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.

Highlights

  • The energy-momentum tensor (EMT) and the associated gravitational form factors [1] have recently attracted significant interest in the hadron physics community

  • The form factors appearing in matrix elements of the EMT encode spatial densities via Fourier transforms

  • The three-dimensional Breit frame density was originally obtained by erroneously assuming that the hadron can be spatially localized [26,27], though it has been rehabilitated as a quasidensity through the Winger phase space formalism [16,28]

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Summary

INTRODUCTION

The energy-momentum tensor (EMT) and the associated gravitational form factors [1] have recently attracted significant interest in the hadron physics community. The form factors appearing in matrix elements of the EMT encode spatial densities via Fourier transforms. When performing these Fourier transforms, it is important to keep perspective about the actual, physical meaning of the densities that are obtained. The three-dimensional Breit frame density was originally obtained by erroneously assuming that the hadron can be spatially localized [26,27], though it has been rehabilitated as a quasidensity through the Winger phase space formalism [16,28]. The Abel transform has recently been proposed as a means of connecting the light front and Breit frame formalisms [30,31].

ABEL TRANSFORMS OF PHYSICAL DENSITIES
RELATIVISTIC AND NONRELATIVISTIC DENSITIES OF THE EMT
Relativistic light front densities
Transversely polarized hadrons
Breit frame densities
Nonrelativistic densities
Nonrelativistic reduction
Nonrelativistic mass density
Nonrelativistic stress tensor
Nonrelativistic form factors
MODEL CALCULATIONS
B M ð54aÞ
Breit-frame densities and transverse pseudodensities
Nonrelativistic limit
Selected examples
Deuteronlike kinematics
Pionlike kinematics
SUMMARY AND CONCLUSIONS
Full Text
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