Top quark mass in supersymmetric SO(10) unification

The successful prediction of the weak mixing angle suggests that the effective theory beneath the grand unification scale is the minimal supersymmetric standard model (MSSM) with just two Higgs doublets. If we further assume that the unified gauge group contains SO(10), that the two light Higgs doublets lie mostly in a single irreducible SO(10) representation, and that the t, b, and \ensuremath{\tau} masses originate in renormalizable Yukawa interactions of the form 1${6}_{3}$scrO1${6}_{3}$, then also the top quark mass can be predicted in terms of the MSSM parameters. To compute ${\mathit{m}}_{\mathit{t}}$ we present a precise analytic approximation to the solution of the two-loop renormalization group equations, and study supersymmetric and GUT threshold corrections and the input value of the b quark mass. The large ratio of top to bottom quark masses derives from a large ratio, tan\ensuremath{\beta}, of Higgs vacuum expectation values. We point out that when tan\ensuremath{\beta} is large, so are certain corrections to the b quark mass prediction, unless a particular hierarchy exists in the parameters of the model. With such a hierarchy, which may result from approximate symmetries, the top mass prediction depends only weakly on the spectrum. Our results may be applied to any supersymmetric SO(10) model as long as ${\ensuremath{\lambda}}_{\mathit{t}}$\ensuremath{\simeq}${\ensuremath{\lambda}}_{\mathit{b}}$\ensuremath{\simeq}${\ensuremath{\lambda}}_{\mathrm{\ensuremath{\tau}}}$ at the GUT scale and there are no intermediate mass scales in the desert.