# Analytic Coupling Structure of Large $N_f$ (Super) QED and QCD

## TL;DR

This paper analyzes the convergence and singularity structure of the large-$N_f$ expansion of the beta-function in QED, QCD, Super QED, and Super QCD, revealing common analytic features and the physical origin of singularities.

## Contribution

It provides a detailed analysis of the convergence properties and singularity structures of the large-$N_f$ beta-function expansions across multiple gauge theories, including supersymmetric cases.

## Key findings

- The expansions are convergent up to roughly 30 orders.
- Singularities are linked to iterated bubble diagrams.
- A common analytic structure is identified across theories.

## Abstract

We study the analytic properties of the 't Hooft coupling expansion of the beta-function at the leading nontrivial large-$N_f$ order for QED, QCD, Super QED and Super QCD. For each theory, the 't Hooft coupling expansion is convergent. We discover that an analysis of the expansion coefficients to roughly 30 orders is required to establish the radius of convergence accurately, and to characterize the (logarithmic) nature of the first singularity. We study summations of the beta-function expansion at order $1/N_f$, and identify the physical origin of the singularities in terms of iterated bubble diagrams. We find a common analytic structure across these theories, with important technical differences between supersymmetric and non-supersymmetric theories. We also discuss the expected structure at higher orders in the $1/N_f$ expansion, which will be in the future accessible with the methods presented in this work, meaning without the need for resumming the perturbative series. Understanding the structure of the large-$N_f$ expansion is an essential step towards determining the ultraviolet fate of asymptotically non-free gauge theories.