Critical two-point correlation functions and "equation of motion" in the phi^4 model

TL;DR

This paper investigates the asymptotic behavior of critical two-point correlation functions in phi^4 models, challenging some assumptions of conformal field theory and highlighting issues with its formalism and equations of motion.

Contribution

The study provides non-perturbative analysis of correlation functions in phi^4 models and questions the validity of conformal field theory assumptions in this context.

Findings

Critical correlation functions are proportional to <phi(0) phi(x)> at large distances.

Questions the applicability of conformal field theory to phi^4 models.

Identifies issues with the 'equation of motion' in the CFT framework.

Abstract

Critical two-point correlation functions in the continuous and lattice phi^4 models with scalar order parameter phi are considered. We show by different non-perturbative methods that the critical correlation functions <phi^n(0) phi^m(x)> are proportional to <phi(0) phi(x)> at |x| --> infinity for any positive odd integers n and m. We investigate how our results and some other results for well-defined models can be related to the conformal field theory (CFT), considered by Rychkov and Tan, and reveal some problems here. We find this CFT to be rather formal, as it is based on an ill-defined model. Moreover, we find it very unlikely that the used there "equation of motion" really holds from the point of view of statistical physics.