Higher-Loop Calculations of the Ultraviolet to Infrared Evolution of a Vectorial Gauge Theory in the Limit $N_c\to \mathrm{\infty }$, $N_f\to \mathrm{\infty }$ with $N_f\mathrm{/}{N}_c$ Fixed

We consider an asymptotically free vectorial SU($N_c$) gauge theory with $N_f$ fermions in the fundamental representation and analyze higher-loop contributions to the evolution of the theory from the ultraviolet to the infrared in the limit where $N_c\to \mathrm{\infty }$ and $N_f\to \mathrm{\infty }$ with $r=N_f\mathrm{/}{N}_c$ a fixed, finite constant. We focus on the case where the $n$-loop beta function has an infrared zero, at $\xi ={\xi }_{IR,n\mathrm{\ell }}$, where $\xi=\alpha N_c$\mathrm{.}Wegiveresultson$\xi_{IR,n\ell}$, the anomalous dimension of the fermion bilinear evaluated at ${\xi }_{IR,n\mathrm{\ell }}$, denoted ${\gamma }_{IR,n\mathrm{\ell }}$, and certain structural properties of the beta function, ${\beta }_{\xi }$. The approach to this limit is investigated, and it is shown that the leading correction terms are strongly suppressed, by the factor $1\mathrm{/}{N}_c^2$. This provides an understanding of a type of approximate universality in calculations for moderate values of $N_c$ and $N_f$, namely that ${\alpha }_{IR,n\mathrm{\ell }}{N}_c$, ${\gamma }_{IR,n\mathrm{\ell }}$, and structural properties of the beta function are similar in theories with different values of $N_c$ and $N_f$ provided that they have similar values of $N_f\mathrm{/}{N}_c$. We give results up to four loops for nonsupersymmetric theories and up to three loops for supersymmetric theories.