On the trace identity in a model with broken symmetry

TL;DR

This paper demonstrates a trace identity in a chiral fermion meson model with explicit symmetry breaking, linking dilatation breaking to the energy-momentum tensor trace via a Ward identity, valid to all perturbation orders.

Contribution

It establishes a trace identity acting as a Callan-Symanzik equation in a model with broken symmetry, connecting dilatation breaking to the energy-momentum tensor trace through a Ward identity.

Findings

Validates the trace identity to all orders of perturbation theory.

Expresses the conservation of energy-momentum via Ward identities.

Links dilatation breaking directly to the trace of the energy-momentum tensor.

Abstract

Considering the simple chiral fermion meson model when the chiral symmetry is explicitly broken, we show the validity of a trace identity -- to all orders of perturbation theory -- playing the role of a Callan-Symanzik equation and which allows us to identify directly the breaking of dilatations with the trace of the energy-momentum tensor. More precisely, by coupling the quantum field theory considered to a classical curved space background, represented by the non-propagating external vielbein field, we can express the conservation of the energy-momentum tensor through the Ward identity which characterizes the invariance of the theory under the diffeomorphisms. Our ``Callan-Symanzik equation'' then is the anomalous Ward identity for the trace of the energy-momentum tensor, the so-called ``trace identity''.