Lattice Chiral Symmetry and the Wess-Zumino Model

TL;DR

This paper explores the challenges of implementing supersymmetric Wess-Zumino models on a lattice using Ginsparg-Wilson operators, highlighting conflicts with chiral symmetry and Yukawa couplings, and demonstrating partial preservation of supersymmetry at one-loop level.

Contribution

It identifies fundamental conflicts between lattice chiral symmetry and Yukawa couplings in Ginsparg-Wilson formulations and shows how to preserve a SUSY-like symmetry at finite lattice spacing.

Findings

One-loop superpotential non-renormalization is maintained at finite lattice spacing.

Finite wave function renormalization deviates from supersymmetric values.

The properties are independent of the specific Ginsparg-Wilson operator used.

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 (K\"{a}hler) 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.