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

Abstract

Exact expressions have been proposed for correlation functions of 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. The short-distance form of the two-point function of the scaling field $\Phi (x)$, was found to be $N^{-1}⟨\mathrm{Tr}\Phi (0{)}^{†}\Phi (x)⟩=C_2{\ln }^{2}mx$, 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 Lévy flight in one dimension. This observation enables us to calculate $C_2=1/16\pi$.

Comments: Text lengthened from 3 to 6 pages, to include discussion of previous results and directions for further work. Some references added. Accepted for publication in Phys. Rev. D