Revisiting the local potential approximation of the exact renormalization group equation

TL;DR

This paper revisits the local potential approximation (LPA) in the exact renormalization group framework, showing that incorporating a non-zero ta without momentum dependence aligns better with the derivative expansion logic.

Contribution

It introduces a modified LPA with ta  that does not rely on momentum dependence, challenging the conventional ta=0 assumption and highlighting reparametrization invariance issues.

Findings

LPA with ta  is incompatible with the derivative expansion logic.

A version of LPA with ta  can break reparametrization invariance.

Restoration of invariance occurs at higher orders of the derivative expansion.

Abstract

The conventional absence of field renormalization in the local potential approximation (LPA) --implying a zero value of the critical exponent \eta -- is shown to be incompatible with the logic of the derivative expansion of the exact renormalization group (RG) equation. We present a LPA with \eta \neq 0 that strictly does not make reference to any momentum dependence. Emphasis is made on the perfect breaking of the reparametrization invariance in that pure LPA (absence of any vestige of invariance) which is compatible with the observation of a progressive smooth restoration of that invariance on implementing the two first orders of the derivative expansion whereas the conventional requirement (\eta =0 in the LPA) precluded that observation.