The universal coefficient of the exact correlator of a large-$N$ matrix field theory

TL;DR

This paper links the universal coefficient of a two-point correlator in a large-N matrix field theory to a Lévy flight mean first-passage time, enabling an exact calculation of the coefficient.

Contribution

It reveals a novel connection between the correlator coefficient and Lévy flight statistics, providing an exact value for the universal coefficient.

Findings

Universal coefficient C_2 = 1/16π calculated.

C_2 proportional to Lévy flight first-passage time.

Short-distance correlator matches perturbative RG predictions.

Abstract

Exact expressions have been proposed for correlation functions of the large-$N$ (planar) limit of the $(1+1)$-dimensional ${\rm SU}(N)\times {\rm SU}(N)$ principal chiral sigma model. These were obtained with the form-factor bootstrap. The short-distance form of the two-point function of the scaling field $\Phi(x)$, was found to be $N^{-1}\langle {\rm Tr}\,\Phi(0)^{\dagger} \Phi(x)\rangle=C_{2}\ln^{2}mx$, where $m$ is the mass gap, in agreement with the perturbative renormalization group. Here we point out that the universal coefficient $C_{2}$, is proportional to the mean first-passage time of a L\'{e}vy flight in one dimension. This observation enables us to calculate $C_{2}=1/16\pi$.