CP breaking in lattice chiral gauge theories

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.

A construction of the Glashow-Weinberg-Salam model on the lattice with exact gauge invariance

“…CP symmetry, however, is not manifest even with the real Yukawa couplings. But it is possible to show that at the quantum level both the partition function and the on-shell amplitudes respect the CP symmetry [68,69,70,71]. With the three families, then, the breaking of CP symmetry comes from the Kobayashi-Maskawa phase [72] as in the continuum theory.…”

General chiral gauge theories

“…Only requiring that the Dirac operator D allows a decomposition into Weyl operators we still extend the large class of operators decribing massless fermions on the lattice which we have found recently [3]. In addition to Ginsparg-Wilson (GW) fermions [4] this class includes the ones proposed by Fujikawa [5] and the extension of the latter [6].…”

CP breaking in lattice chiral gauge theory

“…In conclusion, the relation between the CP transformed generating functional (14) and the original generating functional (1) is written as 5…”

“…To add the Yukawa coupling to the present formulation, we introduce the right-handed Weyl 5 One can show that measure terms L {w,w} ξ and L {v,v} ξ are invariant under t → 2 − t [13], which means…”