Coupling-constant renormalization in unified gauge theories containing the Pati-Salam model

The relationship between ${{sin}^2\theta }_W$ and the renormalized quark-gluon constant is found for unified models containing the Pati-Salam model with either fractional or integral quark charges, if $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}\times \mathrm{SU}\mathrm{}\left(3\right)\mathrm{}$ or $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}\times \mathrm{SU}\mathrm{}\left(3\right)\mathrm{}$ is preserved down to the final stage of symmetry breaking ${{sin}^2\theta }_W$ is reduced from a bare value of $\frac{3}{8}$ to asymptotic values of $\frac{1}{6}$ and ¼, respectively, in the limit of a large quark-gluon coupling constant. The relevance of this limit is discussed for Pati-Salam models with fractional and integral quark charges. The experimental range for ${{sin}^2\theta }_W$ is shown to be consistent with an asymptotic value of ¼, but not $\frac{1}{6}$ if the Weinberg-Salam constraint ${m_Z}^2=\frac{{m_W}^2}{{{cos}^2\theta }_W}$ is relaxed.