Lorentz Contracted Proton

TL;DR

This paper investigates relativistic effects on proton charge and magnetization densities derived from form factors, showing that including Lorentz corrections significantly reduces the proton radius obtained from scattering data.

Contribution

The authors extend the Breit equation to higher orders in 1/c^2 to incorporate relativistic corrections in the Fourier transform relating form factors and densities.

Findings

Relativistic corrections decrease the proton radius from scattering data.

The corrected proton radius is approximately 0.8404 fm.

Inclusion of corrections aligns scattering results closer to other measurements.

Abstract

The proton charge and magnetization density distributions can be related to the well known Sachs electromagnetic form factors $G_{E,M}({\bm q}^{2})$ through Fourier transforms, only in the Breit frame. The Breit frame however moves with relativistic velocities in the Lab and a Lorentz boost must be applied to the form factors before extracting the static properties of the proton from the corresponding densities. Apart from this, the Fourier transform relating the densities and form factors is inherently a non-relativistic expression. We show that the relativistic corrections to it can be obtained by extending the standard Breit equation to higher orders in its $1/c^2$ expansion. We find that the inclusion of the above corrections reduces the size of the proton determined from electron proton scattering data. Indeed the central value of the latest proton radius of $r_p = 0.879$ fm as determined from e-p scattering changes to $r_p = 0.8404$ fm after applying corrections.