Wegner-Houghton equation and derivative expansion

TL;DR

This paper investigates the derivative expansion in the Exact Renormalization Group framework for scalar theories, deriving coupled equations for potential and kinetic terms, and numerically analyzing the anomalous dimension at non-Gaussian fixed points.

Contribution

It introduces a coupled differential equation approach for potential and kinetic functions in the derivative expansion, including new approximations for the kinetic term's flow.

Findings

Sharp and smooth cut-offs yield identical results.

Two different approximations for the kinetic coefficient Z_k are proposed.

Numerical analysis determines the anomalous dimension at non-Gaussian fixed points.

Abstract

We study the derivative expansion for the effective action in the framework of the Exact Renormalization Group for a single component scalar theory. By truncating the expansion to the first two terms, the potential $U_k$ and the kinetic coefficient $Z_k$, our analysis suggests that a set of coupled differential equations for these two functions can be established under certain smoothness conditions for the background field and that sharp and smooth cut-off give the same result. In addition we find that, differently from the case of the potential, a further expansion is needed to obtain the differential equation for $Z_k$, according to the relative weight between the kinetic and the potential terms. As a result, two different approximations to the $Z_k$ equation are obtained. Finally a numerical analysis of the coupled equations for $U_k$ and $Z_k$ is performed at the non-gaussian fixed point in $D<4$ dimensions to determine the anomalous dimension of the field.