hep-ph/9611409 25 Nov 1996

APCTP-5 KEK-TH-501 November 1996

An Introduction to Supersymmetry

Manuel Drees Asia-Pacific Center for Theoretical Physics, Seoul, Korea

Abstract A fairly elementary introduction to supersymmetric field theories in general and the minimal supersymmetric Standard Model (MSSM) in particular is given. Topics covered include the cancellation of quadratic divergencies, the construction of the supersymmetric Lagrangian using superfields, the field content of the MSSM, electroweak symmetry breaking in the MSSM, mixing between different superparticles (current eigenstates) to produce mass eigenstates, and the embedding of the MSSM in so-called minimal supergravity.

1. Introduction In the last 20 years the SLAC Spires data base has registered almost 10,000 papers dealing with various aspects of supersymmetric field theories. This is quite remarkable, given that there is no direct experimental evidence for the existence of any of the numerous new particles predicted by such theories. This apparent discrepancy between theoretical speculation and experimental fact has even caught the public eye, and led to charges that modern particle physics resembles medieval alchemy.

I will therefore start these lecture notes by reviewing in some detail the main argument for the existence of supersymmetric particles "at the weak scale" (i.e., with mass very roughly comparable to those of the heaviest known elementary particles, the W and Z bosons and the top quark). This argument rests on the observation that supersymmetric field theories "naturally" allow to chose the weak scale to be many orders of magnitude below the hypothetical scale MX of Grand Unification or the Planck scale MP l. This is closely related to the cancellation of quadratic divergencies [1] in supsersymmetric field theories; such divergencies are notorious in non-supersymmetric theories with elementary scalar particles, such as the Standard Model (SM). In Sec. 2 this question will be discussed in more detail, and the cancellation of quadratic divergencies involving Yukawa interactions will be demonstrated explicitly (in 1-loop order).

This explicit calculation will indicate the basic features that the proposed new symmetry has to have if it is to solve the "naturalness problem" [2] of the SM. In particular, we will need equal numbers of physical (propagating) bosonic and fermionic degrees of freedom; also, certain relations between the coefficients of various terms in the Lagrangian will have to hold. In Sec. 3 we will discuss a method that allows to quite easily construct field theories that satisfy these conditions, using the language of superfields. This will be the most formal part of these notes. At the end of this section, the Lagrangian will have been constructed, and we will be ready to check the cancellation of quadratic divergencies due to gauge interactions. This involves a far greater number of diagrams and fields than the case of Yukawa interactions; it seems quite unlikely that one could have hit on the necessary set of fields and their interactions using the kind of guesswork that will be used (with hindsight) in Sec. 2. At the end of Sec. 3 the problem of supersymmetry breaking will be discussed briefly.

Having hopefully convinced the reader that supersymmetric field theories are interesting, and having shown how to construct them in general, in Sec. 4 I attempt to make contact with reality by discussing several issues related to the phenomenology of the simplest potentially realistic supersymmetric field theory, the Minimal Supersymmetric Standard Model or MSSM. I will begin this section with a review of the motivation for considering a supersymmetrization of the SM. The absence of quadratic divergencies remains the main argument, but the MSSM also has several other nice features not shared by the SM. In Sec. 4a the field content of the model will be listed, and the Lagrangian will be written down; this is an obvious application of the results of Sec. 3. In Sec. 4b the breaking of the electroweak gauge symmetry will be discussed. This plays a central role both theoretically (since without elementary scalar "Higgs" bosons the main argument for weak-scale supersymmetry collapses) and phenomenologically (since it will lead to a firm and, at least in principle, easily testable prediction). Next, mixing between various superparticles ("sparticles") will be discussed. This mixing, which is a direct consequence of SU (2) \Theta U (1)Y gauge symmetry breaking, unfortunately makes the correspondence between particles and sparticles less transparent. However, an understandig of sparticle mixing is essential for an understanding of almost all ongoing work on

the phenomenology of the MSSM.

The least understood aspect of the MSSM concerns the breaking of supersymmetry. A general parametrization of this (necessary) phenomenon introduces more than 100 free parameters in the model. Fortunately not all of these parameters will be relevant for a given problem or process, at least not in leading order in perturbation theory. Nevertheless, it is of interest to look for schemes that attempt to reduce the number of free parameters. The most popular such scheme is (loosely) based on the extension of global supersymmetry to its local version, supergravity, and is hence known as "minimal supergravity" or mSUGRA. This model is attractive not only because of its economy and resulting predictive power, but also because it leads to a dynamical explanation (as opposed to a mere parametrization) of electroweak symmetry breaking. This will be discussed in Sec. 4d. I will in conclude Sec. 5 by briefly mentioning some areas of active research.

2. Quadratic Divergencies This section deals with the problem of quadratic divergencies in the SM, and an explicit calculation is performed to illustrate how the introduction of new fields with judicioulsy chosen couplings can solve this problem. In order to appreciate the "bad" quantum behaviour of the scalar sector of the SM, let us first briefly review some corrections in QED, the best understood ingredient of the SM.

The examples studied will all be two-point functions (inverse propagators) at vanishing external momentum, computed at one-loop level. The calculations will therefore be quite simple, yet they suffice to illustrate the problem. Roughly speaking, the computed quantity corresponds to the mass parameters appearing in the Lagrangian; since I will assume vanishing external momentum, this will not be the on-shell (pole) mass, but it is easy to see that the difference between these two quantities can at most involve logarithmic divergencies (due to wave function renormalization).

e\Gamma e+ fl fl

Fig. 1: The photon self-energy diagram in QED. Let us first investigate the photon's two-point function, which receives contributions due to the electron loop diagram of Fig. 1:

ss_*flfl (0) = \Gamma Z d

(2ss)4 tr "(\Gamma iefl