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

We make a numerical study of gauge and Yukawa unification in supersymmetric grand unified models and examine the quantitative implications of fermion mass ans\"{a}tze at the grand unified scale. Integrating the renormalization group equations with $\alpha _1(M_Z)$ and $\alpha _2(M_Z)$ as inputs, we find $\alpha _3(M_Z)\simeq 0.111 (0.122)$ for $M_{SUSY}^{}=m_t$ and $\alpha _3(M_Z)\simeq 0.106 (0.116)$ for $M_{SUSY}^{}=1$ TeV at one-loop (two-loop) order. Including $b$ and $\tau $ 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 $\beta$, one with $\tan \beta > m_t/m_b$, and one with $\sin \beta \simeq 0.78(m_t/150\;{\rm GeV})$. 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 $\alpha _3(M_Z)=0.11 (0.12)$ and an upper limit on the supersymmetry parameter $\tan \beta \ltap 50$ if $\alpha _3(M_Z)=0.11$. The evolution of the quark mixing matrix elements is also evaluated.