Four-point renormalized coupling constant and Callan-Symanzik beta-function in O(N) models

TL;DR

This paper examines the zero-momentum four-point coupling constant and beta-function in O(N) models, revealing non-analytic behavior at the fixed point and comparing various expansion methods for accurate determination.

Contribution

It demonstrates the non-analyticity of the Callan-Symanzik beta-function at the fixed point and extends the epsilon-expansion to improve estimates of g^* across dimensions.

Findings

Good agreement among g-, epsilon-, and strong-coupling expansion results.

Analytic assumptions in beta-function resummation may overestimate g^* for N=0,1.

Non-analyticity affects the accuracy of fixed-point coupling constant determinations.

Abstract

We investigate some issues concerning the zero-momentum four-point renormalized coupling constant g in the symmetric phase of O(N) models, and the corresponding Callan-Symanzik beta-function. In the framework of the 1/N expansion we show that the Callan- Symanzik beta-function is non-analytic at its zero, i.e. at the fixed-point value g^* of g. This fact calls for a check of the actual accuracy of the determination of g^* from the resummation of the d=3 perturbative g-expansion, which is usually performed assuming analyticity of the beta-function. Two alternative approaches are exploited. We extend the \epsilon-expansion of g^* to O(\epsilon^4). Quite accurate estimates of g^* are then obtained by an analysis exploiting the analytic behavior of g^* as function of d and the known values of g^* for lower-dimensional O(N) models, i.e. for d=2,1,0. Accurate estimates of g^* are also obtained by a reanalysis of the strong-coupling expansion of lattice N-vector models allowing for the leading confluent singularity. The agreement among the g-, \epsilon-, and strong-coupling expansion results is good for all N. However, at N=0,1, \epsilon- and strong-coupling expansion favor values of g^* which are sligthly lower than those obtained by the resummation of the g-expansion assuming analyticity in the Callan-Symanzik beta-function.