# A critical look at $\beta$-function singularities at large $N$

## TL;DR

This paper introduces a self-consistency approach for the $eta$-function at large $N$, demonstrating that singularities in critical exponents do not always indicate $eta$-function singularities, and finds no evidence of a UV fixed point.

## Contribution

The authors develop a novel method using the Wilson-Fisher critical exponent to analyze $eta$-function singularities at large $N$, clarifying their physical significance.

## Key findings

- Singularities in critical exponents do not necessarily imply $eta$-function singularities.
- Applying the method to gauge theories removes singularities in the $eta$-function and anomalous dimensions.
- No evidence of a UV fixed point in the large-$N$ limit is found.

## Abstract

We propose a self-consistency equation for the $\beta$-function for theories with a large number of flavours, $N$, that exploits all the available information in the Wilson-Fisher critical exponent, $\omega$, truncated at a fixed order in $1/N$. We show that singularities appearing in critical exponents do not necessarily imply singularities in the $\beta$-function. We apply our method to (non-)abelian gauge theory, where $\omega$ features a negative singularity. The singularities in the $\beta$-function and in the fermion mass anomalous dimension are simultaneously removed providing no hint for a UV fixed point in the large-$N$ limit.