Non-analyticity of the Callan-Symanzik beta-function of two-dimensional O(N) model

TL;DR

This paper investigates the non-analytic behavior of the Callan-Symanzik beta-function in the two-dimensional O(N) model, revealing slow convergence issues in perturbative expansions due to nonanalytic corrections.

Contribution

It provides a detailed analysis of the non-analyticity of the beta-function and derives its asymptotic behavior at the fixed point for different N values, supported by explicit calculations.

Findings

Beta'(g) exhibits nonanalytic behavior near the fixed point g^* for N=1 and N>2.

Explicit calculations support the derived asymptotic behaviors.

Nonanalytic corrections may cause slow convergence of perturbative series.

Abstract

We discuss the analytic properties of the Callan-Symanzik beta-function beta(g) associated with the zero-momentum four-point coupling g in the two-dimensional phi^4 model with O(N) symmetry. Using renormalization-group arguments, we derive the asymptotic behavior of beta(g) at the fixed point g^*. We argue that beta'(g) = beta'(g^*) + O(|g-g^*|^{1/7}) for N=1 and beta'(g) = beta'(g^*) + O(1/\log |g-g^*|) for N > 2. Our claim is supported by an explicit calculation in the Ising lattice model and by a 1/N calculation for the two-dimensional phi^4 theory. We discuss how these nonanalytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g.