Gradient formula for the beta-function of 2d quantum field theory

TL;DR

This paper provides a non-perturbative proof of a gradient formula for beta functions in two-dimensional quantum field theories, linking the beta functions to the c-function and metric corrections.

Contribution

It introduces a non-perturbative proof of the gradient formula for 2D QFT beta functions, including metric corrections and conditions for conformal symmetry.

Findings

Proves the gradient formula under specific infrared conditions.

Establishes a correspondence between RG fixed points and c-function critical points.

Extends the formula to non-linear sigma models.

Abstract

We give a non-perturbative proof of a gradient formula for beta functions of two-dimensional quantum field theories. The gradient formula has the form \partial_{i}c = - (g_{ij}+\Delta g_{ij} +b_{ij})\beta^{j} where \beta^{j} are the beta functions, c and g_{ij} are the Zamolodchikov c-function and metric, b_{ij} is an antisymmetric tensor introduced by H. Osborn and \Delta g_{ij} is a certain metric correction. The formula is derived under the assumption of stress-energy conservation and certain conditions on the infrared behaviour the most significant of which is the condition that the large distance limit of the field theory does not exhibit spontaneously broken global conformal symmetry. Being specialized to non-linear sigma models this formula implies a one-to-one correspondence between renormalization group fixed points and critical points of c.