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 kμ carried by any field variable satisfies |akμ|<δ in the limit a→0 for arbitrarily small but finite δ. 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)×U(1)R symmetry and holomorphicity. We show that this model reproduces continuum theory in the limit a→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.
