# Large-$N_c$ and Large-$N_F$ Limits of SU($N_c$) Gauge Theories with Fermions in Different Representations

## TL;DR

This paper analyzes the behavior of SU(N_c) gauge theories with multiple fermion representations in the large-N_c and large-N_f limits, providing scheme-independent calculations of key operator dimensions and beta function derivatives.

## Contribution

It introduces new calculations of anomalous dimensions and beta function derivatives in large-N_c and N_f limits for theories with fermions in different representations, extending previous analyses.

## Key findings

- Derived scheme-independent series expansions for anomalous dimensions.
- Calculated the derivative of the beta function at infrared fixed points.
- Explored limits with multiple fermion representations in SU(N_c) gauge theories.

## Abstract

We present calculations of certain limits of scheme-independent series expansions for the anomalous dimensions of gauge-invariant fermion bilinear operators and for the derivative of the beta function at an infrared fixed point in SU($N_c$) gauge theories with fermions transforming according to two different representations. We first study a theory with $N_f$ fermions in the fundamental representation and $N_{f'}$ fermions in the adjoint or symmetric or antisymmetric rank-2 tensor representation, in the limit $N_c \to \infty$, $N_f \to \infty$ with $N_f/N_c$ fixed and finite. We then study the $N_c \to \infty$ limit of a theory with fermions in the adjoint and rank-2 symmetric or antisymmetric tensor representations.