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We consider an asymptotically free vectorial SU($N_c$) gauge theory with$N_f$ fermions in the fundamental representation and analyze higher-loopcontributions to the evolution of the theory from the ultraviolet to theinfrared in the limit where $N_c \to \infty$ and $N_f \to \infty$ with$r=N_f/N_c$ a fixed, finite constant. We focus on the case where the $n$-loopbeta function has an infrared zero, at $\xi=\xi_{IR,n\ell}$, where $\xi=\alphaN_c$. We give results on $\xi_{IR,n\ell}$, the anomalous dimension of thefermion bilinear evaluated at $\xi_{IR,n\ell}$, denoted $\gamma_{IR,n\ell}$,and certain structural properties of the beta function, $\beta_\xi$. Theapproach to this limit is investigated, and it is shown that the leadingcorrection terms are strongly suppressed, by the factor $1/N_c^2$. Thisprovides an understanding of a type of approximate universality in calculationsfor moderate values of $N_c$ and $N_f$, namely that $\alpha_{IR,n\ell}N_c$,$\gamma_{IR,n\ell}$, and structural properties of the beta function are similarin theories with different values of $N_c$ and $N_f$ provided that they havesimilar values of $N_f/N_c$. We give results up to four loops fornonsupersymmetric theories and up to three loops for supersymmetric theories.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

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