Seeing asymptotic freedom in an exact correlator of a large-$N$ matrix field theory

TL;DR

This paper derives an exact nonperturbative expression for the two-point correlation function in a large-$N$ principal chiral sigma model, confirming known perturbative results at short distances.

Contribution

It provides an exact expression for the correlation function in the large-$N$ limit of the model, using the spectrum of a specific operator, which was previously not known.

Findings

Exact two-point function expressed via operator spectrum

Matching with perturbative RG at short distances

Nonperturbative solution for large-$N$ asymptotically-free model

Abstract

Exact expressions for correlation functions are known for 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, an entirely nonperturbative method. The large-$N$ solution of this asymptotically-free model is far less trivial than that of O($N$) sigma model (or other isovector models). Here we study the Euclidean two-point correlation function $N^{-1}< {\rm Tr}\,\Phi(0)^{\dagger} \Phi(x)>$, where $\Phi(x)\sim Z^{-1/2}U(x)$ is the scaling field and $U(x)\in SU(N)$ is the bare field. We express the two-point function in terms of the spectrum of the operator $\sqrt{-d^{2}/du^{2}}$, where $u\in (-1,1)$. At short distances, this expression perfectly matches the result from the perturbative renormalization group.