Dynamics of the Peccei Quinn Scale

Invoking the Peccei-Quinn (PQ) solution to the strong CP problem substitutes the puzzle of why θqcd is so small with the puzzle of why the PQ symmetry is of such high quality. Cosmological and astrophysical considerations raise further puzzles. This paper explores this issues in several contexts: string theory and field theory, and theories without and with low energy supersymmetry. Among the questions studied are whether requiring axion dark matter can account for the quality of the PQ symmetry, to which the answer is sometimes yes. In non-supersymmetric theories, we find fa = 10 12 GeV is quite plausible. In gauge mediation, cosmological constraints on pseudomoduli place fa in this range , and require that the gravitino mass be of order an MeV. 1 Axions: Their Virtues and Deficiencies Nuclear physics is almost indifferent to the QCD angle[1], yet for some reason θqcd is incredibly small. In thirty years, only three persuasive solutions of this puzzle have been put forward. 1. mu = 0: This could result as an accident of discrete flavor symmetries[2], or a result of “anomalous” discrete symmetries as in string theory[3]. 2. Spontaneously broken CP: Here one postulates that CP is an exact symmetry of the microscopic theory, which is spontaneously broken. θ is then calculable, in principle, and under certain circumstances, might be small[4, 5]. In critical string theories, CP is an exact (gauge) symmetry[6, 7], spontaneously broken at generic points in typical moduli spaces. So this would seem a plausible framework. 3. Axions: in the presence of an approximate, global symmetry (Peccei-Quinn (PQ) symmetry) with a QCD anomaly, the pseudo-Goldstone boson which arises from symmetry breaking (the axion) adjusts to yield θ ≈ 0. This raises two puzzles. First, the symmetry must be extremely good if it is to solve the strong CP problem, and second, with simple, but strong, cosmological assumptions, the decay constant of the axion must be small compared to scales such as the unification scale and the Planck scale. Critical string theories typically exhibit an extremely good (but approximate) global symmetry; these symmetries are exact in perturbation theory, broken only by non-perturbative effects[8]. Moreover, in the string theory framework, the standard cosmological assumptions do not hold[9]. So the axion solution, also, has a certain plausibility. Each of these solutions, however, poses problems. 1. mu = 0. While the work of [2] suggests that this possibility is often realized in models of flavor, lattice computations appear to rule out mu = 0[10]. 2. Spontaneous CP: To assess the plausibility of this idea, it is necessary to consider some sort of underlying structure. If nature is described by theories which resemble critical string theory, one needs to consider fixing of moduli. If we take, as a model, moduli fixing in flux vacua, one often speaks of 10 states as arising from turning on many different fluxes. But typically only half of the fluxes are invariant under CP. This means that, say 10 → 10 states. So only an extremely tiny fraction of states preserve CP. It is not