Less naive about supersymmetric lattice quantum mechanics

TL;DR

This paper investigates the issues with naive lattice discretization of supersymmetric quantum mechanics, demonstrating the necessity of counterterms for correct continuum limits and showing improved actions and simulations.

Contribution

It provides a nonperturbative proof that counterterms are needed for naive discretization and introduces an improved action that better approximates the continuum theory.

Findings

Counterterms are necessary for naive lattice discretization to reach the continuum limit.

An improved O(a) action reduces discretization errors.

Supersymmetric lattice actions inherently cancel certain divergences, avoiding counterterms.

Abstract

We explain why naive discretization results that have appeared in [hep-lat/0006013] do not appear to yield the desired continuum limit. The fermion propagator on the lattice inevitably yields a diagram with nonvanishing UV degree D=0 contribution in lattice perturbation theory, in contrast to what occurs in the continuum. This diagram gives a finite correction to the boson 2-point function that must be subtracted off in order to obtain the perturbation series of the continuum theory, in the limit where the lattice spacing $a$ vanishes. Using a transfer matrix approach, we provide a nonperturbative proof that this counterterm suffices to yield the desired continuum limit. This analysis also allows us to improve the action to O(a). We demonstrate by Monte Carlo simulation that the spectrum of the continuum theory is well-approximated at finite but small $a$, for both weak and strong coupling. We contrast the above situation for the naive lattice action to what occurs for the supersymmetric lattice action, which preserves a discrete version of half the supersymmetry. There, cancellations between D=0 diagrams occur, obviating the need for counterterms.