Symmetry breaking and intermediate mass scales in the SO(10) grand unified theory

An attempt is made to fill in the particle-mass gap between the ordinary masses of a few hundred GeV and the grand unification mass of ${10}^{15}$ GeV by incorporating two and three stages of spontaneous symmetry breaking in the various descents of SO(10) to the low-energy symmetry $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}\times \mathrm{SU}{\mathrm{}\left(3\right)\mathrm{}}_c$. It is shown that intermediate masses of order ${10}^{10}$ GeV can exist in the particle-mass gap. Values lower than ${10}^{10}$ GeV can only be achieved at the expense of pushing the grand unifying mass to the Planck mass. Also similarities in the gauge structure and differences in the Higgs-scalar and fermion representations of the $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R\times \mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_c$-based Pati-Salam model and its synthesis into SO(10) are pointed out.