Unified formalism for electromagnetic and gravitational probes: densities

TL;DR

This paper investigates the relationships between relativistic light front densities and non-relativistic three-dimensional densities, highlighting the limitations of inverse Abel transforms and differences among various density definitions.

Contribution

It clarifies the connections and differences between light front, Breit-frame, and non-relativistic densities, and demonstrates the non-invertibility of the inverse Abel transform in this context.

Findings

Inverse Abel transform is not physically meaningful for light front densities.

Breit-frame densities differ significantly from true light front densities.

Numerical examples illustrate the discrepancies among density definitions.

Abstract

The use of light front coordinates allows a fully relativistic description of a hadron's spatial densities to be obtained. These densities must be two-dimensional and transverse to a chosen spatial direction. We explore their relationship to the three-dimensional, non-relativistic densities, with a focus on densities associated with the energy momentum tensor. The two-dimensional non-relativistic densities can be obtained from the light front densities through a non-relativistic limit, and can subsequently be transformed into three-dimensional non-relativistic densities through an inverse Abel transform. However, this operation is not invertible, and moreover the application of the inverse Abel transform to the light front densities does not produce a physically meaningful result. We additionally find that the Abel transforms of so-called Breit-frame densities generally differ significantly from the true light front densities. Numerical examples are provided to illustrate the various differences between the light front, Breit frame, and non-relativistic treatment of densities.