N=1 Euler Anomaly from RG-dependent metric-Background

Vladimir Prochazka,Roman Zwicky

doi:10.1051/epjconf/201713710009

Abstract

We consider ${\cal N}=1$ supersymmetric gauge theories in the conformal window. By applying a suitable matter superfield rescaling and a Weyl-transformation the renormalisation group running (matter and gauge field $Z$-factors) are absorbed into the metric. The latter becomes a function of the $Z$-factors. The Euler flow $\Delta a \equiv a_{\rm UV} - a_{\rm IR} |_{{\cal N}=1}$ is then obtained by free field theory computation with the non-trivial dynamics coming from expanding the Euler invariant in the flow dependent metric. The result is therefore directly obtained in terms of the infrared anomalous dimension confirming an earlier result using the matching of conserved currents.

Highlights

For a theory on a curved space, with no explicit scale symmetry breaking, the trace of the energy momentum tensor TEMT is parametrised by [1, 2]Θ = a E4 + b W2 + c H2 + c H, H≡ R, d−1 (1)where E4, W2 and R are the topological Euler term, the Weyl tensor squared and the Ricci scalar respectively
The difference of the Euler anomaly is computed using the conformal anomaly matching and dilaton effective action techniques used by Komargodski and Schwimmer (KS) [8, 9] to prove the a-theorem but at the same time differs substantially from it
The main result of this work is the computation of ∆a|N=1 (22) in the new framework where the dynamics, i.e. the Z-factor and γ∗ the anomalous dimension at the IR fixed point, is absorbed into the metric (20)

Summary

Introduction

For a theory on a curved space, with no explicit scale symmetry breaking, the trace of the energy momentum tensor TEMT is parametrised by [1, 2]. The exact expression for the difference of the ultraviolet (UV) and infrared (IR) Euler anomaly ∆a ≡ aUV − aIR was derived for N = 1 supersymmetric gauge theories [3] almost twenty years ago. Ever since it has served as a fruitful laboratory for testing different techniques by rederiving the result. We derive ∆a|N=1 by absorbing the Z-factor and the IR anomalous dimension γ∗ into the background metric This renders the theory, in the vacuum sector, equivalent to a free field theory in a curved background carrying the information of the dynamics gρλ = f (Z, γ∗)δρλ. The difference of the Euler anomaly is computed using the conformal anomaly matching and dilaton effective action techniques used by Komargodski and Schwimmer (KS) [8, 9] to prove the a-theorem but at the same time differs substantially from it

Framework

Findings

Discussion