Universal coefficient of the exact correlator of a large-Nmatrix field theory

Abstract

Exact expressions have been proposed for correlation functions of the large-N (planar) limit of the (1 + 1)-dimensional SU(N) × 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 Φ(x), was found to be N −1 Tr Φ(0) † Φ(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 = 1/16π.

Exact Results for the SU(∞) Principal Chiral Model

“…Form factors of local operators and exact expressions for correlation functions have been found. The short-distance two-point function of the scaling field [9], [10], matches the renormalization group result [11]. The short-distance forms of other quantities are known, although the derivations are not yet published.…”

“…We recovered this result from entirely nonperturbative methods [9]. With E. Katzav [10], we found the universal constant C in (4), which is impossible to calculate with perturbation theory. Form factors of currents and the stress tensor have also been found (including some form factors for currents for finite N ) [8], [12], [13].…”

“…Our nonperturbative calculation has revealed the asymptotic freedom of the PCM. This is the only method yielding the coefficient C [10].…”

“…q n are the corresponding quantum numbers. Other matrix elements, besides (10), may be found by crossing particle states from the ket to the bra.…”

Lattice study of continuity and finite-temperature transition in two-dimensional SU(N) x SU(N) Principal Chiral Model

“…Unfortunately, not much is known about thermodynamic properties of PCM in general, although this model is integrable and many exact results can be obtained using bootstrap techniques [16][17][18][19][20] > . One of the reasons is that for PCM there is no obvious local parameter which can be used to characterize the "deconfinement" phase transition, rendering analytic and lattice studies very difficult.…”