On and in the Gross–Neveu and O(N) models

Abstract

We apply large N diagrammatic techniques for theories with double-trace interactions to the leading corrections to CJ, the coefficient of a conserved current two-point function, and CT, the coefficient of the stress–energy tensor two-point function. We study in detail two famous conformal field theories in continuous dimensions, the scalar O(N) model and the Gross–Neveu (GN) model. For the O(N) model, where the answers for the leading large N corrections to CJ and CT were derived long ago using analytic bootstrap, we show that the diagrammatic approach reproduces them correctly. We also carry out a new perturbative test of these results using the O(N) symmetric cubic scalar theory in 6 − ϵ dimensions. We go on to apply the diagrammatic method to the GN model, finding explicit formulae for the leading corrections to CJ and CT as a function of dimension. We check these large N results using regular perturbation theory for the GN model in dimensions and the Gross–Neveu–Yukawa model in dimensions. For small values of N, we use Padé approximants based on the and expansions to estimate the values of CJ and CT in d = 3. For the O(N) model our estimates are close to those found using the conformal bootstrap. For the GN model, our estimates suggest that, even when N is small, CT differs by no more than 2% from that in the theory of free fermions. We find that the inequality applies both to the GN and the scalar O(N) models in d = 3.