Probing Large $N_f$ Through Schemes

TL;DR

This paper examines the scheme dependence of the large $N_f$ expansion in gauge-fermion theories, showing that only one scheme maintains the dominance of leading terms, thus questioning the reliability of perturbative predictions of UV fixed points.

Contribution

It demonstrates that higher-order corrections in the large $N_f$ expansion introduce singularities and that only one renormalization scheme preserves the leading order dominance, highlighting limitations of the expansion.

Findings

Higher-order corrections are increasingly singular.

Only one scheme preserves leading order dominance.

The reliability of UV fixed points predictions is scheme-dependent.

Abstract

We investigate the reliability of the large $N_f$ expansion of four-dimensional gauge-fermion quantum field theories, focusing on the structure and scheme dependence of the beta function. While the existence of a nontrivial UV fixed point at leading order in $1/N_f$ suggests the possibility of asymptotic safety, the absence of higher-order terms precludes robust conclusions. We analyze the impact of renormalization scheme transformations and show that higher-order corrections inevitably introduce increasingly singular contributions. We prove that at most one renormalization scheme can preserve the dominance of the leading contribution, rendering the truncation trustworthy; in all other schemes, higher-order terms dominate and the expansion becomes unreliable. This result places strong constraints on the physical interpretation of UV fixed points in large $N_f$ theories and emphasizes the need for resummation or non-perturbative control to establish asymptotic safety.