Supersymmetry, chiral symmetry and the generalized BRS transformation in lattice formulations of 4D $\mathcal{N}=1$ SYM

TL;DR

This paper develops a unified BRS framework for lattice 4D N=1 SYM, proving that chiral symmetry implies supersymmetry and ruling out exotic SUSY breaking operators, thus strengthening the theoretical foundation of lattice supersymmetric theories.

Contribution

It introduces a generalized BRS transformation unifying gauge, SUSY, translation, and axial symmetries, and proves key symmetry relations to all orders in perturbation theory.

Findings

Chiral symmetric limit implies supersymmetric limit.

No emergence of exotic three-fermion SUSY-breaking operators.

Provides a rigorous theoretical basis for lattice 4D N=1 SYM.

Abstract

In the context of the lattice regularization of the four-dimensional $\mathcal{N}=1$ supersymmetric Yang--Mills theory (4D $\mathcal{N}=1$ SYM), we formulate a generalized BRS transformation that treats the gauge, supersymmetry (SUSY), translation and axial U(1) ($U(1)_A$) transformations in a unified way. A resultant Slavnov--Taylor identity or the Zinn-Justin equation gives rise to a strong constraint on the quantum continuum limit of symmetry breaking terms with the lattice regularization. By analyzing the implications of the constraint on operator-mixing coefficients in the SUSY and the $U(1)_A$ Ward-Takahashi (WT) identities, we prove to all orders of perturbation theory in the continuum limit that, (i) the chiral symmetric limit implies the supersymmetric limit and, (ii) a three-fermion operator that might potentially give rise to an exotic breaking of the SUSY WT identity does not emerge. In previous literature, only a naive or incomplete treatment on these points can be found. Our results provide a solid theoretical basis for lattice formulations of the 4D $\mathcal{N}=1$ SYM.