A better large $N$ expansion for chiral Yukawa models

We consider the most general renormalizable chiral Yukawa model with $SU(3)_{\rm color}$ replaced by $SU(N_c)$, $SU(2)_{\rm L}$ replaced by $SU(N_w )$ and $U(1)_{Y}$ replaced by $U(1)^{N_w -1}$ in the limit $N_c \rightarrow\infty$, $N_w \rightarrow\infty$ with the ratio $\rho=\sqrt{{N_w}\over{N_c}} \ne 0,\infty$ held fixed. Since for $N_w \ge 3$ only one renormalizable Yukawa coupling per family exists and there is no mixing between families the limit is appropriate for the description of the effects of a heavy top quark when all the other fermions are taken to be massless. The large $N=\sqrt{N_{c} N_{w}}$ expansion is expected to be no worse quantitatively in this model that in the purely scalar case and the $N=\infty$ limit is soluble even when the model is regularized non--perturbatively. A rough estimate of the triviality bound on the Yukawa coupling is equivalent to $m_t \le 1~TeV$.