SU(4)c-breaking scale and bounds on the top-quark mass

We discuss a scenario in which the SO(10) grand unified symmetry breaks via $\mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_c\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R$ to $\mathrm{SU}{\mathrm{}\left(3\right)\mathrm{}}_c\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_Y$. We determine the $\mathrm{SU}{\mathrm{}\left(4\right)\mathrm{}}_c$-breaking scale $M_c$ through standard (one-loop) renormalization-group evolution of the gauge couplings, as well as by demanding that the ratio $\frac{m_b}{m_{\tau }}$, which is unity above $M_c$, lies in the "observed" range. Compatibility between these two independent determinations of $M_c$ leads to bounds on the top-quark mass. We study separately the cases of having one or two Higgs doublets in the electroweak region. In both cases $m_t$ is found to be less than about 180 GeV.