QED confronts the radius of the proton

TL;DR

This paper examines discrepancies in proton radius measurements from muonic hydrogen and electron scattering, highlighting the importance of proper renormalization and the uncertain third Zemach moment in resolving the tension.

Contribution

It identifies the invalid approximation in muonic hydrogen analysis and emphasizes the role of the third Zemach moment in proton radius determinations.

Findings

Correcting the approximation shifts the radius estimate closer to other data.

The third Zemach moment is not empirically well-determined.

No significant tension remains when considering form factor analyticity.

Abstract

Recent results on muonic hydrogen [1] and the ones compiled by CODATA on ordinary hydrogen and $ep$-scattering [2] are $5\sigma$ away from each other. Two reasons justify a further look at this subject: 1) One of the approximations used in [1] is not valid for muonic hydrogen. This amounts to a shift of the proton's radius by $\sim 3$ of the standard deviations of [1], in the "right" direction of data-reconciliation. In field-theory terms, the error is a mismatch of renormalization scales. Once corrected, the proton radius "runs", much as the QCD coupling "constant" does. 2) The result of [1] requires a choice of the "third Zemach moment". Its published independent determination is based on an analysis with a $p$-value --the probability of obtaining data with equal or lesser agreement with the adopted (fit form-factor) hypothesis-- of $3.92\times 10^{-12}$. In this sense, this quantity is not empirically known. Its value would regulate the level of "tension" between muonic- and ordinary-hydrogen results, currently {\it at most} $\sim 4\sigma$. There is no tension between the results of [1] and the proton radius determined with help of the analyticity of its form factors.