The proton charge radius

Nucleons (protons and neutrons) are the building blocks of atomic nuclei and are responsible for more than 99% of the visible matter in the Universe. Despite decades of efforts in studying its internal structure, there are still a number of puzzles surrounding the proton such as its spin and charge radius. Accurate knowledge about the proton charge radius is not only essential for understanding how QCD works in the nonperturbative region but also important for bound state QED calculations of atomic energy levels. It also has an impact on the Rydberg constant, one of the most precisely measured fundamental constants in nature. This review examines the latest situation concerning the proton charge radius in light of the new experimental results from both atomic hydrogen spectroscopy and electron-scattering measurements, with particular focus on the latter. Theoretical backgrounds and recent developments concerning the determination of the proton charge radius using different experimental techniques are also presented. Upcoming experiments are discussed, and the deuteron charge radius puzzle is mentioned at the end.

Figure 1

One-photon-exchange diagram describing the elastic electron-proton scattering. From Jingyi Zhou.

Figure 2

Parton distribution family. From [172].

Figure 3

Coupling of a spacelike virtual photon to a relativistic many-body system such as a proton. Upper images: diagonal transition where the photon couples to a quark in the leading $3q$ Fock component (left image) or in a higher $5q$ Fock component (right image). Lower image: process where the photon creates a $q\overline{q}$ pair leading to a nondiagonal transition between an initial $3q$ state and a final $5q$ state in the proton.

Figure 4

Schematic view of the projection of the charge density along the line of sight (perpendicular to the image) for a hadron polarized along the direction of ${\mathbf{S}}_{\perp }$. The position of the quark charge inside the hadron is denoted by $\mathbf{b}$.

Figure 5

Transverse charge densities for a proton (left panel) and neutron (right panel). The curves show the density along the $y$ axis for an unpolarized nucleon (dashed blue curves), and for a nucleon polarized along the $x$ axis (solid red curves). For the nucleon form factors, the empirical parametrization of [253] was used.

Figure 6

Compilation of recent lattice QCD results for the isovector charge radius (left panel) and the proton charge radius (right panel) obtained from ensembles at the physical pion mass. Results shown are from LHPC ([118]) and ETMC, both using a form factor fit ETMC 18 ([12]), as well as the direct calculation of the radius ETMC 20, avoiding an extrapolation through a form factor fit ([13]); PNDME ([139]); PACS ([230]); and CLS ([78]). Inner error bars display the statistical errors, whereas outer error bars display the full errors. The vertical bands show the empirical result extracted from muonic hydrogen spectroscopy and the CODATA-2014 recommended value, as discussed in Secs. 5 and 6. From Jingyi Zhou.

Figure 7

One-photon-exchange diagram for spin-dependent electron-proton scattering. From Jingyi Zhou.

Figure 8

The one-photon-exchange diagram for polarization transfer from longitudinally polarized electron to unpolarized proton. From Jingyi Zhou.

Figure 9

Two-photon-exchange diagram for elastic electron-proton scattering. The blob denotes the doubly virtual Compton subprocess on the proton.

Figure 10

Hydrogen $1S$, $2S$, and $2P$ energy levels. From Jingyi Zhou.

Figure 11

The box diagram for the $\mathcal{O}({\alpha }^5)$ corrections to $l=0$ energy levels in muonic hydrogen. The blob denotes all possible hadronic intermediate states.

Figure 12

Top image: schematics of the three-spectrometer setup of the A1 experiment at Mainz. From [42]. Bottom: photo showing spectrometers A, B, and C in red, blue, and green, respectively. The electron beam progresses from right to left. From [247].

Figure 13

Feynman diagrams showing electron-proton scattering with an electron or proton radiates a real photon in the initial state or final state. In the electron case, the two diagrams are labeled for Bethe and Heitler (BH-i and BH-f), while the diagrams for the proton are Born-i and Born-f, where i and f stand for the initial-state and final-state radiation, respectively. From [187].

Figure 14

Schematics of the PRad experiment in Hall B at Jefferson Lab. The electron beam progresses from left to right. From E. Pasyuk, J. Brock, K. Gnanvo, P. Hemler, D. Kashy, N. Liyanage, and G. Swift.

Figure 15

The proton electric form factor $G_E^p$ from the PRad experiment together with results from the Mainz experiment ([35]; [37]) in the overlapping $Q^2$ region. Both datasets include statistical and systematic uncertainties; see the text. Two fits of the PRad data ([9]; [250]) and a fit of the Mainz data ([36]) are also shown. From Weizhi Xiong.

Figure 16

The proton electric form factor $G_E^p$ from the PRad experiment together with those from [117], [197], [234], and [37] normalized to the standard dipole form in the overlapping $Q^2$ region, on linear scale. From Weizhi Xiong.

Figure 17

Sample fits using rational functions of pseudodata generated with nine proton form factor models including the projected PRad statistical and systematic uncertainties. From [252].

Figure 18

The proton charge radius $⟨r_{Ep}^2{⟩}^{1/2}$ as extracted from electron-scattering and spectroscopic experiments since 2010 and before 2020 together with CODATA-2014 and CODATA-2018 recommended values. Note the reinterpreted result from the Mainz ISR experiment was scheduled for publication in 2021. From Jingyi Zhou.

Figure 19

The proton charge radius values determined from electron-scattering experiments since 2010 together with the results from various analyses of electron-proton scattering data (see the text). From Jingyi Zhou.

Figure 20

Muonic hydrogen energy levels relevant to the proton charge radius measurement. From Jingyi Zhou.

Figure 21

The latest proton charge radius results from ordinary hydrogen spectroscopic measurements together with muonic hydrogen results and the CODATA-2014 recommended value based on ordinary hydrogen spectroscopy. From Jingyi Zhou.

Figure 22

Schematic of the MUSE experiment at PSI. From Steffen Strauch.

Figure 23

Top panel: schematics of the AMBER setup for the proton charge radius measurement. Bottom panel: entire setup in the AMBER spectrometer with relevant parts shown. From Christian Dreisbach, Technical University of Munich.

Figure 24

Schematic of the setup for the proposed PRad-II experiment. The incident electron beam progresses from left to right. From Dipangkar Dutta.

Figure 25

PRad-II projection for $⟨r_{Ep}^2{⟩}^{1/2}$ with all proposed upgrades and improvements shown with a few selected results from other experiments and CODATA-2018 recommendations (see the text). From Jingyi Zhou.

Figure 26

Bethe-Heitler (left image) direct and (right image) crossed diagrams of the $\gamma p\to l^{-}{l}^{+}p$ process, where the four-momenta of the external particles are $k$ for the photon, $p(p^{\prime })$ for initial (final) protons, and $l_{-},l_{+}$ for the lepton pair.

Figure 27

The MAGIX high-resolution dual-spectrometer setup at the MESA accelerator. The gas-jet target in the center is also visible. From [227].

Figure 28

Schematics of the ${\mathrm{ULQ}}^2$ experimental setup. From Toshimi Suda.

Figure 29

Existing results on the deuteron charge radius $⟨r_{Cd}^2{⟩}^{1/2}$; see the text for details. From Randolf Pohl.

Figure 30

Momentum transfer $t$ dependence of the $\gamma d\to e^{+}{e}^{-}d$ cross-section ratio $R(t,t_0)$, defined in Eq. (82), for reference value $t_0=-0.01{\mathrm{GeV}}^2$, at fixed deuteron lab angle, and for beam energy 0.65 GeV. For convenience, the ratio is normalized to the result using the form factors of [1]. The curves and associated error bands are for different extractions of the deuteron charge radius; see the text for details. From [62].