Supersymmetric grand unified theories: Two-loop evolution of gauge and Yukawa couplings

We make a numerical study of guage and Yukawa unification in supersymmetric grand unified models and examine the quantitative implications of fermion mass Ansa$iuml---tze at the grand unified scale. Integrating the renormalization group equations with α1(MZ) and α2(MZ) as inputs, we find α3(MZ)congruent 0.111(0.122) for MSUSY=mt and α3(MZ)congruent 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 mt approx-lt 200 GeV from Yukawa unification. For given mt approx-lt 175 GeV, there are two solutions for β: one with tanβ>mt/mb, and one with sinβ congruent 0.78(mt/150 GeV). Taking a popular Ansatz for the mass matrices at the unified scale, we obtain a lower limit on the top-quark mass of mt approx-gt 150(115) GeV for α3(MZ)=0.11(0.12) and an upper limit on the supersymmetry parameter tanβ approx-lt 50 if α3(MZ)=0.11. The evolution of the quark mixing elements is also evaluated