Gauged R-symmetry, Fermion and Higgs Mass Problem

We consider the simplest model of $SU(3) \times SU(2) \times U(1)_Y \times U(1)_R$ gauge symmetry with one extra singlet field whose vacuum expectation value breaks the horizontal $R$-symmetry $U(1)_R$ and gives rise to Yukawa textures. The $U(1)_R$ symmetry is able to provide both acceptable fermion mass hierarchies and a natural solution to the $μ$ problem only if its mixed anomalies are cancelled by the Green-Schwarz mechanism. When the canonical normalization $g_3^2=g_2^2={5\over3}g_1^2$ of the gauge coupling constants is assumed, the Higgs mass parameter $μ\sim m_{3/2}$ can arise taking into acount the uncertainty in the ultraviolet relation $m_e m_μm_τ/m_d m_s m_b \simeq λ^q$ with $q \neq 0$. When $q=0$ is taken only a suppressed value of $μ\sim λm_{3/2}$ can be obtained.

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