Higher-loop calculations of the ultraviolet to infrared evolution of a vectorial gauge theory in the limit N_c→∞, N _f→∞ with N_f/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→∞ and N _f→∞ with r=N_f/N_c a fixed, finite constant. We focus on the case where the n-loop beta function has an infrared zero, at ξ=ξ_IR,nℓ, where ξ=αN_c. We give results on ξ_IR,nℓ, the anomalous dimension of the fermion bilinear evaluated at ξ_IR,nℓ, denoted γ _IR,nℓ, and certain structural properties of the beta function, β_ξ. The approach to this limit is investigated, and it is shown that the leading correction terms are strongly suppressed, by the factor 1/Nc2. This provides an understanding of a type of approximate universality in calculations for moderate values of N_c and N_f, namely, that α_IR,nℓN_c, γ_IR,nℓ, 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/N_c. We give results up to four loops for nonsupersymmetric theories and up to three loops for supersymmetric theories.