can be relatively light, and therefore differ significantly from the standard results which usually assume heavy sleptons. We conclude that with 100 pb-’ of integrated luminosity and optimal detection efficiencies,
0556-2821/93/48(5)/2062(11)/$06.00
2062
I. INTRODUCTION
-
4s
x:
@ 1993 The
American Physical Society
25 8 48
SUPERSYMMETRY TESTS AT FERMILAB: A PROPOSAL
2063
only occurs for go>> 1. The chargino and neutralino masses depend on tar$?, mg,and the Higgs mixing parameter p, whose magnitude is calculable from the radiative 11. THEMODELS breaking constraints, but its sign remains undetermined. The Higgs boson masses receive a tree-level contribution Both models under study have the same light matter which depends on tan@ and m A , and a one-loop correccontent as the minimal supersymmetric extension of the tion which depends most importantly on m, and the standard model. That is, in addition to the ordinary par( A ,tan@, m, ) squark mass. For any given set ( m /2, to, ticles, we have (i) twelve squarks ii,,,a,,,,?,,,, one can compute all particle masses and couplings and ~L,R,61,~,i;,*, (ii) nine sleptons ~ ~ , ~ , j i ~ , ~(iii) ,7~,~,~~,~,” reject sets which violate the present experimental bounds and four neutralinos x:,~,~,.,, and (iv) two charginos on m g , m q ,m,,, m A , m *, etc. Only those sets which two CP-even neutral Higgs bosons h,H, one CP-odd neuXI satisfy all present phenomenological constraints (as detral Higgs boson A , and one charged Higgs boson H*. scribed in detail in Ref. [ l 11) are kept for further analysis. The masses of all these particles depend on a set of paAll the above remarks apply to the two models under rameters which can be significantly reduced by imposing consideration. We now turn to the differences between universal soft-supersymmetry breaking at the unification them. For reference, in Table I we collect the gist of the scale and then radiative electroweak symmetry breaking following discussion. [l 11. Once this is done the masses depend on only five parameters: m , / 2 , c o ~ m O / m 1 /g2 A, s A / r n 1 I 2 , t a d , m,. For reference, the squark and slepton masses can be A. The minimal SU(5) supergravity model approximated by m?=(fmg )?ci +gi,)+di C O S ~ P M ~ , where ci and di are calculable coefficients with The model is based on the gauge group SU(5) and its minimal matter content implies that it unifies at 0 < cI <
xt2
rp,
TABLE I. Comparison of the most important features describing the minimal SU(5) supergravity model and the no-scale flipped SU(5)supergravity model. Minimal SU(5) supergravity model
No-scale flipped SU(5) supergravity model
Not easily string derivable, no known examples Symmetry breaking to standard model due to the VEV of 24 and independent of supersymmetry breaking No simple mechanism for doublet-triplet splitting No-scale supergravity excluded mT,mp< 1 TeV by ad hoc choice: naturalness Parameters 5 : m ,,2,mo, A,tan@,m, Proton decay: d = 5 large, strong constraints needed Dark matter: Q,hi >> 1 for most of the parameter space, strong constraints needed
Easily string derivable, several known examples Symmetry breaking20 standard model due to VEV’s of 10,lO and tied to onset of super. symmetry breaking Natural doublet-triplet splitting mechanism
15tanp53.5, m, < 180 GeV, & 2 6 mp 6 500 GeV mp > mi > 2m-
No-scale supergravity by construction m7,mp < 1 TeV by no-scale mechanism
,
Parameters 3: m ,2, ta@, m, Proton decay: d = 5 very small Dark matter: 0 , h i 60.25, acceptable with cosmology and big enough for dark matter problem 26tan/?532,m,<190 GeV, &,=O mp 5 1 TeV, m7 =mp m- =m,=0.3mp6300 GeV ‘L
miR =O. 18mp 5 2 0 0 GeV 2 m r ~ - m ~ ~ - m r ~ - 0 . 3 m5150 p GeV
*-+I
2m
8
-mr;=mrf
-0.3mp 5285 GeV
- - - IP I
mh 5 100 GeV
mx; mx; mh 6 135 GeV
No analogue
m, 6 135 GeV -,u>O,
mx;-mr:-m
y2
Strict no scale: t a d = t a n p ( m Z , m , 1 mh 6 100 GeV m, k 140 GeV -p
259
2064
48 -
LOPEZ, NANOPOULOS, WANG, AND ZICHICHI
sons. The light Higgs doublets come together with colored Higgs triplets ( H , ) in SU(5) pentaplets. The H, fields mediate dangerous dimension-five proton decay operators and must be heavy ( M H 3? M u ) . This needed “doublet-triplet splitting” is not explained in a simple way in this model and requires either an ad hoc choice of parameters or the introduction of additional large GUTmass representations. Requiring that M H 3< 3Mu so that the high-energy theory remains perturbative, and imposing the naturalness condition mg,mp< 1 TeV, it has been shown that the five-dimensional parameter space of this model is severely constrained by the experimental lower bound on the proton lifetime [2]. Adding on the require1, where llx is ment of a not too young universe, ll,h I the present abundance of the lightest neutralino and h, I 1 is the scaled Hubble parameter, constrains 0.5 I the model significantly more [4,5]. Yet stricter constraints follow from the recalculation of the unification mass to a higher level of precision [3]. The finally allowed set of points in parameter space is highly restricted and entails the following constraints: (i) parameter space variables
:
15tan/353.5, m,5180 G e V , i 0 2 6 ,
(2.1)
mg 5 500 GeV , mp > m, > 2mg ,
(2.2)
(2.5)
(ii) gluino, squark, and slepton masses (2.6a)
(2.3a)
m =m,=0.3mg 5 3 0 0 GeV ,
(2.6b)
(2.3b)
m =0.18mg 5 2 0 0 GeV ,
(2.6~)
5 150 GeV ,
.
’L
’R
(iii) chargino and neutralino masses
(iv) the one-loop corrected lightest Higgs boson mass mh 5 100 GeV
, g A =O ,
mg 5 1 TeV , mq =mg ,
(iii) chargino and neutralino masseslV2 X2
*,
2 5 t a n P 5 3 2 , m , 5 1 9 0 GeV , i o = O
(ii) gluino, squark, and slepton masses
2mxy-m o-rnx:-0.3mg
on this gauge group [I21 and at least one with the additional matter particles [13]. Symmetry breaking down to the s t g d a r d model gauge group occurs through VEV’s of 10,lO representations along flat directions of the scalar potential, and thus it is tied to the onset of supersymmetry breaking. The doublet-triplet splitting of the Higgs pentaplets alluded to in Sec. I1 A is realized naturally in this model through gauge symmetry allowed couplings which occur in all known examples. This mechanism also ensures that the potentially dangerous dimension-five proton decay operators are highly suppressed and innocuous. The no-scale supergravity component of the model implies that m o = A =O [I41 and therefore the model depends only on three parameters: m I ,2, tano, m,. Furthermore, consistency of the no-scale model requires m I 5 1 TeV [ 151 which explains the naturalness requirement which otherwise would need to be imposed by hand. The relic abundance of the lightest neutralino is found to be R x h i 50.25, which is well within cosmological requirements and may be large enough to explain the dark matter problem. This model also entails constraints on its parameters and correlations among the various particle masses: (i) parameter space variables
(2.4)
B. The no-scale flipped SU(5) supergravity model
2 m x ~ - m x ~ = m x ~ - 0 . 3 m5285 g GeV ,
(2.7a)
(iv) the one-loop corrected lightest Higgs boson mass
This recently proposed model (61 is based on the gauge group SU( 5 ) X U ( 1) and has additional intermediate scale matter particles that delay unification until M u 10’’ GeV, as expected to occur in string-derived models. The minimal choice of the S t r a particles is a pair of vectorlike quark doublets Q, Q with mQ 10l2 GeV and a pair of vector-like charge -133 quark singlets D , D with mD-106 GeV. There exist several string models based
-
-
-
‘The signs in Eqs. (2.3) and (2.7) indicate that these relations are only qualitative, although the majority of points in the allowed parameter space follow them closely. 2The result m < 150 GeV is the weakest possible bound
*
XI
which applies to both signs of p. For p>O ( p< O ) the upper bound is 150 GeV (120 GeV). In the subsequent figures [l(a), 3,5], for p > 0, there is an isolated point shown for m = 145
*
XI
GeV and no other points appear until = 105 GeV. This gap is sparsely populated by fine-tuned points in parameter space [3].
m h 5 135 GeV
.
(2.8)
In addition, a strict version of the no-scale scenario allows tanb to be determined as a function of mg and m , . This special case of “no-scale” has two very interesting consequences: (i) determination of the sign of ,u and (ii) determination of whether mh is above or below 100 GeV. One finds that p > O and mh 5 100 GeV if m, 5 135 GeV, whereas p < 0 and mh k 100 GeV if rn, 2 140 GeV. 111. THE TRILEPTON SIGNAL
The set of diagrams that needs to be calculated is the same for both models, only the input masses and couplings differ, and so do the resulting signals. Two diagrams contribute to pjT-x:xyX: (i) s-channel virtual W exchange ud+ W*-X:X’, and (ii) t-channel squark exchange. The second diagram has been neglected since m g2 2 0 0 GeV in the no-scale flipped SU(5) model and mg 2 6 0 0 GeV in the minimal SU(5) model, and the
260
48
2065
(b)
10-3
1-0 3-
50
100
p'>o
150
200
250
300
100
m,: (cev)
150
200
250
300
m,; (Gev)
no-scale flipped SU(5) s u p e r g r a v i t y model
FIG. 1. The cross section for pjJ-xF&' vs the chargino mass at V'y = I. 8 TeV for all points in the allowed parameter space of (a) the minimal SU(5)supergravity model and (b) the no-scale flipped SU(5) supergravity model.
x! xFx!
W x f x ! coupling only vanishes if is a pure b-ino which does not occur in practice. In Figs. l(a) and I(b) we show and for the cross section (summed over ds = 1.8 TeV, computed using the parton distribution functions of Ref. [ 161 [fit S-MS scheme]. This set of parton distribution functions is given in a convenient, compact, analytical form and describes well the small-x behavior relevant for our present purposes ke., x 20.01). For the points in parameter space in common with Ref. [lo], we have checked that our numerical results agree well with theirs.3 The scatter plots include all allowed points in parameter space as obtained in Ref. [3] for the minimal SU(5) model and in Ref. [6] for the no-scale flipped model.4 In the former case m , takes values
x;x$
3Note that our sign convention for p is opposite to that used in Ref. [lo]. 4For the no-scale flipped model, in Ref. [6] a J M Z )=O. I18 was used. For the minimal SU(5) model, in Ref. [3] a 3 ( M z ) = 0 . 1 2 6 was instead chosen in order to maximize the proton lifetime and therefore the size of the allowed parameter space. Larger values ) reduce the fraction of points which give calof a 3 ( M Z quickly culated values of sin20wwithin the present experimental bounds ~31.
throughout the interval 100-160 GeV, whereas in the latter case only the reference values m,=100, 130, 160 GeV are shown (since there are many more allowed points in parameter space). From the figure one can see that, in both models,
One can also show that for m + < 100 GeV the maxXi
imum indirectly explorable gluino masses are given by (i) 320 (460) GeV [for p > O ( p
xf
x!
XI
neglected. If the sleptons and squarks are heavy enough, the W-exchange diagrams dominate and the branching ratio into e * and p* should be close to ( 1 1 ) A 1 1 1 + 3 + 3 )=2/9=22%, independently of
+
++
26 1 2066
48 -
LOPEZ, NANOPOULOS, WANG, A N D ZICHICHI
the model parameters.5 This is precisely the result we obtained for the minimal SU(5) supergravity model:
many contributing amplitudes. A concern is the case when the 0 goes on shell ( m , < m +) and dominates the XI
(3.2)
decay amplitude. If m v = m x * , then the daughter lepton
The results for the no-scale flipped SU(5) model are shown in Fig. 2 for (a) m , = 100 GeV, (b) m , = 130 GeV, and (c) m , = 160 GeV. We find
cal configuration ke., m < m +) occurs for 18% (1%) of
*,xyv,,p*) , i , i , , , = 0 . 2 2 2 ~ 0 . ~ 8.
B(x:+xyv,e
This upper bound is model independent and occurs when the slepton exchange diagrams dominate (e.g., when the sleptons go on-shell), in particular the sneutrino one since m B< m for low chargino masses. (In the flipped model PL the squarks are always significantly heavier than the sleptons and therefore basically irrelevant for the present discussion.) As m , , 2 grows, all masses grow and the sleptons become heavier than the W boson. In fact, the twobody decay mode , Y - W*xy becomes kinematically allowed for m + 2 2 M w . At the same time, the two-body XI
chargino decay into sleptons xF-1 *t3 becomes kinematically suppressed since m p + m in this limit [see Eqs.
*
XI
(2.6b) and (2.7a)I. Therefore, the leptonic branching generally shows a transition from 2 / 3 (slepton dominance) down to 2 / 9 ( W dominance). We note that the transition may be delayed somewhat if the W x f x y coupling is dynamically suppressed as it occurs in some “pure” limits, i.e., when , y - W * and xy-B, which occurs for (pl >>M2=0.3mg. In fact, Figs. 2(b) and 2(c) show that as m , grows, the transition is delayed a bit. This is understandable since Ipl grows with m , (for fixed m g ) and the “pure” limits are then approached. Figure 2 (especially for m , = 100 GeV) also shows that during the transition between the two regimes, destructive interference between the slepton-exchange and W-exchange diagrams can be important and the branching ratio can drop down to a few percent. It is important to investigate the energy spectrum of the daughter lepton in chargino decays in order to identify possible regions of parameter space where the lepton may be too soft to be detectable. In the minimal SU(5) model the W-exchange diagram dominates the decay amplitude. For m > m x 7 + m w (two-body
*
XI
decay) the daughter lepton energy in the W rest frame is E l = t m , . For m + < m o+rnw (three-body decay) the XI
XI
lepton energy peaks for the largest W fourmomentum, p & = ( m + - m o ) 2 < m & , which in the XI
XI
chargino rest frame corresponds to E , = $ ( m + - m x p ) = a m + > l o GeV. XI
XI
xy
at rest and In the flipped
SU(5) model things are more complicated because of the 5This assumes that the ,y:-xyr6 channel is closed Le., * < m o + m , + m b ) which is certainly true for m + < I 0 0
I
will be soft, i.e., E l = ( m ~ : - m ~ ) / 2 mf. This kinematiXI
XI
the points in parameter space for p > 0 ( p < 0). Assuming that only the +-exchange amplitude contributes, we find that - f of these points lead to soft leptons ( E , < 5 GeV). More generally (i.e., including all two-body and threebody decay amplitudes) we have computed the probability that El < 5 GeV, and find it over 75% for each.of these points; i.e., soft leptons are indeed likely for m i . < m f. For m , > m
XI
the two-body decay
xF-
inates when kinematically accessible ( m
XI
W*xy dom-
* 2 2Mw), and
XI
no soft leptons occur. If this channel is closed or suppressed, three-body decays are the norm and for each of these points the lepton has > 75-90% probability of having El > 5 GeV. Finally, when soft leptons are likely to occur [ < 6%( < 1 % ) of the total number of allowed points in parameter space], no range of chargino masses is preferentially singled out and therefore exploration of points in parameter space for all possible chargino masses should not be a problem. The x! decay channels which are possibly kinematically open are several: (i) x!-xyI+l-, through virtual Z,TL,?, exchanges, (ii) x!+xyv,SI, through Z,V,, (iii) ,y:+xyqq, through Z , q L , q R , and (iv) the “spoiler mode” x ! - ~ y h . In analogy with the chargino decay, if the sleptons and squarks are heavy enough, and the Zxyx! coupling is not too small, then the Z-exchange diagrams dominate and B(X!--fXye +e-,,yy,u+p- ) =B(Z-e+e-,p+p-)=6.6%. In the minimal SU(5) supergravity model this situation occurs quite often (especially for p < O), see Fig. 3. In fact, from Fig. 3 (bottom row) in Ref. 5 one can see that the composition is away from a pure b-ino state (which would drive the Zxyx; coupling to zero) for most of the parameter space. Exceptions to this situation occur for p > 0 when m g is low (and thus so is m + ), while for p < 0 when m g 300 GeV
xy
-
XI
=mr:-90
GeV. Figure 3 here shows that in these
cases the branching ratio is higher, since the Z-exchange channels are highly suppressed and the slepton and squark exchanges play a role. The largest value the branching ratio could take is =2/3, although in this model it remains 5 2 5 % . This result agrees well with that in Ref. [lo] for the relevant case ( m g = 2 m g )in that paper. When the spoiler mode is open ( m > m h or x2
approximately m
XI
+
2 2 m h ) the dilepton branching ratio
x: vanishes, although this occurs only for a small portion ( 5 10%)of the allowed parameter space.6
m
XI
YI
XI
GeV. Using the approximate relations in Eqs. (2.3a)and (2.7a) this channel will not be open until m 2 2 ( r n , + m b ) > 190 GeV.
x;
61n fact, it has been recently shown [17] that improved lower bounds from the CERN e’e- collider LEP on the lightest Higgs boson mass in this model ( m h2 60 GeV) actually exclude all points in parameter space where the spoiler model is open.
262 2067
SUPERSYMMETRY TESTS AT FERMILAB: A PROPOSAL
48
(a)
pU>O
m , = l 00 1 .o
1.o h
*,
0.8 -
+'
0.6 --
-
-\ ..
-
06
-
04
>
J
0-
0.8
.............................. . .
0-
0.4
-
, .,. ... .
+I..
x D
0.2
02
0.0
0.0
50
100
m,: ( c e v )
150
200
250
300
m,; ( c e v )
no-scale flipped S U ( 5 ) supergravity model
0.8
1 0.2
0.2
0.0
0.0
50
100
150
m,:
200
(cev)
250
300
50
100
150
200
250
300
m,: ( c e v )
no-scale flipped SU(5) supergravity model
no-scale flipped SU(5) supergravity model
FIG. 2. The branching ratio for X ~ - + x ~ v e e * , x ~ v #in * the no-scale Ripped SU(5) supergravity model for (a) m,= 100 GeV, (b) m,= 130 GeV, and (c) m,= 160 GeV. Note that the branching ratio is bounded above by 2/3 (when the slepton exchange diagrams dominate) and can be lower than 2/9 (when the W-exchange diagrams dominate) due to destructive interference effects.
263 2068
LOPEZ, NANOPOULOS, WANG, A N D ZICHICHI
P>O 0l '.O8
o.6 0.4
48
P
I minimal SU(5) supergravity model
FIG. 3. The branching ratio for ,y-xye +e -,,&+pin the minimal SU(5) supergravity model. The points on the horizontal axis occur when the spoiler mode is open. The accumulation of points around 6% occurs when the 2-exchange diagrams dominate.
x:
The results for the dilepton branching ratio for the no-scale flipped model are shown in Fig. 4 and exhibit quite a bit of structure with values ranging from zero up t o =2/3. For a given chargino mass, the various points correspond to different values of tanp. For example, in Fig. 4(a) for fixed m 5 2 100 GeV, tanB starts at the botXI
tom of the pack of curves at 28 and then decreases in steps of two until it becomes 4 (8) for p > 0 ( p < 0)at the top. (The isolated curve for p>O corresponds t o tanB=2.) A new spoiler mode opens when the sneutrino is sufficiently light, but this one and the original spoiler mode can be overtaken by an "antispoiler mode" when the ?L,R are sufficiently light. In fact, the branching ratio approaches its maximum value in this case. Note that if m o-m is small, then the daughter lepton will be soft ~2
~L,R
[E,= ( m 2o - m ~2
)/2m ~L,R
and difficult to detect. Since X2
in the flipped model m
-0.3mg, m =0.3mg, and x: 'L m =0.18mg (see Table I), the two-body decay ampliTR
tude &-l*?: dominates and it is not likely to lead to soft leptons. However, because of uncertainties in the above mass relations, -4% (-6%) of the points for p > O ( p < O ) do give E , < 5 GeV. We have checked though that the soft leptons always occur for chargino masses below = 110 GeV. For large values of the chargino mass, the analogue of the W dominance in chargino decays occurs here, as the two-body decay x!-xyZ becomes kinematically allowed for m >2Mz. Furthermore, the usual spoiler mode x: X:-Xyh also turns on and the branching ratio drops quickly. Since mh 5 105 (120) GeV for m, = 130 (160) GeV (see Fig. 6 in Ref. [6]), one would expect m o>m ,+mh to be always satisfied for x2
m
X:
XI
2 2m, 2 210 (204) GeV, which is in good agreement
with the actual results in Fig. 4. The opening of these two channels is seen clearly in Figs. 4(b) and 4(c) which
show a successive two-step drop in the neutralino branching ratio for large chargino masses. Finally we compute the number of trilepton events per 100 pb-' of integrated luminosity for both models, summed over all possible e and p combinations. In the minimal SU(5) supergravity model (see Fig. 5 ) we find at least one event for all allowed points in parameter space and as high as 127 (129) events for p > O ( p < O ) for low chargino masses. The actual fraction of points which could be probed at Fermilab depends on the ultimate integrated luminosity achieved and on the experimental efficiencies for the detection of these signals. Statistically speaking, only points in parameter space for which three or more events are predicted could be experimentally verified or excluded. An across-the-board 30% efficiency cut appears reasonable, with lower (higher) efficiencies expected for lighter (heavier) chargino masses [18]. This situation will probe points in the parameter space with 10 or more predicted events, i.e., about half of the allowed parameter space in this model. An idealized situation would occur with 200 pb-' (e.g., combining the data from both detectors) and a 60% efficiency for the heavier chargino masses. In this case, values down to 2.5 in Fig. 5 could be probed which constitutes a large portion of the allowed parameter space (all points for p < 0)and chargino masses as high as = l o 0 GeV. Bearing in mind the mass correlations in this model, this probe would explore indirectly gluino masses as high as 320 (460) GeV for p > 0 ( p
gino masses. The number of events for light charginos can be quite large, as high as 443 for m, = 160 GeV and p < 0. In the conservative scenario discussed above one could probe as high as m = 120- 150 GeV depending
*
XI
on m , , although a large unexplored region with chargino
264 48 -
SUPERSYMMETRY TESTS AT FERMILAB: A PROPOSAL
(a)
50
100
Pu>O
150
200
P
m,=100 G e V
250
300
50
2069
100
150
200
250
300
250
300
250
300
no-scale flipped SU(5) supergravity model
-rt +
(b)
P’>O
P
m,=130 G e V
1.0
0.8
:o
x
I
0.6
+
aJ
x
0-
t x
-
0.4
O N
m
0.2 0.0
50
100
150
200
250
300
50
100
m,: (Gev)
150
200
m,; ( c e v )
no-scale flipped SU(5) supergravity model
(C) ., 3
+%
P>O
P
m,=160 G e V
1.o 0.8
0 -
x
I
0.6
+
aJ
0-
x t
0.4
3 in
0.2
O N
0.0
50
100
150
200
m,: (Gev)
250
300
50
100
150
200
m,; (Gev)
no-scale flipped SU(5) supergravity model
FIG. 4. The branching ratio for x:+xye+e-,xyp+p- in the no-scale Ripped SU(5) supergravity model for (a) m,=100 GeV, (b) m,=130 GeV, and (c) m,=160 GeV. The accumulation of points near the horizontal axis correspond to a new spoiler mode x;-x@,. The maximum values are reached when the “antispoiler mode” ~ ~ - + ~ opens ~ 1 *up.7 For ~ ~large chargino masses, the usual spoiler mode Xq-Xyh dominates.
265 48 -
LOPEZ, NANOPOULOS, WANG, A N D ZICHICHI
2070
1
e a 0
3
h
a c) vi
W c W
c d
-
W a
.-
4
5
I,, I,,
'," l1 o 40
, I - .,
..
.
:.
,#
.
. ,
,
,
60
j
, I , , .4 ,.. I , , , , I , , , , I ,
1.
/,,
80
I
.... .
I,,,,I,,,,I
100
120
I
140
minimal SU(5) s u p e r g r a v i t y model
FIG. 5 . The number of trilepton events per 100 pb-l for the minimal SU(5) supergravity model. Note that with 200 pb-' and 60% efficiency it should be possible to probe as high as m = 100 GeV.
*
XI
(a)
wo
102
102
101
101
100
100
0-1
10-1
0-2
P
m t = 100 G e V
10-2
50
100
150
200
250
300
50
100
m,; (Gev)
150
m,;
200
250
300
250
300
(cev)
no-scale flipped S U ( 5 ) s u p e r g r a v i t y model
(b)
P
m,=130 G e V
PU>O
102
102
101
100 C
4
-5e
10-1
."
10-2
I V
50
100
150
200
250
300
50
100
150
200
m,; ( G e V ) no-scale flipped S U ( 5 ) s u p e r g r a v i t y model
FIG. 6. The number of trilepton events per 100 pbKi for the no-scale flipped SU(5)supergravity model (excluding points for which the spoiler modes are open) for (a) m , = 100 GeV, (b) m , = 130 GeV, and ( c ) m , = 160 GeV. Note that with 200 pb.-' and 60% efficiency it should be possible to probe up to m = 175 GeV.
*
xi
266 SUPERSYMMETRY TESTS AT FERMILAB: A PROPOSAL
48
-
(c)
I
n a
pu;O
m,=160 G e V
PL,O
50
207 1
100
150
200
250
300
m,; ( c e v )
m,: (Gev)
no-scale flipped SU(5) supergravity model
. FIG. 6. (Continued).
masses all the way down to the LEP limit will remain. In the idealized situation, the unexplored regions will diminish considerably and the reach could be extended up to mx:-175 GeV. Note that in this model the range of
SU(5) supergravity model. As discussed in Sec. I1 B, this allows us to determine ta@ and the sign of p. The values for the cross section and branching ratios are more precise here, since they only depend on m, for fixed m
Fermilab for chargino masses could exceed that of LEP 11. This statement needs to be viewed from the proper perspective since at LEP I1 the region m S 100 GeV
but they still fall within the limits found above. In Fig. 7 we show the results. The same general remarks as for the regular no-scale model apply here, although this time one could easily exclude ranges of rn +, since the spoiler
X:
*
XI
will be studied quite thoroughly, whereas at the Tevatron some of this region will remain unexplored. The indirect reach for mg due to the mass correlations in this model could be as high as mg=600-750 GeV. As discussed above, a small fraction of points in parameter space are likely to lead to soft daughter leptons. However, these points are not concentrated on any particular values of chargino masses and thus should not impair the search for charginos up to the highest possible masses. Before we conclude let us also present the number of trilepton events for the strict case of the no-scale flipped
a
-
I'
0
a
.
~
101
.'
c
:
100
C
-3
' I
"
' I'
I
'
'
I
"
' ' I
"
x:
'
10-1
.. ... .
'..: .
',::,
...-.: .
130
100
7
:::.,
1,,,,1,/,/11/111111
I
100
710-1
.. >.I.,
10-2
102
4 101
.:.:..,
al
.I
"
4
E .: .
I L I I
7
'-10-2
50
m,:
For the conser-
tion the upper end could be pushed up to 175 GeV. Note that any excluded range of chargino masses or more generally any excluded portion of the parameter space, implies constraints on the sparticle masses. In particular, the chargino mass ranges discussed above apply also to the second-to-lightest neutralino masses [see Eqs.
.. . . . .. ..
.
* 5 75 GeV.
XI
vative scenario, it should be possible to fully explore 75 GeV S m S 150 GeV, whereas in the idealized situa-
102
al v] 4
"
XI
modes are only open for m
100
(cev)
150
200
250
300
m,: (Gev)
s t r i c t no-scale flipped SU(5) supergravity model
FIG. 7. The number of trilepton events per 100 pb-' for the strict case of the no-scale flipped SU(5)supergravity model for the indicated values of the top-quark mass in GeV. The spoiler modes are only open for rn 5 75 GeV. Note that with 200 pb-' and 60%
*
XI
efficiency it should be possible tofully probe 75 GeV 5 m f + 5 175 GeV.
267
2072
48 -
LOPEZ, NANOPOULOS, WANG, AND ZICHICHI
(2.3a) and (2.7a)I. This correspondence is quite uccurute for the no-scale flipped model, and more qualitative for the minimal SU(5) model. IV. CONCLUSIONS
We have studied the signal for the best prospect to detect supersymmetry at Fermilab, short of actually observing the gluino and squarks. The neutralino-chargino sector in the class of unified, supersymmetric models we have explored has the potential of probing a very large portion of the whole parameter space of these models, without actually seeing the traditionally sought for strongly interacting supersymmetric particles. We have considered the minimal SU(5) supergravity model and a recently proposed no-scale flipped SU(5) supergravity model. These are well motivated models which can be taken as typical examples of nonstringlike and stringlike models, respectively. Results in the class of models we consider, are expected to differ from those obtained in the minimal supersymmetric standard model (MSSM) since there (i) the large dimension of the parameter space does not allow its complete exploration, (ii) important theoretical constraints which exclude large regions of parameter space are not taken into account, and (iii) the interdependence of the various sparticle and Higgs boson masses is absent. Our computations of the branching ratios in the
[l] For a review, see A. Lahanas and D. V. Nanopoulos, Phys. Rep. 145, 1 (1987). [2] R.Arnowitt and P. Nath, Phys. Rev. Lett. 69, 725 (1992); P. Nath and R. Arnowitt, Phys. Lett. B 287, 89 (1992); 289, 368 (1992). [3] J. L. Lopez, D. V. Nanopoulos, H. Pois, and A. Zichichi, Phys. Lett. B 299,262 (1993). [4] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 291, 255 (1992). [5] J. L. Lopez, D. V. Nanopoulos, and H. Pois, Phys. Rev. D 47,2468 (1993). [6] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Texas A&M University Report Nos. CTP-TAMU-68/92, CERN-TH.6667/92, and CERN-PPE/92-188 (unpublished). [7] V. Barger, W.-Y. Keung, and R. J. N. Phillips, Phys. Rev. Lett. 55, 166 (1985); H. Baer and X. Tata, Phys. Lett. 155B, 278 (1985); H. Baer, K. Hagiwara, and X. Tata, Phys. Rev. Lett. 57, 294 (1986);R. Arnowitt and P. Nath, Phys. Rev. D 35, 1085 (1987). [8] P. Nath and R. Arnowitt, Mod. Phys. Lett. A 2, 331 (1987).
no-scale flipped model provide a clear example of this situation. We have considered two experimental scenarios which could be realized in due time and have shown that a large fraction of the parameter space for both models could be probed. For the minimal SU(5) supergravity model it should be possible to explore as high as m 2 100 GeV
*
XI
(and indirectly m g as high as 460 GeV), whereas in the no-scale flipped model it may be possible to reach up t o mx: = 175 GeV (and indirectly m g as high as 750 GeV) and therefore greatly exceed the reach of LEP I1 for chargino and neutralino masses, at least in some regions of parameter space.
ACKNOWLEDGMENTS We would like to thank J. White for very helpful discussions. This work has been supported in part by D O E Grant DE-FG05-91-ER-40633. The work of J.L. has been supported by the SSC Laboratory. The work of D.V.N. has been supported in part by a grant from Conoco Inc. We would like t o thank the H A R C Supercomputer Center for the use of their NEC SX-3 supercomputer and the Texas A&M Supercomputer Center for the use of their Cray-YMP Supercomputer.
[9] R. Barbieri, F. Caravaglios, M. Frigeni, and M. Mangano, Nucl. Phys. B367, 28 (1991). [lo] H. Baer and X. Tata, Phys. Rev. D 47, 2739 (1993). [ll] For a detailed description of this general procedure see, e.g., S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Nucl. Phys. B398, 3 (1993). [I21 For a review, see J. L. Lopez and D. V. Nanopoulos, in Proceedings of the 15th Johns Hopkins Workshop on Current Problems in Particle Theory, Baltimore, Maryland, 1991, edited by G. Domokos and S. Kovesi-Domokos (World Scientific,Singapore 1992),p. 277. [I31J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Nucl. Phys. B399,654 (1993). [I41J. Ellis, C. Kounnas, and D. V. Nanopoulos, Nucl. Phys. B247,373 (1984). [I51 J. Ellis, A. Lahanas, D. V. Nanopoulos, and K.Tamvakis, Phys. Lett. 134B,429 (1984). [16] J. F. Morfin and W. K. Tung, Z. Phys. C 52, 13 (1991). [I71J. L. Lopez, D. V. Nanopoulos, H. Pois, X.Wang, and A. Zichichi, Phys. Lett. B 306, 73 (1993). [I81 J. White (private communication).
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269
Jorge L. Lopez, D.V. Nanopoulos, Xu Wang and A. Zichichi
SUSY SIGNALS AT DESY HERA IN THE NO-SCALE FLIPPED SU(5) SUPERGRAVITY MODEL
From Physical Review D 48 ( 1993) 4029
I993
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27 1 PHYSICAL REVIEW D
1 NOVEMBER 1993
VOLUME 48, NUMBER 9
SUSY signals at DESY HERA in the no-scale flipped SU(5) supergravity model Jorge L. Lopez,’,2 D. V. Nanopoulos,”’ Xu Wang,’,’ and A. Zichichi3 ‘Centerfor Theoretical Physics, Department of Physics, Texas A&M Uniuersity, College Station, Texas 77843-4242 ’Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, Texas 77381 ’CERN, Geneva, Switzerland
(Received I3 April 1993) Sparticle production and detection at DESY HERA are studied within the recently proposed no-scale flipped SU(5) supergravity model. Among the various reaction channels that could lead to sparticle production at HERA, only the following are within its limit of sensitivity in this model: e-p-+FL~Rx~+X,V,x;+X where , x: (i=1,2) are the two lightest neutralinos and x; is the lightest chargino. We study the elastic and deep-inelastic contributions to the cross sections using the Weizsacker-Williams approximation. We find that the most promising supersymmetric production channel is right-handed selectron ( F R ) plus first neutralino (xp),with one hard electron and missing energy signature. The V e x ; channel leads to comparable rates but may also allow jet final states. A righthandedly polarized electron beam at HERA would shut o f fthe latter channel and allow preferentially the former one. With an integrated luminosity of L=IW pb-’, HERA may be able to extend the present CERN LEP I lower bounds on mZR,m g , m by ~ 1 5 - 2 0GeV, while L=1000 pb-’ would x?
make HERA competitive with LEP 11. We also show that the Leading Proton Spectrometer at HERA is an excellent supersymmetry detector which can provide indirect information about the sparticle masses by measuring the leading proton longitudinal momentum distribution. PACS numbeds): 14.80.Ly, 12.10.Gq,.13.60.Fz, 13.60.Hb
I. INTRODUCTION
The search for supersymmetric (SUSY) particles using existing facilities is the crucial problem for particle physicists nowadays. One of the most important reasons to study detailed spectra and properties of the expected SUSY particles on the basis of well motivated theoretical concepts is that quite a few particle accelerators are either running [Fermilab Tevatron, CERN e +e - collider LEP I, SLAC Linear Collider (SLC), DESY HERA] or will become operational in the near future (LEP 11) and their center-of-mass energy is within the range of the sparticle masses. Using two well motivated supersymmetric (the minimal SU(5) [ l] and the no-scale flipped SU(5) [2] supergravity) models, we have previously discussed the possible SUSY production channels and detection signatures at the Tevatron 131 and at LEP I1 [4]. In this paper we continue this program applying it to the H E RA e - p collider within the context of the same two models. Fortunately or unfortunately, the minimal SU(5) supergravity model is out of the reach of H E R A because the slepton and squark masses ( 2 500 GeV) are too large t o be kinematically accessible. On the other hand, in the no-scale flipped SU(5) supergravity model, the slepton and squark masses are much lighter and part of the parameter space can be explored at HERA. However, since the squark masses are always above 200 GeV, the much studied production channels involving squarks [5] are highly suppressed and are neglected in this paper. Therefore, we focus on the production of sleptons, charginos, and neutralinos at H E R A within the predictions of the no-scale flipped SU(5) supergravity model. It is in0556-2821/93/48(9)/4O29(8)/%06.M)
48
teresting to remark that in contrast with “generic” supersymmetric models where the squarks can arbitrarily be taken t o be light or heavy, this is not an option in this model; that is, H ERA should not produce squarks if this model is correct. The production processes of interest at H ERA are (1.la)
e-p-+T,X;+X,
( l .l b )
both of which have small standard model backgrounds. W+ei7,)=3X10-3 pb and Indeed, d e p - + v , W p , a ( e p - + e Z p ,Z + v F ) = 2 X lo-’ pb [6,7]. Moreover, by measuring the total Y , W and eZ cross sections through the other decay modes of the W and Z one could in principle subtract off these backgrounds [8]. A more serious background is deep-inelastic ep -+ e WX, where the scattered electron is lost in the beam pipe (-50% of the time) and the W decays leptonically: a ( e p - - t e W X ) B ( W - + e ~ , ) = 0 . 0 5pb [7]. The processes in Eq. (1.1) receive elastic (relevant only for Q25 mj),deepinelastic ( Q 2 >4 GeV’), and inelastic contributions, where - Q 2 is the exchanged virtual photon mass squared. It has been shown [9] that the cross section for the inelastic processes, whereby the proton gets excited into various resonances, is smaller than that for the other two. We neglect its contribution in our calculations. This makes our results conservative as far as the sparticle mass lower bound explorable at H ERA is concerned. Also, the exact calculation of the total cross section for the processes mentioned above usually involves the numerical evaluation of a three- (or more) body phase space which is rath4029
@ 1993 T h e American Physical Society
272 LOPEZ,NANOPOULOS, ?TANG,AND ZICHICHI
4030
48 -
er time-consuming because of the large size of the parameter space to be scanned. For this reason we use the Weizsacker-Williams (WW) [ 101 approximation scheme proposed in Refs. [6,8]. In this method the e y reaction is treated as a subprocess with a real (on-shell) photon. By incorporating the density distribution of photons inside protons or quarks, one can get reasonable approximations to the total cross sections. The signature for selectron-neutralino production is dominated by iiRx: and consists of one outgoing hard electron plus missing transverse momentum (fir). There is a small contribution from pXx! production which can produce trilepton (x!-Z+Z-+x:) or mixed (x!-+2 jets +x’$ signals. Chargino-sneutrino production can also lead to one outgoing hard lepton since the chargino is likely to decay leptonically and the sneutrino decays mostly invisibly (t,-+ye +x:). This paper is organized as follows. First we discuss the features of the no-scale flipped SU(5) supergravity model (Sec. 11). Then we give the exact formulas for the relevant tree-level cross sections in the WW approximation (Sec. III), followed by the results of the calculation (Sec. IV). Finally we discuss the phenomenological implications of our work (Sec. V).
Also, most of the weakly interacting sparticles cannot be too heavy. In fact, we take the “no-scale inspired” condition mg,q5 1 TeV to hold. One finds
II. THE NO-SCALE FLIPPED SU(5) SUPERGRAVITY MODEL [2]
III. THE ALLOWED PRODUCTION PROCESSES
This supersymmetric model can be viewed as a specific subset of the minimal supersymmetric standard model (MSSM),in that its three-dimensional parameter space is contained in the 2 1-dimensional parameter space of the MSSM. This subset is not arbitrary, but determined by the application of several well-motivated theoretical constraints. In this model it is assumed that below the Planck scale the gauge group is flipped SU(5),with some special properties expected from a superstring-derived model; that is, it is a string-inspired model. For example, gauge coupling unification occurs at the scale M u= lois GeV, in contrast with 10l6 GeV for the minimal SU(5) model. Moreover, the usual supergravity-induced universal soft-supersymmetry-breaking parameters are assumed to obey m o = A = O , as is the case in typical no-scale supergravity models [ 111. Thus the only three parameters in this model are the top-quark mass (m,), the ratio of Higgs vacuum expectation values ( t a d ) , and the gluino mass (mg )’ Through the running of the renormalization group equations and the minimization of the one-loop effective potential, one can obtain the whole set of masses and couplings (including the one-loop-corrected Higgs boson masses) in this model for each allowed point in parameter space [12]. In what follows we take m,=100, 130,160 GeV, for which we find 2 < t a d < 32. Clearly, the several sparticle masses will be correlated, and are found to scale with the gluino mass. Of great relevance is the fact that the present body of phenomenological constraints on the sparticle masses disallows certain combinations of the parameters, in particular one obtains mg=mg 2200 GeV .
(2.1)
m
ZR
rn 71
< 190 GeV , mq < 305 GeV , m, <295 GeV , <185 GeV , m- <315 GeV , mh
m ~ < l 4 5 G e V ,m 0 < 2 8 5 G e V , m + < 2 8 5 G e V . XI
Xl
(2.2) There are also simple approximate relations that these masses obey, namely m -0.3mg , m -0.18mg, EL
(2.3a)
ZR
(2.3b) For low mg, the sneutrino mass is close to mFR;as mg grows, the sneutrino mass approaches m Note that 9* mzR/mFL=O. 6, in sharp contrast to usual approximation of degenerate selectron masses. For more details on the construction of this model we refer the reader to Ref. [2].
The relevant Feynman diagrams for the sparticle production channels in Eq. (1.1) are shown in Fig. 1 for the elastic contributions. The deep-inelastic processes receive contributions analogous to those shown in Fig. 1 with the replacement proton for parton, plus additional production diagrams involving squark exchanges. Since it has been shown [9] that the squark contributions to the
d.2
pWton
proion
x:.2
proton
proton
FIG.1. The Feynmann diagrams contributing to the production channels e-p+.PL,R&+p and, e - p + V , x ; + p , through the elastic processes. The relevant deep-inelastic diagrams can be obtained by simply replacing proton by parton.
273 SUSY SIGNALS AT DESY HERA IN THE NO-SCALE . . .
48
403 1
are highly nondegenerate (see Sec. II), in sharp contrast with the approximation of degenerate selectron masses usually made in the literature. The differential cross section for the subprocess V-+z& (for UnPolarized incident electrons) is given by (see also [13])
cross sections for this type of deep-inelastic processes are negligible for m4 2 200 GeV, which is the case in this model (see Set. II), in what follows we neglect all diagrams involving squarks. We also remark that in this model the masses of the right- and left-handed selectrons
(3.1) where h 2 ( u , b , c ) = u 2 + b 2 + c 2 - 2 u b - 2 u ~-2bc, p , ( p , ) is the electron (selectron) momentum, and 3,?=(p, -p2)2, are the Mandelstam variables for this subprocess. The coupling factors fL,R are (3.2a)
with
N:, =Ni,cosBw+Ni,sinB,
, Ni;= -Nilsin0,+N,,cosBw
,
(3.3)
where Nil, N,, are elements of the matrix diagonalizing the neutralino mass matrix. Here we follow the conventions of Ref. [14]. The differential cross section for the subprocess ey-+S,X; is given by (see also [15]) e ‘fL2 (ey-.p,X;)=-ih(a,m,-,m2 d cod 32379 XI
) ye
I
m2-(mt -mZ-) XI
vc
(2-m2-
XI
)2
-
( m t -*t ) ye
?(ii-m’-) XI
+
(mt -m2-)(m:;-?) ve
XI
?(ii-m2-) XI
1
,
(3.4)
wherep, ( p 2 )is the electron (chargino) momentum, and f; = g V , , , with V , , an element of the matrix diagonalizing the chargino mass matrix [14]. The Weizsacker-Williams (WW) approximation [lo] is now used to simplify the calculation. For elastic processes we use the following photon distribution in the proton [8] (3.5)
(3.6) I
The total elastic cross section for ep-+Xp can then be written as
cesses we use the photon distribution in the quark of Ref. PI,
where b(3) is the total subprocess cross section for the real ye-X process [i.e., Eqs. (3.1), (3.4) integrated over COSB],and z =a/s, where 3 is the center-of-mass energy of the subprocess. For a two-body final state X with particles of masses f i and r?i one has
(3.9)
,
,,
z m i n =1- ~ -m , + r ? i ~ ~ ’ . Also, zmax=( 1-m pD’;
)2.
(3.8) For the deep-inelastic pro-
where t , , , , = ~ s - ( f i , + i i l , ) ~ and tCut=4 GeV2 are the limits put on Q 2 for the deep-inelastic process. Also, e q/ is the electric ch: 5~ of the qf quark, x is the parton density distribution variable, and v = z / x . The total cross section for the deep-inelastic processes is thus given by [61
274
4032
48 -
LOPEZ,NANOPOULOS, WANG, A N D ZICHICHI mostly in the following ways
(4.2a)
FL-Lx?,
(3.10) where the parton distribution functions of Ref.[16] ["fit S-modified minimal subtraction scheme (MS)"] have been used, with the energy scale Q2=(tm,-tfeut)/
In(?,,
).
/?,"t
It has been observed that by using the WW approximation, the results are usually larger than the exact results by 20-30% for the elastic case [8]. However, for the deep-inelastic processes the WW results are smaller than the exact ones [6]. Consequently the WW approximation will not enhance the effects and is thus good enough in light of the inherent uncertainties in this type of calculations. Moreover, these shifts in the cross sections are equivalent to shifts in the selectron or chargino masses of 5 GeV or less. IV. RESULTS A. Selectron-neutralinoproduction
There are four possible production channels at HERA e~
+
FR
x?,ZR x;, ~ L x ? zLx;+X , .
(4.1)
By far the largest cross section is for the Z R x y channel. The main reason for this is that in this model the ER mass is much smaller than the EL mass [see Eq. (2.3a)],and the is the lightest SUSY particle (LSP), first neutralino which escapes detection. By the same kinematical reasons the F R x ! and ZLx? cross sections are smaller but still observable, whereas the Z L x ; contribution is negligipb). This pattern holds for both elastic and ble ( < deep-inelastic processes. The above sparticles decay
x?
P> 0
ERReRX1
0
(4.2b)
,
x ; - + y / i i / x ~ , I + z - ~ ? , q ~ x.?
(4.2~)
However, in this model there are some points in the parameter space that also allow the rare decay channels -eL-+e& and F R + e R ~ ; . These only contribute for a small region of parameter space ( 012% of the allowed points) and are phase-space suppressed. The cross section for the dominant elastic ep-FRxy-ep+$ and deep-inelastic ep +FRx?-eX +$ processes are shown in the top row of Figs. 2 and 3 respectively. Note that for increasingly larger selectron masses, the cross section for the deep-inelastic process drops faster than that for 'the elastic one. (Also, the deep-inelastic cross section suffers from a much larger SM background than the elastic one does.) The analogous results for the smaller Z R x ; and ZLx? channels are shown in the bottom row of Figs. 2 and 3. Let us consider the four elastic cross sections O ( F ~ , ~ & ) in order to disentangle the best signal to be experimentally detected. According to Ref. [8], the cross section for the elastic processes [Q. (4.1)] peaks at a value (p:) of the daughter electron transverse momentum given by m2
-m20
F ~ R . ~ XI,Z
(4.3)
Pb = 2mzR.L
Moreover, a Monte Carlo study shows that the average transverse momentum is close to ( p ; ) =p:. To get an idea of the most likely values of p ; , we have computed the average p: (weighed by the four elastic cross sec-
flipped SU(5)
FIG. 2. The elastic cross section for e-p-P,yp-ep+$ versus mlR (top row) and e p -+i? ;,L,y!, -ep +$ (bottom row). Note the dominance of
the former. The corresponding cross section for pL,y! is negligible.
275 SUSY SIGNALS AT DESY HERA IN THE NO-SCALE
48
-G n
W O
4033
P
flipped SU(5)
0.100 0.050
5 0-
0.010
'y
0.005
5 (1.
-
v
t?
1 1 1 1 , 1 1
* : d ,
0.001 40
n
2
60
80
I
I
I
100
I
I
I
I
I
120
,ool
I
I
I
40
$,I
I
60
I
, ,
I , I'b, \ . , I , , , 80
!OO
0.100 0.050
2
tions) and the results are shown in Fig. 4. Clearly, the daughter electrons will be hard and with large p T . This is an excellent signal to be detected at HERA. For elastic processes, another measurable signal is the slowed down outgoing proton. Since the transverse momentum of the outgoing proton is very small, the relative energy loss of the proton energy z=(E;-E,oU')/E: is given by r = l - - x L , where x L is the longitudinal momentum of the leading proton. It has been pointed out [8] that the t distribution is peaked at a value not much larger than its minimal value,
I
I
J
120
FIG. 3. The deep-inelastic (DI) cross section for e - p -F~~+&+# versus mZR
(top row) and e-p+F;.,& +ep +# (bottom row). Note the dominance of the former. The corresponding cross section for F& is negligible.
weighed by the different elastic cross sections U ( F ~ , ~ X ~ , ~ ) . The results are shown in the top row of Fig. 5 versus the total elastic cross section. These plots show the possible values of Zmin for a given sensitivity. For example, if elastic cross sections could be measured down to pb, then Tmin could be fully probed up to -0. 17. Now, Zmin can be computed from Eq. (4.4) and be plotted against, say m as shown in the bottom row of Fig. 5. For the
-
2R
'
example given above (Zmin5 0.17) one could indirectly probe U, masses as high as -108 GeV (as Fig. 2 also shows). Note that a useful constraint on m is possible ZR
because the correlation among the various sparticle masses in this model makes these scatter plots be rather well defined. This indirect experimental exploration still Therefore, the smallest measured value in the z distriburequires the identification of elastic supersymmetric tion should be a good approximation to zmin. Since the events with signature (in order to identify proLeading Proton Spectrometer (LPS) of the ZEUS detectons that contribute to the relevant z distribution), but tor at HERA can measure this distribution accurately, does not require a detailed reconstruction of each such one may have a new way of probing the supersymmetric event. spectrum, as follows. We calculate the average Imin One interesting phenomenon in selectron-neutralino zmin=-(m )2 . s %,L + m x O 1,2
w0
(4.4)
flipped SU(5)
FIG. 4. The most likely value
c
of the transverse momentum of
the daughter electron (weighed by the various elastic cross sections) versus the total elastic
cross section for selectronneutralino production. The daughter electron will be hard and with largepT.
276 4034
v 0 r
LOPEZ, NANOPOULOS,WANG, AND ZICHICHI
100 l 80
Z
E
-
;
1
-
~
-+a,:>-
.-
60 40
-;>? O '..-_ -.,..' - . i .. ._ --.I +- .. ..._. .. -*.. --:
48
FIG. 5 . The most likely value of the relative proton energy loss in elastic processes (weighed by the various elastic cross sections) versus the total elastic cross section for selectron-neutralino production (top row) and mT (bottom row). The Leading Proton Spectrometer (LPS)will allow determination of Z,,,, and thus ~ an indirect measurement of mPR
.
60
0
0.05
0.1
0.15
0.2
40
0
0.05
Zmin
0.1
0.15
0.2
Zrnl"
production at H E M is the possibility of using polarized electron beams. Since we have seen that m(i?&) >>u(F&), right-handed beams are expected to be much more active in producing SUSY signals than left-handed beams. To compare the results obtained with R and L polarized beams is a further selection power to disentangle a genuine signal at HERA.
nal, sneutrino-chargino production can only occur when the electron beam is nof completely right-handedly polarized, because J e couples only to left-handed electrons. The allowed decay modes for the channel in Eq. (I.lb) are
B. Sneutrino-charginoproduction
Since the masses of x! or x ; are usually larger than the sneutrino mass, pe can rarely decay t o x; or x; and thus decays mostly invisibly. To contribute to the desired
Unlike selectron-neutralino production, where righthandedly polarized beam electrons yield the largest sig-
D
P>O
(4.5a) (4.5b)
P
flipped SU(5)
0.100
a 0.050 z + 0
4-
0.010
'$
0.005
x. a
-
v
0
0.001
40
60
80
100
m,:(GeV)
120
40
60
80
100
m,:(GeV)
120
FIG. 6. The elastic and deepinelastic (DI) cross sections for ep-+V,x;-tep(X) +$ versus m ?+. Note the faster drop off of .-I
i; 0.100 n
-
0.050
? i
%
x.
0.010
t
0005
1 -
I>
-
4 8 v
t?
0 001
the deep-inelastic cross section. This same phenomenon occurs for selectron-neutralino production.
277 48
SUSY SIGNALS AT DESY HEM IN THE NO-SCALE . . .
4035
jlo-' 40
60
80
100
120
40
60
80
100
120
eX+dT signal, the chargino must decay leptonically. In this model this branching ratio is quite sizeable (see Fig. 2 in Ref. [3]). Moreover, for most points in the allowed parameter space of the model, the daughter electron from the decay of the chargino is hard (E, > 5 GeV). For a detail discussion of this point, we refer the reader to Ref. [3]. The cross section for this process, including branching ratios, is shown in Fig. 6 [top (bottom) row for elastic (deep-inelastic)contribution], and can be seen to be of the same order as that for selectron-neutralino production (cf. Figs. 2 and 3).
t
FIG. 7. The elastic selectronneutralino cross section versus mFR(top row). This signal will be the dominant one for a righthandedly polarized electron beam. Also (bottom row) the total elastic supersymmetric cross section (including selectronneutralino and sneutrinochargino channels) versus m'R ' showing the discovery potential at HERA on this mass variable.
The signature for this production channel is different from the selectron-neutralino channel in the following ways: it only produces left-handed daughter leptons (compared to dominantly right-handed ones); and (ii) the daughter leptons can equally likely be of any flavor (as can also decay into opposed to only electrons). Since hadronically noisy jets, in general, sneutrino-chargino detection is more complicated than selectron-neutralino detection. However, in practice such events can be highly suppressed: for p > O , B(X;-+X~q~')
x;
1
FIG. 8. The discovery potential at HERA (i.e., the total elastic supersymmetric cross section) for the lightest neutralino (top row) and the sneutrino (bottom row).
27 8 4036
LOPEZ, NANOPOULOS, WANG,AND ZICHICHI
elastic channel suffers from a more manageable S M background. Note that since a right-handedly polarized electron beam would shut off this channel completely, the signal could in principle be studied in a elastic ZR,Ld hadronically quiet environment.
V. DISCUSSION AND CONCLUSION We have investigated the relevant SUSY production channels at HE R A within the no-scale flipped SU(5) supergravity model, where direct squark production is highly suppressed. Because of the different masses of ZL and ZR , the production rate is dramatically different when the incident electron beam is polarized left or right handedly. If it is right-handedly polarized, then the Z R x y , 2 channels will be the only ones allowed, with a hard electron with large p r as the dominant signal. If the beam is left-handedly polarized, only the much smaller ZL& channels will contribute, as well as the hadronically quite (noisy) for p > 0 (p < 0) S e x ; channel. This tuning of the matching would be relevant only after positive sparticle identification. Before that the unpolarized beam will allow for a larger total supersymmetric signal. In order t o estimate the discovery potential at H E M , in Fig. 7 we consider the elastic contribution to the ZR,LXy-+ep+$ signal versus m as well as the total 'R
'
elastic supersymmetric contribution (including also S e x ; +ep +$ production). The total elastic supersymmetric signal versus m and m - is shown in Fig. 8. The X1
"e
deep-inelastic contributions to these processes are less important and not easily assessed without a careful background study which is beyond the scope of this paper. The S,x; contribution to the total supersymmetric signal has been included since at least for p>O the decay into hadronically quiet leptons is highly probable.
[l] R. Arnowitt and P. Nath, Phys. Rev. Lett. 69, 725 (1992); P. Nath and R. Arnowitt, Phys. Lett. B 287, 89 (1992); 289,368 (1992);J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, ibid. 291, 255 (1992);J. L. Lopez, D. V. Nanopoulos, and H. Pois, Phys. Rev. D 47, 2468 (1993);J. L. Lopez, D. V. Nanopoulos, H. Pois, and A. Zichichi, Phys. Lett. B 299,262 (1993). [2] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Texas A&M University Report No. CTP-TAMU-68/92, CERN-TH.6667/92, and CERN-PPE/92-188 (unpublished). [3] J. L. Lopez, D. V. Nanopoulos, X. Wang, and A. Zichichi, Phys. Rev. D 48,2062 (1993). [4] J. L. Lopez, D. V. Nanopoulos, H. Pois, X. Wang, and A. Zichichi, this issue, Phys. Rev. D 48,4062 (1993). (51Physics ct HERA, Proceedings of the Workshop, Hamburg, Germany, 1991, edited by W. Buchmuller and G. Ingelman (DESY, Hamburg, 1992). [6] G.Altarelli, G. Martinelli, B. Mele, and R. Ruckl, Nucl. Phys. B262, 204 (1985).
48 -
Assuming optimal experimental efficiencies and a suppressed or subtracted-off background, with an integrated luminosity of L= 100 (1000) pb-', and demanding at least five fully identified ep -+ep +$ events [i.e., D > 5X (5X lop3)pb], one could probe as high as m -65 (90)GeV, m -35 (60)GeV, and m - =60 (120) ZR
x?
VC
GeV. The analogous plots versus m
* are not very inforXI
mative in pinning down the discovery limit in this variable, since it ranges widely m 550-115 (120-170)
*
XI
-
GeV for f =lo0 (1ooO)pb-'. The short term discovery limits U=l00 pb-') may then extend the present LEP I lower bounds on these sparticle masses by 15-20 GeV. The long term discovery limits would be competitive with those foreseeable at LEP I1 [4]. We have also shown that the Leading Proton Spectrometer (LPS) at HERA is an excellent supersymmetry detector which can provide indirect information about the sparticle masses by measuring the leading proton longitudinal momentum distribution in elastic e$ + p processes, without the need t o reconstruct all such events. We conclude that H ERA is an interesting supersymmetric probe in the no-scale flipped SU(5)supergravity model. ACJCNOWLEDGMENTS This work has been supported in part by D O E Grant DE-FG05-91-ER-40633. The work of J.L. was supported by the SSC Laboratory. The work of D.V.N. was supported in part by a grant from Conoco Inc. The work of X.W. was supported by the World-Laboratory. We would like to thank the H A RC Supercomputer Center for the use of their NEC SX-3 supercomputer and the Texas A&M Supercomputer Center for the use of their CRAY-YMP supercomputer.
[7] U. Baur, J. A. M. Vermaseren, and D. Zeppenfeld, Nucl. Phys. B375,3 (1992),and references therein. [8]M. Drees and D. Zeppenfeld, Phys. Rev. D 39, 2536 (1989). [9] H. Tsutsui, K. Nishikawa, and S. Yamada, Phys. Lett. B 245,663 (1990). [lo] C. F. Weizsacker, Z. Phys. 88, 612 (1934);E. J. Williams, Phys. Rev. 45, 729 (1934). [ I l l For a review see A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. 145, 111987). [12] S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Nucl. Phys. B398, 3 (1993). [13]M. K. Gaillard, L. Hall, and I. Hinchlire, Phys. Lett. 116B,279 (1982);T. Kobayasi and M. Kuroda, ibid. 134B, 271 (1984). [14] J. F. Gunion and H. E. Haber, Nucl. Phys. B272, 1 (1986). [I51 A. Grifols and R. Pascual, Phys. Lett. 135B, 319 (1984); G. Eilam and E. Reya, ibid. 145B, 425 (1984); 148B, 502(E)(1984). [16] J. G. Morfin and W. K. Tung, 2. Phys. C 52, 13 (1991).
279
Jorge L. Lopez, D.V. Nanopoulos, H. Pois, Xu Wang and A. Zichichi
SPARTICLE AND HIGGS-BOSON PRODUCTION AND DETECTION AT CERN LEP I1 IN TWO SUPERGRAVITY MODELS
From Physical Review D 48 ( 1993) 4062
I993
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28 1 PHYSICAL REVIEW D
VOLUME 48, NUMBER 9
1 NOVEMBER 1993
Sparticle and Higgs-boson production and detection at CERN LEP I1 in two supergravity models Jorge L. Lopez,’s2D. V. Nanopo~los,’-~ H. Pois,’32Xu Wang,’” and A. Zichichi4 ‘Centerfor Theoretical Physics, Department of Physics, Texas A&M University, College Station, Texas 77843-4242 ’Astroparticle Physics Group, Houston Advanced Research Center IHARC), The Woodlands, Texas 77381 ’CERN Theory Diuision, 1211 Geneva 23, Switzerland 4CERN, Geneua, Switzerland
(Received 10 March 1993) We study the most promising signals for supersymmetry at CERN LEP I1 in the context of two wellmotivated supergravity models: (i) the minimal SU(5) supergravity model including the stringent constraints from proton stability and a not too young universe and (ii)a recently proposed string-inspired no-scale flipped SU(5) supergravity model. Our computations span the neutralino, chargino, slepton, and Higgs sectors together with their interconnections in this class of models. We find that the number
of “mixed” (I-lepton+Z-jets+#) events occurring in the decay of pair-produced charginos (x:) is quite significant (per L= 100 pb-I) for both models and that these predictions do not overlap. That is, if m < 100 GeV then LEP I1 should be able to exclude at least one of the two models. In the no-scale
*
XI
flipped SU(5) model we find that the number of acoplanar dielectron events from selectron pair production should allow for exploration of selectron masses up to the kinematical limit and chargino masses indirectly as high as 150 GeV. We find that the cross section e + e - - Z * h deviates negligibly from the SM result in the minimal model, whereas it can be as much as f lower in the flipped model. The usually neglected invisible mode h -+flflcan erode the preferred h 4 2 jets signal by as much as 40% in these models. We conclude that the charged slepton sector is a deeper probe than the chargino neutralino, or Higgs sectors of the flipped SU(5)model at LEP 11, while the opposite is true for the minimal SU(5) model where the slepton sector is no probe at all. PACS numbeds):14.80.Ly, 12.10.Gq, 14.80.Gt
I. INTRODUCTION
The quest for a theoretical understanding of supersymmetry and its phenomenological consequences has been going on for over a decade. So far no supersymmetric particle has been directly observed in accelerator experiments or indirectly in proton decay or dark matter detectors. However, the recent precise measurements of the gauge coupling constants at the CERN e + e - collider LEP can be taken in the context of supersymmetric grand unification as indirect evidence for virtual supersymmetric corrections [ 11. This observational situation may appear discouraging to some. However, it really should not since from a totally unbiased point of view, most sparticle masses could lie anywhere up to a few TeV, with no particular correlations among them. This means that existing facilities (Fermilab, LEP I,II, the DESY ep collider HERA, Gran Sasso) as well as future ones [CERN Large Hadron Collider (LHC), Superconducting Super Collider (SSC)] are needed in order to truly explore the bulk of the supersymmetric parameter space. On the other hand, specific supergravity models incorporating well-motivated theoretical constraints can be very predictive, and perhaps even fully tested in the next few years with the present generation of collider experiments at Fermilab, HERA, and LEP 11. We have recently focused our attention on two such models: (i) the minimal SU(5) supergravity model including the severe constraints of proton decay [2-61 and a not too young universe [4,6-81, and hi) a recently proposed no-scale 0556-2821/93/48(9)/4062(14)/$06.00
48 -
flipped SU(5) supergravity model 191. The parameter spaces of these models have been scanned and a set of allowed points has been identified in each case. Several results then follow for the sparticle masses. These are summarized in Table I and discussed in detail in Refs. [4,6,7,10,11] for the minimal SU(5) model and in Refs. [9,10,11] for the flipped model. As far as the sparticle masses are concerned, perhaps the most striking difference between the two models is in the slepton masses which are below =300 GeV in the flipped SU(5) model, while they are out of reach of existing facilities, i.e., above 300 GeV, in the minimal SU(5) model. T h e study of the specific models such as the two we are pursuing singles out small regions of the vast 21-dimensional parameter space of the MSSM (minimal supersymmetric extension of the standard model). We have already shown [6,10] that experimental predictions for these models can be so precise that potential discovery or exclusion in the next few years is a definite challenge. In a previous paper [ 101 we have studied the prospects for supersymmetry detection at Fermilab in the neutralino-chargino sector. Here we continue our general program by exploring the supersymmetric signals for charginos, neutralinos, sleptons, and the lightest Higgs boson at LEP I1 in the two models. For charginos we study the reaction e+e--+,y:X; and the subsequent “mixed” (1 lepton plus 2 jets plus fl and dilepton decay signatures. We show that the predicted number of mixed events for both models are experimentally significant up to the kinematical limit, and do not overlap. Therefore, if 4062
@ 1993 The American Physical Society
282
48 m
SPARTICLE AND HIGGS-BOSON PRODUCTION AND DETECTION A T . .
* < 100 GeV, then LEP I1 should be able to exclude at
XI
least one of the models. For neutralinos we analyze e f e - + x y x 2 and the dilepton signature, as a means to indirectly probe chargino masses above 100 GeV. The charged slepton sector appears very interesting for LEP I1 in the predictions of the flipped SU(5) supergravity model. We compute the number of dilepton events exjip, and -, and conclude pected from pair-produced that these also should be accessible up to the kinematical limit. Finally, we explore the Higgs sector and stud1 e + e - - - t Z * h production, the branching ratios h +bb, T+T-, and cT,gg and the "invisible" mode h -xy,yy. We show that the latter can have a branching ratio as large as 30%, therefore significantly eroding the preferred h +2 jets mode. Nonetheless, detection is possible in a large fraction of parameter space for both models at LEP 11. Throughout this paper we emphasize the interconnections among the various sectors of the models and their experimental consequences. For example, charged slepton pair production should indirectly probe chargino masses as high as 150 GeV in the flipped model.
=,
.
4063
sections for this process for both models are shown in Fig. 1 for 6 =200 GeV. The reason the cross sections are lower in the no-scale flipped model is due to a wellknown destructive interference between the s and t channels, which is relevant for light PL masses, or more properly for mcL-mx?. In addition, for (pl >>M2=0.3mg
xt
[M2is the SU(2), gaugino mass] the mass eigenstate is predominantly gaugino and therefore its coupling to lepton-slepton is not suppressed by the small lepton masses. In the minimal SU(5) the model, m - > 500 GeV VL
and the contribution of the t channel is small. In the and the destructive interferflipped model m - - m VL
*
XI
ence is manifest. The best signature for this process is presumed to be the one-charged lepton (e* or pi) 2 jets +$ or "mixed" mode, where one chargino decays leptonically and the other one hadronically [15,16]. In the minimal SU(5),since the sleptons and squarks are heavy, the chargino decays are mediated dominantly by the W-exchange channels [ 101 and one gets
+
II.CHARGINOS AND NELJTRALINOS Among the various supersymmetric neutralinochargino production processes accessible at LEP 11, the which one with the largest cross section is e + e - + X : x ; proceeds through s-channel y * and Z * exchange and tchannel P L exchange. This cross section has been calculated in the literature [12-141 for various limiting cases of the chargino composition and for a general composition (i.e., an arbitrary linear combination of FV-ino and charged Higgsino components) as well. We have independently calculated the cross section in the general case, and our result agrees with, e.g., Ref. 1141. The cross
P>O
minimal SU(5)
and
For the flipped case things are more complicated due to the light slepton-exchange channels. There are three regimes which one can identify: (i) when the slepton exchange channels dominate, the leptonic branching ratio (into I = e + p ) is = f and the hadronic one goes to zero; (ii) when the W-exchange channels dominate [as in the minimal SU(5) case], the leptonic branching ratio drops down to =3 and the hadronic one grows up to = f; and
.p
FIG. 1. The cross section for e+e--+X:X; at 6 =200 GeV as a function of chargino mass (rn * ) for the minimal SU(5)suXI
pergravity model (top row) and the no-scale Sipped SU(5)supergravity model (bottom row). The smaller size of the latter is due to destructive interference effects in the presence of a light sneutrino.
v
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283 48 -
LOPEZ,NANOPOULOS,POIS,WANG,AND ZICHICHI
4064
(iii) in the transition region between these two regimes, destructive interference between the W-exchange and slepton-exchange amplitudes can suppress the leptonic branching ratio and enhance the hadronic one beyond their values at the end of the transition. In Fig. d we show an example of this phenomenon for m,=100 GeV; for larger values of m, the effect is less pronounced (see Fig. 2 in Ref. [lo]). In Fig. 3 we show the number of “mixed” events to be expected per L = 100 pb-’ for both models, i.e.,
integrated luminosity for u ( e +e --x:x;
) k 0.40 (0.17) pb
(Ref. [181). These calculations assume W-exchange dominance in chargino decays [case (ii) above] and are therefore applicable to the minimal SU(5) model. In this case Fig. 3 shows that one could explore all allowed points in parameter space with m < 100 GeV, since uk0.40
*
XI
(0.17) pb for L = 1 0 0 (500) pb-’ would require 40 ( 8 5 ) observed events. Moreover, since in this model m i < 104 XI
where the factor of 2 accounts for summing over the two charges of the outgoing lepton. The very small numbers for the flipped model which occur mostly for p > 0 correspond to points in the parameter space where the slepton-exchange channels dominate the chargino decays and the hadronic branching ratio goes to zero [case (i) in the previous paragraph]. Perhaps the most interesting feature of these results is that the predicted number of events for both models do nor overlap. Therefore, if m < 100 GeV, then LEP I1 should be able to exclude at
*
XI
least one of the models (and possibly even both). As far as the backgrounds are concerned, the dominant one is e +e - -+ W + W - with one W decaying leptonically an_d the other one hadronically. (To a lesser extent, the ff(y ), Z e ( e ) ,W v ( e ) ,ZZ, and Z w backgrounds also apply [ 171.) Several features of the chargino decays (such as an isolated lepton, missing mass, hadronic mass, etc.) allow for suitable cuts to be made which reduce the WW background to very small levels [16-183. Modeldependent studies indicate that a 5u effect (i.e., S/* 2 5u) can be observed with L= 100 (500) pb-’ of
(92) GeV for p > 0 (/I < 0) [ll], only a few points in parameter space should remain unexplored in this direct way at LEP 11. For the flipped model the experimental study referred to above may not apply since the W-exchange dominance assumption is not likely to hold for m + < 100 GeV (see, XI
e.g., Fig. 2). Assuming that the results apply at least approximately, we can see that a fraction of the parameter space for m < 100 GeV could be explored. If no signal
*
XI
is observed, this would imply that m
* > 100 GeV in the
XI
minimal SU(5) model, but not necessarily in the flipped model because of possible highly suppressed hadronic decay channels. To probe the remaining unexplored regions of the flipped model for m < 100 GeV we show in
*
XI
Fig. 4 the predicted number of events for e + e --+,y:X;-+dileptons which does not suffer from small chargino hadronic branching ratios. However, the e5ciency cut needed to suppress the backgrounds to this process is not known at present. Nevertheless, the signal is significant and should encourage experimental scrutiny. For the minimal SU(5) case the dilepton signal is
+
’
flipped SU(5) 1 .o
1.0
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150 200 250 m,: ( c e v )
300
FIG. 2. The leptonic and ha-, dronic branching ratios for the chargino in the flipped model and m,=100 GeV. The former ranges from =$ when the slepton-exchange channels dominate, down to =; when the Wexchange channels dominate, and through very small values when the two channels interfere destructively. A complementary effect is seen to happen for the hadronic branching ratio.
‘In the next section we show that the chargino-dilepton signal is, in general, a “background” to dileptons from charged slepton decays. Therefore experimental isolation of the chargino-dilepton signal may be required anyway.
284
48
SPARTICLE AND HIGGS-BOSON PRODUCTION AND DETECTION AT.
..
4065
a
t
FIG. 3. The number of "mixed" (1 lepton + 2 jets P, events p e r f = 100 pb-' for both models. Note that the predictions on the top row do not ouerfup with those on the bottom one, and therefore if mxf < 100
+
50
" ' ~ ~ " " ~ " " " ' ~ ' ' ' ' ~ ' ' '
0
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(cev)
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GeV, LEP I1 should be able to exclude at least one of the mod-
0
els. I
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of that for the mixed mode: a factor of 3 is lost in substituting the hadronic branching fraction (3)for the leptonic one ($), and a further factor of 2 from not needing to sum over the charges of the outgoing lepton. Thus, Fig. since m > 5 0 0 GeV. Even with 100% efficiencies and 9.I( 3 (top row X i ) shows that the minimal model dilepton high branching ratios, it would take at least L=1000 signal is generally smaller than the flipped model one. pb-' to get an observable signal at the largest cross secConcerning neutralino detection at LEP I1 112,191, the tion. For m + > 100 GeV the cross section drops below Xi largest observable cross section occurs for e +e --+xp-& fb and therefore this mode is hopeless for exploration 0.1 which is mediated by Z * s-channel exchange and izL,L,R tof chargino masses above 100 GeV at LEP I1 in the channel exchange. Since in the models we consider minimal SU(5)model. m + =m -2m 0, this process could explore indirectly Xi Xi XI For the flipped model we have m <190 GeV and PR chargino masses up to 130 GeV and may be worth conm <300 GeV and thus the cross section for sidering despite the potentially small rates. The coupling ZL e +e --+xpx: is correspondingly much larger, although ZxYx; depends exclusively on the Higgsino admixture of slightly below 1 pb at most; see Fig. 5 top row. With and and is thus highly suppressed here (and so is L=500 pb-' and a neutralino dilepton branching ratio the s-channel ampIitude) and in any model where as high as 3 (see Fig. 4 in Ref. [lo]), one could get an oblpl >>MI. The cross section then depends crucially on servable number of neutralino-dilepton +I -1 the t-channel amplitude, i.e., on the selectron mass. For events even for m + > 100 GeV; see Fig. 5 bottom row.' the minimal SU(5) model we find
-
~7
x;
(x7-xyZ
XI
-
P>O
F
flipped SU(5)
150
FIG. 4. The number of dilepton events per L=TOO pb-' to be expected from the process e+e--tX:X; for the tlipped model. The corresponding number in the minimal SU(5) model is of that shown on the top row in Fig. 3.
n a
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$:
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L.
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.e
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2Note that neutralino dileptons need still to be distinguished from the chargino dileptons discussed above; the one-sided nature of the former signal may help in this regard.
285 48 -
LOPEZ,NANOPOULOS,POIS, WANG,AND ZICHICHI
4066
The cross section for for the flipped model as a function of the chargin0 mass. Note that chargino masses up to =130 GeV could be explored with this process. Also shown is the number of dilepton events per L=500 pb-‘. FIG. 5. e+e--+,y:d
nn
u 102
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10-1 140
Most of the backgrounds to this process can be reduced, except for the W W one which was shown in Ref. [16] to overwhelm the signal for both leptonic and hadronic decays, at least for the parameters considered by those authors. A reevaluation of this analysis in the light of the flipped model cross section and branching ratios would need to be performed to be certain of the fate of this mode. Since this is one way in which LEP I1 could indirectly explore chargino masses above 100 GeV, it would appear to be a worthy exercise.
III. SLEPTONS The charged sleptons (ifL,R,pL,R,?L,R ) offer an interesting supersymmetric signal through the dilepton decay mode, if light enough to be produced at LEP I1 [12,15,16,20,21]. This is partially the case for the flipped SU(5) model where m S 300 GeV and mzRS 200 GeV. 9 [No such signal exists at LEP I1 for the minimal SU(5) model since mT> 300 GeV.] We have computed the cross sections for
considerably heavier than the right-handed (R) ones (see Fig. 3 in Ref. [9]).In particular,
mTR < m z R , p R < m q ’ p<mTL L
*
The acoplanar dilepton signal associated with selectron pair production has been traditionally assumed to come entirely from F& + e f g decay channels, i.e., purely dielectrons. This IS an idealization which need not hold in specific supergravity models. In the flipped model the following decay channels are allowed:
-* 1 0 * o f eL-+e x1,e x2,yeXl
(3.2a)
I
(3.2b)
F:+efxy,efx;.
If X; decays invisibly (x;-vTxy) and x: leptonically (X:-Z*v&,Z = e , p , T ) , then one has new contributions to the sought-for dilepton signal. Note however that in the latter case the charged leptons ( e , p , T ) will be “nonleading” and thus their spectrum (from three-body x: decay) is likely to deviate from the “leading” lepton spectrum (from two-body ZL,R decay). Because of the details of the model, more than 90% of the points in the allowed parameter space have m < r n and therefore ‘R
x!
B (F:+e*X:)=
100% for these points; the remaining points, which allow m- > mx;, have branching fractions eR
The i?:FL,i?$?i final states receive contributions from s-channel y * and Z* exchanges and t-channel xp exonly proceeds through the r changes, while the channel. The p,$L,pipi and ?:?L,?i?i final states receive only s-channel contributions, since all couplings are lepton flavor conserving, and therefore mixed LR final states are not allowed for smuon or stau production. Our results agree with those in Ref. [21]. Note that in the flipped model the left-handed ( L )slepton masses are
&;
to ef,yy no smaller than 75%. On the other hand, for all points in parameter space we find m > md,x: and the FL
decays of the heavier ZL proceed in all three ways. It is important to realize that in this model, for a given point in parameter space, all slepton masses are determined, and the lighter of the final states in Eq. (3.la) (ZlFi) will dominate the total cross section into selectron pairs. Moreover, the dilepton signal from this dominant contribution will be purely leading dielectrons. The
286 SPARTICLE AND HIGGS-BOSON PRODUCTION AND DETECTION AT . . .
48
L
other final states in Eq. (3.la) involving the heavier ZL(Z2,‘u ~ , P ) have ~ Fsmaller ~ cross sections and contribute mostly leading dielectrons. This is because nonleadin leptons ( e , p , T ) require the production of the heavier i$ and the further branching ratio suppressed decay into vlx:. Therefore, we expect the traditional acoplanar dielectron +$ (missing energy) signature to prevail. We have computed the total (leading) dielectron signal (per L = 1 0 0 pb-’) from all channels in Eq. (3.la). The result is shown in Fig. 6 (top row) as a function of the The thinning of the point distribuselectron mass m R ‘
The slepton dilepton signal could also be used to explore indirectly values of the chargino masses beyond the direct kinematical limit of 100 GeV. In Fig. 7 (top row) we show the P dielectron signal versus the chargino mass, and observe that, in principle, one could probe as high as mw: = 150 GeV. In fact, this indirect method appears to be much more promising than the one suggested in Sec. I1 through the channel. It is important to realize that dileptons also occur in chargino (and to a lesser extent x’$ decays, as discussed for the flipped model in Sec. 11. In Fig. 7 (bottom row) we show the number of chargino-dileptons from Fig. 4 but this time plotted against m This “background” to
x:x;
.
tions for m, 2 80 GeV is due to the kinematical closing of the i?:Z2Rproduction channel. The number of dielectron events is quite significant and with adequate experimental efficiencies to account for the dielectron W , Z Z decay backgrounds [17,18,22] it should be possible to explore the whole kinematical allowed mass range (i.e., m 5 100 GeV and indirectly larger m masses) with *R % L = 5 0 0 pb-’. For example, a study in Ref. [17] indicates that one would need u ( e + e - - - t Z ) 2 0 . 1 pb to observe a 50 effect. From Fig. 6 this will allow exploration up to m =95GeV.
ZR.
slepton-dileptons (the search topology for dileptons is the same) has some features which may allow for it to be sufficiently accounted for. The slepton-dileptons contain only (leading) 2’2- (I=e,p for now) pairs, whereas the chargino-dileptons in Fig. 7 contain a mixture of 25% e’e-, 25% p+p-, and 50% e*p’. Moreover, the compairs have a different energy spectrum (c.f. mon Z’ZT*--+Z*x: with x:+Z*v,x:). Of course, if charginos are observed through the mixed signal, one could simply “subtract out” the ensuing chargino dilepton signal from the total observed (chargino slepton) dilepton signal. The previous two paragraphs exemplify the interconnections among the various sectors of this class of models. These correlations allow experimental exploration of one sector to probe indirectly other sectors. They also allow for a reliable computation of all contributing sectors to a particular physics signal (such as acoplanar dielectrons). We have not considered the production of sneutrinos since because of their masses (in between the R and L charged lepton masses) the rates will be lower than for selectron production. Moreover, their visible decay
PIP
+
The analysis in the previous paragraphs for selectron pair production can be carried over to p and 7 pair prodominate the production. In this case ,iL:jii and ?;: duction cross section and the leading dimuons and di-& constitute the bulk of the dilepton signal, respectively. Furthermore, nonleading leptons ( e ,p,T ) are even less likely to occur here since the larger contributions from LR final states (compared to LL h a 1 states) in the selectron case are not present here. In Fig. 6 (bottom row) we show the result for the di-7 case; the dimuon signal is very similar. These signals are smaller (although not by too much) than the dielectron signal in selectron pair production because of the fewer production diagrams.
P>O
4067
P
flipped SU(5)
D a O
0,
102 I
;
-c
.-... ..
.._...
tR
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aJ P
G tc
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80
mTR(cev)
90
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40
50
60
70 80 mi, (Gev)
FIG. 6. The number of dielectron events per L=100 pb-’ from selectron pair production (top row) and the corresponding number of di-7’s from stau pair production (bottom row), as a function of the lightest charged slepton mass of the corresponding flavor. The results for the dimuons from smuon pair production are very similar to the number of di-ys.
287 LOPEZ,NANOPOLXOS,POIS, WANG, AND ZICHICHI
4068 P>O
48
P< 0
flipped SU(5)
. :..: . ., . .
>..SS>,.. . . -.. , , ...-.. *.. <.;y;.-. -..
[,
,: ’
102
-2:.:.
-...-. .. . -. . . .. -..,.-.... __._. ,.... . . . .. .
I
I
, , I , , , , I-,,I‘.;
;:;:
FIG. 7. The number of dielectrons from selectron pair production as in Fig. 6, but plotted against the chargino mass (top row), showing that one could indirectly probe chargino masses as high as 150 GeV. Also shown (bottom row) are the chargino dileptons from Fig. 4 which are comprised of 25%, 25%, and 50% of ee, pp, and ep dileptons, respectively.
101
.. . . _.._.. 40
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channels (Ve +veXi,e *x:) are branching-ratio suppressed (since V e + v e X : is expected to dominate), and will lead to one-sided (likely soft) dileptons.
IV. THE LIGHTEST HIGGS BOSON We now consider the standard alternative to direct sparticle production and decay modes, namely, the SUSY Higgs sector. Since the supergravity models we consider here contain two complex Higgs doublets, after spontaneous symmetry breaking the physical Higgs spectrum contains the usual h,H (CPeven) and A (CP odd) neutral Higgs fields, and the charged H* Higgs field. For a comprehensive review of the SUSY Higgs sector, we refer the reader to Ref. [23]. Our goal in this section is to reformulate the “generic” analysis for SUSY Higgs-boson production and decay in terms of the specific minimal and ilipped SU(5) supergravity models described above. As a result we must include some nonstandard decay channels, such as h + X : S , which are usually not considered in generic analyses since they are so model dependent. Signikantly, B ( h + X y S ) can be quite large in these models, and this modifies the usual assumptions regarding Higgs signals at colliders. We also incorporate the one-loop corrections to the Higgs-boson masses which can be quite significant in large regions of parameter space [24]. From the underlying radiative breaking mechanism in the two supergravity models we consider [25], and the experimental lower bound on the gluino mass, one can show [ l l ] that the Higgs sector of both models approaches a SM-like situation with a light scalar ( h )with SM-like couplings, and a heavy Higgs spectrum (H,A , H * ) which tends to decouple from fermions and gauge bosons for increasing m, (see Ref. [ l l ] for further details). The approach to this limit is accelerated (as a
70
60
90
100
function of m,) in the minimal SU(5) model due to the proton decay constraint which requires large scalar masses. In the flipped model the A Higgs boson can be relatively light for large ta@(mA 5 mh ), implying a slower approach to the limiting situation. For the most part then, we need only consider the h Higgs boson at LEP. Thus, the phenomenological analysis of the Higgs sector for each supergravity model simplifies dramatically, particularly when making contact with previous experimental and Monte Carlo results. With proper care for the nonstandard hZZ coupling and branching ratios (which simply amounts to a rescaling of the SM analysis), we can adopt the definitive SM analysis of Ref. [26] for mHSM 5 80 GeV along with the recent results summarized 5 80 GeV. in Ref. [22] for mHSM With regard to Higgs-boson production, since we take fi =200 GeV, the only relyant mode is the standard schannel- e + e - + Z * h +hff production process, since the Hff,H + H - final states are kinematically forbidden, and the W -fusion ?-channel processes are relevant only for 6 5400 GeV +0.6mh [23]. For the flipped case, there are a few exceptional ( < 1%) points in the allowed parameter space for which the associated e +e - +h A process is kinematically allowed Le., for mh 5 m A S 90 GeV which is possible for large tax$ values, see Fig. 6 in Ref. [9]); we neglect this mode in our analysis. Ln Fig. 8 we show the cross sect@ o ( e + e - + Z * h + h v i j ) vs mh for both models for ds =200 GeV. The values shown for the minimal SU(5) model also correspond to the SM result since one can verify
in this case [ll]. As a reference point, for mh =SO GeV, and the canonical integrated luminosity of 500 pb-’, the
288 SPARTICLE AND HIGGS-BOSON PRODUCTION AND DETECTION AT. . .
48
W O
4069
P
minimal SU(5)
FIG. 8. The cross section at 6+hvV = 2 0 0forGeV thefor minimal e + e - - ZSU(5) *h model (top row) and the flipped
:i-
b,
b
n
23 c
0.05 0.00
40
, ,
, , , , , , ,
60
, , , ,
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, , ,
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,
,, ,,
, ,
, ,, , ,,
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,
T
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‘i0.15
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1::
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model (bottom row) as a function of the Higgs mass mh. In the minimal SU(5) model the cross section differs negligibly from the SM result, whereas in the flipped model decreases of up to =+ are possible for a small ( < 1%) set of points in the allowed parameter space corresponding to relatively light values of rn A . Note the ‘‘tail’’ in the cross section in the flipped case for mh 2 110 GeV when the second Z goes off shell.
0.10
’ ’
.
0.10
.,,
.
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b
0.00 40
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SM [and also the minimal SU(5)] value _Of u =O. 145 pb would correspond to approximately 62 bbvv events (56 if initial state radiation is included) [26]. The “tail” in the cross section when the second 2 is forced off shell is evident in the flipped case for mh 2 110 GeV. In this model, significant deviations from the SM curve are possible (see Fig. 8 and compare top and bottom rows), and these correspond to the small set of points ( < 1%) for which m A can achieve moderately light values ( m , ? m h for large tar$). Since in this case sin(a-P)20.8, a reduction of in the cross section is possible compared to the up to SM case. We should add that these results agree quite well with previous results obtained in Ref. [27] for a slightly different version of the no-scale flipped SU(5) supergravity model. There the reduction in sin2(a-/3)-0.8 is only possible for large tar$ values, and 80 GeV 5 mh 5 100 GeV. In addition, in Ref. [27] it was found that mh 5 120 GeV, and this limit is close to the one obtained in the present version of the flipped model ( m h5 135 GeV). Regarding the decay of the h Higgs boson, it is well known that higher-order QCD corrections can be important and affect the overall results for the relevant decay channels h+bg,cF by up to 20% [28]. This should be particularly true at LEP 11, where mh is much greater than mcTband there will be significant running of the quark masses from the scale Q =mc,b up to Q =mh. We therefore choose to include QCD corrections to O ( a : ) as outlined in detail in Ref. [29]. In addition, significant departures from the usual SM branching ratios result when we include the h decay channel. Statistically speaking, we find that 7% (12%) of the points for the flipped model for p > 0 ( p < 0) kinematically allow for this nonstandard decay mode. In the minimal SU(5)case, this
-+
-xyxy
(cev)
fraction rises to 26% (36%) for p > 0 (p
xyxy
-
I
3There are only a handful of points for which the h-+,yy,y! mode is kinematically accessible in the flipped model [there are none in the minimal SU(5) model] with B ( h +xy,y:) 5 We do not show these points here. 4We use the expressions for h+gg that appear in Ref. [30] along with minor corrections pointed out by those authors.
289 48 -
LOPEZ, NANOPOULOS, POIS, WANG, AND ZICHICHI
4070
0.6
0.4
FIG. 9. The branching ratio for h+b6 as a function of the Higgs-boson mass in,, for the minimal SU(5) model (top row), and the flipped model (bottom row). Note the significant departures from the SM result ( ~ 8 5 %due ) to the opening of the h-dd (and to a lesser extent the h +gg) channel.
k
0.0 40 o'2
60
80
GeV and therefore suppresses the b6 mode significantly [note the-points with uncharacteristically small values of B ( h - t b b ) in Fig. 9, top row, for p < O ] . We expect B ( h - + y y ) to be much smaller, and do not include this mode in our analysis here. Although not shown, we have calculated B ( h-+r+r- ) also, and find that for both models, this channel is confined to a narrow band centered around B ( h-r+r- )=0.08; this agrees well with the results obtained in Refs. [29,31]. In Fig. 12 we show B ( h-+cF) for both models. For the flipped case, there is a large range of values [0.0001 Z B ( h + c F ) S 0 . 0 6 ] , with a noticeable
P> 0
. . .. . .
&= X : : :1
.
100
120
dip in the range 80 GeV S mh Z 100 GeV corresponding t o the deviation from the SM couplings. For the minimal SU(5) case, B(h-+cT)-O0.06. The latter result is predominantly due t o the fact that in the minimal case si n ( a- p ) - l and therefore the h-c-T coupling ( 0:cosa/si$) goes to the SM HsM-c-Tcoupling, for virtually all points. Detectability of the h Higgs boson requires the combination of production and experimentally important decay modes, as well as a detailed treatment of the backgrounds and overall efficiency.' From our previous discussion of h production and decay, it is clear that the
P< 0
minimal SU(5)
10-2
.
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FIG. 10. The branching ratio for h +x:xy as a function of the Higgs-boson mass mh for the minimal SU(5) model (top row) and the flipped model (bottom row).
10-2
0-
-J
10-3
m
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mh (GeV)
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(Qv)
'In what follows we adapt the results of Ref. [26]for the SM Higgs boson to the h Higgs boson for mh 5 80 GeV.
290 SPARTICLE AND HIGGS-BOSON PRODUCTITON A N D DETECTION A T . . .
48
7
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FIG. 11. The branching ratio for h-gg as a function of the Higgs-boson mass mh for the minimal SU(5) model (top row) and the flipped model (bottom row). For the minimal SU(5) model note the few points where the branching ratio can be quite large, corresponding to a very light 7,.
80
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fraction of h Higgs events compared to the SM will be
R, Esin2(a-b)f ,
(4.1)
where
f B ( h -+X)/B ( HsM+ X ) ,
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W + W - , qqy, and Wev, backgrounds are dominant. Considering the SM analysis first, Ref. [26] finds that for mHSM -80 GeV, the efficiency ( E ) is -21% for the dominant e + e - ~ H s M Z * - + ( H s M - b 6 ) v i jchannel. his corresponds to
u ( e +e --+z*HSM -+*€ISM ) L B( H s , and X is a specific Higgs final state. As for the backgrounds, the various SM e + e - - Z Z , W + W - , Ze'e-, We Y , , and qqy modes apply to a different degree depending on the particular production channel. For the ( h-+jj)vij final states we consider here ( j =jet), the Z Z , P>O
-,"y
10-2 1-ii:I
-.
.
.-
-.-.
-
b6 )EEISR
=(O. 145) X (500)X (0.85) X (0.21) X (0.91 ) = 12 (4.2) accounts for the initial state radiaexpected events kISR P
minimal SU(5)
..
-sm 10-3
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y
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FIG. 12. The branching ratio for h-cT as a function of the Higgs-boson mass mh for the minimal SU(5) model (top row), and the flipped model (bottom row).
29 1 48 -
LOPEZ, NANOPOULOS, POIS, WANG,AND ZICHICHI
4072
tion effects at mHsM=80GeV) with a background of four events, leading to a signalhackground of -3. For mHSM2 80 GeV, both the signal and efficiency decrease further. Thus, mHsM-80 GeV has until recently been considered the limit of detectability for the SM Higgs. Turning now to the supersymmetric Higgs analysis, for mh < 80 GeV, we find that R h 2 0 . 7 for both models, where f=B(h+b&+cT+gg)/BSM
.
Thus, additional integrated luminosity would be needed in order to probe up to mh -80 GeV from two-jet reconstruction off the Z via h Z * production. Despite the degradation of the favored two-jet signal, for mh 5 80 GeV detection via recoil against l + l - pairs or two-jets may still be possible through the e + e - - h Z * +(h + X ) l + l - , ( h + X ) j j channels, where X is invisible. Overall, a detailed Monte Carlo study would be needed to determine the experimental mass limits for Higgs boson decaying invisibly. The analysis summarized in Ref. [32] has desonstrated that with b-quark tagging from H s M + b b (and for 6= 190 GeV), a 90 GeV SM Higgs boson should be detectable above background with L = 3 0 0 pb-’ luminosity at the 5 a level [22]. This extends the experimental Higgs boson reach at LEP even further, through the previously troublesome region where mHSM -MZ. For higher beam energies, the Higgs mass reach is expected to be 6- 100 GeV. If the beam energy could be pushed up to the magnet limit of 6 = 2 4 0 GeV, a value of MHsM 140 GeV could, in principle, be explored; however, a reach of mHSM 5 100 GeV is more realistic for the near future, corresponding to; ‘% =200 GeV, For mh 2 80 GeV, we define f E B ( h +bb )/BsM since b tagging is the only source of the signal. For both models we find the value of Rh can be as small as -0.60 for the experimentally accessible region mh S 100 GeV.6 In this case, the irreducible background is certain to obscure the 50-80 % reduction in signal in a b-tagging analysis. Thus, we conclude that for the minimal SU(5) model and for p > 0, the h Higgs will most likely be seen at LEP 11 since mh 2 83 GeV and R h 20.7. For p
-
model that we have considered here, it is possible (but unlikely) that nature could conspire to fall within the SOcalled “tie” region in the ( t a d , m A ) plane where neither process could be seen at LEP 11. In this unlikely event, the h (and/or the A Higgs) could conceivably escape detection. (For the minimal model m A>>Mzand the “tie” region is avoided entirely.) For the flipped model (and for light m A 2 mh) in the mass region mh > 80 GeV, the only possible hope would be looking for the h at a 500 GeV e + e - machine or at the SCC and/or LHC [18,33]. The present lower bound for the h Higgs is mh > 43 GeV [34]. This limit is regarded as model independent, valid for mq < 1 TeV, and assumes SM final-state products. In the models we consider here, we have shown that the h-~yx’ mode should also be considered for some regions of parameter space. One can see, however, , from Fig. 9 that for mh 5 43 GeV the h +xy.x’ mode is relatively unimportant. Even for mh 5 60 GeV, the nonstandard reduction of B ( h--+b&,cT,gg) is less than =15%, and we expect a drop in the upper limit to mh compared to mHsMofonly -1 GeV. Coupled with the very SM-like h production (see Fig. 8) for mh 560 GeV, and R h 20.85, we find that the mHSM> 60 GeV limit also applies to the h Higgs of both the minimal and flipped SU(5) models. For a more detailed discussion of h Higgs mass limits at LEP I in the two models we consider here, see Ref. [ 111.
V. DISCUSSION AND CONCLUSION
In this paper we have studied the most promising signals for supersymmetry at LEP I1 in the context of two well motivated supergravity models: (i) the minimal SU(5) supergravity model including the stringent constraints from proton stability and a not too young universe, and (ii) a recently proposed string-inspired noscale flipped SU(5) supergravity model. These signals involve the neutralino chargino, slepton, and Higgs sectors. Because of the study of specific models, we are led to modifications in the standard assumptions regarding sparticle and Higgs boson decay. In the first sector we 2 jets d, computed the number of “mixed” (1 lepton events occurring in the decay of pair-produced charginos and found that the predictions for both models should lead to detection (with L=100 pb-’) up to the kinematical limit (m 5 100 GeV). Moreover, these pre-
+
+
(x:)
*
XI
dictions do not overlap: the minimal model predictions being larger than the flipped model ones. This result can be directly traced to a characteristically light sneutrino spectrum in the flipped case (m,-0.3mE ). This implies that if m < 100 GeV then LEP I1 should be able to ex-
*
XI
clude at least one of the two models. In fact, in the minimal SU(5) model m < 104 (92) GeV for p > O
*
XI
(‘We exclude from the discussion the very few points for p < 0 in the minimal SU(5)model where B ( h -gg) ~ 0 . and 9 the f ratio drops to values as low as 0.25.
( p < O), assuming mq,E5 1 TeV, while in the flipped case m -+ S 285 GeV ( p > 0,p < 0 ) and the mixed chargino sigXI
nature can be suppressed. Consequently, it is possible to explore nearly all of the allowed parameter space for the
292 SPARTICLE AND HIGGS-BOSONPRODUCTION AND DETECTION AT . . .
48
minimal SU(5)model but only 5 20% of the flipped model. We found significant chargino-dilepton even rates (per L = 5 0 0 pb-' for rn > 100 GeV) in the flipped model,
*
XI
and a negligible signal in the minimal model. The question of backgrounds to this process remains open. The magnitude of the experimental efficiency cut for this dilepton signal is not known at present. In the models we consider, the relations among the neutralino and chargino masses rn = rn 0 -2rn (see Table I) imply that the
-+x?x;
* XI
fl
xi
e'e process could, in principle, explore indirectly chargino masses up to 130 GeV. The slepton sector could be kinematically accessible at LEP I1 only in the flipped SU(5)model. We studied
-
and obtained significant numbers of dielectron events which may allow exploration of the full kinematical range with L=500 pb-'. Smuon and stau production are suppressed but may be observable as well. Correlating the slepton and chargino sectors we observed that slepton-dileptons could probe indirectly chargino masses as high as 150 GeV, and thus -50% of the allowed parameter space. This is especially important for this (the flipped) model since a significant number of points in parameter space for rn < 100 GeV yield negligible mixed
-
*
XI
chargino event signatures. We also discussed the impact
4073
of chargino-dileptons on the slepton-dileptons and the possibilities for experimental discrimination of these signals. For an analysis of the total dileptomignal from all supersymmetric sources in these models see Ref. [35]. In the Higgs sector we found that the cross section e + e - + Z * h deviates negligibly from the SM result in the minimal model, whereas it can be as much as f lower in the flipped model. Also, the usually neglected itvisible can erode the preferred h -+ bb ,cT,gg model h ( h-b6) for mh 5 80 GeV (mh 2 80 GeV) by as much as 30% (15%) [40% (40%)] in the minimal (flipped) model. although there The h +gg mode is usually below ~ 0 . 2 are exceptional points in the minimal model where it can be much larger, because of a very light TI. We have recently shown [ 111 that the current experimental lower bound on the SM Higgs-boson mass (rnHsM> 60 GeV) applies as well to both supergravity models considered here and is therefore more stringent than the supposedly model-independent experimental lower bound rnh >43 GeV. In this connection, we have found it useful to relate the results obtained in the chargino sector (as shown in Fig. 3) with those obtained in the Higgs sector by plotting the number of mixed events in chargino pair production versus the Higgs-boson mass; this is shown in Fig. 13. With this plot it is straightforward to determine which points of interest in the chargino sector become excluded by an increasing lower bound on the Higgs-boson mass. In particular, all points for
-x:x?
TABLE I. Comparison of the most important features describing the minimal SU(5) supergravity model and the no-scale flipped SU(5) supergravity model.
Minimal SU(5) supergravity model
No-scale flipped SU(5) supergravity model
Not easily string derivable, no known examples Symmetry breaking to the standard model due to the vacuum expectation value (VEV) of 24 and independent of supersymmetry breaking No simple mechanism for doublet-triplet splitting No-scale supergravity excluded mq,mg< 1 TeV by ad hoc choice: naturalness Parameters 5 : m,,,,mo,A,tan8,m, Proton decay: d = 5 large, strong constraints needed Dark matter: 0,hi >> 1 for most of the parameter space, strong constraints needed 15tangS3.5, m , < 180 GeV, (026 mg S 400 GeV
Easily string derivable, several known examples Symmetry breaking to standard model due to VEV's of l0,m and tied to onset of supersymmetry breaking Natural doublet-triplet splitting mechanism
m- > mi > 2m-
- - ,:-0.3mg 5 100 GeV
2mxy mx; m m *;
-m ,!-m ,f - IP I
60 GeV < mh 5 100 GeV
No-scale supergravity by construction mp,mP< 1 TeV by no-scale mechanism Parameters 3: m,,,,tan&m, Proton decay: d = 5 very small Dark matter: Oxhi 50.25, ok with cosmology and big enough for dark matter problem 2 5 t a d S 3 2 , m, < 190 GeV, &,=O my 5 1 TeV, mp=mg miL=m,=0.3mpS300 GeV mTD=O. 18m, 5200 GeV 2mxy-mx; =m,f -0.3m, 5 285 GeV
- ,:-
-
m m$ , If4 60 GeV < m h 5 135 GeV
m x;
No analogue
Strict no-scale: tanP=ta@( m,,m,) m, 5 135 GeV -p>O,mh 5 100 GeV m, 2 140 GeV -p
Cosmic baryon asymmetry?
Cosmic baryon asymmetry explained [36]
293 48 -
LOPEZ, NANOPOULOS, POIS, WANG,AND ZICHICHI
4074
minimal SU(5)
n 250 a 200 0
2
........... . ...*,.:. . . .. . . '
,-01
150
'a
100
a
1, : :.:.
. ... .. ,.
'.:;:..: . . .
..
,
50
0
60
40
80
100
120
L'
40
60
80
100
120
80
100
120
I
n a 0 0
Bo
. ,
' ,
'
..
FIG. 13. The number of mixed chargino events as shown in Fig. 3 but versus the lightest Higgs-boson mass instead. All points with m,,< 60 GeV are actually experimentally excluded.
.
'.,
20
40
60
80 mh (Gev)
100
120
40
60
m,,< 60 GeV are actually experimentally excluded. At LEP 11, if no Higgs events are seen for mh 5 80 GeV, Fig. 13 shows that in the minimal model S 1 2 5 ( 1 + 2 j + b ) events are expected. This would allow to unambiguously test these models. In fact, the number of mixed chargino events seen is predicted to be different in the two models for the same Higgs-boson mass limit. We conclude that the charged slepton sector is a deeper probe than the chargino, neutralino, or Higgs sectors of the flipped SU(5)model at LEP 11, while the opposite is true for the minimal SU(5)model where the slepton sector is no probe at all. The interconnections among the various sectors of the models should make them easily falsifiable, or, if verified experimentally, hard to dismiss as coincidences thus providing firm evidence for the underlying structure of these models.
[I] J. Ellis, S. Kelley, and D. V. Nanopoulos, Phys. Lett. B 249, 441 (1990); 260, 131 (1991); Nucl. Phys. B373, 55 (1992);P. Langacker and M.-X. Luo, Phys. Rev. D 44, 817 (1991); U. Amaldi, W. de Boer, and A. Furstenau, Phys. Lett. B 260, 447 (1991);F. Anselmo, L. Cifarelli, A. Peterman, and A. Zichichi, Nuovo Cimento 104A, 1817 (1991); 105A, 581 (1992); H. Arason et al., Phys. Rev. Lett. 67, 2933 (1991);R. Barbieri and L. Hall, ibid. 68, 752 (1992); A. Giveon, L. Hall, and U. Sarid, Phys. Lett. B 271, 138 (1991); G. Ross and R. Roberts, Nucl. Phys. B377, 571 (1992). [2] M. Matsumoto, J. Arafune, H. Tanaka, and K. Shiraishi, Phys. Rev. D 46,3966 (1992). [3] R. Arnowitt and P. Nath, Phys. Rev. Lett. 69, 725 (1992); P. Nath and R. Amowitt, Phys. Lett. B 287, 89 (1992); 289, 368 (1992).
mh (GeV)
ACKNOWLEDGMENTS
We would like to thank M. Felcini, J.-F. Grivaz, J. Hilgart, S. Katsanevas, and J. White for very helpful discussions. J.L. would like to thank the CERN-Theory Division for its hospitality while part of this work was being done. This work has been supported in part by D O E Grant No. DE-FG05-91-ER-40633. The work of J.L. has been supported by the SSC Laboratory. The work of D.V.N. has been supported in part by a grant from Conoco Inc. The work of X.W. has been supported by the World-Laboratory. We would like to thank the H A R C Supercomputer Center for the use of their N E C SX-3 supercomputer and the Texas A&M Supercomputer Center for the use of their Cray-YMP supercomputer.
[4] J. L. Lopez, D. V. Nanopoulos, and H. Pois, Phys. Rev. D 47,2468 (1993). [5] J. Hisano, H. Murayama, and T. Yanagida, Phys. Rev. Lett. 69, 1014 (1992);Nucl. Phys. B402,46 (1993). [6]J. L. Lopez, D. V. Nanopoulos, H. Pois, and A. Zichichi, Phys. Lett. B 299, 262 (1993). [7] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 291, 255 (1992). [8]R. Arnowitt and P. Nath, Phys. Lett. B 299, 58 (1993); 307,408(E)(1993);Phys. Rev. Lett. 70, 3696 (1993). [9] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Texas A&M University Report Nos. CTP-TAMU-68/92, CERN-TH.6667/92, and CERN-PPE/92-188 (unpublished). [lo] J. L. Lopez, D. V. Nanopoulos, X. Wang, and A. Zichichi, Phys. Rev. D 48, 2062 (1993).
294
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SPARTICLE AND HIGGS-BOSON PRODUCTION AND DETECTION A T .
[ l l ] J. L. Lopez, D. V. Nanopoulos, H. Pois, X. Wang, and A. Zichichi, Phys. Lett. B 306, 73 (1993). [12] D. Dicus, S. Nandi, W. Repko, and X. Tata, Phys. Rev. Lett. 51, 1030 (1983);X. Tata and D. Dicus, Phys. Rev. D 35, 2110 (1987); H. Baer, A. Bartl, D. Karatas, W. Majerotto, and Tata, Int. J. Mod. Phys. A 4,41 l l (1989). [13] T. Schimert, C. Burgess, and X. Tata, Phys. Rev. D 32, 707 (1985); A. Bartl, H. Fraas, W. Majerotto, and B. Mosslacher, Z. Phys. C 55,257 (1992). [14] A. Bartl, H. Fraas, and W. Majerotto, Z. Phys. C 30, 441 (1986). [15] C. Dionisi et al., in Proceedings of the ECFA Workshop on LEP 200, Aachen, West Germany, 1986, edited by A. Bohm and W. Hoogland (CERN Report No. 87-08, Geneva, Switzerland, 1987),p. 380. [16] M. Chen, C. Dionisi, M. Martinez, and X. Tata, Phys. Rep. 159, 201 (1988). [17] M. Felcini, in “Ten years of susy confronting experiment,” Report No. CERN-TH.6707/92-PPE/92-180, 1992 (unpublished). [18] J.-F. Grivaz, LAL Report No. 92-64 (unpublished). [19] D. Dicus, S. Nandi, W. Repko, and X. Tata, Phys. Rev. D 29, 1317 (1984); A. Bartl, H. Fraas, and W. Majerotto, Nucl. Phys. B278, 1 (1986). [20] M. Gluck and E. Reya, Phys.Rev. D 31, 1581 (1985); T. Schimert and X. Tata, ibid. 32,721 (1985). [21] A. Bartl, H. Fraas, and M. Majerotto, Z. Phys. C 34, 411 (1987). [22] S. Katsanevas, talk given at the 1993 Aspen Winter Conference (unpublished).
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[23] J. Gunion, H. Haber, G. Kane, and S. Dawson, The Higgs Hunter’s Guide (Addison-Wesley,Redwood City, 1990). [24] Y. Okada, M. Yamaguchi, and T. Yanagida, Prog. Theor. Phys. 85, 1 (1991);Phys. Lett. B 262, 54 (1991);J. Ellis, G. Ridolfi, and F. Zwirner, ibid. 257, 83 (1991); H. Haber and R. Hempfling, Phys. Rev. Lett. 66,1815 (1991). [25] S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Nucl. Phys. B398, 3 (1993). [26] S. L. Wu et al., in Proceedings of the ECFA Workshop on LEP200 [15], p. 312. [27] S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Phys. Lett. B 285, 61 (1992). [28] E. Braaten and J. P. Leveille, Phys. Rev. D 22, 715 (1980); M. Drees and K. Hikasa, Phys. Lett. B 240, 455 (1990); S. G. Gorishny et al., Mod. Phys. Lett. A 5,2703 (1990). [29] V. Barger, M. S. Berger, A. L. Stange, and R. J. N. Phillips, Phys. Rev. D 45,4128 (1992). [30] T. J. Weiler and T. C. Yuan, Nucl. Phys. B318, 337 (1989). [31] Z. Kunszt and F. Zwirner, Nucl. Phys. B385,3 (1992). [32] D. Treille, in “Ten years of susy confronting experiment,” Report No. CERN-TH.6707/92-PPE/92-180, 1992 (unpublished). [33] V. Barger, K. Cheung, R. J. N. Phillips, and A. L. Stange, Phys. Rev. D 46,4914 (1992). [34] J.-F. Grivaz, in “Ten years of susy confronting experiment,” Report No. CERN-TH.6707/92-PPE/92-180, 1992 (unpublished). [35] F. Anselmo et al., CERN report (unpublished). [36] J. Ellis, D. V. Nanopoulos, and K. Olive, Phys. Lett. B 300, 121 (1993).
295
Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi
TOWARDS A UNIFIED STRING SUPERGRAVITY MODEL
From Physics Letters B 319 (1993)451
I993
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PhysicsLenersB 319 (1993)451-456 North-Holland
PHYSICS LETTERS B
Towards a unified string supergravity model Jorge L. LOpezaib,D.V. Nanopoulosa,b~c and A. Zichichid a
Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, The Woodlands, TX 77381, USA CERN Theory Division, 1211 Geneva 23, Switzerland CERN, 1211 Geneva 23, Switzerland
Received 15 June 1993 Editor: R. Gatto We present a unified string supergravity model based on a string-derived SU ( 5) x U ( 1) model and a string-inspired supersymmetry breaking scenario triggered by the F-term of the universally present dilaton field. This model can be described by three parameters: m t , tanp, and m-. We work out the predictions for all sparticle and Higgs masses g and discuss the prospects for their detection at the Tevatron, LEPI, 11, and HEW. We find that the cosmological neutralino relic density is always within current expectations (i.e., S2,hi 5 0.9).We also consider a more constrained version of this supersymmetry breaking scenario where the B-term is specified. In this case tan p can be determined (tanfl x 1.4-1.6) and implies m1 s 155GeV and mh 5 91 GeV. Thus, continuing Tevatron topquark searches and LEPI, I1 Higgs searches could probe this restricted scenario completely.
1. Motivation The ultimate unification of all particles and interactions has string theory as the best candidate. If this theory were completely understood, we would be able to show that string theory is either inconsistent with the low-energy world or supported by experimental data. Since our present knowledge of string theory is at best fragmented and certainly incomplete, it is important to consider models which incorporate as many stringy ingredients as possible. The number of such models is expected to be large, however, the basic ingredients that such "string models" should incorporate fall into few categories: (i) gauge group and matter representations which unify at a calculable model-dependent string unification scale; (ii) a hidden sector which becomes strongly interacting at an intermediate scale and triggers supersymmetry breaking with vanishing vacuum energy and hierarchically small soft supersymmetry breaking parameters; (iii ) acceptable high-energy phenomenology, e.g., gauge symmetry breaking to the Standard Model (if needed), not-too-rapid proton decay, de-
coupling of intermediate-mass-scale unobserved matter states, etc.; (iv) radiative electroweak symmetry breaking; (v) acceptable low-energy phenomenology, e.g., reproduce the observed spectrum of quark and lepton masses and the quark mixing angles, sparticle and Higgs masses not in conflict with present experimental bounds, not-too-large neutralino cosmological relic density, etc. All the above are to be understood as constraints on potentially realistic string models. Since some of the above constraints can be independently satisfied in specific models, the real power of a string model rests in the successful satisfaction of all these constraints within a single model. String model-building is at a state of development where large numbers of models can be constructed using various techniques (so-called formulations) [ 1 ]. Such models provide a gauge group and associated set of matter representations, as well as all interactions in the superpotential, the Kahler potential, and the gauge kinetic function. The effective string supergravity can then be worked out and thus all the above constraints can in principle be enforced. In practice
0370-2693/93/$06.00@ 1993-Elsevier Science Publishers B.V. All rights reserved SSDI 0 3 70-2 69 3 ( 9 3 ) E 1 4 6 9-E
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PHYSICS LETTERS B
this approach has never been followed in its entirety: sophisticated model-building techniques exist which can produce models satisfying constraints (i), (iii), (iv) and part of (v); detailed studies of supersymmetry breaking triggered by gaugino condensation have been performed for generic hidden sectors; and extensive explorations of the soft-supersymmetry breaking parameter space satisfying constraints (iii) , (iv),and (v) have been conducted. It has been recently noted (see refs. [2,3] and references therein) that much model-independent information can be obtained about the structure of the soft supersymmetry breaking parameters in generic string supergravity models. Supersymmetry breaking can generally be triggered by non-zero F-terms for any of the moduli fields of the string model ( ( F T ) ) or a non-zero F-term for the dilaton field ( ( F s ) ) . With some simplifying assumptions one can obtain an expression for the scalar masses 2: = m$2( 1 + ni cos20 ) , with tan0 = (Fs)/ ( F T ) [3]; here 172312 is the gravitino mass and the ni are the modular weights of the respective matter field. There are two ways in which one can obtain universal scalar masses (as strongly desired phenomenologically to avoid large flavor-changing-neutral-currents (FCNCs ) [ 4 ] ) : (i ) setting 0 = a / 2 , that is (Fs) >> ( F T ) ;or (ii) in a model where all ni are the same, as occurs for Z2 x Z2 orbifolds and free-fermionic constructions. In the first scenario, supersymmetry breaking is tnggered by the dilaton F-term and yields universal softsupersymmetry-breaking gaugino and scalar masses and trilinear interactions [2] (in the notation of e.g., ref. [5]) 1 mo = - m l p ,
Js
A = -mlp.
(1 1
In this paper we explore the phenomenological consequences of this supersymmetry breaking scenario which complements the string-derived flipped SU (5) model of ref. [ 6 1. This scenario has been studied recently also in the context of the MSSM in ref. [7]. In the second scenario, in the limit (Fr) >> (Fs) (i.e., 0 .-* 0) all scalar masses at the unification scale vanish, as is the case in no-scale supergravity models [ 51. With the additional ingredient of a flipped SU (5 ) gauge group (to solve the dimension-five proton decay problem [ 81 and be easily derivable from string theory [ 9 ] ) all the above problems are naturally avoided 452
30 December 1993
[ lo], and interesting predictions for direct [ 1 1- 141 and indirect [ 15 ] experimental detection follow. In sum, we see basically two unified string supergravity models emerging as good candidates for phenomenologically acceptable string models, both of which include a flipped SU (5 ) observable gauge group supplemented by matter representations in order to unify at the string scale Mu 10”GeV [ 16,171. Supersymmetry breaking is triggered by either: (a) the dilaton F-term, or (b) a moduli F-term such that the all the scalar masses vanish (as occurs in free-fermionic constructions). In either case, after enforcement of the above constraints, the low-energy theory can be described in terms of just three parameters: the top-quark mass ( m t ) ,the ratio of Higgs vacuum expectation values ( t a n p ) , and the gluino mass ( m0: m1/2). Therefore, measurement of only g two sparticle or Higgs masses would determine the remaining thirty. Moreover, if the hidden sector responsible for these patterns of soft supersymmetry breaking is specified, the gravitino mass (mo) will also be determined and the supersymmetry breaking sector of the theory will be completely fixed. N
2. Phenomenology
The procedure to extract the low-energy predictions of the model outlined above is rather standard by now (see, e.g., ref. [18]): ( a ) the bottom-quark and tau-lepton masses, together with the input values of mr and t a n p are used to determine the respective Yukawa couplings at the electroweak scale; (b) the gauge and Yukawa couplings are then run up to the unification scale MU = 10’’ GeV taking into account the extra vector-like quark doublet ( w 10l2GeV) and singlet ( 1O6 GeV) introduced above [ 17,101; (c) at the unification scale the soft-supersymmetry breaking parameters are introduced (according to eq. ( 1) ) and the scalar masses are then run down to the electroweak scale; ( d ) radiative electroweak symmetry breaking is enforced by minimizing the one-loop effective potential which depends on the whole mass spectrum, and the values of the Higgs mixing term IpI and the bilinear soft-supersymmetry breaking parameter B are determined from the minimization conditions; (e) all known phenomenological constraints on the sparticle and Higgs masses are applied (most importantly the N
299 PHYSICS LETTERS B
Volume 319, number 4
Table 1 The value of the ai,b, coefficients appearing in eq. (3) for a3(Mz)= 0.1 18. The results apply as well to the secondgeneration squark and slepton masses.
30 December 1993
1000
800
i
-
eL
-eYR -U L -UR
ai
b,
0.406 0.329 0.406 1.027 0.994
+Oh16 +0.818 -1.153 -0.110 -0.015
dL
1.027
+0.152
dR
0.989
-0.030
-
600
400
200
0 0
200
400
600
mi (GeV)
LEP lower bounds on the chargino and Higgs masses), including the cosmological requirement of not-toolarge neutralino relic density. We have scanned the parameter space for ml = 130,150,170 GeV, tan p = 2 -+ 50 and m1/2 = 50 -+ 500GeV. Imposing the constraint m-- < 1 TeV we find ml/2 5 465GeV 4 and tan B 5 46. Relaxing the above condition on ml/2 simply allows all sparticle masses to grow further proportional to m-. g The neutralino and chargino masses show a correlation observed before in this class of models [ 19,101, namely
800
1000
0
200
400
600
mi
800
1000
(CeV)
Fig. 1. The first-generation squark and slepton masses as a function of the gluino mass, for both signs of p and rnt = 150GeV. The same values apply to the second generation. The thickness of the lines and their deviation from linearity are because of the small tan dependence.
g 9
The first- and second-generation squark and slepton masses can be worked out analytically, mi (cev)
mi (cev)
-
with the a,, bi given in table 1. The coefficients have been obtained for a3(Mz) = 0.118. These masses are plotted in fig. 1 . The thickness and straightness of the lines shows the small tan B dependence, except for V. The results do not depend on the sign of ,K, except to the extent that some points in parameter space are not allowed for both signs of ,K: the p < 0 lines startoff at larger mass values. Note that m- /m- x 0.81 eR eL and y R , <~ m- < m- - , with the average firstg UrdL generation squark mass m- = l . O l m 7 In fig. 2 we 4
show ?1,2,&,2,K~2 for the choice m, = 150GeV.
Fig. 2. The T1,2, b1,2,and 6.2 masses versus the gluino mass for both signs cf p and m l = 150 GeV. The large variability in the ?,,2, bl,2 masses is because of the large tanp dependence in the off-diagonal elements of the corresponding mass matrices. Note that m- can be light for a fraction of 11 the parameter space because of the non-negligible value of the A-parameter. The large variability on the 71.2 and bl,2 masses is due to the tap-dependence in the off-diagonal element of the corresponding 2 x 2 mass matrices (a m,b (Ar,b + K , tan 8 ) ) . O n the the other hand, the off-diagonal element in the stop-squark mass matrix
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-
,,.,..
;
I jj:,.
...
80
.'.
.. ...
$
3
...... ,.,:..........
. . .
80
1
.................
............
4.'.
500 400 300
E' 200 100 - 0
0
200
400
600
mi (cev)
800
- 0 1000 0
200
400
800
800
1000
mi (GeV)
Fig. 3. The one-loop corrected h and A Higgs masses versus the gluino mass for both signs of p and rnl = 150GeV. Representative values of tan fi are indicated.
(a ml (At + p/ tan j3) ) is rather insensitive to tan j3 but still effects a large z-5 mass splitting (see fig. 2 ) because of the significant A, contribution. The lowest values of the 5 mass go up with ml and can be as low as ";r > 88,112,150 (92,106,150) GeV for p > 0 ( p < 0) and mI = 130,150,170GeV. The one-loop corrected lightest CP-even ( h ) and CP-odd ( A ) Higgs boson masses are shown in fig. 3 as functions of m- for ml = 150 GeV. Following the 8 methods of ref. [ 121 we have determined that the LEP lower bound on mh becomes mh 2 60GeV, as the figure shows. The largest value of mh depends on ml; we find mr" z 107, l l 7,125 GeV for m, = 130,150, 170 GeV. It is interesting to note that the one-loop corrected values of mh for t a n p = 2 are quite dependent on the sign of p. This phemass splitnomenon can be traced back to the ting which enhances the dominant ? one-loop corrections to mh [ 201, an effect which is usually neglected in phenomenological analyses. The 5 . 2 masses for tanj3 = 2 and are drawn closer together than the rest. The opposite effect occurs for p < 0 and therefore the one-loop correction is larger in this case. The sign-of-p dependence appears in the off-diagonal entries in the ?mass matrix 0: m r ( A ,+ p / tan B ) , with At < 0 in this case. Clearly only small tan j3 matters, and p < 0 enhances the splitting. The A-mass grows fairly linearly with m; with a tan j3-dependent slope which decreases for increasing tanp, as shown in
5-<
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30 December 1993
fig. 3. Note that even though mA can be fairly light, we always get mA > mh, in agreement with a general theorem to this effect in supergravity theories [ 2 11. This result also implies that the channel e + e - + hA at LEPI is not kinematically allowed in this model. The computation of the neutralino relic density (following the methods of refs. [22,23]) shows that none of the points in parameter space are constrained by cosmology. In fact, we find QxG 5 0.9, which implies that in this model cosmologically interesting values Qxhi occur quite naturally (cf., the model in ref. [ 101 where eq. (1 ) is substituted by mo = A = 0 and Qz hi 5 0.25 is obtained).
. . .
600
-
PHYSICS LETTERS B
3. A special w e
In our analysis above, the radiative electroweak breaking conditions were used to determine the magnitude of the Higgs mixing term p at the electroweak scale. This quantity is ensured to remain light as long as the supersymmetry breaking parameters remain light. In a fundamental theory this parameter should be calculable and its value used to determine the Zboson mass. From this point of view it is not clear that the natural value of p should be light. In specific models on can obtain such values by invoking nonrenormalizable interactions [ 24,251. Another contribution to this quantity is generically present in string supergravity models [26,25,2]. The general case with contributions from both sources has been effectively dealt with in the previous section. If one assumes that only supergravity-induced contributions to p exist, then it can be shown that the B-parameter at the unification scale is also determined [ 21, (4)
which is to be added to the set of relations in eq. (1 ). This new constraint effectively determines tan B for given ml and 7values and makes this restricted version of the model highly predictive, and even mostly in conflict with experiment, as we now show. From the outset we note that only solutions with p < 0 exist. This is not a completely obvious result, but it can be partially understood as follows. In treelevel approximation, rn; > 0 + p B < 0 at the electroweak scale. Since B ( M u ) is required to be posi-
30 1 Volume 319, number 4
30 December 1993
PHYSICS LETTERS B
Table 2 The range of allowed sparticle and Higgs masses in the restricted supersymmetry breaking scenario discussed in section 3. The topquark mass is restricted to be mt < 155 GeV. All masses in GeV. I
130 150 155
335-1000 260-1000 640-1000
38-140 24-140 90-140
75-270 50-270 170-270
1.57-1.63 1.37-1.45 1.38-1.40
tive and not small, B (Mz) will likely be positive as well thus forcing ,u to be negative. A sufficiently small value of B ( M u ) and/or one-loop corrections to rn; could alter this result, although in practice this does not happen. A numerical iterative procedure allows us to determine the value of tan B which satisfies eq. (4), from the calculated value of B ( M z ) . We find that tanp
= 1.57-1.63,
1.37-1.45,1.38-1.40
for rnl = 130,150,155 GeV
is required. Since tanB is so small (rn;= = 28-41 GeV), a significant one-loop correction to M h is required to increase it above PZ 60 GeV [ 121. This requires the largest possible top-quark masses and a not-too-small squark mass. However, perturbative unification imposes an upper bound on rnl for a given tan /3 [27 1, which in this case implies [ 18]
which limits the magnitude of
(6) mh,
for rnr = 130,150,155GeV .
I
61-74 64-87 84-91
110-400 90-400 210-400
A, H,H +
335-1000 260-1000 640-1000
(7)
Lower values of rnl are disfavored experimentally. In table 2 we give the range of sparticle masses that are allowed in this case. Clearly, continuing top-quark searches at the Tevatron and Higgs searches at LEPI, I1 should probe this restricted scenario completely.
4. Prospects for detection and conclusions
The sparticle and Higgs spectrum shown in figs. 1,2,3 and table 2 can be explored partially at present and near future collider facilities:
> 400 > 400
> 970
(i) Tevatron: The search and eventual discovery of the top quark will narrow down the three-dimensional parameter space considerably; even possibly ruling out the restricted scenario discussed in section 3 if rnl > 150 GeV. The trilepton signal in pF + ,Y;X FX, where x ; and x decay leptonically, is a clean test of supersymmetry [28] and in particular of this class of models [ 11 1. We expect that some regions of parameter space with rn + 5 150 GeV could be probed with XI
(5 1
rnt 5 155GeV,
h
I
lOOpb-'. Therelation rn;= l . O l mg- f o r t h e Z ~ ~ , d L p squark masses should allow to probe the low end of the squark and gluino allowed mass ranges (at least for ,u > 0, see fig. 1 ). The mass can be below 100 GeV for sufficiently low mi. As the lower bound on rnr rises, this signal becomes less accessible. The actual reach of the Tevatron for the above processes depends on its ultimate integrated luminosity, and wiU be explored elsewhere. (ii) LEPI, I t The lower bound on the Standard Model Higgs boson mass could still be pushed up several GeV at LEPI and therefore the strict scenario of section 3 (which requires r n h = 61-91 GeV) could be further constrained at LEPI and definitely tested at LEPII. In the general case, at LEPII only a fraction of the Higgs mass range could be explored, generally for small tanB values (see fig. 3). The e + e - + hA channel will be open only for low Chargino masses below the kinematical limit (rn i 5 100 GeV) should XI not be a problem [ 131, although rnx* canbe as high as I M 285 GeV in this model. Charged slepton pair production is accessible at LEPII for a small fraction of the parameter space (see fig. 2). (iii) H E M : The elastic and deep-inelastic contributions to e - p + Gx; and e p -+ [ 141, should probe a non-negligible fraction of the parameter space since both G and & are light for low r y . We conclude that the well motivated string-
5
T
455
302 Volume 3 19, number 4
PHYSICS LETTERS B
inspiredlderived model presented here could soon be probed experimentally, and a strict version of it even be ruled out. The various ingredients making up our model are likely to be present in actual fully stringderived models which yield the set of supersymmetry breaking parameters in eq. ( 1 ), The search for such a model is imperative, although it may not be an easy task since in traditional gaugino condensation scenarios eq. (1) is usually not reproduced. Moreover, the requirement of vanishing vacuum energy may be difficult to fulfdl, as a model with these properties and all the other ones outlined in section 1 is yet to be found.
Acknowledgement This work has been supported in part by DOE grant
DE-FG05-91-ER-40633. The work of J.L. has been supported by an SSC Fellowship. J.L.L. would like to thank J. White for useful discussions. We thank L. Ibiiiez for pointing out an error in the relation between rno and ml/2 as derived in ref. [ 2 ] .
References [ 1 ] See, e.g., String theory in four dimensions, ed. M. Dine (North-Holland, Amsterdam, 1988); Superstring construction, ed. A.N. Schellekens (NorthHolland, Amsterdam, 1989). [2] V. Kaplunovsky and J. Louis, Phys. Lett. B 306 (1993) 269. [ 3 ] L. Ibaiiez and D. Lust, Nucl. Phys. B 382 (1992) 305; A. Brignole, L. Ibaiiez and C. Muiioz, FTUAM-26/93. [4] J. Ellis and D.V. Nanopoulos, Phys. Lett. B 110 (1982) 44. [5] For a review see A. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1. (61 J.L. Lopez, D.V. Nanopoulos and K. Yuan, Nucl. Phys. B 399 (1993) 654. [7] R. Barbieri, J. Louis and M. Moretti, Phys. Lett. B 312 (1993) 451. [ 81 I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, P h y . Lett. B 194 (1987) 231; J. 1311is, J. Hagelin, S . Kelley and D.V. Nanopoulos, Nu:1. Phys. B 311 (1988/89) 1. 191 I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, Phys. Lett. B 231 (1989) 65; J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 268 (1991) 359:
456
30 December 1993
For a review see, J.L. Lopez and D.V. Nanopoulos, in: Proc. 15th Johns Hopkins Workshop on Current Problems in Particle Theory (August 1991), eds. G. Domokos and S . Kovesi-Domokos (World Scientific, Singapore, 1992) p. 277. J.L. Lopez, D.V. Nanopoulos and A. Zichichi, Texas A & M University prepnnt CTP-TAMU-68/92, CERNTH.6667/92 and CERN-PPE/92-188, Phys. Rev. D, to appear. J.L. Lopez, D.V. Nanopoulos, X. Wang and A. Zichichi, Phys. Rev. D 48 (1993) 2062. [ 12) J.L. Lopez, D.V. Nanopoulos, H. Pois, X. Wang and A. Zichichi, Phys. Lett. B 306 (1993) 73. [ 131 J.L. Lopez, D.V. Nanopoulos, H. Pois, X. Wang and A. Zichichi, Phys. Rev. D 48 (1993) 4062. [ 141 J.L. Lopez, D.V. Nanopoulos, X. Wang and A. Zichichi, Phys. Rev. D 48 (1993) 4029. [ 15 ] J.L. Lopez, D.V. Nanopoulos and G. Park, Phys. Rev. D 48 (1993) R974; J.L. Lopez, D.V. Nanopoulos, G. Park, H. Pois and K. Yuan, Phys. Rev. D 48 (1993) 3297; R. Gandhi, J.L. Lopez, D.V. Nanopoulos, K. Yuan and A. Zichichi, CERN-TH.6999/93. [ 161 I. Antoniadis, J. Ellis, S . Kelley and D.V. Nanopoulos, Phys. Lett. B 272 (1991) 31; D. Bailin and A. Love, Phys. Lett. B 280 (1992) 26. [ 171 S . Kelley, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 278 (1992) 140. [ 181 S . Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois and K. Yuan, Nucl. Phys. B 398 (1993) 3. [19] P. Nath and R. Amowitt, Phys. Lett. B 289 (1992) 368. [20] J. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B 262 (1991) 477. [21] M. Drees and M. Nojiri, Phys. Rev. D 45 (1992) 2482. [ 221 J.L. Lopez, D.V. Nanopoulos and K. Yuan, Nucl. Phys. B 370 (1992) 445. [23] S . Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois and K. Yuan, Phys. Rev. D 47 (1993) 2461. [24] J.E. Kim and H.P. Nilles, Phys. Lett. B 138 (1984) 150; Phys. Lett. B 263 (1991) 79; J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 251 (1990) 73; E.J. Chun, J.E. Kim and H.P Nilles, Nucl. Phys. B 370 (1992) 105. [25] J. Casas and C. Mufioz, Phys. Lett. B 306 (1993) 288. [26] G. Giudice and A. Masiero, Phys. Lett. B 206 (1988) 480. [27] L. Durand and J.L. Lopez, Phys. Lett. B 217 (1989) 463; Phys. Rev. D 40 (1989) 207. [28] P. Nath and R. Amowitt, Mod. Phys. Lett. A 2 (1987) 331; R. Barbieri, F. Caravaglios, M. Frigeni and M. Mangano, Nucl. Phys. B 367 (1991) 28; H. Baer and X. Tata, Phys. Rev. D 47 (1993) 2739.
303
Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi
SIMPLEST, STRING-DERIVABLE, SUPERGRAVITY MODEL AND ITS EXPERIMENTAL PREDICTIONS
From Physical Review D 49 (1994)343
I994
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305 1 JANUARY 1994
VOLUME 49, NUMBER 1
PHYSICAL REVIEW D
Simplest, string-derivable, supergravity model and its experimental predictions Jorge L. Lopez Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, Texas 77843-4242 and Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, Texas 77381
D. V . Nanopoulos Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, Texas 77843-4242 Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, Texas 77381 and CERN Theoiy Division, 1211 Geneva 23, Switzerland
A. Zichichi CERN, Geneva, Switzerland
(Received24 June 1993) We present the simplest, string-derivable, supergravity model and discuss its experimental consequences. This model is a new string-inspired flipped SU(5) which unifies at the string scale M U = loi8 GeV due to the introduction of an additional pair of 1 0 , E flipped SU(5)representations which contain new intermediate scale “gap” particles. We study various model-building issues which should be addressed in string-derived incarnations of this model. We focus our study on the no-scale supergravity mechanism and explore thoroughly the three-dimensional parameter space of the model ( m g , m , ,tad), thus obtaining several simple relationships among the particle masses, such as m q= m-, mq = ms =O.30mg, m- =0.18mp,and m =2m4 = m + . In a strict interpretation of the no-scale suXI
CR
pergravity scenario we solve for tanB as a function of m , and m l , and show that m , determines not only the sign of the Higgs mixing parameter p but also whether the lightest Higgs boson mass is above or below 100 GeV. We also find that throughout the parameter space the neutralino relic abundance is within observational bounds (nxh650.25) and may account for a significant portion of the dark matter in the Universe. PACS number(s):04.65.+e, 11.30.Pb, 12.60.Jv, 14.80.L~
The purpose of this paper is to find out the simplest supergravity model compatible with all boundary conditions imposed by present experimental and theoretical knowledge. The first property of this model is the number of parameters needed, which we restrict to a minimum. In this search we follow string-inspired choices. The most significant is the “no-scale” supergravity condition which, in addition to being the only known mechanism to guarantee the existence of light supersymmetric particles, has the very interesting property of being the infrared limit of superstring theory. The other choices, aimed at the minimum number of free parameters, are at present inspired by string phenomenology and are good candidates to being rigorously derivable from string theory. Our main goal is to produce a model whose basic conceptual choices are attractive, in terms of what we think (and hope) will be the final theory of all particles and interactions. One point needs to be emphasized. In order to put string theory under experimental test, the first step is to construct models with a number of parameters, which is as minimal as possible. Our aim is to propose experimental tests that are steps towards the inclusion or exclusion of our choices needed to build the model. In addition t o the very economic grand unified theory
(GUT) symmetry-breaking mechanism in flipped SU(5) [1,2], which allows it to be in principle derivable from superstring theory [3], perhaps one of the more interesting motivations for considering such a unified gauge group is the natural avoidance of potentially dangerous dimension-five proton decay operators [4]. In this paper we construct a supergravity model based on this gauge group, which has the additional property of unifying a t a scale M u 10’’ GeV, as expected to occur in stringderived versions of this model [5]. As such, this model should constitute a blueprint for string model builders. This string unification scale should be contrasted with the naive unification scale, M u 10l6GeV, obtained by running the standard model particles and their superpartners to very high energies. This apparent discrepancy of two orders of magnitude [6] creates a gap which needs to be bridged somehow in string models. It has been shown [7] that the simplest solution to this problem is the introduction in the spectrum of heavy vectorlike particles with standard model quantum numbers. -The minimal such choice [8], a quark doublet pair Q,Q and a 1/3-charge quark singlet pair D, D, fit snugly inside a 10, pair of flipped SU(5) representations, beyond the usual 3 X ( 1 0 + 5 + 1 ) of matter and 10,lO of Higgs fields. In this model, gauge symmetry breaking occurs due to vacuum expectatioi values (VEV’s) of the neutral components of the 10,lO Higgs representations, which devel-
0556-282 1/94/49( 1)/343(12)/$06.00
343
I. INTRODUCTION
49 -
-
-
a
@ 1994 The American Physical Society
306 JORGE L.LOPEZ,D. V. NANOPOULOS,AND A. ZICHICHI
344
op along flat directions of the scalar potential. There are two known ways in which these VEV's (and thus the symmetry-breaking scale) could be determined: (i) In the conventional way, radiative corrections to the scalar potential in the presence of soft supersymmetry breaking generate a global minimum of the potential for values of the VEV's slightly below the scale where supersymmetry-breaking effects are first felt in the observable sector [4]. If the latter scale is the Planck scale (in a suitable normalization) then M u -Mp1/&10" GeV. (ii) In string-derived models a pseudo U, ( 1 ) anomaly arises as a consequence of truncating the theory to just the massless degrees of freedom, and adds a contribution to its D term, D , = 2 q t I ( ,$i ) l23.e, with ~ = g ~ T r U , ( l ) / 1 9 2 d - ( l O ~GeW2 ' [9]. Toavoid a huge breaking of supersymmetry we need to demand D , =O and therefore the fields charged under U A ( 1) need to get suitable VEV's. Among these one generally finds the symmetry breaking Higgs fields, and thus M u 10l8 GeV follows. In general, both these mechanisms could produce somewhat lower values of M u . However, M u 2 loL6GeV is necessary to avoid too rapid proton decay due to dimension-six operators [lo]. In these more general cases the SUP) and U(1) gauge couplings would not unify at M u (only a2and a3would), although they would eventually "superunify" at the string scale Msu 10" GeV. To simplify matters, below we consider the simplest possible case of Mu=Msu 10" GeV. We also draw inspiration from string model building and regard the Higgs mixing term phh as a result of an effective higher-order coupling [ 1 I], instead of as a result of a light sicglet field getting a small VEV Le., hhF,$-.h( ,$ )hh ) as originally considered [2,4]. For the supersymmetry breaking parameters we consider the no-scale ansatz [12], which ensures the vanishing of the (tree-level)cosmological constant even after supersymmetry breaking. This framework also arises in the low-energy limit of superstring theory [13]. In a theory which contains heavy fields, the minimal no-scale structure SU(1,l) [14] is generalized to SU(N,I) [I51 which implies that the scalar fields do not feel the supersymmetry breaking effects. In practice this means that the universal scalar mass ( m , ) and the universal cubic scalar coupling ( A ) are set to zero. The sole source of supersymmetry breaking is the universal gaugino mass ( m ). We first let the universal bilinear scalar coupling ( B ) float, i.e., be determined by the radiative electroweak symmetry breaking constraints. We also consider the strict no-scale scenario where B(M,)=O.It is worth pointing out that with the no-scale framework the value of m l R should be determined dynamically and explicit calculations [16] show that it should be below 1 TeV. A recent analysis has shown that this result may also occur automatically once all phenomenological constraints on the model have been imposed [ 171. We should remark that a real string model will include a hidden sector in addition to the observable sector discussed in what follows. The model presented here tacitly
-
-
-
49
assumes that such hidden sector is present and that it has suitable properties. For example, the superpotential in Eq. (1)below, in a string model will receive contributions from cubic and higher-order terms, with the latter generating effective observable sector couplings once hidden sector matter condensates develop [I I]. The hidden sector is also assumed to play a fundamental role in triggering supersymmetry breaking via gaugino condensation. This in turn makes possible the first mechanism for gauge symmetry breaking discussed above. Our comments here are of a generic nature because we do not have a specific string model where these assumptions can be tested explicitly. In the known string models of the class we draw inspiration from ( i e , free fermionic flipped SU(5) models [ 9 ] ) , suitable hidden sectors which do not affect the observable sector Yukawa couplings are known to exist [3,11,21]. Finally, no string model has yet been derived which can accommodate all of the phenomenological properties that we know must exist-such an enterprise is clearly beyond the scope of this paper. This paper is organized as follows. In Sec. I1 we present the string-inspired model with all the modelbuilding details which determine in principle the masses of the new heavy vectorlike particles. We also discuss the question of the possible reintroduction of dangerous dimension-five proton decay operators in this generalized model. We then impose the constraint of flipped SU(5) unification and string unification to occur at M u = 10" GeV to deduce the unknown masses. In Sec. I11 we consider the experimental predictions for all the sparticle and one-loop corrected Higgs boson masses in this model, and deduce several simple relations among the various sparticle masses. In Sec. IV we repeat this analysis for the strict no-scale case. This additional constraint allows us to determine tax$ for a given mg and m , (up to a possible twofold ambiguity), and thus to sharpen the most tar$-sensitive predictions. In Sec. V we summarize our conclusions.
II. THEMODEL The model we consider is a generalization of that presented in Ref. [2], and contains the following flipped SU(5) fiess: (i) three generations of quark and lepton fields Fi,fi,lf,i =1,2,3; (ii) two pairs of Higgs 10,mrepresentations H,,Pj,i = 1,2; (iii) one pair of "electroweak" Higgs 5,3 representations h,F; (iv) three singlet '$1,2,3.
Under SU(3)X SU(2) the various flipped SU(5) fields decompose as follows:
h=[H,DJ,
F=(A,b)
(2.lc)
307 SIMPLEST, STRING-DERIVABLE, SUPERGRAVITY MODEL. . .
49
The most general effective' superpotential consistent with SU(5)XU(1)symmetry is given by
345
pressed in terms of these mixed light eigen~tates.~ This low-energy quark-mixing mechanism is an explicit realization of the general extra-vector-abeyance (EVA) mechanism of Ref. [19]. As a first approximation though, in what follows we will set h'(.=O, so that the light eigenstates are di,2,3. B. Neutrino seesaw matrix
The seesaw neutrino matrix receives contributions hyFiifjji-+ m LJvfvj,, h y k F i R j#k -+ A t k Vjvf#k, from pij#i#j. The resulting matrlx is4 Symmetry breaking- is effected by nonzero YEV's (vh, ) = Vi, ( v c ) = Vi, such that V: V:= P: V:. Hi
+
+
(2.5)
A. Higgs doublet and triplet mass matrices
The Higgs doublet mass YaLrix receives-contributions from phE-+pHH and h Y H i f j h -+AYViLjH. The resulting matrix is
C. Numerical scenario
To sjmplify the discussion we will assume, besides5
A? =AY =0, that
hY= 6ijhY1, hi(=8ijhd, ..
(2.3)
..
pv=6vpi,
wij=6iiwi
@k,
gijgikh'il
.
6
9
(2.6)
These choices are likely to be realized in string versions of this model and will not alter our conclusions below. In this case the Higgs triplet mass matrix reduces to
4 To avoid fine tunings of the A$ couplings we must demand A$ -0, so that f;i remains light. The HigKs triple matrix receives several contributions: phh +pDD;- -+hyVid);,D;h y H i H j h sulting matrix is2
d);,
dfr, (2.7)
Regarding the (3,2) states, the scalars get either eaten by the X , Y SU(5) heavy gauge bosons or become heavy Higgs bosons, whereas the fermions interact with the X , P gauginos through the mass matrix [2 11
(2.8)
Clearly three linear combinations of [5,dhl,2 ,di,2,3) will remain light. In fact, such a general situation will induce a mixing in the down-type Yukawa matrix hyFiFjh+hyQid,FH, since the d f will need to be reex-
ITo be understood in the string context as arising from cubic and higher order terms [18,11]. ,The zero entries in A, result from the assumption ( & ) =O in
lakFiPjgk.
3Note that this mixing is on top of any structure that A'( may have, and is the only source of mixing in the typical string model-building case of a diagonal A2 matrix. jWe neglect a possible higher-order contribution which could produce a nonvanishing vtv; entry [20]. Ref. [2] the discrete symmetry H I+ - H , was imposed so that these couplings automatically vanish when H 2 , P , are not present. This symmetry (generalized to H , +-H, ) is not needed here since it would imply w ' j ~ 0 which , is shown below to be disastrous for gauge coupling unification.
308 346
JORGE L. LOPEZ, D. V. NANOPOULOS, AND A. ZICHICHI
The lightest eigenvalues of these two matrices (denoted generally by df; and Q H ,respectively) constitute the new relatively light particles in the spectrum, which are hereafter referred to as the “gap” particles since with suitable masses they bridge the gap between unification masses at 10l6and 10l8GeV. Guided by the phenomenological requirement on the gap particle masses, i.e., M >>M [S],we consider the eH dH
49
TABLE I. The value of the gap particle masses and the unified coupling for a 3 ( M Z)=O. 118+O.M)8. We have taken MU=10” GeV, sin20,=0.233, and a;’=127.9.
a3(Mz)
M d;I
0.110 0.118 0.126
(GeV)
MoH (GeV)
dM“)
2.2x 1Ol2 4.1 x 1Ol2
0.0565 0.0555 0.0547
4.9X104 4.5X106 2.3X10’
7.3 x loi2
explicit numerical scenario Liz)=A5( 2 )=0 ,
vl,V*,v2,V*-v>>w,>>w2>>p, (2.9)
which would need to be reproduced in a viable stringderived model. From Eq. (2.7) we then get Mdh2-Mdc = w z , and all other mass eigenstates V.
-
x2
Furthermore, Atz,,,,) has a characteristic polynomial AS -A2( w 1 w 2 ) -A(2 V 2 - w ,w 2) ( w , w 2) V2=0, which has two roots of O ( V )and one root of 0 ( w ,). The latter corresponds to ( QH,- QH2) and (Q,, - Qg2).
+
+ +
-
-
-
In sum then, the gap particles have masses Me, w and M w 2 , whereas all other heavy particles have masses
-
- v. d;l
D , D admixtures. In the scenario described above, the relatively light eigenstates (df;>,dK ) contain no D , D adH2 mixtures, and the operator will again be a p / V 2 . Note, however, that if conditions (2.9) (or some analogous suitability requirement) are not satisfied, then diagonalization of A, in Eq. (2.7) may reintroduce a sizable dimension-five mediated proton decay rate, depending on the value of the ai,Zi coefficients. To be safe one should demand [22,23] paiEi -5At
1
(2.11)
Id7GeV
For the higher values of M
v,
vf
d;l
in Table I (see below), this
constraint can be satisfied for not necessarily small values of a i , E i .
The see-saw matrix reduces to
4;
E. Gauge coupling unification
(2.10)
for each generation. The physics of this see-saw matrix has been discussed recently in Ref. [20], where it was shown to lead to an interesting amount of hot dark matter ( v,) and a Mikheyev-Smirnov-Wolfenstein (MSW) effect (v,,v,) compatible with all solar neutrino data. D. Proton decay
The dimension-six operators mediating proton decay in this model are highly suppressed due to the large mass of the X,Y gauge bosons ( - M u = 10” GeV). Higgsino mediated dimension-five operators exist and are naturally suppressed in the minimal model of Ref. [2]. Tke reason for this is that the Higgs triplet mixing term phh -+pDD is small ( p - M M z )whereas , the Higgs triplet mass eigenstates obtained from Eq. (2.4) by just keeping the 2 x 2 submatrix in the upper left-hand corner, are always very heavy ( V ) . The dimension-five mediated operators are then proportional to p / V 2 and thus the rate is suppressed by a factor or (p/V)’<
-
-
Since we have chosen V-MMU=Msu=10’8 GeV, this means that the standard model gauge couplings should unify at the scale M u . However, their running will be modified due to the presence of the gap particles. Note that the underlying flipped SU(5) symmetry, even though not evident in this respect, is nevertheless essential in the above discussion. The masses MQ and M can then be dH
determined as [8] 1
1
2a,
3a3
MU
-2In--0.63
sin*8,-0.0029 ae
,
1 (2.12a)
mz
MU
-61n--11.47,
(2.12b)
m2
where ae,a3,and sin%, are all measured at M,. This is a one-loop determination (the constants account for the dominant two-loop corrections) which neglects all lowand high-energy threshold effects,6 but is quite adequate
%ere we assume a common supersymmetric threshold at M,. In fact, the supersymmetric threshold and the d;l mass are anticorrelated. See Ref. [El for a discussion.
309 SIMPLEST, STRING-DERIVABLE,SUPERGRAVITY MODEL.. .
49 -
FIG. 1. The running of the gauge couplings in the flipped SU(5) model for cr3(Mz)=0.118 (solid lines). The gap particle masses have been derived using the gauge coupling renormalization group equations to achieve unification at M U= 10" GeV. The case with no gap particles (dotted lines) is also shown; here M u = 10l6GeV.
for our present purposes. As shown in Table I [and formula (2.12b)I the df, mass depends most sensitively on a,(M,)=O. 118*0.008 [24], whereas the QH mass and the unified coupling are rather insensitive to it. The unification of the gauge couplings is shown in Fig. 1 ). This figure (solid lines) for the central value Of "~(hfz also shows the case of no gap particles (dotted lines), for which M u = 10I6GeV.
In. EXPERIMENTALPREDICTIONS The model presented in the previous section can be analyzed t o determine its low-energy experimental predictions for e.g., the Higgs and sparticle masses. Consistent with the assumption of flipped SU(5) gauge symmetry breaking at -MU=10" GeV, we assume that the onset of universal supersymmetry breaking in the observable sector occurs at this same scale 141. . - This can be parametrized in terms of a universal gaugino mass ( m ), a universal scalar mass ( m , I,, and universal trilinear ( A )and bilinear ( B )scalar couplings. One also needs to specify the fermion Yukawa couplings and the Higgs mixing parameter p. The renormalization group equations then run the relevant parameters to low energies where radiative electroweak symmetry breaking occurs (studied using the one-loop effective potential). When all is said and done, the whole theory can be specified in terms of just five parameters: m1,2,mo, A , the ratio of Higgs vacuum expectation values tar$, and the topquark mass m,. Note that in this scheme p and B are calculated quantities; the sign of p remains undetermined. Our calculations enforce all known experimental bounds on supersymmetric and one-loop corrected Higgs boson masses. We refer the reader to Ref. [25] for a detailed account of this procedure. As discussed in the Introduction, in what follows we consider the typical no-scale su-
,/,
347
pergravity boundary conditions [12], where rno= A =O.' i n this section we let B float and in the following section ) = O is we consider the strict no-scale case where B ( M u required. For each sign of p we have explored a threedimensional grid in this parameter space: tanB =2-50(2), m ,,,=50-5OO(6) GeV, m, =95- 195(5 ) GeV, where the numbers in parentheses indicate the size of the step taken in that particular direction. Larger values of tanb and/or m, violate perturbative unification, and m , / , >500 GeV leads to mq,rng> 1 TeV, which would make the theory "unnatural." As discussed in the Introduction, the correct superstring model will have to provide an explanation for why these masses are light, and if not so, why this is not unnatural. For now u'e just take this to be true realizing that relaxing this assumption will not add regions for parameter space which could be tested experimentally at the next generation of colliders. On the other hand, at least in the realm of supergravity, the condition mg,mg < 1 TeV is granted by the no-scale supergravity mechanism [16] and we believe that the correct superstring model will reproduce this important condition. Our exploration resulted in = 12 K acceptable points for each sign of p , and for all of these we found taI@<32 and m ,S 1 8 5 G e V .
(3.1)
A. Massranges
The restriction of mg,mg < 1 TeV cuts off the growth of the sparticle and ~i~~~ boson maSSeS at 1 TeV. However, the sleptons, the lightest Higgs boson, the two lightest neutralinos, and the lightest chargino are cut off at a much lower mass, as followss:
of
m PR < 190 GeV , rn ZL < 305 GeV , m , < 295 GeV , m Y l< 185 GeV
,
rn O < 145 GeV XI
, m
m
71 O ,
12
<315 GeV
,
mh < 135 GeV ,
<285 GeV , rn + <285 GeV . XT
(3.2) It is interesting to note that due t o the various constraints on the model, the gluino and squark masses are predicted t o satisfy the current experimental bounds automatically. We find mg L 220 GeV and m g 2 200 GeV, except for the lightest stop eigenstate F,, which can be as light as = 150 GeV. Therefore, the TI squark could be the first squark to be possibly observed at Fermilab in the near future.
'In Refs. [17,25] a similar analysis was performed for a model without the gap particles (i.e., where M u 10l6GeV). In Ref. [25] an SU(3)X SU(2)X U(1) version of the model presented in this paper was considered (referred to as the SISM model), although only a rather limited analysis was performed. this class of supergravity models the three sneutrinos ( V ) are degenerate in mass. Also, m - = m- and m - = m z ,
-
PL
'L
PR
R
310 348
JORGE L. LOPEZ, D. V. NANOPOULOS, AND A. ZICHICHI
TABLE 11. The value of the ci coefficients appearing in Eq. (3.3) for a3(MZ)=0.118f0.008. Also shown is the ratio cE= m p/ m
i P,
c, (0.110)
c, (0.118)
c, (0.126)
3.98 3.68 3.63 0.406 0.153 1.95
4.41 4.1 1 4.06 0.409 0.153 2.12
4.97 4.66 4.61 0.413 0.153 2.30
,aL
UR JR
v,FL h C-
The first and second generation squark and slepton masses can be determined analytically: (3.3) di=(T,,-Q)tan28w+T3i
--tan2Ow, d =-tan20w), FR
(e.g.,
and
in
our
d Y_L =I 2
co
case
= m 0 / m j ,2 =O. The coefficients ci can be calculated numerically in terms of the low-energy gauge couplings, and are given in Table 119 for a3(M,)=0.1181ir0.O08. In the table it is also shown ct = m g_ / m , , 2 . The “average” squark mass rn r L ( mG L + m - + m + m - +m ‘R
+m
‘R
+m +m ’L
mq =O. 97m
3R
t ’
so that mb and mb blend into a wide band. The topI
squark masses are separated in the figure. Note the lesser definition of the TI masses due to the relatively larger m, and t a d effects. For all these masses one should note that the average squark mass m1 =0.97m9 runs somewhere in between these mass bands. The sleptons are much lighter than the squarks since roughly m 7 / r n p = ( c / c )‘”20.3. In principle, for small 7 ? mt, one would expect a stronger t a d dependence for sleptons due to the relatively smaller contribution of the first term in Eq. (3.3). This, together with the large implies that an “averdifference between c,, and c ZR ’
‘ L
B. Mass relations
where
49
age” slepton mass [as usually assumed in phenomenological studies of the minimal supersymmetric standard model (MSSM)] is a rather poor approximation to this model. In Fig. 3 we show the and FL,R masses; the inadequacy of the average slepton mass approximation is evident. As expected, the off-diagonal elements in the 7 mass matrix give a broad band of masses for a given m value. g On the other hand, the TL,Rmasses look much sharper as a function of m t . What happens is that for small m g , when the t a d effects are potentially important, t a d is not allowed to become large and thus the D term is suppressed. The t masses start off below m and quickly FR
approach the m
‘L
dR
FL
line. In numbers we find
m = m ,= 0 .3 0 2 m g ,
1 is then determined to be
ZL
(3.4)
within +3%, allowing for a + l u error in a 3 ( M , ) (the dependence on t a d is negligible). The squark splitting around the average is ~ 2 % . The third-generation squarks deviate considerably from the average squark mass and have a non-negligible dependence on tang due to the off-diagonal elements on the squark mass matrix (which are proportional to the corresponding quark mass). Throughout the parameter space we found the following maximal relative deviations of these squark masses relative t o the average squark mass ke., Imp,- m, I / m , ):
m
FR
=0.185mg,
(3.6)
where the small D-term contribution has been neglected; it becomes negligible for increasingly larger values of m E’ The TI (7,)mass approximates OR (FL,G)as a “central value,” but has quite a spread around it, as Fig. 3 shows. We find that throughout the parameter space 1p1 is generally much larger than M , and lpl > M 2 . This is shown on the top row of Fig. 4. Note that (pi 0: mg with the tang-dependent slope growing with the value of m , [25]; the three values of m , used are evident in the figure. This behavior points to a simple eigenvalue structure for the two lightest neutralinos and the lightest chargino [26]:
m o-fmx! XI
; mx~=mx~=M2=cL2/a3mg=0.3m . g
6 , : < 1 4 % ; 6 , : 5 8 % ; r,:547% ; 7 2 : 5 3 5 % . (3.5) In Fig. 2 we plot” the bottom-squark and top-squark masses. The bottom-squark masses are not split enough
(3.7) In practice we find m
x:
=m
rately (see Fig. 5, top row), whereas m x y= i m
0
XI
is only
qualitatively satisfied (see Fig. 5, bottom row). In fact, these two mass relations are much more reliable than the one that links them to mg (not shown). The heavier neutralino and chargino masses are determined by the value of Ipl (shown in Fig. 4); they all approach this limit for large enough / p i . More precisely, m o ap-
(x!,,,)
9These are renormalized at the scale Mz. In a more accurate treatment, the c, would be renormalized at the physical sparticle mass scale, leading to second-order shifts on the sparticle masses. ’OFor all the scatter plots shown in this paper we have restricted the values of the top-quark mass to m,=lM), 130, and 160 GeV to have a manageable number of points.
* to be satisfied quite accu-
XI
(x:)
x3
proaches lpl sooner than m
m
approaches m x4
X4
does. On the other hand,
* rather quickly.
x2
The one-loop corrected lightest Higgs boson mass ( m , 1 is shown in Fig. 6 (top row). The three noticeable bands
311 SIMPLEST, STRING-DERIVABLE, SUPERGRAVITY MODEL . . .
49
349
FIG. 2. Scatter plots of the bottom-squark 200
400
600
m;.
800
1000
200
600 800 (CeV)
1000
600 800 rnp (CeV)
1000
400
m;
(CeV)
(81.11 and the top-squark 1 mass eigenstates vs mz. (The 81.2cannot be resolved in this manner.) The average squark mass (first two generations) is m7-0.97rnz. From this figure onward, all results are shown for both signs of p.
1000 800
600
mr 400 200 0
200
400
600
800
1000
200
400
m;. (GeV)
correspond to the three values of the top-quark mass used (m,=100, 130, 160 GeV), for large values of t a d . For smaller values of tang the tree-level contribution is suppressed and the curves reach down t o low values of m h . In this case one can most easily note the expected logarithmic rise of mh with the squark mass (recall that m = m g ) . For low m , the curves rise very slightly. However, for large m , these rise quite dramatically. This is all in agreement with the expected behavior deduced from the approximate analytical expressions for mh in the literature. In the figure ( m , 5 160 GeV) we get m h S 1 2 0 GeV. Allowing for the whole range of m, values this upper bound gets relaxed to = 135 GeV. The one-loop corrected pseudoscalar mass m A is also shown
P>O
200
400
600
P
a,=O 118
800
1000
200
400
400
600
m p (CeV)
600
800
1000
600 800 rnp (CeV)
1000
m;. (CeV)
rnp (CeV)
200
in Fig. 6 as a function of m h . The three bands correspond to from-left-to-right m , = 100, 130, 160 GeV. The predictions for m A are not very sharp. It is nevertheless true that for all points examined m A > m h , as expected from general considerations [27]. The other two Higgs boson states H and H' are approached from below by m A [28]. For mH,,+L200 (300) GeV the difference is 5 8% (3%). To appreciate the relations among the sparticle masses in this model, in Fig. 7 we show a graphical display of the spectrum for mg 7300 GeV and m , = 130 GeV and both signs of p. Th e masses generally scale with m g . The masses shown are also given in Table 111 where in addi-
800
1000
200
400
FIG. 3. Scatter plots for the sfau ( T ] , ~and ) selectron (or smuon) (FL,R masses. Note the spread in the masses for fixed mi, due to the off-diagonal entries in the stau mass rnatrix. The P mass (not shown) starts off slightly below the FR mass and then quickly joins the FL line.
3 12 JORGE L. LOPEZ,D. V. NANOPOUIBS, AND A. ZICHICHI
350
:::I
600
IF1 400
200
200
200
400
600
BOO
400
200
1000
600
800
I000
m? (GeV)
In;- (CeV)
0 25
n,hZ
49 -
0 20
FIG. 4. Scatter plots of the Higgs mixing term ( p )vs rng and the neutralino cosmic relic abundance (R,ha) vs m,. Note the proportionality lpl a rnz whose slope increases with rn, (rn,=100,130, and 160 GeV shown). Also, for rn,=+Mz, the Z-pole annihilation is quite noticeable.
0 15 0 10 0 05 0 00
0
25
50
75
tion we give the percentage deviations of the masses relative to their central values due to the variation of t a d over all its allowed range ( 2 S t a d 5 32 ). C. Neubalino dark matter In Fig. 4 (bottom row) we plot the result for the cosmic relic abundance of the lightest neutralino, Cl,hg. This has been calculated following the methods of Refs. [29]. Since m x G m grows with m g , and lpl grows with mg, XI
then as m y grows, the pure gaugino region ( ]pi>>M, =0.3mg ) is approached and the neutralino pair-annihilation is suppressed, leading to larger Clxh values. Note the effect of the Z pole for m x = t M z .We
i
LOO
125
150
. result is in find that Qxh: can be as large as ~ 0 . 2 5 This good agreement with the observational upper bound on Cl#; [30] and does not constrain the model any further. Moreover, fits to the Cosmic Background Explorer (COBE) data and the small and large scale structure of cold dark the Universe suggest 1311 a mixture of ~ 7 0 % matter and ~ 3 0 % hot dark matter together with ho=O. 5. The hot dark matter component in the form of massive 7 neutrinos has already been shown t o be compatible with the flipped SU(5) model we consider here [20], whereas the cold dark matter component implies Cl,hi=O. 17 which is reachable in this model for m y 2 100 GeV. It is interesting to note that values of Q X h i 50.25
.i/ 100
I00
50
50 0
50
100
150
200
250
300
0
0
50
LOO
m,:
150 200
250
300
m,; 150
xi
125
100
75 50
25 -
-
0
0
50
100
150
200
250
300
FIG. 5 . Scatter plots of the second-tolightest neutralino (x:) mass vs the lightest chargino (,yF 1 mass and second-to-lightestCx!) to lightest Cxyi neutralino masses. Note the accuracy of the mx;=m + relation.
0 0
50
100
150
200
250
300
313 49
mh
SIMPLEST, STRING-DERIVABLE, SUPERGRAVITY MODEL
:"; 50
50
25
25
200
BOO
600
400
m;
1000
200
400
800
600
m;
(GeV)
1000
(GeV)
1000
800
m, 600 400
...
351
FIG. 6. Scatter plots of the one-loop corrected lightest Higgs boson mass m h vs the gluino mass for m,=100, 130, and 160 GeV (top row), and the pseudoscalar Higgs mass m A vs mh (bottom row). The three noticeable bands (from bottom-to-tor, in the top row and from left-to-right in the bottom row) correspond to m,=100, 130,and 160 GeV. Note that in this model m A > m h always. The heavy Higgs boson masses m H and m H + are approached quickly from below by m A,
200 0
60
40
80 mb
100
120
60
40
100
80
120
mh (Gev)
(cev)
rn,= 130 GeV, rn; = 300 GeV
c
6
iG,?l
j
I
5
4 (charginose 2 (neulralinos) 3
1
0
rn, = 130 GeV,
=0.118, and (a) p > O (b) p
mg = 300 GeV
V
;I, ,
(neutralinos) j
0
100
I
FIG. 7. Central values for the sparticle and one-loop corrected Higgs boson masses for m y= 300 GeV, m,=130 GeV, a & M z )
I 300
41
GeV
3 14 JORGE L. LOPEZ, D. V. NANOPOULOS, AND A. ZICHICHI
352
TABLE 111. Central values of the sparticle and one-loop corrected Higgs boson masses for a , ( M Z)GO. 118, m p= 300 GeV, m , = 130 GeV, and both signs for p, showing the percentage deviations from the central value due to the variation of tanB over its whole allowed range ( 2 5 t a d 5 32). (%)
rTj(p>O)
I
H+
164 290 43 83 178 197 83 20 1 79 183 177 195
FL
100
FR
68 69 56 105 270 286 205 344
P
4 X? X! X! X: X: Xf h H A
G
-TI
-7 2 6, 62
-11 12
17
A,(p
(%)
- 164
17 0.1 18 2.1 12 13 9.3 11 19 32 30 24 2.2 4.2 8.5 18 6.8 3.2
0.1
290 29 62 174 212 53 210 82 185 179 198
8.9 19
12 11 20 8.2 23 35 32 26 2.2 4.2 8.5 21 1.3 3.5 0.5 8.2 3.4
100
68 69 57 105 267 290 170 363
B ( M , is determined by the radiative electroweak symmetry breaking conditions, this added constraint needs to be imposed in a rather indirect way. That is, for given mg and m , values, we scan the possible values of t a d looking for cases where B(M,)=O. The most striking result is that solutions exist only for m , 5 135 GeV if p > 0 and for m , k 140 GeV if p < 0. That is, the value of m , determines the sign of p. Furthermore, for p < 0 the value of tar$ is determined uniquely as a function of m , and m g , whereas for p > 0, ta$ can be double valued for some m , range which includes m , = 1 3 0 GeV but does not include m , = 100 GeV. In Fig. 8 (top row) we plot the solutions found in this manner for the indicated m , values. All the mass relationships deduced in the previous section apply here as well. The tar$-spread that some of them have will be much reduced though. The most noticeable changes occur for the quantities which depend most sensitively on tax$, ix., the neutralino relic abundance and the lightest and pseudoscalar Higgs masses. In Fig. 8 (bottom row) we plot 0,h versus m x for this case. Note that continuous values of m , will tend to fill in the space between the lines shown. In Fig. 9 (top row) we plot the one-loop corrected lightest Higgs boson mass versus m g . The result is that mh is basically determined by m , ; only a weak dependence on m E exists. Moreover, for m , 5 135 G e V = = p > O , m h5 105 GeV; whereas for m , 2 140 GeV-p 0, we just showed that the strict no-scale constraint requires m , S 135 GeV. This implies that p cannot grow as large as it did previously. In fact, for p > 0, pLmax=745GeV before and pL,,,=440- GeV now. This
1.o
7.8 1.3
occur naturally in this model, and in general for m >>mo. This situation is in sharp contrast to, for example, the minimal SU(5) supergravity model, where nZ,h>> 1 occurs naturally instead [23].
A
IV. THE STRICT NO-SCALE CASE We now impose the additional constraint on the theory that B(M,)=O, that is the strict no-scale case. Since
w o
w o
a,=0.118
140. ......
tar$
20
....
.... ............
..............
10
.............. 130 .......
0
400
200
800
600
10
........
.... ......'59.. ...
160 . . . ........... . . . . ...... ....
.?Of?... ...
.............................
I000
200
400
rn? (Gev)
BOO
I000
100 125
150
600
rn; (GeV)
0.30
0.25
n,hz
49
0.20 0.15 0.10
0.05 0.00
0
25
50
75 mx
100
Gev)
125
150
0
25
50
75
rnx (Gev)
FIG. 8. Scatter plots of tanB vs ma for the strict no-scale case [where B ( M u)=O] for the indicated values of m,. Note that the sign of p is determined by m , and that tar$ can be double-valued for p > 0. Also shown are the values of the neutralino relic abundance (Q,h vs m y ) for the same values of the parameters.
315 SIMPLEST, STRING-DERIVABLE,SUPERGRAVITY MODEL . . .
49 -
P>Q
P
a,=O 118
0
mr
rnr (GeV)
(tieV)
1000
m
400
A
200 0
353
~
FIG 9 Same as Fig. 6, but for the strict no-scale case Note the weak dependence of mh on the gluino mass mz. Also, I f p > 0 for m, 5 135 GeV, mh 5 105 GeV, whereas Ifp <0, for m, 2 140 GeV, mh 2 100 GeV.
~
200
0 40
60
80
100
120
mh (GeV)
40
60
80 100 mh (GeW
smaller value of pmax has the effect of cutting off the growth of the X ; , ~ , X : masses at =pm,,=440 GeV (cf. c 750 GeV) and of the heavy Higgs masses at =530 GeV (cf. = 940 GeV).
V. CONCLUSIONS In this paper we have presented the simplest, stringderivable, supergravity model and deduced its experimental predictions. This new string-inspired model has several features that are found in real string-derived models, such as string unification and a unified gauge group which can reduce to the standard model one after spontaneous gauge symmetry breaking. We also demanded that the low-energy supergravity theory be of the no-scale type, since this general framework is supported by superstring theory. The model built this way should be considered to be an idealization of what its string-derived incarnation should be. In the process we have identified several potential model-building problems which would need to be watched for in a string implementation. We have assumed that the various needed mass scales are generated somehow and have fit their values to achieve string unification at M u = 10” GeV. The actual origin of these mass scales will lie within the structure of the successful string model. Known examples include condensates [11,32] and vacuum expectation values [33] of hidden matter fields. As in any nonminimal flipped SU(5) model, non-negligible dimension-five proton decay operators could be reintroduced. In the model presented here these remain highly suppressed. However, in variants of this model or in string-derived versions, these operators could exist at an observable level. This question deserves
[l]S. Barr, Phys. Lett. 112B, 219 (1982);Phys. Rev. D 40, 2457 (1989);J. Derendinger, J. Kim, and D. V. Nanopoulos, Phys. Lett. 139B, 170 (1984). [2]I. Antoniadis, J. Ellis, J. Hagelin, and D. V. Nanopoulos,
120
further study. We have also performed a thorough and accurate exploration of the parameter space of the model and solved for all the sparticle and one-loop corrected Higgs masses. The growth of the supersymmetry breaking parameter is cut off by the no-scale supergravity mechanism which guarantees m p , m g< 1 TeV. We found some general results and upper bounds on the sleptons and lightest neutralino, chargino, and Higgs masses. We have also found several simple relations among squark and gluino masses, among slepton masses, and among the lightest neutralino and chargino masses. The neutralino relic abundance 0,h; never exceeds ~ 0 . 2 5and therefore does not constrain this model. However, it may constitute a significant portion of the dark matter in the Universe in general and in the galactic halo in particular. In the strict no-scale case we find a striking result: if p>O, m , 5 135 GeV, whereas if p <0, m , 2 140 GeV. Therefore the value of m, determines the sign of p. Moreover, the value of tar@ can also be determined. Furthermore, we found that the value of m , also determines whether the lightest Higgs boson is above or below 100 GeV. ACKNOWLEDGMENTS
This work has been supported in part by DOE Grant
No. DE-FG05-91-ER-40633. The work of J. L. has been supported by the SSC Laboratory. The work of D.V.N. has been supported in part by a grant from Conoco Inc. We would like to thank the HARC Supercomputer Center for the use of their NEC SX-3 supercomputer.
Phys. Lett. B 194,231 (1987). [3] I. Antoniadis, J. Ellis, J. Hagelin, and D. V. Nanopoulos, Phys. Lett. B 231,65 (1989). [4] J. Ellis, J. Hagelin, S . Kelley, and D. V. Nanopoulos,
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JORGE L. LOPEZ, D. V. NANOPOULOS, AND A. ZICHICHI
Nucl. Phys. B311, 1 (1988/89). [5] I. Antoniadis, J. Ellis, R. Lacaze, and D. V. Nanopoulos, Phys. Lett. B 268, 188 (1991); S. Kalara, J. L. Lopez, and D. V. Nanopoulos, ibid. 269, 84 (1991). [6] J. Ellis, S. Kelley, and D. V. Nanopoulos, Phys. Lett. B 249, 441 (1990); F. Anselmo, L. Cifarelli, and A. Zichichi, Nuovo Cimento 105A, 1335 (1992). [7] I. Antoniadis, J. Ellis, S. Kelley, and D. V. Nanopoulos, Phys. Lett. B 272, 31 (1991); D. Bailin and A. Love, ibid. 280, 26 (1992). [8] S. Kelley, J. L. Lopez, and D. V. Nanopoulos, Phys. Lett. B 278, 140 (1992);G. Leontaris, ibid. 281, 54 (1992). [9] For a review see, e.g., J. L. Lopez and D. V. Nanopoulos, in Proceedings of the 15th Johns Hopkins Workshop on Current Problems in Particle Theory, Baltimore, Maryland, 1991, edited by G. Domokos and S. Kovesi-Domokos (World Scientific, Singapore, 1992),p. 277. [lo] J. Ellis, S. Kelley, and D. V. Nanopoulos, Phys. Lett. B 260, 131 (1991). [ l l ] J. L. Lopez and D. V. Nanopoulos, Phys. Lett. B 251, 73 (1990);256, 150 (1991);268, 359 (1991). [12] For a review see A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. 145, 1 (1987). [13] E. Witten, Phys. Lett. 155B, 151 (1985). [14] J. Ellis, C. Kounnas, and D. V. Nanopoulos, Nucl. Phys. B241,406 (1984). [15] J. Ellis, C. Kounnas, and D. V. Nanopoulos, Nucl. Phys. B247, 373 (1984). [16] J. Ellis, A. Lahanas, D. V. Nanopoulos, and K. Tamvakis, Phys. Lett. 134B, 429 (1984). [17] S . Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Phys. Lett. B 273,473 (1991). [18] S . Kalara, J. Lopez, and D. V. Nanopoulos, Phys. Lett. B 245,421 (1990);Nucl. Phys. B353, 650 (1991). [I91 S. Kelley, J. L. Lopez, and D. V. Nanopoulos, Phys. Lett. B 261,424 (1991).
49 -
[20] J. Ellis, J. L. Lopez, and D. V. Nanopoulos, Phys. Lett. B 292, 189 (1992). [21] J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Nucl. Phys. B399,654 (1993). [22] M. Matsumoto, J. Arafune, H. Tanaka, and K. Shiraishi, Phys. Rev. D 46,3966 (1992); R. Arnowitt and P. Nath, Phys. Rev. Lett. 69, 725 (1992); J. Hisano, H. Murayama, and T. Yanagida, ibid. 69, 1014 (1992); Nucl. Phys. B402, 46 (1993). [23] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 291, 255 (1992); J. L. Lopez, D. V. Nanopoulos, and H. Pois, Phys. Rev. D 47,2468 (1993). [24] S . Bethke, in Proceedings of the XXVIth International Conference on High Energy Physics, Dallas, Texas, 1992, edited by J. R. Sanford, AIP Conf. Proc. No. 272 (AIP, New York, 1993),p. 81. [25] S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Nucl. Phys. B398, 3 (1993). [26] P. Nath and R. Amowitt, Phys. Lett. B 289, 368 (1992). [27] M. Drees and M. M. Nojiri, Phys. Rev. D 45,2482 (1992). [28] S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan,Phys. Lett. B 285, 61 (1992). [29] J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Nucl. Phys. B370, 445 11992); Phys. Lett. B 267, 219 (1991);S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Phys. Rev. D 47, 2461 (1993). [30]See, e.g., E. Kolb and M. Turner, The Early Uniuerse (Addison-Wesley, Reading, MA, 1990). [31] See, e.g., R. Schaefer and Q. Shafi, Nature (London) 359, 199 (1992); A. N. Taylor and M. Rowan-Robinson, ibid. 359, 393 (1992). [32] S. Kalara, J. L. Lopez, and D. V. Nanopoulos, Phys. Lett. B 275, 304 (1992). [33] I. Antoniadis, J. Rizos, and K. Tamvakis, Phys. Lett. B 278,257 (1992).
3 17
Jorge L. Lopez, D.V. Nanopoulos, Gye T. Park and A. Zichichi
STRONGEST EXPERIMENTAL CONSTRAINTS ON SU(5) x U( 1) SUPERGRAVITY MODELS
From Physical Review D 49 (1994)355
I994
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3 19 PHYSICAL REVIEW D
1 JANUARY 1994
VOLUME 49, NUMBER 1
Strongest experimental constraints on SU(5)X U(1) supergravity models Jorge L. Lopez,’**D. V. N a n o p o ~ lo s,’ -Gye ~ T. Park,’,’ and A. Zichichi4 ‘Centerfor Theoretical Physics, Department of Physics, Texas A&M University, College Station, Texas 77843-4242 ’Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, Texas 77381 ’CERN, Theory Division, 1211 Geneva 23, Switzerland 4CERN, 1211 Geneva 23, Switzerland (Received 16 August 1993)
We consider a class of well-motivated string-inspired flipped SU(5)supergravity models which include four supenymmetry-breakingscenarios: no-scale, strict no-scale, dilaton, and special dilaton, such that only three parameters are needed to describe all new phenomena (m,,tar@,ml). We show that the CERN LEP precise measurements of the electroweak parameters in the form of the el variable and the CLEO I1 allowed range for B ( b - + s y ) are at present the most important experimental constraints on this class of models. For m, 2 155 (165) GeV, the constraint [at 90 (95)% C.L.] requires the presence of light charginos ( m + 5 50-100 GeV depending on rn, ). Since all sparticle masses are proportional to XI
5 100 GeV implies m4 5 55 GeV, rn 5 100 GeV, tnr 5 360 GeV, mq 5 350 (365) GeV,
ml,
mX:
m-
5 80 (125) GeV, m- S 120 (155) GeV, and mp 5 100 (140) GeV in the no-scale (dilaton)flipped SU(5)
8
CL
CR
supergravity model. The B ( b+ s y ) constraint excludes a significant fraction of the otherwise allowed region in the ( m +,tar$) plane (irrespectiveof the magnitude of the chargino mass), while future experiXI
mental improvements will result in decisive tests of these models. In light of the E , constraint, we conclude that the outlook for chargino and selectron detection at LEP I1 and at DESY HERA is quite favorable in this class of models. PACS number(s):04.65.+e, 12.60.Jv, 14.80.L~ I. INTRODUCTION
Models of low-energy supersymmetry which have their genesis in physics at very high energies (i.e., supergravity or superstrings) embody the principal motivation for supersymmetry-the solution of the gauge hierarchy problem. These models can also be quite predictive since their parameters are constrained by the larger symmetries usually found at very high mass scales. In contrast, models of low-energy supersymmetry with no such fundamental basis [such as the minimal supersymmetric standard model (MSSM)] should properly be acknowledged as generic parametrizations of all possible supersymmetric beyond-the-standard-model possibilities, and as such not as real “theories.” Specifically, supergravity models with radiative electroweak symmetry breaking need only five parameters to describe all new phenomena: the top-quark mass ( m , 1, the ratio of Higgs vacuum expectation values (tang), and three universal softsupersymmetry-breaking parameters ( m1,2,mo,A ). The MSSM on the other hand requires over 20 parameters to describe the same phenomena, and has led to the erroneous impression that supersymmetric models necessarily introduce numerous parameters which hamper the analysis of processes of interest. In order to ameliorate this situation, it is routinely assumed that these parameters are related among themselves in some arbitrary way. These ad hoc simplifications can be misleading (e.g., implying “general” conclusions based on special cases) or simply insufficient (e.g., as in the analysis of branching 0556282 1/94/49( 1)/355( 11)/%06.00
49 -
fractions involving particles from all sectors of the model). Perhaps one of the most interesting aspects of supergravity models is the radiative electroweak symmetrybreaking mechanism [ 1,2], by which the electroweak symmetry is broken by the Higgs mechanism driven dynamically by radiative corrections. This mechanism involves several otherwise unrelated physical inputs, such as the top-quark mass, the breaking of supersymmetry, the physics at the high-energy scale, and the running of the parameters from high to low energies. Needless t o say, electroweak symmetry breaking has no explanation in the MSSM ke., the negative Higgs boson mass squared is put in by hand). In practice, this constraint is used to determine the magnitude of the Higgs mixing parameter p 131, which is of electroweak size or larger, and implies (i) a generic correlation among the lighter neutralino and chargino masses (2mxy md m ) which has been ob-
- - * XI
served in a variety of models [4-61, and (ii) the connection between the supersymmetric sector and the Higgs sector such that, as the supersymmetry breaking parameters are increased, the Higgs sector asymptotes quickly to a standard-model-like situation with one light Higgs scalar [7]. Experimental predictions in this class of models for processes at present and near future collider facilities can be obtained and have been determined for the various supergravity models we consider below (CERN e + e - collider LEP I [8,7], Fermilab Tevatron [9],DESY ep collider HERA [lo], LEP I1 [ll]). Unfortunately, the range of sparticle and Higgs boson masses is quite broad, even for 355
01994 The American Physical Society
320 49
LOPEZ, NANOPOULOS, PARK,AND ZICHICHI
356
supersymmetry-breaking parameters at the unification scale. The latter can be greatly simplified by assuming universality. That is, all three gaugino masses are taken to be degenerate (M,=M,=M,=ml,z), all 15 squark and slepton masses and the two Higgs scalar doublets are separately assumed to be universal ( M ~ Q , D C , 6 C , t , E Cand , i =M1H,1.22=,m3o=) ,mand ~ the three trilinear scalar couplings are taken to be equal ( A ) . Of these simplifying assumptions, only the one relating the (first- and second-generation) scalar masses is required experimentally l o keep the flavorchanging neutral currents in the K - K system under control [25]. Nevertheless, these assumptions are seen to hold in large classes of supergravity models. The Higgs mixing parameter p and its associated bilinear softsupersymmetry-breaking mass B do not need to be specified at the unification scale since they do not feed into the renormalization-group equations for the other parameters. They are determined at the electroweak scale via the radiative electroweak symmetry breaking mechanism. In sum then, only five parameters are needed to describe the supergravity models we consider. In contrast, 25 parameters would be needed in the MSSM. The minimization of the electroweak Higgs potential is performed at the one-loop level, and yields the values of lpl and B as well as the one-loop-corrected Higgs boson masses [3]. The five-dimensional parameter space is restricted by several consistency conditions and experimental constraints on the unobserved sparticle masses (most importantly the chargino mass and the lightest Higgs boson mass [7]). Moreover, the cosmological constraint of a not-too-young universe (usually incorrectly referred to as “not overclosing the universe”) is also imposed [26]. Building on the above basic structure one is naturally led to the SU(5)XU(I)[or flipped SU(5)] gauge group since this model has a strong motivation in the context of string theory [27]. Moreover, the additional particles fit snugly into complete 10,lO SU(5 ) XU( 1 ) representations. In this model proton decay and cosmological constraints are satisfied automatically [5,6]. Concerning the softsupersymmetry-breaking parameter space, one assumes two possible string-inspired supersymmetry breaking scenarios: (i) the SU(N, 1 ) no-scale supergravity model [28,2] which implies m o = A =0, with a more constrained case called the strict no-scale scenario where B ( M u 1 =O is also assumed; and (ii) the dilaton supersymmetry breaking scenario [29] where mo= 1 //?m and A =-rnIl2, including a special case where 11. THE SU(5)X U(1) SUPERGRAVITY MODELS B(MU)=2m,. The allowed parameter spaces in these two cases have been determined in Refs. [5,6], respectiveThe SU(5)XU(1) supergravity models we consider inly. It is found that there are no important constraints on clude the standard model particles and their supertar$ (tax$<, 32,46, respectively) and the sparticle masses partners plus two Higgs doublets. These are supplementcan be as light as their present experimental lower ed by one additional vectorlike quark doublet with a mass bounds. More importantly, the parameter spaces are 10’’ GeV and one additional vectorlike (charge - 1 /3) three dimensional ( m , ,tan& m a mg ) and therefore the quark singlet with a mass lo6 GeV [23], such that the spectrum scales with mg [27]. (In what follows we conunification scale is delayed until lo’* GeV, as expected sider only three representative values of m,=130, 150, to occur in string models [24]. 170 GeV.) In Table I we give a comparative listing of the In addition to the two parameters needed to describe most important properties of these models. the (third-generation) Yukawa sector of the models ( m , It is interesting to note that the special cases of these and tar$), one must also specify the values of the soft-
the constrained models we consider. In fact, experimental exploration of all of the well-motivated parameter space of these models would require the large hadron [CERN Large Hadron Collider (LHC) and/or Superconducting Super Collider (SSC)] colliders for the strongly interacting particles (gluino and squarks) and the Next Linear Collider (NLC) for the weakly interacting particles (charginos and neutralios). Of course, the present generation of experiments has probed and will continue to probe part of this allowed parameter space (i.e., the lighter end of the spectrum). Another way of testing the predictions of supergravity models is via indirect experimental signatures, usually originating from virtual (e.g., one-loop) processes. In particular, one has the precise electroweak measurements at LEP (in the form of the E , , ’ , ~ , ~parameters [12-15]) and the rare radiative decay b+sy[ 16-20], as observed by the CLEO I1 Collaboration [21]. Both these probes have been investigated independently in the minimal SU(5) and the no-scale flipped SU(5) supergravity models in Refs. [22,19]. In this paper we expand these studies to a larger class of SU(5)XU(l)supergravity models and determine the allowed region of parameter space where both constraints are satisfied simultaneously. These one-loop processes have the advantage of sidestepping the strong kinematical constraints which afflict the direct production channels, although they require high precision experiments which may not be accountable exclusively in terms of supersymmetric effects in the models we consider. Nevertheless, as we point out in this paper, these experimental constraints are at present the most stringent ones on the class of supergravity models we consider. It is also important to note that the knowledge acquired through the indirect tests will help sharpen the predictions for the direct detection processes, and thus focus the experimental efforts to detect these particles. This paper is organized as follows. In Sec. I1 we describe briefly the class of SU(5)X U(1) supergravity models which we consider, as well as the four softsupersymmetry-breaking scenarios of interest. In Sec. 111 we outline the calculational procedure followed to obtain c1 and B ( b - + s y ) . In Sec. IV we present our results and discuss their phenomenological consequences. Finally in Sec. V we summarize our conclusions. The Appendix contains the explicit expressions used to evaluate B ( b - s y ).
-
-
-
,’
321 STRONGEST EXPERIMENTAL CONSTRAINTS ON SU(5)XU(l). . .
49
357
TABLE I. Major features of the class of SU(S)XU(l)supergravity models, and a comparison of the supersymmetrybreaking scenarios considered. (All masses in GeV.) SU(5)XU(1) Easily string-derivable, several known examples Symmetry breaking to standard model due to VEV’s of 10, 10 and tied to onset of supersymmetry breaking Natural doublet-triplet splitting mechanism Proton decay: d =5 operators very small Baryon asymmetry through lepton number asymmetry (induced by the decay of flipped neutrinos) as recycled by nonperturbative electroweak interactions No-scale
Dilaton
Parameters 3: ml,2,tan&m, Universal soft-supersymmetry-breaking automatic mo=O, A =O
Parameters 3: m l/2,taq5‘,m, Universal soft-supersymmetry-breaking automatic 1 mo=-m,,,, A = - m 1/2
fi
a
mZR=O. 18m-,m, =0.30mp
Dark matter: n,h < 0.90 m I,2 < 465 GeV, tanS < 46 mp > 195 GeV, ml > 195 GeV mii=l.Olmp mi >90 GeV m- =0.33mp,mzL=0.41mp
m- /m- ~ 0 . 6 1
m- /m-
60 GeV < mh < 125 GeV 2mx7=mx;=m +=0.28mp<290 GeV
60 GeV < m h < 125 GeV = m *=0.28mp5290 GeV
Dark matter: Clxh < 0.25 m1,*<475 GeV, tanp<32 mp>245 GeV, mv>240 GeV ml =O. 97mp mT > 155 GeV
eR
eR
‘R
‘L
XI
mx;-mxy-m
*-Id 4
‘L
~0.81
2mrm *--I$ x;
mx;-mx;-m
XI
x2
Spectrum easily accessible soon
Spectrum accessible soon
Strict no-scale: B ( M U) =0 tanB=tanb(m,,mp) m,5135 GeV-p>0,mh5100 GeV m, k 140 GeV-p
Special dilaton: B ( M o )=2m, t a d = tanS(m,,ml ) tanfl=1.4-1.6,m, < 1 5 5 GeV mh=61-91 GeV
two supersymmetry breaking scenarios are quite predictive, since they allow one to determine ta$ in terms of m, and mg. In the strict no-scale case one finds that the value of rn, determines the sign of p ( p > O m , S 135 GeV, p < O m , 2 140 GeV) and whether the lightest Higgs boson mass is above or below 100 GeV. In the special dilaton scenario, t a d z 1 . 4 - 1 . 6 and m,5 155 GeV, 61 GeV S mh S 91 GeV follow. Thus, continuing Tevatron topquark searches and LEP 1,II Higgs boson searches could probe these restricted scenarios completely. 111. ONE-LOOP CONSTRAINTS ( € 1 and b + S Y )
The one-loop corrections to the W* and Z boson selfenergies Le., the “oblique” corrections) can be parametrized in terms of three variables el,’,) [12] which are constrained experimentally by the precise LEP measurements of the Z leptonic width and the leptonic forward-backward asymmetries at the Z pole ( 1, as well as the M w / M z ratio. A fourth observable is the Z - b 6 width which is described by the eb parameter [14]. Of these four variables, at present e l provides the strongest constraint in supersymmetric models at the
(r,),
Ah
90% C.L. [22,15]. However, the eb constraint is competitive with (although at present somewhat weaker than) the E , constraint [15,30], and in fact may impose interesting constraints on supersymmetric models as the precision of the data increases. The expression for el is obtained from the definition ~ 3 1 (1) where are the following combinations of vacuum polarization amplitudes, (2)
e 5 = ~ i ~ i z (, ~ 2 )
(3)
and the q2#0 contributions F i , ( q 2 )are defined by
n p q 2 ) =n g 0) +q ’Fij( q * ) .
(4)
The Sg, in Eq. (1) is the contribution to the axial-vector vertex from form factor at Q’=M; in the Z - l ’ l -
322 49 -
LOPEZ, NANOPOULOS, PARK, AND ZICHICHI
358
proper vertex diagrams and fermion self-energies, and S G , , comes from the one-loop box, vertex and fermion self-energy corrections to the p-decay amplitude at zero external momentum. These nonoblique SM corrections are non-negligible, and must be included in order to obtain an accurate SM prediction. As is well known, the SM contribution to el depends quadratically on m , but only logarithmically on the SM Higgs boson mass ( m , ). In this fashion upper bounds on m, can be obtained which have a non-negligible m, dependence: up to 20 GeV stronger when going from a heavy ( = 1 TeV) to a light ( = 100 GeV) Higgs boson. It is also known (in the MSSM) that the largest supersymmetric contributions to el are expected to arise from the _ t-b sector, and in the limiting case of a very light top squark, the contribution is comparable to that of the t-b sector. The remaining squark, slepton, chargino, neutralino, and Higgs sectors all typically contribute considerably less. For increasing sparticle masses, the heavy sector of the theory decouples, and only SM effects with a fight Higgs boson survive. (This entails stricter upper bounds on m, than in the SM, since there the Higgs boson does not need to be light.) However, for a light chargin0 ( m +fM, ), a Z wave function renormalization
*
21
threshold effect can introduce a substantial q 2 dependence in the calculation, i.e., the presence of e 5 in Eq. (1) [ 131. The complete vacuum polarization contributions from the Higgs sector, the supersymmetric charginoneutralino and sfermion sectors, and also the corresponding contributions in the SM have been included in our calculations [22]. The rare radiative flavor-changing neutral current (FCNC) b +sy decay has been observed by the CLEO I1 Collaboration in the following 95% C.L. allowed range [21]:
B (b-*sy)=(0.6-5.4)X
.
(5)
In Ref. [ 191 we have given the predictions for the branching ratio in the minimal SU(5) supergravity model [B(b-+sy),i,i,,=(2.3-3.6)X10-4] and in the nonscale flipped SU(5)supergravity model. However, in that paper the experimental lower bound on B ( b+ s y ) was not available. Since a large suppression of B ( b + s y ) (much below the SM value) can occur in the flipped SU(5) models, such a bound can be quite restrictive. Below we give the predictions for B ( b+ s y ) in the two variants of the flipped SU(5) model (and their special subcases) described in Sec. 11. The expressions used to compute the branching ratio B ( b-+sy ) are given in the Appendix for completeness. IV. DISCUSSION
( 9 3 % C.L. The experimentally allowed interval for ) is given in Eq. (51, although in what follows we will also consider a less conservative estimate of the We now discuss lower bound, namely, B ( b - t s y ) > the results and ensuing constraints on each model in turn.
B ( b-+sy
A. No-scale flipped SU(5)
At the 90 (95)% C.L., for m, 5 150 (165) GeV there are no restrictions on the model parameters from the el constraint (see Fig. 1). For m,=170 GeV [see Fig. l(c)] the constraint alone implies a strict upper bound on the chargino mass': (i) for p > 0 there are no allowed points at 90% C.L., while m 5 70 GeV is required at 95% C.L.; x:
(ii) for p < O one obtains m
* C 58 (70) GeV at 90 (95)%
XI
C.L. Interestingly enough, for this range of chargino masses the B ( b-+sy 1 constraint is also restrictive. Combining the E , and B ( b+sy constraints we obtain (i) for p > 0, m x f 5 67 GeV at 95% C.L. and t a d = 8- 10; and (ii) for p < O , m
:x
5 5 4 (67) GeV at 90 ( 9 9 % C.L. and
t a n a s 8. No significant improvement is obtained by required B ( b+sy ) > lop4. Analogously, upper bounds on m + up to = l o 0 GeV are obtained for values of m, in XI
the range 150-170 GeV [22]. Larger values of m, ( m ,5 190 GeV is required [ 5 ] ) could only be made consistent with LEP data if the chargino mass is very near its present experimental lower bound. For m, = 130, 150 GeV, the E , constraint is ineffective. However, for p > 0 the B ( b +sy ) constraint is quite restrictive, as shown in Figs. l(a) and l(b). The various (dotted) curves correspond to different values of tar@. For large values of m +, these curves start off at values of XI
B ( b-sy
which decrease with increasing t a d , i.e., the largest value corresponds to tan/3=2, and then tar@ increases in steps of two. As the chargino mass decreases, these curves reach a minimum Le., zero) value and then increase again (except for tan/3=2), and even exceed the upper bound on B ( b + s y ) for large enough tana. To show better the excluded area, in Fig. 2 we have plotted those points in parameter space which survive the B ( b - + s y ) constraint, in the ( m *, tar@) plane. The XI
swath along the diagonal is excluded because B ( b-+sy ) is too small. In fact, if we demand B ( b+sy ) > the points denoted by crosses would be excluded as well. The area to the left of the left group of points is excluded because B ( b+sy ) is too large. Note that no matter what the actual value of B ( b+sy ) ends up being, there will always be some allowed set of points, namely, a subset of both sets of presently allowed points. Another consequence of the B ( b +sy ) constraint is an upper bound on tan@ for m r = 130 (150) GeV, tanB5 26 (20)compared to
The results of our computations for B ( b+sy 1 and e l are shown in Figs. 1, 3,4, and 6 in the various models under consideration, for m , = 130, 150, 170 GeV. (Smaller values of m , are not ruled out experimentally, although they appear ever more unlikely.) The LEP value for el 'An upper bound on the chargino mass implies upper bounds which we use in our analysis is ~ ~ = ( - 0 . 9 i 3 . 7 ) X l O - ~ on all sparticle and Higgs boson masses, since they are all proportional to m p ,see Table I. [31], which implies ~ , < 0 . 0 0 5 1 7 ( 0 . 0 0 6 3 5 )at the 90
323 STRONGEST EXPERIMENTAL CONSTRAINTS ON SU(S)XU(l). . .
49 -
the upper bound of t a d 6 30 (26) which existed prior to the application of the B ( b +sy ) constraint. Finally, note that t a d 2 20 implies m + Z 100 GeV. XI
B. Strict no-scale flipped SU(5)
In this variant Of the is for given mt and mg values and is such that mi = 130 GeV is only allowed for p > 0, whereas m , = 150, 170 GeV are only allowed for pO
10-2
10-3
10-3
(165) GeV at the 90 (95)% C.L. For m ,= 170 GeV there is an upper bound m 5 54 (62) GeV at 90 (95)% C.L.,
*
XI
and there is no further constraint from B ( b +s y ) (see Fig. 3). In fact, B ( b + s y ) is only constraining for rn, = 130 GeV. Note that in this case there are two possible solutions for t a d which are most clearly seen in the B ( b+ s y ) plot (Fig. 3). The lower t a d solution [which asymptotes to the larger value of B ( b - + s y ) ] excludes in the interval 95 G ~ V5 m x F 6 175 chargino
GeV, whereas the larger mx: 150 GeV.
PtO
no-scale flipped SU(5)
10-2
359
CLEO 11
10-4
10-5
10-6
-
w
50
100
150
10-6
250
200
0004
~
50
100
0002
...... .. . . , .. .... . ..
6;
-
,
50
100
200
150
0.000
250
10-3
50
100
150
200
FIG. 1. The values of B ( b + s y ) (top row) and the E , parameter (bottom row) vs the chargino mass in no-scale flipped SUW supergravity model for (a) m,=130 GeV, (b) m,=l5O GeV, and (c) m,=170 GeV. The 95% C.L. CLEO I1 Iimit on B ( b - s y ) and the 90% C.L. LEP upper limit on E , are
250
P
CLEO 11
10-3
3
10-4
10-4
1
u 3
10-5
.
.
..
. ,
50
100
.
.
.
. 10-6
150
200
250
50
100
150
200
250
150
ZOO
250
m,=150 CeV
m,; (Gev)
0.000
0.008
-
,
3-
no-scale flipped SU(5)
P
w
250
r ~ . .
0002
' '
p>o
-
200
0 004
. 5..
0 000
150
0.006 00.004
0
,
,
......0
6
E
{
0 004
0 002
0.000
0 002
50
100
150
200
m,; (cev)
250
0.000
(b)
50
100
m,: (cev)
indicated.
324 360
LOPEZ, NANOPOULOS, PARK, AND ZICHICHI
PcL>o
49
cL< 0
no-scale flipped SU(5)
10-2
10-3
t;
-
...
104
10-4
10-4
10-5
10-5
. . .. ... .
0
-
CLEO I1
2
~
~
t
P
10-8
50
100
150
ZOO
10-6
250
m,; (GeV) 0.008
0006 .+ w
50
100
m,=170 GeV
150
200
250
FIG. 1 (Continued).
m,; (Gev)
0 008
0.006
~~
0.OD4
0.004
0,002
0 002
0.000 50
100
150
ZOO
250
m,: (GeV)
0.000 50
100
150
200
250
m,: (Gev)
(C)
C. Dilaton flipped SU(5)
Figs. 4(a) and 4(b)],although it is more important for p>O. The allowed regions of parameter space in the
The discussion in this model parallels closely that for the no-scale model given above, with some relevant differences. For m,=170 GeV, the constraint alone implies [see Fig. 4(c)]: (i) for p > O,m 5 53 (60)GeV at
x:
the 90 (95)% C.L. and tanP510, and (ii) for p
m,=130 GeV
( m *,tan@) space are shown in Fig. 5. The only qualitaXI
tive difference with the results for the no-scale case is that for m,=150 GeV and p
In this case, only p < 0 is allowed and m, 5 155 GeV is required. It also follows that tanb < 2 (see Table I and
no-scale flipped SU(5) ( p > O ) B(b+sy) allowed region m,=150 GeV
FIG. 2. The region in ( m +,tar@) space which is allowed by the CLEO I1 limit on B ( b - s y ) in the no-scale fllpped SU(5)super-
q
gravity model, for p > 0 and m , = 130, 150 GeV. The value of B ( b - s y ) is too large to the left of the left group of points, and toc small in between the two groups of points. (No such constraints exist for p < 0.) The points denoted by crosses would becomes excludedifthelowerboundonB(b-sy) isstrengthenedto B ( b - s y ) >
325 STRONGEST EXPERIMENTAL CONSTRAINTS ON SU(5)XU(1)
p>o
strict no-scale flipped SU(5)
p
:I
-n
CLEO I1
2. f
FIG. 3. The values of B(b-+ry) (top row) and the
m
10-5
m,: (cev) 0.008
0006
-
w
36 1
LEP
0 006
0.004
0 004
... .
...................... 0002
0 002
0 000
j,
150
3
parameter (bottom row) vs the chargino mass in the strict noscale flipped SU(51 supergravity model, for m,=130 GeV ( p > O ) and m,=150,170 GeV (p0) two tan@ solutions exist which are clearly visible in the B ( b - 5 ~ )plot.
0 000
50
100
150
200
250
50
100
rn,; (Geb')
150
200
250
m,: (GeV)
Ref. [6]). At the moment B ( b-+sy ) does not impose any constraints on the parameter space (see Fig. 6). The same is true for E , , except for m,=155 GeV which at the 90% C.L. cannot be made acceptable since light chargino masses are not allowed (see Fig. 6); there are no constraints at the 95% C.L. V. CONCLUSIONS
We have considered a class of well-motivated stringinspired supergravity models based on the gauge group SU(5)XU(1). The various constraints on the models allow one to predict all new low-energy phenomena in terms of just three parameters ( m l ,tanp, mg ) and as such these models are highly predictive. We have shown that one-loop processes which do not create real sparticles can nonetheless be used to constrain these models in interesting ways. The LEP E , constraint and the CLEO I1 B ( b+sy allowed range are perhaps surprisingly restrictive in the models which we have considered. The constraint requires the presence of light charginos if the topquark mass exceeds the moderate value of = 155 GeV. In fact, m 5 100 GeV is the weakest possible constraint in
in the no-scale (dilaton) flipped SU(5) supergravity model. The B ( b - + s y ) constraint can probe the models irrespective of the mass scales involved because of a large suppression of the amplitude for a wide range of chargino masses (50 GeV 5 mx: 5280 GeV depending on the value of t a g ) . I t should be noted that most of the qualitative aspects of our discussions should apply quite generally to the class of supergravity models with radiative electroweak symmetry breaking, where the parameters in the ways mo and A are not necessarily related to m discussed here. We conclude that future refinement of the allowed B ( b-+sy ) range are likely to result in decisive tests of this class of models. Moreover, if the top-quark mass is too heavy to be easily detectable at the Tevatron, then the constraint would require light charginos. These could nevertheless escape detection at the Tevatron, since the characteristic trilepton events are generally suppressed for light charginos [9]. However, the outlook for chargino and selectron detection at LEP I1 [ 111 and at H ERA [lo] will remain quite favorable in this class of models.
Xf
this case. Since all sparticle masses are proportional to mg, tafi-dependent upper bounds on all of them also follow. The weakest possible upper bounds (i.e., tan B independent) which follow f o r m , k 155 (165) GeV at the 90 (95)% C.L. are mxy5 55 GeV , mx* I
.XI
5 100 GeV ,
mg 2 3 6 0 GeV , mg 5 3 5 0 (365) GeV , m
PR
5 8 0 (125) G e V , m,, 5 1 2 0 (155) G e V ,
m , 5 100 (140) GeV
(6)
ACKNOWLEDGMENTS
This work has been supported in part by DOE Grant No. DE-FG05-91-ER-40633. The work of J.L. has been supported by the SSC Lab.
(7) APPENDIX EXPRESSION FOR B ( b - s y )
The expression used for B ( b-+sy ) is given by [ 181
326
49
LOPEZ,NANOPOUMS, PARK, AND ZICHICHI
362
where q = a , ( M z ) / a , ( m b ) ,I is the phase-space factor I(x)=1-8x2+8x6-xx-24x41nx, and f ( m c / r n b ) = 2 . 4 1 the QCD correction factor for the semileptonic decay. In our computations we have used a , ( M z ) = 0 . 1 1 8 , B(b-+ceV)=10.7%, m b = 4 . 8 GeV, and m c / m b = 0 . 3 . The A,, A , are the coefficients of the effective b s y and bsg penguin operators evaluated at the scale M,. Their simplified expressions are given below, in the justifiable limit of negligible gluino and neutralino contributions [ 161 and degenerate squarks, except for the which are significantly split by m , . The contributions to A,,g from the W -f loop, the H * - f loop, and the xT-Tk loop are given by
p>o
-t;
-
P<0
dilaton flipped SU(5)
10-2
10-2
10-3
10-3
10-4
10-4
10-5
10-5
?
n
10-6
50
100
150
200
10-6
250
50
100
m,=130 GeV
m,; (cev)
150
200
250
m,: (GeV)
4
w
0.004
0 002
0 000
FIG. 4. The values of B ( b - s y ) (top row) and the E , parameter (bottom row) vs the chargino mass in the dilaton flipped SU(5) supergravity model, for (a) m,=130 GeV, (b) m,=150 GeV, and (c) m,=170 GeV. The 95% C.L.CLEO I1 limit on B ( b -+sy ) and the 90% C.L.LEP upper limit on E , are indicated.
0.002
50
100
150
200
m,: (Gev)
P>o
0.000
250
50
100
150
200
250
m,: (GeV)
(a)
P
dilaton flipped S U ( 5 ) 10-2
10-3
s
10-4
P
v
m
10-5
10-6
k50
-! 100
=150 GeV
150
200
250
m,: (GeV)
0 006 +
w
0004
O 0o o Z
50
100
150
200
250
0000
' ~ ~ " ~ " " ~ ' ' ~ ~ ' " ' ' J ' '
50
100
I50
200
250
327 363
STRONGEST EXPERIMENTAL CONSTRAINTS ON SU(5)XU(1)
49 -
/L>O
LOO
50
0.004
/L< 0
dilaton flipped SU(5)
150
250
200
FIG. 4 (Continued).
g
4 50
100
150
200
250
m,; (GeV)
-
a m
i
FIG. 5 . The region in ( m +,tar$) space which is al-
..... .............. ..........* ......... .......... *
20 10 0
"i-
I..=4 ......... .-
....... *
...........*
. . . . . .._ .......... ........*.._
-.
t"
'
...... __m..-. ...........
................. __ -._. ......................... __ ........................................
- ................................................. ............................................ "
50
'
"
(
100
'
"
' "
150
m,: ( G e W 50
I
I
,
,
I
1
I
I
I
,
1 , 1il;
I
j'
3 0 1
2
20
.......... ..... .* ........ .......... * ......... .~ ........ ......... ** ...... .* .*
,..;.-
1'
50
100
150
200
250
30
~
20
-.
.............. -. ............................ ........................................ ..................... .....................
-. _.................................................. ............................................... " " ' iL,L,',, I
100
50
40
--
.......
....
...... * .. ..,
0
1
. . . . . . .~
2
0
I
I ' ' 250
-
40
10
I
"
200
10
150
m,:
200
(cev)
"
250
10 0
lowed by the CLEO I1 limit on B ( b - + s y ) in the dilaton flipped SUW supergravity model, for m,= 130,150 GeV. The value of B ( b+ s y ) is too large to the left of the left group of points, and too small in between the two groups of points [except for m,= 150GeV ( p < O ) ,where it is too small to the left of the single group of points]. The points denoted by crosses would become excluded if the lower B(b-sy) is bound on B ( 6+ s y ) strengthened to
> 50
100
150
200
250
328 364
LOPEZ, NANOPOULOS, PARK, AND ZICHICHI
49 -
special d i l a t o n flipped SU(5) ( p i 0 ) 10-2 F /
,1
-.
.,
,
,, ,, , , ,,
(A6)
I""i""i"
Ir -
CLEO 11
; -
fL"(X
E
f,
)=
2+5x-x2 1 2 (~ 1))
x
Inx,
2 b - 1 )4
(A9)
4
130
032 L
rn,: (CeV)
FIG. 6. The values of B ( h - + s y ) (top) and the E , parameter (bottom) vs the chargino mass in the special dilaton SU(5) supergravity model (only allowed for p
f ~ 3 ' ( x ) = ( l - x ) f ~ ' ) ( x ) - - x f ~ 2 ) ( x 1) - .2 3
(A10)
In these formulas Uij,Vij ar e t h e elements of t h e matrices which diagonalize t h e chargino mass matrix, m + X,
with
are t h e chargino masses, m i are t h e t o p squark mass ei7-5x -8x2
x(3x -2)
(As)
genvalues, an d Tij ar e the elements of t h e matrix which diagonlizes t h e 2 X 2 t o p squark mass matrix.
[ l ] L. Ibaiiez and G. Ross, Phys. Let. llOB, 215 (1982); K. Inoue et al., Prog. Theor. Phys. 68, 927 (1982); I. Ibanez, Nucl. Phys. B218, 514 (1983);Phys. Lett. 118B, 73 (1982); H. P. Nilles, Nucl. Phys. B217, 366 (1983); J. Ellis, D. V. Nanopoulos, and K. Tamvakis, Phys. Lett. 121B, 123 (1983); J. Ellis, J. Hagelin, D. V. Nanopoulos, and K. Tamvakis, ibid. 125B, 275 (1983); L. Alvarez-Gaumk, J. Polchinski, and M. Wise, Nucl. Phys. B221,495 (1983);L. Ibanez and C. Lopez, Phys. Lett. 126B, 54 (1983); Nucl. Phys. B233, 545 (1984); C. Kounnas, A. Lahanas, D. V. Nanopoulos, and M. Quiros, Phys. Lett. 132B, 95 (1983); C. Kounnas, A. Lahanas, D. V. Nanopoulos, and M. Quiros, Nucl. Phys. B236,438 (1984). [2] For a review see A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. 145, 1 (1987). [3] See, e.g., S . Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Nucl. Phys. B398, 3 (1993). [4] R. Arnowitt and P. Nath, Phys. Rev. Lett. 69, 725 (1992); P. Nath and R. Arnowitt, Phys. Lett. B 287, 89 (1992); 289,368 (1992). [5] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys, Rev. D 49, 343 (1994). [6] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Texas A&M University Report Nos. CTP-TAMU-31/93 and CERN-TH.6903/93 [Phys. Lett. B (to be published)].
[7] J. L. Lopez, D. V. Nanopoulos, H. Pois, X. Wang, and A. Zichichi, Phys. Lett. B 306, 73 (1993). [8] J. L. Lopez, D. V. Nanopoulos, and H. Pois, Phys. Rev. D 47, 2468 (1993); J. L. Lopez, D. V. Nanopoulos, H. Pois, and A. Zichichi, Phys. Lett. B 299, 262 (1993). [9] J. L. Lopez, D. V. Nanopoulos, X. Wang, and 2 . Zichichi, Phys. Rev. D 48,2062 (1993). [lo] J. L. Lopez, D. V. Nanopoulos, X. Wang, and A. Zichichi, Phys. Rev. D 48,4029 11993). [ I l l J. L. Lopez, D. V. Nanopoulos, H. Pois, X. Wang, and A. Zichichi, Phys. Rev. D 48,4062 (1993). [I21 G. Altarelli and R. Barbier;, Phys. Lett. B 253, 161 (1990); G. Altarelli, R. Barbieri, and S . Jadach, Nucl. Phys. B369, 3 (1992). [13] R.Barbieri, M. Frigeni, and F. Caravaglios, Phys. Lett. B 279, 169 (1992). [I41 G. Altarelli, R. Barbieri, and F. Caravaglios, Nucl. Phys. B405, 3 (1993). See also, A. Blonde1 and C. Verzegnassi, Phys. Lett. B 311, 346 (1993). [15] G . Altarelli, R. Barbieri, and F. Caravaglios, Phys. Lett. B 314, 357 (1993). [16] S . Bertolini, F. Borzumati, A. Masiero, and G. Ridolfi, Nucl. Phys. B353, 591 (1991); N. Oshimo, ibid. B404, 20 (1993). [17] V. Barger, M. Berger, and R. J. N. Phillips, Phys. Rev.
f:')(x)=
36(x - 1l3
+
6(x - 1 l4
1nx
I
329 49 -
STRONGEST EXPERIMENTAL CONSTRAINTS ON SU(5)XU(1).. .
Lett. 70, 1368 (1993);J. Hewett, ibid. 70, 1045 (1993). [18] R. Barbieri and G. Giudice Phys. Lett. B 309, 86 (1993). [I91 J. L. Lopez, D. V. Nanopoulos, and G. Park, Phys. Rev. D 48, R974 (1993). [20] Y. Okada, Pbys. Lett. B 315, 119 (1993);R. Garisto and J. N. Ng, ibid. 315, 372 (1993). [21] E. Thorndike, Bull. Am. Phys. SOC.38, 922 (1993);CLEO Collaboration, R. Ammar et al., Phys. Rev. Lett. 71, 674 (1993). [22] J. L. Lopez, D. V. Nanopoulos, G. Park, H. Pois, and K. Yuan, Phys. Rev. D 48, 3297 (1993). [23] S. Kelley, J. L. Lopez, and D. V. Nanopoulos, Phys. Lett. B 278, 140 (1992);G. Leontaris, ibid. 281, 54 (1992). [24] V. Kaplunovsky, Nucl. Phys. B307, 145 (1988);ibid. B382, 436(E) (1992);I. Antoniadis, J. Ellis, R. Lacaze, and D. V. Nanopoulos, Phys. Lett. B 268, 188 (1991);S. Kalara, J. L. Lopez, and D. V. Nanopoulos, ibid. 269,84 (1991). [ 2 5 ] J. Ellis and D. V. Nanopoulos, Phys. Lett. llOB, 44 (1982).
365
For a recent reappraisal see J. Hagelin, S . Kelley, and T. Tanaka, Report No. MIU-THP-92-59, 1992 (unpublished). [26] S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Phys. Rev. D 47, 2461 (1993), and references therein. [27] For a recent review see J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Report No. CERN-TH.6926/93; Texas A&M University Report No. CTP-TAMU-33/93 (unpublished). [28] J. Ellis, C. Kounnas, and D. V. Nanopoulso, Nucl. Phys. B247, 373 (1984). [29] V. Kaplunovsky and J. Louis, Phys. Lett. B 306, 269 (1993). [30] J. L. Lopez, D. V. Nanopoulos, G. Park, and A. Zichichi, Report No. CERN-TH. 7078/93, Texas A&M University Report No. CTP-TAMU-68/93 (unpublished). [31] R. Barbieri, Report No. CERN-TH.6659/93 (unpublished).
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33 1
Raj Gandhi, Jorge L. Lopez, D. V. Nanopoulos, Kajia Yuan and A. Zichichi
SCRUTINIZING SUPERGRAVITY MODELS THROUGH NEUTRINO TELESCOPES
From Physical Review D 49 (1994) 3691
1994
This page intentionally left blank
333 PHYSICAL REVIEW D
VOLUME 49,NUMBER 7
1 APRIL. 1994
Scrutinizing supergravity models through neutrino telescopes Jorge L.
Raj
lope^,^.^ D. V. Nanopo~los,'-~Kajia Yuan,lt2
and A. Zichichi4 'Center for Theoretical Physics, Department of Physics, T a w A&M University, College Station, Tezas 77843-4242 'Astrqmrticle Physics Group, Houston Adwarreed Research Centet (HARC), The WoodIand.9, Tezw 77381 Centre Eumpien de Recherche8 Nucliairea Theory Division, 1.211 Geneva 23, Switzerland Centre EumpCen de Recherches Nucl6aires, Geneva, Switzerland (Received 28 September 1993)
'
'
Galactic halo neutralinos ( x ) captured by the Sun or Earth produce high-energy neutrinos as end products of various annihilation modes. These neutrinos can travel from the Sun or Earth cores to the neighborhood of underground detectors ("neutrino telescopes") where they can interact and produce upwardly moving muons. We compute these muon fluxes in the context of the minimal SU(5) supergravity model, and the no-scale and &ton SU(5)xU(l) supergravity models. At present, with the Kamiokande 90% C.L. upper limits on the flux, only a small &action of the parameter space of the SU(5)xU(l) models is accessible for m, m p e , which in turn implies constraints for the lightest chargino m a s around 100 GeV for a range of tanp values. We also delineate the regions of parameter space that would be accessible with the improvements of experimental sensitivity expected in the near future at Gran Sasso, Super-Kamiokande, and other facilities such as DUMAND and AMANDA, currently under construction. We conclude that if neutralinos are present in the halo, then this technique can be used to eventually explore more than half of the allowed parameter space of these specitic models, and more generally of a large class of supergravity models, in many ways surpassing the reach of traditional collider experiments. N
PACS number(s): 12.60.Jv, 04.65.+e, 13.15.+g, 96.40.T~
I. INTRODUCTION Even though there appear to be several indications
that low-energy supersymmetry is indeed a symmetry of nature [l],no more converts should be expected until an actual sparticle is observed experimentally or perhaps unequivocal indirect experimental evidence becomes compelling. For either discovery mode to be efficient, one must provide as accurate experimental predictions as possible. In supersymmetric models where the number of parameters is large, as in the minimal supersymmetric standard model (MSSM), easily falsifiable experimental predictions are hard t o obtain since the various parameters usually allow a myriad of possibilities. One basic step forward is to study specific supersymmetric models wherein the assumptions made are well motivated by physics at very high energies, such as grand unification, supergravity, or superstrings. In this way one can hope to test general theoretical frameworks experimentally, as opposed to just ruling out small regions of a many-dimensional parameter space. This line of thought has been in existence for almost as long as supersymmetric phenomenology has. However, it has only been since the firm establishment of the standard model at the CEFLN e+e- collider LEP that real candidates for the theory beyond the standard model have emerged from a morass of pre-LEP challengers. The aim of this paper is to study a very promisiig indirect experimental procedure to discover supersymmetry, in the context of a class of SU(5)and SU(S)xU(l)supergravity models, via interactions of upwardly moving muons in underground detectors. As we will show, this 0556282 1/94/49(7)/369 1 (13)/SO6.00
49
method of experimental exploration is quite competitive with other indirect probes which have been recently investigated in the context of this same class of models, namely, supersymmetric contributions to the one-loop electroweak LEP observables [2], to the rare radiative b + s7 decay (31, and to the anomalous magnetic moment of the muon [4]. The basic idea is that the neutralinos x (lightest linear combination of the superpartners of the photon, 2 boson, and neutral Higgs bosom), which are stable and weakly interacting massive particles (WIMP'S), are assumed t o make up the dark matter in the galactic halo, and can be gravitationally captured by the Sun or Earth [5,6], after losing a substantial amount of energy through elastic collisions with nuclei. The neutralinos captured in the Sun or Earth cores annihilate into all possible ordinary particles, and the cascade decays of these particles as well as their interactions with the solar or terrestrial media produce high-energy neutrinos as one of several end products. These neutrinos can then travel from the Sun or Earth cores to the vicinity of underground detectors, and interact with the rock underneath producing detectable upwardly moving muons.' Such detectors are
'In this paper, we do not consider the less promisiig %ontained events" in which neutrino interactions take place within the detector, since %heevent rate for such events is proportional to E,, as opposed to E: for the upwardly moving muon events. 3691
@ 1994 T h e American Physical Society
334 3692
GANDHI, LOPEZ, NANOPOULOS, YUAN, AND ZICHICHI
rightfuuy called ‘neutrino telescopes,” and the possibility of indirectly detecting various WIMP candidates has been considered in the past by several authors [7]. More recent analyses can be found in Refs. [8-141. We should emphasize that the assumption of a sizable population of neutralinos in the galactic halo is central to our work and the possible constraints which could be obtained on the models we consider. This assumption is well motivated if the overall neutralino relic abundance in the Universe is sizable (see, e.g., Ref. [15]). The recent observation of possible b q o n i c dark matter components of the halo in the form of massive astrophysical compact halo objects (MACHO’S) [16] has been argued not to help resolve the nonbaryonic halo dark matter problem [17] and therefore does not weaken our assumption of halo neutralinos. Making use of the techniques in the literature [8,10,11], our current work focuses on the predictions for the u p wardly moving muon event rates (fluxes) in two distinct supersymmetric models: namely, the minimal SU(5) supergravity model and the string-inspired SU(5)xU(1) supergravity model. In these models, because of the underlying structure of supergravity, the low-energy couplings and masses are completely determined via renormalization group equations (RGE’s) in terms of only five parameters: the three universal soft-supersymmetrybreaking parameters (ml,z, mo,A), the topquark mass (m),and the ratio of E g g s vacuum expectation values (tanp). The restriction to just five parameters is possible because of the radiative electroweak symmetry-breaking mechanism, which allows us to determine some further parameters, such as the magnitude of the Higgs mixing term p (but not its sign). This mechanism imposes further constraints on the model parameters (e.g., it requires tanp > 1) and involves the value of n~ in a fundamental way [1&20]. These features make it possible to incorporate all currently available experimental constraints, both direct and indirect, as well as some theoretical consistency conditions into a complete determination of the allowed parameter space of these models. We calculate the upwardly moving muon fluxes induced by the neutrinos from the Sun and Earth in the still-allowed parameter space of these models, and compare them with the currently most stringent 90% C.L. experimental upper bounds, obtained at Kamiokande, for neutrinos from the Sun [21] and Earth [12], respectively, i.e.,
Aiming at the next generation of underground experimental facilities, such as the Monopole, Astrophysics, and Cosmic Ray Observatory (MACRO) and other detectors at the Gran Sass0 Laboratory (221, SuperKamiokande [23], the Deep Underground Muon and Neutrino Detector (DUMAND), and the Antarctic Muon and Neutrino Detector (AMANDA) [24], where improvements in sensitivity by a factor of 2-100 are expected, we also delineate the region of the parameter space of these
49 -
models that would become accessible with an improvement of experimental sensitivity by modest factors of 2 and 12. This paper is organized as follows. In Sec. I1 we describe the minimal SU(5) and SU(5)xU(1) supergravity models. In Sec. In we outline the calculation of the c a p ture rates at the Sun and Earth, while in Sec. IV we outline the calculation of the corresponding detection rates. In Sec. V we present the results of our computations for the two supergravity models described in Sec. 11. Finally, we conclude in Sec. VI with some comments. 11. SUPERGRAVITY MODELS
We work in the context of two supergravity models based on the gauge groups SU(5) [25] and SU(5)xU(1) [“flipped SU(5)”] [26]. In these two models, spontaneous supersymmetry breaking takes place in the “hidden” sector, which manifests itself in the “observable” sector Lagrangian as a set of soft-supersymmetry-breaking terms that are universal at the respective unification scales of the models. The low-energy particle content of these two models is the same as that of the MSSM. However, mainly because of the different gauge group structure, these two models have rather different low-energy phenomenologies, which we now briefly describe in turn. An important assumption in both SU(5) and SU(5)xU(l) models is that R parity is unbroken, and therefore the lightest supersymmetric particle (the ‘‘neutralino”) is stable. This assumption is built in by construction in the minimal SU(5) supergravity model, and does not have to hold in generic non-minimal SU(5) scenarios. In the SU(5)xU(l) model, R parity is guaranteed before SU(5)xU(l) breaking, but it can be brc+ ken by precisely the vacuum expectation values which break SU(5)xU(l) [27]. However, R-parity-breaking interactions are absent in the string-derived version of this model even after SU(5)xU(1) symmetry breaking [28], and have been set to zero in the field-theory version [29] to mimic the string model result. This point was not specifically discussed in those papers, but a simple examination of the allowed couplings shows that the above assertion is true. In the minimal SU(5) supergravity model, the unification of the standard model gauge couplings takes place at a scale Mu 10’‘ GeV [30], as a combined result of the SU(5) gauge group and the minimal matter content. This property of the model has been shown to hold even after the most general low-energy and high-energy threshold corrections have been incorporated, as well as the twc+loop expressions for the renormalization group equations (RGE’s) [26]. The unified symmetry implies the existence of additional heavy particles with masses of order Mu. In particular, the Higgs sector includes at least the 24 representation to break SU(5) down to SU(3)xSU(2) x U(1), as well as the 5, 5 superfields, whose scalar doublet components ( H z ) are responsible for the electroweak gauge symmetry breaking and should be kept light, while the scalar triplet components ( H 3 ) must be heavy to avoid fast proton decay through dimension-6
-
335 S C R U T J " G SUPERGRAVITY MODELS THROUGH.
operators. It also follows that the Yukawa couplings of the bottom quark and r lepton must be unified at Mu, which when evolved to low energies entails a constraint on the relation between the ratio of Eggs vacuum expectation values (tanp) and the topquark mass (mt) (311. In our current analysis we do not enforce this constraint since it depends rather strongly on the values assumed to mt.(ma) and Q ~ ( M z ) . Perhaps the most distinguishing feature in the SU(5) model has to do with the genuinely supersymmetric proton decay via dimension-5 operators, mediated by the ex-change of superpartners of the heavy Higgs triplets ( H 3 ) . Even for sufliciently large values of MB,, it is necessary to tune the sparticle spednun so that this type of proton decay remains at an acceptable level [32]. This usually requires light charginos and neutralinos, while squarks and sleptons should be heavy. We study a generic supersymmetry-breaking scenario characterized by the universal parameters (ml,z,mo,A). It turns out that,
..
3693
because of the correlation between the relic density of the lightest neutralino ( f i x h i )and the soft-supersymmetrybreaking patterns (331, the cosmological constraint of not having a too young universe (i.e., "not overclosing the universe," fixhi < 1.0 [15]) together with the proton decay constraint dramatically reduce the parameter space of the minimal SU(5) model, and what remains are essentially points corresponding to the resonances in the neutralino pair annihilation cross section where m, N f M z , fmh [34-361, since otherwise R,hi N 100.0 in this model. In addition, the ratio ( 0 = mo/ml/z must be significantly larger than unity, and tanp must not be too large (tanj3 5 3-5) [32,37]. The assumption of universal soft supersymmetry breaking and radiative electroweak symmetry breaking, together with the proton decay and cosmological constraints, singles out a region of parameter space where simple relations among the different sparticle masses in this model hold, as follows [32,35]:
Higgs boson : 60 GeV (381 < mh < 125 GeV x&,xf : 2m,p mxp mX +I 0.3mj
- -
minimal SU(5) :
N
ma x i
sleptons :
man x i m j d o i i ,
m
i
(3)
m;
squarks :
mE
I
Here 2 3-6 depending on the value of the triplet Higgsino mass used [Mfi, < (3-10)Mul. In this paper we choose Mfi, < 3Mu (we also comment on the more relaxed case of MB, < 1OMu) and therefore ( 0 2 6 is required. The actual five-dimensional parameter space t o be explored in this paper for the minimal SU(5) model is also constrained by the LEP lower bounds on the chargino, slepton, and Higgs-boson masses (as discussed in detail in Ref. [ZO]). The allowed parameter space once all theoretical and experimental constraints have been imposed has been obtained in Ref. [37]. We should note that we have cut off the growth of the sparticle masses by a "naturalness" constraint, i.e., m i , mg 5 1 TeV. Therefore, when ma x 1 TeV (for ( 0 2 6), m i 5 450 GeV and m i 5 140 GeV, which finally implies m, 5 70 GeV. XI
-
In the string-inspired SU(5)xU(l) supergravity model, the preferred unification scale is Mu 10" GeV, as expected in the context of string theory [39,40], which has been realized by the effects of extra vectorlike matter fields at some intermediate scales [41,28]. Contrary to the minimal SU(5) model, the Higgs bosons needed to break the SU(5)xU(l) symmetry belong to the 10 and
Higgs boson : X?,Z,X: : squarks : sleptons :
10 representations, which can be easily accommodated by the simplest string models (with Kac-Moody level k = l ) , while the adjoint representation (24) may only a p pear in more complicated constructions. Because of the SU(5)xU(1) gauge group structure, the doublet-triplet splitting of the pentaplet Higgs superfields occurs in a very simple way, and the dimension-5 proton decay o p erators are automatically suppressed. In the case of the SU(5)xU(l) supergravity model, we study two stringinspired supersymmetry-breakingscenarios: (i) the noscale model [29], where mo = A = 0 [42], and (ii) the = -rnllz dilaton model [43], where % = 'mlp,A 43 [44]. Therefore, the SU(5) xU(1) model (either scenario) is quite predictive since it depends on only three parameters: w , t a n p , m l l z . Also, in these two scenarios the cosmological constraint is automatically satisfied [i.e., fixhi 5 0.25(0.90) in the no-scale (dilaton) model] [29,43]. The same theoretical and experimental constraints applied for the minimal SU(5) case yield in this instance, for the nescale (dilaton) case, the following mass relations:
60 GeV [38] < mh < 125 GeV ; 2mxp x m X a0 mXI* z 0.28mi ; m i z mg ; miR z 0.18(0.33)mg , miL z 0.30(0.41)mg .
(4)
336 3694
GANDHI, LOPEZ,NANOPOULOS, YUAN, AND ZICHICHI
Note that the squark and slepton mass relations in Eq. (3) do not reduce to those in Eq. (4), for values of €0 = 0 (no-scale) and €0 = l/& (dilaton), because of slight changes in the running of the scalar masses down t o low energies from the m e r e n t starting value of MU. The allowed three-dimensional parameter space of this model has been determined in Refs. [29,43] for the noscale and dilaton cases, respectively. The naturalness fmx: = cut imposed above implies in this case % f x 0.3mg = 150GeV. We now comment on some other features about these two models which are particularly relevant t o our current work. First, for mg 5 1 TeV the lightest neutralino can not be a pure Higgsino [45]. In fact, we have found that for most of the points in the allowed parameter spaces of both models, the lightest neutralino is a “mixed” state with large gaugino component (2 75-80%) and small Higgsino component. The gaugino component (mostly bmo) grows with m, and sometimes can be as large as 99%. In the MSSM,the “mixed” neutralinos with gaugin0 admixture less than 99% normally cannot even account for the halo dark matter [46,45,47], because the efficient annihilation via Eggs-boson exchange renders the relic density rather small ( a x h i < 0.05). This situation is improved in the two supergravity models, as a result of the one-loop radiative corrections to the masses of E g g s bosons which we have included in our analysis. Since the one-loopcorrected Higgs-boson masses are normally larger than the tree-level values, the overall effect is an enhancement of the relic density for “mixed” neutralinos. Therefore, the lightest neutralinos in these two models, although mostly “mixed” states, are still good candidates for the major component of the galactic halo. Such neutralinos are mainly captured by the Sun and Earth through their coherent (spin-independent) scattering off nuclei due to the exchange of Higgs bosons. Furthermore, in these two models the neutralinos typically ,have masses in the 20-150 GeV range, and so their c a p ture by the Earth is expected to be enhanced when the neutralino mass closely matches that of the abundant elements in the Earth’s core (Fe) and mantle (Si and Mg), while the same effect is irrelevant for the capture by the Sun [6]. We will discuss the implications of these features in Sec. V.
s
111. CAPTURE RATE
In order t o calculate the expected rate of neutrino production due to neutralino annihilation, it is necessary to first evaluate the rates at which the neutralinos are c a p tured in the Sun and Earth. Following the early work of Press and Spergel [5], the capture of WIMP’S by a massive body was studied extensively by Gould [S]. In this paper, we make use of Gould’s formula, and follow a procedure similar to that of Refs. [10,11]in calculating the capture rate. R o m Eq. (A10) of the second paper in Ref. [6], the capture rate of a neutralino of mass m, by the Sun or Earth can be written as
49
where MB is the mass of the Sun or Earth, px and fix are the local neutralino density and rms velocity in the halo, respectively, u; is the elastic scattering cross section of the neutralino with the nucleus of element i with mass mi, f; is the mass &action of element i, and X;is a kinematic factor which accounts for several important effects: (1) the motion of the Sun or Earth relative to the Galactic center, (2) the suppression due to the mismatching of m, and m;, and (3) the loss of coherence in the interaction due to the h i t e size of the nucleus (see Ref. [6] for details). In the summation in Eq. (5), we only include the ten most abundant elements for the Sun or Earth, respectively, and use the mass fraction fi of these elements as listed in Table A.l of Ref. [lo]. We choose ex = 300 kms-l, a value within the allowed range of the characteristic velocity of halo dark matter particles. To take into account the effect of the actual neutralino relic density, we follow the conservative a p proach of Ref. [lo] for the local neutralino density px: (a) px = Ph = 0.3GeV/cm3, if Qxhi > 0.05, while (b) px = (nxhg/0.05)ph, if nxh$ 5 0.05. As for ui, the dominant contribution is the coherent interaction due to the exchange of two CP-even Higgs bosons h and H and squarks, and we use the expressions (A10) and ( A l l ) of Ref. [1112to compute the spin-independent cross section for all the elements included. In addition, for c a p ture by the Sun, we also evaluate the spin-dependent cross section due t o both 2-boson exchange and squark exchange for the scattering &om hydrogen according to Eq. (A5) IEuropean Muon Collaboration (EMC) model case] of Ref. [ll].It should be noted that in all these expressions the squarks were assumed to be degenerate. In the two supergravity models that we consider here this need not be the case, although for most of the parameter space this is a fairly good approximation. Hence, we simply use the average squark mass m i in this part of the calculation. The kinematic factor X ; in Eq. (5) can be most accurately evaluated once detailed knowledge of the mass density profile as well as the local escape velocity profile is specified for all the elements. In practice, this can be done by performing a numerical integration with the physical input provided by the standard solar model or some sort of Earth model. Instead of performing such an involved calculation, we approximate the integral for each element by the value of the integrand obtained with the average effective gravitational “potential energy” +i times the integral volume. The values of 4; are taken &om Table A.l of Ref. [lo].
’We have corrected a sign error for the
(AlO)of Ref. 1111.
H m coupling in
337 SCRUTINIZING SWERGRAVITY MODELS THROUGH.. .
49
IV.DETECTION RATE We next describe the procedure employed by us to calculate the detection rate of upwardly moving muons, resulting &om the particle production and interaction sub sequent to the capture and annihilation of neutralinos in the two supergravity models. The annihilation process normally reaches equilibrium with the capture process on a time scale much shorter than the age of the Sun or Earth. We assume this is the case, so that the neutralino annihilation rate equals half of the capture rate. The detection rate for neutrineinduced upwardly moving muon events is then given by
In Eq. (6), 0;is a constant, R is the distance between the detector and the Sun or the center of the Earth, and (dN/dE,);F is the differential energy spectrum of neutrino type i as it emerges at the surface of the Sun or Earth due to the annihilation of neutralinos in the core of the Sun or Earth into final state F with a branching ratio BF. It should be noted that in Eq. (6) that i is summed over muon neutrinos and antineutrinos, and that F is summed over final states that contribute to the high-energy neutrinos.-The only relevant fermion pair final states are 77,cE, bb, and [for the SU(5) xU(1) model] tl when 9 > m+.The lighter fermions do not produce high-energy neutrinos since they are stopped by the solar or terrestrial media before they can decay [8]. The branching ratio BF can be easily calculated as the relative magnitude of the thermal-averaged product of annihilation cross section into h a l state F (OF)with the M0ller velocity U M . Since the core temperatures of the Sun and Earth are very low compared with the neutralino 4.31 X lo-’’ mass (Tsun 1.34 X lo-’ GeV; TE&h GeV), only the s-wave contributions are relevant; hence, it is enough here to use the usual thermal average expan~ 0 limit), i.e., sion up to zerworder of T/m, ( v +
-
N
Bp =
(OFVM) ~
(OtotW)
- -. aF Rot
(7)
In Eq. (7), all kinematically allowed final states contribute to atot. In addition t o all the fermion-pair final states, we have also included boson-pair final states WW, 22,and hA in our calculation of BF. Because of the parameter space constraints, these channels are not open for the minimal SU(5) model. But WW and ZZ channels are generally open in the SU(5)xU(l) model, and the hA channel also opens up for large values of tanp in the dilaton case. We should also remark that the annihilation channel into lightest CP-even Higgs pair hh is always allowed kinematically in some portion of the parameter space for both supergravity models we consider, but since its s-wave contribution vanishes, we do not include it in the calculation of Bp. However, this channel is taken into account in the calculation of the neutralino relic density, which does affect the capture rate through the scaling of local density px when R,hi < 0.05 (see
3695
Sec. KU). In addition, we have kept all the nonvanishing interference terms in the evaluation of Rot and a F . The calculation of the neutrino differential energy spectrum is somewhat involved, since it requires a reasonably accurate tracking of the cascade of the particles which result from neutralino annihilation into each of the final state F. This involves the decay and hadronization of the various annihilation products and their interactions with the media of the Sun or the Earth’s cores. In addition, at high energies, neutrinos interact with and are absorbed by solar matter, a fact that affects the spectrum. In Ref. [8], Ritz and Seckel rendered this calculation tractable by their adaptation of the Lund Monte Carlo model for this purpose. Subsequently, analytic approximations to the Monte Carlo procedure outlined in their paper were refined and employed by Kamionkowski to calculate the neutrino energy spectra &om neutralino annihilation for the MSSM in Ref. [ll].The procedure involved is described below for completeness. Since the probability for producing an underground muon that traverses the detector is proportional to the square of the neutrino energy, the primary quantity of interest is the second moment ( N z 2 ) m z ,defined as
Once the second moments are obtained, the detection rate for the neutrino-induced upwardly moving muon events (6) may be conveniently written as
i
F
where K B = 1 . 2 7 lo-’’ ~ (7.11 x for neutrinos from the Sun (Earth), and the scattering coefficients a; = 6.8 (3.1) for neutrinos (antineutrinos), while the muon range constants b; = 0.51 (0.67) for neutrinos (antineutrinos). The approximations of Re&.. (8,111 consist in obtaining expressions for the second moment without detailed knowledge of the functional form of the differential energy spectra. We now list these approximate expressions ~ ~ , for ( N Z ~as) follows. (i) Ferrnaon-pair find states (77, e,b6, tt7. The simplest case is that of fermions injected into the core of the Earth, since all interactions are negligible. We use
Here (N) and (y2) are the rest frame yield and second moments, while ( z ;) is the second moment of the fragmentation function, obtained from Table 2 and Table 3, respectively, in [8]. m f and Ei, are the mass and energy of the injected fermion. Fermions injected into the core of the Sun undergo interactions and decay before 6nal state neutrinos emerge at the surface. These are analytically approximated reli-
338 3696
GANDHI, LOPEZ, NANOPOULOS, YUAN, A N D ZICHICHI
ably for fermion injection energies relevant to our situation by [ll]
where 50 = 155/Eh (275/Ein) and a = 0.056 (0.052) for neutrinos (antineutrinos) kom c quarks, while 5 0 = 185/Eh (275/Eh) and a = 0.086 (0.082) for neutrinos (antineutrinos) from b quarks. For the top quark we have considered three diEerent masses: 130, 150, and 170 GeV in the SU(5)xU(1) model. In this case the approximations-are less reliable [ll],but better at energies below the 100 GeV scale than above it. The appropriate values here are a = 0.18 (0.14) and 20 = IlO/Eh (380/Ein) for neutrinos (antineutrinos). In the above (and in what follows), E h and other energies are taken in GeV when obtaining numerical values. The above expressions in-
Here 7;= 1.01 x GeV-l (3.8 x GeV-’) and ai = 5.1 (9.0) for neutrinos (antineutrinos). For the neutrinos from the Earth, the second moment is approximated by a simpler expression
(~2 = r(w ) -+ p , , ) ( 3 + p2)/12
(14)
due to the absence of interactions. The expressions for the ZZ final state can be similarly obtained [ll]. (iii) Higgs-boson-pair find state ( h A ) . For the annihilation final state hA, each Higgs boson S (= h , A ) with velocity ps and energy E s will decay into fermion pair ff with a branching ratio B(S -+ ff). Assuming such decays are isotropic, the second moment of this final state is
x (NzZ)pf
.
(15)
Here E f is the fermion energy, and ( N z ’ ) is ~ the second moment for the fermion pair fj(77, cE, b6, t f ) as given in Eqs. (10)-(12). V. RESULTS AND DISCUSSION
For each point in the parameter spaces of the two models described in Sec. 11, namely, the minimal SU(5) and the SU(S)xU(l) supergravity models (both the no-scale and dilaton cases), we have determined the relic abundance of neutralinos and then computed the capture rate in the Sun and Earth (as described in Sec. 111) and the resulting upwardly moving muon detection rate (as de-
49
clude the hadronization of b, c, and t quarks in the solar medium. The T lepton decays almost instantly, and does not hadronize, and so must be treated differently. The second moment for the T is approximated by
where a = 0.0204 (0.0223) and E, = 476 (599) for neutrino (antineutrino) production. (ii) Gauge-boson-pairfind states ( W W ,ZZ). The production of us and ps by a Z-boson pair and a Higgsboson pair is related to that by fermions, since the bosons primarily decay into fermions; hence, the subsequent cascade and hadronization remains the same. For the annihilation hal state W W , the W boson with velocity p and energy E will decay into p,,with a fractional width r ( W -+ pijs). The second moment for this annihilation final state for the neutrinos &om the Sun is [Ill
w,
scribed in Sec. W ) . In Figs. 1 and 2, the predicted c a p ture and detection rates in the minimal SU(5) supergravity model are shown, based on the assumption that the mass of the triplet Higgsino, which mediates dimension-5 proton decay, obeys Mfis < 3Mu. We have redone the calculation relaxing this assumption to Mfi, < IOMu, in which case the results for the muon fluxes remain qualitatively the same, except that the parameter space is opened up somewhat. The dashed lines in Fig. 2 represent the current Kamiokande 90% C.L. upper limits (1) and (2). Similarly, the predictions of the SU(5)xU(1) supergravity model are presented in Figs. 3 and 4 for the no-scale scenario, and in Figs. 5 and 6 for the dilaton scenario, again along with the Kamiokande upper limits (dashed lines). In Figs. 3-6, we have taken the representative value of mt = 150 GeV. Similar results are obtained for other values of mt. Several comments on these figures are in order. First, the kinematic enhancement of the capture rate by the Earth manifests itself in all figures as the big peaks near ~ GeV), as well as the smaller the Fe mass ( m =~ 52.0 peaks around Si mass (msi = 26.2 GeV). Second, there is a severe depletion of the rates near m, = f M z in Figs. 3-6, which is due to the decrease in the neutralino relic density. In the case of Earth capture, this effect is largely compensated by the enhancement near the Fe mass. As we mentioned in Sec. 111, in our procedure, the relic density affects the local neutralino density p, only if R,hg < 0.05, while in the minimal SU(5) model this almost never happens; therefore, the effect of the Z pole is not very evident in Figs. 1 and 2. Also, in Figs. 3-6,the various dotted curves correspond to different d u e s of t a n p , starting from the bottom curve with tan@ = 2, and increasing in steps of 2. These curves clearly show that the capture and detection rates in-
339 S C R U T I " G SUPERGRAVITY MODELS THROUGH.. .
49
crease with increasing tanp, since the dominant piece of the coherent neutralino-nucleon scattering cross section via the exchange of the lightest Higgs boson h is proportional to (1 tan'p). The capture rate decreases with increasing m,, since the scattering cross section falls off as m i 4 and mh increases with increasing %. It is expected that the detection rate in general also decreases for a large value of %, since it is proportional to the capture rate. However, the opening of new annihilation channels, such as the WW, 22, and hA channels in the SU(S)xU(l) model, could have two compensating effects on the detection rate: (a) the presence of a new channel to produce high-energy neutrinos which leads to an enhancement of the detection rate and (b) the decrease of the branching ratios for the fermion-pair channels, which
+
3697
makes the neutrino yield from rt, cE, and a smaller and hence reduces the detection rate. Therefore, these new annihilation channels could either increase or decrease the detection rate, depending which of these two effects wins over. We found that, for small values of tanp and p < 0, the WW channel can become dominant if open, basically because in this case the neutralino contains a rather large neutral W-in0 component. This explains the distortion of the detection rate curves in the p < 0 half of Figs. 4 and 6. The effect of the 22 channel turns out to be negligible in the SU(5)xU(1) model, since neutralinos with mx > M z have very small Higgsino components. The same argument applies to the hA channel which sometimes opens up in the dilaton case for rather large values of tanp.
10-1 f
10-2
FIG. 2. The upwardly moving muon flux in underground detectors originating
10-5 10-6 10
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neutralino annihilation in the Sun and Earth, as a function of the neutralino mass in the minimal SU(5) supergravity model. The dashed lines represent the present Kamiokande 90% C.L.experimental upper limits. &om
340 GANDHI, LOPEZ,NANOPOULOS, WAN, AND ZICHICHI
3698
25
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smaller due t o the smaller relic density. It is clear that at present the experimental constraints from the “neutrino telescopes” on the parameter space of the two supergravity models are quite weak. In fact, only the Kamiokande upper bound from the Earth can be used to exclude regions of the parameter space with m, FZ mFe for both models, in particular for the SU(5)xU(l) model, due to the enhancement effect discussed above. However, it is OUT belief that the results presented in this paper will be quite useful in the future, when improved sensitivity in underground muon detection rates become available. An improvement in experimental sensitivity by a factor of 2 should be easily possible when MACRO [22] goes into operation, while a tenfold improvement is envisaged when Super-Kamiokande 1231 announces its results sometime by the end of the decade. More dramatic improve-
-
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7
10-1 10-2
L
7% 10-3 ..E-.
$
c
10-4 10-5 10-6
25
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100
m x (cev)
FIG. 3. The neutralino capture rate for the Sun and Earth as a function of the neutralino mass in the no-scale SU(S)xU(l) supergravity model. The representative value of r~ = 150 GeV has been used. Note the depletion of neutralinos in the halo near the Z resonance, and the enhancement in the Earth capture rate near the iron nucleus mass (52.0 GeV).
mx (Gev)
In the dilaton case, for large values of t a n p the CPodd Higgs boson A can be rather light, and the presence of the A pole when m, f m makes ~ the relic density very small. Qxhi as a function of m, is first lower than 0.05, it increases with m,, and eventually reaches values above 0.05, when neutralinos move away from the A pole. Thus, the capture and detection rates also show this behavior, which can be seen as the few “anomalous” lines in Figs. 5 and 6. For lower values of tanp, Qxhi < 0.05, and there is no such effect. In the minimal SU(5) model, since the allowed points include different supersymmetry-breaking scenarios and several values of mt and tanp, all these features are blurred. Nonetheless, in the same range of m, and for same values of t a n P and mt, we have found the results of these two models comparable, with the rates in the SU(5)xU(l) model slightly
25
49 -
m, ( c 4
FIG. 4. The upwardly moving muon flux in underground detectors originating from neutralino annihilation in the Sun and Earth, as a function of the neutralino mass in the no-scale SU(5) xU(1) supergravity model. The representative value of mt = 150 GeV has been used. The dashed lines represent the present Kamiokande 90% C.L.experimental upper limits.
34 1 SCRUTINIZING S W E R G R A V m MODELS THROUGH.. .
2 lL>o
-
;’
v
:
u”
3699
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1026 1025
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FIG. 5. Same as Fig. 3 but for the dilaton SU(5)xU(1) supergravity model.
1014
0
1013 1012
100
75 m,
m, (GeV)
have been imposed. It can be seen here that the small = 100 GeV and a variety of values voids of points for of tan@are excluded by the constraint from the “neutrino telescopes.” In these figures we have marked by crosses the points in the (max:, tan@)plane that MACRO should be able to probe (assuming an increase in the sensitivity by a factor of 2) for the no-scale and dilaton scenarios, respectively. Finally, Figs. 9 and 10 show that, using this indirect technique, Super-Kamiokande can cover nearly half of the parameter space of the SU(5)xU(1) model, assuming that an improvement by a factor of about 12 can be achieved. As expected, the constraints from future “neutrino telescopes” will be strictest for large values of tanp. To keep the above results in perspective, we now discuss some effects that have not been taken into account
ments in the sensitivity (by a factor of 20-100) may be expected hom DUMAND and AMANDA [24],currently under construction. In addition, as recently argued [48], we think that perhaps the full parameter space of a large class of supergravity models, including the two specific ones considered here, may only be convincingly probed by a detector with an effective area of 1 km2.It is interesting to note that, with a sensitivity improvement by a factor of 100,a large portion of the p < 0 half parameter space of the minimal SU(5) model can be probed. Unfortunately, the remaining portion, with fluxes below can hardly be explored by underground experiments in the foreseeable future. For the SU(5)xU(1) supergravity models, in Figs. 7 and 8 we have plotted the allowed points in $he ( m X ptanp) space. These points are those obtained originally in Refs. (29,431, such that the neutrino telescope constraint is also satisfied; no other constraints
nx.
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p>o
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L
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FIG.6. Same as Fig. 4 but for the dilaton SU(5)xU(1) supergravity model.
3700
GANDHI, LOPEZ, NANOPOULOS, YUAN, AND ZICHICHI
30
25
a c
no-scale S U ( 5 ) X U ( l ) NT allowed region ( x = r 2 )
PL>o p a 0
'"3
"""'"'I"
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,,,, , , , , , , , ,
, ,
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;;
150 200 md (cev)
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.....
...............
........................
a
c
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5
5
49 -
FIG. 7. The allowed parameter space of the nwscale SU(5)xU(1) supergravity model [in the (rn,,,, tanp) plane] after the present "neutrino telescopes" (NT's) constraint has been applied. Two values of mt (130,150 GeV) have been chosen. The crosses denote those points which could be probed with an increase in sensitivity by a factor of 2.
d
0
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150 200 m,: (cev)
0
250
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150 200 m,: (cev)
in our analysis. As we have shown, for the two supergravity models, the capture of the neutralinos by the Earth is more important than that by the Sun. In the Earth case, we have considered only the primary direct capture of the neutralinos bom the galactic halo, in which a neutralino is trapped by the Earth's gravitational field only when its velocity falls below the escape velocity at a point inside the Earth, as a consequence of its interaction with the nuclei around. However, there exists yet another mechanism by which neutralinos can be captured by the Earth, namely, the secondary indirect capture, fkst studied by Gould [6]. In this mechanism, a neutralino fkst loses only enough energy to be bound in a solar orbit ("orbit capture"), but not enough to be directly captured by the Earth. The orbit-captured neutralinos further weakly interact with the nuclei in the Earth, and a fraction of
250
them subsequently can be indirectly captured into the Earth's core. It was found by Gould that, for Dirac neutrinos of mass 10-80 GeV, direct and indirect capture by the Earth are of the same order of magnitude, and the kinematic enhancement of the total capture rate becomes enlarged and broadened [6].We believe that this result should also apply to the case of neutralino c a p ture, which means that the Earth capture rate that we have computed in this work may have been significantly underestimated. Although to our knowledge the indirect capture was also not considered in recent works that dealt with Earth capture [10,12-141,we feel that this important mechanism should be taken into account in future analyses of this type. In fact, bom Figs. 7 and 8, we see that even before MACRO, interesting constraints may already be extracted from the current Kamiokande up-
50 40
40 -.
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.......... .-. .......................................... ...................................................... ..........................................
1,'
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I
"
"
'1'
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1
'
"
I
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'
md (cev)
FIG. 8. Same as Fig. 7 but for the dilaton
343 SCRUTINIZING SUPERGRAVITY MODELS THROUGH..
49
30
'
I
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,
I
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______m.. ...... 20 15
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t 7 " + ~ " ' 1' ~' ' I ' , j
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150 200 m,: (GeV)
250
per limits, had we included the indirect capture in our calculation. Recently, after the calculations in this paper had been completed, we received Refs. [49,50]in which some other issues of particle physics relevant to our work have been addressed. Given the fact that the neutralino in the two supergravity models we study is mostly a "mixed" state, in particular for m, m p e , according to Fig. 7 of Ref. [49], we expect that the results of Ref. [49], when they become appreciable, could lead t o a shift of our detection rates in both directions by about 10% or less. We have not redone our calculations of the capture rate using the cross section of Ref. [50]. The inclusion of the new neutralino-gluon scatterings [50], when they become appreciable, would increase the chance that neutralinos can be captured by both the Sun and the Earth, barring
..... 50
40
100 I
I I
I
I
150 200 m,: ( c e v ) I I
I
-
I I
I
I
250
50
81k;,
F ....................
50
FIG. 9. Same as Fig. 7 but the crosses now denote points which could be probed with an increase in sensitivity by a factor of 12.
the normally small interference effects which sometimes could render the total cross section of Ref. [50] smaller than what it would be without these new scatterings [51]. Therefore, the effect of the results of Ref. [50] on the detection rate could somehow enhance or compensate that of Ref. [49], depending whether the latter is an increase or decrease of the detection rate. However, even as a conservative estimate, we do not expect that the combined effect of Refs. [49,50] would alter our results quantitatively by more than 20%, a change not large enough to affect OUT conclusions qualitatively. We have also looked into the possibility that the high-energy neutrino flux may be altered by MikheyevSmirnov-Wolfenstein (MSW) oscillations [52] in the Sun. This issue has been addressed in a recent paper [53]. From their results we conclude that for the neutrino en-
N
50
3701
I , < I ~ I \ I I I I I ............. II
'i 25
.
..................................... 150 m,: (
100
1 1 , ~ ~
250
200
c4
I , I I
I
FIG. 10. Same as Fig. 9 but for the dilaton SU(5)xU(l) supergravity model.
,,,, 1501
.
.................................
n
........................................
,,,*
50
100
I
,,1
'
5
I
150 200 m,: (GeV)
j
250
'
344 3102
GANDHE, LOPEZ, NANOPOULOS, YUAN, AND ZICHICHI
ergy range relevant to this work, v,, to vc oscillations are negligible. The v,, flux may also be altered by v,, to vr vacuum oscillations during passage &om the Sun to the Earth. This effect will be significant only if the v,,-vr mixing angle is large. Large mixing is disfavored by the general grand unified theory (GUT) based seesaw arguments [54]. An analysis based on these considerations coupled with phenomenological arguments in the context of the flipped SU(5) model [55] supports a value for sin’ 26 u lo-*, which is much too small to affect the flux values predicted here.
VI. CONCLUSIONS
We have explored the possibility of detecting supersymmetry indirectly through an anomalously large flux of upwardly moving muons in underground detectors or “neutrino telescopes.” These muons originate &om highenergy neutrino interactions in the rock below the detector, and these neutrinos result from annihilation of neutralinos in the Earth or Sun cores. The latter would have been captured by these heavenly bodies if they constitute an important part of the galactic halo-an important assumption which should not be overlooked. The present day experimental upper bounds on the muon flux are only w e d y constraining. In fact, this is mostly because the large possible fluxes for light neutralinos (see for example Fig. 4 for small m x ) have already been ruled out by the LEP lower bounds on the neutralino mass. Nonetheless, there is a region of parameter space with mx: u 100 GeV (corresponding to
[l] See, e.g., H. Haber, in Recent Advances in the Superworld, Proceedings of the HARC Workshop, edited by J.
L. Lopez and D.V. Nanopoulos (World Scientific, Singapore, 1994). [2]J.L. Lopez, D.V.N ~ ~ o D o ~ G.T. ~ o s .Park. H. Pois, and K. Yuan, Phys. Rev. D-48,3297 (1993). J.L. Lopez, D.V.Nanopoulos, and G.T. Park, Phys. Rev. D 48,R974 (1993);J. L. Lopez, D.V.Nanopoulos, G.T. Park, and A. Zichichi, ibid. 49,355 (1994). J.L. Lopez, D.V.Nanopoulos, and X. Wang, Phys. Rev. D 49,366 (1994). W.H.Press and D.N. Spergel, Astrophys. J. 296, 679 (1985). A. Gould, Astrophys. J. 321, 560 (1987); 321, 571 (1987);528,919 (1988); 388,338 (1992). J. Silk, K. Olive, and M. Srednicki, Phys. Rev. Lett. 55, 257 (1985); T. Gaisser, G. Steigman, and S. Tilav, Phys. Rev. D 34,2206 (1986); J. Hagelin, K. Ng, and K. Olive, Phys. Lett. B 180,375 (1987); M. Srednicki, K. Olive, and J. Silk, Nucl. Phys. B279,804 (1987);K. Ng, K. Olive, and M. Srednicki, Phys. Lett. B 188,138 (1987);K.Olive and M. Srednicki, ibid. 205,553 (1988); L. Krauss, M. Srednicki, and F. Wilczek, Phys. Rev. D 35, 2079 (1986);K. F’reese, Phys. Lett. 167B,295 (1986). S. Ritz and D.Seckel, Nucl. Phys. BS04,877 (1988). G.F. Giudice and E. Roulet, Nucl. Phys. BS16, 429 ’
49 -
m, u m ~ =and ) a range of values of tanp, which is excluded at the 90% C.L. However, expected increases in experimental sensitivity in the next few years should turn this technique into a very efficient way of probing the parameter space of the specific supergravity models considered here, even surpassing the reach of traditional direct detection collider experiments. We should remark that even though our explicit computations apply only to the SU(5) and SU(5)xU(1) models, the correlations among sparticle and Higgs-boson masses, which play such a fundamental role in the quantitative results, are common to a large class of supergravity models with radiative electroweak symmetry breaking [18-201. In this respect, the specific models considered here include values of €0 = mo/ml/z which are small (€0 = 0, no-scale), moderate (€0 = l/& dilaton), and large [to >> 1, minimal SU(5)], and therefore span the whole allowed range. Thus, our results (i) represents a significant sampling of what would be obtained using an arbitrary selection of parameters in a generic supergravity model, and (ii) will allow to select the correct supergravity model when experimental data start to restrict the parameter space.
ACKNOWLEDGMENTS
This work has been supported in part by DOE Grant No. DEFG05-91-ER-40633. The work of R.G. and K.Y. has been supported by the World Laboratory. The work of J.L. has been supported by the SSC Laboratory.
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B. Pattison, Nucl. Phys. B (to be published). [25] For reviews see, e.g., R. Arnowitt and P. Nath, Applied N=l Supergmvity (World Scientific, Singapore, 1983); H. P. Nilles, Phys. Rep. 110, 1 (1984); J.L. Lopez,
D.V. Nanopoulos, and A. Zichichi, Report No. CERNTH.6934/93 (unpublished); Texas A&M University Report No. CTP-TAMU-34/93 (unpublished). [26] For a recent review, see J.L. Lopez, D.V. Nanopoulos, and A. Zichichi, in From Superstrings to Supergmvity, Proceedings of the INFN Eloisatron Project 26th Workshop, Erice, Italy, 1992, edited by M.J. DufT, s. Ferrara, and R.R. Khuri (World Scientific, Singapore, 1993). [27] S.P. Martin, Phys. Rev. D 46, 2769 (1992). [28] J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Nucl. Phys. B399, 654 (1993). [29] J.L. Lopez, D.V. Nanopoulos, and A. Zichichi, Phys. Rev. D 49, 343 (1994). 1301 J. Ellis, S. Kelley, and D.V. Nanopoulos, Phys. Lett. B 249, 441 (1990); 260, 131 (1991); 287, 95 (1992); Nucl. Phys. B373, 55 (1992); P. Langacker and M.-X. Luo, Phys. Rev. D 44, 817 (1991); U. Amaldi, W. de Boer, and H. Fiirstenau, Phys. Lett. B 260, 447 (1991);
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F. Anselmo, L. Cifarelli, and A. Zichichi, ibid. 105, 1335 (1992); 105,1357 (1992); G. Ross and R. Roberts, Nucl. Phys. B377, 571 (1992); P. Langacker and N. Polonsky, Phys. Rev. D 47, 4028 (1993); R. Barbieri and L. Hall, Phys. Rev. Lett. 68, 752 (1992); L. Hall and U. Sarid, ibid. 70, 2673 (1993). [31] J. Ellis, S. Kelley, and D.V. Nanopoulos, Nucl. Phys. B373, 55 (1992); S. Kelley, J.L. Lopez, and D.V. Nanopoulos, Phys. Lett. B 274,387 (1992); S. Dimopoulos, L. Hall, and S. Raby, Phys. Rev. Lett. 68, 1984 (1992); Phys. Rev. D 45, 4192 (1992); G. Anderson, S. Dimopoulos, L. Hall, and S. Raby, ibid. 47, R3702 (1992); V. Barger, M. Berger, and P. Ohman, ibid. 47, 1093 (1993); P. Langacker and N. Polonsky, ibid. 49,1454 (1994). [32] R. Arnowitt and P. Nath, Phys. Rev. Lett. 69, 725 (1992); P. Nath and R. Arnowitt, Phys. Lett. B 287, 89 (1992). [33] J.L. Lopez, K. Yuan, andD.V. Nanopoulos, Phys. Lett. B 267, 219 (1991); S. Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois, and K. Yuan, Phys. Rev. D 47, 2461 (1993). (341 J.L. Lopez, D.V. Nanopoulos, and A. Zichichi, Phys.
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Proceedings of the Workshop, Stony Brook, New York, 1979, edited by P. van Nieuwenhuizen and D. Fkeedman (North-Holland, Amsterdam, 1979), p. 315. 1551 J. Ellis, J.L. Lopez, and D.V. Nanopoulos, Phys. Lett. B 292, 189 (1992); J . Ellis, J.L. Lopez, D.V. Nanopoulos, and K. Olive, ibid. 308, 70 (1993).
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341
Jorge L. Lopez, D.V. Nanopoulos, Gye T. Park and A. Zichichi
NEW PRECISION ELECTROWEAK TESTS OF
SU(5) x U( 1) SUPERGRAVITY
From Physical Review D 49 ( I 994) 4835
1994
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349 1 MAY 1994
VOLUME 49, NUMBER 9
PHYSICAL REVIEW D
New precision electroweak tests of SU(5) X U(1) supergravity Jorge L. Lopez,'** D. V. Nanopo~los,'-~ Gye T.Park,"' and A. Zichichi4 'Centerfor Theoretical Physics, Department of Physics, Tam ABM University, College Station, Texas 77843-4242 2AstroparticlePhysics Group, Houston Advanced Research Center (HARC), The Mitchell Campus, The Woodlands, Texas 77381 'CERN, Theory Division, 1211 Geneva 23, Switzerland 'CERN, 1211 Geneva 23, Switzerland (Received 15 November 1993) We explore the one-loop electroweak radiative corrections in SU(5)XU(1) supergravity via explicit calculation of vacuum-polarization and vertex-correctioncontributions to the el and eb parameters. Experimentally, these parameters are obtained from a global fit to the set of observables r,,rb, A h , and Mw/Mz.We include q2-dependenteffects, which induce a large systematic negative shift on el for light chargino masses ( m 570 GeV). The (nonoblique) supersymmetric vertex corrections to 2-bb;
*
XI
which define the Eb parameter, show a significant positive shift for light chargino masses, which for taI$=2 can be nearly compensated by a negative shift from the charged Higgs contribution. We conclude that, at the 90% C.L., for m,5 160 GeV the present experimental values of el and eb do not constrain in any way SU(5)XU(l)supergravity in both no-scale and dilaton scenarios. On the other hand, for m, 2 160 GeV the constraints on the parameter space become increasingly more strict. We demonstrate this trend with a study of the m,= 170 GeV case, where only a small region of parameter space, with ta@S4, remains allowed and corresponds to light chargino masses ( m + S70 GeV). Thus XI
SU(5)XU(1) supergravity combined with high-precision CERN LEP data would suggest the presence of light charginos if the top quark is not detected at the Fermilab Tevatron. PACS number(s):12.15.Ji, 04.65.+e, 12.60.Jv, 14.80.L~
I. INTRODUCTION
Since the advent of the CERN e + e - collider LEP, precision electroweak tests have become rather deep probes of the standard model of electroweak interactions and its challengers. These tests have demonstrated the internal consistency of the standard model, as long as the yet-to-be-measured top-quark mass ( m , ) is within certain limits, which depend on the value assumed for the Higgs-boson mass (m,): m,=135*18 GeV for m,-60 GeV and m,= 1 7 4 f l 5 GeV for m H 1 TeV (for a recent review see, e.g., Ref. [l]). In the context of supersymmetry, such tests have been performed throughout the years within the minimal supersymmetric standard model (MSSM) [2-51. The problem with such calculations is well known but usually ignored-there are too many parameters in the MSSM (at least twenty)-and therefore it is not possible to obtain precise predictions for the observables of interest. In the context of supergravity models, on the other hand, any observable can be computed in terms of at most five parameters: the top-quark mass, the ratio of Higgs vacuum expectation values ( t a d ) , and three soft-supersymmetry-breaking parameters universal ( n ~ , , ~m,, , A ) [6]. This implies much sharper predictions for the various quantities of interest, as well as numerous correlations among them. Of even more experimental interest is SU(5) XU( 1) supergravity where string-inspired Ansatze for the soft-supersymmetrybreaking parameters allow the theory to be described in terms of only three parameters: m,,tar@, and m g [7].
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0556-2821/94/49(9)/4835(7)/$06.00
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Precision electroweak tests in the no-scale [8] and dilaton [9] scenarios for SU(5)XU( 1 ) supergravity have been performed in Refs. [10,11], using the description in terms of the parameters introduced in Refs. [12,13]. In this paper we extend these tests in two ways: first, we include for the first time the eb parameter [4] which encodes the one-loop corrections to the Z - b 6 vertex, and second, we perform the calculation of the el parameter in a new scheme [4], which takes full advantage of the latest experimental data. The calculation of Eb iS of particular importance, since in the standard model, of the four parameters at present only Eb falls outside the 10 experimental error (for m,> 120 GeV [4,14]). This discrepancy is not of great statistical significance, although the trend should not be overlooked, especially in the light of the much better statistical agreement for the other three parameters. Within the context of the standard model, another reason for focusing attention on the eb parameter is that, unlike the el parameter, E* provides a constraint on the top-quark mass which is practically independent of the Higgs-boson mass. Indeed, at the 95% C.L., the limits reon Eb require m,< 185 GeV, whereas those from quirem,<177-198GeVformH-100-l000GeV [14]. In supersymmetric models, the weakening of the eldeduced m , upper bound for large Higgs-boson masses does not occur (since the Higgs boson must be light) and both and Eb are expected to yield comparable contraints. In this context, it has been pointed out [5] that if certain mass correlations in the MSSM are satisfied, then the prediction for E* will be in better agreement with the 4835
01994 The American Physical Society
350 LOPEZ, NANOPOULOS, PARK,AND ZICHICHI
4836
data than the standard model prediction is. However, the opposite situation could also occur Le., worse agreement), as well as negligible change relative to the standard model prediction (when all supersymmetric particles are heavy enough). We show that this three-way ambiguity in the MSSM prediction for eb disappears when one considers SU(5 ) XU( 1) supergravity in both no-scale and dilaton scenarios. The SU(5 ) XU( 1) supergravity prediction is practically always in better statistical agreement with the data (compared with the standard model one). This study shows that at the 90% C.L., for m, S 160 GeV the present experimental values of and eb do not constrain SU(5 ) XU( 1) supergravity in any way. On the other hand, for rn, 2 160 GeV the constraints on the parameter space become increasingly more strict. We demonstrate this trend with a study of the m, = 170 GeV case, where only a small region of parameter space, with tax$24, remains allowed and corresponds to a light supersymmetric spectrum, and in particular light chargino masses (m 5 70 GeV). Thus SU(5 )XU( 1) supergravity
*
XI
combined with high-precision LEP data would suggest the presence of light charginos if the top quark is not detected at the Fermilab Tevatron.
II. SU(5 ) XU( 1 ) SUPERGRAVlTY Our study of one-loop electroweak radiative corrections is performed within the context of SU(5)XU( 1) supergravity [7]. In addition to the several theoretical string-inspired motivations that underlie this theory, of great practical importance is the fact that only three parameters are needed to describe all their possible predictions. This fact has been used in the recent past to perform a series of calculations for collider [15,16] and rare [17,10,1I] processes within this theory. The constraints obtained from all these analyses should help sharpen even more the experimental predictions for the remaining allowed points in parameter space. In SU(5 ) XU( 1 ) supergravity, gauge coupling unification occurs at the string scale f i l s GeV 171, because of the presence of a pair of 10,lO representations with intermediate-scale masses. The three parameters alluded to above are: (i) the top-quark mass ( m ,1, (ii) the ratio of Higgs vacuum expectation values ( t a d ) , which satisfies 1 S tar$S40, and (iii) the gluino mass, which is cut off at 1 TeV. This simplification in the number of input parameters is possible because of specific stringinspired scenarios for the universal soft-supersymmetrybreaking parameters ( m o , m A ) at the unification scale. These three parameters can be computed in specific string models in terms of just one of them [18]. In the no-scale scenario one obtains mo= A =0, whereas in the dilaton scenario the result is m0=( l / f i ) m , / 2 , A = - m I n . After running the renormalization group equations from high to low energies, at the low-energy scale the requirement of radiative electroweak symmetry breaking introduces two further constraints which determine the magnitude of the Higgs mixing term p, althoueh its sim remains undetermined. Finally, all the
49 -
TABLE I. The approximate proportionality coefficients to the gluino mass for the various sparticle masses in the two
supersymmetry-breakingscenarios considered.
-
eR, P R V
2x?, xL
xf
FLY P L
B g
No scale
Dilaton
0.18 0.18-0.30 0.28 0.30 0.97 1.M)
0.33 0.33-0.41 0.28 0.41 1.01 1.oo
known phenomenological constraints on the sparticle masses are imposed (most importantly, the chargino, slepton, and Higgs-boson mass bounds). This procedure is well documented in the literature [ 191 and yields the allowed parameter spaces for the no-scale [8] and dilaton [9] scenarios. These allowed parameter spaces in the three defining variables (m,,tar$,mg ) consist of a discrete set of points for three values of m, (ml=130,150,170 GeV), and a discrete set of allowed values for t a d , starting at 2 and running (in steps of two) up to 32 (46)for the no-scale (dilaton) scenario. The chosen lower bound on t a d follows from the requirement by the radiative breaking mechanism of tar@> 1, and because the LEP lower bound on the lightest Higgs-boson mass (m,, 2 60 GeV [16) is quite constraining for 1
*
III. ONE-LOOP ELECTROWEAK RADIATIVE CORRECTIONS AND THE NEW E PARAMETERS
There are different schemes to parametrize the electroweak (EW) vacuum polarization corrections [20-22,121. It can be shown, by expanding the vacuum polarization tensors to order q2, that one obtains three independent physical parameters. Alternatively, one can show that upon symmetry breaking three additional terms appear in the effective Lagrangian [22]. In the (S, i“, U )scheme [21], the deviations of the model predictions from the SM predictions (with fixed SM values for m ,,m H S M) are considered as the effects from “new physics.’’ This scheme is only valid to the lowest order in q2, and is therefore not applicable to a theory with new, light (-Mz)particles. In the E scheme [13,4], on the other hand, the model predictions are absolute and also valid up to higher orders in q 2 , and therefore this scheme is more applicable to the EW precision tests of the MSSM
35 1 NEW PRECISION ELECTROWEAK TESTS OF SU(S)XU(l). . .
49
[3] and a class of supergravity models [lo]. There are two different E schemes. The original scheme [13] was considered in our previous analyses [10,11], where E ~ , are ~ ,defined ~ from a basic set of observables rl,AL and M w / M z . Because of the large m,dependent vertex corrections to rb, the E ~ , parameters ~ , ~ and rb can be correlated only for a fixed value of m,. Therefore, rtot, rhadron, and r bwere not included in Ref. [13]. However, in the new E scheme, introduced recently in Ref. [4], the above difficulties are overcome b_y introducing a new parameter E,, to encode the Z+bb vertex corrections. The four E'S are now defined from an enlarged set of I-,, rb, and M w / M z without even specifying m,. In this work we use this new E scheme. Experimentally, including all LEP data allows one to determine the allowed ranges for these parameters [ 11:
4837
symmetric models at the 90% C.L. [10,5], we discuss below only and eb. The expression for el is given as [3] (2)
where are the combinations of vacuum polarization amplitudes, (3)
Ah,
ePt=(-0.3f3.2)X10-3
, ~$p'=(3.1f5.5)XlO-~ . (1)
Since among
E ~ , only ~ , ~
provides constraints in super-
and the q2#0 contributions F i j ( q 2 )are defined by
~ ~ ( q 2 ) = ~ ~ ( 0 ) + q z F.i j ( q 2 )
(5)
The 6 g A in Eq. (2) is the contribution to the axial-vector vertex from form factor at q 2 = M ; in the Z - t l ' l proper vertex diagrams and fermion self-energies, and
FIG 1. The predictions for the (top row) and cb (bottom row) parameters versus the chargino mass in the no scale SU(5 )XU(1 ) supergravity scenario for m,=170 GeV. In the top (bottom)row, points between (above) the horizontal line(s) are allowed at the 90% C.L. The solid curve (bottom row) represents the tan/3=2 line. E,
352 LOPEZ. NANOPOULOS, PARK, AND ZICHICHI
4838
6 G , , comes from the one-loop box, vertex, and fermion self-energy corrections to the p-decay amplitude at zero external momentum. These nonoblique SM corrections are non-negligible, and must be included in order to obtain an accurate SM prediction. As is well known, the SM contribution to el depends quadratically on rn, but only logarithmically on the SM Higgs-boson mass (rnH ). In this fashion upper bounds on m, can be obtained which have a non-negligible rnH dependence: up to 20 GeV stronger when going from a heavy ( = 1 TeV) to a light (=100 GeV) Higgs boson. It is also known (in the MSSM) that the largest supersyrnmet_riccontributions to are expected to arise from the T-b sector, and in the limiting case of a very light top squark, the contribution is comparable to that of the t-b sector. The remaining squark, slepton, chargino, neutralino, and Higgs sectors all typically contribute considerably less. For increasing sparticle masses, the heavy sector of the theory decouples, and only SM effects with a light Higgs boson survive. (This entails stricter upper bounds on rn, than in the SM, since there the Higgs boson does not need to be light.) However, for a light chargino (rn -+MZ ), a Z-
49 -
tion to eb depends quadratically on m,. In supersymmetric models there are additional diagrams involving Higgs bosons and supersymmetric particles. The charged Higgs contributions have been calculated in Refs. [25-271 in the context of a nonsupersymmetric two Higgs doublet model, and the contributions involving supersymmetric particles in Refs. [23,28]. Moreover, Eb itself has been calculated in Ref. [27]. The additional supersymmetric contributions are: (i) a negative contribution from charged-Higgs-boson-top-quark exchange which grows as rn:/tan’b for tar$<<m,/rnb; (ii) a positive contribution from chargino- top-squark exchange which in this case grows as rn:/sin2g; and (iii)a contribution from neutralino(neutra1-Higgs-bosonl-bottom-quark exchange which grows as rnitan’p and is negligible ex-
*
XI
wave-function renormalization threshold effect can introduce a substantial q ’-dependence in the calculation, i.e., the presence of e 5 in Eq. (2) [3]. The complete vacuum polarization contributions from the Higgs sector, the supersymmetric chargino-neutralino and sfermion sectors, and also the corresponding contributions in the SM have been included in our calculations [lo]. Following Ref. [4], E , is defined from rb, the inclusive partial width for Z -+ bb; as
Eb
-4
-2
0
2
4
6
El
with 2
Pb=
I
1--
,
Eb
-8
’
-4
I -2
2
0
4
6
El Here Fk is an effective sin’6, for on-shell Z,and eb is closely related to the real part of the vertex correction to Z-bb; denoted in the literature by Vb and defined explicitly in Ref. [23]. In the SM, the diagrams for V, involve top quarks and W* bosons [24], and the contribu-
FIG.2. The correlated predictions for the E , and eb parameters in units of lo-’ in the no scale S U ( S ) X U ( l )supergravity scenario. The ellipse represents the lo contour obtained from all LEP data. The values of m , are as indicated.
353 4839
NEW PRECISION ELECTROWEAK TESTS OF SU(5)XU(1). . .
49
cept for large values of tar$ (i.e., t a n a > m , / m b ) [the contribution (iii)has been neglected in our analysis].
IV. RESULTS AND DISCUSSION In Figs. 1-4 we show the results of the calculation of and E~ (as described above) for all the allowed points in SU(5 ) XU(1) supergravity in both no-scale and dilaton scenarios. Since all sparticle masses nearly scale with the gluino mass (or the chargino mass), it suffices to show the dependences of these parameters on, for example, the chargino mass. Table I can be used to deduce the dependences on any of the other masses. We only show the explicit dependence on the chargino mass (in Figs. 1 and 3) for the case m,=170 GeV, since for rn,=130, 150 GeV there are no constraints at the 90% C.L. However, in the correlated plots (Figs. 2 and 4) we show the results for all three values of m,. The qualitative results for are similar to those obtained in Refs. [10,11]using the old definition of el. That is, for light chargino masses there is a large negative shift due to a threshold effect in the 2-wave-function renorE,
malization for m k-++Mz (as first noticed in Ref. [3]).
-
XI
As soon as the sparticle masses exceed 100 GeV the result quickly asymptotes to the standard model value for a light Higgs-boson mass ( 5100 GeV). Quantitatively, the enlarged set of observables in the new E scheme shifts the experimentally allowed range somewhat, and the bounds become slightly weaker than in Refs. [lO,ll]. These remarks apply to both no-scale and dilaton scenarios. In the case of E ~ the , results also asymptote to the standard model values for large sparticle masses as they should. Two competing effects are seen to occur: (i) a positive shift for light chargino masses, and (ii) a negative shift for light charged-Higgs-boson masses and small values of tan/?. In fact, the latter effect becomes evident in Figs. 1 and 3 (bottom rows) as the solid curve corresponding to tar$=2. What happens here is that the charged Higgs contribution nearly cancels the chargino contribution [23], making Eb asymptote much faster to the SM value. We also notice from Fig. 3 (bottom row) that there are lines of points far below the solid curve corresponding to t a n g 7 2 in the dilaton scenario. These correspond to
FIG. 3. The predictions for E , (top row) and E~ (bottom row) parameters versus the char-
the
mass in dilaton SU( 5 ) XU( 1 ) supergravity scenario for m,=170 GeV. In the top (bottom row), points between (above) the horizontal line(s) are allowed at the 90% C.L. The solid curve (bottom row) represents the tar@=2 line. gin0
-0.w 4.0045
4.005 4.0055
%
4.M
4.M5
4.m 4.w5 4.W8 50
IW
1%
ZW
250
3W
50
1W
I50
UIO
250
Mo
354 LOPEZ, N A N O P O W S , PARK, AND ZICHICHI
4840
large t a d ( k m,/rnb ) for which the charged Higgs diagram gets a significant contribution -rnitan2B coming from the charged Higgs coupling to b,. Such large values of t a d are not allowed in the no-scale scenario. It must be emphasized that for such large values of tan@, the neglected neutralino-neutral-Higgs-boson diagrams will also become significant [23], and since especially neutralino diagrams give a positive contribution, their effect could compensate the large negative charged Higgs contributions. For rn, = 170 GeV at the 90% C.L., one can safely exclude values of t a d 5 2 in the no-scale and dilaton (except for just one point for p < 0 ) scenarios. Moreover, as Figs. 1 and 3 show, there are excluded points for all
49 -
values of t a d . In the dilaton scenario, large values of t a d ke., tar@k 32 for p > 0 and t a d 2 24 for p < 0) are also constrained, and even perhaps excluded in the neutralino- neutral-Higgs-boson contributions are not large enough to compensate for these values. It is seen that for light chargino masses and not too small values of tax$, the fit to the Eb data is better in SU(5 ) XU( 1) supergravity than in the standard model, although only marginally so. To see the combined effect of E l , b for increasing values of m,, in Figs. 2 and 4 we show the calculated values of these parameters for m,= 130,150,170 GeV, as well as the lo experimental ellipse (from Ref. [5]). Clearly smaller values of m, fit the data better. V. CONCLUSIONS
4 -
2 -
. -
-2
-4-
. .
-
-6
I
-R
-4
-2
0
2
4
6
El
10
W O I
I
I
9J
-4
-2
0
2
4
We have computed the one-loop electroweak corrections in the form of the and eb parameters in the context of SU(5 ) XU( 1) supergravity in both no-scale and dilaton scenarios. The new E scheme used allows us to include in the experimental constraints all of the LEP data. In addition, the minimality of parameters in SU(5 ) XU(1) supergravity is such that rather precise predictions can be made for these observables, and this entails strict constraints on the parameter spaces of the two scenarios considered. In agreement with our previous analysis, we find that for m,S 160 GeV, at the 90% C.L. these constraints are not restricting at present. However, their quadratic dependence on m, makes them quite severe for increasingly large values of m,. We have studied explicitly the case of rn, = 170 GeV and shown that most points in parameter space are excluded. The exceptions occur for light chargino masses which shift down and Eb up. However, for ta@ 5 2 the eb constraint is so strong that no points are allowed in the no-scale scenario. In the near future, improved experimental sensitivity on the eb parameter is likely to be a decisive test of S U ( 5 ) X U ( l ) supergravity. In any rate, the trend is clear: lighter values of the top-quark mass fit the data much better than heavier ones do. In addition, supersymmetry seems to always help in this statistical agreement. Finally, if the top quark continues to remain undetected at the Tevatron, high-precision LEP data in the context of SU(5)X U ( 1) supergravity would suggest the presence of light charginos. Note added in proof: Since the completion of this paper new LEP data have been released which shift the central values in Eq. (1) such that the l o ellipses in Figs. 2 and 4 now encompass all points for m,= 130 and 150 GeV. See Ref. [29] for an updated analysis.
6
El
FIG.4. The correlated predictions for the el and eb paramein the dilaton SU(S)XU(1) supergavity ters in units of scenario. The ellipse represents the Iu contour obtained from all LEP data. The values of m, are as indicated.
ACKNOWLEDGMENTS
This work has been supported in part by DOE grant DE-FG05-91-ER-40633. The work of G. P. has been supported by the World Laboratory. G. P. thanks Michael Boulware and Donnald Finnell for very helpful discussions.
355 49 -
NEW PRECISION ELECTROWEAK TESTS OF SU(5)XU(1) . . .
[l] G. Altarelli, Report No. CERN-TH.6867/93, 1993 (unpublished). [2] E. Eliasson, Phys. Lett. 147B, 67 (1984); S. Lim et al., Phys. Rev. D 29, 1488 (1984);J. Grifols and J. Sola, Nucl. Phys. B235, 47 (1985); B. Lynn et al., in Physics at LEP, Proceedings of the LEP Jamboree, Geneva, Switzerland, 1985, edited by I. Ellis and R. Peccei (CERN Yellow Report No. CERN86-02, Geneva, 1986), Vol. 1; R. Barbieri et al., Nucl. Phys. B341, 309 (1990); A. Bilal, J. Ellis, and G. Fogli, Phys. Lett. B 246, 459 (1990); M. Drees and K. Hagiwara, Phys. Rev. D 42, 1709 (1990); M. Drees, K. Hagiwara, and A. Yamada, ibid. 45, 1725 (1992). [3] R. Barbieri, M. Frigeni, and F. Caravaglios, Phys. Lett. B 279, 169 (1992). [4] G. Altarelli, R. Barbieri, and F. Caravaglios, Nucl. Phys. B405,3 (1993). [5] G. Altarelli, R. Barbieri, and F. Caravaglios, Phys. Lett. B314, 357 (1993). [6]For a recent review and extensive references see J. L. Lopez, Erice '93 Subnuclear Physics School Lecture, Texas A & M University Report No. CTP-TAMU-42/93 (unpublished). [7] For a recent review see J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Report No. CERN-TH.6926/93 (unpublished). [8] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Rev. D 49, 343 (1994). [9] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 319,451 (1993). [lo] J. L. Lopez, D. V. Nanopoulos, G. T. Park, H. Pois, and K. Yuan, Phys. Rev. D 48,3297 (1993). [ 111 J. L. Lopez, D. V. Nanopoulos, G. T. Park, and A. Zichichi, Phys. Rev. D 49, 355 (1994). [I21 G. Altarelli and R. Barbieri, Phys. Lett. B 253, 161 (1990). [I31 G. Altarelli, R. Barbieri, and S. Jadach, Nucl. Phys. B369, 3 (1992). [14] A. Blonde1 and C. Verzegnassi, Phys. Lett. B 311, 346 (1993). [15] J. L. Lopez, D. V. Nanopoulos, X. Wang, and A. Zichichi, Phys. Rev. D 48, 2062 (1993); J. L. Lopez, D. V. Nanopoulos, H. Pois, X. Wang, and A. Zichichi, ibid. 48, 4062
4841
(1993); J. L. Lopez, D. V. Nanopoulos, X. Wang, and A. Zichichi, ibid. 48,4029 (1993). [16] J. L. Lopez, D. V. Nanopoulos, H. Pois, X. Wang, and A. Zichichi, Phys. Lett. B 306,73 (1993). [17] J. L. Lopez, D. V. Nanopoulos, and G. T. Park, Phys. Rev. D 48, R974 (1993); J. L. Lopez, D. V. Nanopoulos, and X. Wang, Phys. Rev. D 49,366 11994). [I81 See e.g., L. Ibailez and D. Lust, Nucl. Phys. B382, 305 (1992); V. Kaplunovsky and J. Louis, Phys. Lett. B 306, 269 (1993); A. Brignole, L. Ibaiiez, and C. Muiioz, Report No. FTUAM-26/93, 1993 (unpublished). [I91 See e.g., S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Nucl. Phys. B398,3 (1993). [20] D. Kennedy and B. Lynn, Nucl. Phys. B322, 1 (1989);D. Kennedy, B. Lynn, C. Im, and R. Stuart, Nucl. Phys, B321, 83 (1989). [21] M. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990); W. Marciano and J. Rosner, ibid. 65, 2963 (1990); D. Kennedy and P. Langacker, ibid. 65,2967 (1990). [22] B. Holdom and J. Terning, Phys. Lett. B 247, 88 (1990); M. Golden and L. Randall, Nucl. Phys. B361, 3 (1991);A. Dobado, D. Espriu, and M. Herrero, Phys. Lett. B 255, 405 (1991). I231 M. Boulware and D. Finnell, Phys. Rev. D 44, 2054 (1991). [24] J. Bernabeu, A. Pich, and A. Santamaria, Phys. Lett. B 200, 569 (1988); W. Beenaker and W. Hollik, Z. Phys. C 40, 141 (1988);A. Akhundov, D. Bardin, and T. Riemann, Nucl. Phys. B276, 1 (1986);F. Boudjema, A. Djouadi, and C. Verzegnassi, Phys. Lett. B 238,423 (1990). [25] A. Denner, R. Guth, W.Hollik, and J. Kuhn, Z. Phys. C 51, 695 (1991). The neutral Higgs contributions to Z - t b b were also calculated here. [26] G. T. Park, Texas A & M University Report No. CTPTAMU-54/93,1993 (unpublished). [27] G. T. Park, Texas A & M University Report No. CTPTAMU-69/93, 1993 (unpublished). [28] A. Djouadi, G. Girardi, C. Verzegnassi, W. Hollik, and F. Renard, Nucl. Phys. B349,48 (1991). [29] J. Lopez, D. V. Nanopoulos, G. T. Park, X. Wang, and A. Zichichi, Report No. CERN-TH.7139/94 (unpublished).
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357
Jorge L. Lopez, D.V. Nanopoulos and A . Zichichi
THE TOP-QUARK MASS IN SU(5) x U( 1) SUPERGRAVITY
From Physics Letters B 327 (1994)279
1994
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359 19 May 1994
PHYSICS LETTERS B Physics Letters B 327 (1994) 279-286
El SEV I El<
The top-quark mass in SU (5) x U ( 1) supergravity Jorge L. Lopez a,b, D. V. Nanopoulos a,b,c, A. Zichichi Center for Theoretical Physics, Department of Physics, T a m A&M University, College Station, TX 77843-4242, USA krtroparticle Physics Group, Houston Advanced Research Center (HARC) The Mitchell Campus, The Woodlands, TX 77381, USA CEM, Theory Division. I211 Geneva 23, Switzerland CEM, 1211 Geneva 23, Switzerland a
Received 9 February 1994 Editor: L. Alvarez-Gaud
Abstract
We show that the currently experimentally preferred values of the top-quark mass (i.e., 130 d m, d 180 GeV) are naturally understood in the context of string models, where the top-quark Yukawa coupling at the string scale is generically given by Ar = O(g), with g the unified gauge coupling. A detailed study of the Yukawa sector of S U ( 5 ) x U(1) supergravity shows that the ratio of the bottom-quark to tau-lepton Yukawa couplings at the string scale is required to be in the range 0.7 6 A b / A I 5 1, depending on the values of m, and mb. This result is consistent with S U ( 5 ) X u( 1) symmetry, which does not require the equality of these Yukawa couplings in the unbroken symmetry phase of the theory. As a means of possibly predicting the value of mr, we propose a procedure whereby the size of the allowed parameter space is determined as a function of m,,since all sparticle and Higgs-boson masses and couplings depend non-trivially on m,. At present, no significant preference for particular values of m, in S U ( 5 ) x U(1) supergravity is observed, except that high-precision LEP data requires m l5 180 GeV.
1. Introduction
mf = Y (vev) ,
(2)
The origin of elementary particle masses is one of the most profound questions in physics. Modern field theories try to answer this question, in the context of spontaneously broken gauge symmetries, through vacuum expectation values (vevs) of elementary or composite scalar Higgs fields. In general the masses of all particles (scalars, fennions, gauge bosons) are proportional to this (or these) vev( s). The proportionality coefficients are: the quartic couplings ( A ) for the scalars, the Yukawa couplings ( y ) for the fermions, and the gauge couplings (g) for the gauge bosons. Thus, schematically we have:
mg = g (vev) .
(3)
This general picture looks convincingly simple, but its implementation in realistic models is not. At present, there are several reasons that prevent us from a complete and satisfactory solution of the mass problem. The quark and lepton mass spectrum (neglecting neutrinos) spans a range of at least five orders of magnitude, i.e., from me = 0.5MeV to mt 2 13OGeV. If we take as ''normal'' the electroweak gauge boson masses, U (80-90) GeV, then a seemingly "heavy" top quark U ( l50GeV) looks perfectly reasonable, while all other quark and lepton masses look peculiarly small. Clearly, a natural theory cannot support funda-
0370-2693/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDI 0 3 7 0 - 2 6 9 3 [ 94)00370-M
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mental Yukawa couplings extending over five orders of magnitude. The hope has always been [ 11 that several of these Yukawa couplings are naturally zero at the classical level, and that quantum corrections generate Yukawa couplings that reproduce reality. A modern version of this program has arisen in string theory, as we discuss shortly. We should point out that in a softly broken supersymmetric theory, several mass parameters arise beyond those in Eqs. ( 1)-( 3 ) . However, these lead to “normal” sparticle masses, and thus do not relate to the light fermion mass puzzle. Despite the pessimism expressed above, certain features of the fermion mass spectrum have been already explored, most notably in unified theories, where the difference between quark and lepton masses is attributed to the strong interactions that make the quarks much heavier than the leptons (of the same generation). In this context, the successful prediction for the rnb/m, ratio [21 led to the highly correlated prediction of Nf = 3, which was spectacularly confirmed at LEP: Nf = 2.980 f0.027 [ 31. An important feature of supergravity unified models is their ability to trigger radiative spontaneous breaking of the electroweak symmetry [4,5], thus explaining naturally why m W / r n p I = However, this mechanism only works when the theory contains a Yukawa coupling of the order of “g”, i.e., y = O ( g ) , which is naturally identified with the top-quark Yukawa coupling. In other words, in supergravity models, a “heavy” top quark is not only natural, but it is also needed if we want to have a dynamical understanding of electroweak symmetry breaking. Finally, string theory - more precisely its infrared limit, which naturally encompasses supergravity - is characterized by two features of relevance to us here (see e.g., Refs. [.6-91.. ) : (i) Most of the Yukawa couplings are naturally zero at the lowest order, and acquire non-vanishing values progressively at higher orders (through effective “non-renormalizable” terms), consistent with the spectrum of fennion masses observed in Nature. (ii) Non-zero Yukawa couplings, at lowest order, are automatically of O ( g ) . Once more, in string theory a “heavy” top quark is a natural possibility and, for the first time, we may even have a dynamical explanation for the origin of its large Yukawa coupling, i.e., O ( g ) . We should remark that large values of the top-quark Yukawa coupling at very
high energies have long been advocated as the explanation for a “heavy” top quark in connection with the infrared quasi fixed point of the corresponding renormalization group equation [ 10-131. However, the origin of such large values has been usually regarded as a remnant of new non-perturbative physics at very high energies [ lo], or simply left unspecified. In this note we emphasize that string theory provides a natural underlying structure where the experimentally favored values of the top-quark mass can be understood.
2. SU(5) x U(l) supergravity: bottom-up view Here we briefly describe the most salient features of string-inspired SU(5) x U(1) supergravity [ 141, which constitutes our bottom-up approach to the prediction for m,. The SU(5) x U(1) gauge group (also known as “flipped SU(5)”)can be argued to be the simplest unified gauge extension of the Standard Model. It is unified because the two non-abelian gauge couplings of the Standard Model (a2 and ( ~ 3 ) are unified into the SU(5) gauge coupling. It is the simplest extension because this is the smallest unified group which provides neutrino masses. In this interpretation, minimal SU(5) would appear as a subgroup of SO( lo), if it is to allow for neutrino masses. Moreover, the SU(5) x U( 1) matter representations entail several simplifications, such as the breaking of the gauge group via vacuum expectation values of 1 0 , a Higgs fields, the natural splitting of the doublet and triplet components of the Higgs pentaplets and therefore the natural avoidance of dangerous dimension-five proton decay operators, and the natural appearance of a see-saw mechanism for neutrino masses. We supplement the SU(5) x U(1) gauge group choice with the minimal matter content which allows 10I8GeV, as it to unify at the string scale MU expected to occur in the string-derived versions of the model [ 151. This entails a set of intermediate-scale mass particles: a vector-like quark doublet with mass rnQ 10I2GeV and a vector-like charge -1/3 quark singlet with mass r n D lo6GeV [ 161. The model is also implicitly constrained by the requirement of suitable supersymmetry breaking. We choose two stringinspired scenarios which have the virtue of yielding universal soft-supersymmetry-breaking parameters, N
-
N
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J.L Lapez er al. /Physics Letters B 327 (1994)279-286
in contrast with non-universal soft-supersymmetrybreaking scenarios which occur quite commonly in string constructions [ 17,181 and may be phenomenologically troublesome [ 191. These scenarios are examples of the no-scale supergravity framework [ 2031 in which the dimensional parameters of the theory are undetermined at the classical level, but are fixed by radiative corrections, thus including the whole theory in the determination of the low-energy parameters. In the moduli scenario, supersymmetry breaking is driven by the vev of the moduli fields ( T ) , and gives mo = A = 0, whereas in the dilaton scenario [ 181 supersymmetry breaking is driven by the vev of the dilaA = -m1p ton field (S) and entails mo = -)=m1/2, 3 Thus, the supersymmetry breaking sector depends on only one parameter (i.e., mlp). The procedure to extract the low-energy predictions of the models outlined above is rather standard (see e.g., Ref. [ 211 ) : (a) the bottom-quark and taulepton masses, together with the input values of m, and tan p are used to determine the respective Yukawa couplings at the electroweak scale; (b) the gauge and Yukawa couplings are then run up to the unification scale Mu = 10I8GeV taking into account the extra vector-like quark doublet ( 10l2GeV) and singlet (- 106GeV) introduced above [22,16]; (c) at the unification scale the soft-supersymmetry breaking parameters are introduced (i.e., moduli and dilaton scenarios) and the scalar masses are then run down to the electroweak scale; (d) radiative electroweak symmetry breaking is enforced by minimizing the one-loop effective potential which depends on the whole mass spectrum, and the values of the Higgs mixing term lpl and the bilinear soft-supersymmetry breaking parameter B are determined from the minimization conditions; (e) all known phenomenological constraints on the sparticle and Higgs masses are applied (most importantly the LEP lower bounds on the chargino and Kiggs-boson masses), including the cosmological requirement of a not-too-young Universe. The three-dimensional parameter space of this model (i.e., rn1/2,tan p and the top-quark mass) has been explored in detail in Refs. [ 16J and [23J for the moduli and dilaton scenarios respectively. More recently, we have investigated further constraints on the parameter space, including: (i) the CLEO limits on the b + sy rate [ 24,251, (ii) the long-standing N
limit on the anomalous magnetic moment of the muon [ 261, (iii) the electroweak LEP high-precision measurements in the form of the €1, Eb parameters [ 27,25J , (iv) the non-observation of anomalous muon fluxes in underground detectors (“neutrino telescopes”) [ 281, and ( v ) the possible constraints from trilepton searches at the Tevatron [ 291.
3. SU(5) x U(l) supergravity: top-down view
In the context of string model-building, the
SU(5)x U(1) structure becomes even more important, since the traditional grand unified gauge groups (SU(5), SO( lo), &) cannot be broken down to the Standard Model gauge group in the simplest (and to date almost unique) string constructions, because of the absence of adjoint Higgs representations [ 301. This reasoning is not applicable to the S V ( 5 ) x V ( 1) gauge group, since the required 1 0 , s representations are very common in string model building [ 6,8]. As a “descendant” of string theory, SU(5) x U(1) supergravity is characterized by two basic features: (a) a large top-quark Yukawa coupling: O(g), and ( b ) the no-scale structure. Notice that (b) in conjunction with (a), not only triggers radiative electroweak breaking but, in principle, may also determine dynamically the magnitude of the supersymmetry breaking scale [ 20,5]. As mentioned above, string unification occurs at the scale Mu 10l8GeV [ 151, and this has been seen to occur in explicit SU(5) x U(1) string models [8]. Of more relevance to the present discussion is the composition of the Yukawa sector in SU(5) x V ( 1) string models. The usual situation [ 6,7,9] is that at the cubic level of superpotential interactions, string symmetries allow only few couplings among the matter fields containing the quarks, leptons, and Higgs bosons of the low-energy theory. A particularly simple solution to the question of how to assign low-energy fields to the string representations consists of having only the top-quark, bottom-quark, and tau-lepton Yukawa couplings be non-vanishing. Further assumptions lead to a scenario with At = Ab = A, = fig at the string scale, where g is the unified gauge coupling detennined by the vacuum expectation value of the dilaton field in the top-down approach, or by the unification condition in the bottom-up approach. This is however not a robust N
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prediction since various unknown mixing angles could possibly destroy this relation. Moreover, it is possible that the bottom-quark and tau-lepton Yukawa couplings could be suppressed relative to the top-quark Yukawa coupling [ 91. What is a robust prediction is the magnitude of the top-quark Yukawa coupling At (Mu)= J z g cos 6,,
(4)
where cosBt is a possible mixing angle factor. The bottom-quark and tau-lepton Yukawa couplings are not necessarily equal at the string scale, since no obvious symmetry principle is at play in SU(5) x U(1) (as opposed to the case of SU(5 ) ) . Nonetheless, equality of these Yukawa couplings does occur in many explicit SU(5) x V ( 1) shing models 16-81. The Yukawa couplings for the first- and secondgeneration quarks and leptons appear at the quartic or higher non-renormalizable order [31,7] and are naturally suppressed relative to the cubic level Yukawa couplings, in agreement with the observed hierarchical mass spectrum.
4. The Yukawa sector and the value of ml From the bottom-up approach we are able to compute the value of the third-generation Yukawa couplings at the string scale in terms of rn, and tanp. (These string-scale Yukawa couplings also depend on mb and ~ ( M z ) . In ) Fig. 1 we show the top-quark Yukawa coupling at the string scale versus the topquark mass for various values of tanp. As expected, a Landau pole is encountered in the running of the Yukawa coupling if the top-quark mass exceeds a maximum value at low energies. For example, rn, 5 170 GeV is required for tan p = 2. Values of tan p larger than those shown are indistinguishablefrom the tanp = 10 curve. The dependence on a 3 ( M z ) and mb is rather small in this case, i.e., comparable to the thickness of the lines for a3 ( M z ) = 0.118 f 0.007 and mb = 4.25-4.9 GeV. The above are the results of the bottom-up approach. On the other hand, from the top-down approach we expect values of A, as given in Eq. (4), which are shown as dashed lines on Fig. 1 for two typical cases. Here g M 0.84 is obtained from the running of the gauge couplings up to the string scale. These values of At do not exceed the unitarityrequirement ofRef. [ 121
SU(S)xU( I) supergravity 2.5
2.0
-1'
1.5
v
2 1.o
0.5
0.0 . 130
. . . . . . . . . . . . . . . . . . . . . . . . . . 140
150
180
170
180
180
mt (Gev)
Fig. 1. The topquark Yukawa coupling at the string scale in SU(5)x U(1) supergravity versus the topquark mass for fixed values of tan p (larger values of tan p overlap with the tan p = 10 curve). The dashed lines indicate typical string-like predictions for the Yukawa coupling.
( A t < 4.8) or the perturbative criterion of Ref. [ 131 ( A t < 3.3). Thus, the experimentally preferred topquark masses (direct Tevatron limits rn, > 131 GeV [32] and indirect fits to the electroweak data m, M 140 f 20GeV [33]) can be naturally understood in string models, and do not require the existence of new non-perturbative interactions at the unification scale. From the bottom-up approach we also obtain the values for the bottom-quark and tau-lepton Yukawa couplings at the string scale, as shown in Figs. 2 and 3 (for a 3 ( M z ) = 0.118). In Fig. 2, the bottomquark Yukawa coupling is plotted against the topquark Yukawa coupling for various values of tanp (2,6,10,20). Along the (solid) lines the top-quark mass varies as shown. The two sets of curves for each value of tan p correspond to the representative choices of mb = 4.25 and 4.9 GeV. The dashed lines for tan p = 10 show the decrease in & ( M u ) due to a shift in a 3 ( M ~ from ) 0.118 to 0.125. The corresponding shifts for larger (smaller) values of tan p are proportionally larger (smaller). Values of tan @ larger than the ones shown, when allowed by the theoretical constraints on the model, simply yield proportionally larger values of Ab( Mu). In Fig. 3 the bottom-quark yukawa coupling is plotted against the tau-lepton Yukawa coupling for various values of tan p (2,6,10,20). Two representative
363 J.L. Lopez et al. /Physics Letters B 327 (1994)279-286
-$ -
283
from bottom to top. The effect of shifts in a3 ( Mz)is to extend the vertical lines slightly. It is interesting to note that the traditional Ab = A, relation (as would be required in an SU(5) model) can be obtained for the largest values of m, and for the larger values of mb. However, the range 0.7
0075
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2
h(Mu)
Fig. 2. The bottomquark Yukawa coupling versus the topquark Yukawa coupling at the string scale in S U ( 5 ) x U( 1 ) supergravity for various values of tanp (2.6.10.20). two values of mb (4.25 and 4.9 GeV), and ~ ( M z =) 0.118. The topquark mass varies along the curves as indicated. The dashed lines for tanp = 10 show the effect of varying ~ ( M z from ) 0.118 to 0.125. The magnitude of this effect scales with tan p .
0.125
0.100
t I
x' 0.050
0.025
Gr, 0.000
" "
0
I
" "
0.025
I
0.05
'
"
' I 0.075
" "
I
0.1
"
' ' 1
" "
0.125
0.15
%(Mu)
Fig. 3. The bottomquark Yukawa coupling versus the tau-lepton Yukawa coupling at the string scale in SU(5) x U( 1 ) supergravity for various values of tan p (2,6,10,20), two values of t q , (4.25 and 4.9 GeV), and n 3 ( M z ) = 0.118. The value of mt increases from bottom to top dong the vertical lines. Note that 0.7 s &/A7 5 1 is obtained.
values of mb have been chosen (4.25 and 4.9 GeV) which are only visibly distinguished for tanp = 20, as indicated. Also, a3 ( M z )= 0.118 has been chosen. Along the vertical lines the top-quark mass increases
=; &/A, 5 1
(5)
is a more realistic estimate of what would be required from a string model in the top-down approach. Such deviations from the Ab = A, relation have been explored in the literature [ 131 and have been shown to weaken significantly the tight constraint on the (m,,tanP) plane which otherwise results from imposing the Ab = h, relation. As discussed above, the allowable free parameters are reduced to a minimal number in SU(5 ) x U(1) supergravity, allowing severe experimental scrutiny. An interesting exercise along these lines consists of determining the size of the allowed parameter space (in the ( M I / * , tan p ) plane) as a function of m,,hoping that the correlations among the model variables and their intricate dependence on m1may show a preference for particular values of the top-quark mass. The results of this exercise, when only the basic theoretical and experimental LEP constraints are imposed, are shown in Fig. 4 ("theory+LEF"' curves) . The drop in the curves near mt = 190 GeV has been studied in detail (form, = 180,185,187,188,189GeV) andcorresponds to encountering a Landau pole in the topquark Yukawa coupling below the string scale [ 12,211. Imposing in addition all of the direct and indirect experimental constraints mentioned above (i.e.. b -t sy, (g-2),, neutrino telescopes, and E l $ ) we obtain the curves labelled "ALL" in Fig. 4 [ 341. These curves still do not show any obvious preference for particular values of m,. However, m, 5 180 GeV is now required, basically to fit the precise LEP electroweak data [ 341. This exercise is rather interesting and should be repeated as present experimental constraints are tightened or new constraints arise.
364 J.L. Lopez et al./Physics Letters B 327 (1994) 279-286
284
p>o
no-scale SU(5)xU(1) moduli scenario
1200
PCO
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theory+LeP 600 u
2
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200 0 130
140
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180
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200
0 1 30
140
150
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Fig, 4. The number of allowed points in parameter space of no-scale SU(5)x U(1) supergravity in the moduli and dilaton scenarios, as a function of rn, when the basic theoretical and experimental LEP constraints have been imposed ("theory+LEF"'), and when all known direct and indirect experimental constraints have been additionally imposed ("ALL"). Note that rn, 5 180GeV is required.
5. Conclusions
We have shown that the currently experimentally preferred values of the top-quark mass are naturally understood in a top-down approach in the context of string models. We have studied this point explicitly in the context of S U ( 5 ) x U(1) supergravity, which is to be viewed as the bottom-up approach to physics at the string scale. Using the bottom-up approach we have also found that the ratio of the bottom-quark to tau-lepton Yukawa couplings at the string scale is required to be in the range 0.7 5 Ab/& ,S 1, depending on the values of m, and mb. This result is consistent with S U ( 5 ) x U(1) symmetry, which does not require the equality of these Yukawa couplings in the unbroken symmetry phase of the theory.Finally, as a means of possibly predicting the value of ml, we have proposed a procedure whereby the size of the allowed parameter space is determined as a function of m,. Since all sparticle and Higgs-boson masses and couplings, and therefore all observables calculated from them, depend non-trivially on m, (mostly through the radiative breaking mechanism), such procedure could show a preference towards particular values of m,.At
present no such preference is clearly observed, except for the high-precision LEP data requirement of m, 5 180GeV. Nonetheless, future more sensitive experimental constraints may produce more clear effects. This present relative insensitivity to the value of mt should not obscure the fact that all experimentally preferred values of mlare allowed in SU(5)x U(1) supergravity, even after the many theoretical and experimental constraints have been applied to the model. We should remark that this procedure could also be applied to more general classes of supergravity models, which have been recently studied in the literature [ 351, as a means of gauging the experimental viability of these models.
6. Acknowledgements
This work has been supported in part by DOE grant DE-FG05-9 1-ER-40633. References [ I ] D.V. Nanopoulos, Phys. Scripta 24 (1981) 873.
365 J.L. Lopez et al. /Physics Letters B 327 (1994)279-286 [2] A.J. Bum, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B 135 (1978) 66; D.V. Nanopoulos and D.A. Ross, Nucl. Phys. B 157 (1979) 273; Phys. Lett. B 108 (1982) 351; B 118 (1982) 99. [3] The LEP Collaborations (ALEPH, DELPHI, L3. OPAL) and the LEP Electroweak Working Group, in: Proceedings of the International Europhysics Conference on High Energy Physics, Marseille, France, July 22-28, 1993, ed. by J. Cam and M. Perronet (Editions Frontieres. Gif-sur-Yvette, 1993) CERN/PPE/93-157 (August 1993). [4] L. Ibiiiez and G. Ross, Phys. Lett. B 110 (1982) 215; K. Inoue, et. al.. Prog. Theor. Phys. 68 (1982) 927; L. Ibiiiez, Nucl. Phys. B 218 (1983) 514; Phys. Lett. B 118 (1982) 73; H.P. Nilles, Nucl. Phys. B 217 (1983) 366; J. Ellis, D.V. Nanopoulosand K. Tamvakis, Phys. Lett. B 121 (1983) 123; J. Ellis, J. Hagelin, D.V. Nanopoulosand K. Tamvakis, Phys. Lett. B 125 (1983) 275; L. Alvarez-Gaum.5, J. Polchinski and M. Wise, Nucl. Phys. B 221 (1983) 495; L. IbaiiCz and C. Mpez, Phys. Lett. B 126 (1983) 54; Nucl. Phys. B 233 (1984) 545; C. Kounnas. A. Lahanas, D.V. Nanopoulosand M. Quirbs, Phys. Lett. B 132 (1983) 95; Nucl. Phys. B 236 (1984) 438. [51 For a review see A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1. [61 I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, Phys. Lett. B 231 (1989) 65. [71 J.L. Lopez and D.V. Nanopoulos. Phys. Lett. B 251 (1990) 73; B 256 (1991) 150; B 268 (1991) 359. [8] J.L. Lopez, D.V. Nanopoulosand K. Yuan, Nucl. Phys. B 399 (1993) 654. (91 A. Faraggi, D.V. Nanopoulosand K. Yuan, Nucl. Phys. B 335 (1990) 347; A. Faraggi, Phys. Lett. B 278 (1992) 131; B 274 (1992) 47; Nucl. Phys. B 387 (1992) 239. [ l o ] L. Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B 136 (1978) 115; N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B I58 (1979) 295; W. Bardeen, M. Carena, S. Pokorski and C. Wagner, Phys. Lett. B 320 (1994) 110. [ 11I B. Pendleton and G. Ross, Phys. Lett. B 98 (1981) 291; C. Hill, Phys. Rev. D 24 (1981) 691; I. Bagger, S. Dimopoulos and A. Masso, Nucl. Phys. B 253 (1985) 397; C. Hill, C. Leung and S. Rao, Nucl. Phys. B 262 (1985) 517; P. Krawczyk and M. Olechowski, Z. Phys. C37 (1987) 413. [I21 L. Durand and 1.L. Lopez,Phys. Lett. B 217 11989) 463; Phys. Rev. D 40 (1989) 207. [I31 V. Barger, M. Berger and P. Ohmann, Phys. Rev. D 47 (1993) 1093; V. Barger, M. Berger, P. Ohmann and R. Phillips, Phys. Lett. B 314 (1993) 351.
285
[ 141 For a recent review see J.L. Lopez, D.V. Nanopoulosand A.
Zichichi, in From Superstrings to Supergravity, Proceedings of the INFN Eloisatron Project 26th Workshop, ed. by M J . Duff, S.Ferrara and R.R. Khuri (World Scientific,Singapore 1993). [ 151 I. Antoniadis, J. Ellis, R. Lacaze and D.V. Nanopoulos, Phys. Lett. B 268 (1991) 188; S. Kalara, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 269 (1991) 84. [ 161 J.L. Lopez, D.V. Nanopoulosand A. Zichichi, Phys. Rev. D 49 (1994) 343. [ 171 L. Ibiiiez and D. Lust, Nucl. Phys. B 382 (1992) 305. 181 V. Kaplunovskyand J. Louis. Phys. Lett. B 306 (1993) 269; A. Brignole, L. Ibiiiez and C. Muiioz, FTUAM-26/93 (August 1993). [ 191 J. Ellis and D.V. Nanopoulos, Phys. Lett. B 110 (1982) 44. [20] J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B 241 (1984) 406; B 247 ( 1984) 373; J. Ellis, A. Lahanas, D.V. NanopoulosandK. Tamvakis. Phys. Lett. B 134 (1984) 429. [21] S. Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois and K. Yuan, Nucl. Phys. B 398 (1993) 3. 1221 S. Kelley, J.L. Lopezand D.V. Nanopoulos, Phys. Lett. B 278 (1992) 140; G. Leontaris, Phys. Lett. B 281 (1992) 54. [23] J.L. Lopez, D.V. Nanopoulosand A. Zichichi Phys. Lett. B 319 (1993) 451. [24] J.L. Lopez, D.V. Nanopoulosand G.T. Park, Phys. Rev. D 48 (1993) R974. I251 J.L. Lopez, D.V. Nanopoulos, G.T. Park and A. Zichichi, Phys. Rev. D 49 (1994) 355. [261 J.L. Lopez, D.V. Nanopoulosand X. Wang, Phys. Rev. D 49 (1994) 366. [27] J.L. Lopez, D.V. Nanopoulos, G.T. Park, H. Pois and K. Yuan, Phys. Rev. D 48 (1993) 3297; J.L. Lopez, D.V. Nanopoulos, G.T. Park and A. Zichichi, Texas A & M University preprint CTP-TAMU-68/93 (to appear in Phys. Rev. D, May 1). [281 R. Gandhi. J.L. Lopez, D.V. Nanopoulos, K. Yuan and A. Zichichi, Texas A & M University preprint CTP-TAMU48/93 (to appear in Phys. Rev. D, April 1). [291 J.L. Lopez, D.V. Nanopoulos, X. Wang and A. Zichichi. Phys. Rev. D 48 (1993) 2062. [30] See e.g.. J. Ellis, J.L. Lopezand D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font. L. Ibiiiez and E Quevedo, Nucl. Phys. B 345 (1990) 389. 1311 S. Kalara J.L. Lopezand D.V. Nanopoulos, Phys. Len. B 245 (1990) 421; Nucl. Phys. B 353 (1991) 650. 1321 S. Abachi, et. al. (M)C o l l a b o r a t i o n ) , F E R M ~ - p ~ - 9 ~ OWE (January 1994). 1331 G. Altarelli, in Proceedings of the International Europhysics Conference on High Energy Physics, Marseille, France, July 22-28, 1993, ed. by J. Carr and M. Perrottet (Editions Frontieres, Gif-sur-Yvette. 1993) cERN-m.7045/93 (October 1993); J. Ellis, G.L. Fogli and E. Lisi, CERN-TH.7116/93 (December 1993).
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[34]J.L. Lopez, D.V. Nanopoulos, G. Park, X. Wang and A. Zichichi, Texas A & M University preprint CTP-TAMU74/93. [ 351 R. Roberts and L. Roszkowski, Phys. Lett. B 309 (1993) 329; M. Olechowski and S. Pokorski, Nucl. Phys. B 404 (1993) 590; B. de Carlos and I. Casas, Phys. Lett. B 309 (1993)320; M. Carena, L. Clavelli, D. Matalliotakis, H. Nilles, and C. Wagner, Phys. Lett. B 317 (1993) 346; G.Leontaris, Phys. Lett. B 317 (1993) 569; S. Martin and P. Ramond, Phys. Rev. D 48 (1993)5365; D. Castaiio, E. Piard and P. Ramond, UFIlT-HP-93-18 (August 1993);
W. de Boer, R. Ehret and D. Kazakov, IEKP-KA/93-13 (August 1993); A. Faraggi and B. Grinstein, SSCL-Preprint-496 (August
1993); M. Bastero-Gil, V. Manias and J. Perez-Mercader, LAEFF931012 (September 1993); M. Carenn, M. Olechowski, S. Pokorski and C. Wagner, CERN-TH.7060/93(October 1993); V. Barger, M. Berger, and P. Ohmann, MADIPHI801 (November 1993); A. Lahanas, K. Tamvakis, and N. Tracas, cERN-TH.7089/93 (November 1993); G. Kane, C. Kolda, L. Roszkowski, and I. Wells, UM-TH93-24(December 1993).
367
Jorge L. Lopez, D.V. Nanopoulos, Gye T. Park, Xu Wang and A. Zichichi
EXPERIMENTAL ASPECTS OF SU(5) x U( 1) SUPERGRAVITY
From Physical Review D 50 (1994) 2 164
I994
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369 PHYSICAL REVIEW D
VOLUME 50, NUMBER 3
1 AUGUST 1994
Jorge L. Lopez,'.' D. V. Nanop~ulos,'-~ Gye T. Park,lV2Xu Wang,lP2and A. Zichichi4 'Centel.for Theoretical Physics, Department of Physics, Texas A&M University, College Station, Texas 77843-4242 'Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Mitchell Campus, The Woodlands, Texas 77381 'CERN, Theory Division, 1211 Geneva 23, Switzerland 'CERN, I211 Geneva 23, Switzerland (Received 27 January 1994) We study various aspects of SU(5 ) XU(1) supergravity as they relate to the experimental verification or falsification of this model. We consider two string-inspired, universal, one-parameter, no-scale softsupersymmetry-breaking scenarios, driven by the F terms of the moduli and dilaton fields. The model is described in terms of the supersymmetry mass scale (is., the chargino mass m ),M, and the top-
*
XI
quark mass. We first determine the combined effecton the parameter space of all presently available direct and indirect experimental constraints, including the CERN LEP lower bounds on sparticle and Higgs-boson masses, the b + s y rate, the anomalous magnetic moment of the muon, the high-precision electroweak parameters (which imply m , S 180 GeV), and the muon fluxes in underground detectors (neutrino telescopes). For the still-allowed points in ( m *,tanP) parameter space, we reevaluate the XI
experimental situation at the Fermilab Tevatron, LEP 11, and DESY HERA. In the 1994 run, the Tevatron could probe chargino masses as high as 100 GeV. At LEP I1 the parameter space could be explored with probes of dflerent resolutions: Higgs-boson searches, selectron searches, and chargino searches. Moreover, for m,5 150 GeV, these Higgs-boson searches could explore all of the allowed parameter space with 5210 GeV.
<
PACS number(s):12.10.Dm, 04.65.+e, 14.80.L~
I. INTRODUCI'ION
I n the search for physics beyond the standard model, what is needed are detailed calculations to be confronted with experimental data. The starting point is the choice of a model described by the least numbers of parameters, and based on well-motivated theoretical assumptions. Our choice is SU(5)XU(1) supergravity [l], the reasons being twofold: first, because this model is derivable from string theory; second, because the SU(5)XU(1) gauge group is the simplest unified gauge extension of the standard model. It is unified because the two non-Abelian gauge couplings of the standard model (a2and a,)are unified into the SU(5) gauge coupling. It is the simplest extension because this is the smallest unified group which provides neutrino masses. In this interpretation, minimal SU(5) would appear as a subgroup of S0(10), if it is to allow for neutrino masses. Moreover, the matter representations of SU(5)xU(1) entail several simplifications [2]. The most important are (i) the breaking of t& gauge group via vacuum expectation values of 10,lO Higgs fields, (ii) the natural splitting of the doublet and triplet components of .the Higgs pentaplets and therefore the natural avoidance of dangerous dimension-five proton decay operators, and (iii) the natural appearance of a seesaw mechanism for neutrino masses. In the context of string model building, the SU(5)XU(1) structure becomes even more important, since the traditional grand unified gauge groups [SU(5),S0(lO),E6]cannot be broken down to the standard model gauge group in the simplest (and to date
0556-2821/94/50(3)/2 164(28)/$06.00
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almost unique) string constructions, because of the absence of adjoint Higgs representations [3]. This reasonthe SU(5)XU(1) gauge group, ing is not applicable since the required 10,lO representations are very common in string model building [4-61. We supplement the SU(5)X U l ) gauge group choice with the minimal matter content which allows it to unify at the string scale M u 10" GeV, as expected to occur in the string-derived versions of the model [7,8]. This entails a set of intermediate-scale mass particles: a vectorlike quark doublet with mass m Q 1Ol2 GeV and a vectorlike charge - f quark singlet with mass mD lo6 GeV [9,10]. The model is also implicitly constrained by the requirement of suitable supersymmetry breaking. We choose two string-inspired scenarios which have the virtue of yielding universal soft-supersymmetry-breaking parameters ( m l n , m o , A ), in contrast with nonuniversal soft-supersymmetry-breaking scenarios which occur quite commonly in string constructions [ 11- 131 and may be phenomenologically troublesome [ 141. These scenarios are examples of the no-scale supergravity framework [15,16] in which the dimensional parameters of the theory are undetermined at the classical level, but are fixed by radiative corrections, thus including the whole theory in the determination of the low-energy parameters. In the moduli scenario, supersymmetry breaking is driven by the vacuum expectation value (VEV) of the moduli fields ( T ) ,and gives m , = A =0, while in the &laton scenario [ 12,131 supersymmetry breaking is driven by the VEV of the dilaton field ( S ) and entails
to
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@ 1994 The American Physical Society
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JXPERIMENTALASPECTS OF SU(5)XU(1)SUPERGRAVITY
mo=(l/~)mln, A=-m,,. Thus, the supersymmetry-breaking sector depends on only one parameter Le., mln). The parameter space of SU(S)xU(1) supergravity is fully described by just two more quantities: the ratio of Higgs-boson vacuum expectation values (tar@), and the top-quark mass ( m , ). This three-dimensional parameter space (i.e., m l R ,tan& and m,) has been explored in detail in Refs. [10,17] for the moduli and dilaton scenarios, respectively. The allowed points in parameter space are determined by a theoretical procedure (including renormalization-group evolution of the model parameters from the unification scale down to the electroweak scale and enforcement of radiative electroweak symmetry breaking using the one-loop effective potential) and by the further imposition of the basic constraints from the CERN e + e - collider LEP on the sparticle and Higgsboson masses, as described in Ref. [18]. More recently, we have investigated further constraints on the parameter space, including (i) the CLEO limits on the b - u y rates [19,20], (ii) the long-standing limit on the anomalous magnetic moment of the muon [21], (iii) the electroweak high-precision LEP measurements in the form of the parameters [22,20,23] (here we update our analysis including the latest LEP data), (iv) the nonobservation of anomalous muon fluxes in underground detectors (“neutrino telescopes”) [24], and (v) the possible constraints from trilepton searches at the Fermilab Tevatron [25]. In our analysis we combine the most useful elements of the top-down and bottom-up approaches to physics beyond the standard model. The top-down approach consists of selecting particularly well-motivated stringinspired scenarios for supersymmetry breaking (Le., with a single mass parameter), whereas the bottom-up approach aims at imposing all known direct and indirect experimental constraints on the chosen model. In this way, we can comer the high-energy parameter space of the model (bottom-up) and thus focus our search for further realistic supersymmetric models (top-down). On the other hand, the completely phenomenological approach in which the many parameters (more than 20) of the minimal supersymmetric standard model (MSSM) are arbitrarily varied, is neither practical nor illuminating. It is important to note that our advocacy of supersymmetry, as the choice for physics beyond the standard model, seems to be accumulating indirect supporting evidence: (i)global fits to the electroweak sector of the standard model show a preference for a light Higgs boson [26],in agreement with low-energy supersymmetry where a light Higgs boson is always present; (ii) the precisely measured gauge couplings, when extrapolated to very high energies using standard model radiative effects, fail to converge at any high-energy scale [27,28], consistent with the fate of nonsupersymmetric grand unified theories (GUT’S) in light of the gauge hierarchy problem; (iii) on the contrary, in the supersymmetric version of the standard model, the gauge couplings unify at a scale MU-10l6 GeV [28]; (iv) global fits to the electroweak data also imply that m,=140*20 GeV for mH=60 GeV and m, = 180f18 GeV for mH= 1 TeV (see, e.g., Refs. [29,30]), consistent with the radiative electroweak sym-
2165
metry breaking mechanism [31,16]; (v) m, S 190-200 GeV (see, e.g., Ref. [32]) is required in a supersymmetric unified theory, consistent with the electroweak fits to m,; and (vi) the resulting top-quark Yukawa couplings at the unification scale are naturally obtained in supersymmetric string models [4,5]. In this paper we first briefly review the basic SU(S)XU(l) supergravity properties (Sec. IL), and then discuss each of the constraints on the parameter space separately (Sec. 1111,and also their combined effect (Sec. IV). Next we address the prospects for detecting the sparticles and Higgs bosons directly through searches at the Tevatron, LEP 11, and the DESY ep collider HERA (Sec.V). We conclude that with the present generation of collider facilities, direct searches for the lighter weakly interacting sparticles and Higgs bosons probe the parameter space of SU(5)XU(1)supergravity in a much deeper way than direct searches for the heavier strongly interacting sparticles do. Moreover, within the weakly interacting sparticles, the deepest probe is provided by the lightest Higgs boson, followed by the selectrons, and then by the charginos. We also discuss the two most e5cient ways of exploring the parameter space in the near future in an indirect way (Sec. VI), namely, through more precise B ( b +sy ) and (g -2 )a measurements. We summarize our conclusions in Sec. VII.
IL SU(5)X U(1) SUPERGRAVITY A. Model building
The supergravity model of interest is based on the gauge group SU(5)X U(1) and is best motivated as a possible solution to string theory. In this regard several of its features become singularly unique, as discussed in the Introduction. However, string models (such as the one in Ref. [6]) are quite complicated and their phenomenology tends to be obscured by a number of new string parameters (although these could in principle be determined dynamically). It is therefore more convenient to study the phenomenology of a “string-inspired” model [ 101 which contains all the desirable features of the real string model, but where several simplifying assumptions have been made, as “inspired” by the detailed calculations in the real model. The string-inspired model is such that unification of the low-energy gauge couplings of the standard model occurs at the string scale M,-lO’s GeV. This is a simplifying assumption since in the string model there are several intermediate-scale particles which in effect produce a threshold structure as the string scale is approached. Perhaps because of this simplifying assumption, in the string-inspired model one seems to be forced to introduce nonminimal matter representations at intermediate scales: a vectorlike quark doublet with mass mQ lo’* GeV and a vectorlike charge - f quark singlet with mass m, lo6 GeV [9,10]. The low-energy spectrum of the model contains the same sparticles and Higgs bosons as the minimal supersymmetric standard model (MSSM). A very important component of the model is that which triggers supersymmetry breaking. In the string
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LOPEZ, NANOPOULOS, PARK, WANG,AND ZICHICHI
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model this task is performed by the hidden sector and the universal moduli and dilaton fields. Model-dependent calculations are required to determine the precise nature of supersymmetry breaking in a given string model. In fact, no explicit string model exists to date where various theoretical dficulties (e.g., suitably suppressed cosmological constant, suitable vacuum state with perturbative gauge coupling, etc.) have been satisfactorily overcome. Instead, it has become apparent [ll-131 that a more model-independent approach to the problem may be more profitable. In this approach one parametrizes the breaking of supersymmetry by the largest F-term vacuum expectation value which triggers supersymmetry breaking. Of all the possible fields which could be involved (i.e., hidden sector matter fields, various moduli fields, dilaton) the dilaton and three of the moduli fields are quite common in string constructions and have thus received the most attention in the literature. In a way, if supersymmetry breaking is triggered by these fields Le., (F, )#0 or (F,)#O), this would be a rather generic prediction of string theory. There are various possible scenarios for supersymmetry breaking that are obtained in this model-independent way. To discriminate among these we consider a simplified expression for the scalar masses (e.g., m,) iif=m:n( l + n i cos20), with t a d = (F,) /(F,) [13]. Here m,n is the gravitino mass and the ni are the modular weights of the respective matter field. There are two ways in which one can obtain universal scalar masses, as desired phenomenologically to avoid large flavorchanging-neutral currents (FCNC‘s) [ 141: (i) setting O=a/2, that is ( Fs ) >> (F,); or (ii) in a model where all n i are the same, as occurs for Z,X Z , orbifolds [I31 and free-fermionic constructions [8]. In the first (“dilaton”) scenario, supersymmetry breaking is triggered by the dilaton F term and yields universal soft-supersymmetry-breaking gaugino and scalar masses and trilinear interactions [12,13]:
In the second (“moduli”) scenario, in the limit (F,) >>(Fs) he., @-to) all scalar masses at the unification scale vanish, as is the case in no-scale supergravity models with a unified group structure [16]. In this case we have mo=O,
A=O .
(2)
The procedure to extract the low-energy predictions of
the model outlined above is rather standard (see, e.g., Ref. [18]): (a)the bottom-quark and 7-lepton masses, together with the input values of m, and ta~$ are used to determine the respective Yukawa couplings at the electroweak scale; (b) the gauge and Yukawa couplings are then run up to the unification scale M,=lO1* GeV taking into account the intermediate-scale particles introduced above; (c) at the unification scale the softsupersymmetry-breaking parameters are introduced [according to Eqs. (1) and (2)] and the scalar masses are then run down to the electroweak scale; (d) radiative electroweak symmetry breaking is enforced by minimizing the one-loop effective potential which depends on the whole mass spectrum, and the values of the Higgs mixing term lpl and the bilinear soft-supersymmetry-breaking parameter B are determined from the minimization conditions; (e) all known phenomenological constraints on the sparticle and Higgs-boson masses are applied (most importantly the LEP lower bounds on the chargino and Higgs masses), including the cosmological requirement of a not-too-large neutralino relic density (which happens to be satisfied automatically). In either of the supersymmetry-breaking scenarios considered, after enforcement of the above constraints, the low-energy theory can be described in terms of just three parameters: the top-quark mass ( m , 1, the ratio of Higgs vacuum expectation values ( t d ) , and the gaugino mass ( m ). Therefore, measurement of only two sparticle or Higgs-boson masses would determine the remaining thirty. Moreover, if the hidden sector responsible for these patterns of soft-supersymmetry-breaking is specified (as in a string-derived model), then the gravitino mass will also be determined and the supersymmetry breaking sector of the theory will be completely fixed.
B. Massranges We have scanned the three-dimensional parameter space for m,=130,150,170 GeV, tan/3=2-+50, and m =50-+500 GeV. Imposing the constraint mg,mg< 1 TeV we h d moduli: mIl2 <475 GeV, t a n p S 3 2 ,
(3)
.
(4)
dilaton: m 1/2 < 465 GeV, t a g S 46
These restrictions on m 1/2 cut off the growth of most of the sparticle and Higgs-boson masses at = 1 TeV. However, the sleptons, the lightest Higgs boson, the two lightest neutralinos, and the lightest chargino are cut off at a much lower mass, as follows:’
I
r
M
ZR
< 190 GeV, m <305 GeV, m p <295 GeV, ZL
m <185 GeV, mT2<315GeV, TI
‘
(5)
mh < 125 GeV, m
o XI
< 145 GeV, m <290 GeV, m
2
* <290 GeV , XI
‘In this class of supergravity models the three sneutrinos ( 8 )are degenerate in mass. Also, mp,= m l and m p.8
=m-
,
fR
372 2167
m
ZR
<325 GeV, m <400 GeV, m , <400 GeV, ZL
rnTl < 325 GeV, mF2<400 GeV, dilaton: ‘ mh < 125 GeV, m~ < 145 GeV, m
l? <285 GeV,
m
* <285 GeV .
XI
I
=o, 1/d. The coefficients ci can be calculated numerically in terms of the low-energy gauge couplings, and are given in Table I for a&M, )=O. 118f0.008. In the table we also give cg=rng/mln. Note that these values are smaller than what is obtained in the minimal SU(5)supergravity model [where cg=2.90 for a,(MZ)=0.118] and (7) therefore the numerical relations between the gluino mass mg 5 195(235) GeV, and the neutralino masses are m e r e n t in that model. In dilaton: my 5 193235) GeV , the table we also show the resulting values for ui,bi for the central value of a&MZ). for p > O(p< 0 1. Relaxing the above conditions on m The “average” squark mass, m ? = + ( m ’L + m ‘R +mJL simply allows all sparticle masses to grow further propor+ m +m +m +m +m ) = ( m g / c r ) ( F ~ + ~ ; ) I ~ , tional to mg. ’R ‘L ‘R ’L ‘R with Fr given in Table I, is determined to be
It is interesting to note that because of the various constraints on the model, the gluino and (average) squark masses are bounded from below: mg 5 245(260) GeV, moduli: my 5 240(250) GeV ,
I
1
C. Massrelations The neutralino and chargino masses show a correlation observed before in this class of models [33,10]: namely (see Fig. 1, top row),
This is because throughout the parameter space 1111 is generally much larger than M , (see Fig. 1, bottom row) and l p l > M z . In practice we find mS=m to be XI satisfied quite accurately, whereas mp=+ml? is only
*
I
qualitatively satisfied, although the agreement is better in the dilaton case. In fact, these two mass relations are much more reliable than the one that links them to mg. The heavier neutralino ($&) and chargino (x:) masses are determined by the value of IpI; they all approach this limit for large enough ] p [ . More precisely, m p approaches IpI sooner than m m Z approaches m
2 does.
O n the other hand,
* rather quickly.
TABLE I. The value of the ci coefficients appearing in Eq. (9),the ratio cS=rnS/mIn, and the average squark coefficient Fi, for a3(MZ)=0.118*0.008. Also shown are the ni, bi coefficients for the central value of n , ( M Z ) and both supersymmetry-breaking scenarios ( T: moduli, S: dilaton). T h e results apply as well to the second-generation squark and slepton masses. i
T h e first- and second-generation squark and slepton masses can be determined analytically:
(0.110)
ci (0.118)
ci (0.126)
0.499
b
0.406 0.153 3.98 3.68 3.63 1.95 3.82
4.1 1 4.06 2.12 4.07
0.413 0.153 4.91 4.66 4.61 2.30 4.80
i
a;( T )
b;(T)
a;(S)
TL
0.302 0.185 0.302 0.991 0.956 0.991 0.950
3, TL ZR U L , JL
UR JR
FR
3
x2
ci
VL
QR JL
2,
0.153
4.41
+1.115
+2.602 -2.089 -0.118 -0.016 +0.164
-0.033
0.406 0.329 0.406 1.027 0.994 1.027 0.989
b;(S)
+0.616 +0.818
- 1.153 -0.110 -0.015 $0.152 -0.030
1R
(9)
where
di =( T3i-Q )tan2@,+ T3i
-+tan2@,,
d5 =-tan2@,),
and
(e.g.,
dnL=
+
~o=m,/m,n
%ese are renormaliked at the scale M,. In a more accurate treatment, the ci would be renormalized at the physical sparticle mass scale, leading to second-order shifts on the sparticle masses.
313 2168
50 -
LOPEZ, NANOPOULOS, PARK,WANG, AND WCHICHI
I
( 1.00,O. 95,O. 95)m, ml= (1.05,0.99,0.98)mE
moduli,
m -0.18mz, % i
(10)
dilaton
moduli: . m =0.30m,, ZL
m,R/m,L=0.61 for a3(Mz)=O. 110,O. 118,O. 126 (the dependence on t& is small). The squark splitting around the average is
, (11)
-0.33m,,
m
=2%. The first- and second-generation squark and slepton masses are plotted in Fig. 2. The thickness and straightness of the lines shows the small tar$ dependence, except for V . The results do not depend on the sign of p, except to the extent that some points in parameter space are not allowed for both signs of p: the p < 0 lines start off at larger mass values. Note that
dilaton:
% . m =0.41m,, FL
mzR/m,L=0.81
.
1
FIG. 1. T h e correlation between the lightest chargino mass rn and the next-to-lightest
*
XI
0
200
400
mg
600
BOO
(cev)
1000 0 200 q - 1 5 0 GeV
400
600
BOO
mE (GeV)
1000
neutralino mass m
8
(top row) for both signs of
p, m,= 150 GeV, and (a) the moduli and (b)di-
laton scenarios. Also shown (bottom row) is the absolute value of the Higgs-mixing parameter p versus the gluino mass. Two values of tar$ are singled out, larger ones tend to accumulate and are not individually discernible in the figure.
374
50
EXPERIMENTAL ASPECTS OF SU(S)XU(l)SUPERGRAVITY
can be fairly light, we always get m A > m h ,in agreement with a general theorem to this effect in supergravity theories [35]. This result also implies that the channel e +e - +h A at LEP I is not kinematically allowed in this model. The computation of the neutralino relic density (following the methods of Refs. [36,37]) shows that "2; S0.25(0.90) in the moduli (dilaton) scenarios. This implies that in these models the cosmologically interesting values "$; <, 1 occur quite naturally. These results are in good agreement with the observational upper bound on a$; [38]. As we have discussed, in the scenarios we consider all sparticle masses scale with the gluino mass, with a mild dependence (except for the third-generation squark and slepton masses). In Table I1 we collect the approximate proportionality coefficients to the gluino mass for each sparticle mass (not including the third-generation squarks and sleptons). From this table one can (approxi-
and exhibit a large variability for 6xed mg because of the ta@ dependence in the off-diagonal element of the corresponding 2 X 2 mass matrices. The lowest values of the TI mass go up with m yand can be as low as
I
160,170,190(155,150,170) GeV moduli, mi,5 88,112,150,(92,106,150) GeV dilaton
(12)
for m y=130,150,170 GeV and p> O(p
(13)
for rn,=130,150,170 GeV. Note that even though m A
(a 1
p ~ o
1000
no-scale SU(5)XU(l) moduli scenario 1000
BOO
BOO
600
600
400
400
200
200
0 0
200
400
mi
600
BOO
(cev)
I000
0 0
m,=150 GeV
200
P
400
600
BOO
1000
mi (GeV)
1000
BOO
600
400
200
0
2169
t
FIG. 2. T h e first-generation squark and slepton masses as a function of the gluino mass, for both signs of p, m,=150 GeV, and (a) the moduli and (b) dilaton scenarios. The same values apply to the second generation. The thickness of the lines and their deviation from linearity are because of the,small tans dependence.
375 LOPEZ, NANOPOULOS, PARK, WANG, AND ZICHICHI
2170
mately) translate any bounds on a given sparticle mass on bounds on all the other sparticle masses.
D. Specialcases 1. Strict no-scale case
We now impose the additional constraint B(M,)=O to be added to Eq. (2), and obtain the so-called strict noscale case [lo]. Since B ( M , ) is determined by the radiative electroweak symmetry-breaking conditions, this added constraint needs to be imposed in a rather indirect way. That is, for given mg and m, values, we scan the possible values of tar$ looking for cases where B ( M , ) = O . The most striking result is that solutions exist only for m,S 135 GeV i f p > 0 and for m,2 140 GeV if
p>o
no-scale SU(S)XU(I) dilaton scenario
W O
50 -
p < 0. That is, the value of m, determines the sign of p.
Furthermore, for p 0, tar$ can be double valued for some m, range which includes m,= 130 GeV. All the mass relationships deduced in the previous subsection apply here as well. The t d spread that some of them have will be much reduced though. The most noticeable changes occur for the quantities which depend most sensitively on tan& such as the Higgs-boson masses. Figure 5 of Ref. [l] shows that the one-loop-corrected lightest Higgs-boson mass is largely determined by m,, with a weak dependence on mg. Moreover, for m,5 135 GeV - p > 0, mh 5 105 GeV; whereas for m,5 140 GeV - p < 0, mh 2 100 GeV. Therefore, in the strict no-scale
FIG. 3. The TI, 6,.,, and ilmasses ., vs the gluino mass for both signs of p, m,= 150 GeV, and (a) the moduli and (b) dilaton scenarios. The variability in the 6,,2, and TI,* masses is because of the off-diagonal elements of the corresponding mass matrices.
376
50
EXPERIMENTAL ASPECTS OF SU(S)XU(l)SUPERGRAVITY
case, once the topquark mass is measured, we will know the sign of p and whether rnh is above or below 100 GeV.
TABLE 11. The approximate proportionality coefficients to the gluino mass, for the various sparticle masses in the two supersymmetry breaking scenarios considered. The 1p1 coe5cients apply for rn, = 150 GeV only.
2. Special dilaton scenario case
In the analysis described above, the radiative electroweak breaking conditions were used to determine the magnitude of the Higgs mixing term p at the electroweak scale. This quantity is ensured to remain light as long as the supersymmetry-breaking parameters remain light. In a fundamental theory this parameter should be calculable and its value used to determine the Z-boson mass. From this point of view it is not clear that the natural value of p should be light. In specific models one can obtain such
mi (GeV)
zR
-rVpR
2Xk&Xf Z L ~ L
I
Moduli
Dilaton
0.18 0.18-0.30 0.28 0.30
0.33 0.33-0.41 0.28 0.41 1.01 1 .oo 0.6-0.8
g
0.91 1.oo
1P1
0.5-0.7
mi (GeV)
m,=150 CeV
2171
FIG. 4. The one-loop corrected h and A Higgs-boson masses versus the eluino mass for -both signs of p, m,=150 GeV, and (a) the moduli and (b) dilaton scenarios. Representative values of tar@are indicated. I
600
0
200
400
600
BOO
1000
600
500
500
400
400
300
300
200
200
100
100
0
0
200
400
600
BOO
mi (GeV)
1000
0
0
200
400
600
BOO
1000
0
200
400
600
BOO
1000
m,=150 GeV
mi (GeV)
311 ~
2172
~
~~
~
LOPEZ,NANOPOULOS, PARK,WANG, AND ZICHICHI
values by invoking nonrenormalizable interactions [39,40,5]. Another contribution to this quantity is generically present in string supergravity models [41,40,12]. The general case with contributions from both sources has been effectively dealt with in the previous section. If one assumes that only supergravity-induced contributions to p exist, then it can be shown that the B parameter at the unification scale is also determined [12,13],
TABLE 111. The range of allowed sparticle and Higgs-boson masses in the special dilaton scenario. The topquark mass is restricted to be rn, < 155 GeV. All masses in GeV. rn,
130
150
155
g
335-1000 38-140 75-270 1.57- 1.63 61-74 110-400 335-1000 >400
260-1000 24-140 50-270 1.37- 1.45 64-87 90-400 260-1000
640- 1000 90- 140
a3 2,Xf tanS h
(14) which is to be added to the set of relations in Fq. (1). This new constraint effectively determines t a d for given m, and mg values and makes this restricted version of the model highly predictive (171. It can be shown [17] that only solutions with p < 0 exist. A numerical iterative procedure allows us to determine the value of tar$ which satisfies Eq. (14), from the calculated value of B ( M z 1. We find that
T A,H,H+
>400
170-270 1.38-1.40 84-91 210-400 640- lo00 > 970
[34]. This requires the largest possible topquark masses and a not-too-small squark mass. However, perturbative unification imposes an upper bound on m, for a given tar$ [32], which in this case implies [ 181
rn, 5 155 GeV ,
tax$= 1.57-1.63,1.37-1.45,1.38-1.~ for m,=130,150,155 GeV
(15)
is required. Since t a d is so small (mff’z28-41 GeV), a significant one-loop correction to mh is required to increase it above its experimental lower bound of ==60 GeV no-scale SU(5)xu(1) moduli scenario
( a ) P’O
50 -
(16)
which limits the magnitude of mh: mh 574,87,91 GeV for rn,=130,150,155 GeV
.
(17)
In Table 111 we give the range of sparticle and Higgsboson masses that are allowed in this case.
P< 0
................................
50
100
150
200
250
50
100
m,; (Gev)
200
FIG. 5. The parameter space for no-scale
250
SU(5)XU(])supergravity (moduli scenario) in the (rnxf,tan,9) plane for (a) rn,=130 GeV, (b)
m,: (GeV)
. : OK +
150
m,=130 GeV
m,=150 GeV, (c) m,=170 GeV, and (d)
b-sy x
x . (84, x 0:NTx 0 : El-eb x
GeV. T h e periods indicate points that passed all constraints, the plusses fail the B ( b + s y ) constraint, the crosses fail the (g-2), constraint, the diamonds fail the neutrino telescopes (NT) constraint, the squares fail the E , - E ) constraint, and the octagons fail the updated Higgs-boson mass constraint. The reference dashed line highlights rn *=1M) rn, = 180
o:m,r
25L
..
20
#=I::[ I
......
d
................ .......................... ..................................
XI
GeV, which is the direct reach of LEP I1 for chargino masses. Note that when various symbols overlap a more complex symbol is ob-
tained.
50
100
150
200
250
50
100
150
200
mx: (Gev)
m,: (GeV) m,=150 GeV
250
378
50
EXPERIMENTALASP-
OF SU(5)X U(1) SUPERGRAVITY
JIL CONSTRAINTS ON PARAMETER SPACE
k b+sy
In this section we describe the experimental constraints which have been applied to the points in the basic parameter space described in Sec. 11. Each of these constraints leads to an excluded area in the (mx:,tar@) plane for a fixed value of m,. Since all sparticle masses scale with m l R , the lightest chargino mass is as good a choice as any other one, and has the advantage of being readily measurable. Our choices for m,, i.e., m,=130,150,170,180 GeV are motivated by the direct lower limit on the top-quark mass from Tevatron searches ( m , > 131 GeV [42]) and by the indirect estimates of the mass from fits to the electroweak data (mf=140f20 GeV [29,30]). The effect of each of the constraints is denoted by a particular symbol on the parameter space plots in Figs. 5-8 for the various scenarios under consideration. In all these figures there is an eyeguiding vertical dashed line which corresponds to rn = 100 GeV. The purpose of Fig. 7 is to show where
*
XI
such a line lies in the (m,,t@) plane. Kinematically speaking, the weakly interacting sparticles (Le., charginos) are more accessible than the strongly interacting ones Le., gluino and squarks).
no-scale SU(5)W(l) moduli scenario
-__.
50
100
150
200
........................
50
100
m,: (Gev) . : OK + : b-ry x
The rare radiative flavor-changing-neutral-current (FCNC) b +s y decay has been observed by the CLEO11 Collaboration in the 95% C.L. range B ( b - + ~ y ) = ( O . 6 - 5 . 4 ) X l O -[43]. ~ Since large enhancements and suppressions of B ( b - + s y ) , relative to the standard model value, can occur in SU(S)XU(l)supergravity, the above allowed interval can be quite restrictive [ 19,201 (see also Ref. [44,45]). The results of the calculation in the moduli and dilaton scenarios are given in Refs. [19,20]. In both scenarios there exists a significant region of parameter space where B ( b - + s y ) is highly suppressed [19,20]. The points in parameter space which are excluded at the 95% C.L. are denoted by pluses (+) in Figs. 5 and 6 for the moduli and dilaton scenarios, respectively, and for the four chosen values of m , . The strict no-scale scenario [see Fig. 8(a)] is also constrained in this fashion, although only for m, = 130,150 GeV. The special dilaton scenario is not constrained by B ( b - + s y ) [see Fig. 8(b)]because of the small values of ta~$ required in this case. Note that the constraints are generally much stricter for p > 0.
P
......
250
150
ZOO
250
m,: (CeV) m,=170 GeV
x : (6-2)” x
FIG. 5. (Continued).
0:NTx
n : c,-s,
x
0:m.x
30 25 20
2
15
6
d
10
5 0
50
2173
100
150
200
250
50
100
m,: (Gev)
150
ZOO
m,: (Gev) m,=IBO GeV
250
379 50 -
LOPEZ,NANOPOULOS, PARK,WANG,AND ZICHICHI
2174
.....
I
30
30
20
20
10
LO
50
100
150
200
250
.................................
FIG. 6. The parameter space for no-scale SU(5)XU(1) supergravity (dilaton scenario) in the ( m *,tar$) plane for (a) m,= 130 GeV, (b) 50
100
150
200
250
XI
m,=150 GeV, (c) m,=170 GeV, and (d) m,=180 GeV. The periods indicate points
that passed all constraints, the pluses fail the B ( b - t s y ) constraint, the crosses fail the (g-2), constraint, the diamonds fail the neutrino telescopes (NT) constraint, the squares fail the el - eb constraint,and the octagons fail the updated Higgs-boson mass constraint. The reference dashed line highlights m * = I 0 0 XI
GeV, which is the direct reach of LEP I1 for chargino masses. Note that when various symbols overlap a more complex symbol is obtained.
50
100
150
200
250
50
100
150
200
250
m,: (Gev)
m,: (Gev)
m,=150 GeV
B. (g-2),, The supersymmetric contributions to a p = t ( g -2)p in SU(5)XU(1) supergravity have been recently computed in Ref. [21], and have been compared with the presently allowed 95% C.L. interval - 1 3 . 2 X 1 0 - 9 < a ~ <20.8 X In this paper it was noted that a contribution to u p , which is roughly proportional to tar$, leads to enhancements which can easily make a y run in conflict with the present experimental bounds. The points in parameter space which are excluded at the 95% C.L. are denoted by crosses ( X ) in Figs. 5 and 6 for the moduli and dilaton scenarios, respectively, and for the four chosen values of m,. As expected, the ( g -2),, constraint has a similar effect for the two signs of p, and exclude the larger values of tax$ which are allowed for chargino masses up to about 100 GeV. The constraint appears less effective for rn, =170,180 GeV (Le., there are fewer crosses), but this is just because for the larger values of m,,is cutoff at smaller values. The strict no-scale scenario [see Fig. 8(a)] is also constrained in this fashion, although only for rn, = 130,150 GeV. The special dilaton scenario is not constrained by ( g -2 )p [see Fig. 8(b)] because of the small values of tax$ required in this case (i.e., tm$ < 1.64).
C. Neutrino telescopes
Neutralinos in the galactic halo which are gravitationally captured by the Sun or Earth [46,47], annihilate into all possible ordinary particles, and the cascade decays of these particles produce high-energy neutrinos as one of several end products. These neutrinos can then travel from the Sun or Earth cores to the vicinity of underground detectors (“neutrino telescopes”), and interact with the rock underneath producing detectable upwardly moving muons. The calculation of the upwardly moving muon fluxes induced by the neutrinos from the Sun and Earth in SU(5)XU(1) supergravity has been performed in Ref. [24]. The present experimental constraints from “neutrino telescopes” on the parameter space are quite weak, as evidenced by the few excluded points in Figs. 5 and 6 (denoted by diamonds “@*). In fact, the Kamiokande upper bound from the Earth capture is only useful to exclude regions of the parameter space with rnX=rnFc due to the kinematic enhancement in the capture rate. Because of the weakness of this constraint, the effect has not been calculated for the special scenarios in Fig. 8. Nonetheless, future improved sensitivity in underground muon detection rates should make this constraint
380 50 -
EWERZMENTAL ASPECTS OF SU(5)XU(1) SUPERGRAVITY
.......
40
I
2175
..::::::::::::::::::::;d
.... ................... .............
30 Q 6 4
20
10
0
50
100
150
200
250
50
100
150
200
250
+
FIG. 6. (Continued).
X
e 0 0
Q
6
4
50
100
150
200
m,; ( G e v )
250
50
100
m,=lBO GeV
150
ZOO
250
m,: (GeV)
rather important, if neutralinos indeed constitute a significant portion of the dark galactic halo.
D. Updated precision electroweak tests
Among the various schemes to parametrize the electroweak vacuum polarization corrections [ 48-5 11, we choose the so-called E scheme [52,53] which is more suitable to the electroweak precision tests of the MSSM [54] and a class of supergravity models [22]. There are two E schemes. The original scheme [52] was considered in our ~ , defined ~ from a previous analyses [22,20], where E ~ , are basic set of observables rr,AkB and M,/M,. Because of the large m,-dependent vertex corrections to rb, the E , , ~ parameters , ~ and r b can be correlated only for a fixed value of m,. However, in the new E scheme, introduced recently in Ref. [53], the above difficulties are overcome by introducing a new parameter, eb, to encode the Z+b6 vertex corrections. The four 6's are now defined from an enlarged set of rr,r b , AkB and M , / M , without even specifying m,. Here we use this new E scheme. Experimentally, including all of the latest LEP data (complete 1992 LEP data plus preliminary 1993 LEP data) allows one to determine most accurately the
allowed ranges for these parameters [29]: ~YP'=(l.8t3.1) X
,
~P'=(-0.5i5.1)X10-3.
(18)
We only discuss since only these parameters provide constraints in supersymmetric models at the 90% C.L. [22,25]. The expressions for E, and Eb have been discussed in Ref. [23]. Compared with the previous experimental values for the E parameters obtained by including the complete 1992 LEP data [56] (which were used in Ref. [23]) those in Eq. (18)have moved in such a way that the standard model predictions have become in better agreement with LEP data than before [29,30]. In Fig. 9 we present the results of the calculation of and E* (as described above) for all the allowed points in SU(5)XU(l) supergravity in both moduli and dilaton scenarios, and for rn, = 130,150,170,180 GeV. In the figures we include three experimental ellipses representing the 1 - u (from Ref. [29]), 90% C.L., and 95% C.L. experimental limits obtained from analyzing all of the latest LEP electroweak data. The shift in the experimental data corresponds to a shift in the center point of the ellipses towards larger values of el and smaller values of E b . As a consequence, at the 90% C.L. there are no constraints from Eb alone
38 1 LOPEZ, NANOPOULOS, PARK, WANG, AND ZICHICHI
2176
(a)
no-scale s U ( 5 ) x U ( l ) moduli scenorio
PO
50 -
P
I 25
I
I
d..... -t 15 -... ;
............. .................. ......................... ............................ ...........,.......................... .,.,...................... ,..................
-.-===2!;[
-. _.I*. -.
10
.......
ZOO
. -
.,......................... 400
600
. ....................
,,
800
ZOO
400
600
FIG.7. The parameter space for the moduli and dilaton SU(5)XU11 supergravity scenarios in the (m,,ta$) plane for m,=150 GeV. T h e meaning of the various symbols is the same as in Figs. 5 and 6. The dashed line marks the contour of mxF=lOO GeV k.f.
800
Figs. 5 and 6 ) and makes apparent the kinematical disadvantage of searching for the heavier squarks and gluinos, HS opposed to the lighter charginos. 40
30
a
5
20
10
0
200
400
600
800
200
400
600
800
mi=mc (GeV)
mi=m6 (GeV) m,=150 GeV
(cf. Ref. [23]). Nonetheless, the imposition of the correlated constraint Ke., the ellipses), is significantly more restrictive than imposing the c1 constraint by itself. For both scenarios, the effects of light charginos (x:) and stop squarks (?1,2), as described above, are rather pronounced. At the 90% C.L. there are no constraints for m,5 170 GeV, but for m,=180 GeV only very light charginos ( m 5 7 0 GeV) are allowed. Should the top
rather light values of the chargino mass. Moreover, such are very likely to be excluded by othlight values of m
quark be rather heavy, this light-chargino effect would appear to be a sensible explanation. Now we choose to constrain the parameter space by demanding theoretical predictions which agree with experiment to better than 90% C.L.; i.e., we exclude points in parameter space which are outside the 90% C.L. ellipses in Fig. 9. This constraint entails restrictions for m,= 180 GeV only. Note that from our calculations, all m,= 180 GeV points are allowed at the 95% C.L. However, more comprehensive analyses [29,30] already exclude m,=180 GeV at the 95% C.L. Le., mt=140f20 GeV), and thus our restriction is in practice likely to be more statistically significant than can be surmised from our analysis alone. The excluded points in parameter space are shown as squares “El” in Figs. 5(d), 6(d), and 8(a). The effect of this constraint is severe and requires
The process of interest is pp-x%:X, where both neutralino and chargino decay leptonically: x’&xyZ + I -, and x:-+x~Z*v,, with Z=e,p. The production cross section proceeds through s-channel W* exchange and tchannel squark exchange (a small contribution). This signal, first studied in Ref. [57], has been explored in SU(5)XU(1)supergravity in Ref. [25]. The first experimental limits obtained by the DO [58] and the Collider Detector at Fermilab (CDF) [59,60] Collaborations have been recently announced. The irreducible backgrounds for this rocess are very small, the dominant one being pp+W ~ - + ( Z * V , ) ( T + T - ) with a cross section into
*
XI
er constraints, as the figures show. This means that a “light chargino effect” may not be a viable way out from a possible experimentally heavy top quark.
E. Trileptons
*
Xl
P
tdeptons of (-1 ~ b ) ( ~ ) ( 0 . 0 3 3 ) ( 0 . 3 4 ) ~fb. - 1 Much larger “instrumental” backgrounds exist when for example inpp-+Zy, the photon “converts” and fakes a lepton in the detector; with the present sensitivity, suitable cuts
382 2177
30 25 20
2
..__... -. 15
6
4
10
170
5
180
0
50
100
150
ZOO
250
FIG.8. The parameter space for (a) strict no-scale and (b)special dilaton SU(5)X U(1)supergravity in the ( m +,tar$) plane. The XI
meaning of the various symbols is the same as in Figs. 5 and 6.
t
i
155
j
have been designed to reduce this background to acceptable levels [58]. The present experimental limits [58-601 from the Tevatron are rather weak, with sensitivity for rnxF 550 GeV only [59]. In the case of SU(5)XU(1)supergravity, no points in parameter space are excluded by the present experimental limits. However, with the projected increase in integrated luminosity during 1994, this experimental constraint could soon become relevant, as we discuss in Sec. V A below.
F. Updated Eggs-boson mass limit The current LEP I lower bound on the standard model (SM) Higgs-boson mass stands at m H > 63.8 GeV [61]. This bound is obtained by studying the process e +e - +Z * H with subsequent Higgs-boson decay into two jets. The MSSM analogue of this production process leads to a cross section differing just by a factor of sin2(a-8). In Ref. I341 it was shown that in supergravity models with radiative electroweak symmetry breaking, as is the case of SU(5)XU(1) supergravity, the lightest Higgs boson behaves very much like the standard model Higgs boson. In particular, the sin2(a-p) factor approaches unity as the supersymmep mass scale is raised. The branching fraction B ( h +bb ) also approaches the
standard model value, although one has to watch out for new supersymmetric decays, most notably h-xyd. In any event, a straightforward procedure to adapt the experimental lower bound on the standard model Higgsboson mass to the supersymmetric case is described in Ref. [34]. The following condition must be satisfied for allowed points in parameter space [34,62]:
where MFinis the experimental lower bound on the standard model Higgs-boson mass ke., Mgin=63.8 GeV), and we have used the fact that the cross sections differ only by the coupling factor sin2(a-/3) and the Higgsboson mass dependence, which enters through a function P [63]:
- 3(y4-6y 2+4 )lny - +( 1- y 2
)(2y4- 13y $47 ) . (20)
The determination of the basic parameter-space in Sec.
383
no-scale SU(5) x U(l)
(a t
11, includes the LEP experimental limits on sparticle masses and the experimental limit ?&-=61.3 GeV. The updated experimental limit of MHm=63. 8 GeV excludes some further points in parameter space (denoted by octagon symbols in Figs. 5, 6, and 8)for the smallest values of t d a n d m,=130,150 GeV.
moduli scenario
’
-
10
I
5 -
%a
50 -
LOPEZ, NANOPOULOS,PARK,WANG,AND ZICHICHI
2178
O -
-
-5
-10
-I
IV. ALLOWED PARAMETER SPACE n
~
-4
-2
0
2
4
6
8
I 10
95%
10
The constrained parameter spaces shown in Figs. 5, 6, and 8 show some regularities which are worth pointing out. First, the constraints for p < 0 are generally weaker than those for p > 0. It is also clear that the region to the left of the dashed line ( m < 100 GeV) is rather restrict-
*
XI
5 3-10 -50
ed. This region represents the area of sensitivity at LEP I1 from direct chargino searches. (LEP I1 could greatly extend this region through Higgs-boson searches though.) For m,=180 GeV things are very constrained. The most important constraint comes from the E, - E ~ ellipses in Fig. 9. Moreover, the remaining allowed points, which require rather light chargino masses ( m 5 70 GeV), are
-
*
XI
-4
-2
0
2
4
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El
*
no-scale SU(5) x U(l) dilaton scenario
(b
quite often in conflict with other experimental constraints. The few remaining points in parameter space have t a d 5 8 (12)in the moduli (dilaton) scenario. Also, m x f 5 68 (66) GeV and m 5 6 5 (68) GeV for p > 0 XI
( p< 0) in the moduli and dilaton scenarios, respectively.
L
a -4
J -2
0
2
4
6
8
10
-2
0
2
4
6
8
10
I
10
5 t
f
1 -4
4 FIG. 9. The correlated values of el and E~ (in units of for both signs of p, m,=130,150,170,180GeV, and (a) the moduli and (b) dilaton scenarios. The ellipses represent the lo, 90% C.L., and 95% C.L. experimental limits obtained from analyzing all LEP electroweak data.
For both scenarios, a more sensitive measurement of the b + s y branching fraction is likely to probe the remaining allowed points for m,=180 GeV. Also, for p < O in both scenarios, the expected increased sensitivity in trilepton searches is likely to probe about half of the remaining points. It is interesting to wonder if the present experimental constraints show any preference for particular values of the top-quark mass. To explore this question we carry out the following exercise: we count the number of points in parameter space which are allowed for a fixed value of m,. We do this in two steps (see Fig. 10): (i) 6rst imposing only the basic theoretical and LEP experimental constraints Ytheory+LEP”) and (ii)imposing in addition all of the experimental constraints described in Sec. I11 (“all”). The result in Fig. 10 is interesting. The drop in the “theory +LEP” curves near m,= 190 GeV has been studied in detail (for m,=180,185,188,189GeV) and corresponds to encountering a Landau pole in the top-quark-Yukawa coupling below the string scale [32]. The “all” curves show some m, dependence, although at the moment no marked preference for particular values of rn, is apparent (other than the requirement of m, 5 180 GeV). Note that in spite of the intricate dependence of the sparticle and Higgs-boson masses on the top-quark mass (through the running of the renormalization group equations (RGE’s) and the radiative breaking mechanism), the overall size of the parameter space does not depend so critically on m,. One can repeat the above exercise to see if any trends
384
50
EXPERIMENTAL ASPJEl3 OF SU(5)XU(I)SUPERGRAVITY no-scale sV(5)XU(l) moduli scenario
P>O
2
2179
P< 0
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9
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g
800 theory+LEP
theory+LEP
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n
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(GeV)
dilaton scenario
ml
(Gev)
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9 1000
-g
-
BOO
800
0
800
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2
400
400
2
200
theory+LFP
C
FIG. 10. The number of allowed points in parameter space of the moduli and &ton SU(5)XU(I) supergravity scenarios as a function of m, when the basic theoretical and experimental LEP constraints have been imposed (“theory+LEP”), and when all known direct and indirect experimental constraints have been additionally imposed (“all”).
D
n
200 n
~
130
140
150
170
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180
130 140 150 160 170 160 190
190
m, (GeV)
m, ( G e v )
on the preferred value of tan,3 appear. This time we count the number of allowed points in parameter space for a given value of tan& for fixed m,.The resulting distribution is shown in Fig. 11 for m,= 150 GeV. Qualitatively similar distributions are obtained for other values of m,. In this case we discover that for ,u > 0 there is a significant preference towards the smaller values of t a g . This result is apparent from Figs. 5 and 6 also, and is mostly a consequence of the b -+sy constraint. Future improvements in sensitivity on the experimental constraints which we have imposed here, or the advent of new experimental constraints, may sharpen the “predictions” for the preferred values of m, and t a d obtained in this statistical exercise.
p ~ o
no-scale SU(5)XU(l) moduli scenario
V. PROSPEKTS FOR DIRECX EXPERIMENTAL
DETECTION In this section we consider the still-allowed parameter space, i.e., the points marked by dots in Figs. 5, 6, and 8, and the prospects for their direct experimental detection. In this section we consider only the representative value of m,= 150 GeV. A. Tevatron
The Tevatron can explore the supersymmetric spectrum through the traditional missing energy signature in the decay of the strongly interacting gluinos and squarks, or through the trilepton signal in the decay of the weakly
P
0
L
B
6 C
20 0 0
5
10
15
20
tanp pn
25
30
tanp
dilaton scenario 80
80
FIG. 11. The number of good points in parameter space in the moduli and dilaton SU(S)XU(I) supergravity scenarios as a function of tar$ for m,=150 GeV. All known direct and indirect experimental constraints have been imposed. Note that for p > 0 there is preference for not so large values of tar$.
C
8
60
60
2 40
40
‘El
0
0 L
B
20 0
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tan@
30
0 40 0 m,=150 GeV
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40
385 ~
2180
~
LOPEZ, NANOPOULOS, PARK, WANQ, AND ZICHICHI
50 -
Fig. 13 we show the analogous results for the strict noscale and special dilaton scenarios, neither of which show a suppression for light chargino masses. The present experimental limits from CDF have been obtained by analyzing approximately 18 pb-' of data. By the end of the 1994 run it is expected that each detector will be delivered 75 pb-I, of which CDF should be able to collect say 80%. Therefore, CDF could expect to have about 80 pb-' by the end of the run,which is 4 times the present amount of data. A similar situation is expected from DO. Moreover, the center-of-mass energy will be increased to nearly 2 TeV, which implies a = 30% increase in the chargino-neutralino cross section for 100 GeV charginos. Since tougher cuts will be required to suppress the backgrounds with the increased sensitivity, as a working estimate we have assumed that the new lim-
interacting charginos and neutralinos. Here we concentrate on the latter signal, whose calculation has been described in Sec. IIIE. The cross section c r ( p ~ - ~ ~ ~ '(for X :6 x ~=)1.8 TeV) is shown in Fig. 12 for m,=150 GeV in the moduli and dilaton scenarios (top row), and shows little variation from one scenario to the other. Moreover, the results for other values of rn, are qualitatively the same and quantitatively quite similar. On the bottom row of Fig. 12 we show the cross section into trileptons, i.e., with the leptonic branching fractions included. The 95% C.L. experimental upper limit from CDF is also indicated.' As mentioned above, in the moduli scenario the neutralino leptonic branching fraction can be suppressed for light chargino masses. Note that in the dilaton scenario such suppression is not manifest because of the heavier sparticle mass spectrum. In
FIG. 12. The cross section a(pF-xfdX) at the Tevatron (top row) as a function of rn
*
Xl
( b ) p>O
Tevatron
10-3
50
100
150
no-scale SU(S)XV(l) dilaton scenario
200
250
10-3 300
50
100
m,: (Gev)
-
for rn, = 150 GeV in (a) the moduli and (b) the dilaton SU(5)X U(1) supergravity scenarios. Also shown (bottom row) is the cross section into trileptons, with the 95% C.L. experimental upper limit from CDF as indicated.
/I
150
200
250
300
250
300
m,: (GeV)
100
P
....:.....\ . .
i
...........
.. ......
10-3
50
100
150
10-2
.-.
200
250
mx; (Gev)
10-3 300
50
m,=150 CeV
100
150
ZOO
m,: ( G 4
3Note that the experimental numbers in Ref. [59] apply to a single channel (i.e., eee, eep, epp, or ppp), and need to be multiplied by four to be compared with our predictions for the total e + p trilepton rate.
SU(5)XU(U SUPERGRAVITY
--
h
n
3.
......,
v
10-1
r
P
2
'..._
10-2
T
i (a)
-
7
v
n a v
k!b!, 50
m,=150 GeV
, jl,,, I,,,
100
150
1 1
,,,,I,,
200
250
,I 300
m,: (GeV) FIG. 13. The trilepton cross section at the Tevatron as a function of m f for m,= 150 GeV in (a) the strict no-scale and XI
(b) the
special dilaton SU(5)XU(I)supergravity scenarios. The 95% C.L. experimental upper limit from CDF is indicated.
kinematics, still a strong constraint will follow for a large class of supersymmetric models, in particular the ones under consideration here. In Fig. 15 we show the lightest a d for m, = 150 GeV in the Higgs-boson mass versus t moduli and dilaton scenarios. Along each vertical line the chargino mass increases from bottom to top. The dotted portions of the lines are already excluded by the various constraints discussed in Sec. 111. For m,=150 GeV we find mh < 118 GeV. In Fig. 16 we consider the strict no-scale and special dilaton scenarios. Since in these cases the value of tar$ is determined (see Fig. 8), the plot is against the chargino mass. In both Figs. 15 and 16 the horizontal line indicates the limit of sensitivity of LEP 11for 6=200 GeV, as we shortly discuss. First let us note, as pointed out in Ref. [ 171, that the special dilaton scenario (see Fig. 16) should be completely explored at LEP I1 (even with 6 = 190 GeV) since m, S 155 GeV is required in this case (see Sec. I1D 2). In SUW XU(1) supergravity, the dominant Higgsboson production mechanism at LEP I1 is e+e-+Z*+Zh. This cross section differs from its standard model counterpart only by a factor of sinz(a-fO. Here we find that generally sin2(a -~)>0.96,in agreement with a general result to this effect [34].4 The usual analysis of the b-tagged_ Higgs-boson signal at LEP I1 also requires the k+bb branching fraction. If we define f = B ( h +b6)/B(HsM+b6), then the expected limit of sensitivity at LEP 11, u(e+e-+Z*+ZHSM)>0.2 pb [611 becomes a ( e + e - - + Z * + Z h ) X f > 0 . 2 pb
its (if no signal is observed) will be down by a factor of 4 from those shown on Fig. 12. We are then able to identify points in the still-allowed parameter space which could be probed by the end of 1994. (The Tevatron will likely not run again until 1997.) These are shown as pluses (+) in Fig. 14 for the moduli and dilaton scenarios. Note that with the increased sensitivity, chargino masses as high as =I00 GeV could be probed in the moduli scenario. The rates are smaller in the dilaton scenario. Note also that this probe is much more sensitive for p < 0 (seeFig. 12). Searches for squarks and gluinos at the Tevatron are at kinematical disadvantage in the model under consideration. Indeed, compare the relative position of the dashed vertical line on Fig. 14, with the corresponding line on Fig. 7. The near-future trilepton searches correspond mostly to gluino and squark masses in the range (300-400)GeV, which are likely beyond the direct reach of the Tevatron for the same data set.
B. LEPII
2181
.
(21)
This size signal is needed to observe a 3u effect over background with L=500 pb-'. Our results for this quantity are shown in Fig. 17, along with the sensitivity limit in Eq. (21). Note that most points in parameter space accumulate along a well-defined line. This line corresponds to the standard model result. (Deviations from the line are discussed below.) For 6 =200 GeV, the limit of sensitivity in 4. (21) translates into mh S 105 GeV, while for & =210 GeV, mh S 115 GeV is obtained. From Fig. 17 it would still appear that even with 6 = 2 1 0 GeV, some points in parameter space for m, c 150 GeV would remain unreachable. However, detailed studies [61] show that for Higgs-boson masses away from the Z pole (here we are interested in mb 0115-118 GeV) the limit of sensitivity could be improved to (0.05-0.15) pb, and thus the whole parameter space for m,=lSO GeV and all scenarios considered could be explored at LEP I1 with 6=210 GeV. Note that the same conclusion is obtained for m, < 150 GeV, since the values of mh are lower then (e.g., for m,=130 GeV we lind mh 5 105 GeV), and even smaller luminosities or beam energies may suffice.
1. Lightest Higgs boson
Perhaps the single most useful piece of information that could come out of LEP 11 is a measurement of the lightest Higgs-boson mass. Moreover, if the Higgs boson is not observed at LEP 11,because of limited statistics or
41n SU(5)XU(1) supergravity, the e 4 e - - h A channel is rarely kinematically allowed at LEP I1 (since m A > m h )and is further suppressed by the small values of cos2(a-8).
387 LOPEZ,NANOPOULOS,PARK,WANG, AND ZIC!I-KlcHI
2182
no-scale SU(S)xU(l) moduli scenario
(a) w 0
,,
.
50 -
(1
25
105 1
20
.t_
.
..
.
... ..
FIG. 14. The still-allowed parameter space in the ( m *,tar@) plane for m,=150 GeV in
40
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00
120
40
80
60
100
120
XI
(a) the moduli and (b) the dilaton SU(S)XU(l) supergravity scenarios. The pluses ( + I indicate points explorable with near-future trilepton searches at the Tevatron, the crosses ( X ) will be explorable at LEP Jl (with f=500 pb-I) through the mixed mode in chargino pair production, and the diamonds (0) will be explorable at LEP I1 (with f =500 pb-I) through the dilepton mode in selectron pair production. Contours of the lightest one-loop corrected Higgs-boson mass are as indicated (i.e., for mh = 80,90,100,105,110 GeV). With 6 = 2 0 0 (210) GeV it should be possible to explore at LEP I1 up to mh= 105 (115)GeV.
+ trilepton searches at FNAL x chargino searches at LEPII 0
selecthn searches at UP11
no-scale SW(S)XU(1) dilaton scenario
(b) w0
P
40
a
s
c
no-scale SW(S)XW(l) moduli scenario
P> 0
P
120
E'
80
60
5
10
15
20
tanp
25
30
60
5
10
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25
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tan@
dilaton scenario
120
E"
60
60
60
10
tan@
m,=150 GeV
20
tanp
30
40
FIG. 15. The lightest one-loop-corrected Higgs-boson mass vs ta43 in the moduli and dilaton SU(5)XU(1) supergravity scenarios, for m,=150 GeV. The dotted portions of the verticd lines indicate excluded ranges of mh. The horizontal line marks the limit of sensitivity of LEP I1 with 6 =200 GeV.
388 ~
50 -
EXPERIMENTALASPstrict no-scale S U ( ~ ) W ( I )
50
100
150
200
OF SU6)XU(1) SUPERGRAVITY
(p
300
250
special dilaton SU(S)xU(l) (p
120 110 h
%
100
0
v
€7
90 BO
70
t I&=, m,=150 GeV
.....-
60
50
100
150
200
250
,I 300
FIG. 16. The lightest one-loop corrected Higgs-boson mass vs ta~$ in the strict no-scale (for rn, = 150 GeV) and the special dilaton (for m, = 130,150,155 GeV) SU(5)XU(l)supergravity scenarios. The horizontal line marks the limit of sensitivity of LEP I1 with 6 =200 GeV. Note that LEP I1 should be able to explore all of the parameter space in the special dilaton scenario. In Fig. 17 there are some points which ‘‘fall oil” the main c u r v ~ . These correspond to suppressed values of the h -+bb branching fraction Le., f < 1) which occur when the invisible supersymmetric decay channel h -+&& is kinematically open [MI, as shown in Fig. 18. However, the fraction of points in parameter space where this happens is rather small (less than 10%). Nonethe-
0.8
b
2183
less, most of these special points are still within the limit of sensitivity in Eq. (21) and should not escape detection. The scarcity of points in parameter space where the Higgs boson could decay invisibly may discourage detailed studies of such signature in SU(S)XU(l)supergravity. However, when the invisible mode is allowed, its branching fraction can be as large as 60%. We can see the effect on the parameter space of a possible measurement of mh by studying the Higgs-boson mass contours shown in Fig. 14, or for the full parameter space in Fig. 19. In general one would obtain a constraint giving tar$ for a given chargino mass. Moreover, a minimum value of the chargino mass would be required, if mh 2 100 GeV. Furthermore, in the strict noscale and special dilaton scenarios, the chargino mass itself would be determined (see Fig. 16) and thus the whole spectrum. If only a lower bound on mh is obtained, still large portions of the parameter space could be excluded, i.e., all of the areas to the left of the corresponding mass contour. What if m,> 150 GeV? For m,= 170 GeV, one obtains mh S 128 GeV and V‘x =240 GeV would be required for a full exploration of the parameter space at LEP 11. We close this section with a last-minute remark. Twoloop QCD corrections to mh have been recently shown to decrease the Higgs-boson mass by a non-negligible amount [65]. A complete calculation of this effect in SU(5)X U(1) supergravity is beyond the scope of this paper. A rough assessment of the effects indicates that the Higgs-boson mass contours in Fig. 19 (see also Fig. 14) would likely shift to lower values. This downward shift implies an enlarged reach for LEP 11. Equivalently, the above conclusions would require even lower values of the center-of-mass energy or integrated luminosity. 2. Charginos
The cross section for chargino pair production is the largest of all cross sections involving charginos and neu-
4
FIG. 17. The cross section a k + e - - Z h ) X f vs rn, at LEP I1 for d s =200,210 GeV and rn,=150 GeV in the moduli and dilaton SU(5)XU(1) supergravity scenarios. Here f = B ( h + b F ) / B ( H S M - b 6 ) . Except for the relatively few points deviating from the main curves, the result is very close to the standard model one. The dashed line indicates the expected level of sensitivity attainable at LEP 11.
389 LOPEZ, NANOPOULOS, PARK, WANG, AND ZICHICHI
2184
50 -
FIG.18. The branching fraction B ( h + b 6 ) vs mh for m,=150 GeV in the moduli and dilaton SU(5)XU(1) supergravity scenarios. Note the points which deviate significantly from the standard model expectation (of =O. 85 owing to the contribution to the total width from the h +xyd channel.
tralinos at LEP 11. In the context of SU(5)XU(1) supergravity this has been shown in Ref. [a]. The most studied signature is the so-called mixed mode, where one chargino decays leptonically and the other one hadronically. If the chargino decay channels are dominated by Wexchange Le., branching ratio into electron+muon is
50
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no-scale SU(5)xu(l) moduli scenario
N O
250
150
50
100
200
8
and branching ratio into jets is 4) then the mixed channel has a rate six times larger than the dilepton channel. The mixed signature still has to contend with the W'W- background. However, a series of cuts have been designed which take advantage of different values for the missing mass, the mass of the hadronic system,
250
FIG.19. T h e Higgs-boson mass contours in the (mx*,ku$) plane for m , = 150 GeV in the 1
(a) moduli and b) dilaton SU(S)XU(l)supergravity scenarios. The dots represent the stillallowed points in parameter space. For p < 0 in the dilaton case, the labeling of the mass contours is as for p > 0. 40
30
20 10
0
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390
50
EXPERIMENTALASPECTS OF SU(5)XUW SUF'ERGRAVrrY
and the mass of the lepton+neutrino system when one considers the background and the signal separately [66]. In the case of SU(5)XU(1) supergravity, W exchange is not expected to dominate in chargino decay [25]. In fact, in the moduli scenario the sleptons are lighter and can therefore be on shell, thus enhancing the leptonic branching fraction to its maximum value o f f . When this occurs the mixed signal is negligible, because of the much suppressed hadronic branching fraction. In Fig. 20 we show the cross section for the mixed signal at LEP I1 for .\/5 =200 GeV and rn, =150 GeV, for both moduli and dilaton scenarios. As expected, the mixed rate is small (even vanishing for p < O!) in the moduli scenario. In the dilaton scenario the rate is larger, but still much smaller than the corresponding rate in a model where W ex-
2185
change dominates chargino decays. This situation in fact occurs in the minimal SU(5)supergravity model where the rate is typically in the range of (1.5-2) pb, as shown in Fig. 3 of Ref. [25]. The signal is further suppressed in the dilaton scenario because of a negative interference effect between the ?-channel sneutrino-exchange and the s-chamel y' and Z' exchange [64]. Despite all these suppression factors, the mixed signal is still quite observable, as we now discuss. The various cuts on the W + W - background mentioned above manage to suppress it down to 9 fb [66], while the signal (assuming W-exchange dominance) is suppressed by a factor of about ~ = 0 . 4 .Assuming that E is not too d8erent in our case, to observe a 50 effect one would require
I
The sensitivity limit obtained in this way for L=500 pb-' is shown as a horizontal dashed line on Fig. 20. The points in parameter space which would be probed in this way are marked by crosses ( X ) in Fig. 14. In the dilaton scenario one could thus probe nearly all points up to the kinematical limit Le., m + < 100 GeV).
Fig. 21 for both scenarios. The real problem here is the taming of the irreducible dilepton background from W+W- production, i.e., u(e+_e-+W+W- Z + ~ ~ Z - 9 ~ ) = ( 1 8 ) ( ~ ) ( ~ ) = 0pb . 9 at z/s =200 GeV. Cuts are apparently not very e5cient in suppressing this background [67], although a reassessment of this problem needs to be performed to be certain. In any event, demanding that the dilepton signal have a 5u significance over this background implies
Xl
Before concluding this section, let us examine the dilepton mode in chargino pair production, since in the moduli scenario it is likely to have a much larger rate than the mixed mode does. The dilepton rate is shown in I
1
I -
:+
,I
b
J.
t , , , ,, , ,~ ---- - -- -
,,,,,,,,,,,,,
0.2
-
0.0
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00
90
mx: (GeV)
=
0.2
i,,,;,;,:1,; ;*.a.;t#
I,.j,
. _ - . .- -. - -.. .. -. . .-. , . . '
7::
0.0 100 40 50 dilaton SEenanO
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(cev)
mX:
0.6 0.4
*x-
: 0
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0.0 40
50
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70 "X;
80
(cev)
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100 40 50 m,=150 GeV
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mx: (cev)
FIG. 20. The u(e+e--CX:X;-ll+2j)
cross section vs m for chargi-
*
XI
no searches through the mixed mode at LEP I1 ( 6 = 2 0 0 GeV) for m,=150 GeV in the moduli and dilaton SU(5)XU(1) supergravity scenarios. The dashed lines indicate an estimated limit of sensitivity with L =500 pb-I.
39 1 50 -
LOPEZ, NANOPOULOS, PARK, WANG, AND Z I c H I a I
2186
p>o
LEP I1
?
\ -
no-scale SU(5)xU(l) moduli scenario
K O
0.6 . .
+% 0.4
.. .
0.4 .
.
FIG. 21. The cross section u(ete--~:~;-21) vs m + for chargino XI
b
0.0 40
0.0 50
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70
mx:
80
(cev)
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dilaiton scenario
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mX:
searches through the dilepton mode at LEP I1 (?4=200 GeV) for m,=150 GeV in the moduli and dilaton SU(5)XU(l)supergravity scenarios. The dashed lines indicate an estimated limit of sensitivity withL=500 pb-I.
. ... 0.4
0.0
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0.0 70
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100 40 50 q - 1 5 0 CeV
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mx:(Gev)
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mx:(
c4
This sensitivity limit is shown as a dashed line in Fig. 21. The regions of parameter space possibly explorable in this way are not shown in Fig. 14 since the dielectron signal from selectron pair production (discussed next) is much larger. Moreover, only 25% of the dilepton signal from chargino pair production consists of dielectrons. 3. Sleptons
The charged sleptons ( F L L , R , p L , R , ? L , R ) could be pair produced at LEP I1 if light enough, and offer an interesting supersymmetric signal through the dilepton decay mode. In the moduli scenario there is a significant portion of the parameter space where these particles are kinematically accessible at LEP 11, while in the dilaton scenario the accessible region is very small and will be neglected in what follows. The cross sections of interest are e + e - + F + H - B+t7 (24) L
E
Ly R
-*
R *eLzR
The HEHL,Z:Zi final states receive contributions from s-channel y' and Z * exchanges and t-channel xp exchanges, while the ZfHF only proceeds through the rchannel. The pzp,,:!p,, and ?;f?L,?i?i final states receive only s-channel contributions, since all couplings are lepton flavor conserving, and therefore mixed LR h a l states are not allowed for smuon or stau production. In Fig. 22 we show the total selectron and total smuon cross sections, which include all the kinematically accessible final states mentioned above. The results for stau pair production are very similar to those for smuon pair production. The horizontal line represents an estimate of the limit of sensitivity achievable with -C=500 pb-', as given in Eq. (23) to observe a 5 0 signal over the irreducible W +W - dilepton background. The selectron cross section is considerably larger than the smuon one because of the additional production channels. Our discussion in effect assumes that the acoplanar
2
FIG. 22. T h e cross sections cr(e+e--PV) and o(e+e--pp) vs m for selectron and
*
t
XI
muon searches at LEP I1 ( ?4 =200 GeV) for m,=150 GeV in the moduli and dilaton SU(5)XU(1) supergravity scenarios. T h e dashed lines indicate an estimated limit of sensitivity with I =500 pb-'. Note that slepton searches extend the indirect reach of LEP I1 for chargino masses, beyond m = 100 GeV.
D
*
.
XI
. .
.
b
0.0
40
60
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m,: P e V )
120
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0.0 40
60
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m,:
100
(c4
120
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EXPERIMENTAL ASPECTS OF SU(5)XU(1) SWERGRAVITY
2187
FIG. 23. The total elastic supersymmetric cross section (including selectron-neutralino and sneutrino-chargino production) at HERA vs m for rn, = 150 GeV in the moduli and
*
XI
dilaton SU(5)X U(1) supergravity scenarios. The dashed lines indicate limits of sensitivity with L= 100 and lo00 pb-l.
dilepton signal associated with selectron pair production comes entirely from Z& -+e *xy decay channels, i.e., purely dielectrons, and similarly for the m u o n case. This is an approximation which holds fairly well in the moduli scenario [64]. The points in parameter space in the moduli scenario which would be explorable through selectron searches at LEP I1 are shown in Fig. 14 as diamonds (0).The corresponding points explorable through smuon searches are not shown since the signal is smaller than in the selectron case. A rather interesting result is that the indirect reach in the chargino mass can be extended beyond the direct reach (of about 100 GeV). This effect depends on the value of tar$, and is relevant only for ta43 5 6 and p > 0, as Fig. 14 shows. In fact, the three dotted lines for p > 0 in Fig. 22 correspond from left to right to t a d = 6 , 4 , 2 respectively. C. HERA
The weakly interacting sparticles may be detectable at
HERA in SU(5)X u ( 1 ) Supergravity [68]. However, the mass range accessible is rather limited, with only the moduli scenario being partially reachable. The elastic scattering signal, i.e., when the proton remains intact, is the most promising one. The deep-inelastic signal has smaller rates and is plagued with large backgrounds [68]. The reactions of interest are e-p-+ii,T,x:,zp and e - p - + ~ , x ; p . The total elastic supersymmetric signal is shown in Fig. 23 versus the chargino mass. The dashed lines represent limits of sensitivity with L = 100 and 1000 pb- which will yield five “supersymmetric” events. This is a rather small signal. Moreover, considering the timetable for the LEP 11 and HERA programs, it is quite likely that LEP 11 would explore all of the HERA accessible parameter space before HERA does. This outlook may change if new developments in the HERA program would give priority to the search for the right-handed selectron (UR 1 which could be rather light in the moduli scenario of SU(5)XU(1) supergravity.
’
VI. PROSPECTS FOR INDIRECTEXPERIMENTAL DETECTION
In Sec. 111 we discussed four indirect [i.e., B ( b - + s y ) , (g - 2 ) p , neutrino telescopes, and - e b ] and two direct ke., trileptons and the lightest Higgs-boson mass at LEP I) experimental constraints on the parameter space of SU(5)XU(I) supergravity. Of the indirect constraints, the neutrino telescopes probe may become strict in the not-so-distant future [i.e., when the Monopole, Astrophysics, and Cosmic Ray Observatory (MACRO) comes into operation], however the implicit assumption of significant neutralino population in the galactic halo cannot be verified directly, and this diminishes the weight to be assigned to this constraint. The constraint on the top-quark mass should become stricter with the reduction of the present error bars by a factor of 2 by the end of the LEP I program. In this section we examine the two remaining indirect constraints [ B ( b-+s y ) and (g -2)J for the still-allowed points in parameter space. In Fig. 24 we show the values of B ( b-+sy ) calculated for the still-allowed points in parameter space (for m,=150 GeV) in the moduli and dilaton scenarios. For reference, the whole range of possible values before the imposition of the constraints discussed in Sec. I11 is addressed in Refs. [ 19,201. In the moduli case, for p > 0 one obtains a set of orderly lines for the indicated values of tan& which keep increasing in steps of two beyond the values explicitly noted. In the dilaton scenario the qualitative picture is somewhat similar, but for p < 0 there is a somewhat wider range of possible values. For comparison, in the standard model for m,=150 GeV one gets B ( b - + ~ y ) ~ ~ o 4 X(although l O - ~ QCD corrections need to be accounted for carefully). A more precise measurement of this branching fraction should be used to exclude points in parameter space which deviate significantly from the standard model prediction. A detailed calculation of the QCD corrections in the supersymmetric case
393 50 -
L.OPEZ, NANOPOULOS, PARK,WANG, AND ZICHICHI
2188
"o-rcale SU(5)XU(l) moduli scenario 0 0006
P> 0 0.0006
70.0004
w o
0.0004
YI
t
e , m o.0002
0.0002
0.0000
FIG. 24. The value of B ( b + s y ) vs the chargino mass for the still-allowed points in parameter space with rn,=150 GeV in the moduli and dilaton SU(5)XU(l)supergravity scenarios. Wherever possible some values of tar$ have been indicated.
0.0000 50
100
150
250
200
m,; (Gev)
50
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mx:
dilaton scenario 0.0006
200
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(Gev)
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20.0004
t
e, 0.0002
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50
100
150
250
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m,: (Gev)
50 m,=150 GeV
150
100
200
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m,: (GeV)
would be required to make a careful comparison with the standard model predictions. In Fig. 25 we show the values of a y versus the gluino mass for m,= 150 GeV in the moduli and dilaton scenarios. Reference values of tar$ are indicated. The dotted portions of the lines correspond to points in parameter space excluded by the combined constraints in Sec. 111. Note that for p>O in both scenarios there is a range of u y values which is excluded for all values of tar$. The new Brookhaven E821 experiment is expected to achieve a precision of O.4X lo-', which would entail a determination of tar$ as a function of the gluino (or chargino) mass. We remark that the supersymmetric contributions to a,, could be so large that the uncertainty in the standard model prediction ( 1.76X low9)would be basically irrelevant when testing a large fraction of the allowed parameter space.
VII. CONCLUSIONS We have presented an analysis of the several direct and indirect experimental constraints which exist at present on the parameter space of SU(S)XU(I) supergravity in the moduli and dilaton scenarios and their special cases (strict no scale and special dilaton). These scenarios are inspired by possible model-independent supersymmetry breaking scenarios in string models, and have the nonautomatic virtue of implying universal softsupersymmetry-breaking parameters. The scenarios can be described in terms of three parameters ( m +,tar$, m,) XI
which will be reduced down to two once the top-quark mass is measured. This minimality of parameters is very useful in correlating the many experimental predictions and constraints on the model. The ( m +,tar$) plane (for Xl
no-scale SU(5)xU(l) moduli scenario
P>O 25
00
20
-2 5
k 2 a 15
-5 0
PCO
-7 5 "0
+
10
-10 0
5
-12 5
0 200
400
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mi (GeV)
-15 0
1000 200 400 dilaton scenario
600
BOO
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ma (GeV)
25 20
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"2
10 5
0
200
400
600
800
mi (GeV)
1000
200
m,=150 GeV
400
600
800
mi (GeV)
1000
FIG. 25. The value of a;'"" vs the gluino mass for m,=150 GeV in the moduli and diiaton SU(5)X U(1)supergravity scenarios. T h e dotted portions of the curves are excluded. Some values of tan8 have been indicated.
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EXPJZRXMENTAL ASPECTS OF SU(5)X U(1) SUPERGRAVITY
fixed rn,) has been discretized and each point scrutinized to determine if the theoretical and basic LEP experimental constraints are satisfied. For satisfactory points we have then computed B ( b +sy ), (g -2),,, the rate of underground muon fluxes, and the trilepton rate at the Tevatron. Generally we 6nd rn, S 180 GeV to satisfy the -eb constraint, and some excluded regions of parameter space for specific values of rn,. For the still-allowed points in parameter space we have reevaluated the experimental situation at the Tevatron, LEP 11, and HERA. We have delineated the region of parameter space that would be explored in the 1994 Tevatron run,and by Higgs-boson, slepton, and chargino searches at LEP I1 with L = 5 0 0 pb-’. With estimates for the possible sensitivities at these colliders, we conclude that the Tevatron could explore the parameter space with chargino masses as high as 100 GeV. On the other hand, searches for the lightest Higgs boson at LEP I1 could explore all of the allowed parameter space in both scenarios if rn, 5 150 GeV and the beam energy is raised up to &=210 GeV (or lower if the two-loop QCD corrections to mh are accounted for). In fact, a measurement of the Higgs-boson mass in the standard model will almost uniquely determine the mass of the lightest Higgs boson in SU(5)XU(1) supergravity, since the relevant cross section and branching fractions deviate little from their standard model counterparts. Because of the mass correlations in the model, searches for selectrons allow LEP I1 to reach into the parameter space beyond the direct reach for chargino masses (i.e., rn < l o 0 GeV), thus selectrons are the next-deepest
*
XI
probe of the parameter space (after the Higgs boson), and charginos are the third probe. Searches for sparticles at HERA are not competitive with those at LEP 11, although supersymmetric particles in the moduli scenario (in particular the right-handed selectron FR may be light enough to be eventually observed at HERA. Searches for strongly interacting sparticles (squarks and gluinos) are not kinematically favored at the Tevatron since, for example, chargino masses of 100 GeV correspond to gluino and squark masses around 400 GeV. All of these possible constraints from future direct particle searches have been shown in plots of the still-allowed points in parameter space (see Fig. 14). These plots show the regions where the various searches are sensitive and should serve as a “clearing house” where the many experimental constraints are brought in, enforced, and their implications discussed.
[I]For a recent review, see J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, in From Supersfrings to Supergruuiry, Proceedings of the INFN Eloisatron Project 26th Workshop, edited by M. J. DuK, S. Ferrara, and R. R. Khuri (World Scientific,Singapore, 1993). [2]I. Antoniadis, J. Ellis,J. Hagelin, and D. V. Nanopoulos, Phys. Lett. B 194,231 (1987). [3]See, e.g., J. Ellis, J. L. Lopez, and D. V. Nanopoulos, Phys. Lett. B 245, 375 (1990);A. Font, L. Ibailez, and F.
2189
Let us conclude with a few general remarks in the context of SU(5)XU(l)supergravity (see Fig. 14). If the Tevatron sees sparticles (charginos), then almost certainly would LEP I1 see sparticles, too. If the Tevatron does not see sparticles (charginos), not much can be said about the prospects at LEP 11. It is quite possible that LEP I1 would see the lightest Higgs boson but no sparticles, if the Higgs-boson mass exceeds some m,-dependent limit (m,,2 105 GeV for rn,=150 GeV). It is unlikely, although possible that LEP I1 would see sparticles but no Higgs boson. If LEP I1 sees the lightest Higgs boson, then we would get a line in the (m +,tar$) plane, i.e., tar$ as a function XI
of m
* (for fixed or known m,).The measurement would XI
be conclusive by itself only in the strict moduli and special dilaton scenarios. If the Higgs boson, and selectrons or charginos are seen at LEP 11, this should be enough to test the model decisively because of the predicted correlations among the various predictions. In summary, the analytical procedure proposed in this paper could be applied to any supergravity model, and would serve as a standard against which the feasibility of various models could be measured and compared. Nore added in proof: An improved calculation of B ( b + s y ) has been recently found to exclude some more points than those marked by pluses in Figs. 6-8. The subsequent calculations in the paper remain unaffected by this change. We also note that the values of the (“running”) top-quark mass considered here m, = 130, 150, 170, 180 GeV correspond to somewhat higher values of the “pole mass,” i.e., mp0’C=139, 160, 181, 192 GeV. Finally, the recent announcement by the CDF Collaboration of evidence for the top quark (mpOle= 174f17 GeV) appears to disfavor the p > 0 possibility in the strict noscale scenario (Sec. I1 D l ) , which requires m, S 135 GeV m,PlC5 144 GeV.
-
ACKNOWLEDGMENTS
We would like to thank James White and Teruki Kamon for useful discussions. This work has been supported in part by U.S. DOE Grant No. DE-FG05-91ER-40633. The work of G.P. and X.W. has been supported by the World Laboratory.
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Jorge L. Lopez, D. V. Nanopoulos, Xu Wang and A. Zichichi
NEW CONSTRAINTS ON SUPERGRAVITY MODELS FROM b
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sy
From Physical Review D 51 (1995) 147
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399 PHYSICAL. REVIEW D
1 JANUARY 1995
VOLUME 51, NUMBER 1
New constraints on supergravity models from b + sy Jorge L. Lopez,’.’ D. V. Nanop~ulos,’~’*~ Xu Wang,’,’ and A. Zichichi4 ‘Center for Thwreticol Physics, Department of Physics, Tezos A&M University, College Station, Teurs 77843-4242 ’Astroparticle Physics Group, Houston Advanced Reneareh Center (HARC), The Mitchell Campus, The Woodlands, Tezas 77381 CERN Theory Division, 121 1 Geneva 23, Switzerland CERN, 1211 Geneva 23, Switzerland (Received 5 July 1994)
’
‘
We perform a detailed study of the constraints &om b + sy on a large class of supergravity models, including generic four-parameter supergravity models, the minimal SU(5) supergravity model, and SU(5)xU(l) supergravity. For each point in the parameter spaces of these models we obtain a range of B ( b -+ sy) values which should conservatively account for the unknown next-to-leadingorder QCD corrections. We then classify these points into three categories: “excluded” points have ranges of B ( b -+ sy) which do not overlap with the experimentally allowed range, “preferred”points have B ( b -t sy) ranges which overlap with the standard model prediction, and “okay” points are neither “excluded” nor “preferred” but may become “excluded” should new CLEO data be consistent with the standard model prediction. In all cases we observe a strong tendency for the “preferred” points towards one sign of the Higgs mixing parameter p. For the opposite sign of p there is an upper bound on tan@ tanp 5 25 in general, and tan p 5 6 for the “preferred”points. We conclude that new CLEO data will provide a decisive test of supergravity models. PACS number(s): 13.40.Hq, 04.65.+e, 12.10.Dm, 12.60.J~
I. INTRODUCTION
Following the precise verification of the unification of the gauge couplings in supergravity unified models [l], there has been a rekindling of interest in low-energy supersymmetric models and their experimental consequences. Supersymmetry can be tested in high-energy colJider experiments through the direct production of supersymmetric particles, and in dedicated low-energy experiments through the precise measurement of rare standard model looplevel processes. Because of the large energies required to produce real supersymmetric particles, high-precision experiments are likely to explore the parameter space of supersymmetric models in a much deeper although indirect way for some time to come. In the context of B physics, and the b -+ s7 process in particular, there appear to be great prospects for a thorough study of this mode by the CLEO Collaboration at the upgraded Cornell Electron Storage Ring (CESR) facility and at planned B factories at SLAC and KEK. At present there is an experimental upper bound B(b t sy) < 5.4 x and the observation of the B t K*y process imposes a conservative lower bound B(b t sy) > 0.6 x [2]. More precise measurements are expected to be announced soon. Despite the availability of precise experimental data, the large [3] and only partially calculated [4] QCD corrections to this process make it unclear [5-71 that one can use the data effectively to test the standard model or constrain models of new physics. Nonetheless, two alternatives appear possible: either the data deviate greatly from the standard model prediction or the data agree well with the standard model prediction. Since supersymmetric models predict 0556-282 1/95/5 1( 1)/I 47( 11)/$06.oO
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a wide range of b t s7 rates (above and below the standard model prediction) [8-161 in either case important restrictions on the parameter space will follow. Generic low-energy supersymmetric models, such as the minimal supersymmetric standard model (MSSM), are plagued by a large number of parameters (at least 20), whose arbitrary tuning allows a wide range of possible experimental predictions. This fact has discouraged many experimentalists, since any experimental limit would appear to be always avoidable by suitable tuning of the many parameters. These many-parameter models lack theoretical motivation, whereas, theoretically wellmotivated supersymmetric models have invariably much fewer parameters and can be straightforwardly falsified. In this context we consider unified supergravity models with universal soft supersymmetry breaking and radiative electroweak symmetry breaking, thus restricting the number of parameters to four. To sharpen the predictions even more, we also consider string-inspired noscale SU(5)xU(l) supergravity, which can be described in terms of two parameters, or even only one parameter in its strict version. Calculations of B(b t s-y) in supergravity models have been performed in Re&. [8,11,14-161 under various assumptions, such as radiative electroweak breaking using the tree-level Eggs potential [8,14,15], or impos ing relations among the supersymmetry breaking parameters (such as B = A - 1 [8,14-161, or B = 2 1161, or mo = A = 0 Ill]). In this paper we reexamine the B(b t sy) calculation in string-inspired no-scale SU(5)xU(l) supergravity and in the minimal SU(5) supergravity model. For a direct comparison with other possible supergravity models, we extend our calculations to a large class of generic four-parameter supergravity 147
@ 1995 The American
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LOPEZ, NANOFQULOS, WANG,AND ZICHICHI
models. It is found that B(b + 5 7 ) can be enhanced in supersymmetric models in the presence of light sparticles, and more importantly for large values of the ratio of Higgs vacuum expectation values ( t a d ) . Indeed, large values of t a n p enhance the down-type quark and charged l e p ton Yukawa couplings, in particular Ab a 1/ cos p tan p for large t a n p . (An analogous effect also enhances the supersymmetric contribution to the anomalous magnetic moment of the muon, through a large muon Yukawa coupling [17].) We find that when the various decay amplitudes have the same sign, this enhancement results in new upper bounds on t a n p , assuming a supersymmetric spectrum below the TeV sc&le. This paper is organized as follows. In Sec. I1 we describe briefly the models under study. In Sec. 111 we discuss the expression used to calculate B(b + ST) and the various uncertainties involved. In Sec. IV we present and discuss our results, and obtain new upper bounds on t a n p using the present experimentally allowed range. We also delineate the regions of parameter space which are consistent with the standard model prediction, and explore the mt dependence of our results. These show a strong preference for one sign of the Higgs mixing parameter p. Finally, in Sec. V we summarize our conclusions.
-
11. THE SUPERGRAVITY MODELS
We consider unified supergravity models with universal soft supersymmetry breaking and radiative breaking of the electroweak symmetry. At the unification scale the models are described by four soft-supersymmetrybreaking parameters: the universal gaugino mass (mllz), the universal scalar mass (mo), the universal trilinear scalar coupling (A), and the bilinear scalar coupling (B); and by four superpotential couplings: the Higgs mixing parameter p , and the third-generation Yukawa couplings (A,, &,, Ar). At low energies there also arises the ratio of Higgs vacuum expectation values (tan@. The renormalization group equations connect the values of the parameters at the low- and high-energy scales. These nine parameters reduce to five by the imposition of a good minimum of the one-loop effective Higgs potential at the electroweak scale (one condition for each of the two real scalar Higgs fields; 1p1 and B are determined), and by trading the set (At,Xb, A,, tanp) for (mt,mb,m,,tanp) and using the known values of mb and m,. Once the five independent parameters ( m l p , mo, A, mt, t a n p ) are specified, one can obtain the whole low-energy spectrum and enforce all the known bounds on sparticle and Higgsboson masses. We note that, unlike Refs. [8,14-163, we do not impose the additional restriction B = A - 1 at the unification scale. For recent reviews of this procedure, see, e.g., Ref. [18]. For a fixed value of the topquark mass, we therefore have a family of four-parameter supersymmetric models. We further specify this set as ( m x : , ~ o r h r t a n P ) , where m x1* a mllz and we have defmed € 0 m o / m l p and (A = A/mlp. This type of ratios of soft-
supersymmetry-breaking parameters occurs naturally in various string-inspired supersymmetry-breaking scenarios (see below). In what follows, we consider continuous values of m,,, and a grid of values for the other three parameters: t a n p = 2-40 (in steps of 2); &, = 0, l ,2,5,10; (A = o,+
-
where
‘In the “strict no-scale’’ case both signs of 1 are in principle allowed. However, 1 > 0 can occur only for mt 5 135 GeV which appears quite disfavored by the latest experimental information on the top-quark mass.
40 1 NEW CONSTRAINTS ON SUPERGRAVITY MODELS FROM b-sy
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149
tanp [29]. This constraint is also imposed below. In what follows we will present our results in a way that makes apparent the impact of B(b + sy) on the parameter space, irrespective of any additional applicable constraints, and also when all constraints are combined.
Thus we obtain my1‘ z 1.07mt, and our choice mt = 150 GeV corresponds to m y ” = 160 GeV, which is in good agreement with the latest fit to all data q = 162 f 9 GeV [28]. When large values of tanp are allowed, we also consider the constraints from the anomalous magnetic moment of the muon. This constraint is not as restrictive as that from B(b + s 7 ) , but nonetheless can exclude additional points in parameter space for p < 0. Also, in the generic four-parameter models, the cosmological constraint of a not-toc-large neutralino relic density becomes quite restrictive for t 2 2 and not-too-large values
of
where 9 = a,(Mz)/a.(Q),g is the phase-space factor g ( s ) = 1 - 8s’ 8s6 - X* - 24s41nz,C = C(Q) is a QCD correction factor, and Q is the renormalieation scale. The A,,Ag are the coefficients of the effective bs7 and bsg penguin operators evaluated at the scale Mz. These coefficients receive five contributions [8] from: the t - W* loop, the t - H i loop, the - t1.z (the “chargino” contribution), the 5 - Q loop (the “gluino” contribution), and the x: - Q loop (the “neutralino” contribution). The gluino and neutralino contributions are much smaller than the chargino contribution (8,151 and are neglected in the following. In fact, it is when the chargino contribution becomes large (for light sparticles and large t a n s ) that B(b + s 7 ) can greatly deviate from the standard model result. As only the top squarks ( i l , p ) are significantly split, in the chargino contribution we ignore the other squark splittings. The topsquark splitting affects the magnitude of the chargino contribution, and for fixed mt it depends mostly on the parameter A (or at low energies on At). Expressions for A,, A, can be found in Ref. [lo]. As is, this expression for B(b -+ sy) is subject to partially unknown next-teleading-order (NLO) QCD corrections. It has been recently shown [6,7] that the magnitude of the NLO corrections can be estimated by allowing the renormalization scale Q to vary between mb/2 and 2mb. A complete NLO calculation would yield an expression with a much milder Q dependence, with the expectation that the NLO value for B(b + s-y) would be obtained from the LO expression for a choice of Q in the mb/2 -+ 2mb interval. Therefore, in what follows, we use the LO expression’ in Eq. (3) and obtain a range of values for each point in parameter space by taking
mb/2 < Q < 2mb, with ma = 4.65 GeV. Consistent with this procedure we use the one-loop approximation to the running of the strong coupling, i.e.,
111. FORMULA AND UNCERTAINTIES
We use the following leading-order (LO) expression for the branching ratio [lo]:
I
+
xtZ
’In Eq. (3)we have removed from the denominator the QCD
correction factor for the semileptonic decay that was used in OUT previous analyses [Il l . This factor has to be removed in order t o obtain a true LO expression, without any NLO contributions.
71 = aa(Mz)/aa(Q) = 1 - 22/3[aa(M~)/2=]ln(M~/Q)
(we take a.(Mz) = 0.120). In evaluating Eq. (3) we A0 take I ~ ~ b l ’ / ~ v c b = l ’ 0.95f0.04, mc/mb = 0.316f 0.013, and B(b + ceP) = 10.7%[7]. Finally C = C(Q) = x:=l b i 9 4 , with the b;,d; coefficients given in Ref. [lo]. For Q = mb/2, mi,, 2mb we obtain q = 0.486,0.583,0.680 and C = -0.208, -0.160, -0.117. Following tEe above procedure, but keeping only the t - W* contribution to A,,A, we obtain the following range for the standard model contribution
B(b + ~ 7 ) =s (1.97-3.10) ~ x . (4) The above discussion of QCD corrections implicitly assumes that only two mass scales are involved in the problem: the high electroweak scale and the low bquark mass scale. In practice the supersymmetric particles have a spectrum that can be spread above or below the electroweak scale. Recently there has appeared the first study of QCD corrections to B(b -+ s-y) in the supersymmetric case [30], which shows that the running from the electroweak scale down to the bquark mass scale gives the largest QCD correction. However, a proper treatment of the supersymmetric spectrum at the high scale may produce non-negligible effects. We hope that the scale uncertainty that we have introduced above is large enough to effectively encompass the supersymmetric high scale uncertainty as well.
IV. RESULTS AND DISCUSSION We now present the results of the calculations of B(b -+ for the various supergravity models under consideration. We start by examining the results in SU(5)xU(1) supergravity and the minimal SU(5) supergravity model, and then extend our calculations to the generic fourparameter models described in Sec. II. sy)
402 150
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LOPEZ, NANOPOULOS, WANG, AND ZICHICHI
A. SU(5)xU(l) supergravity The calculated values of B(b t sy) in the moduli and dilaton scenarios are shown in Figs. 1 and 2 respectively, for selected values of t a n p ; consistency of the model entails a n upper limit of t a n p 26(40) in the moduli (dilaton) scenario. In these and other figures showing values of B(b t sy) we take Q = mb as a “central” value. The full uncertainty range is considered when discussing whether or not given points in parameter space are excluded. In Figs. 1 and 2 the arrows point into the experimentally allowed region and the dashed lines (SM) delimit the standard model prediction. For p > 0 the values of B(b + sy) increase steadily with increasing tanp and eventually fall outside the experimentally allowed region for all values of mr:. (The largest values of rn shown correspond t o mi,mg NU 1 TeV.) XI For p < 0 the tanp-dependence is different. One sees that B(b t sy) can be suppressed much below the standard model result. This behavior was first noticed in Ref. [Ill and simply shows that the various amplitudes and QCD correction factors in Eq. (3)conspire to produce a cancellation. This phenomenon has been since explained in Refs. [12,16].The idea is that the chargino contribution to k, can have the same sign (negative) or opposite sign (positive) compared to the t - W* and t - Hf contributions which are always negative. In fact, the sign of the chargino contribution is determined by the product Bip: positive for Bip < 0 and negative for Bip > 0 [12].3 Here 0; is the topsquark mixing angle, 0; x */4 in this approximation to the chargino contribution [lo]. Therefore, constructive interference occurs for p > 0 and
s
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FIG. 1. The calculated “central” values of B(b -+ s-y) in no-scale SU(5) XU(1) supergravity-moduli scenario, for r e p resentative values of tanp. The arrows point into the experimentally allowed region. The dashed lines delimit the standard model range.
’Our sign convention for p is opposite to that in Ref. [12], but the same as that in Ref. [16].
m,: (cev)
mx:(cev)
FIG. 2. The calculated “central” values of B(b -i s-y) in no-scale SU(5) xU(1) supergravity-dilaton scenario, for representative values of tanp. The arrows point into the experimentally allowed region. The dashed lines delimit the standard model range.
destructive interference occurs for p < 0, as evident in Figs. 1 and 2. Figure 2 also shows that despite the destructive interference, for p < 0 and su5ciently large Values of t a n p , an enhancement can occur for light chargino masses because the chargino contribution overwhelms the other two contributions. To better appreciate the impact of the present experimental limit on B(b t sy), we classify points in parameter space into three categories. Ezclvded points have B(b -t s-y) ranges which do not overlap with the present experimental allowed range [i.e., (0.6-5.4) x ~ O - ~ ]This . conservative constraint (i.e., use of theoretical racges rather than central values) is imposed so that our excluded points are not dependent on presently unknown NLO QCD corrections. Preferred points have B(b t s-y) ranges which overlap with the standard model range [i.e., (1.97-3.10)x 06 points are neither “excluded” nor “preferred,” and may become “excluded” if new CLEO data are consistent with the standard model prediction. The two-dimensional parameter spaces for the moduli and dilaton scenarios are shown in Figs. 3 and 4 respectively. In these figures crosses (x) represent L‘excluded” points, diamonds ( 0 ) represent “preferred” points, and dots (.) represent “Ok”points. The vertical dashed line a t m Xl = 100 GeV in these and following parameter space plots indicates the approximate reach of the CERN e+e- collider LEPII for chargino masses. As anticipated, there are many “excluded” points for p > 0, especially as t a n p gets large. Also, the “preferred” points occur largely for p < 0, signaling the ability of the B(b t sy) constraint to possibly select the sign of p. For s a c i e n t l y large values of tan@, the anomalous magnetic moment of the muon (g - 2), can be constraining as well. We have calculated this quantity as in Ref. [17]and denote excluded points (i.e., points for which $(g - 2)EuSy falls outside the experimentally allowed
403 NEW CONSTRAINTSON SUPERGRAVITY MODELS FROM b - s y
51 -
no-scale SlJ(5)xU(l) moduli scenario
P>O
P
FIG. 3. The two-dimensional parameter space of no-scale SU(5)xU(1) supergravity-moduli scenario. Points excluded by B(b --t 5 7 ) are denoted by crosses (x), those consistent with the standard model prediction are denoted by diamonds (o), and the rest are denoted by dots (.). Plusses (+) indicate points excluded by (g - Z),,,whereas asterisks (*) indicate points excluded by both constraints.
__I... 20
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range of -13.2 x lo-’ + 20.8 x lo-’ [17]) by plusses (+) in Figs. 3 and 4. We have only shown these excluded points for p < 0, since for p > 0 the (g - 2),, constraint does not exclude any points which are not excluded by the B(b + sy) constraint. The effect of this constraint is most noticeable in the dilaton scenario (Fig. 4) where “preferred” points which are thusly excluded appear as the overlap of a plus sign and a clear diamond (i.e., “filled” diamonds) for tanp = 10-32 and mx: 5 100 GeV. Points excluded by both B(b -+ sy) and (g - 2),, appear as the overlap of a plus sign and a cross, i.e., as asterisks (*). Overall, the B(b -+ sy) constraint is quite restrictive. Statistically we have the following distribution of hactions of parameter space: Moduli P
P>O
48% 9% 43%
7% 66%
73%
21%
6% 21%
33% 46%
27%
P
Note that should the new CLEO data be consistent with the standard model prediction, then only the “prefemed” points would survive. For p > 0 this is 9(6)% of all the points in parameter space in the moduli (dilaton) of scenario, whereas for p < 0 one would still have
3 (g)
1-
x1
2”” t I ................... 1
FIG. 4. The two-dimensional parameter ...A,.. ........... space of no-scale SU(S)xU(l) supergrav-. ......,........... I -..._. _____. .................. ity-dilaton scenario. Points excluded by .... .................... ..... ................... -” 30 B ( b + sy) are denoted by crosses (x), those + .....__.__....................... ......................... j 1 L -.... consistent with the standard model predicI L---I-..... ........................... tion are denoted by diamonds (o), and the 20 rest are denoted by dots (.). Filled diamonds and plusses (+) indicate points excluded by (g-2),, whereas asterisks (*) indicate points excluded by both constraints.
I
~
I
30 m
250
the points allowed-an overwhelming inclination towards p < 0. From Figs. 3 and 4 one can also see that for p > 0 there is an upper bound on t a p : tanp 5 25. For the subset of ”preferred” points this bound drops to t a n p 5 6. We should add that these bounds can be evaded by demanding sufficiently heavy sparticles (in the multi-TeV range), but then supersymmetry would lose most of its motivation. There are no analogous bounds for p < 0. Note that the “preferred” points for p > 0 entail a supersymmetric spectrum inaccessible to direct searches a t LEPII (i.e., mx: 2 150 GeV, mi = mi 2 500 GeV). In contrast, some of the “preferred” points for p < 0 would be directly accessible a t LEPII. In the strict no-scale case, the one-parameter models allow one to plot B(b -+ sy) as a function of mx: only. These values are shown in Fig. 5. Since in both cases p < 0, the plots are reminiscent of the p < 0 plots in the general moduli and dilaton scenarios. In Fig. 6 we show the corresponding one-dimensional parameter spaces, where one can see that in the dilaton scenario there are no excluded points. The constraint kom (g-2),, is ineffective in the dilaton scenario and does not exclude any further points in the moduli scenario. Figure 6 shows the interest of working with a theory where all predictions are given in terms of only one parameter. For example, a measurement of m * would immediately de-
Ddaton
P>O
Excluded Preferred Ok
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404 LOPEZ, NANOPOULOS, WANG, A N D WCHICHI
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lo-‘
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termine tanp. Also, if it is found that 2 5 t a n p 5 12 or t a n p 5 19 are preferred, then this one-parameter theory should be abandoned.
B. Minimal SU(5) supergravity As described in Sec. 2, in the case of the minimal SU(5) supergravity model we constrain the four-dimensional parameter space by the requirements of proton decay and cosmology. These entail t a n p 5 10 and €0 2 4. In Fig. 7 we show the calculated central values of B(b t sy) in
,....--
.__..”
...’
moduli scenario
150
m:,
FIG. 5. The calculated “central” values of B(b t sy) in strict no-scale SU(5) xU(1) supergavity-moduli and dilaton scenarios. The arrows point into the experimentally idowed region. The dashed lines delimit the standard model range.
/”.*-
100
ZOO
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(cev)
50
I00
150
m,:
200
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(cev)
FIG. 7. The calculated “central” values of B(b + sy) in the minimal SU(5) supergravity model. The constraints from proton decay and cosmology restrict the parameter space as indicated. The arrows point into the experimentally allowed region. The dashed lines delimit the standard model range.
this case. Because the values of tanp are not allowed to be large we do not expect large deviations from the standard model prediction, as seen in the figure. Nonetheless, there are deviations, especially for p > 0. Given the uncertainties in the calculation of B(b t sy) described in Sec. 111, we find no “excluded” points, and the following distribution of fractions of parameter space: Minimal SU(5) Excluded Preferred Ok
P>O 0% 11% 89%
P
0% 93%
7%
Again we see a strong tendency among the “preferred” points towards p < 0. Moreover, for p > O(p < 0) “preferred” points exist only for t a n p 5 4(8).
1
C. Generic supergravity models
2 4
I
101
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5t I ..
0
I
dilaton scenario
...................______.
50
I
~
100
~
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~
200
~
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’
~
m,: ( G e V
FIG. 6. The one-dimensional parameter space of strict no-scale SU(5)xU(1) supergravity-moduliand dilaton scenarios. Points excluded by B(b t sy) are denoted by crosses ( x ) , those consistent with the standard model prediction are denoted by diamonds ( 0 ) , and the rest are denoted by dots
(.I.
For completeness, we now turn to the generic fourparameter supergravity models. These models could be viewed as essentially SU(5) supergravity models where the constraint from proton decay is simply ignored (the cosmological constraint is not neglected). In the spirit of Ref. 1291, we consider these models to see if b 4 s7 could shed some light on the structure of the soft supersymmetry breaking sector in supergravity or superstring ~ ~ ~ I ~ ~ ’ ~ J models. As mentioned in Sec. 11, we have considered continuous values of mx: and a grid of values for the other three parameters: tanp = 2-40 ( i steps of 2); €0 = 0,1,2,5,10; [a = 0, + t o , -to. We will concentrate on €0 = 1 , 2 since the SU(5)xU(l) scenarios correspond and larger values of tolead approximately t o €0 = 0, to increasingly heavier sparticle masses. We will comment on the €0 = 5,lO cases later. In Figs. 8 and 9 we
-&,
~
405 153
NEW CONSTRAINTS ON SUPERGRAVITY MODELS FROM b + s y
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FIG. 8. The calculated “central” values of B(b -) sy) in the = 1 and ( A = 0, for repgeneric supergravity model for resentative values of tan@. The dotted curve above (below) the tan@= 20 curve for p > 0 corresponds to € A = -l(+l). The arrows point into the experimentallyallowed region. The dashed lines delimit the standard model range.
FIG.9. The calculated “central”values of B(b + sy) in the generic supergravity model for (a = 2 and ( A = 0, for representative values of tan@. The dotted curve above (below) the tan@= 20 curve for p > 0 corresponds to ( A = -2(+2). The arrows point into the experimentallyallowed region. The dashed lines delimit the standard model range.
show B(b -i 87) for (0 = 1 and 2, respectively, for r e p resentative values of tang. These curves are for ( A = 0. The effect of Werent choices for ( A is shown by the dotted lines in these figures (for the same t a n g = 20 and (0 choice): ( A < 0 enhances B(b -i 87) since negative A values drive At at low energies to even more negative J Increasing ( 0 &om values and thus larger ~ I splittings. 1 to 2 has the expected effect of decreasing B(b -i 57). For fixed values of (0 and ( A we can plot the parameter space in the (mx:,tanP) plane, as was done in the SU(5)xU(1) supergravity case. The points in parameter space are again classified into “excluded,” “preferred,” and “Ok” as discussed in Sec. IVA. These plots for ( 0 = 1,2 and < A = 0, +to, -(o are shown in Figs. 10 and 11. For to= 1 the (g - 2), constraint excludes additional points in parameter space for p < 0. These appear as filled diamonds and plusses (+) in Fig. 10. The effect of ( A discussed above is also evident in this figure, i.e., for ( A = -1 one sees that B(b -i 57) is enhanced. The island of “preferred” points for p < O,mx: 5 100 GeV
and not-so-small values of tang corresponds to the curves in Fig. 8 which “bounce back” for light chargino masses. Note that most of these points are in fact excluded by the (g - Z),, constraint. The cosmological constraint is unrestrictive for (0 = 1. For (0 = 2 the (g - 2), constraint does not exclude any points in parameter space which are not also excluded by the B(b -i 57) constraint. However the cosmological constraint becomes important and excludes points denoted by filled diamonds and plusses in Fig. 11. Note that this constraint excludes most (if not all) of the (few) “preferred” points for p > 0. The right boundary of the parameter space corresponds to the squarks at 1 TeV. The position of this boundary as a function of mx: depends on ( A because A affects the calculation of p, which in turn- affects the value of In analogy with the discussion for the previous models, statistically we have the following distribution of kactions of parameter space (excluding the cosmological constraint):
€A
Excluded Preferred Ok
+1 70% 4% 26%
0
74% 4% 22%
-1 76% 4% 20%
+1 12% 53% 35%
0
15% 43% 43%
In this case we again see the marked tendency of the
“preferred” points for p < 0, i.e., only few percent of the points for p > 0 are “preferred.” Including the cosmological constraint makes this tendency even more pro-
m,,,.
-1 20% 36% 44%
+2 62%
7% 31%
0 72% 6% 23%
-2 78% 5% 17%
+2 9% 51% 40%
0 17% 36% 47%
-2 27% 29% 44%
nounced. From Figs. 10 and 11 we also notice the upper bound on tang for p > 0 : t a n g 5 25 in general and tang 5 6 for the “preferred” points. For ( 0 = 5,lO the parameter space is generally quite
406 154
LOPEZ, NANOPOULOS, WANG, AND ZICHICHI
Limited in the tana direction, i.e., t a n p <, 4(2) for €0 = 5(10). Higher values of tanp are inconsistent with radiative electroweak symmetry breaking. Using the tree-level scalar potential such values of t a n p give pz < 0. In our calculations (using the one-loop effective potential) no minimum can be found. This limitation on tang can be circumvented for sufficiently negative d u e s of € A , e.g., [ A = -5 (-10) for €0 = 5 (10). In any event, the qualitative results obtained above for the strong tendency of the ''preferred'' points towards p < 0, holds also for large values of to.The cosmological constraint exac-
4
51 -
erbates this tendency. For €0 = 10 basically all points in parameter space for ,u > 0 are excluded; for p < 0 a few survive. D. mt dependence
Our calculations above have been performed for a fixed value of mt. Given the present uncertainty on the actual value of n-+, it is appropriate to explore the n-+ dependence of our results. In the standard model we obtain
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.................... =.......*.*... ............... ....................
150
'1
'1 t 1 " " l " " I " " l " " I '
50
100
150
200
250
FIG. 10. The parameter space of the generic supergravity model for € 0 = 1 and (a) ( A = 0, (b) € A = +1, and (c) € A = -1. Points excluded by B(b + sy) are denoted by crosses ( x ) , those consistent with the standard model prediction are denoted by diamonds ( o ) , and the rest are denoted by dots (.). Filled diamonds and plusses (+) indicate points excluded by ( g - Z),,, whereas asterisks (*) indicate points excluded by both constraints.
407
51
155
NEW CONSTRAINTS ON SUPERGRAVITY MODELS FROM b -+sy
We have also redone the calculation of B(b + 87) in SU(5)xU(1) supergravity for various values of mt. As expected, the mt dependence is weak. To better quantify this statement, as a function of we have determined
the following LO ranges for B(b + ~ 7 ) s ~ :
B(b + 8 7 ) S M = (1.77-2.89) X B(b + ~ 7 ) s= ~(1.97-3.10) x B(b + 8 7 ) s ~ = (2.15-3.28) X
mt = 130 GeV, mt = 150 GeV, mt = 170 GeV, mt = 190 GeV,
w4,
the fkactions of parameter space which are "preferred":
B ( b -+ ~ 7 ) = s ~(2.31-3.44) x
Moduli
These results indicate that a measurement of B(b + 87) will not give new information on mt (since all intervals overlap), at least at the LO level. A NLO calculation would be most useful in this regard.
40
Dilaton
mt
P>O
P
P>O
P
150 160 170 180
9%
66% 66% 66% 67%
6% 5%
33% 32% 31% 28%
8% 8% 2%
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-
92
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............ ..................... .......... ........................... ....................................... .......... .................................... ......... b. ............ ........................................ ...................................... ......... .1............... ." .......... ........................................ ................................................. .................................
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........................
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...................................... ...................................... . .;:::::::ti:::::::::: : : :::::::::::::: ..........+......................... :;I ::::::::;::::::::::::::: .................................. ...... ...........................
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........I....................*...*....
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................................ .............................. "' ,,,,',',,?,,,*',, 1
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FIG. 11. The parameter space of the generic supergravity model for €0 = 2 and (a) ( A = 0, (b) ( A = +2, and (c) [ A = -2. Points excluded by B(b + sy) are denoted by crosses ( x), those consistent with the standard model prediction are denoted by diamonds (o), and the rest are denoted by dots (.). Filled diamonds and plusses (+) indicate points excluded by the cosmological constraint, whereas asterisks (*) indicate points excluded by both constraints. At the right boundary of the parameter space the squark masses reach 1 TeV.
-I
~~;~::;:::;;:;;;:: ""._. ,... ............... .". ............................... ........._
......
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'
200
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408 51 -
LOPEZ. NANOPOULOS, WANG, AND ZICHICHI
156
Except for mt = 180 GeV, the &actions of parameter space which are “preferred” are largely % independent. The increased sensitivity for large values of n~ is interesting, although not very useful since for % 190 GeV the allowed parameter space disappears because of the well known phenomenon of the t o p q u a r k Yukawa coupling encountering a Landau pole below the unification scale.
2
We have explored a variety of supergravity models with universal soft supersymmetry breaking at the unification scale and radiative electroweak symmetry breaking. Acsy) counting for the inherent uncertainties in the B(b evaluation, we have nonetheless been able t o exclude a good &action of points in parameter space which differ sigrdicantly &om the experimentally allowed range. We have also identified “preferred” regions of parameter space which would b e singled out should the new CLEO data be consistent with the standard model prediction. These “preferred” regions occur mostly for one sign of p(< 0) (when the chargino contribution has opposite sign relative t o the standard model contribution). For t h e other sign of p(> 0) one can obtain upper bounds on t a n p assuming a sparticle spectrum below the TeV scale. It is worth noticing that constraints from b + sy dis-
+
[l] J. Ellis, S. Kelley, and D. V. Nanopoulos, Phys. Lett. B 2 4 9 , 4 4 1 (1990); P. Langacker and M.-X. Luo, Phys. Rev. D 44, 817 (1991); F. Anselmo, L. Cifarelli, A. Peterman, and A. Zichichi, Nuovo Cimento 104A,1817 (1991). [2] E. Thorndike, Bull. Am. Phys. SOC.38, 922 (1993); CLEO Collaboration, R. Ammar et al., Phys. Rev. Lett. 71. 674 (1993). M. Shifman, A. Vainshtein, and V. Zakharov, Phys. Rev. 18, 2583 (1978); S. Bertolini, F. Borzumati, and A. Masiero, Phys. Rev. Lett. 59,180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam, and P. Singer, ibid. 59, 183, (1987); B. Grinstein, R. Springer, and M. Wise, Nucl. Phys. B339,269 (1990). I
N
+
V. CONCLUSIONS
.
favor t h e l a r g e t a n p solution [31] t o the unification of the Yukawa couplings (A, = A,) in SU(5)-like theories (although those are already disfavored by proton decay constraints). In the case of SO(l0)-like Yukawa unification (At = At, = A,), one requires tanp 50 [32], which are also disfavored by b + sy constraints. However, in this case radiative electroweak symmetry breaking is also difficult and it has been suggested [33]that non-universal soft supersymmetry breaking scenarios may b e able t o solve both these difficulties. At t h e moment, b sy is the most important constraint on the parameter spaces of supergravity models. This standing is expected to b e much reenforced when the new CLEO data are announced. At this time it will become quite appropriate t o reexamine t h e phenomenw logical implications of the still-allowed points in parameter space for direct and indirect searches of supersymmetric particles. We close by remarking that contrary t o direct production experiments, the b + sy process can already explore regions of parameter space with “virtual” mass scales all the way up t o the TeV scale.
,
R. Grigjanis, P. J. O’Donnel, M. Sutherland, and H. Navelet, Phys. Lett. B 213,355 (1988); 233,239 (1989); 237,252 (1990); G. CeUa, G. Curci, G. Ricciardi, and A. VicerC, ibid. 248, 181 (1990); M. Misiak, ibid. 269, 161 (1991); 321,193 1994); Nucl. Phys. B393,23 (1993); K. Adel and Y. P. Yao, Mod. Phys. Lett. A 8, 1679 (1993); Phys. Rev. D 49, 4945 (1994). A. Ali and C. Greub, Phys. Lett. B 293, 226 (1992); Z. Phys. C 60, 433 (1993). A. Buras, M. Misiak, M. Miinz, and S. Pokorski, Nucl. Phys. B424,374 (1994). M. Ciuchini et al., Phys. Lett. B 334, 137 (1994), and references therein. S. Bertolini, F. Borzumati, A. Masiero, and G. Ridolfi, Nucl. Phys. B353,591 (1991).
ACKNOWLEDGMENTS
This work was supported i n part by DOE grant D E FG05-91-ER-40633. The work of X.W. has been s u p ported by the World Laboratory.
[9] N. Oshimo, Nucl. Phys. B404,20 (1993).
[lo] R. Barbieri and G. Giudice, Phys. Lett. B 309,86 (1993). [ll]J. L. Lopez, D. V. Nanopoulos, and G. T. Park, Phys. Rev. D 48,R974 (1993); J. L. Lopez, D. V. Nanopoulos, G. T. Park, and A. Zichichi, ibid. 49,355 (1994). [12] R. Garisto and J. N. Ng, Phys. Lett. B 315,372 (1993). 1131 Y. Okada, Phys. Lett. B 315,119 (1993). [14] M. Diaz, Phys. Lett. B 322,207 (1994). [15] F. Borzumati, Z. Phys. C 63,291 (1994). [IS] S. Bertolini and F. Vissani, Report No. SISSA 40/94/EP, hepph/9403397, 1994 (unpublished). (171 J. L. Lopez, D. V. Nanopoulos, and X. Wang, Phys. Rev. D 49,366 (1994). [18] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Nuovo Cimento Rivista, 17,1 (1994). [19] J. L. Lopez, D. V. Nanopoulos, and H. Pois, Phys. Rev. D 47, 2468 (1993); J. L. Lopez, D. V. Nanopoulos, H. Pois, and A. Zichichi, Phys. Lett. B 299, 262 (1993). [20] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 291,255 (1992); J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Phys. Rev. D 4 8 , 2766 (1993). [ Z l ] For a recent review, see “Status of the Superworld: From
Theory to Experiment,” J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Prog. Part. Nucl. Phys. 33,303 (1994). [22] J. Ellis, C. Kounnas, and D. V. Nanopoulos, Nucl. Phys. B241, 406 (1984); B 2 4 7 , 373 (1984); J. Ellis, A. Lahanas, D. V. Nanopoulos, and K. Tamvakis, Phys. Lett. 134B,429 (1984).
409 51 -
NEW CONSTRAINTS ON SUPERGRAVITY MODELS FROM b+sy
[23] For a review, see A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. 15,1 (1987). [24] V. Kaplunovsky and J. Louis, Phys. Lett. B 308, 269 (1993); A. Brignole, L. Ibbiiez, and C . Muiioz, Nucl. Phys. B 422, 125 (1994). [25] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Rev. D 49, 343 (1994). [26] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 319, 451 )(1993). [27] N. Gray, D. Broadhurst, W. Grafe, and K. Schilcher, Z. Phys. C48, 673, (1990). [28] J. Ellis,G. L. Fogli, and E. Lisi, Phys. Lett. B 333,118 (1994). [29] J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Phys. Lett. B 287,219 (1991); S.Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Phys. Rev. D 47, 2461 (1993).
157
[30] H. Ankuf, Report NO. SLAC-PUB-6525 hepph/9406286,1994 (unpublished). [31] See e.g., J. Ellis, S. Kelley, and D. V. Nanopoulos, Nucl. Phys. B 373, 55 (1992); H. Arason, et al., Phys. Rev. Lett. 87, 2933 (1991); S. Kelley, J. L. Lopez, and D. V. Nanopoulos, Phys. Lett. B 274, 387 (1992); V. Barger, M. Berger, and P. Ohman, Phys. Rev. D 47, 1093 (1993); P. Langacker and N. Polonsky, ibid. 49, 1454 (1994); C. Kolda, L. Roszkowski, J. Wells, and G. Kane, ibid. 50, 3498 (1994). [32] B. Ananthanarayan, G. Lazarides, and Q. Shafi, Phys. Rev. D 44, 1613 (1991); S. Kelley, J. L. Lopez, and D. V. Nanopoulos, in Ref. [31]. [33] L. Hall, R. Ratazzi, and U. Sarid, Phys. Rev. D 5 0 , 7048 (1994); M. Carena, M. Olechowski, S. Pokorski, and C. Wagner, Nucl. Phys. B428, 269 (1994).
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41 1
Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi
EXPERIMENTAL CONSEQUENCES OF ONE-PARAMETER NO-SCALE SUPERGRAVITY MODELS
From lnternational Journal of Modern Physics A 10 (1995)4241
I995
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413 International Journal of Modem Physics A, Vol. 10, No. 29 (1995) 4241-4264 @)World Scientific Publishing Company
EXPERIMENTAL CONSEQUENCES OF ONE-PARAMETER NO-SCALE SUPERGRAVITY MODELS JORGE L. LOPEZ* Center for Theoretical Physics, Department of Physics, Texas A&M university, College Station, T X 77843-4242, USA and AStTopaTtiCk Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, The Woodlands, TX 77381, USA
D. V. NANOPOULOS Center f o r Theoretical Physics, Department of Physics, Texas A&M University, College Station, T X 77843-4242, USA and Astropartide Physics G T O U Houston ~, Advanced Research Center (HARC), Mitchell Campus, The Woodlands, T X 77381, USA and CERN Theory Division, 1211 Geneva 23, Suitzer[and A. ZICHICHI CERN, 1211 Geneva 23, Switzerland
Received 24 March 1995 We consider the experimental predictions of two one-parameter no-scale SU(5) x U(1) supergravity models with string-inspired moduli and dilaton seeds of supersymmetry breaking. These predictions have been considerably sharpened with the new information on the top-quark mass from the Tevatron, and the actual measurement of the B(b + sy) branching ratio from CLEO. In particular, the sign of the Higgs mixing parameter p is fixed. A more precise measurement of the top-quark mass above (below) M 160 GeV would disfavor the dilaton (moduli) scenario. Similarly a measurement of the lightest Higgs-boson mass above 90 GeV (below 100 GeV) would disfavor the dilaton (moduli) scenario. At the Tevatron with 100 pb-', the reach into parameter space is significant only in the dilaton scenario (m 580 GeV) via the trilepton and top-squark signals. At
*
XI
LEPII the dilaton scenario could be probed up to the kinematical limit via chargino and top-squark pair production, and the discovery of the lightest Higgs boson is guaranteed. In the moduli scenario, only selectron pair production looks promising. We also calculate the supersymmetric contribution to the anomalous magnetic moment of the muon.
1. Introduction
Experimental tests of supersymmetric models have become quite topical with the advent of high-energy colliders such as the Tevatron and LEP, and their planned *Present address: Department of Physics, Rice University, 6100 Main Street, Houston, TX 77005, USA.
4241
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J . L. Lopez, D. V. Nanopoulos d A . Zichichi
or proposed upgrades. Since the expected scale of supersymmetry is no lower than the electroweak scale, it is not surprising that supersymmetric particles have yet to be found, even though they may be “just around the corner.” In working out the predictions for experimental measurables, one must resort to sensible models of low-energy supersymmetry, which in all generality are described by a large number of parameters. The explicit or implicit choice of a theoretical “framework” to reduce the size of the parameter space is therefore mandatory. The typical framework consists of a supergravity theory with minimal matter content and universal soft-supersymmetry-breaking parameters, and radiative breaking of the electroweak symmetry. Only four parameters are then needed to describe models within this framework: m1/2,mo, A , tan p.’ Further reduction in the number of model parameters can be accomplished by invoking the physics of superstrings, as the underlying theory behind the effective supergravity model one chooses. However, the superstring framework is likely not consistent with the minimal supersymmetric standard model (MSSM) matter content, since unification of the gauge couplings should not occur until the string scale (Mstring lo1* GeV) is reached. Moreover, traditional GUT models are not easily obtained in string model building. Therefore we are led to consider a prototype GUST (grand unified superstring theory) based on the gauge group SU(5) x U(1),2 with additional intermediate-scale particles to unify at the string scale.3 The motivations for SU(5) xU(1) model building have been elaborated e l ~ e w h e r eOf . ~ particular relevance are the natural suppression of dimension-5 proton-decay operators, the elegant doublet-triplet splitting mechanism, and the novel seesaw mechanism. Also, the usual Yukawa coupling unification is not required in SU(5) x U(1). Concrete string models based on the gauge group SU(5) x U ( l ) have also been obtained and explored in detail.5 Within the string framework one can study ansatze for the soft-supersymmetrybreaking parametem6 We consider such assumptions which are also universal, and entail relations of the form: mo = mo(mlp),A = A(m1p). We thus obtain a twodimensional parameter space ( m l l 2 ,tanP). As a last step in the reduction process, we study two specific scenarios in which the supersymmetry-breaking parameter associated with the Higgs mixing term p is also determined [i.e. B = B ( m l p ) ] , which we call “strict no-scale SU(5) x U( 1)supergravity.” We consider two scenarios:
-
(1) Moduli scenario:’ mo=0,
A=O,
B=O.
(1)
This scenario corresponds to a Kahler function with the structure SU(N, 1)/ U ( l ) x SU(N), and supersymmetry breaking is driven by the moduli fields (a single one in this case). It should be mentioned that in the context of fourdimensional strings, such supersymmetry-breaking parameters are likely to also include mll2 = 0, unless the gauge kinetic function possesses a nonminimal form or the dilaton also participates in the supersymmetry breaking.
415 Experimental Consequences of One-Parameter No-Scale
...
4243
(2) Dilaton scenario:6 1 mo = --1/2
6
,
A = -mlp ,
2 B = -ml12.
&
In this case supersymmetry breaking is driven by the dilaton field (this gives the first two relations). Moreover, the p term in the superpotential is assumed to come exclusively from the Kahler potential (this gives the third relation). These two scenarios are one-parameter models, since we can trade B = B ( m l p ) for t a n p = tanP(ml,2). They also have a dependence on the top-quark mass, and the sign of the Higgs mixing parameter p. We should note that our intention is not to study the general class of string supergravity models, but rather to choose specific oneparameter scenarios which can be then analyzed in detail for testable experimental predictions. The procedure followed here can then be repeated for any one-parameter model that one wishes to consider. The final parameter m112, i.e. the scale of the supersymmetric spectrum, can be determined dynamically in the no-scale supergravity framework.' In this framework one starts with a supergravity theory with a flat direction which leaves the gravitino mass undetermined at the classical level. This theory also has two very healthy properties regarding the vacuum energy: (i) it vanishes at tree levelg and (ii) it has no large oneloop corrections (i.e. Str M 2 = O).7 In this case the vacuum energy is at most O(m&). Minimization of the electroweak effective potential with respect to the field corresponding t o the flat direction determines in principle the scale of supersymmetry breaking." In this spirit we study the present one-parameter models keeping in mind that the ultimate parameter will be determined eventually by the no-scale mechanism in specific string models. This paper is organized as follows. In Sec. 2 we explore the constraints on the parameter space of these models and describe their sparticle and Higgs-boson spectrum. In Sec. 3 we update the calculation of B(b + sy) and contrast it with the latest CLEO results. In Sec. 4 we study the experimental signals for these models at the Tevatron (squark-gluino, trileptons, top squarks, top-quark decays), LEPII (Higgs bosons, charginos, selectrons and top squarks) and HERA (elastic selectronneutralino and chargino-sneutrino production). Finally, in Sec. 5, we summarize our conclusions. 2. Parameter Space and Spectrum The one-dimensional parameter space of the models described above can be represented in the ( m *, t a n p ) plane by the relation t a n p = tanP(mx:). This exercise X1 has been carried out for the moduli and dilaton scenarios first in Refs. 11 and 12 respectively. The results depend on the value of mt and the sign of p. In the moduli scenario one can show" that for mt 5 130 GeV the condition in Eq. (1) ( B = 0) can only be satisfied for p > 0, whereas for mt 2 135 GeV this condition requires p < 0. In the dilaton scenario the corresponding condition in Eq. (2) [B = (2/fi)ml/2] can only be satisfied for p < 0.l2 The reason for these restrictions is that p
416 J . L. Lopez, D . V. Nanopoulos d A . Zichichi
4244
and B are determined by the radiative electroweak symmetry-breaking constraint, and this depends on mt. The above-quoted values of mt are running masses which are related to the experimentally observable pole masses vial3
where Kt=16.11-1.04
( 1 - 2 )
x11.
(4)
mgi <mt
Thus we obtain mYle x 1.07mt. Taking the recently announced CDF measurement at face value, m y i e = 174 f 17 GeV,I4 we can see that mt = 130 GeV tt mYle = 139 GeV is more than 2u too low. On the other hand, fits to all electroweak data prefer a lower top-quark mass, when the Higgs-boson mass is restricted t o be light (as expected in a supersymmetric theory). The latest global fit gives mYle = 162 f 9 GeV,15 and mt = 130 GeV is again more than 2u too low. Thus we conclude that in the moduli scenario one must have p < 0, since p > 0 can only occur for values of mt which are in gross disagreement with present experimental data. Interestingly enough, both one-parameter models are viable only for p < 0. All sparticle and Higgs-boson masses and the calculations based on them have a dependence on mt. As discussed below, the mt dependence is small in the dilaton scenario, but it can be significant in the moduli scenario. Unless otherwise stated, in what follows we take mt = 150 GeV t) m r l e = 160 GeV as a representative value. The calculated value of t a n p , as outlined above, is shown in Fig. 1. The various symbols used to denote the points will be discussed below. (1) D i l a t o n scenario. The dependence on mx: is very mild: t a n p x 1.4. The dependence on mt is also mild; m;Ole = 160 GeV is shown in Fig. 1. However, for t a n p = 1.4 one must have mt 5 155 GeV c) mFole 5 165 GeV in order t o avoid a Landau pole in the evolution of At up to the unification scale. This upper limit on m;Ole is within all presently known limits on mt. (2) Moduli scenario. The results are rather mt-dependent (mFole = 160, 170, 180 GeV are shown in Fig. l ) , with t a n p decreasing with increasing values of mt: t a n p x 17.7 - 23.1 for mYle = 150 GeV , t a n p M 12.4 - 18.4 for mYle = 160 GeV, tan p
M
8.25 - 13.1 for mYle = 170 GeV,
tan p x 5.53 - 8.53 for mYle = 180 GeV. The range of tan ,4 values indicates the monotonic increase with m x : .
417 Experimental Consequences of One-Parameter No-Scale . . . 4245
strict no-scale S U ( 5 ) X U ( I ) 20
I
I
I
I
I
l
l
1
I
l
l
1
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l
l
I
1
I
I
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-
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15
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10
4
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dilaton scenario
1
Fig. 1. The one-dimensional parameter space of strict no-scale SU(5) x U(l)supergravity -moduli and dilaton scenarios. In the moduli scenario results are rather mt-dependent = 160, 170, 180 GeV are shown). In the dilaton scenario = 160 GeV is taken and r n : O l e < 165 GeV is required. Points excluded by B(b --t sy) are denoted by crosses ( x ) , those consistent with the Standard Model prediction are denoted by diamonds (o),and the rest are denoted by dots (.).
my'=
For a chosen value of mt we can then calculate the sparticle and Higgs-boson spectrum as a function of m + . After this is done, the current experimental lower x1 bounds on the sparticle and Higgs-boson masses (most important, mh 2 64 GeV and mx:2 45 GeV) are enforced and the actual allowed parameter space results. The neutralino and chargino masses are shown in Fig. 2 , the slepton and Higgsboson masses and the value of p in Fig. 3 , and the gluino and squark masses in Fig. 4. One can see that most of the masses are nearly linear functions of m x ; , although the slope depends on the scenario. Common results are the relations
418 4246
L. Lopez, D. V. Nanopodos
J.
€4 A . Zichichi
strict no-scale SU(5)xU(1) moduli scenario 1000
I
800
-
600
-
50
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,
, ,,,
, , I
dilalon scenario 1000
-
x:.x:
100
150
200
250
7
300
000
600
50
100
150
200
250
300
Fig. 2. T h e chargino and neutralino masses (in GeV) versus the lightest chargino mass in strict no-scale SU(5) x U ( l ) supergravity - moduli and dilaton scenarios. The following relations hold: m o NN m M 2mxy, mxg,4 M m M [PI. x2
*
XI
*
X2
Fig. 3. T h e slepton ( F ~ , ~ , ? l , z , f iand ) Higgs-boson masses ( h , A ,H , H + ) as a function of the chargino mass in strict no-scale SU(5) x U ( l ) supergravity - moduli and dilaton scenarios. Also shown is the calculated value of the Higgs mixing parameter p.
419 Experimental Consequences of One-Parameter No-Scale . . , strict no-scale
SU(S)XU( 1)
moduli scenario
dilalon scenario
1000
1000
800
800
600
600
400
400
200
200
0
50
LOO
150
200
4247
250
300
50
100
150
200
250
300
Fig. 4. The first- and second-generation average squark mass (@),the gluino mass (ij, dashed lines), the sbottom masses ( b l , ~mi2 , M ma), and the stop masses ( t l , ~as ) a function of the chargino mass in strict no-scale SU(5) x U(1) supergravity - moduli and dilaton scenarios. Note that ma M mi and that il can be quite light in the dilaton scenario (mil > 67 GeV).
Another common feature is the little variability of mh with m
f:
x1
mh x 100 - 115 GeV , mh
M
64 - 88 GeV ,
moduli scenario; dilaton scenario .
(8)
(9)
In fact, the two Higgs-mass ranges do not overlap and thus a measurement of mh would discriminate between the two models. Note also that the three Higgs bosom beyond the lightest one are very close in m a s (especially in the dilaton scenario; see Fig. 3) and not light: mH+ 2 160 (400) GeV in the moduli (dilaton) scenario. In the moduli scenario the stau-mass eigenstates are significantly split from the corresponding selectron (and smuon) masses (see Fig. 3). This is not the case for the dilaton scenario, where they are largely degenerate. These facts are easily understood in terms of the value of mo in the two scenarios. Regarding the gluino and squark masses (see Fig. 4 ) , the “average” squark mass [i.e. m,- = (mfiL+ m + ~ md, +mdR)/4,with about 4 ~ 2 %split around the average] is very close to the gluino mass
2 255 GeV , l.Olmy 2 260 GeV ,
m,- M 0.97mg
moduli scenario ;
m,- M
dilaton scenario ;
~
420
4248
J . L. Lopez, D. V. Nanopoulos d A . Zichichi
and both are higher than the present Tevatron sensitivity. We note in passing that the potentially large difference between the running gluino mass m i and the pole gluino m a s mFle,16
is naturally suppressed in these models because of the near-degeneracy of gluino and squark masses. We find that NN 1.1mG. The situation is quite different for the third-generation squark masses, which are significantly split from rnt, except for 82: mi, x m i . The most striking departure is that of the lightest top squark f ~ The . implications of a light top squark in the dilaton scenario have recently been discussed in Ref. 17, and will be reemphasized below. We should add that in this scenario the small value of t a n p ( x 1.4) and the light top squarks would appear to imply very small tree-level [m cos2 2p] and oneIn practice, the one-loop contribution loop [mln(m;lm;z/m~)] contributions to to mh turns out to be sizeable enough because of the often-neglected top-squark 2 1 sinZ2e mixing effect which adds a large positive term (mil - miJ5---&ln(m$m~2) t o the usual piece.l*
mi.
-
strict no-scale SU(S)xU( 1) 1 .o
0.8
0.6
q-4 0.4
0.2
0.0
50
100
150
200
250
300
Fig. 5 . The calculated value of the relic density of the lightest neutralino ( & h i ) as a function of the chargino mass in strict no-scale SU(5) x U(l) supergravity - moduli and dilaton scenarios for mPole - 160 GeV. The dotted line in the moduli scenario corresponds to mfole = 180 GeV. Note that R,hi is sizeable but within cosmological limits.
42 1 Ezperimental Consequences of One-Parameter No-Scale . . . 4249
As noted above, in the moduli scenario the allowed values of t a n @ depend strongly on The calculated value of p also depends on mt: p(mt) oc mt to good approximation in the range of interest. The lightest Higgs-boson mass also depends on mt: the upper limit in Eq. (8) increases to 120 (125) for m r l e = 170 (180) GeV. The squarks and sleptons are only slightly affected via their (small) t a n p dependence. Even the lightest top squark is not affected by more than f 2 % . We have also calculated the relic abundance of the lightest neutralino Rxhi following the methods of Ref. 19. The results are shown in Fig. 5. We get R,hi 5 0.25 (0.9) in the moduli (dilaton) scenario. These results are automatically consistent with cosmological expectations (i.e. R,hi < 1). The structure on the curves (especially in the dilaton scenario) corresponds to s-channel h and 2 poles in the annihilation cross section. The dotted line in the moduli scenario reflects the mt dependence (mpole = 180 GeV for this line) via the different tan@ and 1-1 values.
3. b - s y There are several indirect experimental constraints which can be applied t o SU(5) x U ( l ) supergravity models. For the case of two-parameter models of this kind, these constraints have been discussed in Ref. 20. It turns out that the only one of relevance for the one-parameter models is that from B(b + sy). An analysis of this constraint in a variety of supergravity models has recently been performed in Ref. 21. Here we update this analysis for the one-parameter models in light of the most recent CLEO experimental result ,22 ~ ( b - , s y ) = ( 1 - 4 ) ~ 1 0 - ~ , at95%CL.
(13)
In Fig. 6 we show the calculated value of B(b + sy) in both scenarios (for 160 GeV). The latest CLEO limits are indicated by the solid lines, with
myle -
the arrows pointing into the allowed region. The Standard Model result is also shown. As explained in Ref. 21, there is significant theoretical uncertainty on the value of B(b + sy), mostly from next-to-leading-order QCD corrections. We have roughly quantified this uncertainty by using a leading-order calculation but allowing the renormalization scale to vary between mb/2 and 2mb. This variation gives the dotted lines above and below the solid lines in Fig. 6 . The same procedure is used to estimate the Standard Model uncertainty, which shows that the data agree well with the Standard Model. In fact, the theoretical uncertainty in the Standard Model prediction is larger than the present la experimental uncertainty.22 In the moduli scenario there is further uncertainty, because the value of mt affects the calculated value of t a n p . From Fig. 1 we see that larger values of mfole decrease t a n p , and this leads t o larger values of B(b + sy): the dotted-dashed line represents the result ( p = mb) for m;’le = 180 GeV. In the one-parameter models, we consider points in parameter space to be “excluded” if their interval of uncertainty does not overlap with that in Eq. (13); these are denoted by crosses ( x ) in Fig. 1. In the moduli scenario, this constraint
422 4250
J . L . Lopez, D. V. Nanopoulos €4 A . Zichichi s t r i c t no-scale SU(S)xU( 1) moduli scenario
dilaton scenario
7
CLEO 95% CL -
--
-,
_. -
.-.
I
L. . L
.A.
- .
6 t
P
v
m
10-4
10-4
50
100
150
200
250
m,; (GeV)
300
50
100
150
250
200
300
m,: (GeV)
Fig. 6. T h e branching fraction B(b -+ sy) as a function of the chargino mass in strict no-scale SU(5) x U ( l ) supergravity - moduli and dilaton scenarios for r n p o l e = 160 GeV. T h e dotted lines above and below the solid line indicate the estimated theoretical error in the prediction. The dashed lines delimit the Standard Model prediction. T h e arrows point into the currently experimentally allowed region. The dotted-dashed line in the moduli scenario corresponds t o mpole = 180 GeV (central value).
requires mx*
2
120 GeV for
mpole
= 160 GeV, but only mx:
2
75 GeV for
= 180 GeV. In the dilaton mPole - 170 GeV, and there is no constraint for scenario, because the allowed values of tan ,B are small, the constraint is rather mild, only requiring m * 2 50 GeV. Stricter constraints can be obtained by allowing less x1 experimental uncertainty (e.g. la) or less theoretical uncertainty, both of which are unwise things to do. We have also identified “preferred” points whose interval of uncertainty overlaps with the corresponding Standard Model interval; these are denoted by diamonds (0)in Fig. 1. From this vantage point, the dilaton scenario or the moduli scenario with a somewhat heavy top quark looks quite promising. 4. Experimental Predictions
We now discuss the experimental signatures of these one-parameter models at the Tevatron, LEPII and HERA. For this analysis we consider only the points still allowed by the b t sy constraint, i.e. those denoted by dots and diamonds in Fig. 1.
423 Experimental Consequences of One-Parameter No-Scale . . .
4251
4.1. Tevatron
We consider the present-day fi = 1.8 TeV Tevatron with an estimated integrated luminosity of 100 pb-' at the end of the ongoing Run IB. Three supersymmetric signals could be observed: trileptons from chargino-neutralino production, large missing energy from squark-gluino production, and soft dileptons from top-squark production. All three signals could be observable in the dilaton scenario; only the squark-gluino signal may be observable in the moduli scenario.
-
strict no-scale SU(?i)XU(I) moduli scenario
1.o
I'13'
dilaton scenario
I .o
I"'~I""I"" L
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 c
0.0
50
100
150
200
250
300
n n "._
-1
u 4
150
100
50
200
250
300
m,; (GeV) 1.o 0.8 0.6
0.4
o.6 k 0.4
0.2
0.2
0.0
50
100
150
200
250
e
+
p
,
300
m,; (GeV) Fig. 7. The chargino and neutralino leptonic and hadronic branching fractions as a function of the chargino maSs in strict no-scale SU(5) x U(1) supergravity - moduli and dilaton scenarios. The sudden drops in the neutralino branching fractions correspond to the opening of the "spoiler" mode x! -+ x y h .
(1) Neutralinos and charginos. These are produced in the reaction p p + X : X : X , ~and ~ when required to decay leptonically yield a trilepton The cross section is basically a monotonically decreasing function of m * , whereas the x1 leptonic (and hadronic) branching fractions (given in Fig. 7) are greatly modeldependent and vary as a function of the single parameter. Our calculations have been performed as outlined in Ref. 25. In the moduli scenario there is an enhancement of the chargino leptonic branching fraction because of the presence of light t*tD). However, this gain is undone by the suppressed neusleptons (eg. xf tralino leptonic branching fraction also because of the light sleptons (i.e. xi + v f i ) . --$
424 4252
J . L. Lopez, D. V. Nanopoulos & A . Zichichi
In contrast, in the dilaton scenario the two branching fractions are comparable. However, in this case for m i 2 165 GeV the spoiler mode x: + Xyh opens up and x1 the neutralino leptonic branching fraction becomes negligible. The trilepton rates are given in Fig. 8, where we indicate by a dashed line the present CDF upper limit obtained with 20 pb-I of data.26 By the end of the ongoing Run IB the integrated luminosity is estimated at 100 pb-' per detector. If no events are observed, one could estimate an increase in sensitivity by a factor of 4 (assuming 80% efficiency in recording data) per experiment. Combining the two experiments we see that the sensitivity would be even higher (say 6 times better). In Fig. 8 we show the estimated sensitivity range as the area between the 80-90 GeV. dotted lines. In the dilaton scenario we estimate the reach a t mx; On the other hand, in the moduli scenario the rates are small, but there may be a small observable window for m,,: M 100 GeV. However, because of the constraints
-
-
N
s
from b + sy, such values of m
x1
Tevatron
are allowed only for m;Ole
2 165 GeV.
strict no-scale S U ( ~ ) X U I) (
100 n
--
P
a
v
10-2
1
50
100
1
1
1
150
200
Fig. 8. The rate for trilepton production at the Tevatron as a function of the chargino mass in strict no-scale SU(5) x U(l) supergravity - moduli and dilaton scenarios. The present CDF upper bound is indicated by the dashed line, and the estimated reach at the end of Run IB is bounded by the dotted lines.
(2) Gluino and squarks. Since in these models we obtain m,- M mg, the multijet missing-energy signal is enhanced. In both scenarios we also obtain a lower bound of 260 GeV, which makes this signal almost kinematically inaccessible. Indeed, the reach with 100 pb-' is estimated at mq = mg 5 300 GeV H mx: 5 60 GeV.2r N
425 Experimental Consequences of One-Parameter No-Scale
...
4253
(3) Top squarks. Direct fl pair production at the Tevatron (via the dilepton mode) has been shown recentlyz8 t o be sensitive t o mi, 5 100 GeV by the end of Run IB, provided the chargino leptonic branching fraction is taken t o be 20%. In the dilaton scenario & can be rather light (mr, 2 70 GeV) and the chargino branching fractions are 40% (see Fig. 7). In Ref. 28, with a “bigness” B = I p ~ ( l + )+ l I p ~ ( l ’ - ) l +TI cut of B < 100 GeV, the tf (with mt = 170 GeV) and W+W- backgrounds are estimated at 14 fb and 10 fb respectively. With 100 pb-’, a 5a signal above this background requires (aB)dileptons 2 75 fb. From Fig. 10 in Ref. 28 it appears then that mi, 5 130 GeV could be probed in this case of enhanced branching fractions. In the moduli scenario the top squarks are too heavy to be detectable (mil 2 160 GeV). (4) Soft dileptons. In the dilaton scenario, if f 1 is light enough, events may already be present in the existing- data sample. The cross section for pair production of the lightest top squarks a(i&) depends solely on mrl ,29 and is given for a sampling of values in Table 1. Since in the dilaton scenario mi, > mx: +mB (SEE BG. 4), ONE GETS B ( & + bx;) = 1 (neglecting the small one-loop i1 + cx: mode3’).“ The charginos then decay leptonically or hadronically with branching fractions shown in ~ lvex!) M 0.4 (l = e p ) for mx1* 5 65 GeV ct mi, 5 100 GeV. Fig. 7, i.e. B ( x + The most promising signature for light topsquark detection is through the dilepton mode.” The ratio of stop dileptons to top dileptons is”
-
-
+
Table 1. Cross sections at the Tevatron (in pb) for p p -+ All masses in GeV.
i1f1X2’and pp 4 tTX.”
mzl
70
80
90
100
112
mt
120
140
160
180
c~(ilf1)
60
30
15
8
4
u(t0
39
17
8
4
-
This ratio indicates that for sufficiently light top squarks there may be a significant number of dilepton events of non-top-quark origin, if the experimental acceptances are tuned accordingly. Perhaps the most important distinction between top dileptons and stop dileptons is their p~ distribution: the (harder) top dileptons come from the two-body decay of the W boson, whereas the (softer) stop dileptons come from the (usually) threebody decay of the chargino with masses (in this case) below m w . Therefore, the top-dilepton data sample is essentially distinct from the stop-dilepton sample. Such a distinction is well quantified by the “bigness” (B) parameter mentioned above. ”This unsuppressed two-body decay mode of to be o b ~ e r v a b l e . ~ ~
implies that scalar stoponium will decay too soon
426 4254
J. L . Lopez, D. V . Nanopoulos & A . Zichichi
(5) Top-quark branching fractions. In the case of a light top squark, the channel &x(: may be kinematically accessible. In the dilaton scenario for mFole = 160 GeV this is the case for m ~ 5, 115 GeV. The calculated values of B(t --$ flxt) and B ( t + bW) are shown in Fig. 9. One can see that if is light enough, one would expect up t o a (0.9)2 M 20% reduction in the number of observed top events relative t o the Standard Model prediction. However, this discrepancy would not be observable until a sizeable top-quark sample is collected. For a 2a effect one would need to measure the ti? cross section t o 10% accuracy, which requires 100 background-subtracted top events. This event sample will not be available before the Main Injector era.
t
+
-
strict no-scale S U ( 5 ) x U ( 1)
t-rbW
dilaton scenario
0.4
0.0
70
80
Fig. 9. Top-quark branching fractions for ,Elole supergravity - dilaton scenario.
90
100
110
= 160 GeV in strict no-scale SU(5) x U ( l )
4.2. LEPII
At present it is uncertain what the LEPII beam energy may ultimately be. It is expected that LEPII will turn on in 1996 at f i x 180 GeV, while the highest possible center-of-mass energy is estimated at f i = 240 GeV. The precise value of fi has two main effects: it determines the kinematical reach for pair-produced particles (such as charginos and selectrons), and it determines the reach in Higgsboson masses. The latter is of more relevance, since for sufficiently high values of fi (< 240 GeV), it may be possible to cover all of the parameter space of these models. For definiteness, unless otherwise stated, in what follows we will set f i = 200 GeV. We consider four signals: Higgs bosons, charginos, charged sleptons,
427 Experimental Consequences of One-Parameter No-Scale
.. .
4255
and top squarks. The calculations of the first three signals have been performed as described in Ref. 33. (1) Higgs bosons. These are produced via e+e- + Zh, with h + b6 and btagging to reduce the background. The cross section for this process differs from the corresponding Standard Model cross section in two ways: by the factor sin2(a - p ) and by the ratio f = B ( h + b 6 ) / B ( H + bb)sM. In the models under consideration, sin2(a - p) is very close to 1, according to a decoupling phenomenon induced by the radiative electroweak breaking mechanism.34In Fig. 10 we show the h + b6 branching fraction, which shows that f is usually close t o 1, except when the supersymmetric decay mode h + x:xy is open. This channel is open for mh > 2m,; M m x * , i.e. only for the lightest values of mh,since mh grows little with as Fig. i0 shows.
mx,,
strict
""""""""""'""
/
60
50
100
150
200
250
300
60
70
80
90
100
110
120
Fig. 10. The lightest Higgs-boson mass as a function of the chargino mass in strict no-scale SU(5) x U(1) supergravity - moduli and dilaton scenarios. Note that the predicted mass ranges do not overlap and that a lower bound on m h translates into a lower bound on the chargino mass. Also shown are the Higgs-boson branching fractions into b6 and xyxy.
The effective cross sections cr(e+e- + Z h ) x f are shown in Fig. 11. The deviations of the curves from monotonically decreasing functions of mh,which coincide with the Standard Model prediction, are due to the h + x:~: erosion of the preferred h + b6 mode. These deviations could be used to differentiate between the Standard Model Higgs boson and the supersymmetric Higgs bosons considered here. The LEPII sensitivity for Higgs-boson detection is estimated at 0.2 pb for a 3a effect in 500 pb-' of data.35 In the moduli scenario this cross-section level is reached for mh x 106 (114) GeV for fi = 200(210) GeV. From Eq. (8) we see that
428 4256
J . L . Lopez, D . V . Nanopoulos d A. Zichichi strict no-scale SU(5)XU(l)
LEP 11
I " " I " " I " " I ' " ' ~
moduli scenario
-
0.0
60
70
80
90 mh
100
120
110
(GeV)
Fig. 11. The effective cross section a ( e + e - + Z h ) x f [f = B ( h + b 6 ) / B ( H + b 6 ) ~ for ~ ] Higgs-boson production a t LEPII (for the indicated center-of-mass energies) as a function of the Higgs-boson mass in strict no-scale SU(5) x U ( l ) supergravity - moduli and dilaton scenarios. The dashed line indicates the estimated experimental sensitivity. Note that the two scenario predictions do not overlap. The deviations of the curves from monotonically decreasing functions of m h (which coincide with the Standard Model prediction) are due to the h + x:x: erosion of the preferred h -+ b6 mode, and could be used to differentiate between the Standard Model Higgs boson and the supersymmetric Higgs boson considered here.
LEPII would need to run at fi M 210 GeV to cover the whole parameter space (for - 160 GeV). On the other hand, in the dilaton scenario for fi = 200 GeV one obtains a ( e + e - 4 Z h ) x f > 0.57 pb, which has a 5a significance for L = 170 pb-'. [For fi = 180 GeV we obtain a(e+e- --f Z h ) x f > 0.22 pb, ie. also observable with sufficiently large C.] Therefore, LEPII should be able to cover all of the parameter space in the dilaton scenario via the Higgs signal. In Fig. 10 we also show a detail of the relation between mh and m i which shows that a lower bound on mh would XI immediately translate into a lower bound on the chargino mass, and in turn into a lower bound on all sparticle masses. (2) Charginos. The preferred signal is the "mixed" decay ( l a 2 j ) in pairproduced charginos. The chargino branching fractions in Fig. 7 indicate that this mode is healthy in the dilaton scenario, but rather suppressed in the moduli scenario. This is revealed in Fig. 12, where we show the "mixed" cross sections in both scenarios. With an estimated 5a sensitivity of 0.12 pb (for 500 pb-1),20 one should be able to reach up to mx: 5 96 GeV in the dilaton scenario. The reach should decrease by N 10 GeV for fi = 180 GeV. In the moduli scenario we obtain (uB),ixed < 0.03, i.e. an unobservable signal. However, the much larger dilepton mode may lead to an observable signal in this case if the W+W- background could somehow be dealt with.
+
429 Experimental Consequences of One-Parameter No-Scale . . . 4257
LEPII
strict no-scale SU(5)XU( 1) 0.5
h
P
a
I I I
0.4
-
0.3
-
I
I
I
I
I
~ 9 I1
~I
11
I I I I I I I I I I I
1
1
-
v h
tu”
+
4
.-I
t
I d
+xx
0.2 -
t
I
_____________-_
Q
+
W
v
b
0.1
- moduli 40
scenario
50
60
70
90
80
100
m,; (GeV] Fig. 12. The chargino pair-production cross section into the “mixed” mode ( l l
+ 2j) at LEPII as
a function of the chargino mass in strict no-scale SU(5) x U(1) supergravity - moduli and dilaton
scenarios. The dashed line indicates the estimated experimental sensitivity. Note that the two scenario predictions do not overlap.
strict no-scale SU(S)xU( 1)
LEPIl
n
P
P4
W
8 la
la
t I
+0)
al
W
b
60
70
80
90
100
ma,=mW(GeV) Fig. 13. The selectron (ERER + Z R Z L ) and smuon ( i i ~ i ipair-production ~) cross sections a t LEPII as functions of the slepton mass in strict no-scale SU(5) x U ( l ) supergravity - moduli scenario. The dashed lines delimit the estimated experimental sensitivity. In the EE case, for mSR > 80 GeV the ERZL channel is closed and thus there is a kink on the curve.
430 4258
J . L. Lopez, D. V. Nanopoulos Ed A. Zachichi
(3) Sleptons. At LEPII, only in the moduli scenario are the sleptons kinematically accessible (see Fig. 3). The processes of interest are e+e- -+ ERER ERZL and e+ei i ~ j iwith ~ , the further decays ER,L ex: and DR -+ px! with near-100% branching fractions.33 The selectron cross section, shown in Fig. 13, is large: a(e+e- -+ EE) > 1 pb for miR < 80 GeV. (Note the kink on the curve when the E R E L channel closes.) The 5a sensitivity for C = 100 (500) pb-’ is estimated at 0.47 (0.21) pb.20 This sensitivity level is reached for ma, z 84 (90) GeV and corresponds to mx: x 92 (103) GeV (see Fig. 3). Thus, the indirect reach for chargino masses is larger than the direct one (via the “mixed” mode). The smuon cross section is much smaller (see Fig. 13) and so are the corresponding reaches in smuon [mp, x 70 (82) GeV] and chargino masses. Again, these reaches should decrease by 10 GeV for ,b = 180 GeV. (4) Top squarks. These are kinematically accessible only in the dilaton scenario. Moreover, since mfl 5 100 GeV for mx: 5 65 GeV, the reach into the parameter space is not very significant, but a new lower bound on mi, could be obtained. We consider the process e+e- + f1f1 -+ (bx:)(bx:) (15) with B(fl -+ bx;) = 1. The cross section cT(e+ef1f1) proceeds through schannel photon and Z exchanges. In the case of Z exchange, the coupling Zf1f1 is proportional to cos2 8, - sin2 8w and vanishes for 60s’ 8, x 0.31. In the dilaton
+
-+
-+
-
-+
strict no-scale S U ( 5 ) X U ( 1)
LEPII 0.5
I
i
1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
-
n
dilaton scenario
P
a
W
n 4
0.3
-
0.1
-
--
Lc, 4
&
t
I
al -k
a,
v
b
60
70
80
90
100
Fig. 14. T h e lightest top-squark pair-production cross section at LEPII as a function of the topsquark mass in strict no-scale SU(5) x U(l) supergravity - dilaton scenario. The higher (lower) dashed line indicates the limit of three 2b+2t, 2b+ l t + 2 j or 2 6 + 4 j events in L = 100 (500)pb-’ of data.
43 1 Ezpen'mental Consequences of One-Parameter No-Scale . . . 4259
scenario, for mi, 5 100 GeV we find that cos2 Bt x 0.60 and this cancellation does not occur. The cross section is shown in Fig. 14, and has been calculated including initial-state radiation and QCD corrections, as described in Ref. 36. Depending on the chargino decays, one can have three signatures: 2b+ 2!, 2b+ 1!+2j, 2b+ 4j, all with the same branching fraction of x (0.4)2 (see Fig. 7). The traditional W+Wbackground is not relevant (unless one !is lost and there is no &tagging), and probably the channel with the least number of jets (i.e. 2b+2!) is preferable. Assuming a suitably cut background, three signal events (of any of the three signatures) would be observed for g ( e + e - + ili1) 2 O.lg(0.04) pb with C = lOO(500) pb-l. From Fig. 14 this sensitivity requirement implies a reach of mi, x 85 (95) GeV. 4.3. HERA
The weakly interacting sparticles may be detectable at HERA if they are light enough and if HERA accumulates an integrated luminosity O(100 pb-'). So far HERA has accumulated a few pb-' of data and it is expected that eventually it will be producing 25-30 pb-' per year. The supersymmetric signals in SU(5) x U(l) supergravity have been studied in Ref. 37, where it was shown that the elastic scattering signal, i.e. when the proton remains intact, is the most promising one. The deep-inelastic signal has smaller rates and is plagued with large backgrounds. p e - p + fie/,x;p. The total elastic The reactions of interest are e - p --f E t , R ~ y , 2and supersymmetric signal is shown in Fig. 15 versus the chargino mass. The dashed HERA
a
strict n o - s c a l e SU(s)xU( 1)
0.04
+
moduli
_ _
t 0.02
0.00
40
50
60
70
80
90
100
Fig. 15. The total elastic supersymmetric cross section (including selectron-neutralino and sneutrino-chargino production) at HERA as a function of the chargino mass in strict no-scale SU(5) x U(1) supergravity - moduli and dilaton scenarios. The dashed line indicates the sensitivity limit t o observe five events in 200 pb-' of data.
432 4260
J . L. Lopez, D. V. Nanopoulos d A. Zichichi
line represents the limit of sensitivity with 12 = 200 pb-’ which will yield five “supersymmetric” events. The signal is very small in the dilaton scenario, but may be observable in the moduli scenario. However, considering the timetable for the LEPII and HERA programs, it is quite likely that LEPII would explore all of the HERA accessible parameter space before HERA does. This outlook may change if new developments in the HERA program give priority to the search for the righthanded selectron ( E R ) , which could be rather light in the moduli scenario. At HERA one could also produce top-squark pairs, if they are light enough, as is possible in the dilaton scenario. However, the cross section in this case is smaller than 0.01 pb for mil > 70 GeV and decreases very quickly with increasing values of mi, .38 5. Conclusions
We have explored the experimental consequences of well-motivated one-parameter nmcale SU(5) xU(1) supergravity in the moduli and dilaton scenarios. Such models are highly predictive and therefore falsifiable through the many correlations among the experimental observables. In fact, the recent information on the top-quark mass has in effect ruled out half of the parameter space in the moduli scenario, selecting the sign of p t o be negative. Interestingly, this is also the required sign in the dilaton scenario. The top-quark-mass dependence is particularly important in these models in other ways as well. In the moduli scenario the calculated values of t a n p depend strongly on mt (see Fig. l ) ,and thus so do the calculated values of B(b 4 sy), which become quite restrictive for 5 160 GeV. On the other hand, in the dilaton scenario values of m;’le 2 165 GeV are not allowed. Therefore, future more precise determinations of the top-quark mass are likely to disfavor one of the scenarios and support the other. A similar (and partially related) dichotomy is present in the Higgs-boson masses, which are below 90 GeV in the dilaton scenario and above 100 GeV in the moduli scenario. We concentrated on the experimental signals at present-day facilities: the Tevatron with the expected integrated luminosity at the end of the ongoing run, the forthcoming LEPII upgrade, and HERA. At the Tevatron the traditional squarkgluino signal is enhanced (since mq M m i ) but the possible reach into parameter space is small since mi M mi 2 260 GeV is required. The trilepton signal is more promising, although only in the dilaton scenario, where a reach of mx1* M 80-90 GeV is expected. In this same scenario light top-squarks could be detected for mi, 5 130 GeV c-) mx: 5 80 GeV. So through different channels the reach into the parameter space should be similar. In the moduli scenario the reach into parameter space is not promising. At LEPII the Higgs boson should be readily detectable in the dilaton scenario (for fi > 180 GeV), in effect covering the whole parameter space of the model. In fact, even an improved lower bound on mh will constrain the parameter space immediately by requiring a lower bound on the chargino mass. In the moduli scenario Higgs detection requires @ 2 200-210 GeV. In both scenarios, for sufficiently
433 Ezperimental Consequences of One-Parameter No-Scale . . . 4261
low values of mh, the supersymmetric channel h + xtx: is open and decreases the usual b& yield in a way which could be used to differentiate between the Standard Model Higgs boson and the supersymmetric Higgs bosons considered here. Charginos should be readily detectable (via the “mixed” mode) almost up to the kinematical limit (&/2) in the dilaton scenario, but will be hard to detect in the moduli scenario. On the other hand, selectrons should be detectable up to near the kinematical limit in the moduli scenario (corresponding to charginos slightly over the direct kinematical limit), and be kinematically inaccessible in the dilaton scenario. Top squarks in the dilaton scenario should also be detectable up to near the kinematical limit, although this corresponds to much lighter chargino masses than in the other detection modes. We conclude that at the Tevatron and LEPII the dilaton scenario is significantly more accessible than the moduli scenario is:
LEPII
Tevatron
4-B Moduli Dilaton
J J
x:
21
h
x:
x
x
J
x
,/
J
J
J
d
i*
J x
x J
Moreover, in the dilaton scenario LEPII is basically assured of the discovery of the Higgs boson. Also, LEPII has the possibility of increasing its reach in both scenarios by increasing its center-of-mass energy. There is one set of experimental observables which we have not discussed here, namely the one-loop corrections to the LEP observables and their dependence on the supersymmetric parameters. In the context of SU(5) x U ( l ) supergravity these observables have been discussed in Refs. 39 and 20, where it was concluded that as long as 5 170 GeV there are no constraints on the model parameters at the 90% CL. One of these observables, namely the ratio R b = I‘(Z + b6)/ I’(Z + hadrons), has been measured more precisely during the past year4’ and its value still remains more than 2a above the Standard Model prediction, for not-toe small values of the top-quark mass. Recently, in Ref. 41, it has been argued that supersymmetry could provide a better fit to this observable should the chargino and the lightest top squark be both light. These conditions could be satisfied in the dilaton scenario discussed above, although an explicit calculation of this observable is required to be certain. As a final experimental consequence of these models, we have calculated the supersymmetric contribution to the anomalous magnetic moment of the muon (as described in Ref. 42), which is shown in Fig. 16. The arrow points into the presently experimentally allowed region. The upcoming Brookhaven E821 experiment (1996)43 aims at an experimental accuracy of 0.4 x lo-’, which is much smaller than the moduli-scenario prediction. This indirect experimental test is likely
434 4262
J . L. Lopez, D. V. Nanopoulos
tY A .
Zichichi
to be much more stringent than any of the direct tests discussed above. To close, we reiterate that these oneparameter no-scale supergravity models would become "no-parameter" models once the no-scale mechanism is implemented in specific string models, thereby determining the value of the ultimate parameter. s t r i c t no-scale S U ( ~ ) X U ( I ) 0.0
-2.5
-5.0
-7.5
. .
X
P 23 (d
- 10.0
m o d u l i scenario
- 12.5
-15.0
50
100
150
200
250
300
Fig. 16. The supersymmetric contribution t o the anomalous magnetic moment of the muon)'"":a( as a function of the chargino mass in strict no-scale SU(5) x U(1) supergravity - moduli and dilaton scenarios. The arrow points into the currently experimentally allowed region. In the moduli scenario, results are rather mt-dependent (mrole = 160, 180 GeV are shown). In the dilaton scenario mrole = 160 GeV is taken and mrole < 165 GeV is required.
Acknowledgments
J. L. would like to thank James White for useful discussions. This work has been supported in part by DOE grant DE-FG05-91-ER-40633. References 1. For a recent review see e.g.: J . L. Lopez, D. V. Nanopoulos and A. Zichichi, Nuovo Cimento Rivista 17,1 (1994). 2. I. Antoniadis, J. Ellis, J. Hagelin a n d D. V. Nanopoulos, Phys. Lett. B194, 231 (1987). 3. I. Antoniadis, J. Ellis, S. Kelley and D. V. Nanopoulos, Phys. Lett. B272,31 (1991); S. Kalara, J. L. Lopez and D. V. Nanopoulos, Phys. Lett. B269, 84 (1991); S. Kelley, J. L. Lopez an d D. V. Nanopoulos, Phys. Lett. B278, 140 (1992).
435 Experimental Consequences of One-Parameter No-Scale . . . 4263 4. For reviews see: D. V. Nanopoulos, “Les Rencontres de Physique de la Vallee d’Aoste,” in proc. Supernova 1987A, One Year Later: Results and Perspectives i n Particle Physics (ed. M. Greco) (Editions Frontieres, 1988), p. 795; J. L. Lopez, D. V. Nanopou10s and A. Zichichi, in From Superstrings to Supergravity - PTOC.26th Workshop of the ZNFN Eloisatron Project, eds. M. J. Duff, S. Ferrara and R. R. Khuri (World Scientific, Singapore, 1993), p. 284. 5. I. Antoniadis, J. Ellis, J . Hagelin and D. V. Nanopoulos, Phys. Lett. B 2 3 1 , 65 (1989); J. L. Lopez and D. V. Nanopoulos, Phys. Lett. B 2 5 1 , 73 (1990); J. L. Lopez, D. V. Nanopoulos and K. Yuan, Nucl. Phys. B399, 654 (1993). For a review see: J. L. Lopez, Surveys an High Energy Physics, 8, 135 (1995). 6. V. Kaplunovsky and J. Louis, Phys. Lett. B 3 0 6 , 269 (1993); A. Brignole, L. Ibbiiez, and C. Muiioz, Nucl. Phys. B422, 125 (1994). 7. J. Ellis, C. Kounnas and D. V. Nanopoulos, Nucl. Phys. B 2 4 1 , 406 (1984) and Nucl. Phys. B247, 373 (1984). 8. For a review see: A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. 145, 1 (1987). 9. E. Cremmer, S. Ferrara, C. Kounnas and D. V. Nanopoulos, Phys. Lett. B133, 61 (1983). 10. J. Ellis, A. Lahanas, D. V. Nanopoulos and K. Tamvakis, Phys. Lett. B 1 3 4 , 4 2 9 (1984). 11. J. L. Lopez, D. V. Nanopoulos and A. Zichichi, Phys. Rev. D49, 343 (1994). 12. J . L. Lopez, D. V. Nanopoulos and A. Zichichi, Phys. Lett. B319, 451 (1993). 13. N. Gray, D. Broadhurst, W. Grafe and K. Schilcher, 2. Phys. C48, 673 (1990). 14. CDF Collaboration (F. Abe et al.), Phys. Rev. Lett. 73, 225 (1994); Phys. Rev. D50, 2966 (1994). 15. J. Ellis, G. L. Fogli and E. Lisi, Phys. Lett. B 3 3 3 , 118 (1994). 16. S. Martin and M. Vaughn, Phys. Lett. B318, 331 (1993); Y. Yamada, Phys. Rev. Lett. 72, 25 (1994). 17. J. L. Lopez, D. V. Nanopoulos and A. Zichichi, Texas A & M University preprint CTP-TAMU-31/94 (hep-phj9406254). 18. J. Ellis, G . Ridolfi and F. Zwirner, Phys. Lett. B 2 6 2 , 477 (1991); J. L. Lopez and D. V. Nanopoulos, Phys. Lett. B 2 6 6 , 397 (1991). 19. J. L. Lopez, D. V. Nanopoulos and K. Yuan, Nucl. Phys. B 3 7 0 , 445 (1992); S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois and K. Yuan, Phys. Rev. D47, 2461 (1993). 20. J . L. Lopez, D. V. Nanopoulos, G. Park, X. Wang and A. Zichichi, Phys. Rev. D50, 2164 (1994). 21. J . L. Lopez, D. V. Nanopoulos, X. Wang and A. Zichichi, Phys. Rev. D 5 1 , 147 (1995). 22. CLEO Collaboration (M. S. Alam et al.), Phys. Rev. Lett. 74, 2885 (1995). 23. J. Ellis, J. Hagelin, D. V. Nanapoulos and M. Srednicki, Phys. Lett. B127, 233 (1983); A. H. Chamseddine, P. Nath and R. Arnowitt, Phys. Lett. B 1 2 9 , 445 (1983); H. Baer and X. Tata, Phys. Lett. B 1 5 5 , 278 (1985); H. Baer, K. Hagiwara and X. Tata, Phys. Rev. Lett. 57, 294 (1986); Phys. Rev. D 3 5 , 1598 (1987). 24. P. Nath and R. Arnowitt, Mod. Phys. Lett. A2, 331 (1987); R. Arnowitt, R. Barnett, P. Nath and F. Paige, Znt. J . Mod. Phys. A2, 1113 (1987); R. Barbieri, F. Caravaglios, M. Frigeni and M. Mangano, Nucl. Phys. B 3 6 7 , 2 8 (1991); H. Baer and X. Tata, Phys. Rev. D47, 2739 (1993); H. Baer, C. Kao and X. Tata, Phys. Rev. D48, 5175 (1993). 25. J. I;.Lopez, D. V. Nanopoulos, X. Wang and A. Zichichi, Phys. Rev. D48,2062 (1993). 26. CDF Collaboration (Y. Kato et al.), in PTOC.9th Topical Workshop on ProtonAntiproton Collzder Physics (Tsukuba, Japan, 1993), eds. K. Kondo and S. Kim (Universal Academy Press, Tokyo), p. 291; CDF Collaboration, Fermilab-Conf-94/
436 4264
27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.
40. 41. 42. 43.
J. L. Lopez, D. V. Nanopoulos d A . Zichichi 149-E (1994), to appear in proc. 27th Int. Conf. on High Energy Physics (Glasgow, 20-27 July 1994). J. T. White, private communication. H. Baer, J. Sender and X. Tata, Phys. Rev. D50, 4517 (1994). H. Baer, M. Drees, J. Gunion, R. Godbole and X. Tata, Phys. Rev. D44, 725 (1991). E. Laenen, J. Smith and W. L. van Neerven, Phys. Lett. B321, 254 (1994). I. Bigi and S. R u d a , Phys. Lett. B153, 335 (1985); K. Hikasa and M. Kobayashi, Phys. Rev. D36, 724 (1987). See e.g.: M. Drees and M. Nojiri, Phys. Rev. D49, 4595 (1994) and references therein. J. L. Lopez, D. V. Nanopoulos, H. Pois, X. Wang and A. Zichichi, Phys. Rev. D48, 4062 (1993). J. L. Lopez, D. V. Nanopoulos, H. Pois, X. Wang and A. Zichichi, Phys. Lett. B306, 73 (1993). See e.g.: A. Sopczak, Int. J . Mod. Phys. A9, 1747 (1994). M. Drees and K. Hikasa, Phys. Lett. B252, 127 (1990). J. L. Lopez, D. V. Nanopoulos, X. Wang and A. Zichichi, Phys. Rev. D48,4029 (1993). T. Kobayashi, T. Kon, K. Nakamura and T. Suzuki, Mod. Phys. Lett. A7,1209 (1992). J. L. Lopez, D. V. Nanopoulos, G. Park, H. Pois and K. Yuan, Phys. Rev. D48, 3297 (1993); J. L. Lopez, D. V. Nanopoulos, G. Park, and A. Zichichi, Phys. Rev. 49, 355 (1994); J. L. Lopez, D. V. Nanopoulos, G. Park and A. Zichichi, Phys. Rev. D49,4835 (1994); J. Kim and G. Park, Phys. Rev. D50, 6686 (1994). D. Schaile, to appear in proc. 27th Int. Conf. on High Energy Physics (Glasgow, 20-27 July 1994). J. Wells, C. Kolda and G. Kane, Phys. Lett. B338, 219 (1994). J. L. Lopez, D. V. Nanopoulos and X. Wang, Phys. Rev. D49, 366 (1994). M. May, in AIP Conf. Proc. USA, Vol. 176 (AIP, New York, 1988), p. 1168; B. L. Roberts, 2. Phys. C56, SlOl (1992).
437
S. Kelley, Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi
CONSTRAINTS ON NO-SCALE SUPERGRAVITY MODELS
From Modern Physics Letters A 10 ( 1995) I787
I995
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439 Modern Physics Letters A, Vol. 10, No. 24 (1995) 1787-1794 @World Scientific Publishing Company
CONSTRAINTS ON NO-SCALE SUPERGRAVITY MODELS
S. KELLEYS, JORGE L. LOPEZ*>t,D. V. NAN0POULOS'~t~tand A. ZICHICHIt * Center f o r Theoretical Physics, Department of Physics, Texas A&M University, College Station, T X 77843-4242, USA t Astroparticle Physics Group, Houston Advanced. Research Center (HARC), The Mitchell Campus, The Woodlands, T X 77381, USA CERN, Theory Division, CH-1211 Geneva 23, Switzerland !i Department of Physics, Maharishi International University, Fairfield, ZA 52557-1 069, USA Received 30 May 1995 We revisit the no-scale mechanism in the context of the simplest no-scale supergravity extension of the standard model. This model has the usual four-dimensional parameter m 3 / 2 / m 1 / 2We . show how predictions of space plus an additional parameter €312 the model may be extracted over the whole parameter space. A necessary condition for the potential to be stable is S t r M 4 > 0. In a class of no-scale supergravity models this condition implies m3/2 5 2mp. We also calculate the residual vacuum energy at the unification scale and find that in typical models one must require Co > 10. Such constraints may be important in the search for a realistic string no-scale supergravity model.
The search for a unified theory of everything has two boundaries. At one extreme the standard model matches flawlessly with experiment, and at the other extreme string theory promises quantum consistent unification of gravity with the fields of lower spin. In connecting these extremes, two vital clues have inspired major progress over the years: the gauge hierarchy problem, and the problem of vanishing cosmological constant. Global supersymmetry provides a first solution to the gauge hierarchy problem and ensures the cancellation of A4 divergent one-loop contributions to the vacuum energy. However, global supersymmetry is not enough: we ultimately need local supersymmetry - supergravity. In addition, to simultaneously preserve the gauge hierarchy and respect the present experimental nonobservation of sparticles, supergravity must be broken to a globally supersymmetric theory with soft breaking terms of order of the electroweak scale. Supergravity is described in terms of two functions: the Kahler function (G) and the gauge kinetic function ( f a p ) , plus the gauge and (hidden plus observable) matter content of the theory. Furthermore, the supersymmetry breaking scale (ii) is not assured to be comparable to the electroweak scale. Moreover, in the process a large vacuum energy is usually generated (O(ii2M:) = O(10-32M$)). An important exception to this typical and unsatisfactory situation occurs in a class 1787
440 1788
5’. Kelley et al.
of supergravity models with a distinct Kahler function endowed with some noncompact symmetries (e.g., SU(1,l) or SU(N, This class of theories have two remarkable properties after supersymmetry breaking: (i) the minimum of the (treelevel) scalar potential is at zero vacuum energy,l (ii) at the minimum there is one or more flat directions (moduli) which leave the gravitino mass undetermined.’>’ The no-scale mechanism must, however, be protected against possible large oneloop corrections to the vacuum energy :.( StrM’) which would destroy the flat directions.’ The first property implies that the vacuum energy could first arise at the electroweak scale (i.e. as small as O ( M 2 ) = O(10-64M:)). The second property provides a natural solution to the gauge hierarchy problem via the no-scale me~hanism,’1~ whereby the minimization of the electroweak potential determines the vacuum expectation value of the Higgs fields and of the moduli fields associated with the flat directions, thus determining the scale of supersymmetry breaking to be comparable to the electroweak scale. The above class of models was first proposed in the context of supergravity per se, and have since been found to be strongly supported by the low-energy effective theories from In fact, in string model-building of the class of “string noscale supergravities” is much extended7 to include new kinds of moduli fields and more definite forms for the Kahler function.6 Moreover, in string models one can accommodate the usual nonperturbative (e.g., gaugino condensation) supersymmetry breaking mechanism, as well as tree-level breaking via coordinate-dependent compactifi~ations.~ All these features make string no-scale supergravity a very rich, interesting and well motivated subset of all possible string supergravities. Furthermore, the properties mentioned above are expected to be very discriminating in the selection of phenomenologically appealing string vacua. Consider the simplest supergravity extension of the standard model. Numerous previous studies have identified the parameter space of this model, shown how to solve the model, and extracted its predictions as a function of this four-dimensional parameter space which can be taken as (tanp, m l p , t o , < A ) , with 50 = mo/mlp and & = A / m l p . However, the naive construction of the one-loop effective potentialg has a major flaw: it is not formally independent of the renormalization scale, although derivatives with respect to the various Higgs fields are. This problem was studied in Ref. 10 and more recently in Ref. 11, where a simple and well-motivated ansatz for a Q-independent one-loop effective potential was proposed: one should subtract the field-independent contribution to the potential. That is, one should use the following expression Vl = vo
1 S t r M 4 (InFM’ +2 6 4 ~
i)
-Vl(0),
where VO is the usual RGE-improved tree-level Higgs potential, Vl(0) is the fieldindependent contribution to V1, and the supertrace ( S t r M n = xj(-1)’j(2j+1) x TrMy) includes a term for the gravitino. Although the field-independent term
44 1 Constraints on No-Scale Supergravity Models
1789
and the contribution of the gravitino in the supertrace are irrelevant to minimization with respect to the Higgs fields, these two terms are crucial to dynamically determining supersymmetry breaking. l 2 Since the field-independent contribution to Vl could contain an unknown Qdependent piece, we parametrize it more generally as Cm$12.The RGE satisfied by C is easily derived by demanding that V1 be scale-independent to one-loop order dC dV1 dVo 1 Str M4 mil2 dt (“two-loop”) = 0 , (2) dt dt 32r2 where t = ln(Q/Qo), with Q the renormalization scale and QO a reference scale (e.g., QO = M z ) . Moreover, the resulting relation must hold for all values of the fields. Taking the Higgs fields (hl, h2) to zero gives VO= dVo/dt = 0 , and the RGE follows~2
+
+
with (in the notation of Ref. 3, and ignoring any possible hidden sector contributions)
+ 12(m4,, + m i c +
27726)
+ 2(m& + 2rn4L3) + 4(m4Ee + 2m4L)
- 16Mi - GM; - 2M; - 8p4 - 4m$2.
(4)
Interestingly enough, there has been a recent effort to identify the value of C at the scale of supersymmetry breaking (CO)as the remnant vacuum energy after supersymmetry breaking. l 2 Moreover, COis in principle calculable in specific string models and in specific supersymmetry breaking mechanisms. For example, in treelevel supersymmetry breaking via coordinate-dependent compactifications one has CO ( n-~ n ~ )where , ~ B ( F is ) the number of massless bosonic (fermionic) degrees of freedom after supersymmetry breaking.8>13 The essence of the no-scale approach is that the gravitino mass is a function of the real part of an additional scalar field, TR = Re(T), whose vev is undetermined at tree level, i.e. there is a flat direction. Generally speaking, in a supergravity theory the gravitino mass is given by m:/2 = (IWl)’ e ( K ) ,where W is the superpotential and K is the Kahler potential. Consistent with the no-scale supergravity framework, we restrict the form of K by demanding vanishing of the vacuum energy at tree-level. In Ref. 14 such restrictions have been studied in the case of realistic string-derived free-fermionic models, where it was found that e ( K ) = ((S S)(T T)’)-l is required. That is, in addition to the dilaton field (S),only one modulus field ( T )is allowed. Moreover, duality invariance of m3/2 is guaranteed by the transformation properties of the matter fields in the superpotential ( W ) ,without introducing a T dependence in W (which would destroy the flatness of T).14 Therefore, in what follows we assume the following well-motivated functional form N
+
+
442 1790 S. Kelley et al.
with a single modulus field. Here (Y and p are dimensionless parameters and A is some appropriate mass scale.a We note that the dilaton dependence of m 3 / 2 should not affect the implementation of the no-scale mechanism, since presumably the value of (S) is determined by hidden sector dynamics at a high scale, whereas the no-scale mechanism entails electroweak scale phenomena. To implement the no-scale mechanism one can take one of two approaches: (i) a top-down approach, where COis "given" and minimization of V1 with respect to TR (i.e. the no-scale mechanism) gives m 3 / 2 ; or (ii) a bottom-up approach, where one uses the no-scale mechanism to determine C ( M z ) for a given value of m 3 / 2 and then obtains COby RGE evolution. Since COis in practice rather unknown, here we follow the bottom-up approach with the hope of finding constraints on the calculated value of CO(as a function of the usual soft-supersymmetry-breaking parameters), which should help guide string model builders in their quest for models with phenomenologically acceptable values of CO. We now calculate the first and second derivatives of V1 with respect to TR. Following Ref. 12, we scale out the m 3 / 2 dependence in V1 (here X = m$2)
vI=
xV ' O
X2 + X'C + strM4 64n2
(
XM2
-t)
In Q2
where VO= Vo/X2, etc. Thus we obtain
= 0 then implies
and the no-scale condition
v1 =
I Str M4 , 128n2
--
which allows one to determine C = C(m3/2)in the bottom-up approach.b Is this a true minimum or just an extremum? A necessary condition is d2V1/dTi > 0 at the minimum. We obtain
and at the minimum (10) min
Therefore, if Str M4 > 0, then the minimum in the modulus direction is stable, and the vacuum energy is negative. "In the case of Ref. 14: p = 2, a 1, and A = (4)3/2/Mi/2,where we have written ( W )= (4)3. bOr equivalently m312= m3/z(Co)in the top-down approach. N
443 Constraints on No-Scale Supergravity Models
1791
This necessary condition for stability of the no-scale mechanism may impose constraints on m312as a function of the usual soft-supersymmetry-breaking parameters. If we define the ratios
6312
m3/2 =, m1/2
to=-
m0
m1/2
,
A
then the constraint would be on the parameter c3l2.To get a qualitative idea of the kind of constraints that may arise, we first consider a simple model of supersymmetry breaking where the dominant contributions to Str M4 come from the three generations of (degenerate) squarks and the gravitino: Str M4 M 6 x 1 2 m i - 4mi12> 0 .
(12)
Therefore. m3/2
-< ( 1 8 ) l I 4M 2.06.
mi Quantitatively we expect the factor to be slightly higher because of the neglected contributions to Str M4 (this is confirmed in the numerical calculations presented below). Since to good approximation one can write mi z with CH 4-6, an RGE-dependent constant, we also have
mod-,
-
The above example would be accurate if supersymmetry breaking is somehow negligible in the hidden sector or the hidden sector is absent. In a more general situation we expect the coefficient -4 in the second term in Eq. ( 1 2 ) to be still of O(1). If the sign of this coefficient is positive, the analog of Eq. ( 1 3 ) would be m3/2/m~ 5 O(1), since a fourth root is involved in the calculation. On the other hand, if the sign of the coefficient is positive, the stability of the no-scale mechanism would be automatic with no restriction on m312. In addition to this necessary requirement for stability of the potential, the absence of negative mass eigenstates in the neutral scalar sector provides more stringent although model-dependent constraints. The model dependence comes in two ways: (i) the usual dependence on the supersymmetry breaking parameters, and (ii) a new dependence through the Kahler potential of the no-scale model. The latter is important in determining the proper normalization of the physical fields. Without an explicit model we cannot address this point any further, although we expect a new physical scalar (TR)with a mass typically mi12/Mp.8 Turning to a quantitative analysis, the unknown parameter space of our (supersymmetric or absent hidden sector) model may be taken as (tanp, m 1 / 2 , (0, ( A , ( 3 1 2 ) . Given these parameters the Higgs mixing terms p and B can be calculated from the usual minimization conditions. The parameter C can be calculated
-
444 1792
S. Kelley et al.
from the requirement that the potential is a minimum with respect to TR. Although this may be accomplished by taking derivatives numerically, a more efficient (and accurate) method results from setting Eqs. (1) and (8) equal and solving for Cl/2 = CJ,4/2. 1
StrM4] .
(15)
The problem may be further simplified by separating out the contribution (in the supertraces) of the gravitino to from the other contributions,
Note that 6 1 1 2 depends on the four parameters (tanp, rn1j2, (0, ,$A) and is independent of (312. On the other hand, Cf/2 depends only on ((312, ml12) and is independent of the other three parameters:
Next, consider the scaling of Cl/2 (from MZ to the unification scale Mu) which obeys the same RGE as C, Eq. (3), with ml12 replacing m3/2
Since the RGE is linear, it can also be separated into a part due to the contribution of the gravitino and a part due to all the other particles, and we can write
with a simple expression for ACf/2
We have given the explicit analytic expressions for CfI2and AC$, , whereas 6 1 / 2 and A(?,/, must be calculated numerically as a function of the four-dimensional parameter space (tanp, m1/2,t o , ( A ) . However, within the phenomenologically viable area of this four-dimensional parameter space, we find the results nearly independent of t a n p and ,$A, with a small dependence on ml12. Therefore, we present our final results for Co = C ( M u )in the ((0, ,$3/2) plane as shown in Fig. 1. The particular plot shown is for t a n p = 2, ( A = 0, and ml/2 = 100 GeV although changing these variables shifts the contours only slightly (also mtPole = 160 GeV).
445 Constraints on No-Scale Supergravity Models
1793
tar@+?, m1/2=100 GeV 25
2o
"w2 63/2=G
a
t
15
-
10
-
Excluded Str M4<0
4
,
0
e
10
plane in a class of noFig. 1. Contours of constant values of CO = C ( M u ) in the scale supergravity models. The dependence on mllz is very mild; t a n p is also immaterial. The area above the dashed line is excluded by the stability of the no-scale mechanism which requires S t r M 4 > 0.
Note that the stability of the potential puts an upper bound on J3/2 which is well approximated by Eq. (14). The parameter COhas not been calculated explicitly in realistic string models although from Ref. 15 one can infer Co ~ ( n -g n ~ where ) , n ~ ( n pis)the number of leftover massless bosonic (fermionic) degrees of freedom after supersymmetry breaking, and c is a small one-loop factor [0(1/100) in Ref. 151. In the string models one is familiar with the massless spectrum having 5 1000 fields (counting all gauge degrees of freedom), that is n g - n F 5 1000, and thus we expect Co 5 10. If we require lC0l < 10, then for 50 = 0, 1.2 < J3/2 < 4.6. For JO = 4 the interval becomes 4.1 < 5312 < 9.9. Turning things around, typical supersymmetry breaking scenarios entail mo = m3/2 (i.e. 5312 = or m l p = m3/2 (i.e. 53/2 = 1).13 In both cases Fig. 1 shows that CO> 10 is required. The solution to the gauge hierarchy problem via the no-scale mechanism selects a special class of string-derived supergravities. We have studied some simple examples of these very appealing models and have shown that the no-scale mechanism leads to a stable minimum of the electroweak potential for large regions of the softsupersymmetry-breaking parameter space. Moreover, in certain class of models this stability requirement can be neatly encoded in the upper bound m3/2 5 2mq. We have also studied the dependence on the soft-supersymmetry-breaking parameters of the residual vacuum energy at the unification scale (Comi12). We find that in typical models one must require CO> 10. Our results should be useful to string model builders searching for string no-scale supergravity models with phenomenologically viable values of Co. N
446 1794
S. Kelley
et al.
Acknowledgment This work was supported in part by DOE grant DE-FG05-91-ER-40633.
References 1. E. Cremmer, S. Ferrara, C. Kounnas and D. V. Nanopoulos, Phys. Lett. B133, 61 (1983). 2. J. Ellis, C. Kounnas and D. V. Nanopoulos, Nucl. Phys. B241,406 (1984); B247,373 (1984). 3. For a review see A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. 145,1 (1987). 4. J. Ellis, A. Lahanas, D. V. Nanopoulos and K. Tamvakis, Phys. Lett. B134,429(1984). 5. E. Witten, Phys. Lett. B155, 151 (1985); S. Ferrara, C. Kounnas, M. Porrati and F . Zwirner, ibid. B194,366 (1987). 6. I. Antoniadis, J. Ellis, E. Floratos, D. V. Nanopoulos and T . Tomaras, Phys. Lett. B191,96 (1987); S. Ferrara, L. Girardello, C. Kounnas and M. Porrati, ibid. B193, 368 (1987); B194,358 (1987); J. L. Lopez, D. V. Nanopoulos and K. Yuan, Phys. Rev. D50,4060 (1994). 7. S. Ferrara, C. Kounnas, M. Porrati and F . Zwirner, Nucl. Phys. B318, 75 (1989); M. Porrati and F . Zwirner, ibid. B326, 162 (1989). 8. S. Ferrara, C. Kounnas and F. Zwirner, Nucl. Phys. B429,589 (1994). 9. C. Kounnas, A. Lahanas, D. V. Nanopoulos and M. Quirbs, Nucl. Phys. B236,438 (1984); G. Gamberini, G. Ridolfi and F. Zwirner, ibid. B331,331 (1990). 10. M. Einhorn and D. Jones, Nucl. Phys. B211,29 (1983); B. Kastening, Phys. Lett. B283, 287 (1992); C. Ford, D. Jones, P. Stephenson and M. Einhorn, Nucl. Phys. B395, 17 (1993). 11. S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois and K. Yuan, Nucl. Phys. B398,3 (1993). 12. C. Kounnas, I. Pave1 and F. Zwirner, Phys. Lett. B335, 403 (1994). 13. I. Antoniadis, Phys. Lett. B246,377 (1990); I. Antoniadis, C. Muiioz and M. Quirbs, Nucl. Phys. B397,515 (1993); I. Antoniadis and K. Benakli, Phys. Lett. B326,69 (1994); B331,313 (1994). 14. J. L. Lopez and D. V. Nanopoulos, CERN-TH.7519/94 (hep-ph/9412332). 15. H. Itoyama and T. Taylor, Phys. Lett. B186,129 (1987). 16. L. IbGiez and D. Lust, Nucl. Phys. B382,305 (1992); V.Kaplunovsky and J. Louis, Phys. Lett. B306, 269 (1993); A. Brignole, L. IbGiez and C. Muiioz, Nucl. Phys. B422,125 (1994).
447
Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi
A LIGHT TOP-SQUARK AND ITS CONSEQUENCES AT HIGH ENERGY COLLIDERS
From Modern Physics Letters A 10 ( I 995) 2289
1995
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449 Modern Physics Letters A, Vol. 10, No. 30 (1995) 2289-2296 @World Scientific Publishing Company
A LIGHT TOP-SQUARK AND ITS CONSEQUENCES AT HIGH ENERGY COLLIDERS
JORGE L. LOPEZ and D. V. NANOPOULOS’ Center for Theoretical Physics, Department of Physics, Texas A 6 M University, College Station, TX 77843-4342,USA Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Mitchell Campus, The Woodlands, TX 77381, USA A.ZICHICHI CERN, 1211 Geneva 23, Switzerland
Received 15 July 1995 An interesting prediction of a string-inspired one-parameter SU(5) x U(1) supergravity model, is the fact that the lightest member (tl) of the top-squark doublet (PI, &), may be substantially lighter than the t o p quark. This sparticle (Pi) may be readily pair-produced at the Tevatron and, if mil 5 130 GeV, even be observed at the end of Run IB. Topsquark production may also be a source of top-quark-like signatures in the dilepton and l + j e t s channels, if the experimental acceptances are tuned t o look for soft leptons. Such a light top-squark is linked with a light supersymmetric spectrum, which can certainly be searched for at the Tevatron through trilepton and squark-gluino searches, and at LEPII through direct il pair-production and via chargino and Higgs-boson searches.
The CDF and DO collaborations have recently announced their discovery of the top-quark with mass mt 175 GeV.’I2 There also exists plenty of indirect evidence for the top-quark from precise electroweak measurements at LEP,3 when contrasted with the corresponding theoretical calculation^.^ In the analysis leading to the discovery of the top-quark, the Monte-Carlo simulations which are compared with the data, assume the validity of the standard model, and no other processes beyond it contribute to the sought-for signal. In this letter we would like t o point out that, in the context of supersymmetric models, the pair-production of the lightest top-squark (“stop”) may lead to very similar experimental signatures as the pairproduction of top quarks. This fact by itself is not new, since it is well known that one can always adjust arbitrarily the parameters of the minimal supersymmetric In the context of the minstandard model (MSSM) to have a light top-~quark.~-* imal SU(5) supergravity model? i.e. the simplest model underlying the MSSM, the constraints from the proton lifetimelo typically force all the squarks to be N
*Also at: CERN, Theory Division, 1211 Geneva 23, Switzerland. 2289
450 2290
J. L. Lopez, D. V. Nanopoulos & A . Zichichi
heavier than the top-quark, although a light top-squark may still be allowed in some regions of parameter space. On the other hand, a light top-squark may be the natural consequence of a one-parameter string-inspired SU(5) x U( 1) supergravity model," with the dilaton field being the dominant source of supersymmetry breaking,12 and the electroweak-size Higgs mixing parameter p obtained naturally from supergravity-induced Our model'' is a special case of a generic supergravity model with universal soft supersymmetry breaking, which is described in terms of four parameters: m112, mo, A0 and t a n p . In the "special dilaton" scenario one has1' 1
mo = --m
d3
1/21
A0
= -m1/21
Bo = 2mo I
(1)
where Bo is the soft-supersymmetry-breaking parameter (at the unification scale) associated with p. These conditions determine all but one parameter, taken here t o be ml/2 DC mx* oc my. The requirement of radiative electroweak symmetry breaking," which determines p up to a sign, can only be satisfied here for p < 0, in light of the last condition Bo = 2mo. Moreover, this condition determines tan p as a function of m112; one finds that t a n p must be small: t a n p x 1.4, with little dependence on m1p.l' In what follows we take mrole= 175 GeV. We have verified that for = (160-180) GeV, the calculated value of tanD remains close to 1.4 and 1p1 varies by 5 f 4%. (Details of the following analysis can be found elsewhere.l5 ) For our present purposes, the main result, i.e. a light top-squark, is a consequence of the small value of tan 0.Indeed, the lightest top-squark mass is given by (see e.g. Ref. 8; our convention for the sign of p is opposite) 1
1
m;, = -(miL 2 +m;R)+-M;cos2p+m: 4
1' +
1
-(m? - m? ) + - cos 2P(8M& - 5 M i ) 2 tL tR 12
m:(At
+ p / tan P)2 ,
where m;L,Rare the running top-squark masses. In the present case there is a large cancellation between the first term $(rnFL rn$ and the last term in the square root m:(At p / tanP)2, which leads to light top-squark masses, i.e.
+
+
We find mt, 2 67 GeV (c.f., the LEP limit16 mi, > 45 GeV). (This result has a strong t a n 0 dependence, e.g., mi, 2 90(120) GeV for t a n 0 fis 1.5(2.0), but here t a n 0 is fixed and cannot be varied at will,) We also find md M mg 2 260 GeV, aFor recent reviews of this general procedure see e.g. Ref. 14.
45 1 A Light Top-Squark and i t s Consequences at High Energy Colliders 2291
where mt is the average first- or second-generation squark mass. In Fig. 1we present a collection of spectra plots versus the lightest chargino mass (mx;) for the lighter supersymmetric particles. (The spectrum depends slightly on mt and will have to be recalculated once mt is known more precisely.) We note in passing that in this model we find Br(6 -+ sy) M (1-3) x which is in very good agreement with the present experimental r e ~ u 1 t s . Also, l ~ the relic density of the lightest neutralino satisfies Rxhg 5 0.85, which is in natural agreement with cosmological observations and includes the possibility of a universe with a cosmological constant." 800
500
40C
300
200
100
0
Fig. 1. The relevant lighter sparticle masses vs the chargino mass. The il top-squark mass (with m i l > 67 GeV) is shown by the dashed line. Note that mfl > m f . Here mq X my, XI
with m i the average first- or second-generation squark mass. Also, mx;x mx: mA
x m H x mHf
x 2mx;, and
> 400 GeV.
The cross-section for pair-production of the lightest top-squarks a(~ I T I )depends solely one mil and is given for a sampling of values in Table 1. Since in this model mi, > mx: + mb (see Fig. l),one gets Br(f1 -+ 6~:) = 1. The charginos Table 1 . Cross-sections at the Tevatron (in pb) for6 p p in GeV. mil
~(tltl)
70
80
90
100
112
60.
30
15
8
4
+
tli1X and p p
-+
t i X . l g All masses
mt
120
140
160
180
a(ti)
39
17
8
4
452 2292
J . L. Lopez, D. V. Nanopoulos €4 A . Zichichi
1.0
0.8
&
0.6
2
“M 0.4
2
5
0.2 0.0
50
100
150
200
250
300
m,; ( G 4 0.5 0.4 L
m 0
0.3
E
3 0.2 4
5 0.1 0.0
50
100
150
200
250
300
m,: (GeV) Fig. 2. The leptonic and hadronic branching fractions of the chargino
(x:)
and the neutralino
(x;)(other channels are not shown). The sudden drop in the leptonic neutralino branching ratio at m i M 170 GeV corresponds to the opening of the “spoiler mode” x$ -+ xy + h. X1 then decay leptonically or hadronically with branching fractions shown in Fig. 2, i.e. Br(Xf -t f!vtxy) M 0.4 (f! = e p ) for mx: 5 65 GeV * mi, 5 100 GeV. The most promising signature for light top-squark detection is through the dilepton mode.* The number of stop-dileptons is:
+
2
Nljil = C T ( ~ I ~ I [Br(& )X + b&)]’x [Br(X? -t lvtx;)] X L M O.lGo(ili1)xL. (4) The dilepton mode is also paramount in top-quark searches:
Nit: = c ( t f ) x [Br(t
--t
bW)I2 x [Br(W -t t ! ~ ! ) x] ~C M 0 .0 5 c ( tf)x L .
(5)
Here we have taken Br(t --+ bW) = 1, although one should account for th_e t --+ &x’: mode which is also open for light top-squarks. Moreover, pjj + ti% ii&xyx’:x is another source of top-squarks, although suppressed because of the small branching fraction: we find Br(t -t rlx!) 5 10%. Combining Eqs. (4) and (5) we obtain --+
45 3 A Light Top-Squark and its Consequences at High Energy Colliders
2293
This ratio should open the eyes of experimenters because the number of observed dilepton events depends strongly on the experimental biases. This ratio ( 6 ) indicates that for sufficiently light top-squarks there may be a significant number of dilepton events of non-top-quark origin, if the experimental acceptances are tuned accordingly. Perhaps the most important distinction between top-dileptons and stopdileptons is their p~ distribution: the (harder) top-dileptons come from the twebody decay of the W boson, whereas the (softer) stop-dileptons come from the (usually) three-body decay of the chargino with masses (in this case) below m w . Therefore, the top-dilepton data sample is essentially distinct from the stopdilepton sample. Such distinction is well quantified by the “bigness” ( B )parameter of Ref. 8. Another distinction between the two B = Ip~(1+)l Ip~(l’-)l sources of dileptons are the bjets, which are probably softer in the decay f1 + 6 ~ : (for light top-squarks) compared to those from t --.$ 6W. The above discussion suggests that the CDF and DO top-dilepton data samples should be carefully studied to see if softer stop-dileptons are present: an important new lower bound on the top-squark mass may follow. However, detailed simulations of the stop-dilepton signal and a reanalysis of the top-dilepton data are required before drawing more concrete conclusions. We also note that in the C +jets channel, the ratio analogous to Eq. ( 6 ) is Affiets/lVi:jets M u(f1Tl)/g(tf), since Br(W + 2j) . Br(W --.$ 1) = (2/3)(2/9) M Br(Xf + 2j) Br(Xf --t 1) (see Fig. 2). In this case, the top-squark C jets events still have softer bjets and a softer lepton.
+
+
-
+
Table 2. Upper limits on sparticle masses which follow from mzl relevant in top-quark searches. All masses in GeV.
< 100 GeV, such that
Xi
*
Xy
X!
h
ER
5
EL
El
51
q+ 4
65
35
70
70
108
120
130
100
275
310
il may be
The light top-squarks which may be relevant for the top-quark and top-squark searches at the Tevatron (i.e. mi, 5 100 GeV) entail a light supersymmetric spectrum, as can be seen from Fig. 1. For mt, < 100 GeV, we get the corresponding upper limits as shown in Table 2. We now explore the possibilities for direct detection of these light sparticles a t the Tevatron and LEPII. 0
Tevatron. One could detect these light sparticles in three ways: -
The trilepton signal in pp --.$ xfxiX is the most promising avenue for deas evidenced in tection of weakly interacting sparticles at the Tevatron,zO~zl the context of SU(5) x U ( l ) supergravity in Ref. 22. The leptonic chargino and neutralino branching fractions are given in Fig. 2, and the trilepton rate at the 1.8 TeV Tevatron is given in Fig. 3, where we indicate by a dashed line the present CDF upper limitz3 and by a dotted line the expected reach
454 2294
J . L. Lopex, D. V. Nanopoulos €4 A. Zichichi
-
by the end of Run IB (with 100 pb-' of accumulated data). This reach corresponds to m * 5 80 GeV * mrl 5 130 GeV. Therefore, the light sector XI
of this model - that relevant to top-quark searches - could be definitively falsified in the near future. - Direct i~ pair-production at the Tevatron has been shown recently8 t o be sensitive t o mE1 5 100 GeV by the end of Run IB, provided the chargino leptonic branching fraction is taken to be 20%. For the chargino branching fractions in our model (- 40%, see Fig. 2) the reach through the stop-dilepton channel is extended to mfl 5 130 GeV. - The standard quark-gluino searches may also be able to reach up t o rnq NN my = 310 GeV with the Run IB data. N
LEPII. One could detect these light sparticles in three ways: - Charginos would be readily pair-produced, and best detected through the "mixed" mode (i.e. 2 j ) . For m 65 (80) GeV, we find ( a x B)mixed 2
e+
*5
Xl
0.34(0.27) pb, which is much larger than the estimated 50 sensitivity at 100 pb-', i.e. 0.12 pb.24 - The lightest Higgs-boson should be easily detectable through the standard process e+e- -+ 2' -+ Zh. For mh 5 70 GeV (from Table 2), we find a cross-section in excess of 0.92 pb, which is much larger than the expected sensitivity limit of 0.2 pb for a 3u effect a t 500 pb-1.25 In fact, a 0.92 pb signal corresponds to a significance of 6 . 2 ~at 100 pb-'. - The light top-squark may also be produced directly e+e- + ilfl via s-channel y, Z exchange, and be probed up to mt, M &/2 - 10 GeV.
Tevatron
a
2
8
10-1
n
mb
-
v
2 ' 0 1
50
100
150
200
250
300
Fig. 3. The rate for trilepton events at the Tevatron. The present CDF limit is indicated. The dotted line indicates the expected sensitivity at the end of Run IB (- 100 pb-') equivalent to a reach m f < 80 GeV. XI
455 A
Light Top-Squark and its Consequences at High Energy Colliders 2295
In summary, we have discussed the prediction of a light top-squark in a stringinspired one-parameter SU(5) x U( 1) supergravity model. This sparticle (il)may be readily pair-produced at the Tevatron and, if mil 5 130 GeV, even be observed with the present run accumulated data. Top-squark production may also be a source of top-quark-like signatures in the dilepton and C +jets channels, if the experimental acceptances are tuned to look for soft leptons. Another prediction of this model is a direct link between the light top-squark and a light supersymmetric spectrum, which can certainly be searched for at the Tevatron through trilepton and squarkgluino searches, and at LEPII through direct fl pair-production and via chargino and Higgs-boson searches.
Acknowledgments This work has been supported in part by DOE grant DEFG05-91-ER-40633. We would like to thank T. Kamon, P. McIntyre and J. White for useful discussions.
References 1. F. Abe et al., C D F collab., Phys. Rev. Lett. 74,2626 (1995); see also ibid. 73,225 (1994); Phys. Rev. D50,2966 (1994). 2. S. Abachi et al., D O collab., Phys. Rev. Lett. 74,2632 (1995). 3. T h e L E P collabs.: ALEPH, DELPHI, L3 and OPAL, Phys. Lett. B276,247 (1992); T h e L E P collabs.: ALEPH, DELPHI, L3 and OPAL, and the LEP Electroweak Working Group, CERN preprint CERN/PPE/93-157. 4. See e.g., J. Ellis, G. L. Fogli and E. Lisi, Phys. Lett. B292,427 (1992); G. Altarelli, R. Barbieri and F. Caravaglios, Nucl. Phys. B405, 3 (1994); A. Blondel and C. Verzegnassi, Phys. Lett. B311, 346 (1993); P. Langacker, in Recent Directions in Particle Theory: From Superstrings to the Standard Model, eds. J . Harvey and J. Polchinski (World Scientific, 1993), p. 141; G. Montagna, 0. Nicrosini and G . Passarino, Phys. Lett. B303,170(1993); V. A. Novikov, L. B. Okun, A. N. Rozanov and M. I. Vysotsky, Mod. Phys. Lett. A9,2641 (1994). 5. J. Ellis and S. Rudaz, Phys. Lett. B128, 248 (1983); G. Altarelli and R. Ruckl, ibid. B144, 126 (1984); I. Bigi and S. Rudaz, ibid. B153, 335 (1985); K. Hikasa and M. Kobayashi, Phys. Rev. D36, 724 (1987); H. Baer and X. Tata, Phys. Lett. B167, 241 (1986); M. Drees and K. Hikasa, ibid. B252, 127 (1990). 6. H. Baer, M. Drees, J. Gunion, R. Godbole and X. Tata, Phys. Rev. D44,725 (1991). 7. T. Kon and T. Nonaka, Phys. Lett. B319, 355 (1993); preprint ITP-SU-94/02 (hepph/9404230); Phys. Rev. D50,6005 (1994); M. Fukugita, H. Murayama, M. Yamaguchi and T. Yanagida, Phys. Rev. Lett. 72,3009 (1994). 8. H. Baer, J. Sender and X. Tata, Phys. Rev. D50,4517 (1994). 9. A. Chamseddine, R. Arnowitt and P. Nath, Phys. Rev. Lett. 49,970 (1982); for reviews see e.g., R. Arnowitt and P. Nath, Applied N = 1 Supergravity (World Scientific, 1983); H. P. Nilles, Phys. Rep. 110,1 (1984). 10. R. Arnowitt and P. Nath, Phys. Rev. Lett. 69, 725 (1992); Phys. Rev. D49, 1479 (1994); P. Nath and R. Arnowitt, Phys. Lett. B287, 89 (1992); J. L. Lopez, D. V. Nanopoulos and H. Pois, Phys. Rev. D47,2468 (1993); J. L. Lopez, D. V. Nanopoulos, H. Pois and A. Zichichi, Phys. Lett. B299,262 (1993) and references therein. 11. J. L. Lopez, D. V. Nanopoulos and A. Zichichi, Phys. Lett. B319,451 (1993).
45 6 2296
J . L. Lopez, D. V. Nanopoulos €4 A . Zichichi
12. V. Kaplunovsky and J. Louis, Phys. Lett. B306,269 (1993); A. Brignole, L. Ibiiiez and C. Muxioz, Nucl. Phys. B422,125 (1994). 13. R. Barbieri, J. Louis and M. Moretti, Phys. Lett. B312, 451 (1993); B316, 632(E) (1993). 14. J. L. Lopez, D. V. Nanopoulos and A. Zichichi, Nuovo Cimento Riv. 17,1 (1994); R. Arnowitt and P. Nath, in Proc. of the VII J . A . Swieca Summer School (World Scientific, 1994). 15. J. L. Lopez, D. V. Nanopoulos and A. Zichichi, Texas A&M University preprint CTPTAMU-27/94, to appear in Int. J . Mod. Phys. A . 16. Particle Data Group, Phys. Rev. D46,S1 (1992). 17. E. Thorndike, Bull. A m e r . Phys. SOC.38,922 (1993); R.Ammar et al., CLEO collab., Phys. Rev. Lett. 71,674 (1993); M.S. Alam et al., CLEO collab., ibid. 74,2885 (1995). 18. J. L. Lopez and D. V. Nanopoulos, Mod. Phys. Lett. A9,2755 (1994). 19. E. Laenen, J. Smith and W. L. van Neerven, Phys. Lett. B321,254 (1994). 20. J. Ellis, J. Hagelin, D. V. Nanopoulos and M. Srednicki, Phys. Lett. B127,233 (1983); A. H. Chamseddine, P. Nath and R. Arnowitt, ibid. B129, 445 (1983); H.Baer and X. Tata, ibid. B156,278 (1985); H. Baer, K. Hagiwara and X. Tata, Phys. Rev. Lett. 57,294 (1986); Phys. Rev. D35, 1598 (1987). 21. P. Nath and R. Arnowitt, Mod. Phys. Lett. A2,331 (1987); R.Arnowitt, R. Barnett, P. Nath and F. Paige, Int. J . Mod. Phys. A2,1113 (1987); R.Barbieri, F. Caravaglios, M. F’rigeni and M. Mangano, Nucl. Phys. B367,28(1991); H.Baer and X. Tata, Phys. Rev. D47, 2739 (1993); H. Baer, C. Kao and X. Tata, ibid. D48, 5175 (1993). 22. J. L. Lopez, D. V. Nanopoulos, X. Wang and A. Zichichi, Phys. Rev. D48,2062 (1993). 23. Y. Kato, CDF collab., in Proc. of the 9th Topical Workshop on Proton-Antiproton Collider Physics, Tsukuba, Japan, October 1993; The CDF collab., “Search for supersymmetry at CDF”, Fermilab-Conf-94/149-E (1994), submitted to the 27th ICHEP (G lasgow , Scotland). 24. J. L. Lopez, D. V. Nanopoulos, G. Park, X. Wang and A. Zichichi, Phys. Rev. D50, 2164 (1994). 25. See e.g., A. Sopczak, Int. J . Mod. Phys. A9,1747 (1994).
457
Jorge L. Lopez, D.V. Nanopoulos, Xu Wang and A. Zichichi
SUPERSYMMETRY DILEPTONS AND TRILEPTONS AT THE FERMILAB TEVATRON
From Physical Review D 52 ( I 995) I42
1995
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459
PHYSICAL =VIEW
D
1m
VOLUME 52, NUMBER 1
Y
1995
Supersymmetry dileptons and trileptons at the Fermilab Tevatron Jorge L. Lopez,’B2 D. V. Nanop~ulos,l-~ Xu Wang,’.’ and A. Zichichi4 Center for Theoretical Physics, Department of Physics, T a w A6M University, College Station, T a w 77843-4242 a Astroparticle Physics Group, Houston Advanced Resenrch Center (HARC), The Mitchell Campus, The Woodlands, T a w 77381 ’CERN Theory Division, 1211 Geneva 23, Switzerland ‘CERN, 1211 Geneva 23, Switzerland (Received 28 December 1994) We consider the production of supersymmetry neutralinos and charginos in p p collisions at the Fermilab Tevatron, and their subsequent decay via hadronically quiet dileptons and trileptons. We perform OUT computations in the context of a variety of supergravity models, including generic four-parameter supergravity models, the minimal SU(5) supergravity model, and SU(5)xU(1) supergravity with string-inspired two- and one-parameter moduli and dilaton scenarios. Our results are contrasted with estimated experimental sensitivities for dileptons and trileptons for integrated luminosities of 100 pb-’ and 1 fb-’, which should be available in the short and long terms. We find that the dilepton mode is a needed complement to the trilepton signal when the latter is suppressed by small neutralino branching ratios. The estimated reaches in chargino masses can be as large as 100 (150) GeV for 100 pb-‘ (1 fb-’). We also discuss the task left for CERN LEP I1 once the Tevatron has completed its short-term search for dilepton and trilepton production. PACS number(s): 14.80.Ly, 04.65.+e, 12.10.Dm, 13.85.Qk
Experimental searches for supersymmetric particles have come a long way since the commissioning of the Tevatron p$ collider at Fermilab and the LEP e+e- collider a t CERN. The strengths and weaknesses of these two types of colliders are well known. A hadron collider is best suited for searching the highest accessible mass scales since a sharp kinematical limit does not exist, but discoverability depends on the event rate and the cleanliness of the signal. An e+e- collider is capable of discovery essentially up to the kinematical limit, but this is much lower than what would be accessible in a hadron collider. In fact, it has become apparent that an e+elinear collider with a center-of-mass energy in the multihundred GeV range would be ideal for what has been termed “sparticle spectroscopy.” At present though, in the search for new physics we have to make the best possible use of existing facilities, since information gathered there would illuminate the path towards higher energy machines. One such effort is being conducted a t the Tevatron, where the search for weakly interacting sparticles (charginos and neutralinos) has become quite topical, in view of the fact that the reach of the machine for the traditional strongly interacting sparticles has been nearly reached. This effort will benefit &om an integrated luminosity in excess of 100 pb-’ by the end of the ongoing run IB,and possibly 1-2 fl-’ during the Main Injector era around the year 2000. In this paper we reexamine the prospects for supersymmetry discovery at the Tevatron via the hadronically quiet trilepton signal’ which occurs in the production
’This signal contains no hadronic activity, except for initial state radiation effects, and is thus distinct fkom the usual multilepton signals in squark and gluino production.
and decay of charginos and neutralinos in pp collisions [l-31. Our previous study [3]considered the trilepton signal with an estimated 100 pb-’ of accumulated data. Here we update this analysis by incorporating the latest experimental information on the trilepton signal and 1 fb-’ data. We also prospects for its detection with extend our analysis of this signal to a much broader class of supergravity models than those considered in Ref. [3]. In this paper we also discuss (for the first time) the dilep ton signal which arises in chargino pair production at the Tevatron. This signal has been recently shown t o be experimentally extractable [4], and as we discuss, has the advantage of allowing a significant exploration of the parameter space for chargino masses in the LEP I1 accessible range. Our calculations are performed in the context of a broad class of unified supergravity models, which have the virtue of having the least number of kee parameters and are therefore highly predictive and straightforwardly testable through a variety of correlated phenomena a t different experimental facilities. Our study should give a good idea of the range of possibilities open to experimental investigation, and allow quantitative checks of specific models which yield the largest rates. We consider unified supergravity models with universal soft supersymmetry breaking a t the unification scale, and radiative electroweak symmetry breaking a t the weak scale. Following a standard procedure [5] we evolve numerically the coupled set of renormalization group equations &om the &cation scale down to the electroweak scale. At the electroweak scale we enforce radiative electroweak symmetry breaking by minimizing the one-loop effective potential, as described in Ref. [6). These constraints reduce the number of parameters needed to describe the models to four, which can be taken to be , t o E m o / m l j z , <E ~ A / m l j 2 ,tanp, with a specified value for the togquark mass (mt). In what follows we
0556-2821/95/52(1)/142(7)/$06.00
142
52 -
-
mx.
@ 1995
The American Physical Society
460 SUPERSYMMETRY DILEPTONS AND TRILEPTONS AT T H E . . .
52
take mYle = 160 GeV which is the central value obtained in fits to all electroweak and Tevatron data in the context of supersymmetric models [7]. We should note that when we consider “string-inspired” models below, the unification scale is taken to be the string scale (- 10l8 GeV). This is accomplished by inserting intermediate scale particles, as discussed in Ref. [8]. In all the other models the unification scale is the usual grand unified theory (GUT) scale (- 10’’ GeV). Of relevance t o OUT discussion, we note that in all models considered the following relation holds to various degrees of approximation: m X iI x m x o2 x 2mx:.
Among these four-parameter supersymmetric models we consider generic models with continuous values of mx: and discrete choices for the other three parameters: t a n p = 2,10,
t o = 0,1,2,5,
(A
= 0.
(2)
The choices of tanp are representative; higher values of tax@ are likely to yield values of B(b -+ sy) in conflict with present experimental limits [9]. The choices of ( 0 correspond to m6 x (0.8,0.9,1.1,1.9)m6. The choice of A has little impact on the results. We also consider the case of minimal SU(5) supergravity, where the parameter space is still four dimensional but restricted by the additional constraints from proton decay and cosmology (a not too young Universe). In this case we sample a wide range of tanp,(o,[A discrete values and only keep points in parameter space which satisfy these two constrahts. One can show that the parameter space becomes bounded by tanp lo,& 4, and mx: 120 GeV [lo]. We also consider the case of no-scale SU(5)xU(l)supergravity [5]. In this class of models the supersymmetrybreaking parameters are related in a string-inspired way. In the two-parameter moduli scenario ( 0 = ( A = 0 [ l l ] , whereas in the dilaton scenario ( 0 = l / & , ( ~= -1 [12]. We also compute the rates in the one-parameter moduli [ B ( M v )= 01 and dilaton [B(Mv) = 2mo] scenarios, where Mu is the string unification scale, and with this extra condition tanp is determined as a function of mx: , which is the only kee parameter in the model. A series of experimental constraints and predictions for these models have been given in Refs. [El and [13],respectively. The processes of interest are the following. Trileptons. p p -+ x;xf, where the next-to-lightest neutralino decays leptonically (x; -+ x!Z+l-), and so does the lightest chargino (xf -+ x:l*vl). The cross section proceeds via s-channel exchange of an off-shell W and (small) t-channel squark exchange, and thus peaks a t mx: z ~ M wand , otherwise falls off smoothly with increasing chargino masses with a small tanp dependence. Dileptons. pp + x:x;, where both charginos decay leptonically. The cross section proceeds via s-channel exchange of off-she! 2 and y and t-channel squark ex. could also change, and peaks for mx: w ~ M z Dileptons come from p p -+ xyx&xgx$, with the appropriate l e p tonic or invisible decays of xg. Both of these processes are negligible [2] because the couplings of the Z and y to
<
>
<
143
neutralinos are highly suppressed when the neutralinos have a high gaugino content, as is the case when Eq. (1) holds. Yet another source of dileptons via pp -+ suffers from small rates for selectron masses above the LEP limit [14]. The more important factors in the dilepton and trilepton yields are the leptonic branching fractions which can vary widely throughout the parameter space [3]. If all sparticles are fairly heavy, the decay amplitude is dominated by W or 2 exchange. In this case the branching fractions into electrons plus muons are ~ ( x :-+ x:l*vl) = 2/9 and B ( x ~-+ x!l+l-) x 6%. On the other hand, if some of the sparticles are relatively light, most likely the sleptons, the branching fractions are altered. The extreme, although not unusual, case occurs when the sleptons are on shell. These two-body decays then dominate and the chargino leptonic branching fraction is maximized, i.e., B ( x -+ ~ ~ : Z * v l )=~ 2/3. ~ Light sleptons’ also aEect the neutralino leptonic branching ratio. When the sneutrino is on shell and is lighter than the corresponding right-handed charged slepton (Fa,pa), the channel xg -+ ul4 dominates the amplitude, and the neutralino leptonic branching ratio is suppressed. This situation is reversed when the charged slepton is on shell and is lighter than the sneutrino, which leads to an enhancement of the neutralino leptonic branching ratio. For sufficiently high neutralino masses, both l e g tonic branching ratios decrease because the W and Z go on shell and dominate the decay amplitudes. In the case of the neutralino, the spoiler mode xg -+ x:h also becomes kinematically allowed. These high-mass suppressions do not kick in until chargino and neutralino masses mx: x mxp 2Mz,2mh 200 GeV. We should note that in computing the production cross sections we have used the parton distribution functions o m Ref. [15][fit S in the modified minimal subtraction (MS) scheme] with a scale Q equal to the sum of the masses of the final state particles. The results are not very sensitive to this choice because this scale enters as h [ l n ( Q / A ~ c ~ ) ] . The product of the total hadronic cross section times the relevant leptonic branching fractions (i.e., crB with no cuts) is shown for the models of interest in Figs. 1-6. The various curves in the figures terminate at the low end because of theoretical and experimental (i.e., LEP) constraints on the parameter space. At the high end the curves are cut off when the yields fall below the foreseeable ~ensitivity.~In the case of the minimal SU(5) supergravity model (Fig. 3) we do not have curves, but rather discrete points because of the sampling of parameter space discussed above. The separation into different curves at the low mass end is an artifact of the limited sampling statistics; the whole range spanned by the shown points should be considered as viable.
-
-
’In supergravity models msR = mpR < miL = m Q L ’ 3 F ~€r0 = 5, radiative electroweak symmetry breaking is only possible for tan p 5 4. This is why there i s no curve for [o = 5 in Fig. 2 (tanp = lo), whereas there is such a curve in Fig. 1
(tan0 = 2).
46 1
LOPEZ, NANOPOULOS, WANG, AND ZICHICHI
144
Tevatron
generic supergravity tanp=2, ~,=0.1.2.5. A=O
P”
52 -
P
FIG. 1. The dilepton and trilepton rates at the Tevatron versus the chargino mas in a generic unified supergravity model with tanp = 2,.5 = 0,1,2,5(as indicated), and A = 0. The upper (lower) dashed lines r e p resent estimated reaches with 100 pb-’ (1 fb-’) of data.
tio for light sleptons, i.e., when x; --t VB is allowed. In more detail, in Fig. 5 the trilepton rates for p > 0 and low values of mx: are very suppressed because mxp 2 me for mx: 5 50 GeV, and thus xi -+ VB is kinematically allowed. For p < 0, mxg < me and no sup pression occurs. The same phenomenon is responsible for the behavior of the trilepton rates in Fig. 6, as can be seen by studying the spectrum of these models shown in Fig. 3 of Ref. [13]. The behavior of the trilepton rates in Fig. 4 can also be understood by the above mechanism, which is effective for both signs of p . For p < 0 there is no large suppression for tanp = 2 because m,; < me. Larger values of tanp (6 and 10 in Fig. 4) decrease m p and allow mx; > me, with the corresponding large s u p
In most cases we note that the rates are higher for
< 0. This is a consequence of suppressed branching fractions for p > 0, but also of a generally smaller allowed p
parameter space which requires minimum values of the chargino mass which may exceed significantly the present experimental lower limit. We can also observe that the dilepton rates indeed peak near !jMz, whereas the trilep ton rates are not as large for light chargino masses, since they peak at $ M w . It is also evident in the figures that for chargino masses below 100 GeV, the trilepton rates can be highly suppressed, while the dilepton rates are not, thus producing a rather complementary effect. This “threshold” phenomenon is most evident in the bottom panels of Figs. 4-6 and, as discussed above, corresponds to a suppression of the neutralino leptonic branching ra-
-
Tevatron
~ ’ 0
generic supergravity tanp=lO. ~,=0.1.~.A=O
~ ‘ 0
101 h
P
a
v
j
100
P
.? m
h
lo-l
v
10-2
40
60
80
100
120
140
120
140
m,: (GeV) 101 h
P
a v
J
100
L
t
A 10-1
m
v
10-2
40
60
80
100
m,: (GeV)
FIG. 2. The dilepton.and trilepton rates at the Tevatron versus the chargino mass in a generic unified supergravity model with tan@ = 10,t o = 0,1,2 (as indicated), and A = 0. The upper (lower) dashed lines represent estimated reaches with 100 pb-’ (1 fb-’) of data.
462 SUF'ERSYMMETRY DILEPTONS AND TRILEPTONS AT THE. . .
52
145
FIG. 3. The dilepton and trilepton rates at the Tevatron versus the chargino mass in the minimal SU(5) supergravity model (where tan@ < 10, SO > 4). The upper (lower) dashed lines represent estimated reaches with 100 pb-' (1 fb-') of data.
pression. The significance of our results is quantified by the horizontal dashed lines in the figures, which represent estimates of the experimental sensitivity to be reached with 100 pb-' (upper limits) and 1 fb-' (lower lines). The lesser sensitivity should be achievable at the end of Run IB (i.e., prior to the LEP I1 upgrade), whereas the higher sensitivity should be available with the Main Injector u p grade [i.e., after the LEP I1 shutdown but before the commissioning of the CERN Large Hadron Collider (LHC)]. The trilepton sensitivity with 100 pb-' (i.e., 0.4 pb) has been estimated by simply scaling down by a factor of 5 the present Collider Detector at Fermilab (CDF) experimental limit of 2 pb obtained with 20 pb-l of recorded data [lS]. The factor of 5 is the expected increase in recorded luminosity, and a simple L scaling is appropriate assuming the trilepton signal has no standard model backgrounds at this level of sensitivity. We should note that the experimental sensitivities are actually chargino-mass depender~t,~ following a curve shaped similarly to the signal (i.e., improving with larger masses) and which asymptotes to the indicated dashed lines for mx: 2 100 GeV. The decrease in sensitivity for the lower masses is typically compensated by a corresponding increase in signal. The sensitivity a t 1 fb-' requires a study c.f the background since small standard model processes a d detector-dependent instrumental backgrounds become important at this level of sensitivity [17]. The sensitivity in the figures (i,e., 0.07 pb) is obtained by scaling up by f l the value given in Table I1 of Ref. [17]. The dilepton (plus $ ), signal suffers bom several standard model backgrounds, most notably Z + T T and WW
-
4Thepresent CDF upper l i t as a function of chargino mass is shown in Fig. 1 2 of Ref. [8].
production. A study based on the DO detector [4] reveals that with suitable cuts, in 100 pb-' an estimated background of eight events is expected, which would require eight signal events at 3a significance. The efficiencies for dilepton detection have also been studied [4], and they improve with increasing chargino masses; 8% is a typical value. All this implies a sensitivity of 1 pb for dilepton detection. With 1 fb-l one can scale down the sensitivobtaining a sensitivity of 0.3 pb. As in the ity with
a,
TABLE I. Estimated chargino mass reaches in various supergravity models for chargino-neutralino production in p p collisions at the Tevatron via dilepton and trilepton modes for integrated luminosities of 100 pb-' and 1 fb-'. All masses in GeV. Dashes (-) indicate negligible sensitivity. Generic tanp 2
10
to 0 1 2 5 0 1 2
Model
P>O
100 pb-'
70
120
100
125 100 80 105 95
75
P>O
2 6 10 2 6 10
160
moduli (1-par) dilaton (1-par)
NJA N/A
N/A N/A
minimal SU(5)
50
80
dilaton (Z-Par)
65
55
70 65
1 fb-' 145 115 100 80 135 100 70
P
1 fb-' 115
75 75 95 80 80
@-Par)
100pb-'
70
tan@ 100pb-' moduli
P
1 fb-'
160 135 130 125
100pb-'
1 fb-'
100
1.50 150 150 120 120 120 150 125 75
100 70 80 80 80
70 80 50
463
LOPEZ, NANOPOULOS, WANG, AND ZICHICHI
146
no-scale SU(S)xU(l) moduli acenario
P>O
Tevatron
52 -
PL
h
n
a
Y
10-1
10-2
10-1
40
60
80
100
120
10-2 140 40
60
80
100
120
60
100
120
m,: (GeV) h
P
a
v
-_-__---__
FIG.4. The dilepton and trilepton rates at the Tevatron versus the chargino mass 140 in two-parameter SU(5)XU(l) supergravity-moduli scenario ( ( 0 = (A = 0) for the indicated values of tanp. The upper (lower) dashed lines represent estimated reaches with 100 pb-’ (1 fb-’) of data.
10-1
10-2
40
60
80
100
120
10-2 140 40
60
m,: (GeV)
trilepton case, the actual sensitivities are chargino-mass dependent, reaching the indicated asymptotic values for s d c i e n t l y high masses. The reaches in chargino masses in the various models can be readily obtained fiom the figures by considering both dilepton and trilepton signals, and are summarized in Table I €or the two integrated luminosity scenarios. The reaches in Table I translate into indirect reaches in every other sparticle mass, since they are all related. In particular, mx: 0.3mc and mi % ( m g / 2 . 9 ) m[in the S U ( 5 ) x U ( 1 ) models the numerical coefficients in this relation are slightly different, implying mg FZ. mi]. It is also interesting to point out that the pattern of yields for the various models is quite different; therefore observa-
-
Tevatron
P>O
140
m,: (GeV)
no-scale SU(S)xU(i) dilaton scenario
tion of a signal will disprove many of the models, while supporting a small subset of them. It has been pointed out that the dilepton and trilepton data sample may be enhanced by considering presumed trilepton events where one of the leptons is either missed or has a p~ below 5 GeV (“2-out-of-3”) [18]. Such enhancements would alter our reach estimates above, making them even more promising. From Table I it is clear that in some regions of parameter space, the reach of the Tevatron for chargino masses is quite significant. With 100 pb-I it should be possible to probe chargino masses as high as 100 GeV in the generic models for t a d = 2, l o = 0, and p < 0, and in the two-parameter SU(5)xU(l) moduli scenario for
pCo
FIG. 5. The dilepton and trilepton rates at the Tevatron versus the chargino mass in two-parameter SU(5) xU(1) supergravity-dilaton scenario ( t o = I/&,(,, = -1) for the indicated values of tanp. The upper (lower) dashed lines represent estimated reaches with 100 pb-’ (1 fb-’) of data.
464
52
SUPERSYMMETRY DILEF'TONS AND TRILEF'TONS AT THE.
..
147
strict no-scale SU(5)XU(I) moduli scenario dilaton scenario
Tevatron 101 h
n
a
v
: 4
2 m
100
10-1
v
10-2
40
60
80
100
120
140
40
60
80
100
120
140
101 h
n
a
v
J
FIG. 6. The dilepton and trilepton rates at the Tevatron versus the chargino mass in oneparameter SU(5)xU(1) supergravity-moduli and dilaton scenarios ( p < 0 in both cases). The upper (lower) dashed lines represent estimated reaches with 100 pb-' (1 fb-') of data.
100
-
D.
2 m
10-1
k - - - - - - - -\
v
10-2
40
60
80
100
120
m,; (GeV)
140
40
60
80
tar@ 5 10. More generally, the accessible region of parameter space should overlap with that within the reach of LEP 11, although i t would have been explored before LEP I1 turns on. However, LEP I1 has a n important task: chargino searches at LEP I1 will not be hindered by small branching fractions, and thus a more modelindependent lower limit on the chargino mass should be achievable, i.e., mx: We would like t o conclude with Fig. 6, where we show the predictions for the dilepton and trilepton rates in our chosen one-parameter
= f4.'
5 ~LEP t 11 it might be possible to extend the indirect reach for charginos by studying the process e+e- -+ xyx; with x i + x: 2 j . Equation (1) implies a kinematical reach of mX iI o $4.
+
J. Ellis, J. Hagelin, D. V. Nanopoulos, and M. Srednicki, Phys. Lett. 127B, 233 (1983); P. Nath and R. Arnowitt, Mod. Phys. Lett. A 2, 331 (1987); R. Barbieri, F. Caravaglios, M. Rigeni, and M. Mangano, Nucl. Phys. B367, 28 (1991). H.Baer and X. Tata, Phys. Rev. D 47,2739 (1993); H. Baer, C. Kao, and X. Tata, ibid. 48,5175 (1993). J. L. Lopez, D. V. Nanopoulos, X u Wang, and A. Zichichi, Phys. Rev. D 48,2062 (1993). J. White, D. Norman, and T. Goss, DO Note 2395 (unpublished). For a recent review see, e.g., J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Prog. Part. Nucl. Phys. 33, 303 (1994). S. Kelley, J. L. Lopez, D. V. Nanopoulos, H. Pois, and K. Yuan, Nucl. Phys. B398,3 (1993). J. Ellis, G. L. Fogli, and E. Lisi, Phys. Lett. B 333,118 (1994); J. Erler and P. Langacker, Phys. Rev. D 52, 441 (1995).
100
120
140
m,: (Gev) SU(5)xU(l) models, which are the most predictive supersymmetric models t o date. It is interesting to note that in the moduli scenario, the mass reach for charginos could be as high as 150 GeV with a n integrated luminosFurther proposed increases in luminosity ity of 1 fi-'. or center-of-mass energy of the Tevatron collider have the potential of probing even deeper into the parameter space [17]. We conclude that detection of weakly interacting sparticles at the Tevatron may well bring the first direct signal for supersymmetry. We would like t o thank James White for motivating this study and for providing us with valuable information about the sensitivity of the dilepton signal. This work has been supported in part by DOE Grant No. DE-FG05-91ER-40633. The work of X.W. has been supported by the World Laboratory.
[8]J. L. Lopez, D. V. Nanopoulos, G. Park, Xu Wang, and A. Zichichi, Phys. Rev. D 50, 2164 (1994). [9] J. L. Lopez, D. V. Nanopoulos, Xu Wang, and A. Zichichi, Phys. Rev. D 51, 147 (1995).
[lo] J. L. Lopez, D. V. Nanopoulos, and H. Pois, Phys. Rev. D 47, 2468 (1993); J. L. Lopez, D. V. Nanopoulos, H. Pois, and A. Zichichi, Phys. Lett. B 299,262 (1993); J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Phys. Rev. D 48,2766 (1993). [ll]J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Rev. D 49,343 (1994). 1121 3. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 319,451 (1993). 1131 J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Int. J. Mod. Phys. A (to be published). [14] H. Baer, C. Chen, F. Paige, and X. Tata, Phys. Rev. D 49,3283 (1994). [15] J. Morfin and W. K. Tung, Z. Phys. C 52, 13 (1991). 1161 CDF Collaboration, Y. Kato, in Proceedings of the
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LOPEZ, NANOPOULOS, WANG, AND ZICHICHI 9th Topical Workshop on Proton-Antiproton Collider Physics, Tsukuba, Japan, 1993, edited by K. Kondo and S. Kim (Universal Academy Press, Tokyo, 1994), p. 291; CDF Collaboration, F. Abe et al., in Proceedings of the 27th International Conference on High Energy Physics,
52 -
Glasgow, Scotland, 1994 (unpublished). [17] T. Kamon, J. L. Lopez, P. McIntyre, and J. White, Phys. Rev. D 50, 5676 (1994). [18] J. White (private communication).
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Jorge L. Lopez, D. V. Nanopoulos and A . Zichichi
STRING NO-SCALE SUPERGRAVITY MODEL AND ITS EXPERIMENTAL CONSEQUENCES
From Physical Review D 52 (1995) 41 78
I995
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469 PHYSICAL REVIEW D
VOLUME 52, NUMBER 7
1 OCTOBER 1995
String no-scale supergravity model and its experimental consequences Jorge L. Lopez' and D. V. Nanopoulos Department of Physics, Tezos A&M University, College Station, Teuu,77843-4242 and Astropartiele Physics Group, Houston Advanced Reseamh Center (HARC), The Mitchell Campus, The Woodlands, Team 77381
A. Zichichi
CERN,1211 Geneva 23, Switzerland (Received 1 March 1995) W e propose an SU(S)xU(l) string-derived model that satisfies the constraints from no-scale supergravity. All supersymmetric ob-bles are calculated in terms of a single parameter, with consistency determining the rest, including tan@= 2.2-2.3 and mt z 175 GeV. A small nonuniversality of the scalar masses at the string scale makes the right-handed sleptons the lightest supersymmetric particles for mllz 2 180 GeV. This cutoff in the parameter space entails the guaranteed discovery of charginos at the Fermilab Tevatron and CERN LEP 11, and right-handed sleptons and Higgs bosons
at LEP 11. PACS number(s): 12.60.Jv, 04.65.+e, 12.10.Dm, 14.80.L~ Despite all the experimental evidence in support of the standard model, many physicists believe that it must be extended so that its many ad hoe parameters may find an explanation in a more fundamental theory. Among the various avenues that lead away from the standard model, the ideas of supersymmetry, supergravity, and superstrings are particularly compelling in tackling the shortcomings of the standard model. Low-energy supersymmetry predicts the existence of a superpartner for each of the standard model particles with well-determined interactions but undetermined supersymmetry-breaking masses, although these should not exceed the TeV scale if the gauge hierarchy problem is to remain at a tolerable level. Supergravity provides an effective theory of supersymmetry breaking in terms of two input functions: the K a e r function and the gauge kinetic function. With these inputs all superspnmetry-breaking masses can be calculated in terms of a single parameter: the gravitino mass ( m 3 p ) . Superstrings provide the final link by allowing a fist-principles calculation of these ,two input functions in any given string model. At low energies a new parameter arises, namely, the ratio of vacuum expectation values (tan/?) of the two Eggs-boson doublets minimally required in supersymmetric models. However, minimization of the electroweak scalar potential with respect to the two neutral E g g s fields provides two additional constraints which effectively reduce the number of parameters to zero, and a no-parameter model is obtained. In this paper we describe one such model obtained in the context of string no-scale supergravity [1,2]. Since there are so many possible string models, we guide our search for a realistic model by a few principles. (i) A string-unified gauge group which can break down to the standard model gauge group. In the context of level-
'Present address: Department of Physics, Rice University, 6100 Main Street, Houston, TX 77005.
one Kac-Moody algebra constructions there is only one known choice: SU(5)xU(l). (ii) A matter content which reduces to the supersymmetric standard model at low energies and that allows unification of the gauge couplings at the string scale (Mu lo1' GeV). This requires nonminimal vectorlike representations with intermediatescale masses. (iii) A low-energy effective theory with the nescale supergravity structure with vanishing vacuum energy [3]. One can search for such models within the free-fermionic formulation of the heterotic string in four dimensions. This search entails looking a t a large number of string models which are specified by a set of n basis vectors and an n x n matrix of Gliozzi-Scherk-Olive (GSO) projections. For each choice of these two inputs one obtains a complete model with a specific gauge group, matter representations, and superpotential and K a e r potential interactions. In Ref. [4] one such model was constructed by demanding that the fifst two constraints above are satisfied. In the present paper we show that the third constraint is also satisfied in this model. This last step has only recently become possible because of new developments in the understanding of the Kahler function in this class of models [5,1]. As a by-product we are able to calculate the soft-supersymmetry-breaking parameters in this model and thus we also present a phenomenological analysis of the low-energy consequences of the model. For our present purposes we only need to remark that this model shares many of the appealing properties of the class of SU(5)xU(1) models which have appeared before [6,7], including three generations of quark and l e p ton superfields, and two light E g g s doublet superfields. The full gauge group is S U ( ~ ) X S O ( ~ ) X S O ( ~ O ) X with the observable gauge symmetry [SU(5)xU(1)] broken down to the standard model gauge group via vacuum expectation d u e s of scalar E g g s fields in 10,iTj representations. The model also predicts the existence of intermediate-scale vectorlike particles (Q, Q and D c ,D') contained in one set of (additional) l0,m representations. The masses of these particles are not additional
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@ 1995 The American
Physical Society
470 STRING NO-SCALE SUPERGRAVITY MODEL AND ITS.. .
52
parameters to the model, since they are in principle selfconsistently determined by condensates of hidden-sector matter fields [4]. However, at the present stage of string model building these calculations are not well understood. Thus, here we choose these masses such that they allow unification of the gauge couplings at the string scale (i% mQ lol' GeV, mDc lo6 GeV [81)* Deeper model building studies will need to reproduce these mass
4179
parameters. In Ref. [l]a study was performed of the constraints imposed by n-scale supergravity on free-fermionic string model building. In these modeis the matter fields come in two kinds: untwisted [a!'),i = 1 + nu,] and twisted i = 1 + n T I ] , with three uSetsn I = 1 , 2 , 3 for each kind of field. The K a e r potential is given by [I]
[fly),
where S is the omnipresent dilation field and T is the modulus field which parametrizes a flat direction of the scalar potential, along which the gravitino mass is undetermined. A model with this Kahler potential possesses mnishing vacuum energy a t the tree level if some subsidiary conditions on the vacuum expectation values IVEV's) of the [I]. In our model, the matter fields arrange themselves as nonmoduli fields are satisfied (e.g., all VEV's &h) follows: Set First
Untwisted[af)] @O, @ l@23, , @23
hl,h Second
% r f l l r @ 3 1 , @31
@31
h2, h2 @Sr @I23 a12
where F, f,1' represent fields transforming as 10,I, 1 under SU(5) (including the symmetry-breaking decaplets); h, represent 5 , s Higgs pentaplets; 'P, +,9 represent singlet fields; and D, p,F,T represent hidden matter fields transforming as 6,4,4 of SU(4) and 10 of SO(lO), respectively. Except for the @0,1,3,5 fields, all other fields in Eq. (2) are charged under the various U( 1) gauge symmetries of the model. The $0,1,3,5 are absolute gauge singlets and are promising candidates for moduli fields (51. However, only can be identified as a modulus field (i.e., T ) since @0,3,5 appear explicitly in the cubic superpotential (given in Ref. [4]) and therefore do not correspond to flat directions. Thus our model corresponds to the Kahler potential in Eq. (1) with the nonmoduli fields arranged in three sets as in Eq. (2); i.e., the model possesses the no-scale supergravity structure. Minimization of the electroweak-scale scalar potential with respect to the gravitino mass (the no-scale mechanism) determines in principle its otherwise undetermined value [3]. At the one-loop level, this mechanism becomes unstable if the quantity Q 0: S t r M Z does not vanish a t the scale of supersymmetry breaking. A nontrivial calculations shows that the K a e r potential in Eq. (1)gives [l]Q = nu, nus TIT, - nul - d f - 3, where d f is the dimension of the gauge group. From Eq. (2) we find nul = 13, nu, = 14, nus = 16, TIT, = 80; also d f = 90, and thus Q = 4. We find this tree-level result tantalizingly close to zero. Indeed, if taken as a reference scale the sum of the absolute values of the various contribu-
+
+
I tions, our result is off by 2%. We hope that an all-orders calculation would produce a vanishing result. In fact, it has been recently pointed out that analogous two-and higher-loop quadratic divergencies are generically present [9], and would need to be canceled in a model-dependent fashion. This model dependence is already present at the one-loop level in our calculation of Q. New modelbuilding constraints that may arise &om the cancellation of these higher-loop divergencies w i l l be considered elsewhere. Various.other properties also follow generally for models with a Kahler potential given in Eq. (l), and thus, for our model in particular [l],(a) the Goldstino field is a fited admixture of dilaton'and modulus fields fj 0: S d%, (b) the soft-supersymmetry-breaking scalar masses vanish for fields in the Ul, Tz, T3 sets and are equal to m3p for fields in the U2,U3,T1 sets, (c) A = m3/2 for all cubic superpotential terms, and (d) the gaugino masses are universal mllz = m312. In order to obtain the actual soft-supersymmetrybreaking terms, one needs to identify some of the states in Eq. (2) with the three light generations and the two light Higgs doublets. Here we simply consider the identification made in Ref. [4], which assigns the first, second, and third generation quark doublets to F3, Fz,and F4 (recall that F = {Q, dc, v ' } ) , respectively, and the light Higgs doublets to hl and &5. With the pattern of soft-breaking scalar masses indicated above, such limited identification is enough to determine all the observable-sector scalar
+
47 1 52 -
JORGE L. LOPEZ, D.V. NANOPOULOS, AND A. ZICHICHI
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masses at the string scale. That is (e.g., HIC hl E UI+ mg1 = 0; Q3 C F4 E 2'1 + ma3 = mi/,), for gaugino masses (universal), ml12 = m3/2, for scalar masses of the first generation, m&l,Ui,D;,L1,E;- 0, of the second = 0, of the third generation generation m&u;,D;,La,E,' = mil2, m$g,LJ,E;= 0, for Eggs boson masses, mg1 = mga = 0, and for trilinear scalar couplings (universal), & = m3/2. In this scenario, the remaining several states in Eq. (2) become massive via grand unified theory (GUT) symmetry-breaking interactions or via interactions in the doublet and Eggs triplet mass matrices, which receive contributions &om nonrenormalizable terms in the superpotential [4]. To complete the picture one also needs to determine the Eggs mixing parameter p and its associated scalar coupling B. In this model we find that the Eggs mixing term pH1H2 arises as an effective coupling at the quintic level in superpotential interactions: chl&.(@lD)/MZ, i.e., p (q5DD)/M2 lo3 GeV for M lo1" GeV and the plausible values (c#J)/M 1/10, (DD) (lo1' GeV)'. Our present inability to reliably estimate the value of p makes this the single parameter of the model. We should remark that in this class of models the K a e r potential allows for additional contributions to the p parameter [lo], but only if the Eggs doublets belong to the same untwisted set [l], which is not the case in our specific model. The specific form of the coupling generating the p term allows one to determine B (even if the magnitude of p itself is not well known) [I]: BO= [1+ (T ?)d2(~)/n]m3/22 m3p, where the minimum of the Eisenstein function occurs for T = 1. Also of great interest, the topquark Yukawa coupling in the superpotential is A t = g f i [4], where g M 0.83 is the unified gauge coupling at the string scale, obtained by running up to the string scale the standard model gauge couplings [8].The normalized topquark Yukawa coupling depends however also on the K a e r potential, and we obtain At = g2 x 0.7, once the proper normalization factor (g/& [l])is inserted. The topquark mass itself cannot be yet determined since it also depends on the low-energy parameter tanp (i.e., a sir$). The low-energy theory, obtained by renormalization group evolution &om the string scale down to the electroweak scale, thus depends on only one parameter (m3/2 or ml/2) since the magnitude of the Eggs mixing term 1p1 and tan@ can be self-consistentlydetermined &om the minimization of the one-loop electroweak effective potential. Moreover, we fmd that the constraint Bo 2 m3/2 can only be satisfied for p < 0. This general procedure has been carried out before in supergravity models [12]; however, with the further specification of Bo,the numerical computations which determine the value of tanfl bem3/2 depends come rather elaborate [8,13]. Since Bo on the undetermined value of T, and below we show that BO 2 l.2m3/z is phenomenologically excluded, for concreteness we take & = m3/2 (i.e., T = 1). In our model, a novelty arises because the scalar masses given above are not universal at the string scale. This nonuniversality entails a modikation of the usual renormalization group equations [14], which amounts to shifts
-
N
- -
-
+
in the squared scalar mass parameters at low energies, Amf = -c2Xf, where yi is the hypercharge, cz = m"R, - m&,
= 2m:/,,
(3)
is the nonuniversality coefficient at the string scale, and f m 0.060 is a renormalization group equation (RGE) coefficient [14]. These shifts are most significant for the right-handed sleptons (iR = ZR, f i ~- 7 , ~whose ) mases are
m? IR = am2,/,
+ tanzOwM$(tan2P - l)/(tan2P+ I), (4)
with a = 0.153 is the usual universal case, but a = 0.153 - 0.120 = 0.033 in our nonuniversal case. For the other scalars the (usual) coefficient a is much larger and the effect of the shift (- 0.1) is relatively small. The significance of the downward shift on mie relates to the lightest supersymmetric particle, which IS stable and should be neutral and colorless [15]. For the lowest allowed values of m112,this particle is the lightest neutralino x; with mx; x 0.25m1/2. From Eq. (4) we see that as m , p increases there is a critical value mil2 above become the lightest supersymmetric partiwhich the cles. Since this is phenomenologically unacceptable, we cut off the single parameter of the model at this critical value. Another novelty in our model is that the topquark maw is self-consistentlydetermined by the value of tan0 which results &om the various other constraints. Once the numerical calculations described above are performed we find mil2 M 180 GeV and tan@ = 2.2-2.3. The latter result allows a precise determination of the ("pole") topquark mass mt x 175(179) GeV for
a3
= 0.118(0.130),
(5)
as shown in Fig. 1. With such small value of t d , the
>
tang FIG. 1. The topquark mass vs ta@. Self consistency of the model requires tan@= 2.2-2.3 and thus mt 175 (179) GeV for 03 = 0.118 (0.130).
472 STRING NO-SCALE SUPERGRAVITY MODEL AND ll3. . .
52
Tevatron ,
3.0
500
.
, ,
4181
, , , ,
,
, , ,
, , , ,
h
e a v
5 N
m v
300
0.0
60
50
200
70
MI
90
mx: (GeV)
FIG. 3. The dilepton and trilepton rates at the Fermilab Tevatron vs the chargino mass originating fiom neutralino and chargino production. The indicated reaches are expected with 100 pb-' of accumulated data.
FIG. 2. The full sparticle and Eggs-boson mass spectrum vs the chargino mass. Here m
dashed lines,l which show guaranteed discovery via the trilepton mode. At the CERN e+e- collider LEP 11 with a centerof-mass energy of f i % 180 GeV, it should be possible to observe the lightest E g g s boson (mh < 90 GeV), the lightest chargino, and the right-handed s l e p tons (mi, < 50 GeV). The Higgs-boson coupling to gauge bosom is indistinguishable from the Standard Model prediction [&(a - p) M 11, although its branching ratio into b6 may be eroded somewhat for the lightest allowed masses, because of the supersymmetric decay channel with B(h + < 0.18. The reach of LEP II for Higgs boson masses is estimated a t f i - 95 = 85 GeV [19],and thus an increase of center-of-mass energy to f i = 190 GeV would allow fuJl discovery potential for the Higgs boson. The charginos would be pair produced (efe3 x:x;) and in the preferred "mixed" decay mode (1 lepton 2 jets) one chargino decays leptonically and the other one hadronically. We find a cross section into the mixed mode as large as 2.1 pb and decreasing down to 0.62 pb at the upper end of the allowed interval. This signal should be readily detectable [20]. One would also produce x:x: with a cross section from 1.4 pb down to 0.6 pb, which could only be detected via the dilep ton model [since B(x: + 2j) G 01. The right-handed selectrons (smuons) have a pair-production cross section exceeding 2 pb (0.9 pb) and should be easily detectable over the WW background [20]. One can also test the model via rare processes. The
~"1x01)
bottom and tau Yukawa couplings at the string scale A, 0.01 [Ill. This matshould be comparable Ab ter will be addressed in the context of this model elsewhere. The full mass spectrum is shown in Fig. 2. For I30 > m3/2, the resulting values of tanp decrease and, consequently, the allowed window in ml/z also decreases. In fact, for Bo 2 l.2rn3l2no points in parameter space are phenomenologically allowed. At this extreme tanp N 1.9 and the prediction for the topquark mass is only slightly (few GeV) decreased. At the present-day Fermilab Tevatron the strongly interacting gluino and squarks are not accessible (mg,g 2 300 GeV). O n the other hand, the weakly-interacting neutralinos and charginos (mx: M m,p < 90 GeV) are quite reachable in this model via the trilepton signal in pp --t x;x:X [16] and the dilepton signal in p p + x:x;X [17]. The chargino (x:) branching ratio into leptons (e p ) is N 0.5, whereas that into jets is % 0.25, for all allowed points in parameter space. Also, the neutralino (x:) decays exclusively to dileptons because of the dominant two-body decay mode & + iZl7. The corresponding rates (no cuts) are shown in Fig. 3. The expected experimental sensitivities with 100 pb-' of accumulated data (end of 1995) are indicated by the N
N
+
+
'The sensitivitiesare actually chargino-massdependent, following a curve shaped similarly to the signal and which asymptotes to the indicated dashed lines. With the Collider Detector at Fermilab (CDF) data from run I A (a20 pb-'), this asymptote is at w 2 pb [18].
473 JORGE L. LOPEZ, D. V. NANOPOULOS, AND A. ZICHICHI
4182
52 -
prediction for B(6 -+ 5 7 ) (211 is subject t o large QCD uncertainties, accounting for these as described in Ref. [22] we get a range (4.2-5.3)~10-~ for the lower end and (3.9-5.1)~10-~ for the upper end of the spectrum. These predictions are in fair agreement with the present experimentally-allowed range of (14)x [23]. For the supersymmetric contribution t o the anomalous magnetic moment of the muon [24] we get = (-2.4 -+ -1.7) x which is not in conflict with present experimental limits b u t could be easily observable at the new E821 Brookhaven experiment which aims at a sen-
sitivity of 0.4 x Turning t o cosmology, the relic abundance of the lightest neutralino has been calculated following the methods of Ref. [25] and determined to be n,h2 5 0.025. Such small cold dark matter density would be of interest in models of the Universe with a s i g d i c a n t cosmological constant [26].
[l] J. L. Lopez and D. V. Nanopoulos, Report No. CERNTH.7519/94, hepph/9412332 (unpublished). [2] S. Ferrara, C. Kounnas, M. Porrati, and F. Zwirner, Phys. Lett. B 194, 366 (1987); Nucl. Phys. B318, 75 (1989); S. Ferrara, C. Kounnas, and F. Zwirner, ibid. B429, 589 (1994). [3] E. Cremmer, S. Ferrara, C. Kounnas, and D. V. Nanopoulos, Phys. Lett. 133B, 61 (1983); J. Ellis, C.
[16] P. Nath and R. Amowitt, Mod. Phys. Lett. A 2, 331 (1987); R. Barbieri, F. Caravaglios, M. Frigeni, and M. Mangano, Nucl. Phys. B387, 28 (1991); H.Baer and X. Tata, Phys. Rev. D 47,2739 (1993); J. L. Lopez, D. V. Nanopoulos, X. Wang, and A. Zichichi, ibid. 48, 2062 (1993). [17] J. L. Lopez, D. V. Nanopoulos, X. Wang, and A. Zichichi, Phys. Rev. D 52,142 (1995). [18] CDF Collaboration, Y. Kato, in Promdings of the 9th T o p i d Workshop on Proton-Antiproton Collider Physics, Tsukuba, Japan, 1993, edited by K. Kondo and S. Kim (Universal Academy Press, Tokyo, 1994), p. 291; CDF Collaboration, F. Abe et al., Fer&b-Cod-94/149E, 1994 (unpublished). [19] See,e.g., A. Sopczak, Int. J. Mod. Phys. A 9,1747 (1994). [ZO] See, e.g., J.-F. Grivaz, in Properties of SUSY Particles, Proceedings of the 23rd INFN Eloisatron Project Workshop, Erice, Italy, 1992, edited by L. C. C i e l l i and V. A. Khoze, Science and Culture Series: Physics Vol. 6 (World Scientific, Singapore, 1993); J. L. Lopez, D. V.
aiw
Kounnas, and D. V. Nanopoulos, Nucl. Phys. B241, 406 (1984); B247, 373 (1984); J. Ellis, A. Lahanas, D. V. Nanopoulos, and K. Tamvakis, Phys. Lett. 134B, 429 (1984). For a review, see A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. 145,1 (1987). [4] J. L. Lopez, D. V. Nanopoulos, and K.Yuan, Nucl. Phys. B399,654 (1993). [5] J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Phys. Rev. D 50,4060 (1994). [6] I. Antoniadis, J. Ellis, J. Hagelin, andD. V. Nonopoulos, Phys. Lett. B 194, 231 (1987); 231, 65 (1989); J. L. Lopez and D. V. Nanopoulos, ibid. 251,73 (1990). [7] For a review, see J. L. Lopez, Surv. High Energy Phys. 8,135 (1995). [8] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Rev. D 49,343 (1994). [9] J. Bagger, E. Poppitz, and L. Randall, Report No. EFI95-21, hepphj9505244 (unpublished). [lo] G. Giudice and A. Masiero, Phys. Lett. B 208, 480 (1988); V. Kaplunovsky and J. Louis, ibid. 308, 269 (1993).
[ll] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 327,279 (1994). [12] For a review, see J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Prog. Part. Nucl. Phys. 33,303 (1994). [l3] 3. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Lett. B 319,451 (1993). [14] See, e.g., A. Lleyda and C. Muioz, Phys. Lett. B 317, 82 (1993). [15] J. Ellis et ul., Nucl. Phys. B238,453 (1984).
We would like t o thank T. Kamon and J. T. White for useful discussions. This work has been supported in part by U.S. DOE Grant No. DEFG05-91-ER-40633.
Nanopoulos, H. Pois, X. Wang, and A. Zichichi, Phys. Rev. D 48,4062 (1993). [21] See, e.g., S. Bertolini, F. Borzumati, A. Masiero, and G. Ridolfi, Nucl. Phys. B353, 591 (1991); J. L. Lopez, D. V. Nanopoulos, and G. T. Park, Phys. Rev. D 48,974 (1993); F. Borzumati, 2. Phys. C 83,291 (1994). [22] A. Buras, M. Misiak, M. M k , and S. Pokorski, Nucl. Phys. B424, 374 (1994); M. Ciuchini et ul., Phys. Lett. B 334,137 (1994). [23] CLEO Collaboration, M. S. Alam et al., Phys. Rev. Lett. 74,2885 (1995). [24] J. L. Lopez, D. V. Nanopoulos, and X. Wang, Phys. Rev. D 49,366 (1994), and references therein. [25] J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Nucl. Phys. B370, 445 (1992). [26] See, e.g., G. Ehtathiou, W. Sutherland, and S. Maddox, Nature (London) 348,705(1990); J. L. Lopez and D. V. Nanopoulos, Mod. Phys. Lett. A 9,2755 (1994).
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475
Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi
EXPERIMENTAL CONSTRAINTS ON A STRINGY SU(5) x IJ( 1) MODEL
From Physical Review D 53 ( 1996) 5253
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477 PHYSICAL REVIEW D
VOLUME 53, NUMBER 9
1 MAY 1996
Experimental constraints on a stringy SU(5) x U(1) model Jorge L. Lopez D e p a m e n t of Physics, Bonner Nuclear Lab, Rice University, 6100 Main Street, Houston, Texas 77005
D.V. Nanopoulos Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, Texas 77843-4242; Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Mitchell Campus, The Woodlands, Texas 77381; and Universita di Bologna and INFNs Bologna, Italy
A. Zichichi CERN, I211 Geneva 23, Switzerland and Universita di Bologna and INFN, Bologna, Italy
(Received 15 January 1996) We reexamine the phenomenological aspects of a recently proposed string-derived SU(5)X U(1) oneparameter supergravity model, and explore the sensitivity of the model predictions to variations in the saong coupling and the string unification scale. We also perform an analysis of the constraints on the parameter space of the model in light of the recent Fermilah Tevatron trilepton data and the CERN LEP 1.5 chargino and slepton data. We obtain mx;>70 GeV, which excludes one-third of the parameter space of the model. The remainder of the parameter space should be probed by ongoing analyses of the Tevatron trilepton data and forthcoming LEP 2 runs. [S0556-2821(96)06709-4] PACS number(s): 12.60.J~.04.65.+e, 12.10.Dm, 14.80.L~ I. INTRODUCTION
String model building has come a long way since its inception about a decade ago, making the calculation of experimental predictions in specific string models an ever more refined art.For instance, in free-fermionic string models, after specifying the requisite two-dimensional world sheet inputs that define a model [i.e., basis vectors and GliozziScherk-Olive (GSO) projections], it was originally only known how to obtain the spectrum of states and the corresponding gauge group. Later one learned how to calculate the superpotential interactions at cubic and nonrenomalizable level. More recently one has been able to identify the fields that parametrize flat directions of the scalar potential (moduli) in this class of models, a necessary step in the calculation of the K d e r potential. All of this stringy “technology” has been applied to search for models with vanishing vacuum energy at the tree- and one-loop levels [l]. In Ref, [Z] we presented a string-derived model based on the gauge group SU(S)XU(l) that incorporated these theoretical advances. This model predicted the top-quark mass to be m f - 175 GeV, and a rather light spectrum of superparticle masses expressed as a function of a single mass parameter. Here we discuss recent theoretical and experimental developments that have increased the motivation for such a class of models, and refine and update the corresponding experimental predictions in light of the recent data from the Fermilab Tevatron and CERN e + e - collider LEP 1.5. First let us discuss the theoretical developments. The model in Ref. 123 possessed vanishing vacuum energy (at the tree level) before and after spontaneous supersymmetry breaking. It has since been shown that this essential property survives the shift in vacuum expectation values of scalar fields required to cancel the anomalous UA(1) contribution 0556-2821/96/53(9)/5253(5)/$10.00
53
to the D term in realistic string models [3].Also, the model presented in Ref. [2] suffered from a quadratically divergent contribution to the one-loop vacuum energy: (1116~’)Qom$$4;,, with Q0=4, although it was argued that such value was “small” enough to be possibly shifted towards zero by high-order effects. It has now been shown that in the process of canceling the anomalous UA(l ) , Qo is shifted to Q = Qo A Q , such that Q = 0 can be naturally obtained in this model [4,3]. This shift is only effective when VO=0 and Qo is sufficiently small. These developments put the model in Ref. [2] (the only known one with VO=Q = O ) on much firmer theoretical ground, making exploration of its consequences a better motivated and pressing endeavor. On the experimental side, the prediction for the top-quark mass in this model (m,= 175 GeV) agrees rather well with experimental observations. Also, the predicted light charginos and neutralinos ( m X 2I - m x20<90 GeV) entail dilepton and trilepton signals at the Tevatron that are now entering the range of experimental sensitivity. These could also be probed at LEP 1.5 (the first upgraded phase of LEP with &=130- 136 GeV), as could the light right-handed sleptons (mFX<50 GeV). It has also become apparent that the predicted value for ct5(Mz) in the traditional minimal SU(5) grand unified theory (GUT) i s uncomfortably large [us(MZ)>0.123 [5]], and certainly cannot explain the lowenergy determination of c t 5 ( M Z ) (around 0.11). On the other hand, it has been recently shown [6] that SU(S)XU(l) can naturally explain the whole range of a, values being considered, motivating further studies of models with SU(5) xU(1) gauge symmetry. In what follows we first study the robustness of the scenario presented in Ref. [2] under variations in the smng uni-
+
5253
0 1996 The American Physical Society
478 JORGE L. LOPEZ, D.V. NANOPOULOS,AND A. ZICHICHI
5254
MU
two-loop
53
t ' " " ' I " " " " 4
-1
5g101'
10'6
0.11
0.116
0.12
as (MZ) as (MZ) FIG. 1. The masses of the Q and D intermediate-scaleparticles as a function of a,(MZ) using one-loop (dashed lies) and two-loop
FIG. 2. The gauge coupling at the string unification scale obtained using one-loop (dashed lines) and two-loop (solid lines) contributions to the renormalization group equations. Two choices of the string unification scale ( M u )are shown.
(solid lines) contributions to the renormalization group equations. Two choices of the string unification scale ( M u ) are shown.
fication scale, the strong gauge coupling [ a S ( M Z ) ]and , twoloop effects in the running of the gauge couplings (Sec. II). We then turn to the constrain? on the parameter space of the model that follow from recent Tevatron trilepton data (Sec. ID) and LEP 1.5 chargino and slepton data (Sec. IV). We summarize our conclusions in Sec. V.
tively. In the case of M , , the largest effect is the shift in the string unification scale, with small LY,and two-loop dependences. In the case of M D , the largest effect is the dependence on a,, then comes the effect of shifting the string unification scale, and lastly the two-loop effects. The effects on g also follow a hierarchy, although they do not amount to more than a few percent: the largest is the effect of shifting the unification scale, then comes the effect of varying as, and finally the two-loop effects. The shifts in the above quantities impact the calculation of the top-quark mass itself
II. RUNNING REFINEMENTS
In the present scenario of gauge coupling unification the standard model gauge couplings are run up to the string scale, where SU(S)XU(l) is assumed to break down to the standard model gauge group [7]. This scenario predicts the existence of intermediate-scale vector-like particles: a pair of (Q.f% [(3,2),6,2)1 and apairof P,E) [(3,1),6,1)1, such that string unification occurs at the string scale. The masses of these particles can be in principle derived from the model [8], although in practice they are adjusted to fit this scenario. These masses, and the whole superparticle spectrum, depend on the value of a S ( M z ) used in the calculations. They also depend on the value chosen for the string scale. Previously we simply set M,,,=1018 GeV. The value of the topquark mass also depends on these two inputs. Here we refine our calculations of these observables by allowing a wide range of a , ( M Z ) values consistent with low-energy determinations (0.108-0.110) and the world average (0.11820.006). We also study the effects of using the proper string scale M su;ng= 5 X g X lOI7 GeV, where g is the gauge coupling at the unification scale. Our previous choice for M gives an indication of the effects of string thresholds, which in this class of models tend to yield a slightly larger effective unification scale [9]. Recent more refined analyses also entail small shifts in the effective string unification scale [lo]. First we study the dependence of M , , M D , and g, as we use the proper string scale, as we take ~ ~ ~ ( M ~ ) = 0 . 1 0 8 - 0 . 1 and 2 4 , as we allow for two-loop corrections to the running of the gauge couplings. The results for M , , M Dand for g are shown in Figs. 1 and 2, respec-
where X,(m,) is obtained from the string-scale prediction of X , ( M u ) = g 2 [2]. The above refinements affect rn, through the shifts in g , the variations in the running (starting from a different M u , going through the different Q and D thresholds), and the calculated value of tanp. The range of (pole) m, values is shown in Fig. 3 for three values of a,, and M u = 5 X g X 1017 GeV. For a,=0.118 we also show the result for Mu= 10l8 GeV. We see that without specifying the value of tanp we obtain m,-( 160- 190) GeV,
(2)
irrespective of the various uncertainties in the calculation. In Ref. [2] we showed how the value of tanp can be determined in terms of the one parameter in the model: one adjusts tanp until the predicted value of Bo is reproduced by the calculated value of Bo (obtained from the radiative electroweak breaking constraint). This procedure gives values of tanp which vary with the one model parameter, but only slightly. For a,=0.118 and M u = 10l8 GeV we obtained tanp-2.2- 2.3. The tanp range is cut off, as are the ranges of all the sparticle masses, by a cutoff in the one parameter in the model. This cutoff is obtained when the masses of the right-handed charged sleptons (7;)become lighter than the mass of the lightest neutralino, entailing a cosmologically unacceptable lightest supersymmetric particle (LSP) [111.
479 EXPERIMENTAL CONSTRAINTS ON A STRINGY SU(5)XU(l) MODEL
53 -
t
- 4k
_ _ _ _ - - - - - =
\
5255
'... ~
D O 12.5 pb-'
P
a u w h
a,=0.108-0.124
v
m,=170-176 GeV
tanp FIG. 3. The calculated values of the top-quark mass as a function of tanp for a,=0.108, 0,118, 0.124 (the bottom, central, and top solid lines) and Mu=5XgX 10" GeV. Also shown (for (u,=0.118) is the effect of taking M u = lo'* GeV (dashed line). The vertical lines indicate the dynamically determined value of tanp. Note the stability of the m, prediction. Recalculating the values of tanp that are obtained for a,=0.108,0.118,0.124,andMLI=5XgX10'7GeV we find that the previous results remain qualitatively unchanged, and quantitatively only slightly modified. The tanp range is now a little higher: tanp-2.35-2.45, but the top-quark mass as a function of tanp is a little lower (cf. central solid versus dashed line in Fig. 3). The result of these two compensating effects is a rather stable prediction for the top-quark mass in this model: rnr- (170- 176) GeV.
(3)
FIG.4. Trilepton rates (solid line) at the Tevatron (summed over e e e , e e p , epp, p p p ) versus the chargino mass originating from chargino-neuualino production. The dashed (dot-dashed) line represents the DO (CDF) upper limit based on 12.5(19.1) pb-' of data. The bends on the experimental curves reflect changes in trigger rates and detection efficiencies.
-0.5, whereas that into jets is -0.25, for all poink in parameter space. Also, the neutralino (,& decays exclusively to dileptons because of the dominant two-body decay mode xi-+?;/", and therefore the trilepton signal is nearly maximized. The trilepton rates in Fig. 4 have been summed over the four channels e e e , e e p , epp, and p p p . The DO Collaboration has released its first official results on these searches based on 12.5 pb-' of data [12]. Their results are conveniently expressed as an upper bound on the trilepton rate (into any one of the four channels) as a function of the chargino mass. The upper limits range from' 12.4pb for m =45 GeV down to 2.4pb for rnx; = 100 GeV. The corXi responding curve is shown in Fig. 4 (dashed line), and does not constrain the model in any way. The Collider Detector at Fermilab (CDF) Collaboration has also released an upper limit on the trilepton rate based on 19.11 pb-' of data [13], that is also shown in Fig. 4 (dot-dashed line): and which comes quite close to the model predictions. Ongoing analyses by CDF and DO should be able to probe some of the parameter space of the model. Indeed, since few background events are expected, to estimate the reach obtained by examining the full data set (- 100 pb-') one could simply scale down the present DO (CDF) upper limit by a factor of 100/12.5( 19)==8 ( 5 ) . Such sensitivity would appear to be enough to falsify the model. However, this will likely not be the case, because one of the leptons from decay (,$-+F;/', Fz+!zxy) becomes increasingly softer as the edge of parameter space (i.e., m&=m 0 ) is XI approached. +
The spectrum of superparticle and Higgs boson masses is quite close to that obtained previously in Ref. [2], and definitely indistinguishable from it given the inherent uncertainties in this type of calculations. In what follows we simply use the spectrum obtained in Ref. [2]. In particular one finds rn
180 GeV,
r n x f = mx;= (60- 90) GeV,
mh-(80-90)
(4) (5)
GeV,
(6)
r n ~ ~ - ( 4 5 - 5 0 ) GeV,
(7)
rn? -0.98rn,-.
(8)
Note that for specific values of a,,subsets of the mass intervals above will be realized.
xi
III. TEVATRON CONSTRAINTS
The cross section for production of chargino-neutralino pairs and subsequent decay into trileptons (pF-+x;x!X; x;-+/'vl.xy,&-!+!+ix in this model has been given in Ref. [2], and is reproduced in Fig. 4. We recall that the chargino (xf) branching ratio into leptons ( e + p ) is
'These values are the result of multiplying by four the explicit limits given in Ref. [12], in order to account for our summing over the four possible channels. 2The CDF upper limit shown here is an (adequate) approximation of the actual experimental result, which has some dependence on the decay kinematics through the detection efficiencies.
480 5256
JORGE L. LOPEZ, D. V. NANOPOULOS, AND A. ZICHICHI
Assuming negligible experimental detection sensitivity for rn&-rnxyS6 GeV, only m x ; S 7 0 GeV could be probed. Such kinematical accidents remind us that “the absence of evidence is not evidence of absence.” Analogous dilepton searches [ 141 are not expected to be as sensitive [2], although they will likely boost the supersymmetric signal should trilepton events be observed.
The intermediate-energy upgrade of LEP (LEP 1.5) accumulated close to 3 pb-’ of data (per experiment) at both &=130 GeV and 136 GeV. Preliminary physics results have been announced recently [15]. Two of the new physics searches are relevant to the present analysis: searches for chargino pair production and searches for charged-slepton pair production. Chargino production proceeds via s-channel y , Z exchange and t-channel sneutrino (Fj exchange, with the two amplitudes interfering destructively if the sneutrino mass is not too large. In our model we find rn,-=(85- 115) GeV, and the negative interference is not very pronounced, yielding chargino cross sections of a few pb. Experimentally one finds that as long as m 2 - m 0>5 GeV and the sneutrino XI XI effect is not exceptionally large, then chargino masses up to the kinematical limit are excluded [15]. In our case mx;-mxy= (28-39) GeV. Therefore we conclude that rn Xi* > 68 GeV (LEP 1.5 charginos).
for 10 GeV or larger mass differences [15]. Taking our calculated cross sections, times the integrated luminosity (- 2.8 pb-’), times these efficiencies, we can exclude points in parameter space that predict three or more events. We obtain the lower bound rnrR>46 GeV, which can be translated into a (more useful) lower bound on the chargino mass in this model m X *>70 i GeV (LEP 1.5 selectrons).
IV. LEP 1.5 CONSTRAINTS
(9)
Comparing our calculations of the chargino cross section with the plots in Ref. [15], we have explicitly verified that this generic result indeed applies for our values of the sneu@ino mass. Future runs at LEP 2 energies of &‘-160(175) GeV should be able to probe chargino masses up to 80 (87) GeV, which would amount to probing two-thirds (nearly all) of the parameter space. However, signal extraction will be complicated by the significant WW background expected. Right-handed selectron pair production proceeds via s-channel y , Z exchange and t-channel neutralino exchange, with the subsequent decay TR- e x : . Our cross sections at LEP 1.5 energies range from 2.5 down to 1.2 pb for the allowed mass range mrR<50GeV. At the same time the selectron-neutralino mass difference is in the range rn rR-mX;= (12- 2) GeV. Efficiencies for selectron detection are negligible for mass differences below 5 GeV, around 55% for mass differences of 5-6 GeV, and climb up to 75%
[l] J. L. Lopez and D. V. Nanopoulos, Int. J. Mod. Phys. A (to be
published), and references therein. [2] J. L. Lopez, D. V. Nanopoulos, and A.Zichichi, Phys. Rev. D 52, 4178 (1995).
53 -
(10)
Smuon and stau production have smaller cross sections and do not produce any new constraints on the parameter space. Running at higher energies increases the selectron cross section. With sufficient integrated luminosity one should be able to probe indirectly somewhat higher chargino masses ( rnx: < 75 GeV) . To overcome the soft-daugther-lepton problem when m 0, with sufficient integrated luminosXI ity one could resort to the radiative process e + e - - + T : T i y , employing a hard photon tag [16].
V. CONCLUSIONS We have demonstrated the robustness of the experimental predictions of the model presented in Ref. [2] under changes in various GUT-scale parameters. We then applied the recent Tevatron and LEP 1.5 data and discovered that one-third of the parameter space of the model has since become excluded, i.e., rnxi>70 GeV is required. The most immediate prospects for further exploration of the model are to be found in the ongoing analyses of the Tevatron trilepton data and future higher-energy runs at LEP 2. Note added in proof. Since the completion of this paper, the CDF Collaboration has performed an analysis of its trilepton data (- 100 pb-I) in the context of the present model and concluded that mx;Z70 GeV is required [17]. As anticipated, the soft nature of the daughter leptons from x: decay constitute the limiting factor in the chargino mass reach. Also, the DO Collaboration has updated its estimate for the top-quark mass ( m , = 1 7 0 Z 1 5 t 1 0 GeV [18]), which is now in good agreement with our theoretical prediction in Eq. (3). ACKNOWLEDGMENTS We would like to thank John Ellis, Teruki Kamon, Lee Sawyer, and James White for helpful discussions. The work of J. L. was supported in part by DOE Grant No. DE-FG0593-ER-40717. The work of D.V.N. was supported in part by DOE Grant No. DE-FG05-91-ER-40633.
[3] J. L. Lopez and D. V. Nanopoulos, hep-pM9511266 (unpublished); and (in preparation). [4]J. L. Lopez and D. V. Nanopoulos, Phys. Lett. B 369, 243 (1996).
48 1 53 -
EXPERIMENTAL CONSTRAINTS ON A STRINGY SU(5)xU(l) MODEL
[5] L. Clavelli and P. Coulter, Phys. Rev. D 51, 3913 (1995); Report No. hep-pM9507261 (unpublished); J. Bagger, K. Matchev, and D. Pierce, Phys. Lett. B 348, 443 (1995). [6] J. Ellis, J. L. Lopez, and D. V. Nanopoulos, Phys. Lett. B 371, 65 (1996). [7] J. L. Lopez, D. V. Nanopoulos, and A. Zicbichi, Phys. Rev. D 49,343 (1994). [8] J. L. Lopez, D. V. Nanopoulos, and K. Yuan, Nucl. Phys. B399, 654 (1993). [9] I. Antoniadis, J. Ellis, R. Lacaze, and D. V. Nanopoulos, Phys. Lett. B 268, 188 (1991); S. Kalara, J. L. Lopez, and D. V. Nanopoulos, ibid. 269, 84 (1991); K. Dienes and A. Faraggi, Phys. Rev. Lett. 75, 2646 (1995); Nucl. Phys. B457, 409 (1995). [lo] E. Kiritsis and C. Kounnas, Nucl. Phys. B442,472 (1995); P. Petropoulos and J. Rizos, hep-thl9601037 (unpublished). [ll] J. Ellis, J. S . Hagelin, D. V. Nanopoulos, K. A. Olive, and M. Srednicki, Nucl. Phys. B238, 453 (1984).
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[12] DO Collaboration, S . Abachi et al., Phys. Rev. Lett. 76, 2228 (1996). [13] CDF Collaboration, T. Kamon, presented at the High Energy e’e- Workshop, Batavia, Illinois, 1995 (unpublished). [14] J. L. Lopez, D. V. Nanopoulos, X. Wang, and A. Zichichi, Phys. Rev. D 52, 142 (1995); J. White, D. Noman, and T. Goss, DO Note 2395 (unpublished). [15] ALEPH, DELPHI, L3, and OPAL Collaborations, represented by L. Rolandi, H. Dijkstra, D. Strickland, and G. Wilson, “Joint Seminar on the First Results from LEP 1.5,” CERN report, 1995 (unpublished). [16] See, e.g., C.-H. Chen, M. Drees, and J. Gunion, Phys. Rev. Lett. 76,2002 (1996). [17] CDF Collaboration, T. Kamon, presented at the 1996 Moriond Conference (unpublished). [18] DO Collaboration, M. Narain, presented at the 1996 Lathuile Conference (unpublished).
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Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi
SUPERSYMMETRIC PHOTONIC SIGNALS AT THE CERN e'e- COLLIDER LEP IN LIGHT GRAVITINO MODELS
From Physical Review Letters 77 (1996) 5168
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485 VOLUME 77, NUMBER 26
PHYSICAL REVIEW LETTERS
Supersymmetric Photonic Signals at the CERN e'e-
23 DECEMBER 1996
Collider LEP in Light Gravitino Models
Jorge L. Lopez,' D. V. N a n o p o ~ l o s A. , ~ ~Zichichi4 'Department of Physics, Bonner Nuclear Lnb, Rice University, 6100 Main Street, Houston, Texas 77005 'Center for Theoretical Physics, Department of Physics, T e r n A&M Universiry. College Sta!ion, Texas 77843-4242 'Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Mitchell Campus, The Woodlands. Texas 77381 4University and INFN-Bologna, Italy and CERN, 1211 Geneva 23, Switzerland
(Received 30 September 1996) We explore and contrast the single-photon and diphoton 9gnals expected at the CERN e'e- collider LEP 2 that arise from neutralino-gravitino (e'e,yG y + EmiSs)and neutralino-neutralino (e'e,y,y y y + Emisr)production in supersymmetric models with a light gravitino. LEP 1 limits imply that one may observe either one, but not both, of these signals at LEP 2, depending on the eV; values of the neutralino and gravitino masses: single photons for mx ? Mz and mc 5 3 X diphotons for mx 5 Mz and all allowed values of mc. [SOO31-9007(96)01938-21
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PACS numbers: 14.80.Ly,12.60.J~.13.40.Hq
Searches for supersymmetry at colliders take on a new look in the case of models with a very light gravitino, where the lightest neutralino (xp = x) is no longer the lightest supersymmetric particle and instead decays dominantly (in many models) into a photon and the gravitino (i.e., x -, y c ) [l]. The y-7-G effective interaction is inversely proportional to the gravitino mass [2] and yields an observable inside-the-detector decay for rng < I d eV [3]. On the other hand, the gravitino mass cannot be too small, otherwise all supersymmetric particles would be strongly produced at colliders [4,5] or in astrophysical events [6]: rng > 10@ eV appears to be required. Light gravitino scenarios were considered early on [ 1,2] but have recently received considerably more attention because of their natural ability to explain ~ ~ [7]~ via ~ selectron the puzzling CDF ee y y + E T , event or chargino pair production [3,8,9]. Such scenarios have distinct experimental signatures that often include one or more photons, which may be readily detected at the CERN e + e - collider LEP [3,9,10]. Theoretically, light gravitinos are expected in gaugemediated models of low-energy supersymmetry [8], where the gravitino mass is related to the scale of supersymmetry breaking via mg = 6 X lo-' eV (Asusy/500 GeV)'. Special cases of gravity-mediated models may also yield light gravitinos, when the scale of local and global breaking of supersymmetry are decoupled, as in the context of no-scale supergravity [1,9], in which case rng (rnl/2/MP#' M p [ , with m1/2 the gaugino mass scale and p 2 a model-dependent constant. Our discussion here, though, should remain largely model independent. In the light gravitino scenario, the most accessible ,ye -, supersymmetric processes at LEP are e + e y + Emissande+e,y,y y y + Emiss. Thesinglephoton and diphoton processes differ in their dependence on the gravitino mass: the rate for the first process is proportional to rnc2, whereas the second is independent of the gravitino mass. These processes also differ in their kinematical reach: m, < f i vs rn, < ;&. However,
one must also consider their threshold behavior, which for the single-photon process goes as p8 [4], whereas for the diphoton process goes as p3 [ 111, thus compensating somewhar the different kinematical reaches. In this note we explore and contrast the single-photon and diphoton signals at LEP 2. The diphoton process has been considered in detail previously [3,9,10]. The singlephoton process was originally considered by Fayet [4] in the restricted case of a very light photinolike neutralino. This process was revisited in the context of LEP 1, although only in the restricted case of a non-negligible Z-ino component of the neutralino, where the resonant Z-exchange diagram dominates [ 121. We have recently generalized the single-photon calculation to arbitrary center-of-mass energies and neutralino compositions, details of which appear elsewhere [13]. Let us start by considering the limits that LEP 1 imposes on the single-photon process. At f i = M z , this process proceeds dominantly through s-channel Z exchange via the coupling Z - 2 4 , which is proportional to the Z-ino component of the neutralino N12. (In the notation of Ref. [14], the lightest neutralino can be written as y, = ,yp = N[,Y + N[2Z + N13Rp 4- N 4a; or alternatively as ,yp = N l l B N'2@3 + NI3HI - b + N14Ht,
5 168
0 1996 The American Physical Society
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0031 -9007/96/77(26)/5168(4)$10.00
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where Nil = NIIcosOw + N12sinBw and Ni2 = -N11 sin Ow + Nl2 cos Ow.) The nonresonant contributions, s-channel photon exchange and t-channel ZR,L exchange, are negligible unless the Z-ino component of the neutralino is small (N12 < 0.2), in which case one must include all (resonant and nonresonant) diagrams in the calculation. The explicit expression for the cross section in the general case is given in Ref. [13]. Here we limit ourselves to note its dependence on mg and its threshold behavior, which is valid for all values of f i and 0: p ' / m g , all neutralino compositions: a ( e + e - -, where p = (1 - m:/s)'/2. This threshold behavior results from subtle cancellations among all contributing amplitudes and was first pointed out by Fayet [4] in the case of pure-photino neutralinos. Dimensional
,ye)
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VOLUME77, NUMBER26
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analysis indicates that this cross section is of electroweak strength (or stronger) when M i / ( M ; , m i ) 1 or rnc M ; / M ~ ! 10-~eV (or smaller). A numerical evaluation of the single-photon cross section versus the neutralino mass for mc = 1 0 - ~eV is shown in Fig. 1, for different choices of neutralino composition (“Z-ino”: N12 r= 1; “B-ino”: N I I= 1, and photino: NII = l), and where we have assumed the typical result B(,y & = 1 [which assumes a (possibly small) photino admixture]. In the photino case the Z-exchange amplitude is absent ( N i l = 1 * NI2 = 0) and one must also specify the selectron masses which mediate the t-channel diagrams: we have taken the representative values mlR = mEL= 75, 150 GeV. In Fig. 1 we also show [dotted line LNZ (LopezNanopoulos-Zirichi)] the results for a well-motivated one-parameter no-scale supergravity model [9,15], which realizes the light gravitino scenario that we study here. In this model the neutralino is mostly gaugino, but has a small Higgsino component at low values of m x , which disappears with increasing neutralino masses; the neutralino approaches a pure B-ino at high neutralino masses. The selectron masses also vary (increase) continuously with the neutralino mass and are not degenerate (i.e., 1.5maR 2mx). ma, eV in Fig. 1 leads Our particular choice of me = to observable single-photon cross sections for f i > Mz; otherwise the curves scale with l / m & The dashed line indicates our estimate of the LEP 1 upper limit on the single-photon cross section of 0.1 pb [16]. This estimate
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23 DECEMBER 1996
is an amalgamation of individual experiment limits with partial LEP 1 luminosities (-100 pb-’) and angular acceptance restrictions (I cos 0,l < 0.7). Imposing our estimated LEP 1 upper limit one can obtain a lower bound on the gravitino mass as a function o f the neutralino mass, which in some regions of parameter space is as strong as me > lo-’ eV but, of course, disappears for mx > M Z [13]. In gauge-mediated models, such gravitino masses correspond to h s u s y 3 TeV. As of this writing there are no reported excesses in the single-photon cross sections measured at fi > Mz. However, as it is not clear what the actual experimental sensitivity to these processes is, we refrain from imposing further constraints from LEP 1.5 (fi = 130-140 GeV) and LEP 161 (fi = 161 GeV) searches. To stimulate the experimental study of this process, in Fig. 2 we show the single-photon cross sections calculated at f i = 161 GeV. Note that the cross sections increase with increasing selectron masses (saturating at values somewhat larger than the ones shown) and conversely decrease with decreasing selectron masses. This behavior is expected in the limit of unbroken supersymmetry (i.e., for massless selectrons and photinos) the gravitino loses its longitudiI nal spin-5 component, and therefore amplitudes involving it must vanish. This is the case in our calculations, as only I the spin-7 “goldstino” component of the gravitino becomes enhanced for light gravitino masses. Alternatively, coupling is proportional to rn: and the effective e-& the t-channel amplitude goes as & / ( t - ma). showing
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4s = 161 GeV 1
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mX (GeV) FIG.1. Single-photon cross sections (in pb) from neutralinogravitino production at LEP 1 versus the neutralino mass (m,) for me = lo-’ eV and various neutralino compositions. The photino curves depend on the selectron mass (75, 150). The cross sections scale like u a m i 2 . The dashed line represents the estimated LEP 1 upper limit.
mX ( G W FIG.2. Single-photon cross sections (in pb) from neutralinogravitino production at LEP 161 versus the neutralino mass (m,) for m~ = eV and various neutralino compositions. The solid curves have a fixed value for the selectron mass (75, 150). whereas the dotted curve corresponds to a one-parameter no-scale supergravity model, where the selectron masses vary continuously with the neutralino mass. The cross sections scale like u a m i 2 . 5169
487 VOLUME77, NUMBER26
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the dependence on m@ and its saturation for large values of mz; at threshold r 0 and the r-channel amplitude becomes independent of mi;and combines with the other amplitudes to yield the ,B8 threshold behavior [13]. In the case of the one-parameter model (LNZ) a peculiar bump appears. This bump is understood in terms of the selectron masses that vary continuously with the neutralino mass: at low values of mx the selectron masses are light and the cross section approaches the light fixed-selectron mass curves (‘75”); at larger values of mx the selectron masses are large and the cross section approaches (and exceeds) the heavy fixed-selectron mass curves (“150”). This example brings to light some of the subtle features that might arise in realistic models of lowenergy supersymmetry. We now turn to the diphoton signal, which proceeds via s-channel Z-exchange and r-channel selectron ( Z R J ) exchange and does not depend on me. The 2-exchange contribution is present only when the neutralino has a Higgsino component, whereas the t-channel contribution is present only when the neutralino has a gaugino component (the Higgsino component couples to the electron mass). The numerical results for the diphoton cross section at f i = 161 GeV for various neutralino compositions are shown in Fig. 3 and exhibit the expected p 3 behavior [ll]. (In Fig. 3 the Higgsino curve corresponds to the choice N13 = 1, which maximizes the Higgsino contribution. Otherwise the cross section scales as [(N13)’ - (N14)’]*.)In the absence of published LEP 1 limits on the diphoton cross section (especially in the presence of substantial Emiss),we turn to higher LEP energies. Limits on acoplanar photon pairs at LEP 161 have been
2.5
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1.5
.-.-._ Dhotino 11501
0
20
40
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mx (Gev) FIG.3. Diphoton cross sections (in pb) from neuWinoneutralino production at LEP 161 versus the neutralino mass (m,) for various neutraiino compositions. The dependence on the selectron mass is indicated (75, 150) when relevant. The dashed line represents the preliminary LEP 161 upper bound.
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recently released by the DELPHI, ALEPH, and OPAL Collaborations [17], implying an upper bound of 0.4 pb on the diphoton cross section. Imposing this limit on the LNZ model entails mx > 60 GeV, with analogous limits in other regions of parameter space (see Fig. 3). Comparing Fig. 2 with Fig. 3, it is amusing to note that the dependence on the selectron masses is reversed from one case to the other: the single-photon (diphoton) rate increases (decreases) with increasing selectron masses. The former behavior was explained above, the latter behavior is the usual one. The dependence on the neutralino composition is also reversed from one case to the other: Z-ino’s dominate the single-photon rate because of their 2-pole enhancement, B-ino’s have some Z-in0 component and come close, while photinos have no 2-ino component and come in last. The diphoton rate for gauginolike neutralinos proceeds only via r-channel selectron exchange and depends crucially on the coupling of left- and right-handed selectrons to neutralinos, which when examined in detail, explain the relative sizes of the photino, B-ino, and 2-in0 results in Fig. 3. The striking point of this paper is obtained by comparing the single-photon versus diphoton cross sections at, for example, f i = 190 GeV, once the LEP 1 limit on the single-photon cross section is imposed. To exemplify the result we take as a representative example the one-parameter (LNZ) model [9] and plot both cross sections in Fig. 4, for two values of the gravitino mass. For me = eV (top panel), in principle, both the singlephoton ((+;”) and diphoton (a;?) processes may be observable at LEP 190. However, the LEP 1 limit on the can only be satisfied for mx > single-photon rate 85 GeV, and in this region the diphoton process becomes negligible. Thus in this case one may observe only single photons. Increasing the gravitino mass to ameliorate the eV, botLEP 1 constraint on u? (to me = 5 X tom panel) suppresses the single-photon rate at LEP 1 by a factor of (50)’. but it suppresses the single-photon rate at LEP 190 by the same factor, rendering it unobservable. However, the diphoton process at LEP 190 now is allowed for any value of the neutralino mass (consistent with LEP 1 and LEP 1.5 limits), and this time one may observe only diphotons. Requiring a minimum observable single-photon cross section of 0.1 pb, we obtain two mutually exclusive scenarios: single photons for m, 2 Mz and me 5 3 X eV; diphotons for mx 5 Mz and all allowed values of me. We have verified that the same general result holds for the various other neutraho compositions that we have explored above, although in some small regions of parameter space there is a small overlap region where both single-photon and diphoton signals may be simultaneously observable. However, this may only occur for the highest LEP energies and smallest gravitino masses (me eV), and only very near the diphoton kinematical limit, where the diphoton cross section is small.
((+?)
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VOLUME77, NUMBER 26 1G I , , ,
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PHYSICAL REVIEW LETTERS
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LNZ
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has been supported in part by DOE Grant No. DE-FGOS9 1-ER-40633.
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3 100
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FIG. 4. Comparison of single-photon (u$%)versus diphoton (u;:) signals (in pb) at LEP 190 as a function of the neutralino mass, for two choices of the gravitino mass. The dashed lines represent the single-photon cross section ( u y )and upper at LEP 1. The one-parameter LNZ model is limit taken here as a representative example of the two mutually exclusive scenarios that may be realized: either single photons or diphotons may be observed, but not both.
(uFmx)
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We should mention in passing that single-photon signals are also expected at the Tevatron ( p p x G ) , and at even higher rates. However, large instrumental backW ev, with e faking a photon) grounds (e.g., p p may hamper such searches considerably. In sum, we have explored the photonic signals that may be observed at LEP in models with a light gravitino, where single-photon and diphoton signals play a complementary role, and have the advantage over any other supersymmetric signal of the largest reach into parameter space. The work of J.L. has been supported in part by DOE Grant No. DE-FG05-93-ER-40717. The work of D. V. N.
--
[ l ] J. Ellis, K. Enqvist, and D. Nanopoulos, Phys. Lett. B 147, 99 (1984). 121 P. Fayec'Phys. Lett. B 69,489 (1977); 70,46 I (1977); 84, 421 (1979). [31 S. Ambrosanio et al., Phys. Rev. Lett. 76, 3498 (1996); Phys. Rev. D 54, 5395 (1996). [41 P. Fayet, Phys. Lett. B 175,471 (1986). [5] D. Dicus, S. Nandi, and J. Woodside, Phys. Rev. D 41, 2347 (1990); 43,2951 (1991). [6] See, e.g., T. Gherghetta, Report No. hep-ph/9607448; J. Grifolds, R. Mohapatra, and A. Riotto, Report No. hepph/9610458. [7] S. Park, in Proceedings of the 10th Topical Workrhop on Proton-Antiproton Collider Physics. Fermilab, 1995, edited by R. Raja and J. Yoh (AP,New York, 1995). p. 62. [8] S. Dimopoulos, M. Dine, S. Raby, and S. Thomas, Phys. Rev. Lett. 76, 3494 (1996); S. Dimopoulos, S. Thomas, and J. Wells, Phys. Rev. D 54, 3283 (1996); K. Babu, C. Kolda, and F. Wilczek, Phys. Rev. Lett. 77, 3070 (1996). [9] J. L. Lopez and D. V. Nanopoulos, Mod. Phys. Lett. A 11, 2473 (1996); Report No. hep-ph/9608275. [I01 D. Stump, M. Wiest, and C.-P. Yuan, Phys. Rev. D 54, 1936 (1996). [ l l ] J. Ellis and J. Hagelin, Phys. Lett. B 122, 303 (1983). 1121 D. Dicus, S. Nandi, and J. Woodside, Phys. Lett. B 258, 231 (1991). 1131 J.L. Lopez, D.V. Nanopoulos, and A. Zichichi, Report No. hep-ph/9611437. [14] H. Haber and G. Kane, Phys. Rep. 117.75 (1985). [15] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Rev. D 49, 343 (1994); Int. J. Mod. Phys. A 10, 4241 (1995). [16] 0. Adriani et al. (L3 Collaboration), Phys. Lett. B 297, 469 (1992); R. Akers et al. (OPAL Collaboration), 2. Phys. C 65, 47 (1995); P. Abreu et al. (DELPHI Collaboration), Report No. CERN-PPE/96-03. [17] Joint Particle Physics Seminar and LEPC Open Session, CERN, October 8, 1996. Presentations by W. de Boer (DELPHI Collaboration), R. Miquel (ALEPH Collaboration), M. Pohl (L3 Collaboration), and N. Watson (OPAL Collaboration).
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Jorge L. Lopez, D.V. Nanopoulos and A . Zichichi
SINGLE-PHOTON SIGNALS AT CERN LEP IN SUPERSYMMETRIC MODELS WITH A LIGHT GRAVITINO
From Physical Review D 55 ( I 997) 5813
I997
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49 1
PHYSICAL REMEW D
VOLUME 55, NCTMBER 9
1 MAY 1997
Single-photon signals at CERN LEP in supersymmetric models with a light gravitino Jorge L. Lopez* Bonner Nuclear Lob, Depanment of Physics, Rice Universiry. 6100 Main Street, Houston, Texas 77005
D. V. Nanopoulos Center for Theoretical Physics, Department of Physics, Texas A&M Universiv, College Station, Texas 77843-4242 and Astropam'cle Physics Group, Houston Advanced Research Center (HARC). The Mitchell Campus, The Woodlands, Texas 77381
A. Zichichi University and INFN-Bologna, Italy and CERN, I211 Geneva 23, Switzerland
(Received 26 November 1996) We study the single-photon signals expected at CERN LEP in models with a very light gravitino ( m e 5lo-' eV). The dominant process is neutralino-gravitino production ( e + e - + , y g ) with subsequent neu75, giving a y+ Em,,, signal. We first calculate the cross section at arbitrary center-oftralino decay via
x+
mass energies and provide new analytic expressions for the differential cross section valid for general neutralino compositions. We then consider the constraints on the gravitino mass from LEP 1 and LEP 161 single-photon searches, and possible such searches at the Fermilab Tevatron. We show that it is possible to evade the stringent LEP 1 limits and still obtain an observable rate at LEP 2, in particular, in the region of parameter space that may explain the CDF e e y y + E T , h S event. As diphoton events from neutralino pair production would not be kinematically accessible in this scenario, the observation of whichever photonic signal will discriminate among the various light-gravitino scenarios in the literature. We also perform a Monte Car10 simulation of the expected energy and angular distributions of the emitted photon, and of the missing invariant mass expected in the events. Finally we specialize the results to the case of a recently proposed one-parameter no-scale supergravity model. [SO556-2821(97)01809-2] PACS number(s): 14.80.L~.12.60.Jv, 13.40.Hq I. INTRODUCTION Supersymmetric models with a light gravitino ( E ) have been considered for some time [I-51, but interest on them has recently surged [6-81 because of their ability to explain naturally the puzzling e C e - y y + E T , m i s sevent observed by the Collider Detector at Fermilab (CDF) Collaboration [9]. If the gravitino is the lightest supersymmetric particle (LSP),' the next-to-lightest supersymmetric particle [NLSP, typically the lightest neutralino (x),as we will assume here] becomes unstable and eventually decays into a photon plus a gravitino (x+ y e ) [2]. This decay becomes of experimental interest when it happens quickly enough for the photon to be observed in the detector. Because the interaction of the gravitino with matter is inversely proportional to the gravitino mass, the neutralino lifetime will be short enough for a sufficiently light gravitino: m ~ 5 2 5 0eV [7]. On the other hand, the gravitino may not be too light, as otherwise it would be copiously produced leading to distinctive signals at colliders that have not been observed [3,5] or cosmological [ll] and eV. Searches astrophysical [I21 embarrassments: me> at the CERN e'e- collider LEP 1 strengthen this limit to m ~ s 1 0 - 3eV in large regions of parameter space, when m,<Mz [13].
We should emphasize that even though we are encouraged by the natural interpretation of the CDF event within certain light-gravitino scenarios, we believe that such scenarios are interesting in their own right and should be fully explored irrespective of the status of the CDF event. This is the motivation for this analysis. One may parametrize models with very light gravitinos by the relation m c -(rnl12/MPI)PM~I,with m l R the gaugino mass scale and p-2 a model-dependent constant. Specific forms of such mass relation have been obtained in the literature in the context of no-scale supergravity [2,8]. Light gravitinos are also expected in gauge-mediated models of low-energy supersymmetry [6,14], where the gravitino mass is related to the scale of supersymmetry breaking via m ~ - - 6 X l O - ~eV ( h s ~ s + 0 0 GeV)'. However, the presently known models in this category appear unable to accomodate gravitino masses light enough to yield observable single-photon signals. Our analysis of direct experimental limits on the gravitino mass also complements analyses of indirect constraints on from, e.g., cosmological [ I l l and astrophysical considerations [121. Experimental searches for supersymmetry are considerably more sensitive in this type of neutralino-unstable supersymmetric model. First of all, the lightest easily observable supersymmetric channel is no longer a pair of charginos (x'x-), but instead a pair of (the usually lighter) neutralinos or if the gravitino is light enough ( r n ~ s l O -eV) ~ the neutralino-gravitino channel ( ~ 5 ) . These new channels allow a deeper exploration into parameter space. Furthermore, because of the photonic signature\k all supersymmetric processes, it becomes possible to over-
rnc
(xx),
*Permanent address: Shell E&P Technology Company, 3737 Bellaire Boulevard, Houston, TX 77025. 'We assume that R parity is conserved, as otherwise the decay p + G K + may occur at an unsuppressed rate [lo]. 0556-2821/97/55(9)/5813(13)/$10.00
55
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0 1997 The American Physical Society
492 ~
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JORGE L. LOPEZ, D.V. NANOPOULOS, AND A. ZICHICHI
come the loss of experimental sensitivity that occurs when the daughter leptons become too soft (as in chargino pair production when rnxe - mx< 10 GeV or mx+>m ,+rn,= - 3GeV), and therefore absolute lower bounds on sparticle masses become experimentally attainable in this class of models. Indeed, diphoton searches at LEP 161 (i.e., ,6=161 GeV) [I51 have been recently shown [16] to exclude a significant fraction of the parameter space that is preferred by the supersymmetric interpretations of the CDF event within light gravitino modek2 Ongoing runs at LEP 2 should be able to probe even deeper into the remaining preferred region of parameter space. Our purpose here is to consider in detail a complementary signal in light-gravitino models, namely the associated production of gravitinos with neutralinos? which may be observable in couider experiments for rn$ eV. The resulting single-photon signal has been recently shown to be observable at LEP 2 in certain range of gravitino masses, but only when the diphoton signal from neutralino pair production is itself not kinematically accessible [13]. Therefore, experimental observation of whichever photonic signal will provide very useful information in sorting out the various light-gravitino scenarios in the literature. The gravitino mass plays a central role in gravitino-productionprocesses, whose rate is inversely proportional to the gravitino mass squared 2 (Urn,=). In contrast, the precise value of the gravitino mass plays a minor role in the production of the traditional supersymmetric particles, as it determines only the decay length of the neutralino. The neutralino-gravitinoprocess of interest at LEP is
55
FIG.1. Feynman diagrams for neueralino-gravidno production at LEP: (a) s-chaunel y and Z exchange and (b) t-channel selectron exchange (TR,').Additional u-channel diagram not shown. give new analytic expressions for the corresponding differential cross section (Sec. 11). Next we reassess the constraints on the gravitino mass in view of the full LEP 1 data set and imposing the preliminary limits obtained recently from runs at LEP 161 (&=161 GeV), for general neutralino compoWe also comment on the potential of analositions (Sec. gous searches at the Tevatron. We then perform a Monte Car10 simulation of the production and decay processes leading to the single-photon signal and obtain energy (E,) and angular (cos8,) distributions for representative points in parameter space (Sec. IV),and also discuss the missing invariant mass distribution expected in the events. We show that one may evade the LEP 1 limits and still obtain observable single-photon signals at LEP 2, although only when the diphoton signal from neutralino pair production is kinematically inaccessible. Finally we specialize our results to the case of our proposed one-parameter no-scale supergravity model [8,19](Sec. V). Our conclusions are summarized in
a).
Sec. VI.
JI.THE e + e - + X G PROCESS
which provides an experimental handle on the gravitino mass. This process was considered originally by Fayet [3] (in the restricted case of a very light photinolike neutralino) who noted that the P8 threshold behavior in Eq. (1) results from subtle cancellations among all conpibuting amplitudes. Dimensional analysis indicates that this cross section exceeds 2 electroweak strength when M",l(M&nE))s a,& or rnc" a ~ ~ A 4 ~ / M R - eV. In the context of LEP 1, this process was reexamined in the restricted case of a neutralino with a non-negligible 2-ino component, where the resonant Z-exchange diagram dominates [5]. In this paper we first calculate the cross section for the e + e - + X c process at arbitrary center-of-mass energies and
'We should point out that alternative supenymmehic interpretations of the CDF event have been proposed, involving a one-loop radiative decay of the next-to-lightest neueralino [17] or a light axino [18]. Analyses of the impact of LEP searches on the allowed parameter spaces of these models have not yet appeared. exceeds neutralino3Gravitino pair production (e+e--+&?) gravibho production ( e + e - - + x 6 ) only for gravitino masses that have already been excluded experimentally ( m C S eV). The single-photon signal is further suppressed by a factor of LI from the radiated photon.
The Feynman diagrams for neutralino-gravitino associated production at LEP are shown in Fig. 1, and include s-channel y and Z exchange, and t - and u-channel selectron (ITR~L) exchange. At the Z peak one expects the s-channel 2 exchange diagram to dominate, and one may simply calculate the amplitude for Z + x d decay [5]. This result is accurate as long as the neutralino has a non-negligible Z-in0 component However, for photinolike neutralinos the other diagrams become important. This is also the case for any neutralino composition for center-of-mass energies away from the Z peak ( , 6 > M z ) . To deal with all cases at once, we perform the complete calculation of all diagrams contributing to e + e - + x C . We first present the general form ofthe differential cross section and later specialize the result for the particular case of a photinolike neutraho in order to expound on certain theoretical issues and generalizations of our results. In calculating interactions of gravitinos with matter one can proceed in one of two ways. One may calculate with the full couplings in the supergravity Lagrangian, in which case the vertices of interest are given by [1,11]
where the (spin-f Goldstino component of the) gravitino field is ~ , ~ d , + / r n ~ and , M = 2 . 4 X 10l8GeV is the appro-
493 SINGLE-PHOTONSIGNALS AT CERN LEP IN
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. ..
5815
priately scaled Planck mass. Alternatively one may calculate using a set of much-simplified effective Goldstino couplings 131
The full and effective couplings give the same results for the cross sections of processes where the typical bad high-energy behavior of the gravitational amplitudes is cancelled completely by the diagrams involving only gravitinos and regular supersymmetric particles. This is the case for the e + e - + x z process in hand, where we have verified (in the pure photino limit) that both ways of doing the calculation give identical results. For a derivation and explanation of the meaning of the effective couplings see Ref. [113. The simplification of using the effective couplings is not beneficial in other processes, such as gravitino pair production, where diagrams including graviton exchanges must also be included to cancel the bad high-energy behavior of the amplitudes [3,11]. Also, it is not clear whether this simplificationmay be used in the case of broken gauge symmetries, and therefore we have used the full couplings in the case of neutralino composit ions other than pure photino, where the s-channel Z-exchange amplitude must be taken into account. A. General case
The differential cross section for a general neutralino composition is given by
du -- (s-mi) dcose
F(s,t,u) 32~s' 6 ( ~ ~ , - ) 2 '
(4)
where as usual we define
In these expressions we have used the following e-ZR.,-X (XR,L) and e-e-Z ( C R , L ) couplings [20]: The function F ( s , t , u ) receives contributions from each amplitude squared and various interference terms (some of which vanish). In an obvious notation, we 6nd F=F,,+ F,,+ F,,+ Fzz+F,+ F,,+Fz,+Fz,+F,z,
(7) where c,=-1+
sin2ew,
(20)
where Nil and Ni2 denote the photino and Z-ino components of the neutralino, respectively. Indeed, the lightest neutralino may be written in two equivalent ways [20]:
related by
494 5816
JORGE L. LOPEZ, D.V. NANOPOULOS. AND A. ZICHICHI
55 -
combined with the (s - m i ) term from the phase space integration [see Eq. (4)] yields a cross section proportional to p8 with p= The above results were originally obtained by Fayet [3] based on a calculation of the cross section using the effective couplings [Q. (3)], and have been obtained here for the first time using the full couplings [Eq. (2)]. Such an equivalent expression for the cross section makes more evident some further properties of the results, and we thus give it explicitly here too. The expression for F Y using the effective couplings becomes
JG@.
B. Pure photino case This special case is useful in order to expose various subtleties in the calculation that become less apparent (although they are still present) in the case of a general neucase is also important tralino composition. The e + e - + because the result can be readily taken over to the case of gluino-gravitino production in quark-antiquark annihilation at hadron colliders (qF+g^G. In this special case the couplings of the neutralino (pho-
j G
v-
42s(s-rni)+4uts/rn:
Feff- mx
S2
1
+
(rni-t)(-t)
With this relatively simple expression we can verify certain expected behaviors of the cross section. First, in the limit of unbroken supersymmetry rnx+my=O, rn;L,R +rn,-O, s+t+u=rn:+O, one can readily verify from the above equations that FY+ 0. The vanishing of the cross section in this limit is expected as the spin-; component of the gravitino (the Goldstino) becomes an unphysical particle when supersymmetry is unbroken, as it is no longer absorbed by the gravitino to become massive. A related manifestation of this phenomenon can be exposed by studying the threshold behavior of the cross section. The spin-; (Goldstino) component of the gravitino is essentially obtained by taking the derivative of the full gravitino field, thus making the Goldstino couplings proportional to the Goldstino momentum (k') . At threshold kp+ 0 and there is an additional suppression of the cross section besides the kinematical one. Threshold corresponds to the limit s+mi and therefore from Eqs. (5). ( 6) t,u go to zero as (s - m i ) . In the above expression for F ?, one can see that near threshold each term is proportional to which
+ ( m i - u)(-
u)
1
Despite the seemingly different appearances of Feg and F f it can be verified (at least numerically) that they give identical results. Using F~$ it is immediately apparent that the cross section vanishes in the unbroken supersymmetry limit (i.e., rnx,rn,-R.L 4 0 ) . as it should. [Note also that for amassless photino (a case of interest in the early literature) the s-channel diagram does not contribute.] The p8 threshold behavior is not so apparent this time. One can fist note that near threshold FZff becomes independent of rn";p.L and depends only on m i . A little algebra then shows that indeed, near threshold, F & K ( ~ - r n i ) ~and thus the same p8 threshold behavior results, although this time as a result of a cancellation among all of the contributing amplitudes. The F g form is also useful in exhibiting the dependence of the cross section on the selectron masses. As is evident from Eq. (28). the cross section increases with increasing selectron masses, eventually saturating for very large values of rnr. Thus, the decoupling theorem still holds (i.e., large values of the sparticle masses have no effect), although its specific implementation here is rather peculiar. Before moving on to numerical evaluations of the cross sections, let us note that the above expressions for the photino cross section (using either the full or effective couplings) can be adapted very easily to describe gluinogravitino production in quark-antiquark collisions at the Tevatron or CERN Large Hadron Collider (LHC) (qF+EG). In this case the process is mediated by s-channel gluon exchange and t-channel FL,Rexchange. One needs to replace the e - r R , L - X ( X R , L ) couplings in Eqs. (17), (18) by those appropriate for q - r R , L - X , one needs to replace
495 SATCERNLEPIN
...
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gauginolike neutralinos is motivated by the explanation of the CDF evenf which would become rather unnatural for Higgsino-like neutralinos with a very small gaugino admixture. (Otherwise possible Higgsino admixtures weaken the single-photon signal studied here.) In the photino case the 2-exchange amplitude is absent (Nil= l*N;,=O) and one must also specify the selectron masses which mediate the t - and u-channel diagrams; we have taken the representative values m,-R =m,=75,150 GeV. Increasing the selectron L masses further leads to only a small increase in the cross section, e.g., at m,=O one finds 1.48,2.09, 2.36 pb for m,-= 150,300,1000 GeV, signalling the reaching of the decoupling limit for large selectron masses discussed in Sec.
103
102
101
CT?=
100
10-1
IIB.
10-2
FIG. 2. Single-photon cross sections (in pb) from neutrahogravitino production at LEP 1 versus the neutralino mass (m,)for mc= lo-’ eV and various neutralin0 compositions.The “photino” curves depend on the selectron mass (75, 150). The cross sections scale like a m m z 2 . The dashed h e represents the estimated LEP 1 upper limit. Also shown is the result for a one-parameter no-scale supergravity model (LNZ).
the e - e - y coupling [ez in Eq. (24)] by the strong coupling (g:), and one needs to insert the appropriate color factor. Of course the integration over parton distribution functions also needs to be implemented. (A realistic calculation would also include the gluon-fusion channel, which becomes quite relevant at LHC energies.)
IlI.EXPERIhlENTAL CONSTRAINTS A. LEP 1
The single-photon signal ( y+ E-) has been searched for at LEP 1 by various LEP Collaborations [21]. We estimate an upper bound of 0.1 pb on this cross section. This estimate is an amalgamation of individual experimental limits with partial LEP 1 luminosities (- 100 pb-’) and angular acceptance restrictions (lcos6’,/<0.7). Note that the singlephoton background at the 2 peak (mostly from e+e-’ui;y) is quite si@cant, as otherwise one would naively expect upper bounds of order 3/L<0.03 pb. To be conservative, in what follows we apply the 0.1 pb upper limit to our uncut theoretical cross sections. A numerical evaluation of the single-photon cross section at LEP 1 versus the neutralino mass for m6=10-’ eV is shown in Fig. 2, for different choices of neutralino composition (“2-ino:” N;,-l, “B-ino:” N,,= 1, and “photino:” N i l = l ) , and where we have assumed the typical result B(x+ yG) = 1 !We should remark that our emphasis on
-
4N0te that this implies a nonvanishing (possibly small) photino component of the neutralino, as would be required in the “Z-ino” case discussed above.
In Fig. 2 we also show (dotted line Lopez-NanopoulosZichichi (LNZ)] the results for a well-motivated oneparameter no-scale supergravity model [8,19], which realizes the light gravitino scenario that we study here. In this model the neutralino is mostly gaugino, but has a small Higgsino component at low values of m,, which disappears with increasing neutralino masses; the neutralino approaches a pure B-ino at high neutralino masses. The selectron masses also vary (increase) continuously with the neutralino mass and are not degenerate (i.e., m;,- 1.5m;R-2m,). This figure makes apparent the constraint on the gravitino mass that arises from LEP 1 searches: in some regions of parameter space one must require m c 2 1 0 - ~eV if m,<MZ. To make this result more evident, in Fig. 3 we display the lower bound on the gravitino mass versus the neutralino mass that results from the imposition of our estiM mated upper bound ayZ
Recent runs of LEP at higher center-of-mass energies have so far yielded no excess of single photons over standard model expectations. The latest searches at &=161 GeV have produced upper limits on the single-photon cross section a:61slpb [22]. We have evaluated the single-photon cross sections for the neutralino compositions used in Fig. 2 at &=161 GeV. This time all cases depend on the choice of selectron masses. The numerical results are shown in Fig. 4, with the experimental upper bound denoted by the dashed line. Note that this line extends only for m,>MZ, as for m,<MZ, the much stronger limits discussed in the previous section apply. Moreover, for mE= eV (as used in Fig. 4). LEP 1 limits require m x F M z . As the figure makes evident, for m,<MZ the sensitlvity to single-photon signals at LEP 2 is not competitive with that at LEP 1. As discussed above, the cross sections in Fig. 4 increase with increasing selectron masses (saturating at values somewhat larger than the ones shown), and conversely decrease with decreasing selectron masses. The choice of selectron masses also affects the near-threshold behavior of the cross section, with light selectron masses “delaying” the onset of the /I8threshold dependence (see Fig. 4). Note also that the photino, B-ino, and 2-in0 cross sections become comparable
496 JORGE L. LOPEZ, D. V. NANOPOULOS,AND A. ZICHICHI
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Lower limit on rnG (eV) 10-3
10-3
10-4
10-4
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“-r””
10-6
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0
50
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0
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100
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10-3
10-4
photino 10-4
10-5 10-5
10-6
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m
x
(GeV)
m
x (GeV)
FIG.3. Lower bounds on the pvitino mass (in eV) as a function of the neutralino mass ( mx) that result from single-photon searches (y+E,,,,) at LEP 1 and LEP 161. In the “photino” case at LEP 1 and the “photino,” “Z-Lo,” and “B-ino” cases at LFP 161, the selectron mass influences the results. We have chosen m,-=75, 150, 300 GeV, denoted by solid, dashed, and dotted lines,respectively. Also shown are the bounds in a one-parameter no-scale supergravity model (LNZ). above the 2 pole, when the 2-exchange diagram becomes comparable to the other diagrams. In the case of the oneparameter model (LNZ) a peculiar bump appears. This bump is understood in terms of the selectron masses that vary continously with the neutralino mass: at low values of m, the selectron masses are light and the cross section approaches the light fixed-selectron mass curves (75); at larger values of m, the selectron masses are large and the cross section approaches (and exceeds) the heavy fixed-selectron mass curves (150). This example brings to light some of the subtle features that might arise in realistic models of low-energy supersymmetry. In spite of their apparent weakeness, LEP 161 limits on the single-photon cross section are useful in constraining the gravitino mass in a neutralino-mass range inaccessible at LEP 1. Indeed, decreasing the gravitino mass in Fig. 4 by a factor of 3 will make the cross sections some ten-times larger. The resulting lower bounds on the gravitino mass from LEP 161 searches are shown in Fig. 3. This figure shows that, as expected, LEP 1 limits dominate for 5 M Z . However, because of the b8threshold behavior at s = M Z , LEP 161 limits “take over” for neutralino masses slightly below M Z ,and in the “photino” case, considerably below M,.
7
C. Single photons versus diphotons
It has been made apparent in Fig. 3 that for m , s M z the gravitino mass is constrained to m c S l O - ’ eV. If this was indeed an absolute requirement on the gavitino mass (is., for all values of m,) then the cross section for neutralinogravitino production at LEP 2 would be highly suppressed: Fig. 4 shows that c+i6’slpb for m ~ = l O -eV ~ and u,. -2 “ m c . In other words, if the minimum observable singlephoton cross section at LEP 2 is -0.1 pb (i.e., for L-IOpb-’), then m 6 2 3 X 1 0 - 5 eV appears to be the limit of the sensitivity of LEP 2. On the other hand, the process e+e--+,yx-+ Y ~ + E is~ ~ , sensitive to m, M Z , in which case diphotons may not be observed simultaneously (as they require &>2 M Z = I90 GeV). This dichotomy between single-photon and diphoton signals at LEP was first presented in Ref. [131.
497 SINGLE-PHOTON SIGNALS AT CERN LEP IN . . .
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(e+e-+
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5819
tional light and strongly interacting particles (S and P), an assumption that depends on the detailed name of the mechanism that leads to a very light gravitino.
161 GeV
Bin0 (150)
E. New channels
Another set of channels of interest at the Tevatron consist of the associated production of gravitinos with neutralinos or charginos
2.0
photino (150)
FIG. 4. Single-photon cross sections (in pb) from neutralinogravitino production at LEP 161 versus the neutralino mass (m,) for m c = eV and various neutralin0 compositions. The solid curves have a fixed value for the selectron mass (75, 150). whereas the dotted curve corresponds to a one-parameter no-scale supergravity model where the selectron masses vary continuously with the neutralino mass. The cross sections scale like v a m j ' . me preliminary LEP 161 upper limit is indicated. D. Other limits
The above lower limits on m c are rather significant and improve considerably on previous limits from collider experiments [3-5,211 and astrophysical considerations [12], as long as mx<Mz. There has also been a recent reassessment [23] of the hadron collider limits obtained via associated and via indirect gluino-gravitino production (pF+&, gluino pair production (pF+gX where in addition to the usual supersymmetry QCD diagrams the gravitino is exchanged in the t and u channels. The multijet signature of these processes has been contrasted with experimental limits from the most recent Tevatron run to show that if gluino pair production is accessible at the Tevatron (i.e., eV rem p 2 0 0 GeV) then a lower limit of m p 3 X sults. This limit and our limits in Fig. 3 may be compared by relating the gluino and neutralino masses, as occurs in supergravity models with universal gaugino masses at the unifica(as tion scale m,-3mx~-6mx. Therefore, m ~ 5 2 0 GeV 0 required for the bound in Ref. [23] to apply) corresponds to m,S35 GeV. Consulting Fig. 3, we see that the Tevatron limits are stronger in this neutralino mass range. However, the LEP 1 (161) limit extends up to mx5Mz(2Mw), which corresponds to m+550 (960) GeV, which is far from the direct reach of the Tevatron. By considering the further processes p F + g S , g P , where S and P are very light scalar and pseudoscalar particles associated with the gravitino, the lower bound on the graviho mass becomes much less dependent on the gluino mass and eV [23]. This lower bound can be taken to be mg>3 X is comparable with those obtained above by considering LEP 1 data. However, this result assumes the existence of addi-
which have the advantage over pF-+rc of much less phase space suppression. The most basic channel is qT+xG, which leads to a y+E,,,,, signal. The cross section for this process can be readily obtained from the expressions given in Sec. II by replacing the initial state electron-positron pairs by quark-antiquark pairs, the exchanged selectrons by squarks, and by integrating the resulting expression over parton distribution functions. We have estimated this cross section and find that it may be quite significant: up to 85,25,15 pb for mx=50, 75, 100 GeV and mg= lo-' eV, in favorable regions of parameter space. In the best case scenario of a Tevatron upper limit of 0.1 pb (i.e., 10 events in L= 100 pb-'), one may conclude that m,3(3, 1.6, 1.2) X eV for m x = 50,75,100 GeV. Taken at face value, these limits are quite competitive with those obtained in Ref. [23]. At the moment there are no single-photon limits available from CDF nor DO. To improve the visibility of the signal, one may want to consider the qT+x=Z channel which, depending on the chargino decay channel, may lead to /" + y + ET,,,, or 2 j + y+E,,,,, signals. The leptonic signal appears particularly promising. For all these processes there are some important instrumental backgrounds that need to be overcome. For instance pF+ W+ e v, , where the electron is misidentified as a photon (i.e., because of limitations in tracking efficiency), leading to a very large "single-photon" signal aB(W+eve)=2.4X l@pb [24], which may be reduced significantly by optimizing the tracking efficiency and making suitable kinematical cuts. The other channels mentioned above face similar, although perhaps less severe, instrumental backgrounds ( e.g., WW+e+"y"+ET,miss).
rV. TEIE SINGLE-PHOTON SIGNAL The total cross section for neutralino-gravitino production has been displayed in Fig. 4 for a specific center-of-mass energy (&=161 GeV) and for some illustrative choices of parameter values. The analytic expressions given in Sec. IIA. allow one to calculate these cross sections for arbitrary values of the parameters. In this section we would like to explore some characteristics of the actual signal, i.e., the energy and angular distributions of the observable photon and the missing invariant mass distribution in the events. We should note that there is a different kind of singlephoton signal that arises for gravitino masses 100 eV in neutralino pair production. Because the neutralinos are fairly long lived in this case, only one neutralino may decay to a photon inside the detector [7]. This kind of heavier-gravitino single-photon signal may be distinguished from our present
-
498 JORGE L. LOPEZ,D. V. NANOPOULOS, AND A. ZICHlCHl
5820
,
55
q=ioCI GeV. &=lQO
1.0 b-,
0.2 0.0
I
, ,
I
I
, ,
,GeV, , , (B-ino) , , ,
light-gravitino signal by the presence of a visible displaced vertex where the neutralino decayed.
, A
0-5
cosex
1
(B-ino)
L .
FIG. 5. Single-photon cross sections (in pb) from neumlinogravitino production at LEP 190 versus the neutralino mass (m,) for rnc= lo-' eV and various neutralino compositions. The solid curves have a Exed value for the seleceon mass (75, 150). whereas the dotted curve corresponds to a one-parameter no-scale supergravity model where the selectron masses vary continously with the neutralin0 mass. The cross sections scale like (rmm:*.
,
K ,, , , , , , , , , , , , , , y -1 -0.5 0 m,.=125 GeV. &=lQO GeV
0.6
I
-1
-0.5
0
.
O5
cos8,
1
FIG. 6 . Normalized angular distribution of neuhdinos of B-ino composition in neutralino-gravitino production at LEP 190 for mx= 100, 125, 150 GeV and m,-=75 (solid), 150 (dashed), 300 (dots) GeV.
a "homemade" Monte Carlo event generator. We start in the rest frame of the decaying neutralino, where we generate y + C events that are isotropic in this reference frame. Energy-momentum conservation requires E t =' ; 1 I, = fm,, which leaves two components of the photon momentum to be generated at random (i.e., ); . We then boost the photon momentum back to the laboratory frame using the neutralino four-momentum ( E x, p x ) , whose components are constrained by the kinematics of the e + e - + , y G process:
m x > M Z (to avoid the stringent LEP 1 lower limits on m c ), we concentrate on the following three neutralino mass choices: mx= 100, 125, 150 GeV. To gain some insight into the final distributions, we star&by displaying the normalized neutralino angular distributions (l/cr)(da/dcos6Jx) as a function of cosSx for mg= 75, 150,300 GeV, and B-ino (Fig. 6). Z-in0 (Fig. 7). and photino (Fig. 8) neutraho compositions. The total cross sections for each of the curves can be read off Fig. 5 and for convenience have been tabulated in Table I. As can be seen from Figs. 6 , 7, and 8, the angular distribution varies quite a bit with neutralino mass, although mostly for light selectron masses. Note that the angular distributions always remain finite, and generally show a preference for the central region.
Here we have two components of the neutralino momentum unconstrained (cos6Jx,rjx). For fixed values of these angles we obtain E , and cosSy distributions, which are purely kinematical effects. The observable distributions are obtained by varying (cos6Jx,4x)and weighing these kinematical distributions with the corresponding dynamical (l/a)(du//dcos6Jx) factors calculable from the expressions given in Sec. II. In what follows we focus on the case of LEP 190 (&=190 GeV). First we display in Fig. 5 the total cross sections for single-photon production at this center-of-mass energy. These should give an idea of the reach in neutralino masses that may be accessible at LEP 190. As we expect neutralino-gravitino production to be allowed only for
The observable photonic energy and angular distributions can be quite unwieldly once we allow for the many choices of parameters that we have considered above. An examination of all parameter combinations shows that both energy and angular distributions are largely insensitive to the neutralino composition, being much more sensitive to the mass parameters (i.e., mx,mF). This result is perhaps not surprising as the observable distributions of relativistic particles are dominated by kinematical effects which depend crucially on the mass parameters. Thus, for brevity we show only the result in the B-ino case which, in any event, is representative of typical supergravity models. The energy (E,) and angular (cos0,) distributions for mx= 100,125,150 GeV are shown 9 and Fig. 10, respectively, for in Fig. m ~ = 7 5 , 1 5 0 , 3 0 0GeV. These distributions are obtained by
A. Monte Carlo technique
Our simulation proceeds in a standard way, making use of
-
B. Energy and angular distributions
499 55 -
SINGLE-PHOTON SIGNALS AT CERN LEP IN . . .
5821
1+=100 GeV, &=l90 GeV
f
zx
j
5a
8
-:
h
L V
V
-1
-0.5 0 q = 1 2 5 GeV, &=la0 GeV
cosex 1 (Z-ino)
f 4
0
0.8
:
0.2 0.0
4
3 a
h
h
Y
x
5 V
-1 -0.5 0 nt(=150 GeV, 6 = 1 9 0 GeV
0.5
-0.5
0
0.5
codx
OS
COS8,
1
(photino)
1.0 0.8 0.6
0.4 0.2 0.0
case,
(z-ino)
-1
(photino)
. . . . . . _ _ _ _ .....-- - -
~~
q = 1 2 5 GeV, &=190 GeV 0 "
3 a
1.0 0.8 0.8 0.4 0.2 0.0
-1
-0.5
q = 1 5 0 GeV, &=l90
x
0 GeV
1
(photino)
0
0
a
0
4 3 a
5a
h
h
:
v
0.4 0.2
: n"."n -1
Y
-0.5
0
cos8,
1
FIG 7. Normalized angular distribution of neutralinos of 2-in0 composition in neutralino-gravitino production at LEP 190 for mx= 100, 125, 150 GeV and m,-=75 (solid), 150 (dashed), 300 (dots) GeV.
FIG. 8. Normalized angular distribution of neutralinos of photino composition in neutralino-gravitino production at LEP 190 for mx= 100, 125, 150 GeV and m,-=75 (solid), 150(dashed), 300 (dots) GeV.
generating a total of 100 K events, which are then binned in 2 GeV E , bins (Fig. 9) and 0.1 cos6, bins (Fig. 10). For simplicity, the histograms have been scaled by a factor of 0.01, thus roughly corresponding to 1 K generated events. mx The energy distributions (Fig. 9) show a si-&cant and m,- dependence. As the neutralino mass grows, it tends to produce harder photons. In fact, it is not hard to show that in the decay y c , with a neutralino energy and momentum as in Eq. (30), the photon energy is restricted to the interval
as faithfully reproduced in the simulations. [Near threshold (mxthe photon carries away half of the center-of-mass energy.] These distributions show that any given singlephoton candidate energy (E,) implies an upper bound on the possible neutralino masses consistent with the candidate event,
6)
x--'
TABLE I. Total cross sections corresponding to the differential cross sections shown in Figs. 6,7,8 at LEP 190. All masses in GeV, all cross sections in pb. ~~
composition
m,
m,-=75
m,-=150
m,-=300
B-in0
100 125 150
0.34 0.32 0.20
0.54 0.19 0.04
1.61 0.60 0.1 1
Z-ino
100 125 150
0.19 0.10 0.06
0.49 0.17 0.03
1.12 0.42 0.08
Photino
100 125 150
0.37 0.37 0.23
0.52 0.18 0.04
1.64 0.61 0.12
The photonic angular distributions (Fig. 10) are peaked in the forward and backward directions, even more so as the neutralino becomes heavier. The selectron mass has an interesting effect. In the case of mx= 100 GeV, from Fig. 6 we see that for heavy selectron masses the neutralino angular distribution is fairly flat, and therefore the photonic distributions should reflect only kinematical effects, as they do (i.e., peaked in the forward and backward directions). For light selectron masses, the neutralino avoids the forward and backward directions, and the kinematical effect on the photons is diminished. C. Missing mass distribution
The dominant background to the neutralino-gravitino signal is a single radiative return to the Z: e ' e - 4 yZ- yvV, where the photon is radiated off the initial state and the Z boson tends to be on shell. The most distinctive signature for this background process appears in the missing invariant distribution, which is strongly mass M,,=J= peaked at M m i s s - M Z .
500 JORGE L. LOPEZ, D. V. NANOPOULOS,AND A. ZCHICHI
5822
m,=100 GeV
55 -
m,=l50 GeV
FIG.9. Photonic energy distributions in neutralino- (B-ino-)gravitino production at LEP 190 for m x = 100, 125, 150 GeV and m,-= 75 (solid), 150 (dashed), and 300 (dots) GeV. (Corresponds to 1 K events binned in 2 GeV Ey bins.) The missing mass distribution for the signal can be easily determined, as in this case the missing energy and missing momentum are given by Emis=
pmiss= 1 -Prl=Ey,
&-E,.,
(33)
Mfiss=
-l.
(34)
The allowed range of M d s s is obtained by inserting the range of photonic energies in Eq. (31); we obtain (35)
and therefore
m x = l O O GeV
m,=125 GeV
60
50
40
30 20
10 L
0
-1
A
u -1 1
0 -0.5
0
0.5
cos By
1
-0.5
0
0.5
cos 8,
FIG. 10. Photonic angular distributions in neutralino- (B-ino-)gravitino production at LEP 190 for mx= 100,125, 150 GeV and m,-=75 (solid), 150 (dashed), and 300 (dots) GeV. (Corresponds to 1 K events binned in 0.1 cos8, bins.)
501 55 -
SINGLE-PHOTON SIGNALS AT CERN LEP IN
Histograms showing missing mass distributions fall in the range specified by Eq. (35) and otherwise favor the upper end of the Mmiss range (corresponding to the lower end of the photonic energy range). For brevity, we display these distributions only in the one-parameter model example discussed in Sec. V below. We note in passing that the complementary diphoton events from neutralino pair production have a dominant background (e+e-+yyZ-+yyvg that also peaks for M f i s s - M z [7]. In this case the missing mass in the &photon signal varies from zero up to a maximum value of f( &+J-1, in contrast with the result for the singlephoton signal in Eq. (35). The Mdss distributions of the single-photon and diphoton signals differ not only in range, but also in shape.
...
5823
0.55
0.50
1 0.40
-1
"
"
I
-0.5
"
"
It should have become clear from the discussion in Sec. have a wide range of characteristics because of the variations in the underlying parameters describing the neutralinogravitino process. In reality, the model of supersymmetry that describes nature will have all its mass parameters correlated in some way, and the actual observations may be a bewildering composite of the many curves shown above. To exemphfy this situation, in this section we specialize our results to the case of the one-parameter no-scale supergravity model that has been mentioned at various places in the preceding discussion. The motivation, construction, and experimental consequences of this model have been expounded on in detail elsewhere [19,8]. Perhaps here it would be fit to just mention that from the point of view of unified supergravity models with universal soft-supersymmetry breaking parameters at the unification scale, consistency conditions within the model require that almost all of these parameters vanish (i.e., mo=Ao=Bo=O), leaving the universal gaugino mass ( m l n ) as the seed of supersymmetry breaking. This choice essentially determines the spectrum of sparticle masses and the various mass relations that have been commented on above. This model also requires that the gravitino be the lightest supersymmetric particle, thus leading to the photonic decay signature for sufficiently light gravitinos. We have already shown the single-photon cross sections for the LNZ model at LEP 1, LEP 161, and LEP 190 as the dashed lines in Figs. 2, 4, 5 , respectively. The cross sections for &>Mz show a peculiar bump that, as discussed in Sec. IIIB, can be traced back to the fact that the selectron masses vary with the neutralino mass. As a first step towards obtaining the angular and energy photonic distributions, in Fig. 11 we show the normalized neutralino angular distributions ( l/u)(du/dcos~x) for &=190 GeV and m x = 6 0 , 80,100,120,140 GeV. The total cross sections in each of these cases are u= 1.2, 1.3, 1.0, 0.6,0.2 pb. Note how relatively flat the angular distributions are: no more than a 10% variation. This is to be contrasted with the wide range of variability observed in the generic cases shown in Figs. 6, 7, and 8. In fact, the results in the one-parameter model resemble those in the generic models when the selectron mass is large (i.e., the dotted
'
'
' I
0.5
"
'
'
cos@,
V. ONE-PARAMETER MODEL EXAMPLE
IV,that the signals to be searched for experimentally can
"
FIG. 11. Normalized angular distribution of neutralinos in neumho-gravitino production at LEP 190 for m,=60,80,100, 120, 140 GeV in a one-parameter no-scale supergravity model.
lines in those figures, which correspond to m p= 300 GeV) . This is to be expected as in the one-parameter model one has m 1.5m.;2m,, indicating increasingly heavier selectrons. Following the method outlined in Sec. IV,in Figs. 12 and 13 we display the photonic energy and angular distributions at LEP 190 for three representative neutraho masses ( m x = 100,120,140 GeV). The energy distributions show the same restrictive photon energy behavior as predicted by Eq. (31). The angular distributions are also peaked in the forward and backward directions. Finally we consider the missing mass distributions, which are obtained from Eq. (34), and are shown in Fig. 14. We note the range of M f i s s , as prescribed by Eq. (35), and the tendency to favor missing mass values toward the upper end of the allowed interval. For the neutralino mass choices
;,-
1 " " I " " l " " &=190GeV 140
LNZ
i
il
10
20
40
-
00
.-
80
100
.^
FIG. 12. Photonic energy distributions at LEP 190 in a oneparameter no-scale supergravity model for mx= 100, 120, 140 GeV. (Corresponds to 1 K even& binned in 2 GeV Ey bins.)
502 55 -
JORGE L. LOPEZ, D.V. NANOPOULOS, AND A. ZICHICHI
5824
&=
I " " I " " I " "
I " " I " "
LNZ
190 GeV
20-
6=190GeV
140
I LNZ
... ..:
120
140 -1
-0.5
0.5
I
cos ey FIG. 13. Photonic angular distributions at LEP 190 in a oneparameter no-scale supergravity model for m x = 100, 120, 140 GeV.
shown, the missing mass shows a distinct preference to be larger than M,. (This is in contrast with the M- distribution in diphoton events, which is more evenly distributed.) VI. CONCLUSIONS
In this paper we have attempted to study in some detail the physics of supersymmetric models with a gravitino light enough that it can be produced directly at collider experiments: rn,+ eV accompanied by neutralinos with nonnegligible gaugino admixture. Our discussion has centered mainly around LEP, from where the strongest constraints can be obtained at the moment. We have nonetheless outlined the corresponding program to be followed at the Tevatron, where instrumental backgrounds make identification of the single-photon signal a more challenging task. We have provided new and explicitly analytical expressions for neutralino-gravitino differential cross sections at e'e- colliders and have discussed some of the theoretical
[l] P. Fayet, Phys. Lett 69B,489 (1977); 70B,461 (1977); 84B,
421 (1979); 86B,272 (1979). [2] J. Ellis, K Enqvist, and D. Nanopoulos, Phys. Lett. 147B,99 (1984); see also 151B,357 (1985). [3] P. Fayet, Phys. Lea 175B.471 (1986). [4] D. Dicus, S. Nandi, and J. Woodside, Phys. Rev. D 41,2347 (1990); 43,2951 (1991). [5] D. Dicus, S. Nandi,and J. Woodside, Phys. Lett. B 258, 231 (1991). 161 D. Stump, M. Wiest, and C.-P. Yuan, Phys. Rev. D 54, 1936 (1996); S. Dimopoulos, M. Dine, S. Raby, and S. Thomas, Phys. Rev. Lett 76,3494 (1996). [7] S. Ambrosanio, G. Kane, G. Kribs, S. Martin, and S. Mrenna, Phys. Rev. Lett. 76, 3498 (1996); Phys. Rev. D 54, 5395 (1996).
FIG. 14. Missing invariant mass distributions at LEP 190 in a one-parameter no-scale supergravity model for m y = 100, 120, 140 GeV. (Correspondsto 1 K events binned in 2 GeV Mmi, bins.) subtleties involved in the calculation and some of the peculiar parameter dependences of the cross section. We have used our expressions to obtain new lower bounds from LEP 1 data on the gravitino mass for m x < M z . Weaker limits from LEP 2 are obtained at higher neutralino masses. Our study includes a Monte Car10 simulation of the single-photon signal, which should be helpful in the experimental analyses that are just now getting underway. We have also specialized the results to our one-parameter no-scale supergravity model, where the signals can be analyzed much more simply because of the tight correlations between the model parameters. ACKNOWLEDGMENTS
J.L. would like to thank G. Eppley, T. Gherghetta, and T. Moroi for useful discussions. The work of J.L. has been supported in part by U.S.DOE Grant No. DE-FG05-93-ER40717 and that of D.V.N. by U.S. DOE Grant No. DE-FG059 1-ER-40633.
[8] J. L. Lopez and D. V. Nanopoulos, Mod. Phys. Lett.A 11, 2473 (1996); Phys. Rev. D 55, 4450 (1997). [9] S. Park in Proceedings of the 10th Topical Workshop on Proton-Antiproton Collider Physics, Fermilab, 1995, AIP Conf. Proc. No. 357, edited by R. Raja and J. Yoh (AIP,New York 1995). p. 62. [lo] K. Choi, E. Chun, and J. Lee, Report No. hep-phl9611285 (unpublished). [ l l ] See, e.g., T. Moroi, Ph.D. thesis, 1995; T. Gherghetta, Nucl. Phys. B485, 25 (1997). [12] See, e.g., J. Grifolds, R. Mohapatra, and A. Riotto,Report No. hep-phl9610458 (unpublished); Report No. hep-phl9612253 (unpublished). [13] J. L. Lopez, D. V. Nanopoulos, and A. Zichichi, Phys. Rev. Lett.77,5168 (1996).
503 55
SINGLE-PHOTON SIGNALS AT CERN LEP IN . . .
[14] S. Dimopoulos, S. Thomas, and J. Wells, Phys. Rev. D 54, 3283 (1996); K. Babu, C. Kolda, and F. Wilczek, Phys. Rev. Lett. 77, 3070 (1996). [15] Joint Particle Physics Seminar and LEPC Open Session, CERN, 1996 (unpublished). Presentations by W. de Boer (DELPHI Collaboration). R. Miquel (ALEPH Collaboration), and M. Pohl (L3 Collaboration) http://hpl3sn02.cern.ch/, N. Watson (OPAL Collaboration) http://www.cem.ch/OpaV. [16] J. Ellis, J . L. Lopez, and D. V. Nanopoulos, Report No. hep-pW9610470 (unpublished). [I71 S. Ambrosanio, G. Kane, G. Kribs, S . Martin, and S. Mrenna, Phys. Rev. D 55, 1372 (1997). [18] J. Hisano, K. Tobe, and T. Yanagida, Phys. Rev. D 55, 411 (1997).
5825
[19] 1. L. Lopez, D. V. Nanopoulos, and A. Zichichi,Phys. Rev. D 49, 343 (1994); Int. 1. Mod. Phys. A 10,4241 (1995). [20] H. Haber and G. Kane, Phys. Rep. 117, 75 (1985). [21] L3 Collaboration. 0.Adriani et al., Phys. Lett. B 297, 469 (1992); OPAL Collaboration, R. Akers er al., Z. Phys. C 65, 47 (1995); DELPHI Collaboration P. Abreu et al., Report No. CERN-PPU96-03 (unpublished). [22] W. de Boer, in Joint Particle Physics Seminar and LEPC Open Session, CERN, 1996 [15]. [23] D. Dicus and S. Nandi, Report No. hep-pM9611312 (unpublished). [24] DO Collaboration, S. Abachi et a[.,Phys. Rev. Lett. 75, 1456 (1995); CDF Collaboration F. Abe et al., ibid. 76, 3070 (1996).
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505
Jaewan Kim, Jorge L. Lopez, D. V. Nanopoulos, Raghavan Rangarajan and A. Zichichi
LIGHT-GRAVITINO PRODUCTION AT HADRON COLLIDERS
From Physical Review D 57 ( I 998) 373
I998
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507 PHYSICAL REVIEW D
VOLUME 57, NUMBER 1
1 JANUARY 1998
Light-gravitino production at hadron colliders Jaewan Kim,'** Jorge L. Lopez?+ D. V. Nanopo~los,'.~.~ Raghavan Rangarajan,'-O and A. Zichichi4 'Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Mitchell Campus, The Woodlands, Texas 77381 'Bonner Nuclear Lab, Department of Physics, Rice University, 6100 Main Street, Houston, Texas 77005 'Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, Texas 778434242 and Academy of Athens, Chair of Theoretical Physics, Division of Natural Sciences 28 Panepistimiou Avenue, 10679 Athens, Greece 4University and lNFN, Bologna, ltaly and CERN, 1211 Geneva 23, Switzerland (Received 14 July 1997; published 4 December 1997) We consider the production of gravitinos (G) in association with gluinos (3 or squarks (3 at hadron and qg+FC?, and gg+gG. These channels colliders, including the three main subprocesses: qF+&, become enhanced to the point of being observable for sufficiently light gravitino masses (m;<10-4 eV), as motivated by some supersymmetric explanations of the Collider Detector at Fermilab e e y y + Er,mas event. The characteristic signal of such events would be monojets, as opposed to dijets obtained in the more traditional supersymmetric process p F + g g . Searches for such events at the Fermilab Tevatron can impose lower limits on the gravitino mass. In the appendixes, we provide a complete set of Feynman rules for the gravitino interactions used in our calculation. [SO556-2821(98)02101-81
PACS number(s): 12.60.J~.04.65.+e, 14.80.L~
I. INTRODUCTION
(G)
Supersymmetric models where the gravitino is the lightest supersymmetric particle have been considered in the literature for some time [I-31. More recently they have enjoyed a resurgence motivated by new models of dynamical supersymmetry breaking [4] and by possible explanations [5,6] of the puzzling Collider Detector at Fermilab (CDF) e e y y + E T , m i s s event [7]. Most of the recent effort has been devoted to studying the new signals for supersymmetry via electroweak-interaction processes, that accompany such scenarios by virtue of the newly allowed x - y G decay (i.e., diphoton signals from e+e--+,yX and analogous signals in ppproduction of a variety of supersymmetric channels) [461, or via direct gravitino production (i.e., single photon signals from e + e - - X G [8,9] or pp=t&,x"G [9]). Much less emphasis has been placed on strong-interaction processes at hadron colliders [3,10]. An important part of this phenomenological effort should be directed at obtaining direct experimental information on the gravitino mass.However, this information will not come from the kinematical effects of such a very light particle, but instead from the dynamical effect that each interaction vertex involving gravitinos is inversely proportional to the gravitino mass. Lower bounds on the gravitino mass from singlephoton searches at LEP already exist, but are limited by kinematics at the CERN e + e - collider LEP 1 [8,9] and by dynamics at LEP 2 [9]. Limits on the gravitino mass based
*Electronic address: [email protected] 'Present address: Shell E&P Technology Company, Bellaire Technology Center, P. 0. Box 481, Houston, TX 77001-0481. Electronic address: [email protected] *Electronic address: [email protected] 'Electronic address: [email protected] 0556-2821/97/57(1)/373(lO)/SlO.OO
57 -
on monojet and multijet final states at the Tevatron are obtained in Ref. [lo]. Limits also exist from astrophysical and cosmological considerations [111. These limits are further revised in Ref. [12,13]. More stringent constraints from the Z decay width have also been obtained in Ref. [13]. The purpose of this paper is to examine the manifestation of light gravitinos in the context of hadron colliders.' Two gravitino-mass-dependent processes involving the strong interactions come to mind
_ -
PP-+gg,
(1)
The first process (pF-+gg)proceedsvia the usual supersymmetric QCD diagrams, but receives in addition contributions from new gravitation-induced diagrams involving exchange of gravitinos in the f and u channels [lo]. For sufficiently eV), the gravitationlight gravitino masses ( m c s induced contributions dominate the supersymmetric QCD ones, otherwise the reverse is true. This process is also kinematically constrained at the Tevatron to m,-s 250 GeV. The second process (pF*gE,
'For definiteness, in what follows we will concentrate on the Fercollider, although the discussion applies with milab Tevatron little modification to the case of the CERN Large Hadron Collider (LHC) ( P P ) too.
(pa
373
0 1997 The American Physical Society
508 JAEWAN KIM et al.
374
57 -
FIG. 1. Feynman diagrams for subprocess q+@. p1 and p 2 are incoming momenta and p s and p 4 are outgoing momenta. Arrows on scalars indicate direction of momentum and flow of particle flavor.
FIG. 2. Feynrnan diagrams for subprocess qg+FE. p , and p 2 are incoming momenta and p s and p 4 are outgoing momenta. Arrows on scalars indicate direction of momentum and flow of particle
X l O W 5 ) eV, the second process has a larger cross section even within the kinematical reach of the first process (mg 5250 GeV). Some of our results disagree with those of Ref. [lo]. It is also important to consider the gluino decay modes. For sufficiently light gravitinos (m,-S10-3 eV [14]) these are dominated by g-+gc; otherwise they proceed in complicated ways of which E-+qFx is an example. We then obtain a partitioning of the ( mc, m c) space into four regions of “low or high” values of the parameters, roughly delimeV and mg-250 GeV. Note that in each ited by rncof these regions we expect a signal dominated by a different number of jets. These facts are summarized in the following sketch:
numerical results. In the appendixes we provide a derivation of the interaction Lagrangian involving gravitinos, stating clearly the conventions we use. We include the Feynman rules relevant to our calculation and also provide comparisons with others using different conventions. In what follows we have not included processes involving the scalar partners of the goldstino (S and P). Such processes can also give rise to monojet signals in a hadron collider, if the S and P are light, and shall be addressed in future work. Limits on the gravitino mass that may be obtained using our results will thus be conservative. Processes involving gravitinos suffer unitarity violation at high energies due to the nonrenormalizability of the supergravity Lagrangian [151. To preserve unitarity until energies
low m;,
high mE.
flavor.
(3)
The sparse literature on this subject contains explicit expressions only for the gg initiated subprocesses (gg+gT,& [lo] and phenomenological analyses [3,10] only for the left half (i.e., “low m i ’ )of the above table. The above comments and processes generally apply to squarks as well, with gluinos replaced by squarks and gluons replaced by quarks ( g d q ) . A sample squark decay for heavier gravitinos is F-+q,y. In Sec. II we present the cross sections for different channels at a hadron collider giving a monojet signal and an outgoing gravitino. In Sec. III we present a discussion of our
FIG. 3. Feynrnan diagrams for subprocess gg-+FE. p I and p z are incoming momenta and p s and p 4 are outgoing momenta. Arrows on scalars indicate direction of momentum and flow of particle flavor.
509 57 -
LIGHT-GRAVITINO PRODUCTION AT HADRON COLLIDERS
315
eV. of the order of a few TeV typically requires mC2 Hence, we shall take eV as our lower bound on the gravitino mass in what follows.
II. ANALYTICAL RESULTS In ppcollision there are three parton sub-processes giving rise to a single jet along with a gravitino. As we mention above, the jet is initiated by either an outgoing gluino or squark and the undetected gravitino is associated with missing transverse energy. The different parton subprocesses are qq+Eg, qg-+FG, and gg+& Below we present the differential cross section d&/dt for each subprocess. The Feynman diagrams for each sub-process are provided in Figs. 1, 2, and 3, respectively. The hadronic cross section u(S)is obtained by convoluting the subprocess cross section Gi,j of partons i,j with parton distribution functionsfi(x,Q). (Because of the relation between Mandelstam variables s, t and u , the partonic cross sections below may be expressed differently.) The differential cross section for the subprocess qq+rZ is given by
4t2u(s- m2-) qj 4u2s(s-m2-) qj +
-
+
3s[u -m i ]
3 s [ t - m2-1
-z j=1,2
4st3
3 s[t - m -21
]
-z j=1.2
4su3 3S~u-rn2-1 ’
qj
(4)
“j
Above M = (87rGN)-1’2,where GN is Newton’s gravitational constant. The sums over j run over the two squark mass eigenstates. Left-right squark mixing is not relevant for t h ~ schannel (nor are they for the other two channels), as explained in Appendix C. The total partonic cross section is obtained by integrating the above from - (s - m i ) to 0. We have verified that ow above result agrees with the differential cross section obtained in the second reference in Ref. [9] for e + e - + X G (modulo color factors), where x represents a photino. The differential cross section for the subprocess qg+qjG is given by G= 1,2 represents the two squark mass eigenstates)
I
4stu(s - m2-) ‘‘j +
4t2U2
3 [ t- m2- I [ U - m i ]
3 sz
qj
qj
+ 3 s[t8 t-3 Um2- 1- 3 s 4StU2 [ u -m i ] qj
The above cross section is new and has not been obtained elsewhere in the literature. The total partonic cross section is obtained by integrating the above from - (s-rn?) to 0. When obtaining the total hadronic cross section for this channel we qj
sum over both squark masses and then multiply the result by a factor of 2 to include the subprocess g q + c E , The differential cross section for the subprocess g g - + E Gis given by d&ldt=
1 4 X 64X 1 6 ~ s ’
24g:
[
8(s-rni)sru 3s’
+-i 4
t3(u+mi)
3
[t-mi]2
The total partonic cross section is obtained by integrating the above from -(s-m;) to 0. The above differential cross section disagrees with the result in Ref. [lo] by a factor of fi. However, if one replaces the definition of K =(4.rrGN)’” in Ref. [ l o ] by the “standard” definition K = (8 aGN) In then our results agree. While obtaining the above differential cross sections one must be careful while summing over polarization states of the incoming gluon(s). We have used
+( m p s )
c,
h=1,2
+
t (t - m i ) u ( u -
mi)
[t-m;]’
E’(PI.X)E”(PI
,A)=
c,
E’(PzrX)EY(P2.X)
h=1 . 2
2
= -g’v+
;(Pl;P;+P;Pp5). (7)
where p1.2 are the momenta of the incoming particles [16].
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(For example, for q g - @ , we pick the incoming quark momentum as p 2 . ) For g g + F c one could have alternatively used
a,b,c,d: m@(5,2,1,0.5)~10-~eV
and included the contribution of ghosts as in Ref. [16]. The hadronic cross section corresponding to any parton subprocess is given by
300
100
Above i, j run over all partons, valence and sea, that participate in the subprocess. x i = p , l P i is the ratio of the parton momentum to the hadron momentum. fi,the center of mass energy of the Tevatron, is 1.8 TeV. 7 = x 1 x z , ~ ~ = 2 r n ? Sfor the q q a n d g g channels, and ~ ~ = 2 r n ' - /for S the q g channel. gj For the q q a n d g g channels we set the factorization scale Q in the parton distribution functions f i to be the gluino mass, while for the q g channel we set Q to be the squark mass. We set the renormalisation scale p equal to mZ and use the world average a,(mz) = 0.118 in the modified minimal subtraction scheme.
Ill.EXPERIMENTAL PREDICTIONS The three parton-level cross sections given in the previous section have been integrated over the parton distribution functions as indicated above. We assume both squark masses are equal for -simplicity. We start with the gg initiated process: g g - g ' c , shown by the dashed lines in Fig. 4, as a function of nz,-, for a few choices of the gravitino mass (the cross section scales with For reference we also show the gluino pair-production cross section obtained in Ref. [ 101 (solid lines), which gives the order of magnitude of the traditional supersymmetric signals at hadron colliders. For low gluino masses the latter process dominates whle for larger gluino masses the single gluino cross section dominates. Here we disagree with Ref. [ 101 that the p p = ' g g cross section is greater than the cross section for p ~ - +via ~ gluon g fusion for nicz 1 0 - ~eV. The gluino-gravitino channel may also proceed from a quark-antiquark initial state (qT-+Fc).The hadronic cross section in this case depends on the squark mass in addition to the gluino mass. These are shown in Fig. 5 for m c eV and for various choices of the squark mass (solid lines), and also for the special case of mb=m,- (dotted line). In this figure we also show (dashed line) the additional contribution to this channel discussed above (i.e., from g g - + F G ) .It is evident that the qq=tKg channel generally dominates over the g g + & channel for nz,-Z200GeV. This is at variance with the observation in Ref. [lo] that
mi2).
400
500
FIG. 4. Hadronic cross sections at the Tevatron that arise from the parton-level processes g g - E (solid lines) and g g - g g (dashed lines), as a function of rnp for the indicated choices of the
gravitino mass. gluon fusion is the dominant subprocess. As noted in Ref. [lo], the p F - + g T cross section via ghon fusion is fairly independent of the gravitino mass for mc>10-5 eV (see Fig. 4). Therefore, as the pF+Fg cross section scales with -2 n z ,~ we may conclude from Fig. 5 that the latter process dominates over the former for gravitino masses as high as m p 0 . 3 - 1.OX eV, depending on the gluino (and squark) mass. [Note that the results of Ref. [ 101 that we use for a ( p F - + g x include only the gluon fusion subprocess.] The last channel to consider is that which originates from
102
101
100
10-1
10-2
200
400
800
800
1000
FIG.5. Hadronic cross sections at the Tevatron that arise from the parton-level process qq=t& (solid lines), as a function of mp for the indicated values of the squark mass, and of the gravitino mass (cross section scales as rn:2). Also shown for comparison are the corresponding cross sections via the gg-& (dashed line) and gg+gg
(dot-dashed line) subprocesses.
511 51
LIGHT-GRAVITWOPRODUCTION AT HADRON COLLIDERS
A,
processes is beyond the scope of this paper, and in fact has been considered previously in the literature [3,10].These authors have shown that on imposing suitable cuts, a sizeable signal may be observable over background. They then use limits on the multijet cross section at the Tevatron to exclude certain regions of the (rn,-,mg) parameter space. Here we simply comment that our signal calculations can exceed those in Ref. [lo], and therefore one could expect an even more detectable signal than previously anticipated. The task of determining the actual observable signal is best left to the experimentalists. We hope that our calculations of the total cross sections will help in deciding whether these signals are worth pursuing in earnest.
a,b,c: mg=200,500,1000
~
311
1000
FIG.6 . Hadronic cross sections at the Tevatron that arise from the parton-level processes qg,qgiqG,q'YG, as a function of m, for the indicated values of the gluino mass and of the gavitino mass (cross section scales as m a 2 ) . the parton-level processes qg+qG, qg-+pG. me corresponding cross section is shown in Figs. 6 and 7. For squark masses greater than 500 GeV and gluino masses less than 500 GeV the qgchannel dominates over the q g channel. However for low squark masses the qg channel dominates. Putting all light-gravitino signals together, one sees that these cross sections are higher than the traditional gluino pair-production one for gravitino masses as high as m g -0.3- 1.OX eV, depending on the gluino (and squark) eV, these cross sections have a kinemass. For r n p matical reach in gluino/squark mass about twice as deep as the traditional gluino pair-production one. This reach decreases quickly with increasing gravitino mass, being surpassed by the traditional process for mg-10-4 eV and higher. The study of standard model backgrounds to the above
ACKNOWLEDGMENTS
The work of J.K. and R.R. has been supported by the World Laboratory. The work of J.L. has been supported in part by U.S. DOE Grant No. DE-FG05-93-ER-40717 and that of D.V.N. by U.S. DOE Grant No. DE-FGO5-91-ER40633. J.K. and R.R. would like to thank Takeo Moroi for many useful conversations. J.K. would like to thank J. Bagger for useful clarifications with respect to Ref. [ 171. APPEMlIX A: COVARIANT DERNATIYES
We find it necessary to include the exact recipe for obtaining the Feynman rules from the relevant terms in the supergravity Lagrangian, since there are many conventions available in standard references. Problems arise when one uses Feynman rules from different sources without properly adapting them to a single scheme. For example, the conventions of Refs. [17-191 all differ from each other. For the future convenience of the reader, as well as ours, we provide a self-consistent set of Feynman rules and present the detailed steps involved in obtaining them, along with necessary comparisons with other references. Our goal is to provide a set of Feynman rules for interactions involving gravitinos that is consistent with Ref. [18]. We define our covariant derivative as
D,=d,fig,TAA;,
(All
with commutation relation of SU(3) generators defined as
[ T A,T B ]= ifABCTC,
(A21
which lead to the tensor field
F;,,= %,At- %yA:-gsfABCA:AAf. These definitions follow those in Ref. [20], and differ from those in other texts. For example, in Ref. [21] the authors elect to use anti-Hermitian T matrices, while Refs. [22] and [23] use Hermitian T's which, however, are negative of what we use. These different conventions result in different Feynman rules. For example, the Feynman rules for the quarkquark-gluon vertex are
FIG.7. Hadronic cross sections at the Tevatron that arise from the parton-level processes qg,qg+qG,TG, as a function of rnp for the indicated values of the squark mass and of the gravitino mass (cross section scales as m i 2 ) .
- ig,TAA;,
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g,TAA$,
( Fq"= ( I , - a').
(A6)
in Refs. [20], [22, 231, and [21], respectively. Similarly, the Feynman rule for the three-gluon vertex depends on the conventions used.
(B10)
The generators of the Lorentz group in the spinor representation are given by
APPENDIX B: CONVENTIONS Throughout the article, we use the flat space metric of $"=diag(
+ 1,- 1,-
1,- 1).
We may form a numerically invariant tensor
(B 1) up:
(up),,j=( I ,ui), i = 1.. .3,
(B2)
With this choice of y matrices, Dirac spinors contain two Weyl spinors,
that transforms as a vector in O( 1,3) using Pauli matrices u',
while Majorana spinors contain only one: These are the Clebsch-Gordan coefficients which relate the ( f , $ )of SL(2,C) to the vector of 0(1,3). Here dotted indices transform under the (0,;) of the Lorentz group, while those with undotted indices transform under the (f,O) conjugate representation. Spinors with upper and lower indices are related through the e tensor: *,=
*a= E % p ,
where the antisymmetric tensor
E'S
%=( We define
Xm p)
J as
034) Projection operators are defined accordingly:
are normalized as
035)
which holds true for &-tensors with dotted indices. The advantage of this scheme is that the mixed tensors are symmetric: & aB&@y=
sy,.
037)
These tensors are used to raise and lower the indices of the u matrices: ( Y am) -&
u
rrp& LIP ( U " ) @ j .
(B8)
It is then straightforward to relate two-component spinors to four-component spinors through the realization of the Dirac y matrices:
where
APPENDIX C: LAGRANGIAN AND FEYNMAN RULES We start with the general supergravity Lagrangian given in Chap. XXV of Ref. [17] and adapt it to the conventions listed in Appendix B. (Note that Ref. [17] uses the flat space metric diag(-l,l,l,l), amongst other differences.) Below we explicitly write down the terms of the Lagrangian relevant to our calculation:
513 379
LIGHT-GRAVI'I"0 PRODUCTION AT HADRON COLLIDERS
FIG.8. Feynman rules relevant to the processes discussed in the text. Arrows on scalars indicate direction of momentum, where relevant, and flow of particle flavor. Arrows on Majorana fermions also indicate direction of momentum, where relevant. Rules for primed diagrams are for Hermitian conjugates. Alternate rules for (a), (b), (e), (f), (g), and (h) using the gravitino couplings of Ref. [19] are provided in Table I. where g i j * is the Kihler metric. A"s are scalar superpartners of chiral fermion x i ' s , FA are the usual field strength tensors of the gauge fields u$ whose superpartners are gauginos hA, and $, is the gravitino field. Covariant derivative 5,'s are
5 , A i = d,Ai- g,u;XiA,
-
.
A
~,x'=~,x'-gsu,
axiA
x',
where the Killing vectors X i A are defined in terms of the
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JAEWAN KIM et al.
380
TABLE I. Alternate Feynman rules for gravitino couplings of Ref. [19] using our conventions. The index in parentheses refers to the diagram in Fig. 8. The rules for (e), (0,(el), and (f'), which were not included in Ref. [19], are unchanged from Fig. 8. ~
~~
~
~
~~
~~~
Feynman rule
Vertex Quark-squark,-gravItno (a)
-(m3 - m 3Y ' , "1
(cos BPL+sin OP,)
4flMm3n quark-squark2-gravitino (b)
quark-squark,-gluon-gravitino(e) c p g ( c o s ePL+sin ep,)
-2gs
flMm3n quark-squark2-gluon-gravitino(0
-2gs
c p g ( c o s ePR-sin BP,)
fiMm3.Q gluon-gluino-gravitino (8)
mh &Mm
3,
PP[ ye YPI r' 7
-im,
gluon-gluon-gluino-gravitino (h)
2&4m3n
g J A S C [Yp
9
Yml .J
quark-squark,-gravitino (a' )
quark-squark,-gravitino (b' )
quark-squark,-gluon-gravitino(e')
quark-squark,-gluon-gravitino(f')
gluon-gluino-gravitino (g' )
gluon-gluon-gluino-gravitino (h')
Killing potential D A . For the minimal K&er potential K =A'*A', the Killing vectors and the Killing potential take
where superscript ( D ) is for Dirac spinors, and (M) for Majorana spinors. The extra i's in gauginos are introduced following Ref. [18]. After addmg appropriate mass terms to Eq. (Cl), we obtain the Lagrangian in four-spinor notation: (C3)
L=Lo+L , + L,+ Lc3+ L4+L5 + L,+ Lc,+L*, (C7) where Lo includes kinetic terms
5 15
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LIGHT-GRAVITINOPRODUCTION AT HADRON COLLIDERS
and others include three- and four-particle interaction terms
38 1
;
components, and it interacts with matter as a massless goldstino with derivative couplings. Thus we may use an effective Lagrangian with a massless Goldstino (I, by making the substitution
After simple rearrangements, terms (C13)-(C15) become
from which our Feynman rules will be derived. Note that an extra factor of 2 is necessary in the Feynman rules to account for the Majorana nature of gauginos and gravitinos. Mixing between left- and right-squarks is proportional to the mass of their quark counterpart. In our calculation of cross sections at the Fennilab p - pcollider, the effect of mixing is practically nonexistent. Therefore we replace A, and AR with the mass eigenstates A and A,. Relative signs between terms within Eqs. (C13)-(C15) are related to the definition of the fermions in Eqs. (C5) and (C6), and to the fact that AR’s transform as antitriplets of S U ( ~ ) , ,i.e., their generators are - T ~ * instead of TA (see pp. 208 and 223 of Ref. [18]). Terms (C13)-(C15) include interactions with gravitinos. If the gravitino is very light, which is the scenario we are pursuing, spin-; components of the gravitino decouple from the spin-
where subscripts 1, 2, G , and g of the momenta are for squark,, squarkz, gravitino, and gluino, respectively. Resulting Feynman rules are listed in Fig. 8. For the sake of completeness, we include the effects of left-right squark mixing in the Feynman rules, where O= 0 implies no mixing. In Sec. 4.5 of Ref. [19], the author uses the on-shell condition of external particles to replace derivatives in the Lagrangian by masses of external particles. This replacement is valid for internal particles as well as off-shell contributions cancel, which is characteristic of the effective Lagrangian. In Table I we provide the Feynman rules for gravitino couplings obtained by applying this prescription. Our rules appear different from those in Ref. [19] because of the different conventions that we use (note the differences in the definitions of yp and ?/S, and gluinos). Note that the four-point interaction quark-squark-gluon-gravitino was overlooked in Ref. [19].
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Hadron Collider Workshop, Aachen, Germany, 1990, edited by G. Jarlskog and D. Rein (CERN Report No. 90-10, Geneva, Switzerland, 1990), p. 681. [15] T. Bhattacharya and P. Roy, Phys. Lett. B 206, 655 (1988); NUCl. PhyS. B328, 469 (1989); B328, 481 (1989); R. C a d buoni, S. de Curtis, D. Dominici, F. Feruglio, and R. Gatto, Phys. Lett. B 216, 325 (1989). [16] J. Babcock, D. Sivers, and S. Wolfram, Phys. Rev. D 18, 162 (1978), see Appendix B. [17] J. Wess and J. Bagger, Supersymmetry and Supergravity, 2nd ed. (Princeton University Press, Princeton, 1992).
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[18] H. Haber and G. Kane, Phys. Rep. 117, 75 (1985). [19] T. Moroi, Ph.D. thesis, Tohoku University, Japan. [20] I. J. R. Aitchison and A. J. G. Hey, Gauge Theories in Parficle Physics, 2nd ed. (Adam Hilger, Philadelphia, 1989). [21] C. Itzykson and J. Zuber, Quantum Field Theory (McGrawHill, New York, 1985). [22] S . Pokorski, Gauge Field Theories (Cambridge University Press, Cambridge, England, 1987). [23] T. Cheng and L. Li, Gauge Theory of Elementary Particle Physics (Oxford University Press, Oxford, 1984).
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LISTOF PUBLICATIONS ON SEARCHING FOR THE SUPERWORLD BY A. ZICHICHI AND COLLABORATORS
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5 19
LISTOF PUBLICATIONS ON SEARCHING FOR THE SUPERWORLD BY A. ZICHICHI AND COLLABORATORS
SUPERSYMMETRY AND SU(2), x U( A. Zichichi Proceedings of the XVI Course of the International School of Subnuclear Physics: “The New Aspects of Subnuclear Physics”, Erice, Italy, 31 July-1 1 August 1978, Plenum Press - The Subnuclear Series 16 (1980) 763. NEWDEVELOPMENTS IN ELEMENTARY PARTICLE PHYSICS A . Zichichi Plenary Lecture given at the Closing Session of the 4th General Conference of the EPS on “Trends in Physics”, York, UK, 25-29 September 1978, La Rivista del Nuovo Cimento 2-14 (1 979) 1. DESERTS AREOF THE IMAGINATION A . Zichichi Introductory Lecture in Proceedings of the EPS International Conference on “HighEnergy Physics”, Geneva, Switzerland, 27 June-4 July 1979, CERN (1980) 3. THEEFFECTIVEEXPERIMENTAL CONSTRAINTS ON M,,,, F . Anselmo, L. Cifarelli, A. Peterman and A. Zichichi I1 Nuovo Cimento 104 A (1991) 1817.
AND M,
THEEVOLUTION OF GAUGINO MASSES AND THE SUSY THRESHOLD F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi II Nuovo Cimento 105 A (1992) 581. THE CONVERGENCE OF THE GAUGE COUPLINGS AT E,, FOR a,(M,) AND SUSY BREAKING F . Anselmo, L. Cifarelli, A. Peterman and A. Zichichi I1 Nuovo Cimento 105 A (1992) 1025.
AND
ABOVE:CONSEQUENCES
THE SIMULTANEOUS EVOLUTIONOF MASSESAND COUPLINGS: CONSEQUENCES SUPERSYMMETRY SPECTRA AND THRESHOLDS F. Anselmo, L. Cifarelli, A. Peterman and A. Zichichi I1 Nuovo Cimento 105 A (1992) 1179. SCALE AT TWO LOOPS ANALYTIC STUDY OF THE SUPERSYMMETRY-BREAKING F . Anselmo, L. Cifarelli, A. Peterman and A. Zichichi II Nuovo Cimento 105 A (1992) 1201.
ON
520 A STUDY OF THE VARIOUS APPROACHES TO MG, F. Anselmo, L. Cifarelli and A. Zichichi I1 Nuovo Cimento 105 A (1992) 1335.
AND acVT
LEP DATA, A X2-TEST TO STUDY THE a,, a2,a3CONVERGENCE FOR HIGH-PRECISION IN MIND THE SUSY THRESHOLD HAVING F. Anselmo, L. Cifarelli and A. Zichichi I1 Nuovo Cimento 105 A (1992) 1357. TROUBLES WITH THE MINIMAL s u ( 5 ) SUPERGRAVITY MODEL Jorge L. Lopez, D.V. Nanopoulos and A . Zichichi Physics Letters B 291 (1992) 255. UNDERSTANDING WHERE THE SUPERSYMMETRY THRESHOLD SHOULD BE A . Zichichi Proceedings of the Workshop on “Ten Years of SUSY Confronting Experiments”, CERN, Geneva, Switzerland, 7-9 September 1992, CERN-PPE/92-149, CERN/LAA/MSL/92-017 (7 September 1992), and CERN-TH.6707/92 - PPE/92- 180 (November 1992) 94.
CLOSING REMARKS A . Zichichi Proceedings of the Workshop on “Ten Years of SUSY Confronting Experiments”, CERN, Geneva, Switzerland, 7-9 September 1992, CERN-TH.6707/92 - PPE/92- 180 (November 1992) 1026. PROPOSED TESTS FOR MINIMALs u ( 5 ) SUPERGRAVITY AT FERMILAB, GRANSASSO, SUPERKAMIOKANDE AND LEP Jorge L. Lopez, D.V. Nanopoulos, H . Pois and A. Zichichi Physics Letters B 299 (1993) 262. LEP LOWERBOUNDON THE LIGHEST SUSY HIGGSMASSFROM RADIATIVE ELECTROWEAK BREAKING AND ITS EXPERIMENTAL CONSEQUENCES Jorge L. Lopez, D.V. Nanopoulos, H . Pois, Xu Wang and A . Zichichi, Physics Letters B 306 (1993) 73. IMPROVED
AND STRING VACUA: ON A CLASS OF FINITE SIGMA-MODELS EXTENSION A. Peterman and A . Zichichi I1 Nuovo Cimento 106 A (1993) 719.
A
SUPERSYMMETRIC
WHEREWE STANDWITH THE REALSUPERWORLD A. Zichichi Proceedings of the XXX Course of the International School of Subnuclear Physics: “From Superstrings to the Real Superworld”, Erice, Italy, 14-22 July 1992, World Scientific - The Subnuclear Series 30 (1993) 1.
52 1 OF s u ( 5 ) AND s u ( 5 ) X u(1): A CRITICAL ASSESSMENT AND THE SUPERWORLDS OVERVIEW Jorge L. Lopez, D.V. Nanopoulos and A . Zichichi Proceedings of the XXX Course of the International School of Subnuclear Physics: “From Superstrings to the Real Superworld”, Erice, Italy, 1 4 2 2 July 1992, CERN-TH.6934193, ClT-TAMU-34/93, ACT-13/93 Revised, hep-ph/9307211 (June 1993), and World Scientific - The Subnuclear Series 30 (1993) 3 11.
SUPERSYMMETRY TESTSAT FERMILAB: A PROPOSAL Jorge L. Lopez, D.V. Nanopoulos, X u Wang and A . Zichichi Physical Review D 48 (1993) 2062. THEABDUSSALAMDREAM A . Zichichi Proceedings of the Conference on “Highlights of Particle and Condensed Matter Physics (Salamfest)”, Trieste, Italy, 8- 12 March 1993, CERN/LAA/93-32/a (21 October 1993), and World Scientific - Salamfestschrift: A Collection of Talks (1993) 186. SUSY SIGNALS AT DESY H E M I N THE NO-SCALEFLIPPEDs u ( 5 ) SUPERGRAVITY MODEL Jorge L. Lopez, D.V. Nanopoulos, X u Wang and A . Zichichi Physical Review D 48 (1993) 4029. SPARTICLE A N D HIGGS-BOSON l’RODUCTlON AND DETECTIONAT CERN LEP 11 IN TWO SUPERGRAVITY MODELS Jorge L. Lopez, D.V. Nanopoulos, H. Pois, X u Wang and A . Zichichi Physical Review D 48 (1993) 4062. FRONTIERS IN PHYSICS A . Zichichi Invited Lecture presented at the TWAS - X Anniversary, ICTP, Trieste, Italy, 2 November 1993, CERN/LAA/93-32 (2 November 1993). FIRSTCONSTRAINTS ON s u ( 5 ) X u(1) SUPERGRAVITY FROM TRILEFTONSEARCHES AT THE TEVATRON Jorge L. Lopez, D.V. Nanopoulos, Gye T. Park, Xu Wang and A . Zichichi CERN-TH.7 107/93, CPT-TAMU-72/93, ACT-25/93, CERN-LAA/93-42, hep-ph/93 12206 (November 1993). TOWARDS A UNIFIED STRING SUPERGRAVITY MODEL Jorge L. Lopez, D.V. Nanopoulos and A . Zichichi Physics Letters B 319 ( 1993) 45 1.
522 STATUS OF THE SUPERWORLD FROM THEORY TO EXPERIMENT Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi Progress in Particle and Nuclear Physics 33 (1994) 303. SIMPLEST, STRING-DERIVABLE, SUPERGRAVITY MODEL AND PREDICTIONS Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi Physical Review D 49 (1994) 343.
ITS
EXPERIMENTAL
STRONGEST EXPERIMENTAL CONSTRAINTS ON s u ( 5 ) X u(1) SUPERGRAVITY MODELS Jorge L. Lopez, D.V. Nanopoulos, Gye T. Park and A. Zichichi Physical Review D 49 (1994) 355. A SEARCH FOR EXACT SUPERSTRING VACUA A. Peterman and A. Zichichi
I1 Nuovo Cimento - Note Brevi 107 A (1994) 333. A LAYMAN’S GUIDETO SUSY GUTS J.L. Lopez, D.V. Nanopoulos and A. Zichichi La Rivista del Nuovo Cimento 17-2 (1994) 1. PROOF OF THE EQUIVALENCE BETWEEN DOUBLESCALING LIMITAND FINITE-SIZE SCALING HYPOTHESIS A . Peterman and A. Zichichi I1 Nuovo Cimento - Note Brevi 107 A (1994) 507. s u ( 5 ) X u( 1): A STRING PARADIGM OF A TOE AND ITS EXPERIMENTAL CONSEQUENCES Jorge L. Lopez, D.V. Nanopoulos and A . Zichichi Proceedings of the 26th Workshop of the INFN Eloisatron Project: “From Superstring to Supergravity”, Erice, Italy, 5- 12 December 1992, CERN-TH.6926/93, CPT-TAMU-33/93, ACT-12/93 Revised (June 1993), and World Scientific -The Science and Culture Series 7 (1994) 284. BASICRESULTS AND EXPERIMENTAL PREDICTIONS FROM SUSY Jorge L. Lopez, D.V. Nanopoulos and A . Zichichi Proceedings of the International Workshop on “Recent Advances in the Superworld”, Texas A&M and HARC, The Woodlands, Texas, USA, 14-16 April 1993, World Scientific (1994) 108. UNIFICATION OF STANDARD MODELGAUGECOUPLINGS: MEANINGS, STATUS AND PERSPECTIVES A . Zichichi Proceedings of the XI11 Course of the International School of Cosmology and Gravitation: “Cosmology and Particle Physics”, Erice, 3- 14 May 1993, EMCSC/93-04, CERN/LAA/93-35 (25 November 1993), and Plenum Press - NATO AS1 Series C 427 (1994) 301.
523 SCRUTINIZING SUPERGRAVITY MODELSTHROUGH NEUTRINOTELESCOPES Raj Gandhi, Jorge L. Lopez, D.V. Nanopoulos, Kajia Yuan and A. Zichichi Physical Review D 49 (1994) 3691.
NEW h E C l S I O N ELECTROWEAK TESTSOF s u ( 5 ) X u(1) SUPERGRAVITY Jorge L. Lopez, D.V. Nanopoulos, Gye T. Park and A. Zichichi Physical Review D 49 (1994) 4835. THE TOP-QUARK MASSIN s u ( 5 ) X u ( 1) SUPERGRAVITY Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi Physics Letters B 327 (1994) 279.
SUPERSYMMETRY IN UNDERGROUND LABSAND NEUTRINOTELESCOPES F. Anselmo, L. Cifarelli, D.V. Nanopoulos, A. Peterman and A. Zichichi Proceedings of the 6th International Symposium on “Neutrino Telescopes”, Venice, Italy, 22-24 February 1994, CERN-PPE/94-95, C€T-TAMU-35/94, ACT- 12/94, CERN-LAAI94- 16 (28 June 1994), and Tipografia Franch - University of Padua, Italy (1994) 69. DARKMATTERCOMPACTIFICATION R. Gandhi, I . Giannakis, D.V. Nanopoulos and A. Zichichi CTP-TAMU-12/94 (March 1994). EXPERIMENTAL ASPECTSOF s u ( 5 ) X u(1) SUPERGRAVITY Jorge L. Lopez, D.V. Nanopoulos, Gye T. Park, Xu Wang and A. Zichichi Physical Review D 50 (1994) 2164. NEW PHENOMENA IN THE STANDARD NO-SCALESUPERGRAVITY MODEL S. Kelley, Jorge L. Lopez, D.V. Nanopoulos and A. Zichichi CERN-TH.7433/94, CPT-TAMU-40/94, ACT- 13/94, MIU-THP-69/94 (August 1994), hep-ph/9409223 (5 September 1994). WHERECANSUSY BE? A. Zichichi
Proceedings of the XXXI Course of the International School of Subnuclear Physics: “From Supersymmetry to the Origin of Space-Time”, Erice, Italy, 4-12 July 1993, World Scientific -The Subnuclear Series 31 (1995) 218. EXPLICITSUPERSTRING VACUAI N A BACKGROUND OF GRAVITATIONAL WAVESAND DILATON A. Peterman and A. Zichichi II Nuovo Cimento 108 A (1995) 97. NEW CONSTRAINTS ON SUPERGRAVlTY MODELSFROM b + Sy Jorge L. Lopez, D.V. Nanopoulos, X u Wang and A. Zichichi Physical Review D 51 (1995) 147.
524 EXPERIMENTAL CONSEQUENCES ON ONE-PARAMETER NO-SCALE SUPERGRAVITY MODELS Jorge L. Lopez, D. V. Nunopoulos and A . Zichichi International Journal of Modern Physics A 10 (1995) 4241. CONSTRAINTS ON NO-SCALE SUPERGRAVITY MODELS S. Kelley, Jorge L. Lopez, D.V. Nunopoulos and A . Zichichi Modern Physics Letters A 10 (1995) 1787. A LIGHTTOP-SQUARK AND ITS CONSEQUENCES AT HIGHENERGY COLLIDERS Jorge L. Lopez, D.V. Nunopoulos and A . Zichichi Modern Physics Letters A 10 (1995) 2289. THEBOTTOM-UP APPROACH TO SUPERSYMMETRY A . Zichichi Proceedings of the XXXII Course of the International School of Subnuclear Physics: “From Superstring to Present-Day Physics”, Erice, Italy, 3-1 1 July 1994, EMCSC/94-02 (20 November 1994), and World Scientific - The Subnuclear Series 32 (1995) 60. SUPERSYMMETRY DILEFTONS AND TRILEFTONS AT THE FERMILAB TEVATRON Jorge L. Lopez, D.V. Nunopoulos, X u Wung and A . Zichichi Physical Review D 52 (1995) 142. STRING NO-SCALESUPERGRAVITY MODELAND ITS EXPERIMENTAL CONSEQUENCES Jorge L. Lopez, D.V. Nunopoulos and A. Zichichi Physical Review D 52 (1995) 4178. EXPERIMENTAL CONSTRAINTS ON A STRINGY s u ( 5 ) X Jorge L. Lopez, D. V. Nunopoulos and A . Zichichi Physical Review D 53 (1996) 5253.
u( 1) MODEL
NO-SCALE SUPERGRAVITY CONFRONTS LEP DIPHOTON EVENTS Jorge L. Lopez, D. V. Nunopoulos and A . Zichichi DOE/ER/407 17-34, CE-TAMU-50/96, ACT-15/96, hep-phi9610235 (October 1996). SUPERSYMMETRIC PHOTONIC SIGNALS AT THE CERN e+e- COLLIDER LEP GRAVITINO MODELS Jorge L. Lopez, D.V. Nunopoulos and A . Zichichi Physical Review Letters 77 (1996) 5168.
IN
SINGLE-PHOTON SIGNALS AT CERN LEP IN SUPERSYMMETRIC MODELSWITH GRAVITINO Jorge L. Lopez, D.V. Nunopoulos and A . Zichichi Physical Review D 55 (1997) 5813.
A
LIGHT
LIGHT
525 m O D U C r I O N AT HADRON COLLIDERS 5 5 ) LIGHTGRAVITINO Jaewan Kim, Jorge L. Lopez, D.V. Nanopoulos, Raghavan Rangarajan and A. Zichichi Physical Review D 57 (1998) 373.
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SUBNUCLEAR PHYSICS - THEFIRST50 YEARS:HIGHLIGHTS FROM ERICE TO ELN A . Zichichi (Edited by 0. Barnabei, P. Pupillo, F. Roversi Monaco) Joint publication by University and Academy of Sciences of Bologna, Italy (1 998), and World Scientific - 20th Century Physics Series 24 (2000).
