High-precision determination of the electric and magnetic radius of the proton

Using dispersion theory with an improved description of the two-pion continuum based on the precise Roy-Steiner analysis of pion-nucleon scattering, we analyze recent data from electron-proton scattering. This allows for a high-precision determination of the electric and magnetic radius of the proton, $r_E = (0.838^{+0.005}_{-0.004}{}^{+0.004}_{-0.003})\,$fm and $r_M = (0.847\pm{0.004}\pm{0.004})\,$fm, where the first error refers to the fitting procedure using bootstrap and the data while the second one refers to the systematic uncertainty related to the underlying spectral functions.


Introduction
The electric radius, r E , and the magnetic radius, r M , of the proton are fundamental quantities of low-energy QCD, as they are a measure of the probe-dependent size of the proton. While the electric radius of the proton has attracted much attention in the last decade (see, e.g., Refs. [1,2,3] for recent reviews), this is not true for its magnetic counterpart, which is not probed in the Lamb shift in electronic or muonic hydrogen. A major source of information on the proton form factors (and the corresponding radii) is elastic electron-proton (ep) scattering. These data can be best analyzed in the time-honored framework of dispersion theory [4,5,6,7], which includes all constraints from unitarity, analyticity and crossing symmetry and is consistent with the strictures from perturbative QCD at very large momentum transfer [8]. Of particular importance for the proper extraction of the radii is the isovector two-pion continuum on the left shoulder of the ρ-resonance [9,10], which can be worked out model-independently using dispersively constructed pion-nucleon scattering amplitudes combined with data of the pion vector form factor. Based on the recent Roy-Steiner analysis of pion-nucleon scattering [11], an improved determination of the two-pion continuum was given in Ref. [12], which also includes thorough error estimates. Using a sum rule for the isovector charge radius, in that paper a squared isovector charge radius, (r v E ) 2 = 0.405(36) fm 2 , was obtained which is in perfect agreement with a recent state-of-the-art lattice QCD calculation at physical pion masses, (r v E ) 2 = 0.400(13) sta (11) sys fm 2 [13]. This underscores the importance of the isovector two-pion continuum for the form factors and demonstrates the consistency of the new and improved representation from Ref. [12] with QCD. This new representation of the two-pion continuum has so far not been employed in any dispersion-theoretical analysis of form factor data.
With the advent of new and precise electron-scattering data at low momentum transfer from Jefferson Laboratory (PRad collaboration) [14], it is timely to analyze these together with the precise data from the A1 collaboration at the Mainz Microtron (MAMI) [15] using the improved two-pion continuum contribution. Given the precision of these data and of the underlying formalism, this will allow for a high-precision determination of both the electric and the magnetic form factors and the corresponding radii, r E and r M , respectively. Clearly, this is an important step in pinning down these fundamental quantities with high precision. Other recent analyses of the PRad and Mainz data can be found in Refs. [16,17,18] and will be discussed below.

Formalism
Here, we briefly summarize the underlying formalism, which is detailed in Refs. [19,20]. The differential cross section for ep scattering can be expressed through the electric (G E ) and magnetic (G M ) Sachs form factors (FFs) as where ǫ = [1 + 2(1 + τ ) tan 2 (θ/2)] −1 is the virtual photon polarization, θ is the electron scattering angle in the laboratory frame, τ = −t/4m 2 N , with t the four-momentum transfer squared and m N the nucleon mass. Moreover, (dσ/dΩ) Mott is the Mott cross section, which corresponds to scattering off a point-like spin-1/2 particle. Since −t ≡ Q 2 > 0 is spacelike in ep scattering, the form factors are often displayed as a function of Q 2 . Equation (1) will be our basic tool to analyze the data together with the two-photon corrections from Ref. [20].
The electric and magnetic radii of the proton, which are at the center of this investigation, are given by For the theoretical analysis, it is advantageous to use the Dirac (F 1 ) and Pauli (F 2 ) FFs, which are linear combinations of the Sachs FFs: The FFs for spacelike momentum transfer, t < 0, are given in terms of an unsubtracted dispersion relation, with t 0 = 4M 2 π (9M 2 π ) the isovector (isoscalar) threshold, and M π is the charged pion mass. The spectral functions are expressed in terms of (effective) vector meson poles and continua, which leads to the following representation of the FFs: with i = 1, 2, in terms of the isoscalar (s) and isovector (v) components, F This representation is advantageous for dispersion analyses since the intermediate states contributing to the spectral function have good isospin. In the isoscalar spectral function, the first two poles correspond to the ω(782) and the φ(1020) mesons, so these masses are fixed and the residua are bounded as in Ref. [20]. Furthermore, we take into account the πρ andKK continua as explained in detail in Ref. [19]. The last term in the isovector form factor corresponds to the parameterization of the two-pion continuum taken from Ref. [11]. This is the essential new theory input compared to earlier dispersive analyses. The higher mass poles are effective poles that parameterize the spectral function at large t. We explicitly check in our analysis that the radii are insensitive to the details of this parameterization. The fit parameters are therefore the various vector meson residua a V i and the masses of the additional vector mesons s i , v i .
In addition, we fulfill the normalization conditions F 1 (0) = 1 (in units of the elementary charge e) and F 2 (0) = µ p , with µ p the anomalous magnetic moment of the proton. To ensure the stability of the fit [21], we demand that the residua of the vector meson poles are bounded, |a V i | < 5 GeV 2 , and that no effective poles with masses below 1 GeV appear. Finally, the FFs must satisfy the superconvergence relations with n = 0 for F 1 and n = 0, 1 for F 2 , corresponding to the fall-off with inverse powers of Q 2 at large momentum transfer as demanded by perturbative QCD [8].
These parameterizations (5) are used in Eq. (1) and the number of isoscalar and isovector poles is determined by the condition to obtain the best fit to the data. The quality of the fits is measured in terms of the traditional χ 2 , where C i are the cross section data at the points Q 2 i , θ i and C(Q 2 i , θ i , p ) are the cross sections for a given FF parameterization for the parameter values contained in p. Moreover, n k are normalization coefficients for the various data sets (labelled by the integer k), while σ i and ν i are their statistical and systematical errors, respectively. A more refined definition of the χ 2 is given by [20] in terms of the covariance matrix V ij = σ i σ j δ ij + ν i ν j . Theoretical errors will be calculated on the one hand using a bootstrap method. We simulate a large number of data sets by randomly varying the points in the original set within the given errors assuming their normal distribution. We then fit to each of them separately, derive the radius from each fit, and analyze the distribution of these radius values (see App. D of Ref. [20] for details). On the other hand theoretical errors are estimated by varying the number of effective vector meson poles. The first error thus gives the uncertainty due to the fitting procedure and the data while the second one reflects the accuracy of the spectral functions underlying the dispersion-theoretical analysis.

Results
As a first validation of our method, we only consider the PRad data [14]. These can be best fitted with the lowest two isoscalar mesons (the ω and the φ) and two additional isovector ones (2s + 2v poles). Fitting with statistical errors only (as in Ref. [14]), we have a χ 2 /dof = 1.33, completely consistent with the results reported there. Including also the systematic errors, the reduced χ 2 is slightly improved and we find as central values consistent with the PRad result, r E = (0.831 ± 0.007 stat ± 0.012 syst ) fm for the electric radius. In our case, the first error is obtained by bootstrap using 1000 samples and the second error is obtained by varying the number of poles from two isoscalar and two isovector ones (which gives the best solution) up to 5 isoscalar plus 5 isovector poles. While the absolute χ 2 of these 8 different solutions is almost the same, the χ 2 /dof increase from 1.33 to 1.61. Note also that the uncertainty in the magnetic radius is sizeably larger than the one of the electric radius, which is due to the fact that at the very low Q 2 probed by PRad, the electric form factor dominates the cross section. Next, we turn to the combined analysis of the Mainz and the PRad data. We note that we increase the weight of the PRad data by a factor ten in the combined χ 2 . This is legitimate as the PRad data probe much smaller momentum transfer than the Mainz data and thus should be enhanced. Changing this  weight by a factor of two leads to changes in the proton radii that are well covered by the uncertainties discussed below. The best fit to these data is shown in Fig. 1 for the PRad data and the Mainz data (normalized to the dipole cross section σ dip ). To best describe these combined data sets requires 5s + 5v poles (as in Ref. [20] for the MAMI data alone) with a χ 2 /dof = 1.25. The corresponding vector meson parameters (masses, residua) and the normalization constants of the various data sets are collected in Appendix A. In such a combined fit, the PRad data are described slightly worse than before, as the
Mainz data set is much larger and has also smaller error bars. The resulting radii are: These results are consistent with our earlier dispersion-theoretical determinations [22,20] but have much improved and smaller uncertainties. In particular, the fit error based on bootstrap (using 5000 samples) is considerably improved as compared to Ref. [20] which is related to the new PRad data, while the fit error for the magnetic radius is similar (as it is dominated by the Mainz data). The theory error estimate is obtained performing 11 different sets of fits with varying numbers of isoscalar and isovector poles, from 2s + 2v to 7s + 7v poles, where the reduced χ 2 /dof only varies by less than 3% from the optimal value obtained for 5s + 5v poles. Note further that the theory uncertainty is improved compared to the one in Ref. [22], with the differences that we do not include neutron data here and a much better and precise determination of the so important isovector two-pion continuum is employed.
In Table 1 we have collected these results together with other recent determinations of r E and r M employing both the PRad and the Mainz data. Within the quoted errors all extracted radii are consistent. The analysis in Ref. [16] is similar to ours. In contrast to our approach, however, it employs a dispersively improved chiral perturbation theory representation of the two-pion continuum. This approach is subject to uncertainties in the ρ-region as stressed in Ref. [23], different from the exact representation used here. The work of Ref. [17] can not be directly compared, as it is based on a continued fraction approach which has no relation to the dispersion-theoretical method used here. Also, in that paper no value for the magnetic radius is given. Similar remarks hold for the work of Ref. [18], which applies various fit functions (not guided by unitarity) to the flavor-dependent Dirac form factors with Q 2 ≤ 1 GeV 2 to extract the proton and the neutron charge radii. We note that our charge radius is also consistent with the current CODATA value, r E = 0.8414 (19) fm [26]. Moreover, it is in agreement with the value r E = 0.827 (20) fm obtained by combining the recent precise lattice result by Djukanovic et al. for the isovector radius with the experimental neutron charge radius [13,27]. Another recent lattice study calculated the proton charge radius directly but neglected disconnected contributions to the isoscalar current, which are computationally very expensive [28]. Because of this, the charge radius comes out smaller but their isovector result is in agreement with Ref. [13] and the experimental value. 1 Thus finally a consistent picture for the proton charge radius appears to emerge [2].
In Fig. 2 (left panel), we show the resulting electric and magnetic FF of the proton normalized to the dipole FF, These show a behavior similar to what was found in earlier dispersion-theoretical studies, and we note that again physically constrained fits do not produce oscillations as seen, e.g., the the magnetic form factor in Ref. [15]. We refrain here from displaying the corresponding uncertainties estimates to avoid clutter. In the right panel of Fig. 2, the FF ratio µ p G E (Q 2 )/G M (Q 2 ) is displayed together with the data of Refs. [24,25]. Our ratio is consistent with these data, which were not included in the fits.

Summary
In this paper, we have continued the dispersion-theoretical analysis of the proton form factors triggered by two main developments. On the theoretical side, a much improved representation of the two-pioncontinuum contribution to the isovector spectral function based on the precision results from the Roy-Steiner analysis of pion-nucleon scattering has been presented [11]. On the experimental side, new ep scattering data at very low Q 2 from the PRad collaboration [14] have become available. Using our improved spectral functions and employing the two-photon corrections worked out in Ref. [20], we have analyzed these new data as well as the combination of the PRad and the Mainz data [15] which allowed us to extract the proton's electric and magnetic radius with unprecedented precision, as given in Eq. (10). Theoretical uncertainties from the fit procedure and from variations in the spectral functions have been worked out. In the future, Bayesian methods will be used to further improve these uncertainty estimates. Furthermore, fits including also the time-like proton form factor data should be performed. Finally, data for the scattering off the neutron should be included, as these can be used to precisely pin down the corresponding neutron radii using chiral effective field theory for few-nucleon systems [27].

Appendix A. Fit parameters
We collect the various vector meson masses and couplings that appear in the spectral functions Eqs. (5) and the normalization constants of the various data sets (see Ref. [20] for precise definitions) in Table A