Comment on "A critical look at $\beta$-function singularities at large $N$" by Alanne, Blasi and Dondi

TL;DR

This paper critiques a previous claim about the absence of UV fixed points in large-N gauge-fermion theories, arguing that the original truncation was flawed and did not account for matter field contributions properly.

Contribution

It demonstrates that the earlier conclusion about the non-existence of UV fixed points is invalid due to an incorrect truncation of the beta function.

Findings

The critique shows the original truncation violates matter field counting.

Reevaluation suggests potential UV fixed points may exist.

Highlights importance of proper truncation in large-N analyses.

Abstract

We first briefly review the state-of-the-art of the large $N_f$ gauge-fermion theories and then show that the claim made in the paper by Alanne, Blasi and Dondi that "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" is unwarranted. This is so since the author's truncation of the beta function violates the large number of matter fields counting.