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

TL;DR

This paper investigates the higher-loop behavior of the beta function and anomalous dimensions in large-N_c, large-N_f SU(N_c) gauge theories, revealing approximate universality and suppressed corrections in the infrared regime.

Contribution

It provides new higher-loop calculations of the infrared zero and anomalous dimensions in the large-N_c, large-N_f limit, highlighting structural properties and universality.

Findings

Infrared zero of the beta function is analyzed at higher loops.

Corrections to the large-N_c limit are suppressed by 1/N_c^2.

Results show approximate universality across different N_c and N_f values.

Abstract

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 $N_c \to \infty$ and $N_f \to \infty$ 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 $\xi=\xi_{IR,n\ell}$, where $\xi=\alpha N_c$. We give results on $\xi_{IR,n\ell}$, the anomalous dimension of the fermion bilinear evaluated at $\xi_{IR,n\ell}$, denoted $\gamma_{IR,n\ell}$, 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 $1/N_c^2$. This provides an understanding of a type of approximate universality in calculations for moderate values of $N_c$ and $N_f$, namely that $\alpha_{IR,n\ell}N_c$, $\gamma_{IR,n\ell}$, 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 $N_f/N_c$. We give results up to four loops for nonsupersymmetric theories and up to three loops for supersymmetric theories.