A compared analysis of the susceptibility in the O(N) theory

TL;DR

This paper compares three theoretical approaches to analyze the longitudinal susceptibility in the O(N) model's broken phase, examining their agreement and behavior across different dimensions and N values.

Contribution

It provides a comparative analysis of the susceptibility using 1/N expansion, Functional Renormalization Group, and Callan-Symanzik methods, extending to four dimensions.

Findings

Agreement among approaches in the large N limit.

Susceptibility vanishes as J^{ε/2} for ε>0.

In d=4, susceptibility inverse vanishes as (ln J)^{-1}.

Abstract

The longitudinal susceptibility $\chi_L$ of the O(N) theory in the broken phase is analyzed by means of three different approaches, namely the leading contribution of the 1/N expansion, the Functional Renormalization Group flow in the Local Potential approximation and the improved effective potential via the Callan-Symanzik equations, properly extended to $d=4$ dimensions through the expansion in powers of $\epsilon=4-d$. The findings of the three approaches are compared and their agreement in the large $N$ limit is shown. The numerical analysis of the Functional Renormalization Group flow equations at small $N$ supports the vanishing of $\chi_L^{-1}$ in $d=3$ and $d=3.5$ but is not conclusive in $d=4$, where we have to resort to the Callan-Smanzik approach. At finite $N$ as well as in the limit $N\to\infty$, we find that $\chi^{-1}_L$ vanishes with $J$ as $J^{\epsilon/2}$ for $\epsilon>0$ and as $(\ln (J))^{-1}$ in $d=4$.