Transverse charge density and the radius of the proton

A puzzling discrepancy exists between the values of the proton charge radius obtained using different experimental techniques: elastic electron-proton scattering and spectroscopy of electronic and muonic hydrogen. The proton radius is defined through the slope of the electric form factor, ${G}_{E}({Q}^{2})$, at zero four-momentum transfer, which is inaccessible in scattering experiments. We propose a novel method for extracting the proton radius from scattering data over a broad ${Q}^{2}$ range rather than attempting to directly determine the slope of ${G}_{E}$ at ${Q}^{2}=0$. This method relates the radius of the proton to its transverse charge density, which is the two-dimensional Fourier transform of the Dirac form factor, ${F}_{1}({Q}^{2})$. We apply our method to reanalyze the extensive data obtained by the A1 Collaboration [J. C. Bernauer et al. Phys. Rev. Lett. 105, 242001 (2010)] and extract a radius value, ${r}_{E}=0.889(5{)}_{\mathrm{stat}}(5{)}_{\mathrm{syst}}(4{)}_{\text{model}}\text{ }\text{ }\mathrm{fm}$, that is consistent with the original result. We also provide new parametrizations for the Dirac and Pauli form factors and the transverse charge and magnetization densities of the proton. Our reanalysis shows that the proton radius discrepancy cannot be explained by issues with fitting and extrapolating the A1 data to ${Q}^{2}=0$.