Borel resummation of the ε-expansion of the Fisher exponent up to the sixth order of perturbation theory

Abstract

It is well known that a series obtained in the framework of the renormalization group and ε-expansion in ϕ4-model are asymptotic. Resummation of the Fisher exponent η using Borel transform with conformal mapping up to six loop approximation of the anomalous field dimension γϕ(u) was demonstrated in a recent paper of one of the co-authors (u — coupling constant). In this paper such a resummation of γϕ(u) and beta function β(u) as a function of u was performed in two stages. First, u∗ value was found as zero of the resummed beta function, then resummation of the anomalous field dimension was performed at u = u∗. In this paper we perform resummation of the Fisher exponent η(ε) as a function of ε at six loop level. We performed comparison of our results with results obtained in cited paper as well as with estimations of the Fisher exponent from the high temperature expansions and Monte Carlo simulations for 3D systems and Onsager’s exact solution for two dimensional Ising model. Refs 12. Fig 1. Table 1.