Abstract.We analyze the recent electron-proton scattering data from Mainz using a dispersive framework that respects the constraints from analyticity and unitarity on the nucleon structure. We also perform a continued fraction analysis of these data. We find a small electric proton charge radius, r p E = 0.84 +0.01−0.01 fm, consistent with the recent determination from muonic hydrogen measurements and earlier dispersive analyses. We also extract the proton magnetic radius, r The proton charge radius is a fundamental quantity of physics. It is truly remarkable that, despite decadelong experimental and theoretical efforts, its precise value is not yet determined. The recent controversy about the size of the proton was triggered by the precision measurement of the Lamb shift in muonic hydrogen that led to a "small" charge radius, r p E = 0.84184(67) fm [1]. This result came as a big surprise as it was in stark contrast to the commonly accepted "large" CODATA value of 0.8768(69) fm [2], based on the measurements of the Lamb shift in electronic hydrogen and the analysis of electron-proton scattering data. The large value was further strengthened by the high-precision electron-proton scattering measurements at MAMI-C [3,4]. The analysis of these data including two-photon corrections led to r p E = 0.876(8) fm. These authors also found a magnetic radius of the proton that came out much smaller than commonly accepted values, r p M = 0.803 (17) Given this puzzling situation, in this letter we will reanalyze the MAMI data of Bernauer et al.[3] using dispersion relations. Our main focus will be on the correct treatment of the analytical structure of the nucleon form factors that is driven by the two-pion continuum. Its important role was already stressed by Frazer and Fulco, who were able to predict the ρ-resonance and its influence on the nucleons' structure a long time ago [10]. What has often been overlooked since this seminal work was the large enhancement of the two-pion continuum on the left wing of the ρ-resonance due to a close-by pole on the second Riemann sheet in the elastic pion-nucleon scattering amplitude. This enhancement amounts for roughly half of the nucleon isovector size [6]. This important effect is also recovered in chiral perturbation theory, the effective field theory of QCD at low energies [11]. We consider it, therefore, of utmost importance to include this effect in the re-analysis of the MAMI data. Of course, in light of the muon g − 2 measurement at BNL [12], one might speculate about the influence of new physics as the muon data are more sensitive to such effects, but first one has to exclude possible conventional explanations -and we will offer such a possibility here.

We calculate the two-photon exchange corrections to electron-proton scattering with nucleon and ∆ intermediate states. The results show a dependence on the elastic nucleon and nucleon-∆-transition form factors used as input which leads to significant changes compared to previous calculations. We discuss the relevance of these corrections and apply them to the most recent and precise data set and world data from electron-proton scattering. Using this, we show how the form factor extraction from these data is influenced by the subsequent inclusion of physical constraints. The determination of the proton charge radius from scattering data is shown to be dominated by the enforcement of a realistic spectral function. Additionally, the third Zemach moment from the resulting form factors is calculated. The obtained radius and Zemach moment are shown to be consistent with Lamb shift measurements in muonic hydrogen.

We show that in previous analyses of electron-proton scattering, the uncertainties in the statistical procedure to extract the proton charge radius are underestimated. Using a fit function based on a conformal mapping, we can describe the scattering data with high precision and extract a radius value in agreement with the one obtained from muonic hydrogen. PACS numbers:1. Two principally different methods are commonly used to determine the proton charge radius r P E . On the one hand, it enters the QED calculations of atomic energy splittings (electronic and muonic [1] hydrogen) and can thus be obtained from measurements of these. On the other hand, r P E can be obtained from elastic electron-proton scattering. The corresponding cross sections can be parameterized in terms of the electric and magnetic Sachs form factors G E (Q 2 ) and G M (Q 2 ), respectively, that depend on the invariant momentum transfer squared Q 2 = −t. Positive Q 2 -values refer to the scattering process, negative to annihilation/creation. The reduced cross section, here in the one-photon approximation, describes the deviation from the scattering off a point-like particle: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 and τ = −t/4m 2 N , with m N the nucleon mass. Both methods refer to the same quantity, the slope of the proton form factor at the origin:The form factor obtained from the cross sections has to be extrapolated from the data at lowest momentum transfer to the origin. The most precise electron-proton scattering data from Ref.[2] analyzed using spline and polynomial fit functions lead to a proton charge radius that differs by ∼ 7 σ from the muonic hydrogen radius of Ref.[1], when averaged with measurements in electronic hydrogen [3]. The purpose of this letter is to illustrate that such extrapolations lack precision in purely statistical analyses with arbitrary fit functions. For example, the fit functions quoted in the final results of Ref.[2] are polynomials and splines. In this letter, we construct a simple function, that describes the data equally well and corresponds to a small radius r P E in agreement with the one obtained from muonic hydrogen spectroscopy. This function is based on a conformal mapping and thus obeys the analytic structure of the form factors. The following function maps the cut in the t-plane onto the unit circle in a new variable z:where t cut = 4M 2 π is the lowest singularity of the form factors with M π the charged pion mass. The Sachs form factors can then be expanded in the new variable z:Here, the form factors are normalized to the charge and anomalous magnetic moment of the proton, respectively. Conformal mapping techniques are a standard tool in hadron physics. So far, they have not been applied to the electronproton scattering data by the A1-collaboration, the data for this process with the highest quoted precision. A previous elaborate analysis of world form factor data in a similar approach was carried out ...

In the most recent measurements of the reaction e + e − → pp by the BABAR collaboration, new structures have been found with unknown origin. We examine a possible relation of the most distinct peak to the recently observed φ(2170). Alternatively, we analyse possible explanations due to the nucleon∆ and ∆∆ thresholds. The latter could explain a periodicity found in the data.

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