Hence, ${sin}^2{\theta }_W=\mathrm{Tr}(T_3^2)/\mathrm{Tr}(Q^2)=N_c/[2(N_c+1)]$ at the GUT scale, $M_{\mathrm{GUT}}$. Although one could investigate restrictions due to an associated change in the low-energy value of ${\alpha }_{\mathrm{em}}$, we will not need to do this to derive our constraints on $N_c$.

We assume that any color-nonsinglet Higgs chiral superfields have masses of order $M_{\mathrm{GUT}}$; lighter masses could produce excessive proton decay.

Our notation for the beta function for a gauge group $G_j$ is ${\beta }_j=d{\alpha }_j/dt=-(2\pi {)}^{-1}{\alpha }_j^2[b_0^{(j)}+b_1^{(j)}{\alpha }_j/(4\pi )+O({\alpha }_j^2)]$, where ${\alpha }_j=g_j^2/(4\pi )$ and $t=\ln \mu$.

If one were to relax condition (C3) and use $N_g=2$, one would obtain the same conclusion from Eq. (15) for the MSESM, while an asymptotically free ESS model would exist for $N_c=5$; however, we maintain (C3) because for $N_g=2$, the quark mixing matrix has no $CP$-violating phase and hence fails to fit experimental data on $CP$ violation.

Here, $[G_{\mathrm{ESM}},G_{\mathrm{ETC}}]=0$. The situation in a one-doublet TC model is more complicated because $[G_{\mathrm{ESM}},G_{\mathrm{ETC}}]\ne 0$, so ETC gauge bosons carry color and/or charge; we do not consider such models here.