On the Large-$N_c$ Limit and Electroweak Interactions: Some Properties of the $N_c$-Extended Standard Model

We discuss properties of QCD with variable and large $N_c$, taking into account electroweak interactions; i.e., we analyze the generalization of the standard model based on the gauge group $G={\rm SU}(N_c) \times {\rm SU(2)}_L \times {\rm U(1)}_Y$. General classes of solutions to anomaly constraints are given, and it is shown that these allow a $T_3=\pm 1\mathrm{/}2$ quark or $T_3=-1\mathrm{/}2$ lepton rather than the neutrino as an electrically neutral fermion. The issue of grand unification is addressed, and it is shown that $G$ cannot be embedded in the usual way in a simple SO($N$) or SU($N$) gauge group unless $N_c=3$. The ratio of strengths of QCD and electroweak interactions is discussed.