Bottom quark mass prediction in supersymmetric grand unification: Uncertainties and constraints

Grand unified theories often predict unification of Yukawa couplings (e.g., $h_b=h_{\tau }$), and thus certain relations among fermion masses. The latter can distinguish these from models that predict only coupling constant unification. The implications of Yukawa couplings of the heavy family in the supersymmetric extension of the standard model (when embedded in a GUT) are discussed. In particular, uncertainties associated with $m_t$ and $m_t$, threshold corrections at the low-scale, and threshold and nonrenormalizable-operator corrections associated with a grand-unified sector at the high-scale are parametrized and estimated. The implications of these and of the correlation between $m_t$ and the prediction for ${\alpha }_s$ are discussed. Constraints on the $tan\beta$ range in such models and an upper bound on the $t$-quark pole mass are given and are shown to be affected by the ${\alpha }_s-m_t$ correlation. Constraints on the low-scale thresholds are found to be weakened by uncertainties associated with the high scale.