A natural solution to the $\mu$ problem

We propose a simple mechanism for solving the $\mu$ problem in the context of minimal low--energy supergravity models. This is based on the appearance of non--renormalizable couplings in the superpotential. In particular, if $H_1H_2$ is an allowed operator by all the symmetries of the theory, it is natural to promote the usual renormalizable superpotential $W_o$ to $W_o+\lambda W_o H_1H_2$, yielding an effective $\mu$ parameter whose size is directly related to the gravitino mass once supersymmetry is broken (this result is maintained if $H_1H_2$ couples with different strengths to the various terms present in $W_o$). On the other hand, the $\mu$ term must be absent from $W_o$, otherwise the natural scale for $\mu$ would be $M_P$. Remarkably enough, this is entirely justified in the supergravity theories coming from superstrings, where mass terms for light fields are forbidden in the superpotential. We also analyse the $SU(2)\times U(1)$ breaking, finding that it takes place satisfactorily. Finally, we give a realistic example in which supersymmetry is broken by gaugino condensation, where the mechanism proposed for solving the $\mu$ problem can be gracefully implemented.