Proton charge radius extraction from electron scattering data using dispersively improved chiral effective field theory

We extract the proton charge radius from the elastic form factor (FF) data using a novel theoretical framework combining chiral effective field theory and dispersion analysis. Complex analyticity in the momentum transfer correlates the behavior of the spacelike FF at finite $Q^2$ with the derivative at $Q^2=0$. The FF calculated in the predictive theory contains the radius as a free parameter. We determine its value by comparing the predictions with a descriptive global fit of the spacelike FF data, taking into account the theoretical and experimental uncertainties. Our method allows us to use the finite-$Q^2$ FF data for constraining the radius (up to $Q^2\approx 0.5 {\mathrm{GeV}}^2$ and larger) and avoids the difficulties arising in methods relying on the $Q^2\to 0$ extrapolation. We obtain a radius of 0.844(7) fm, consistent with the high-precision muonic hydrogen results.

Figure 1

Light shaded bands between solid lines (labeled “Theory”): $\mathrm{DI}\chi \mathrm{EFT}$ predictions of the proton charge FF $G_E^p\left(Q^2\right)$ [26] for several values of the proton radius, $r_E^p=(0.80,0.82,0.84,0.86,0.88,0.90)$ fm (the values are indicated on the panels). The bands show the theoretical uncertainty resulting from the effective description of high-mass states in the spectral function (see text). Dark shaded bands (labeled “Global Fit”):$G_E^p\left(Q^2\right)$ determined from global fits of the elastic FF data with constrained proton radius (see text) [27]. The bands show the experimental and fit uncertainties. Each panel's global fit was restricted to reproduce the indicated proton radius.

Figure 2

Light shaded band (labeled “Theory Uncertainty”): Theoretical uncertainty of the $\mathrm{DI}\chi \mathrm{EFT}$ predictions of $G_E^p\left(Q^2\right)$ for fixed $r_E^p$ (here $r_E^p=0.84$ fm; the uncertainty does not depend on the specific value of $r_E^p$; cf. Fig. 1) [26]. Dark shaded band and dashed area above (labeled “Fit Uncertainty”): Experimental uncertainty of the global FF fit [27] (68% confidence level, cf. Fig. 1). The dark shaded band shows the uncertainty of the fits with a fixed proton radius, as shown in the individual panels of Fig. 1. The dashed area above shows in addition the variation of the global fit under changes of the radius. Solid lines: Variation of the $\mathrm{DI}\chi \mathrm{EFT}$ predictions of $G_E^p\left(Q^2\right)$ under changes of $r_E^p$ by $\mathrm{\Delta }r=\pm 0.01$ fm, $\pm 0.02$ fm, etc. (the variation is computed at $r_E^p=0.84$ fm). The variation quantifies the sensitivity of the theoretical FF predictions to the proton radius parameter (cf. Fig. 1).

Figure 3

Reduced ${\chi }^2$, Eq. (1), as a function of the proton radius $r_E^p$. Shown are the results corresponding to three choices of the upper limit $Q_{\max }^2$.

Figure 4

Low-$Q^2$ behavior of the $\mathrm{DI}\chi \mathrm{EFT}$ parametrization of $G_E^p\left(Q^2\right)$ and its power expansion. Solid line: $\mathrm{DI}\chi \mathrm{EFT}$ parametrization ($r_E^p=0.844$ fm). Dashed line: First-order term in $Q^2$. Dashed-dotted line: Sum of first- and second-order terms. Shaded band: Uncertainty of the global fit with the given fixed proton radius.

Figure 5

Proton FF parametrization Eq. (A15) as function of $Q^2=-t$. Dashed line: $A_E^p\left(t\right)$. Dashed-dotted line: $A_E^p\left(t\right)+{\left(r_E^p\right)}^2{B}_E\left(t\right)$. Solid line: $A_E^p\left(t\right)+{\left(r_E^p\right)}^2{B}_E\left(t\right)+{\left(r_E^n\right)}^2{\overline{B}}_E\left(t\right)$ (full result).