(Borel) convergence of the variationally improved mass expansion and the O(N) Gross-Neveu model mass gap

TL;DR

This paper develops a Borel-convergent alternative perturbative expansion for asymptotically free models, enabling the calculation of non-perturbative quantities like the mass gap, and demonstrates its effectiveness on the O(N) Gross-Neveu model.

Contribution

It introduces a Borel-convergent expansion framework for asymptotically free models, improving convergence and enabling non-perturbative quantity estimation from perturbative data.

Findings

Successfully re-evaluated the O(N) Gross-Neveu mass gap using the new method.

Achieved reasonable agreement with exact results across different N values.

Enhanced convergence when combining with variational perturbation techniques.

Abstract

We reconsider in some detail a construction allowing (Borel) convergence of an alternative perturbative expansion, for specific physical quantities of asymptotically free models. The usual perturbative expansions (with an explicit mass dependence) are transmuted into expansions in 1/F, where $F \sim 1/g(m)$ for $m \gg \Lambda$ while $F \sim (m/\Lambda)^\alpha$ for $m \lsim \Lambda$, $\Lambda$ being the basic scale and $\alpha$ given by renormalization group coefficients. (Borel) convergence holds in a range of $F$ which corresponds to reach unambiguously the strong coupling infrared regime near $m\to 0$, which can define certain "non-perturbative" quantities, such as the mass gap, from a resummation of this alternative expansion. Convergence properties can be further improved, when combined with $\delta$ expansion (variationally improved perturbation) methods. We illustrate these results by re-evaluating, from purely perturbative informations, the O(N) Gross-Neveu model mass gap, known for arbitrary $N$ from exact S matrix results. Comparing different levels of approximations that can be defined within our framework, we find reasonable agreement with the exact result.