Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications

TL;DR

This paper introduces a dynamical formulation of stationary scattering (DFSS) that provides a singularity-free approach to delta-function potentials in two dimensions, avoiding the need for renormalization and clarifying the origin of the standard singularity.

Contribution

The paper presents a new dynamical formulation of stationary scattering that inherently regularizes delta-function potentials in two dimensions, eliminating the logarithmic singularity without renormalization.

Findings

DFSS avoids the logarithmic singularity in 2D delta-function scattering.

The standard singularity arises from unphysical contributions parallel to detectors.

Renormalization subtracts these unphysical contributions, while DFSS achieves this intrinsically.

Abstract

In two dimensions, the standard treatment of the scattering problem for a delta-function potential, $v(\mathbf{r})=\mathfrak{z}\,\delta(\mathbf{r})$, leads to a logarithmic singularity which is subsequently removed by a renormalization of the coupling constant $\mathfrak{z}$. Recently, we have developed a dynamical formulation of stationary scattering (DFSS) which offers a singularity-free treatment of this potential. We elucidate the basic mechanism responsible for the implicit regularization property of DFSS that makes it avoid the logarithmic singularity one encounters in the standard approach to this problem. We provide an alternative interpretation of this singularity showing that it arises, because the standard treatment of the problem takes into account contributions to the scattered wave whose momentum is parallel to the detectors' screen. The renormalization schemes used for removing this singularity has the effect of subtracting these unphysical contributions, while DFSS has a built-in mechanics that achieves this goal.