Essential renormalisation group

TL;DR

This paper introduces a new renormalisation group scheme that simplifies calculations by focusing only on essential couplings and maintaining unrenormalised propagators, demonstrated through the Wilson-Fisher fixed point.

Contribution

The novel scheme reduces computational complexity by specifying conditions for inessential couplings and using non-linear field reparameterisations, streamlining the analysis of physical theories.

Findings

Simplifies the Wilson-Fisher fixed point analysis in three dimensions.

Removes all order ∂² operators except the canonical term at second order.

Maintains unrenormalised propagators at constant field values.

Abstract

We propose a novel scheme for the exact renormalisation group motivated by the desire of reducing the complexity of practical computations. The key idea is to specify renormalisation conditions for all inessential couplings, leaving us with the task of computing only the flow of the essential ones. To achieve this aim, we utilise a renormalisation group equation for the effective average action which incorporates general non-linear field reparameterisations. A prominent feature of the scheme is that, apart from the renormalisation of the mass, the propagator evaluated at any constant value of the field maintains its unrenormalised form. Conceptually, the simplifications can be understood as providing a description based only on quantities that enter expressions for physical observables since the redundant, non-physical content is automatically disregarded. To exemplify the scheme's utility, we investigate the Wilson-Fisher fixed point in three dimensions at order two in the derivative expansion. In this case, the scheme removes all order $\partial^2$ operators apart from the canonical term. Further simplifications occur at higher orders in the derivative expansion. Although we concentrate on a minimal scheme that reduces the complexity of computations, we propose more general schemes where inessential couplings can be tuned to optimise a given approximation. We further discuss the applicability of the scheme to a broad range of physical theories.