Supersymmetry. Theory, Experiment, and Cosmology

Introduction xi

advanced notions which will be necessary to the more theoretically oriented reader, and the bulk of the appendix is a presentation of the Standard Model of electroweak unification. The reader is urged to browse through it in order to get acquainted with the notation used throughout the book, as well as to make sure that he (she) feels at ease with the notions discussed there.

In parallel, Appendix D provides a brief introduction to the Standard Model of cosmological evolution which should prove useful to the non-expert to follow the chapters or sections on cosmology.

Appendices B and C introduce the notion of superfield and superspace. They lie on the “Theoretical introduction” track.

Finally, Appendix E provides a summary of the renormalization group equations which are scattered through the main text.

References and index

In addition to the general bibliography, a set of a few key references is presented at the end of each chapter that may provide further reading material.

Use of this book in teaching

It is di cult to discuss the interest of supersymmetry in the context of extensions of the Standard Model of electroweak interactions without having in mind the details of this Model. This is why a rather extended presentation of this Standard Model is presented in Appendix A. Depending on the origin and level of the students, it may provide the basis for some introductory lectures. Indeed, this represents the bulk of a half-semester course which I have given several times on the Standard Model.

Then, according to the nature of the audience, one of the three tracks described in the table above, or a mixture of them, may be followed. Each chapter is followed by exercises that develop some technical points or introduce applications not included in the main text. Exercises with a larger scope are called “Problems.” They may provide material for more extended homework. The purpose of most exercises is not to challenge the students but rather to help them work out some details as well as to open their perspectives. This is why hints of solutions, or sometimes detailed solutions, are provided. They should also be of much help to the student who is studying this book by her(him)self.

Acknowledgments

This book owes much to my collaborators on the subject. It is through their own perception of the field that I have acquired some familiarity with the many methods and tools necessary to grasp the central ideas. Let me mention in particular Mary K. Gaillard, I. Hinchli e, G. Girardi, R. Grimm, P. Ramond, S. Lavignac, and E. Dudas.

It was based originally on a set of lectures given in 1995 at Laboratoire de l’Acc´el´erateur Lin´eaire in Orsay and has evolved through lectures at Ecole Normale Sup´erieure for several years. Let me thank all those who attended these various lectures and contributed one way or another to their improvement, in particular B. Delamotte, D. Langlois and Y. Mambrini. And Mireille Calvet for typing expertly some of these lectures.

xii Introduction

I wish to extend my gratitude to all my colleagues in the Groupement de Recherche (GDR turned to Euro-GDR) SUSY who have shared over the years the excitement of working on the various aspects of supersymmetry. This has been a very enriching experience. Thanks to all, especially J.-F. Grivaz.

A preliminary version was used in class by several colleagues, in particular M.K. Gaillard and G. Kane, whom I thank for their many suggestions. In particular, I am especially indebted to Mary K. for her many major and minor corrections and her invaluable comments.

This book owes to much people but also to places, through which I have carried my ever growing manuscript over the years. I may mention the Institute of Fundamental Physics in Santa Barbara (several times!), the Aspen Center for Physics (several times as well), the University of Michigan, the Perimeter Institute and the Laboratoire d’Annecy-le-Vieux de Physique Th´eorique. Not to forget my own lab for more than ten years, the Laboratoire de Physique Th´eorique d’Orsay!

Very warm thanks to Sonke Adlung, at Oxford University Press, whose patience with the progress of the manuscript has been immense. His repeated encouragements have been the main reason for my not dropping the seemingly impossible task, especially in the last few years when I was becoming increasingly busy.

Finally, I would like to pay tribute to my family and my friends for their heroic patience over all these weekends, holidays, and vacations which were supposed to see the book completed in a few weeks and then days. This book is dedicated to them: they are the only ones who will not need to read it but will be satisfied with it being there on the shelf.

Pierre Bin´etruy

Paris, December 2005

General discussion 3

two eigenvalues: m1 M and m2 m2/M . This is the famous seesaw mechanism [177, 383]. If we interpret the latter eigenvalue as the small neutrino mass observed (mν 10−1 to 10−2 eV), this gives an indication that new physics appears at a scale M 1014 GeV, under the form of a superheavy neutrino (the second eigenstate).

1.1.2Theoretical issues

The Standard Model is in any case not completely satisfactory from a theoretical point of view. For example, one may list the free parameters:

•three gauge couplings

•two parameters in the Higgs sector: m and λ

•nine quark (u, d, c, s, t, b) and charged lepton (e, µ, τ ) masses

•three mixing angles and one CP-violating phase for the quark system

which amounts to 19 free parameters. One may wish fewer parameters for a fundamental theory.

Indeed, extensions of the Standard Model are proposed which relate some of these parameters. For example, grand unified theories unify the three gauge couplings at a scale MU of order 1016 GeV. This works only approximately in a nonsupersymmetric framework. They also classify the quark and lepton fields in larger representations, which induces relations among the mass parameters. For example, grand unified theories predict with success the ratio mb/mτ .

The theory is also determined by the quantum numbers of each field. There are open questions in this respect: why are all the electric charges a multiple of e/3, where −e is the electron charge? This is often called the problem of the quantization of charge. Why are the quantum numbers of quarks and leptons such that all anomalies cancel? Again, grand unified theories give a first answer to these problems by including the electric charge among the nonabelian gauge symmetry generators.

The gap observed in quark and lepton masses when one goes from one family to another is not satisfactorily explained in the Standard Model. This is often referred to as the problem of mass or flavor problem. The Standard Model of electroweak interactions clearly establishes the breaking of the SU (2) × U (1) gauge symmetry as the origin of mass. This is summarized in the formula:

where GF is the Fermi constant. Thus (1.1) means that the only fermion with a “natural” mass scale is the top quark of mass mt = 175 GeV: the corresponding

4The problems of the Standard Model

Yukawa coupling is of order 1. But it does not account for the diversity of quark and lepton masses, especially from one family of quarks and leptons to another. This question must be addressed by a theory of Yukawa couplings yet to come. If complete, this theory should also predict the number of families.

Similarly for mixing angle and phases. It is well-known that, in the quark sector, mass eigenstates do not coincide with interaction eigenstates (i.e. the fields with definite quantum numbers under SU (3) × SU (2) × U (1)). This results in mixing angles and relative phases among the di erent quarks. The mixing angles are found to be very di erent in size. Again, this diversity has yet to be explained in the framework of a more fundamental theory. As for phases, the Standard Model provides a single one, the Cabibbo–Kobayashi–Maskawa phase, and thus a unique source of CP violation. This, however, fails to explain the baryon number of the Universe. Similarly, the strong CP problem seems to indicate that we are missing in the Standard Model some important ingredient, whether the axion or some other solution.

The final question is how to treat gravity. Because Newton’s constant is dimensionful ([GN ] = M −2), the theory of gravity is nonrenormalizable. Quantum e ects become important at a scale (¯hc/GN )1/2 = MP 1.22 × 1019 GeV, known as the Planck scale. How does one obtain a quantum theory of gravity, that is how does one put gravity on the same level as the theory of the other fundamental interactions? Most probably, answering such a question requires some drastic changes, such as the ones proposed in the string approach where the fundamental objects are no longer pointlike.

We will return to these questions in later chapters. For the time being, we will address the question of the coexistence at the quantum level of two vastly di erent scales: the electroweak scale MW and the Planck scale MP (or for that matter any of the superheavy scales we have discussed above: M 1014 GeV, MU 1016 GeV, etc.).

1.2Naturalness and the problem of hierarchy

The central question that we will address in this section is the existence of quadratic divergences associated with the presence of a fundamental scalar field, such as the Higgs field in the Standard Model. Before dealing with this, we must have a short presentation of the notion of e ective theory.

1.2.1E ective theories

In the modern point of view, a given theory (e.g. the Standard Model) is always the e ective theory of a more complete underlying theory, which adequately describes physics at a energy scale higher than a threshold M . This threshold is physical in the sense that the complete physical spectrum includes particles with a mass of order M . For example, in the case of the seesaw mechanism for neutrino masses, discussed in Section 1.1.1, there is a neutrino field of mass M .

The description in terms of an e ective theory, restricted to the light states, is obviously valid only up to the scale M . The heavy fields (of mass M or larger) regulate the theory and therefore the scale M acts as a cut-o Λ on loop momenta.

In quantum field theory, the renormalization procedure allows us to deal with infinities, i.e. contributions that diverge when the cut-o is sent to infinity. However,

Naturalness and the problem of hierarchy 7

valid at all energy scales is known as trivial, i.e. is a free field theory in the infrared (low energy) regime. In practice, this only means that at some scale Λ smaller than

the scale ΛLandau where the coupling would explode (known as the Landau pole, see below), some new physics appears.

The exact value of the Landau pole requires a nonperturbative computation since the running coupling explodes at this scale. Complete calculations show that it is not unreasonable to use the one-loop result (1.13) to obtain an order of magnitude for the Landau pole. Thus, solving for λ the di erential equation (1.13),

Since λ can be expressed in terms of mh itself through (1.6), this is used to put an upper bound, a triviality bound, on the scale of new physics. Keeping the same level of approximation as before, this gives roughly

The right-hand side is a monotonically decreasing function of mh. Alternatively, one may say that, for a given value of Λ, the Higgs mass is bounded by

a decreasing function of Λ.

The reader should be reminded that our expressions are only rough estimates which give the general trend and that they can be refined. In any case, such limits should be taken with a grain of salt: obviously, in the case where Λ is not much greater than mh, the Higgs mass receives new corrections arising from e ective operators (scaling like some negative powers of Λ) which will induce possibly large corrections to the triviality bound.

Vacuum stability

An ever-increasing λ(µ) coupling leads to the constraints just discussed. An everdecreasing λ(µ) coupling leads to di culties of another kind: as soon as the quartic coupling λ(µ) turns negative, the potential (1.3) becomes unbounded from below at large values of the field φ and the theory su ers from an instability. Notwithstanding considerations regarding the cosmological stability of the false vacuum, this instability is a sign that some new physics will take charge.

This situation is faced in particular when the scalar field that we have considered is light (λ is small) but has other interactions that contribute to lower λ(µ) as µ increases. For example, if the Standard Model Higgs is light, the dominant term comes from the top Yukawa interaction: