Ferrara--Zumino supermultiplet and the energy-momentum tensor in the lattice formulation of 4D $\mathcal{N}=1$ SYM

Abstract

It is well-known that Noether currents in the classical four-dimensional $\mathcal{N}=1$ supersymmetric Yang--Mills theory (4D $\mathcal{N}=1$ SYM), i.e., the $U(1)_A$ current, the supersymmetry (SUSY) current and the energy-momentum tensor, form a multiplet under SUSY, called the Ferrara--Zumino supermultiplet. Inspired by this structure, we define the energy-momentum tensor in the lattice formulation of 4D $\mathcal{N}=1$ SYM by a renormalized super transformation of a lattice SUSY current. By using a renormalized SUSY Ward--Takahashi relation, the energy-momentum tensor so constructed is shown to be conserved in the quantum continuum limit. Our construction of the energy-momentum tensor is very explicit and usable in non-perturbative numerical simulations.