A Ginsparg-Wilson approach to lattice $\mathcal{CP}$ symmetry

TL;DR

This paper proposes a modified lattice $ ext{CP}$ symmetry approach using Ginsparg-Wilson fermions, resolving longstanding issues with chiral symmetry and $ ext{CP}$ invariance in lattice QCD, and constructs a $ ext{CP}$ invariant chiral gauge theory.

Contribution

It introduces a Ginsparg-Wilson based method to correctly realize $ ext{CP}$ symmetry on the lattice, ensuring proper continuum limit behavior and gauge invariance.

Findings

Modified lattice $ ext{CP}$ symmetry with smooth continuum limit

Construction of a $ ext{CP}$ invariant lattice chiral gauge theory

Resolution of charge conjugation symmetry issues on the lattice

Abstract

There is a long standing challenge in lattice QCD concerning the relationship between $\mathcal{CP}$-symmetry and lattice chiral symmetry: na\"ively the chiral symmetry transformations are not invariant under $\mathcal{CP}$. With results similar to a recent work by Igarashi and Pawlowski, I show that this is because charge conjugation symmetry has been incorrectly realised on the lattice. The naive approach, to directly use the continuum charge conjugation relations on the lattice, fails because the renormalisation group blockings required to construct a doubler free lattice theory from the continuum are not invariant under charge conjugation. Correctly taking into account the transformation of these blockings leads to a modified lattice $\mathcal{CP}$ symmetry for the fermion fields, which, for gauge field configurations with trivial topology, has a smooth limit to continuum $\mathcal{CP}$ as the lattice spacing tends to zero. After constructing $\mathcal{CP}$ transformations for one particular group of lattice chiral symmetries, I construct a lattice chiral gauge theory which is $\mathcal{CP}$ invariant and whose measure is invariant under gauge transformations and $\mathcal{CP}$.