Third Zemach moment of the proton

Abstract

Modern electron scattering experiments have determined the proton electric form factor ${G}_{\mathit{Ep}}({Q}^{2})$ to high precision. We utilize this data, represented by the different empirical form-factor parametrizations, to compute the third Zemach moment of the proton charge distribution. We find that existing data rule out a value of the third Zemach moment large enough to explain the current puzzle with the proton charge radius, determined from the Lamb shift in muonic hydrogen. This is in contrast to the recent paper of De R\'ujula. We also demonstrate that the size of the third Zemach moment is largely governed by the fourth moment of the conventional charge distributions $\ensuremath{\langle}{r}^{4}\ensuremath{\rangle}$, which enables us to obtain a rigorous upper bound on the magnitude of the third Zemach moment of the proton.