Fine-tuning problem in minimal SO(10) supersymmetric grand unified theories

In grand unified theories (GUT’s) 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 τ lepton have a universal Yukawa coupling at the GUT scale. This implies a very large ratio of Higgs vacuum expectation values, tanβ≃${\mathit{m}}_{\mathit{t}}$/${\mathit{m}}_{\mathit{b}}$. 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 is excessive. The GUT threshold correction to the universal Higgs boson mass parameter can significantly reduce the fine-tuning required for such large values of tanβ. 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 electroweak scale and, hence, the naturalness of large values of tanβ.