Chiral Yukawa models in the planar limit

We consider the most general renormalizable chiral Yukawa model with $SU(3{)}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{r}}$ replaced by $SU(N_c)$, $SU(2{)}_{\mathrm{L}}$ replaced by $SU(N_w )$and$U(1)_{Y}$replacedby$U(1)^{N_w -1}$inthelimit$N_c \rightarrow\infty$,$N_w \rightarrow\infty$withtheratio$\rho=\sqrt{{N_w}\over{N_c}} \ne 0,\infty$heldfixed\mathrm{.}Sincefor$N_w \ge 3$ 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. A rough estimate of the triviality bound on the Yukawa coupling is equivalent to $m_t\le 1TeV$.