The Top Quark Mass in Supersymmetric SO(10) Unification

The successful prediction of $\sin^2\theta_W$ suggests that the effective theory beneath the GUT scale is the two-Higgs MSSM. 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 $\tau$ masses originate in renormalizable Yukawa interactions of the form $16_3 O 16_3$, then also the top quark mass can be predicted in terms of the MSSM parameters. To compute $m_t$ we present a precise analytic approximation to the solution of the 2-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, $\tan\beta$, of Higgs vacuum expectation values. We point out that when $\tan\beta$ 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)-like model as long as $\lambda_t\simeq \lambda_b\simeq\lambda_\tau$ at the GUT scale and there are no intermediate mass scales in the desert.