Supersymmetry on the lattice and the Leibniz rule

TL;DR

This paper investigates how to preserve supersymmetry on the lattice by analyzing the Leibniz rule failure, proposing conditions under which supersymmetry can be maintained in the continuum limit, and constructing a lattice Wess-Zumino model.

Contribution

It introduces a method to maintain supersymmetry on the lattice by controlling the Leibniz rule failure and constructs a lattice Wess-Zumino model that reproduces continuum results perturbatively.

Findings

Leibniz rule can be preserved if momentum satisfies |akμ|<δ as a→0.

A lattice Wess-Zumino model maintaining key symmetries is constructed.

The model reproduces continuum theory up to any finite order in perturbation theory.

Abstract

The major obstacle to a supersymmetric theory on the lattice is the failure of the Leibniz rule. We analyze this issue by using the Wess-Zumino model and a general Ginsparg-Wilson operator, which is local and free of species doublers. We point out that the Leibniz rule could be maintained on the lattice if the generic momentum $k_{\mu}$ carried by any field variable satisfies $|ak_{\mu}|<\delta$ in the limit $a\to 0$ for arbitrarily small but finite $\delta$. This condition is expected to be satisfied generally if the theory is finite perturbatively, provided that discretization does not induce further symmetry breaking. We thus first render the continuum Wess-Zumino model finite by applying the higher derivative regularization which preserves supersymmetry. We then put this theory on the lattice, which preserves supersymmetry except for a breaking in interaction terms by the failure of the Leibniz rule. By this way, we define a lattice Wess-Zumino model which maintains the basic properties such as $U(1)\times U(1)_{R}$ symmetry and holomorphicity. We show that this model reproduces continuum theory in the limit $a\to 0$ up to any finite order in perturbation theory; in this sense all the supersymmetry breaking terms induced by the failure of the Leibniz rule are irrelevant. We then suggest that this discretization may work to define a low energy effective theory in a non-perturbative way.