Consistency of electron scattering data with a small proton radius

We determine the charge radius of the proton by analyzing the published low momentum transfer electron-proton scattering data from Mainz. We note that polynomial expansions of the form factor converge for momentum transfers squared below $4m_{\pi }^2$, where $m_{\pi }$ is the pion mass. Expansions with enough terms to fit the data, but few enough not to overfit, yield proton radii smaller than the CODATA or Mainz values and in accord with the muonic atom results. We also comment on analyses using a wider range of data, and overall obtain a proton radius $R_E=0.840(16$) fm.

Figure 1

Linear plus quadratic fit to all Mainz (2010) data with $Q^2<0.02{\mathrm{GeV}}^2$.

Figure 2

${\chi }^2$/dof vs $R_E$ for polynomial fits to the data set, showing fits with first-, second-, and third-order polynomials in $Q^2$ using data with $0.01<Q_{max}^2<0.08{\mathrm{GeV}}^2$. The small numbers near each data point gives $Q_{\max }^2$ in multiples of $0.01{\mathrm{GeV}}^2$. A good fit should have a good ${\chi }^2$ and a sufficient but not oversufficient number of parameters, as further discussed in the text. The quadratic fit “2” and the cubic fit “4” satisfy the criteria.

Figure 3

Four-parameter continued fraction fit to the full Mainz data set using ${\mu }_p{G}_E/G_M=1-Q^2/\left(8{\mathrm{GeV}}^2\right)$.

Figure 4

Minimum ${\chi }^2/\mathrm{dof}$ values for continued fraction fits of $G_E$ extracted using ${\mu }_p{G}_E/G_M=1-Q^2/Q_0^2$ with different values of $Q_0^2$, when using the Mainz data. Each point shows the value of $R_E$ extracted for a given $Q_0^2$. The quantity $R_E$ remains stable over the wide range $4<Q_0^2<20{\mathrm{GeV}}^2$, indicating that $R_E$ is not sensitive to the Ansatz for $G_E/G_M$.

Figure 5

Fits to world polarization transfer data. The data are from GEp I [21, 24], GEp II [22, 23], GEp III [25], Zhan et al. [30], Ron et al. [28], Crawford et al. [34], RSS [35], Paolone et al. [36], and Strauch et al. [37]. Some data with larger uncertainty limits have been omitted. The solid orange line uses ${\mu }_p{G}_E/G_M=1-Q^2/Q_0^2$ with $Q_0^2=8.02{\mathrm{GeV}}^2$, the black dashed line is the Bernauer et al. [6, 7, 8] fit up to about $0.3{\mathrm{GeV}}^2$ and a hybrid produced by other Mainz workers beyond that [38], and the green dotted line is a fit from Punjabi et al. [39].

Figure 6

$G_E$ vs $\epsilon$. The six sets, from top to bottom, correspond to the six beam energies of the experiment, 180, 315, 450, 585, 720, and 855 MeV.

Figure 7

Dependence of the extracted $G_E$ values on $\epsilon$ for 20 bins in $Q^2$. The data show little or no dependence on $\epsilon$ within statistics. (If desired, the actual $Q^2$ values may be inferred from the next figure.)

Figure 8

$\epsilon$ slopes for 14 bins in $Q^2$.

Figure 9

Differences between various fit forms and the standard continued fraction fit. All the power-series fit forms show undulations when enough parameters are included. The ${\chi }^2$/dof drops accordingly to about 1.37, but at the expense of erratic behavior at the origin and above $Q^2=1{\mathrm{GeV}}^2$. The lower panel expands the low-$Q^2$ region of the upper panel.

Figure 10

Renormalization constants for the 34 Mainz normalization sets. Red inverted triangles, black dots, and purple upright triangles correspond to spectrometers A, B, and C, respectively. The numbers are the six beam energies in MeV, and the points to the left of the corresponding arrow are the sets at that energy. The average and standard deviation of these normalization constants are 1 and 0.015%, respectively.

Figure 11

Average differences ${\left(G_E\right)}_i-f\left(Q_i^2\right)$ for 14 $Q^2$ bins before (a) and after (b) renormalizing the 34 data sets.

Figure 12

Differences, $G_E\left(Q_i^2\right)-f\left(Q_i^2\right)$, averaged over 14 $Q^2$ ranges for spectrometers A (a), B (b), and C (c), individually, after renormalizations of the 34 subsets.

Figure 13

Final global fit with the 34 data sets renormalized and the point-to-point uncertainties on $G_E$ scaled up by 15%.

Figure 14

Distribution of $[G_E\left(Q_i^2\right)-f\left(Q_i^2\right)]/{\sigma }_i$ for all data points. Here the individual uncertainties on each point ${\sigma }_i$ have all been rescaled by a factor of 1.15.