The Supersymmetric Ward-Takahashi Identity in 1-Loop Lattice Perturbation Theory. I. General Procedure

TL;DR

This paper develops a general method to compute one-loop corrections to the supersymmetric Ward-Takahashi identity in lattice N=1 SYM, addressing supersymmetry breaking effects and operator mixing, with implications for restoring SUSY in the continuum limit.

Contribution

It introduces a systematic procedure to calculate renormalization constants and operator mixing for the supercurrent in lattice supersymmetric theories.

Findings

Supercurrent mixes with gauge invariant operators and others from the Ward-Takahashi identity.

Extra operator mixing persists off-shell but cancels on-shell when the gluino mass is zero.

Comparison with numerical results supports the theoretical framework.

Abstract

The one-loop corrections to the lattice supersymmetric Ward-Takahashi identity (WTi) are investigated in the off-shell regime. In the Wilson formulation of the N=1 supersymmetric Yang-Mills (SYM) theory, supersymmetry (SUSY) is broken by the lattice, by the Wilson term and is softly broken by the presence of the gluino mass. However, the renormalization of the supercurrent can be realized in a scheme that restores the continuum supersymmetric WTi (once the on-shell condition is imposed). The general procedure used to calculate the renormalization constants and mixing coefficients for the local supercurrent is presented. The supercurrent not only mixes with the gauge invariant operator $T_\mu$. An extra mixing with other operators coming from the WTi appears. This extra mixing survives in the continuum limit in the off-shell regime and cancels out when the on-shell condition is imposed and the renormalized gluino mass is set to zero. Comparison with numerical results are also presented.