Discrete R Symmetries and Low Energy Supersymmetry

If nature exhibits low energy supersymmetry, discrete (non-Z2) R symmetries may well play an important role. In this paper, we explore such symmetries. We generalize gaugino condensation, constructing large classes of models which are classically scale invariant, and which spontaneously break discrete R symmetries (but not supersymmetry). The order parameters for the breaking include chiral singlets. These simplify construction of models with metastable dynamical supersymmetry breaking. We explain that in gauge mediation, the problem of the cosmological constant makes “retrofitting” particularly natural – almost imperative. We describe new classes of models, with interesting scales for supersymmetry breaking, and which allow simple solutions of the μ problem. We argue that models exhibiting such R symmetries can readily solve not only the problem of dimension four operators and proton decay, but also dimension five operators. On the other hand, in theories of “gravity mediation”, the breaking of R symmetry is typically of order Mp, R parity is required to suppress dimension four B and L violating operators, and dimension five operators remain problematic. 1 What Makes R Symmetries Special While it has long been argued that theories which incorporate general relativity cannot exhibit exact global continuous symmetries, discrete symmetries are another matter. When studying classical solutions of critical string theories, one often encounters such symmetries on submanifolds of the full moduli space of solutions[1]; these are believed, in general, to be discrete gauge transformations. Within spaces of supersymmetric solutions, a particularly interesting set of symmetries are the discrete R symmetries. These can often be thought of as unbroken subgroups of the rotation group in higher dimensions; in such cases, the fact that they transform the supercharges is immediate. We will reserve the term R symmetry, in this paper, for symmetries other than Z2 which rotate the supercharges. Any symmetry which multiplies all of the supercharges by −1 can be redefined by adding a rotation by 2π, leaving an ordinary (non-R) Z2. Conventional R parity in this sense, is not an R symmetry. There are several reasons to think that, if supersymmetry plays some role in low energy physics, discrete R symmetries might be relevant: 1. Perhaps the most important comes from the question of the cosmological constant (c.c.). In order that the c.c. be small, it is necessary that any constant in the superpotential be far smaller than M3 p . The only type of symmetry which can suppress such a constant is an R symmetry.1 2. In gauge-mediated models, supergravity effects should be unimportant for understanding SUSY breaking (we will make this statement more precise shortly), and the theorem of Nelson and Seiberg[2] requires a global R symmetry in order to obtain supersymmetry breaking (in a generic fashion); correspondingly, an approximate global R symmetry seems a requirement for metastable supersymmetry breaking[3]. Discrete R symmetries are a particularly plausible way in which to account for such approximate continuous symmetries. Supersymmetric critical string theories have vanishing c.c. classically, and the superpotential is protected in higher orders of perturbation theory by non-renormalization theorems. In many cases, the vanishing of W appears an accident, from the point of view of the low energy theory (it is not accounted for by symmetries). In flux vacua, vanishing of the cosmological constant is not typical, requiring R symmetries, or accidental cancelations. More generally, as Banks has repeatedly stressed, the only Minkowski space gravity theories of which we can claim any complete understanding exhibit supersymmetry and R symmetries. Similar remarks apply to would-be μ terms.