Gauge Unification, Yukawa Hierarchy and the $\mu$ Problem

The hierarchy in the Yukawa couplings may be the result of a gauged horizontal $U(1)_H$ symmetry. If the mixed anomalies of the Standard Model gauge group with $U(1)_H$ are cancelled by a Green-Schwarz mechanism, a relation between the gauge couplings, the Yukawa couplings and the $\mu$-term arises. Assuming that at a high energy scale $g_3^2=g_2^2={5\over3}g_1^2$ and $(m_e m_\mu m_\tau)/(m_d m_s m_b)\sim \lambda$ (where $\lambda$ is of the order of the Cabibbo angle), the $U(1)_H$ symmetry solves the $\mu$-problem with $\mu\sim\lambda m_{3/2}$.