Two neutral $Z$ bosons in an SO(10) supergravity model

We consider here the supergravity model of the grand unification gauge group SO(10), which breaks to $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_R\times \mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_{B-L}\times \mathrm{SU}{\mathrm{}\left(3\right)\mathrm{}}_C\left(\equiv G_{2113}\right)$ at a mass scale of ${10}^{15}$ GeV. The low-energy breaking of the gauge group $G_{2113}$ to $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_Q\times \mathrm{SU}{\mathrm{}\left(3\right)\mathrm{}}_C$ is induced by supergravity. The model has the following distinctive features: (i) There should be another neutral gauge boson, whose mass could be less than 1 TeV. (ii) The proton lifetime for the decay mode $p\to e^{+}{\pi }^0$ is very much enhanced, but, under some circumstances, it could be barely observable. (iii) Weak-interaction phenomenology in the charged-current sector remains unaltered.