Non-perturbative regularization and renormalization: simple examples from non-relativistic quantum mechanics

TL;DR

This paper compares regularization and renormalization methods in non-relativistic quantum mechanics, showing that dimensional regularization can fail to reproduce cutoff regularization results in certain non-perturbative cases involving zero-range potentials.

Contribution

It provides explicit examples demonstrating the limitations of dimensional regularization in non-perturbative quantum mechanical problems with zero-range potentials.

Findings

Dimensional regularization yields nontrivial scattering amplitudes in certain high odd dimensions.

Cutoff regularization results in trivial scattering amplitudes for dimensions greater than or equal to four.

Discrepancies between the two methods arise from the treatment of power-law divergences.

Abstract

We examine several zero-range potentials in non-relativistic quantum mechanics. The study of such potentials requires regularization and renormalization. We contrast physical results obtained using dimensional regularization and cutoff schemes and show explicitly that in certain cases dimensional regularization fails to reproduce the results obtained using cutoff regularization. First we consider a delta-function potential in arbitrary space dimensions. Using cutoff regularization we show that for $d \ge 4$ the renormalized scattering amplitude is trivial. In contrast, dimensional regularization can yield a nontrivial scattering amplitude for odd dimensions greater than or equal to five. We also consider a potential consisting of a delta function plus the derivative-squared of a delta function in three dimensions. We show that the renormalized scattering amplitudes obtained using the two regularization schemes are different. Moreover we find that in the cutoff-regulated calculation the effective range is necessarily negative in the limit that the cutoff is taken to infinity. In contrast, in dimensional regularization the effective range is unconstrained. We discuss how these discrepancies arise from the dimensional regularization prescription that all power-law divergences vanish. We argue that these results demonstrate that dimensional regularization can fail in a non-perturbative setting.