Manifestly Finite Perturbation Theory for the Short-Distance Expansion of Correlation Functions in the Two Dimensional Ising Model

Abstract

In the spirit of classic works of Wilson on the renormalization group and operator product expansion, a new framework for the study of the theory space of euclidean quantum field theories has been introduced. This formalism is particularly useful for elucidating the structure of the short-distance expansions of the $n$-point functions of a renormalizable quantum field theory near a non-trivial fixed point. We review and apply this formalism in the study of the scaling limit of the two dimensional massive Ising model. Renormalization group analysis and operator product expansions determine all the non-analytic mass dependence of the short-distance expansion of the correlation functions. An extension of the first order variational formula to higher orders provides a manifestly finite scheme for the perturbative calculation of the operator product coefficients to any order in parameters. A perturbative expansion of the correlation functions follows. We implement this scheme for a systematic study of correlation functions involving two spin operators. We show how the necessary non-trivial integrals can be calculated. As two concrete examples we explicitly calculate the short-distance expansion of the spin-spin correlation function to third order and the spin-spin-energy density correlation function to first order in the mass. We also discuss the applicability of our results to perturbations near other non-trivial fixed points corresponding to other unitary minimal models.