A Natural Solution to the MU Problem

TL;DR

The paper addresses the μ problem in minimal low-energy supergravity by promoting a nonrenormalizable coupling $W = W_o + \lambda W_o H_1H_2$, which generates an effective $μ$ parameter proportional to the SUSY-breaking scale via $μ = λ ⟨W_o⟩$. It argues that $μ$ must be absent from $W_o$—a feature naturally realized in string-inspired SUGRA where mass terms are forbidden—while nonrenormalizable terms are allowed and essential. The authors show that this mechanism yields $μ$ of order $m_{3/2}$, preserves electroweak symmetry breaking, and remains robust under generalizations to multiple $W_o$ terms and Kähler potential variations. They illustrate a realistic realization with gaugino condensation in the hidden sector, where the nonrenormalizable coupling enables the μ mechanism to operate within a UV-consistent and phenomenologically viable framework.

Abstract

We propose a simple mechanism for solving the $μ$ 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 $W_o+λW_o H_1H_2$, yielding an effective $μ$ 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 $μ$ term must be absent from $W_o$, otherwise the natural scale for $μ$ 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 $μ$ problem can be gracefully implemented.