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In the context of the lattice regularization of the four-dimensional$\mathcal{N}=1$ supersymmetric Yang--Mills theory (4D $\mathcal{N}=1$ SYM), weformulate 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 toa strong constraint on the quantum continuum limit of symmetry breaking termswith the lattice regularization. By analyzing the implications of theconstraint on operator-mixing coefficients in the SUSY and the $U(1)_A$Ward-Takahashi (WT) identities, we prove to all orders of perturbation theoryin the continuum limit that, (i) the chiral symmetric limit implies thesupersymmetric limit and, (ii) a three-fermion operator that might potentiallygive rise to an exotic breaking of the SUSY WT identity does not emerge. Inprevious literature, only a naive or incomplete treatment on these points canbe found. Our results provide a solid theoretical basis for latticeformulations of the 4D $\mathcal{N}=1$ SYM.

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