Can the Supersymmetric $\mu$ parameter be generated dynamically without a light Singlet?

Abstract

It is generally assumed that the dynamical generation of the Higgs mass parameter of the superpotential, $\mu$, implies the existence of a light singlet at or below the supersymmetry breaking scale, $\msusy$. We present a counter-example in which the singlet field can receive an arbitrarily heavy mass (\eg, of the order of the Planck scale, $M_{\rm P}\approx 10^{19}~\gev$). In this example, a non-zero value of $\mu$ is generated through soft supersymmetry breaking parameters and is thus naturally of the order of $\msusy$.