On the fine-tuning problem in minimal SO(10) SUSY-GUT

In grand unified theories (GUT) based on SO(10) all fermions of one generation are embedded in a single representation. As a result, the top quark, the bottom quark, and the $\tau$ lepton have a universal Yukawa coupling at the GUT scale. This implies a very large ratio of Higgs vacuum expectation values, $\tanb \simeq m_t/m_b$. We analyze the naturalness of such a scenario quantitatively including all the relevant radiative corrections and find that in minimal unified supergravity models with universal soft supersymmetry breaking parameters at the GUT scale, the necessary amount of fine-tuning needed is excessive. GUT threshold correction to the universal Higgs mass parameter can significantly reduce the fine-tuning required for such large values of $\tanb$. We also point out that the top quark mass can play a crucial role in explaining the hierarchy between the SUSY breaking scale and the electro-weak scale and, hence, the naturalness of large values of $\tanb$.