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

Abstract

Exact expressions for correlation functions are known for the large-$N$ (planar) limit of the $(1+1)$-dimensional $\mathrm{SU}(N)\times \mathrm{SU}(N)$ 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 $N^{-1}<\mathrm{Tr}\Phi (0{)}^{†}\Phi (x)>$, 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 $\sqrt{-d^2/d{u}^2}$, where $u\in (-1,1)$. At short distances, this expression perfectly matches the result from the perturbative renormalization group.

Comments: Small corrections in text and references. Published in PRD. Still 4 pages, revtex