Third Zemach moment of the proton

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 Rújula. We also demonstrate that the size of the third Zemach moment is largely governed by the fourth moment of the conventional charge distributions $\langle r^4\rangle$, which enables us to obtain a rigorous upper bound on the magnitude of the third Zemach moment of the proton.

Figure 1

Plots (a) and (b) illustrate, for various $Q^2$ ranges, the proton $G_{\mathit{Ep}}$ form factor from the parametrizations of Kelly [5] (solid line), fit II from Alberico et al. [6] (dashed line), and the dipole form factor of Eq. (6) (dot-dashed line). The toy model of Ref. [3], with $\sin {}^2\theta =0.3$, $m=15$ MeV, and $M=750$ MeV, is illustrated as the dotted line. Note that to aid clarity, we do not plot the results from Arrington et al. [7] since they are similar to the results from Refs. [5, 6].

Figure 2

Plot of the integrand in Eq. (12) (solid line) and the dominant piece at moderate to large $q$ given by Eq. (14) (dashed line).