Implications of exclusive photon leptoproduction measurements for the proton charge-radius puzzle

Technical Summary: Implications of Exclusive Photon Leptoproduction Measurements for the Proton Charge-Radius Puzzle

Problem Statement
The proton charge-radius puzzle remains a significant discrepancy in nuclear physics, characterized by conflicting measurements of the proton charge radius ($r_E$) derived from elastic electron–proton scattering versus muonic hydrogen spectroscopy and the PRad experiment. While traditional extraction of electromagnetic form factors (FFs) relies on elastic scattering (Rosenbluth separation and polarization transfer), these methods face challenges at higher momentum transfers and require corrections for two-photon exchange effects. Exclusive photon leptoproduction (EP), specifically in kinematic regions dominated by the Bethe–Heitler (BH) mechanism, offers a complementary approach. In the BH process, a real photon is emitted from the lepton line, and the cross-section amplitude depends explicitly on the proton's Dirac ($F_1$) and Pauli ($F_2$) FFs. The authors aim to extend their previous analysis by incorporating a broader set of high-precision EP data to extract proton FFs and radii, thereby providing an independent cross-check on the proton's electromagnetic structure and the charge-radius puzzle.

Methodology
The study utilizes a phenomenological framework identical to that of the authors' previous work [66], applied to a significantly expanded dataset.

• Data Selection: The analysis incorporates 1,249 data points from the CLAS (2015 and 2018) and Hall A (2015 and 2017) collaborations at Jefferson Lab. The kinematic coverage spans $0.125 \le x_B \le 0.5$, $0.108 \le -t \le 0.909 \text{ GeV}^2$, and $1.11 \le Q^2 \le 3.982 \text{ GeV}^2$.
• BH Dominance Criterion: To ensure the extraction relies on the BH mechanism, data points are selected where the relative difference between the pure BH prediction and the full EP cross-section (calculated via the Gepard framework) is $\le 5\%$. An additional uncorrelated systematic uncertainty of 5% is assigned to account for this selection procedure.
• Parametrization: The Dirac ($F_1$) and Pauli ($F_2$) FFs are modeled using both dipole and P-pole parametrizations. Given the limited sensitivity of EP data to $F_2(t)$, alternative fits are performed where $F_2(t)$ is fixed to the Kelly parametrization [23], allowing only $F_1(t)$ to vary.
• Analysis Strategy: A simultaneous $\chi^2$ minimization is performed using the iminuit interface to CERN MINUIT. The study investigates the consistency of different datasets, specifically examining the impact of the CLAS 2018 data [75] by comparing fits that include all data ("CLAS18inc") against those excluding it ("CLAS18exc") or applying low-$|t|$ cuts to the CLAS 2018 set.
• Radius Extraction: Proton charge ($r_E$) and magnetic ($r_M$) radii are derived from the slopes of the Sachs FFs ($G_E$ and $G_M$) at $t=0$.

Key Results

• Data Tension: A significant tension is observed between the CLAS 2018 measurements and the remaining EP data (CLAS 2015 and Hall A). Including the full CLAS 2018 dataset results in a poor fit quality ($\chi^2/\text{d.o.f.} = 2.26$) and yields unrealistic FFs. Excluding CLAS 2018 or applying low-$|t|$ cuts ($|t|_{cut} \approx 0.265\text{--}0.343 \text{ GeV}^2$) restores fit stability, with $\chi^2/\text{d.o.f.}$ approaching unity (e.g., 1.02 for the exclusion case).
• Form Factors:
  • Electric FF ($G_E$): In all stable fitting scenarios (excluding full CLAS 2018), the extracted $G_E(t)$ is systematically larger than the YAHL18 parametrization (based on global elastic scattering), with deviations increasing as $|t|$ increases.
  • Magnetic FF ($G_M$): The extracted $G_M(t)$ shows better agreement with the YAHL18 parametrization, with deviations generally not exceeding 7%, except when the full CLAS 2018 dataset is included.
• Proton Radii:
  • Charge Radius ($r_E$): All analyses yield a proton charge radius smaller than the Particle Data Group (PDG) average and most elastic scattering determinations. The extracted values are consistent, within uncertainties, with the PRad measurement, muonic hydrogen spectroscopy, and recent lattice QCD calculations. For instance, the "CLAS18exc" dipole fit yields $r_E = 0.819 \pm 0.011 \text{ fm}$.
  • Magnetic Radius ($r_M$): The extracted magnetic radius is compatible with the current world average ($r_M \approx 0.851 \pm 0.026 \text{ fm}$) across all fitting scenarios, showing no systematic shift due to the inclusion or exclusion of CLAS 2018 data.

Significance and Claims
The paper claims that BH-dominated EP measurements provide an independent and complementary method for determining the proton's electromagnetic structure. The primary conclusion is that these measurements favor a "small-radius" solution to the proton charge-radius puzzle, aligning with muonic hydrogen and PRad results rather than traditional elastic scattering averages. The study demonstrates that while EP data offers strong constraints on the Dirac form factor ($F_1$), the sensitivity to the Pauli form factor ($F_2$) remains limited. The authors assert that future high-precision EP measurements, particularly at lower $|t|$, will further refine these constraints and that combined analyses of EP and elastic scattering data within a unified framework could yield the most reliable determination of proton radii.