Some issues concerning the proton charge radius puzzle

TL;DR

This paper investigates the proton charge radius discrepancy by analyzing the effects of vacuum polarization on the Lamb shift, emphasizing the importance of proper wave function propagation in QED calculations.

Contribution

It highlights the necessity of propagating modified wave functions with the vacuum polarization propagator to accurately explain the Lamb shift and resolve the proton radius puzzle.

Findings

The Uehling potential correction is consistent with the observed radius difference.

Proper wave function propagation affects the theoretical Lamb shift calculation.

Neglecting propagation leads to conflicting relativistic results.

Abstract

An explanation of the difference of the charge radius of the proton as determined from the Lamb shift in electronic hydrogen and from elastic electron scattering off the proton on the one side and the recent high precision determination with muonic hydrogen on the other side is presented. It is shown that the modification of the $2S_{1/2}$ and $2P_{3/2}$ wave functions by the "Uehling potential" yields a correction to the theoretical Lamb shift of $\delta (\Delta E_{\textrm{Lamb}}) = 0.302$ meV which has to be compared to $\delta (\Delta E_{\textrm{Lamb}}) = 0.322(46)$ meV equivalent to the stated radius difference. The explanation is based on the realization that the bound state wave functions modified by the external "Uehling potential" have to be propagated by the vacuum polarization propagator in order to give the correct leading order Lamb shift. It is argued that a conflicting relativistic calculation neglects this propagation aspect. The explanation demonstrates that the Lamb shift is dynamically induced through the QED vacuum polarization and is not only the result of a static external "Uehling potential" probed by a test charge.