Ginsparg–Wilson operators and a no-go theorem

Abstract

If one uses a general class of Ginsparg-Wilson operators, it is known that CP symmetry is spoiled in chiral gauge theory for a finite lattice spacing and the Majorana fermion is not defined in the presence of chiral symmetric Yukawa couplings. We summarize these properties in the form of a theorem for the general Ginsparg-Wilson relation.

Domain wall fermion andCPsymmetry breaking

“…The appearance of the singular factor 1/(γ 5 Γ 5 ) is consistent with our "no-go theorem" [2]. The question related to the existence of the inverse 1/(γ 5 Γ 5 ) is discussed later.…”

“…Namely, it is impossible to maintain the manifest invariance of the local and chiral Lagrangian under the CP transformation [1,2].…”

“…This is again interpreted as a change of representation of chiral projection operators, from P ± andP ± to γ 5P± γ 5 and P ± , constructed from a vector-like Ginsparg-Wilson theory if one introduces two sets of fermion fields ψ (1) and ψ (2) in eq. (4.1)…”

“…It has been recently shown that CP symmetry in chiral gauge theory [1,2,3] and also the Majorana reduction in the presence of chiral symmetric Yukawa couplings [4] have a certain conflict with lattice chiral symmetry, doubler-free and locality conditions in the framework of Ginsparg-Wilson operators [5]- [19]. There exists a closely related formulation of lattice fermions which is called the domain wall fermion [20]- [26].…”

“…Although the above simplest form of the Ginsparg-Wilson relation is relevant to our analysis of the domain wall fermion, the generality of the conflict with CP (or C) symmetry is best understood if one considers a more general algebraic relation [2] γ 5 H + Hγ 5 = 2H 2 f (H 2 ) (1.3)…”