CP breaking in lattice chiral gauge theories

TL;DR

This paper analyzes the origins and nature of CP symmetry breaking in lattice chiral gauge theories with Ginsparg-Wilson operators, identifying where effects appear and discussing their implications in different scenarios.

Contribution

It precisely identifies the sources and characteristics of CP breaking in lattice chiral gauge theories using Ginsparg-Wilson operators, including effects on the fermion functional and propagators.

Findings

CP breaking appears in the fermion generating functional and propagators.

The measure transformation accounts for some CP breaking effects without anomalies.

Non-local CP breaking persists with Higgs vacuum expectation value, affecting continuum limits.

Abstract

The CP symmetry is not manifestly implemented for the local and doubler-free Ginsparg-Wilson operator in lattice chiral gauge theory. We precisely identify where the effects of this CP breaking appear. We show that they appear in: (I) Overall constant phase of the fermion generating functional. (II) Overall constant coefficient of the fermion generating functional. (III) Fermion propagator appearing in external fermion lines and the propagator connected to Yukawa vertices. The first effect appears from the transformation of the path integral measure and it is absorbed into a suitable definition of the constant phase factor for each topological sector; in this sense there appears no ``CP anomaly''. The second constant arises from the explicit breaking in the action and it is absorbed by the suitable weights with which topological sectors are summed. The last one in the propagator is inherent to this formulation and cannot be avoided by a mere modification of the projection operator, for example, in the framework of the Ginsparg-Wilson operator. This breaking emerges as an (almost) contact term in the propagator when the Higgs field, which is treated perturbatively, has no vacuum expectation value. In the presence of the vacuum expectation value, however, a completely new situation arises and the breaking becomes intrinsically non-local, though this breaking may still be removed in a suitable continuum limit. This non-local CP breaking is expected to persist for a non-perturbative treatment of the Higgs coupling.