Evaluation of the strength of electron-proton scattering data for determining the proton charge radius

Precisely measured electron-proton elastic scattering cross sections [Phys. Rev. Lett. 105, 242001 (2010)] are reanalyzed to evaluate their strength for determining the rms charge radius ($R_{\mathrm{E}}$) of the proton. More than half of the cross sections at lowest $Q^2$ are fit using two single-parameter form-factor models, with the first based on a dipole parametrization, and the second on a linear fit to a conformal-mapping variable. These low-$Q^2$ fits extrapolate the slope of the form factor to $Q^2=0$ and determine $R_{\mathrm{E}}$ values of approximately 0.84 and 0.89 fm, respectively. Fits spanning all $Q^2$, in which the single constants are replaced with cubic splines at larger $Q^2$, lead to similar results for $R_{\mathrm{E}}$. We conclude that the scattering data are consistent with $R_{\mathrm{E}}$ ranging from at least 0.84 to 0.89 fm, and therefore is consistent with both of the discrepant determinations of $R_{\mathrm{E}}$ made using muonic and electronic hydrogen-atom spectroscopy.

Figure 1

Fits to the low-$Q^2$ data of Ref. [4] using single-parameter models for the form factors in Eq. (1). In (a), the dipole model of Eq. (3) is used, and in (b) the linear model of Eq. (5) is used. Separate plots are required since the fits return different normalization constants, and therefore the cross sections take on slightly $(\sim 1\%)$ different values for the two plots. The $E=180$ MeV data and fits are continued in the insets, and the other energies are offset for clarity of presentation.

Figure 2

The range of $R_{\mathrm{E}}$ predicted from the dipole fits (lower bands) and conformal-mapping fits (upper bands) as a function of the cutoff value $Q_{\max }^2$. The red bands correspond to single-parameter dipole and linear fits of Eqs. (3) and (5), and the other colors show extensions to these fits for which the single parameters are replaced with cubic splines (with the number of nodes, $N_{\mathrm{kn}}$, as indicated) at larger $Q^2$. All fits shown have a reduced ${\chi }^2<1.14$.

Figure 3

Electric form factor $G_{\mathrm{E}}$ determined from the single-parameter fits of Fig 1. The blue curve is from the dipole-model fit, and the red curve is from the linear-model fit. Also shown on the plots are the low-$Q^2{G}_{\mathrm{E}}$ data from other experiments (from [23]), and these data are consistent with the $G_{\mathrm{E}}$ obtained from either fit.

Figure 4

The sensitivity of the fit results of Fig. 2 to two-photon exchange corrections is illustrated by redoing the analysis with poorer approximations for these corrections. The change in $R_{\mathrm{E}}$ when the Feshbach corrections (solid curves) and a low-$Q^2$ approximation [35] (dashed curves) are applied shows that the radius extracted from both the dipole model (a) and conformal-mapping model (b) is not very sensitive to these corrections.