Elastic scattering of electron by a Yukawa potential in non-commutative spacetime

TL;DR

This paper explores how non-commutative geometry modifies electron scattering by a Yukawa potential, revealing enhanced cross sections at small angles and establishing bounds on the non-commutative parameter.

Contribution

It derives the leading-order non-commutative correction to the Yukawa potential and analyzes its impact on scattering cross sections and energy bounds.

Findings

Non-commutativity enhances differential cross section at small angles.

The NC correction leads to a Kratzer-like potential.

A new lower bound on the NC parameter of about 10^{-28} meters.

Abstract

In this paper, we investigate the elastic scattering of an electron by a Yukawa potential within the framework of non-commutative (NC) geometry. We first derive the NC correction to the Yukawa potential at leading order in the NC parameter, resulting in a modified potential resembling a screened Kratzer potential. This potential reduces to the standard Kratzer form when considering the NC correction to the Coulomb potential. Subsequently, we calculate the NC correction to the electron scattering amplitude using the first-order Born approximation. We then analyze the effects of NC geometry on both the differential and total cross sections for elastic scattering. Our results indicate that non-commutativity enhances the differential cross section at small scattering angles and naturally gives rise to a Kratzer-like potential, reflecting the quantum nature of spacetime. Additionally, we establish a direct relationship between the system's energy level and the bound on the NC parameter. Specifically, for an ultra-relativistic incident electron scattering by a heavy molecule, we derive a new lower bound on $\Theta$ of the order of $10^{-28}\,\text{m}$.