Symmetry breaking of SO(10) by the 210-dimensional Higgs boson and Michel’s conjecture

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Abstract
We study the symmetry-breaking pattern of SO(10) grand unified theories using the {210}-dimensional Higgs representation. We conclude that there exist finite ranges of parameters of the Higgs potentials such that the minima of the potentials break SO(10) to either SU(2)×SU(2)×SU(4) or SU(2)×SU(2)×U(1)×SU(3). In the first symmetry-breaking pattern, D parity (or charge conjugation) is necessarily broken and this reflects in the mass spectrum of the Higgs boson. In the second case, D parity is conserved as expected. When supersymmetry is imposed on the Higgs superpotential, {210} can no longer break SO(10) to either SU(2)×SU(2)×SU(4) or SU(2)×SU(2)×U(1)×SU(3) without breaking supersymmetry. The implication of our result to the ‘‘Michel’s conjecture’’ is discussed.