Optimisation of the exact renormalisation group

TL;DR

This paper introduces a generic optimization criterion for coarse-grainings in exact renormalisation group equations, aiming to enhance the convergence of approximate solutions by adjusting the propagator and threshold functions.

Contribution

It provides explicit optimal parameters for various coarse-grainings and explains why the sharp cut-off regulator performs poorly in amplitude expansions.

Findings

Optimal parameters for coarse-grainings are explicitly given.

Sharp cut-off regulator is shown to be suboptimal for amplitude expansions.

The criterion improves convergence of approximate solutions.

Abstract

A simple criterion to optimise coarse-grainings for exact renormalisation group equations is given. It is aimed at improving the convergence of approximate solutions of flow equations. The optimisation criterion is generic, as it refers only to the coarse-grained propagator at vanishing field. In physical terms, it is understood as an optimisation condition for amplitude expansions. Alternatively, it can be interpreted as the requirement to move poles of threshold functions away from the physical region. The link to expansions in field amplitudes is discussed as well. Optimal parameters are given explicitly for a variety of different coarse-grainings. As a by-product it is found that the sharp cut-off regulator does not belong to the class of such optimal coarse-grainings, which explains the poor convergence of amplitude expansions based on it.