Divergence of the $1/N_f$ series expansion in QED

TL;DR

This paper investigates the divergence of the $1/N_f$ series expansion in QED, providing evidence that it also diverges, which has implications for understanding the true nature of perturbative series in quantum electrodynamics.

Contribution

The paper demonstrates that the $1/N_f$ series expansion in QED diverges, extending the understanding of divergence beyond the standard perturbative series.

Findings

The $1/N_f$ series in QED diverges.

The divergence is shown using arguments similar to Dyson.

Implications for the nature of perturbative series in QED.

Abstract

The perturbative expansion series in coupling constant in QED is divergent. It is either an asymptotic series or an arrangement of a conditionally convergent series. The sum of these types of series depends on the way we arrange partial sums for successive approximations. The $1/N_f$ series expansion, where $N_f$ is the number of flavours, defines a rearrangement of this series, and therefore, its convergence would serve as a proof that the perturbative series is, in fact, conditionallyconvergent.Unfortunately, the $1/N_f$ series also diverges.We proof this usingarguments similar to those of Dyson. We expect that some of the ideas and techniques discussed in our paper will find some use in finding the true nature of the perturbative series in coupling constant as well as the $1/N_f$ expansion series.