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\"atze at the grand unified scale. Integrating the renormalization group equations with α₁(MZ) and α₂(MZ) as inputs, we find α₃(MZ)0.111(0.122) for MSUSY=mₜ and α₃(MZ)0.106(0.116) for MSUSY=1 TeV at one-loop (two-loop) order. Including b and τ Yukawa couplings in the evolution, we find an upper limit of mₜ200 GeV from Yukawa unification. For given mₜ175 GeV, there are two solutions for β: one with tanβ>mₜmb, and one with sinβ0.78(mₜ150 GeV). Taking a popular Ansatz for the mass matrices at the unified scale, we obtain a lower limit on the top-quark mass of mₜ150(115) GeV for α₃(MZ)=0.11(0.12) and an upper limit on the supersymmetry parameter tanβ50 if α₃(MZ)=0.11. The evolution of the quark mixing matrix elements is also evaluated.