Lattice chiral symmetry and the Wess–Zumino model

Abstract

A lattice regularization of the supersymmetric Wess-Zumino model is studied by using Ginsparg-Wilson operators. We recognize a certain conflict between the lattice chiral symmetry and the Majorana condition for Yukawa couplings, or in Weyl representation a conflict between the lattice chiral symmetry and Yukawa couplings. This conflict is also related, though not directly, to the fact that the kinetic (Kähler) term and the superpotential term are clearly distinguished in the continuum Wess-Zumino model, whereas these two terms are mixed in the Ginsparg-Wilson operators. We illustrate a case where lattice chiral symmetry together with naive Bose-Fermi symmetry is imposed by preserving a SUSY-like symmetry in the free part of the Lagrangian; one-loop level non-renormalization of the superpotential is then maintained for finite lattice spacing, though the finite parts of wave function renormalization deviate from the supersymmetric value. All these properties hold for the general Ginsparg-Wilson algebra independently of the detailed construction of lattice Dirac operators.

Domain wall fermion andCPsymmetry breaking

“…In fact this formulation agrees with a past suggestion [4,30] of the Wess-Zumino action in terms of the GinspargWilson operators.…”

“…It was first shown that the variables q L andq L cannot describe the topological properties, and the full physical contents are only described by the local Ginsparg-Wilson variables ψ L andψ L . The domain wall variables q andq in the infinite flavor limit, which themselves exhibit nice CP and charge conjugation properties, cannot help to resolve the difficulty associated with CP symmetry in chiral gauge theory [1] and the failure of the Majorana condition in the presence of chiral symmetric Yukawa couplings [4]. The conflict among the good chiral property, strict locality and CP (or charge conjugation) symmetry thus persists.…”

“…We then have 4 We define the CP operation by W = Cγ 0 = γ 2 with hermitian γ 2 and the CP transformed gauge field by U CP , and then…”

“…The above complication of CP symmetry (or equivalently charge conjugation symmetry since the parity is normal in the above model) for the vector-like theory with the chiral invariant Yukawa coupling gives rise to a difficulty in defining Majorana fermions in a Euclidean sense [4,2]. Following the standard procedure, we replace the field variables [36,37,38] …”

“…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].…”