Realistic Transverse Images of the Proton Charge and Magnetic Densities

TL;DR

This paper introduces the finite radius approximation (FRA), a method based on the Shannon-Nyquist sampling theorem, to accurately determine proton transverse charge and magnetic densities from experimental form factors, including uncertainty analysis.

Contribution

The paper develops the FRA technique that relates form factors to transverse densities with quantified uncertainties, providing a new way to interpret experimental data in proton structure studies.

Findings

Proton transverse charge density is well known for separations >0.1 fm.

The Pauli form factor allows reasonable magnetic density determination up to 10 GeV$^2$.

FRA relates form factors to coordinate space regions precisely.

Abstract

We develop a technique, denoted as the finite radius approximation (FRA), that uses a two-dimensional version of the Shannon-Nyquist sampling theorem to determine transverse densities and their uncertainties from experimental quantities. Uncertainties arising from experimental uncertainties on the form factors and lack of measured data at high $Q^2$ are treated. A key feature of the FRA is that a form factor measured at a given value of $Q^2$ is related to a definite region in coordinate space. An exact relation between the FRA and the use of a Bessel series is derived. The proton Dirac form factor is well enough known such that the transverse charge density is very accurately known except for transverse separations $b$ less than about 0.1 fm. The Pauli form factor is well known to $Q^2$ of about 10 GeV$^2$, and this allows a reasonable, but improvable, determination of the anomalous magnetic moment density.