Supersymmetric Grand Unified Theories: Two Loop Evolution of Gauge and Yukawa Couplings

Abstract

We make a numerical study of gauge and Yukawa unification in supersymmetric grand unified models and examine the quantitative implications of fermion mass ansätze at the grand unified scale. Integrating the renormalization group equations with $α_1(M_Z)$ and $α_2(M_Z)$ as inputs, we find $α_3(M_Z)\simeq 0.111 (0.122)$ for $M_{SUSY}^{}=m_t$ and $α_3(M_Z)\simeq 0.106 (0.116)$ for $M_{SUSY}^{}=1$ TeV at one-loop (two-loop) order. Including $b$ and $τ$ Yukawa couplings in the evolution, we find an upper limit $m_t\ltap 200$ GeV from Yukawa unification. For given $m_t\ltap 175$ GeV, there are two solutions for $β$, one with $\tan β> m_t/m_b$, and one with $\sin β\simeq 0.78(m_t/150\;{\rm GeV})$. Taking a popular ansatz for the mass matrices at the unified scale, we obtain a lower limit on the top quark mass of $m_t\gtap 150 (115)$ GeV for $α_3(M_Z)=0.11 (0.12)$ and an upper limit on the supersymmetry parameter $\tan β\ltap 50$ if $α_3(M_Z)=0.11$. The evolution of the quark mixing matrix elements is also evaluated.