# Defining the Proton Radius: a Unified Treatment

## TL;DR

This paper unifies hydrogen spectroscopy and parton distribution approaches to define the proton radius as a relativistically proper quantity, clarifying its measurement and theoretical interpretation.

## Contribution

It introduces a relativistically correct, two-dimensional charge density and a new moment expansion (RME) to accurately define the proton radius.

## Key findings

- Hydrogen spectroscopy and elastic lepton scattering measure the same proton radius.
- The traditional derivation of the mean-square radius as a 3D density moment is incorrect.
- A new relativistically invariant moment expansion (RME) is proposed.

## Abstract

Background: There is significant current interest in knowing the value of the proton radius and also its proper definition. Purpose: Combine the disparate literatures of hydrogen spectroscopy and diverse modern parton distributions to show that the quantity $r_p^2\equiv -6 G_E'(0)$ is the relativistically proper definition that originates from the separate bodies of work. Methods: Use perturbation theory, light-front dynamics and elementary techniques to find relativistically correct definitions of the proton radius and charge density. Results: It is found that the very same proton radius is accessed by measurements of hydrogen spectroscopy and elastic lepton scattering. The derivation of the mean-square radius as a moment of a spherically symmetric three-dimensional density is shown to be incorrect. A relativistically-correct, two-dimensional charge density is related to the diverse modern literature of various parton distributions. Relativistically invariant moments thereof are derived in a new moment expansion, the RME.