Supersymmetry. Theory, Experiment, and Cosmology
.pdfSupersymmetry
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Supersymmetry:
Theory, Experiment, and Cosmology
P. Bin´etruy
AstroParticule et Cosmologie
Universit´ Paris 7
1
3
Great Clarendon Street, Oxford OX2 6DP
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First published 2006
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British Library Cataloguing in Publication Data
Data available
Library of Congress Cataloging in Publication Data
Bin´etruy, P.
Supersymmetry : theory, experiment, and cosmology / P. Bin´etruy. p. cm.
Includes index.
ISBN-13: 978-0-19-850954-7 (alk. paper) ISBN-10: 0-19-850954-5 (alk. paper)
1.Supersymmetry. I. Title.
QC174.17.S9B56 2006 539.7 25—dc22 2006017203
Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain
on acid-free paper by Biddles Ltd. www.biddles.co.uk
ISBN 0–19–850954–5 978–0–19–850954–7 (Hbk)
1 3 5 7 9 10 8 6 4 2
Contents
Introduction |
ix |
||
1 The problems of the Standard Model |
1 |
||
|
1.1 |
General discussion |
1 |
|
1.2 |
Naturalness and the problem of hierarchy |
4 |
|
1.3 |
Supersymmetry as a solution to the problem of naturalness |
12 |
2 The singular rˆole of supersymmetry |
20 |
||
|
2.1 |
Why supersymmetry? |
20 |
|
2.2 |
The supersymmetry algebra |
23 |
|
2.3 |
Supersymmetry breaking |
25 |
|
2.4 |
Supersymmetric quantum mechanics |
27 |
|
2.5 |
Witten index |
32 |
3 |
Basic supermultiplets |
37 |
|
|
3.1 |
Chiral supermultiplet |
37 |
|
3.2 |
Vector supermultiplet and gauge interactions |
45 |
4 The supersymmetry algebra and its representations |
53 |
||
|
4.1 |
Supersymmetry algebra |
53 |
|
4.2 |
Supermultiplet of currents |
55 |
|
4.3 |
Representations of the supersymmetry algebra |
57 |
|
4.4 |
Multiplets of N = 2 supersymmetry |
62 |
|
4.5 |
BPS states |
65 |
5 The minimal supersymmetric model |
86 |
||
|
5.1 |
Why double the number of fundamental fields? |
86 |
|
5.2 |
Model building |
88 |
|
5.3 |
The Minimal SuperSymmetric Model (MSSM) |
91 |
|
5.4 |
Baryon and lepton number |
102 |
|
5.5 |
The LSP and dark matter |
102 |
|
5.6 |
Nonminimal models |
117 |
6 |
Supergravity |
120 |
|
|
6.1 |
Local supersymmetry is supergravity |
120 |
|
6.2 |
Coupling of matter to supergravity |
123 |
|
6.3 |
Supersymmetry breaking |
126 |
|
6.4 |
The gravitino and Goldstino fields |
132 |
vi Contents |
|
|
6.5 |
Radiative breaking of SU (2) × U (1) |
134 |
6.6 |
Gaugino masses |
136 |
6.7 |
Scalar masses |
137 |
6.8 |
The minimal supergravity model |
138 |
6.9Infrared fixed points, quasi-infrared fixed points and focus
|
points |
139 |
6.10 |
The issue of fine tuning |
144 |
6.11 |
The µ problem |
147 |
6.12 |
No-scale models |
149 |
7Phenomenology of supersymmetric models:
supersymmetry at the quantum level |
154 |
|
7.1 |
Why does the MSSM survive the electroweak precision tests? |
154 |
7.2 |
The Higgs sector |
164 |
7.3Avoiding instabilities in the flat directions of the scalar
|
potential |
169 |
7.4 |
High-energy vs. low-energy supersymmetry breaking |
173 |
7.5 |
Limits on supersymmetric particles |
184 |
7.6 |
R-parity breaking |
187 |
7.7 |
The issue of phases |
191 |
8 Dynamical breaking. Duality |
195 |
|
8.1 |
Dynamical supersymmetry breaking: an overview |
195 |
8.2 |
Perturbative nonrenormalization theorems |
200 |
8.3 |
Key issues in dynamical breaking |
205 |
8.4Example of supersymmetric SU (Nc) with Nf flavors. The rˆole
|
|
of R-symmetries |
209 |
|
8.5 |
N = 2 supersymmetry and the Seiberg–Witten model |
215 |
9 |
Supersymmetric grand unification |
224 |
|
|
9.1 |
An overview of grand unification |
225 |
|
9.2 |
Gauge coupling unification |
228 |
|
9.3 |
The minimal supersymmetric SU(5) model |
236 |
|
9.4 |
The SO(10) model |
246 |
|
9.5 |
E6 |
251 |
10 |
An overview of string theory and string models |
254 |
|
|
10.1 |
The general string picture |
254 |
|
10.2 |
Compactification |
263 |
|
10.3 |
String dualities and branes |
275 |
|
10.4 |
Phenomenological aspects of superstring models |
286 |
11 |
Supersymmetry and the early Universe |
312 |
|
|
11.1 |
The ultimate laboratory |
312 |
|
11.2 |
Cosmological relevance of moduli fields |
313 |
|
11.3 |
Inflation scenarios |
319 |
|
11.4 |
Cosmic strings |
324 |
|
11.5 |
Baryogenesis |
328 |
|
|
Contents |
vii |
12 The challenges of supersymmetry |
332 |
||
12.1 |
The flavor problem |
332 |
|
12.2 |
Cosmological constant |
349 |
|
Appendix A A review of the Standard Model and of various |
364 |
||
notions of quantum field theory |
|||
A.1 |
Symmetries |
364 |
|
A.2 |
Spontaneous breaking of symmetry |
372 |
|
A.3 |
The Standard Model of electroweak interactions |
381 |
|
A.4 |
Electroweak precision tests |
397 |
|
A.5 |
Dilatations and renormalization group |
403 |
|
A.6 |
Axial anomaly |
411 |
|
Appendix B |
Spinors |
419 |
|
B.1 |
Spinors in four dimensions |
419 |
|
B.2 |
Spinors in higher dimensions |
424 |
|
Appendix C |
Superfields |
429 |
|
C.1 |
Superspace and superfields |
429 |
|
C.2 |
The chiral superfield |
432 |
|
C.3 |
The vector superfield |
441 |
|
C.4 |
The linear superfield |
445 |
|
Appendix D An introduction to cosmology |
454 |
||
D.1 |
Elements of general relativity |
454 |
|
D.2 |
Friedmann–Robertson–Walker Universes |
457 |
|
D.3 |
The hot Big Bang scenario |
465 |
|
D.4 |
Inflationary cosmology |
475 |
|
D.5 |
Cosmic strings |
481 |
|
Appendix E Renormalization group equations |
485 |
||
E.1 |
Gauge couplings |
485 |
|
E.2 |
µ parameter |
485 |
|
E.3 |
Anomalous dimensions |
486 |
|
E.4 |
Yukawa couplings |
486 |
|
E.5 |
Gaugino masses |
486 |
|
E.6 |
Soft scalar masses |
487 |
|
E.7 |
A-terms |
488 |
|
E.8 |
Dimensional reduction |
488 |
|
Bibliography |
|
490 |
|
Index |
|
|
509 |
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Introduction
The turn of the century witnesses a somewhat unprecedented situation in the study of fundamental interactions. On one hand, we have a theory, the Standard Model, which has been tested successfully to a few per mil: present data shows no deviation from this theory. On the other hand, we have a symmetry, supersymmetry, which seems to be the necessary ingredient to discuss the questions left aside by the Standard Model: diversity of masse scales, vacuum energy, etc. Supersymmetry is obviously not realized in the spectrum of fundamental particles. If it is realized at some deeper level, it must therefore be spontaneously broken. But even though it might be hidden in this way, there must be a level of precision to which experiments should be able to test deviations from the Standard Model. Clearly, this has not happened yet. But the ideas developed within the supersymmetric framework have lead to a general picture which seems supported by recent experimental data: gauge coupling unification, small neutrino masses, relatively light Higgs, a Universe with a density close to the critical density, etc.
It is the purpose of this book to give the tools necessary to discuss these issues. Because fundamental interactions provide, at some deeper level, a general picture of our own world, these issues may be solved at di erent levels: consistency of the theory, experimental discoveries, observation of the Universe. It would be preposterous to say at this point which will be the most decisive approach. It is therefore important to develop as much as possible a common language and to be able to follow the progress of each path: this will ease the way to a very enriching and most probably fruitful exchange between the di erent communities involved in the search for supersymmetry. This is why this book is conceived for readers with di erent backgrounds and varied interests: it may provide an introduction to the concepts and methods of supersymmetry for theorists; it can also be used as an introduction to the phenomenology of supersymmetric models for high energy experimentalists involved in supersymmetry searches; finally it targets the community of cosmologists involved in unravelling the properties of the early Universe. Of course supersymmetry, as we understand it now, may not be found in the end but one may be confident that the theory that will emerge eventually will feed upon the concepts, the methods and the results developed in a supersymmetric context.
Regarding experimental results, the present times represent a turning point. The precision tests at the LEP collider have successfully confronted the Standard Model and have started constraining the bulk of the parameter space of supersymmetric models. If supersymmetry is realized as we think now, the discovery of supersymmetric
xIntroduction
particles should wait for the turning on of the LHC collider. This does not mean that there is nothing to expect besides: searches at the Tevatron collider and precise measurements in B factories have been confronted with supersymmetry. Moreover, astrophysics and cosmology will most probably provide another perspective to study supersymmetric models: searches for dark matter will reach their maturity and cosmology which has become a quantitative science will provide a unique window on the very high mass scale regime of the theory.
On a parallel track, fundamental theories are being developed. It is probable that string theories provide, as they did from the beginning, the logical framework for supersymmetry. They are still under construction but it is important to be aware of the general picture they present us with. For example, recent ideas about extra spacetime dimensions have enriched the phenomenology of high energy colliders. Therefore, one chapter of this book (Chapter 10) presents in a non-technical way the general string and brane picture.
Roadmap
The text is organized in such a way that it can be read using di erent tracks depending on the interests of the reader. It can provide: (i) a theoretical introduction to supersymmetry; (ii) a presentation of supersymmetric models for high energy experimentalists; and (iii) an introduction to supersymmetry emphasizing its rˆole in the early Universe. The sections that should be read in each case are summarized in the following table.
Theoretical Introduction |
High Energy |
Astrophysicist or |
|
Experimentalist |
Cosmologist |
|
|
|
1 |
1.1, 1.2, 1.3.1 |
1.1, 1.2, 1.3.1 |
2 |
2.1–2.3 |
2.1–2.3 |
3 and App. B, C |
3 |
3 |
4 |
— |
— |
5.1–5.4 |
5 |
5, esp. 5.5 |
D.1 and 6 |
D.1 and 6.1–6.8 |
6.1–6.4 |
7.1–7.4 |
7 |
7.4, 7.5 |
8 |
8.1 |
8.1 |
9 |
9 |
9.1, 9.4.1 |
10 |
10 (no box) |
10.1–10.4.2 (no box) |
— |
— |
11 |
12 |
12.1 |
12.2 |
|
|
|
In the course of the general text, some comments intended solely for readers who are following the “Theoretical introduction” track are put between square brackets: [...].
One should note that some very elementary knowledge of quantum field theory is assumed. Appendix A provides a sketch of the basic notions which are needed, which might prove useful to some readers to refresh their memory. It also describes some more