Strong coupling asymptotics of the \beta-function in \phi^4 theory and QED

TL;DR

This paper develops a modified Borel-conformal mapping algorithm to determine the strong coupling asymptotics of the eta-function in irc^4 theory and QED, revealing linear asymptotics eta(g) or g or all dimensions and its relation to the zero charge problem.

Contribution

The paper introduces a modified summation algorithm to analyze the strong coupling behavior of the eta-function, establishing linear asymptotics across dimensions and in QED.

Findings

eta(g) or g or all dimensions d=2,3,4.

Asymptotics eta(g) or g or QED.

Connection to the zero charge problem.

Abstract

The well-known algorithm for summing of divergent series is based on the Borel transformation in combination with the conformal mapping (Le Guillou and Zinn-Justin, 1977). Modification of this algorithm allows to determine a strong coupling asymptotics of the sum of the series through the values of the expansion coefficients. Application of the algorithm to the \beta-function of \phi^4 theory leads to the asymptotics \beta(g)=\beta_\infty g^\alpha at g\to\infty, where \alpha\approx 1 for space dimensions d=2,3,4. The natural hypothesis arises, that asymptotic behavior is \beta(g)\sim g for all d. Consideration of the "toy" zero-dimensional model confirms the hypothesis and reveals the origin of this result: it is related with a zero of a certain functional integral. Generalization of this mechanism to the arbitrary space dimensionality leads to the linear asymptotics of \beta(g) for all d. The same idea can be applied for QED and gives asymptotics \beta(g)=g, where g is the running fine structure constant. Relation to the "zero charge" problem is discussed.