Symmetry breaking of SO(10) and constraints on the Higgs potential: Adjoint {45} and spinorial {16} representations

Abstract

The symmetry breaking of SO(10) is studied in the tree approximation of the potential for an adjoint {45} representation and a spinorial {16} representation. The potential can break SO(10) down to $\mathrm{SU}{(3)}_{c}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$. To break $\mathrm{SU}{(3)}_{c}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ successively down to $\mathrm{SU}{(3)}_{c}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}_{\mathrm{em}}$ due to the {16} requires a cubic {16} {16*} {45} coupling; otherwise, the vacuum arranged by the {16}, which contains (1,${3}^{c*}$) under $\mathrm{SU}{(3)}_{c}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ belonging to the ${5}_{\mathrm{\ensuremath{\sim}}}$ representations of SU(5), ceases to be stable---a signal of the instability of the $\mathrm{SU}{(3)}_{c}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}_{\mathrm{em}}$-symmetric vacuum. The masses for the physical Higgs scalars are calculated in $\mathrm{SU}{(3)}_{c}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$. The dynamically allowed region of the vacuum expectation values of the {45} is found to be strongly restricted. As a result, SO(6) and SO(4) cannot show up in the course of the breaking in the Higgs system of the {45} and {16}.