Renormalization group and diffusion equation

TL;DR

This paper explores the connection between the renormalization group and diffusion equations, showing how diffused field correlations relate to bare and effective actions through a generalized diffusion process.

Contribution

It generalizes previous results by incorporating arbitrary cutoff functions and seed actions into the exact renormalization group framework for scalar fields.

Findings

Correlation functions of diffused fields match those of bare fields under effective actions.

The diffused field obeys a generalized diffusion equation influenced by cutoff and seed actions.

The results extend prior work by Sonoda and Suzuki to more general settings.

Abstract

We study the relationship between the renormalization group and the diffusion equation. We consider the exact renormalization group equation for a scalar field that includes an arbitrary cutoff function and an arbitrary quadratic seed action. As a generalization of the result obtained by Sonoda and Suzuki, we find that the correlation functions of diffused fields with respect to the bare action agree with those of bare fields with respect to the effective action, where the diffused field obeys a generalized diffusion equation determined by the cutoff function and the seed action and agrees with the bare field at the initial time.