The electric and magnetic form factors of the proton

TL;DR

This work presents a highly precise measurement of elastic electron-proton scattering at MAMI, combining Mainz data with world data to extract the proton's electric and magnetic form factors GE and GM via a global, model-flexible cross-section fit. It emphasizes a consistent treatment of radiative corrections, employs an empirical two-photon-exchange term to reconcile Rosenbluth and polarization data, and uses high-quality low-Q^2 data to determine the proton radii. The analysis reveals deviations from the classic dipole form and hints at structure on top of the gross form-factor shape, including possible pion-cloud effects, while obtaining r_E ≈ 0.879 fm and r_M ≈ 0.777 fm. The results contribute to ongoing discussions about the proton's internal structure and highlight tensions with muonic-hydrogen determinations, motivating further theoretical and experimental work on TPE and low-Q^2 behavior.

Abstract

The paper describes a precise measurement of electron scattering off the proton at momentum transfers of $0.003 \lesssim Q^2 \lesssim 1$\ GeV$^2$. The average point-to-point error of the cross sections in this experiment is $\sim$ 0.37%. These data are used for a coherent new analysis together with all world data of unpolarized and polarized electron scattering from the very smallest to the highest momentum transfers so far measured. The extracted electric and magnetic form factors provide new insight into their exact shape, deviating from the classical dipole form, and of structure on top of this gross shape. The data reaching very low $Q^2$ values are used for a new determination of the electric and magnetic radii. An empirical determination of the Two-Photon-Exchange (TPE) correction is presented. The implications of this correction on the radii and the question of a directly visible signal of the pion cloud are addressed.

Figures (21)

• Figure 1: The accessible kinematical region in the $\varepsilon$-$Q^2$-plane. Shaded regions are excluded because of minimum or maximum beam energy (darkest gray), maximum detectable momentum of spectrometer A and B (medium grays), and minimum angle of B and maximum angle of C (lightest gray). The centers of the acceptances of the different kinematical settings are denoted with symbols (squares, circles, and triangles correspond to spectrometer A, B, and C). The symbols are slightly shifted vertically to distinguish overlapping measurements. The upper black lines represent a possible future extension using MAMI C energies. To stretch the low-$Q^2$ part, the (left) $y$ axis is presented linear in $Q=\sqrt{Q^2}$.
• Figure 2: The kinematic parameters for the elastic scattering of an electron on a target initially at rest.
• Figure 3: Feynman graphs of leading and next to leading order for elastic scattering. (b) Leading order, (v1--v5) next to leading order with an additional virtual photon, and (r1--r4) leading-order graphs with a radiated real photon.
• Figure 4: (a) Comparison of experimental (black) and simulated (gray) $\Delta E^\prime$ histogram in the region of the tail. A cut on the vertex position was applied to suppress background from reactions off the walls. (b) Ratio of the integral of the experimental data to the integral of the simulation integrated up to the cut-off energy $\Delta E^\prime$. The ratio varies by less than 0.1% for cut-off energies up to 75 MeV. The ratio is scaled to start at 1. Data: Spectrometer A, $53^\circ$, and 855-MeV incident beam energy.
• Figure 5: Typical $\Delta E_\mathrm{exp}^\prime$ spectrum for a measurement with spectrometer A at $90.5^\circ$ at an incident energy of $E=585$$\mathrm{MeV}$. (a) Spectrum without any cut. (b) After all cuts except a cut in $\Delta E_\mathrm{exp}^\prime$. (c) Events that are rejected by the cuts. Random events give rise to the nearly constant level between $-$30 and 15 MeV. The bump around 30 MeV originates from events detected near the edges of the detector plane.