Analytical asymptotics of \beta-function in \phi^4 theory (end of the "zero charge" story)

TL;DR

This paper analytically investigates the asymptotic behavior of the eta-function in  theory, confirming it grows linearly with the coupling constant for all space dimensions, and clarifies its relation to the zero charge problem.

Contribution

It provides an analytical derivation of the eta-function's asymptotics in  theory, extending the understanding across all space dimensions and linking it to the zero charge issue.

Findings

 eta(g) o ext{const} imes g as g o \u221e

Asymptotic behavior of anomalous dimensions is constant

Confirms linear eta(g) asymptotics for all dimensions d

Abstract

Reconstruction of the \beta-function for \phi^4 theory, attempted previously by summation of perturbation series, leads to 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 zero-dimensional case confirms the hypothesis and reveals the origin of this result: it is related with a zero of a certain functional integral. Consideration can be generalized to the arbitrary space dimensionality, confirming the linear asymptotics of \beta(g) for all d. Asymptotical behavior for other renormalization group functions (anomalous dimensions) is found to be constant. Relation to the "zero charge" problem is discussed.