Constraints on proton lifetime andsin2θWin SO(10) grand unified theory

Simple constraints on the proton-decay lifetime are derived in SO(10) grand unified theory for major descents to the standard group $\mathrm{SU}{\mathrm{}\left(3\right)\mathrm{}}_C\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_Y$. It is shown that the only unambiguous pattern of symmetry breaking that predicts a longer proton lifetime [than SU(5)] goes through the Pati-Salam group $\mathrm{SU}\mathrm{}\left(4\right)\mathrm{}\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R$ and distinguishes quarks from leptons [i.e., SU(4) breaks down to $\mathrm{SU}{\mathrm{}\left(3\right)\mathrm{}}_C\times \mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_{B-L}$] before parity is broken [i.e., $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R$ is broken]. The lifetime depends on $M_R$ [the mass scale associated with $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R$ breaking] but neither on the other intermediate mass scales nor on ${{sin}^2\theta }_W$ in the limit where Higgs-boson contributions are neglected.