Polynomial fits and the proton radius puzzle

The proton radius puzzle refers to the $\approx 7\sigma$ discrepancy that exists between the proton charge radius determined from muonic hydrogen and that determined from electronic hydrogen spectroscopy and electron-proton scattering. One possible partial resolution to the puzzle includes errors in the extraction of the proton radius from $ep$ elastic scattering data. This possibility is made plausible by certain fits that extract a smaller proton radius from the scattering data consistent with that determined from muonic hydrogen. The reliability of some of these fits that yield a smaller proton radius was studied. We found that fits of form factor data with a truncated polynomial fit are unreliable and systematically give values for the proton radius that are too small. Additionally, a polynomial fit with a ${\chi }_{\text{reduced}}^2\approx 1$ is not a sufficient indication for a reliable result.

Figure 1

A summary of some recent proton charge radius determinations: Sick [16], CODATA 2006 [17], Pohl et al. [1], Bernauer et al. [4, 5], CODATA 2010 [3], Zhan et al. [6], Hill & Paz [8], Sick Gaussians [9, 10], Lorenz et al. [13], Griffioen & Carlson [12], Antognini et al. [2], and Mart & Sulaksono [14]. The dashed and dotted lines are drawn at 0.88 and 0.84 fm, respectively, for reference.

Figure 2

Parametrizations of the proton electric form factor used to generate pseudodata, relative to the dipole form factor, $G_{\text{dipole}}={(1+Q^2/0.71{\mathrm{GeV}}^2)}^{-2}$.

Figure 3

The truncation offset versus $Q_{\text{max}}^2$ for the AMT parametrization. The lines indicate the size of the truncation offset and the bands the r.m.s. width of the radius distribution from the 5000 fits.

Figure 4

Reduced ${\chi }^2$ for the fits of the pseudodata generated from the AMT form factor parametrization versus $Q_{\text{max}}^2$.

Figure 5

The truncation offset divided by the fit uncertainty as a function of the fit ${\chi }_{\text{reduced}}^2$, for the AMT parametrization.

Figure 6

The truncation offset versus $Q_{\text{max}}^2$ for the third-order fits of the pseudodata generated from the different form factor parametrizations.

Figure 7

Reduced ${\chi }^2$ versus $Q_{\text{max}}^2$ for the third-order fits of the pseudodata generated from the different form factor parametrizations.

Figure 8

The truncation offset divided by the fit uncertainty as a function of the fit ${\chi }_{\text{reduced}}^2$, for the six different parametrizations.

Figure 9

The truncation offset versus $Q_{\text{max}}^2$ for liner and quadratic fits in the low $Q^2$ region.

Figure 10

Reduced ${\chi }^2$ versus $Q_{\text{max}}^2$ for linear and quadratic fits in the low $Q^2$ region.