Proton radius from electron-proton scattering and chiral perturbation theory

We determine the root-mean-square proton charge radius, $R_p$, from a fit to low-$Q^2$ electron-proton elastic-scattering cross-section data with the higher moments fixed (within uncertainties) to the values predicted by chiral perturbation theory. We obtain $R_p=0.855\left(11\right)$ fm. This number falls between the value obtained from muonic hydrogen analyses and the CODATA value (based upon atomic hydrogen spectroscopy and electron-proton scattering determinations).

Figure 1

Shown is the fit of the MAMI data [22] up to $Q_{\max }^2=0.023$ ${\mathrm{GeV}}^2$ (the largest $Q_{\max }^2$ value considered in this work). The quantity plotted is the experimental cross section, scaled as in Eq. (4), with the contribution from the magnetic form factor (as predicted from chiral perturbation theory) subtracted out. The derivative of this quantity with respect to $Q^2$ at $Q^2=0$ gives the value of $R_p$ [Eq. (3)]. The fit uses the functional forms of Eqs. (5) and (6), with $R_p$ floating and the higher moments set to the values of Table 1. The different symbols (colors) represent separate data groups, with the closed circles (blue), open circles (red), and open squares (green) data taken at an energy of 180 MeV, the closed squares (magenta) at 315 MeV, and the crosses (gray) at 450 MeV. Each group has a separate normalization constant that must also float in the fit. Repeated measurements within a data group at identical (or nearly identical) $Q^2$ values have been averaged only for the purpose of aiding the clarity of the presentation.

Figure 2

(a) The range of $R_p$ allowed by the fits as a function of $Q_{\max }^2$. The central (blue) region is the range of uncertainty from the fits. The expanded (pink) range includes the uncertainty in ${\langle r^2\rangle }_M$. The largest (green) band includes the uncertainties in all of the higher-order moments, and represents the full uncertainty in the determination. The dashed lines show the 0.855(11)-fm range of Eq. (9), which represents the final result of this work. (b) A comparison of the result of this work to the original (Refs. [1, 2]${}^a)$ and chiral perturbation theory motivated (Ref. [21]${}^b)$ determinations of $R_p$ from muonic hydrogen [1, 2] and from muonic deuterium (Ref. [46]${}^c$, along with the hydrogen-deuterium $1S\text{−}2S$ isotope shift of Ref. [47]), CODATA 2010 (Ref. [3]${}^d)$ and 2014 (Ref. [4]${}^e)$, and the MAMI analysis of their electron-proton elastic-scattering data (Ref. [23]${}^f)$.