The Proton Radius Problem and Point-Particle Effective Field Theory

Abstract: We argue that the proton's charge-radius contributes differently to shifts of Hydrogen-like energy levels than naively expected due to an incorrect choice for the boundary condition at the proton's position in standard calculations. In particular we show how to obtain the correct boundary condition, which depends on the charge radius itself in a predictable way. We argue this difference in boundary conditions only matters when they are imposed at a radius $r=\epsilon < Z \alpha/m$ where $m$ is the orbiting-particle mass, because only then is the particle relativistic at these distances. The boundary condition difference is therefore important for ordinary Hydrogen while not for muonic Hydrogen. The boundary condition can be interpreted in terms of a second type of nuclear moment, and a prediction is made for the proton-radius energy shift as a function of charge-radius, $r_p$, this second nuclear moment, $h$, and orbiting particle mass, $m$. The observed difference between electronic and muonic contributions to the Lamb shift is accounted for with $r_p \simeq 0.87$ fm similar to its traditional value, and $2mh$ of order a few fm.