The strong CP problem versus Planck scale physics

We discuss conditions that should be satisfied by axion models for solving the strong CP problem. It has been observed that Planck scale effects may render the axion models ineffective if there are gauge invariant operators of dimension less than 10 which break explicitly the Peccei-Quinn (PQ) symmetry. We argue that only those operators formed of fields which have vacuum expectation values are dangerous. Supersymmetric axion models fail to prevent even this restricted class of operators. Furthermore, the models that relate the PQ scale and the supersymmetry breaking scale are particularly sensitive to gauge invariant PQ-breaking operators. By contrast, in non-supersymmetric composite axion models the PQ scale arises naturally, and the dangerous operators can be avoided. However, the composite axion models contain heavy stable particles which are cosmologically ruled out. Another problem is a Landau pole for the QCD coupling constant. Both these problems may be solved if the unification of color with the gauge interactions which bind the axion could be achieved.