Supersymmetry on the lattice and the Leibniz rule

Kazuo Fujikawa

doi:10.1016/S0550-3213(02)00443-1

paper

Supersymmetry on the lattice and the Leibniz rule

arXiv:hep-th/0205095 · doi:10.1016/S0550-3213(02)00443-1

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 $k_{μ}$ carried by any field variable satisfies $∣ a k_{μ} ∣ < δ$ in the limit $a \to 0$ 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 $U (1) \times U (1)_{R}$ 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.

19 pages. References were up-dated, and a brief footnote on Wilson fermion was added. Nucl. Phys. B (in press)

Supersymmetry on the lattice and the Leibniz rule

Abstract

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