Supersymmetry in Field Theory

Supersymmetric theories are reviewed in the context of field theories. The gauge hierarchy problem in attempting the unification of all fundamental interactions is the strongest motivation of modern development of supersymmetry. Starting from the general notion of supersymmetry as a symmetry between bosons and fermions, we explain how the supersymmetry becomes a part of the space-time symmetry if we wish to maintain the relativisitic invariance. The precise idea of supersymmetry is then introduced and the supersymmetric field theories are formulated. There has been a significant breakthrough in the study of nonperturbative effects in supersymmetric field theories using the holomorphy and symmetry arguments. Some of these ideas and results are briefly reviewed. 1 e-mail: [email protected] 1 Motivations for Supersymmetry 1.1 Gauge Hierarchy 1. Standard model Many efforts have been devoted to study the fundamental constituents of matter and the fundamental interactions between them. At present, the experimental efforts have reached the energy scales of several 1000GeV in collisions between protons and/or antiprotons, and that of a few 100GeV in collisions between electrons and positrons. It has been found that all the available experimental data up to these energies can be more or less adequately described by the so-called standard model. In the standard model, the fundamental constituents of matter are quarks and leptons and the three known fundamental forces in nature, strong, weak, and electromagnetic interactions are described by a gauge field theory with the SU(3) × SU(2) × U(1) gauge group. The standard model succeeded to describe the three fundamental interactions by a common unifying idea called the gauge principle and gave many successful predictions. The most striking confirmation of standard model is the discovery of the weak bosons, W and Z with the mass of the order of MW ≈ 100GeV. However, there are three different gauge coupling constants for each of these gauge groups SU(3), SU(2), and U(1). In that sense, the three different strengths of the three fundamental interactions are parametrized nicely, but are not quite unified. Moreover, the standard model has many input parameters that can only be determined from the experimental measurements. There are also other conceptually unsatisfactory points as well. For instance, the electric charge is found to be quantized in nature, but this phenomenon is just an accident in the standard model. 2. Grand Unified Theories Because of qunatum effects, the effective gauge coupling constants change logarithmically as a function of energy scale. Then there is a possibility that the different gauge couplings for the three fundamental interactions can become the same strength at very high energies MG. This means that the three gauge interactions can be truly unified into a single gauge group if we choose an appropriate simple gauge group. This idea was proposed by Georgi and Glashow [1], and these models are called the grand unified theories. The grand unified theories achieved at least two good points: • Because of simple gauge group, the electromagnetic charge is now quantized. • Two coupling can meet at some point provided they are in the right direction. Since the grand unified theory unifies all three couplings at high energies, it gives one constraint for three couplings. Taking the two measurements of coupling constants at low energies as inputs, one can then predict the third coupling. With the simplest possibility for the unifying gauge group, this prediction was found to be not very far from the 2 experimental data. On the other hand, the unification energy MG is now very large compared to the mass scale MW of the weak boson in the standard model [2] M W M G ≈ ( 1