D ec 1 99 2 Family Symmetry , Gravity , and the Strong CP Problem ∗

We show how in a class of models Peccei–Quinn symmetry can be realized as an automatic consequence of a gauged U (1) family symmetry. These models provide a solution to the strong CP problem either via a massless u–quark or via the DFSZ invisible axion. The local family symmetry protects against potentially large corrections to θ induced by quantum gravitational effects. In a supersymmet-ric extension, the 'µ–problem' is shown to have a natural solution in the context of gravitationally induced operators. We also present a plausible mechanism which can explain the inter–generational mass hierarchy in such a context. 1 I. Global symmetries have lately come under suspicion. There are arguments that quantum gravitational effects violate global symmetries and will induce in the effective low–energy theory all possible operators that respect the local symmetries of the theory. 1 The magnitudes of the coefficients of these operators are expected to be set by the appropriate powers of the Planck scale, but beyond that no quantitative statements are possible at present. In superstring theory there are firmer arguments against the possibility of exact global continuous symmetries. 2 This suspicion has led various authors to reevaluate theoretical ideas that involve global symmetries, such as Peccei–Quinn symmetries, 3 CP, 4 baryon 5 and lepton numbers 6 and cosmological texture. 7 The idea has been to see if such global symmetries can arise as an automatic consequence of local symmetries. In this paper we will reexamine Peccei–Quinn symmetry. 8 Peccei–Quinn symmetry can be realized in two ways: either in the Wigner–Weyl way which leads to a massless quark (or quarks) usually taken to be the u–quark; or in the Nambu–Goldstone way giving rise to an axion. 9 We will first show how local family symmetries can lead to a u–quark light enough to solve the strong CP problem. Then we will show how, in a closely related fashion, very simple DFSZ axion models 10 can arise from local family symmetry realizing an idea of Wilczek. 11 Generalization to grand unification and supersymmetry will be presented. A natural solution to the 'µ–problem' in SUSY models is found in the context of quantum gravity induced operators. We also construct a scheme in this context which can explain the inter–generational mass– hierarchy. We adopt the philosophy in this paper that the operators presumably induced by quantum gravitational effects are not further highly suppressed (beyond the powers of M Pl …