Gauge theories induced by scalars in the fundamental representation of the $\mathrm{U}{(N}_c{)}_{\mathrm{gauge}}\times \mathrm{U}{(N}_f{)}_{\mathrm{global}}$ group are investigated in the large $N_c$ and $N_f$ limits. A master field is defined from bilinears of the scalar field following an Eguchi-Kawai type reduction of spacetime. The density function for the master field satisfies an integral equation that can be solved exactly in two dimensions $\mathrm{}(D=2\mathrm{})\mathrm{}$ and in a convergent series of approximations at $D>\mathrm{}2\mathrm{}$. While at $D=2\mathrm{}$ the system is in the same phase at all $\epsilon {=N}_c{/N}_f,$ it undergoes a phase transition at a critical value, ${\epsilon }_c\mathrm{}\left(D\right)\mathrm{}$, for $D>\mathrm{}2\mathrm{}$.

• Received 24 September 1997

DOI:https://doi.org/10.1103/PhysRevD.57.5084