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Charged-fermion masses in SO(10): analysis with scalars in 10+120

We consider the scenario in which the mass matrices of the charged fermions in the SO(10) Grand Unified Theory are generated exclusively by renormalizable Yukawa couplings to one $\mathbf{10} \oplus \mathbf{120}$ representation of scalars. We analyze, partly analytically and partly numerically, this scenario in the three-generations case. We demonstrate that it leads to unification of the $b$ and $τ$ masses at the GUT scale. Testing this scenario against the mass values at the GUT scale, obtained from the renormalization-group evolution in the minimal SUSY extension of the Standard Model, we find that it is not viable: either the down-quark mass or the top-quark mass must be unrealistically low. If we include the CKM mixing angles in the test, then, in order that the mixing angles are well reproduced, either the top-quark mass or the strange-quark mass together with the down-quark mass must be very low. We conclude that, assuming a SUSY SO(10) scenario, charged-fermion mass generation based exclusively on one $\mathbf{10} \oplus \mathbf{120}$ representation of scalars is in contradiction with experiment.