The third Zemach moment and the size of the proton

TL;DR

This paper investigates whether a small, localized feature in the proton's electric form factor at very low momentum transfer could explain the proton size discrepancy observed in muonic hydrogen, suggesting current data does not fully exclude this possibility.

Contribution

It proposes that a localized 'thorn' in the proton's electric form factor at low Q^2 could account for the large third Zemach moment, and recommends improved global fitting methods.

Findings

Current ep scattering data do not fully exclude the 'thorn' hypothesis.

A 'thorn' could lead to a larger third Zemach moment as suggested by De Rújula.

A more sophisticated fitting procedure is proposed for future analysis.

Abstract

To resolve the puzzle of the proton size raised from the recent result of muonic hydrogen Lamb shift, De R\'{u}jula has proposed that a large value of the third Zemach moment $< r^3_{p}>{(2)}$ of the proton to be the solution. His suggestion has been criticized by many groups based on the $ep$ scattering data at low $Q^2$ regime. However, if there is a "thorn" or "lump" in the electric form factor of the proton $G_{E}(Q^2)$ at extremely low $Q^2$ regime, then the third Zemach moment $< r^3_{p}>{(2)}$ would be as large as De R\'{u}jula suggested. In this article, we show that the existence of such a "thorn" or "lump" has not been completely excluded, although tightly restricted, by the current data of $ep$ elastic scattering. We also suggest a more sophisticated global fitting procedure of $G_{E}(Q^2)$ for the future fitting.