SEARCHING FOR THE SUPERWORLD: A Volume in Honor of Antonino Zichichi on the Occasion of the Sixth Centenary Celebrations of the University of Turin, Italy ... (World Scientific in 20th Century Physics)

SERRCHING FOR THE SUPERWORlO World Scientific Series in 20th Century Physics Published Vol. 20 The Origin of the Thir...

Author: Sergio Ferrara (Editor) | Rudolf M. Mossbauer (Editor)

8 downloads 1314 Views 26MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

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

*

LONDON

6 World Scientific 1 : SINGAPORE

*

BElJlNG

*

SHANGHAI

*

HONG KONG

*

TAIPEI

*

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224

USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library

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

This page intentionally left blank

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


1

INTRODUCTORY PAPERS


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. ”


25

SELECTED PAPERS ON SEARCHING FOR THE SUPERWORLD BY A. ZICHICHI AND COLLABORATORS

PARTA


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


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 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


and

A. ZICHICHI

7

Fig. 14. - Fitted Msusv and EGUT as functions of as(MZ) according to the 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


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 - - 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 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 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


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


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


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 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


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.


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


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


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


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 =

(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


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


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@.


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


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


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


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


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.


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


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.


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


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.


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


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


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 0.25 (dotted line) is excluded.

259

236 Volume 29 1, number 3

PHYSICS LETTERS B

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.

260

24 September 1992

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.

237 Volume 29 I , number 3

PHYSICS LETTERS B

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

-

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.

26 1


PHYSICS LETTERS B

24 September 1992

[ 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.

262

239

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

I993


24 1

Physics Letters B 299 ( 1993) 262-266 North-Holland

PHYSICS L E T T E R S B

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.

242 Volume 299, number 3,4

PHYSICS LETTERS B

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 ) 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

+

264

"3

P
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 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

250 PHYSICSLETTERS B

Volume 306, number 1,2

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.

25 1 Volume 306, number 1,2

PHYSICS LETTERS B

27 May 1993

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 )

75


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

76

21 May 1993

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.

25 3 PHYSICS LETTERS B

Volume 306, number 1,2

21 May 1993

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


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

1993


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 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,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 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

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 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 O

150

200

PO

PO

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-

.-

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 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~') 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


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 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 O

F 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

.

D-

7 r m

10-3

10-3

10-4

10-4

10-5

40

60

40

60

80

100

10-5 120 40

60

80

100

120

60

80

100

120

80

100

120

100

10-1

5

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

10-4

10-5

mh (GeV)

40

mh

(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

4071

1

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

60

40

100

I20

60

40

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 ) ,

80

100

120

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 0, the h Higgs will most likely be seen at LEP 11 since mh 2 83 GeV and R h 20.7. For p>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 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


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


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)’ 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) pO)

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 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 0 for m, 5 135 GeV, mh 5 105 GeV, whereas Ifp O, m , 5 135 GeV, whereas if p 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 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 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;

-

,

SEARCHING FOR THE SUPERWORLD: A Volume in Honor of Antonino Zichichi on the Occasion of the Sixth Centenary Celebrations of the University of Turin, Italy ... (World Scientific in 20th Century Physics)

Philosophy of Language in the 20th Century

History of Constructivism in the 20th Century

Selected Papers of K C Chou (20th Century Physics) (World Scientific Series 20th Century Physics)

Ettore Majorana Scientific Papers. On occasion of the centenary of his birth

The Agricultural Revolution of the 20th Century

A Century of Ideas: Perspectives from Leading Scientists of the 20th Century (Fundamental Theories of Physics)

A Century of Ideas: Perspectives from Leading Scientists of the 20th Century (Fundamental Theories of Physics)

In the Name of Honor

In the Name of Honor

In the Name of Honor

In the Name of Honor

In the Name of Honor

In the Name of Honor

In The Name Of Honor

Groups of Diffeomorphisms: In Honor of Shigeyuki Morita on the Occasion of His 60th Birthday

In the Name of Honor

In the Name of Honor

In the Name of Honor

in the Name Of Honor

In the Name of Honor

In the Name of Honor

In the Name of Honor

The Civilization of the Renaissance in Italy

Physics of the Impossible, A Scientific Exploration

Poland in the 20th Century

Physics of the Impossible, A Scientific Exploration

Great Inventions of the 20th Century (Scientific American)

The Saint-Napoleon: Celebrations of Sovereignty in Nineteenth-Century France

Justification in the 20th Century

SEARCHING FOR THE SUPERWORLD: A Volume in Honor of Antonino Zichichi on the Occasion of the Sixth Centenary Celebrations of the University of Turin, Italy ... (World Scientific in 20th Century Physics)

Philosophy of Language in the 20th Century

History of Constructivism in the 20th Century

Selected Papers of K C Chou (20th Century Physics) (World Scientific Series 20th Century Physics)

Ettore Majorana Scientific Papers. On occasion of the centenary of his birth

The Agricultural Revolution of the 20th Century

A Century of Ideas: Perspectives from Leading Scientists of the 20th Century (Fundamental Theories of Physics)

A Century of Ideas: Perspectives from Leading Scientists of the 20th Century (Fundamental Theories of Physics)

In the Name of Honor

In the Name of Honor

In the Name of Honor

In the Name of Honor

In the Name of Honor

In the Name of Honor

In The Name Of Honor

Groups of Diffeomorphisms: In Honor of Shigeyuki Morita on the Occasion of His 60th Birthday

In the Name of Honor

In the Name of Honor

In the Name of Honor

in the Name Of Honor

In the Name of Honor

In the Name of Honor

In the Name of Honor

The Civilization of the Renaissance in Italy

Physics of the Impossible, A Scientific Exploration

Poland in the 20th Century

Physics of the Impossible, A Scientific Exploration

Great Inventions of the 20th Century (Scientific American)

The Saint-Napoleon: Celebrations of Sovereignty in Nineteenth-Century France

Justification in the 20th Century

Recommend Documents