Renormalization effects in gauge theories with color-flavor symmetry

Abstract

We discuss renormalization effects in color-flavor-symmetric gauge theories of strong, weak, and electromagnetic interactions based on the gauge groups SU${\mathrm{}\left(4\right)\mathrm{}}_{\mathrm{flavor}}$ × SU${\mathrm{}\left(4\right)\mathrm{}}_{\mathrm{color}}$ and ${[\mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_A\times \mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_A\times \mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_B]}_{\mathrm{flavor}}\times {\left[\mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_B\right]}_{\mathrm{color}}$. Under the assumption that the currently observed gauge group for these interactions is the subgroup $\mathrm{SU}{\mathrm{}\left(3\right)\mathrm{}}_c\times \mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$, and invoking the quark-lepton-unification hypothesis of Pati and Salam, we show the following. (i) For models based on SU${\mathrm{}\left(4\right)\mathrm{}}_{\mathrm{flavor}}$ × SU${\mathrm{}\left(4\right)\mathrm{}}_{\mathrm{color}}$, the renormalized value of the weak mixing angle is predicted to be approximately 90°. A possible way out, which is unattractive, is that the quark-lepton unification occurs at an energy scale well beyond the Planck mass. (ii) For models based on ${[\mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_A\times \mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_B]}_{\mathrm{flavor}}\times {[\mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_A\times \mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_B]}_{\mathrm{color}}$, ${{sin}^2\theta }_W$ is predicted to be less than 0.5, where ${\theta }_W$ is the Salam-Weinberg mixing angle. Further, assuming that at present energies (∼GeV say) the quark-gluon coupling constant $\frac{g_{30}^2}{4\pi }\gtrsim 0.05$ and ${{sin}^2\theta }_W\approx 0.38$, one predicts that the quark-lepton unification occurs at an energy scale greater than, or of the order of, ${10}^7$ GeV. (iii) The color-flavor-unification scale depends on the total number of fermions in the theory. In particular, if this scale is to lie well below the Planck mass, one needs to introduce many new quarks and leptons. Most of our conclusions apply also to theories based on arbitrary gauge groups $G_{\mathrm{flavor}}\times G_{\mathrm{color}}$, provided they contain $\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}\times \mathrm{SU}{\mathrm{}\left(3\right)\mathrm{}}_c$ as a subgroup and implement the quark-lepton unification in the Pati-Salam sense.