Potential Flow Of The Renormalisation Group In A Simple Two Component Model

TL;DR

This paper investigates the renormalisation group flow in a simple two-coupling model involving scalar and fermion fields, revealing it can be derived from a potential when considering two-loop corrections and a specific metric.

Contribution

It demonstrates that the RG flow in a two-coupling model can be expressed as a potential flow using the Zamalodchikov metric at two-loop order.

Findings

RG flow is potential to sixth order in couplings.

Zamalodchikov metric induces potential structure.

Two-loop calculations are essential for this potential form.

Abstract

The renormalisation group (RG) flow on the space of couplings of a simple model with two couplings is examined. The model considered is that of a single component scalar field with $\phi^4$ self interaction coupled, via Yukawa coupling, to a fermion in flat four dimensional space. The RG flow on the two dimensional space of couplings, ${\cal G}$, is shown to be derivable from a potential to sixth order in the couplings, which requires two loop calculations of the $\beta$-functions. The identification of a potential requires the introduction of a metric on ${\cal G}$ and it is shown that the metric defined by Zamalodchikov, in terms of two point correlation functions of composite operators, gives potential flow to this order.