Proton radius from electron scattering data

Background: The proton charge radius extracted from recent muonic hydrogen Lamb shift measurements is significantly smaller than that extracted from atomic hydrogen and electron scattering measurements. The discrepancy has become known as the proton radius puzzle.

Purpose: In an attempt to understand the discrepancy, we review high-precision electron scattering results from Mainz, Jefferson Lab, Saskatoon, and Stanford.

Methods: We make use of stepwise regression techniques using the F test as well as the Akaike information criterion to systematically determine the predictive variables to use for a given set and range of electron scattering data as well as to provide multivariate error estimates.

Results: Starting with the precision, low four-momentum transfer (Q²) data from Mainz (1980) and Saskatoon (1974), we find that a stepwise regression of the Maclaurin series using the F test as well as the Akaike information criterion justify using a linear extrapolation which yields a value for the proton radius that is consistent with the result obtained from muonic hydrogen measurements. Applying the same Maclaurin series and statistical criteria to the 2014 Rosenbluth results on G_E from Mainz, we again find that the stepwise regression tends to favor a radius consistent with the muonic hydrogen radius but produces results that are extremely sensitive to the range of data included in the fit. Making use of the high-Q² data on G_E to select functions which extrapolate to high Q², we find that a Padé (N =M =1 ) statistical model works remarkably well, as does a dipole function with a 0.84 fm radius, G_E(Q²) =(1^{+Q2/0.66 GeV2) -2} .

Conclusions: Rigorous applications of stepwise regression techniques and multivariate error estimates result in the extraction of a proton charge radius that is consistent with the muonic hydrogen result of 0.84 fm; either from linear extrapolation of the extremely-low-Q² data or by use of the Padé approximant for extrapolation using a larger range of data. Thus, based on a purely statistical analysis of electron scattering data, we conclude that the electron scattering results and the muonic hydrogen results are consistent. It is the atomic hydrogen results that are the outliers.