Determination of proton electromagnetic form factors from DVCS measurements

This paper presents a detailed analysis of proton electromagnetic form factors using exclusive photon leptoproduction data in the Beth-Heitler dominant region, demonstrating that such measurements can provide complementary constraints on the Dirac form factor and yield a charge radius consistent with recent high-precision PRad results.

The Gist

Imagine the proton as a tiny, bustling city inside an atom. For decades, physicists have been trying to draw a perfect map of this city: Where is the electric charge? How big is the city? How is the magnetic "wind" blowing inside it?

This paper is about a new, clever way to draw that map, using a technique that acts like a "magnifying glass" for specific parts of the city.

The Problem: The "Proton Radius Puzzle"

First, there's a mystery. When scientists measure the size of the proton (specifically its "charge radius") using traditional methods (bouncing electrons off it like billiard balls), they get one answer. But when they use other methods (like looking at how muons orbit the proton), they get a slightly smaller answer. This disagreement is called the Proton Radius Puzzle. It's like two surveyors measuring the same house and getting different square footage.

The New Tool: The "Shadow" Technique

The authors of this paper decided to try a different approach. Instead of just bouncing electrons off the proton, they looked at a process called Exclusive Photon Leptoproduction (EP).

Think of it like this:

• The Setup: You shoot a high-speed electron at a proton.
• The Event: Sometimes, the electron hits the proton, and a real photon (a particle of light) pops out.
• The Confusion: There are two ways this light can be produced:
  1. The "Deep" Way (DVCS): The electron hits a tiny quark inside the proton, and the proton gets excited and spits out a photon. This tells us about the messy, complex interior of the city.
  2. The "Surface" Way (Bethe-Heitler or BH): The electron misses the quarks entirely and just interacts with the proton's overall electric charge, emitting a photon from its own path. This is like a shadow cast by the building.

The Key Insight:
Usually, the "Deep" way is what scientists want to study to understand the proton's inner structure. However, in certain conditions (specifically at low energy and specific angles), the "Surface" way (the BH shadow) becomes overwhelmingly dominant. It's like standing in a bright spotlight where your shadow is huge, but the object casting it is hard to see.

The authors realized: If the shadow is so big and clear, maybe we can use the shadow to measure the object itself!

How They Did It

1. Finding the Sweet Spot: They used computer simulations to find the exact "angles" and energies where the "Surface" shadow (BH) makes up 95% to 99% of the total signal. In these zones, the messy "Deep" signal is almost invisible.
2. The Data: They took real data from the CLAS experiment at Jefferson Lab (a giant particle accelerator).
3. The Filter: They filtered the data, throwing away everything that didn't fit their "Shadow Dominance" criteria.
4. The Extraction: Since the "Shadow" is directly determined by the proton's Form Factors (mathematical descriptions of how charge and magnetism are spread out), they could reverse-engineer the data to find the shape of the proton's electric and magnetic fields.

What They Found

• The Electric Map (Charge): When they mapped the proton's electric charge using this new "shadow" method, they found the proton looks smaller than what traditional methods suggest.
• The Magnetic Map: The magnetic size came out looking very similar to the traditional measurements.
• The Connection to the Puzzle: Their "smaller" electric size matches very well with a famous experiment called PRad, which also found a small proton. This suggests that the "small proton" might be the correct answer, and the traditional "large proton" measurements might have had some hidden issues.

The Analogy: The Foggy Mirror

Imagine trying to see your reflection in a mirror covered in thick fog.

• Traditional Method: You try to clean the whole mirror to see your face clearly. Sometimes you miss a spot, or the cleaning changes the glass.
• This Paper's Method: They realized that in one specific corner of the room, the fog is so thick that you can't see your face at all, but you can see the outline of your body perfectly against the wall. By studying that perfect outline (the BH shadow), they could deduce exactly how wide your shoulders are, even without seeing your face.

Why This Matters

This paper doesn't just give a new number; it gives a new tool.

1. Complementary View: It proves that we can learn about the proton's structure using a completely different physical process than the old way.
2. Solving the Puzzle: It adds weight to the idea that the proton is indeed smaller than we thought, helping to solve the "Proton Radius Puzzle."
3. Future Roadmap: It provides a blueprint for future experiments. If we can measure even smaller angles (getting closer to the "shadow" edge), we might get an even more precise map of the proton.

In short: The authors found a way to use the "noise" (the dominant shadow effect) in particle collisions as a signal to measure the proton's size, and it confirms that the proton is likely smaller than previously believed.

Technical Summary

1. Problem Statement

The extraction of proton electromagnetic form factors (FFs), specifically the Dirac ($F_1$) and Pauli ($F_2$) form factors, is traditionally performed using elastic electron-proton scattering data. However, a significant discrepancy known as the "proton charge radius puzzle" exists between measurements derived from elastic scattering and those from muonic hydrogen spectroscopy. Furthermore, recent high-precision measurements (e.g., PRad) suggest a smaller charge radius than the Particle Data Group (PDG) average.

The paper addresses whether Exclusive Photon Leptoproduction (EP) data—comprising Deeply Virtual Compton Scattering (DVCS) and the Bethe-Heitler (BH) process—can serve as a complementary tool to extract these form factors. The core challenge is that DVCS observables are typically dominated by Generalized Parton Distributions (GPDs), making the extraction of FFs difficult. However, the BH process is purely electromagnetic and directly sensitive to the elastic FFs. The authors investigate if there are kinematic regions where the BH contribution dominates the total EP cross-section, allowing for a direct extraction of $F_1(t)$ and $F_2(t)$ without relying on complex GPD models.

2. Methodology

Theoretical Framework:

• Process: The study focuses on the reaction $e + p \to e' + p' + \gamma$. The total amplitude is a coherent sum of the DVCS amplitude, the BH amplitude, and their interference.
• BH Dominance: The authors utilize the fact that the BH amplitude depends explicitly on the nucleon elastic FFs ($F_1, F_2$). They employ the Gepard package and the KM15 model to calculate the relative contributions of BH, DVCS, and interference terms.
• Kinematic Selection: They identified that at low momentum transfer ($|t|$) and specific values of the Bjorken variable ($x_B$) and azimuthal angle ($\phi$), the BH term dominates the cross-section (accounting for >95% in some regions).
• Data Selection: Using CLAS collaboration data (Jefferson Lab, 5.75 GeV beam), the authors constructed five distinct datasets (Set 1 to Set 5). Each set includes only data points where the theoretical BH cross-section differs from the full EP cross-section by less than 1%, 2%, 3%, 4%, or 5%, respectively. This isolates the "BH-dominated" kinematic region ($0.11 < |t| < 0.45 \text{ GeV}^2$).

Fitting Strategy:
The authors performed $\chi^2$ minimizations using the iMinuit library to extract FFs under several scenarios:

1. Dipole Parametrization: Fitting $F_1(t)$ and $F_2(t)$ using standard dipole forms with free parameters $a_E$ and $a_M$.
2. P-pole Parametrization: Using a more flexible form $(1 - t/a)^{-P}$ to test stability.
3. Fixed $F_2$ Scenario: Given the limited sensitivity to $F_2$ at small $|t|$, a specific fit (Fit 8) was performed where $F_2(t)$ was fixed to the well-established Kelly parametrization, allowing only $F_1(t)$ to vary.

3. Key Contributions

• Novel Extraction Method: The paper establishes a rigorous framework for extracting proton electromagnetic FFs directly from EP data by isolating the BH-dominated kinematic region, offering an alternative to traditional elastic scattering analysis.
• Kinematic Analysis: It provides a detailed map of the kinematic dependencies ($x_B, \phi, Q^2, t$) required to ensure BH dominance, demonstrating that for $Q^2 \approx 1 \text{ GeV}^2$ and specific $x_B$, the BH contribution can exceed 99% of the total cross-section.
• Parametrization Independence: The study compares results across different parametrization schemes (dipole vs. P-pole) and fitting strategies (free vs. fixed $F_2$) to assess the robustness of the extracted radii.
• Systematic Uncertainty Handling: The authors assign additional systematic uncertainties (1–5%) to the selected data points to account for the selection bias introduced by the BH-dominance cut.

4. Results

• Sensitivity to Form Factors:
  • The EP data in the range $0.11 < |t| < 0.45 \text{ GeV}^2$ provides strong constraints on the Dirac form factor $F_1(t)$.
  • Sensitivity to the Pauli form factor $F_2(t)$ is limited due to the suppression factor $\tau = -t/(4M^2)$ in the BH cross-section at small $|t|$. Consequently, fits allowing $F_2$ to vary freely show larger uncertainties and deviations from standard parametrizations.
• Comparison with Global Fits (YAHL18):
  • The extracted $F_1(t)$ values are systematically larger than those from the global YAHL18 fit (based on elastic scattering).
  • The extracted $F_2(t)$ values are systematically smaller.
  • These differences lead to a significant deviation in the Sachs electric form factor $G_E(t)$, while the magnetic form factor $G_M(t)$ shows better consistency due to partial cancellation of errors.
• Proton Radii:
  • Charge Radius ($r_E$): All fitting scenarios yield a charge radius smaller than the PDG 2024 average ($0.8409 \pm 0.0004$ fm).
    • Fit 5 (Dipole, free $F_2$): $r_E = 0.779 \pm 0.013$ fm.
    • Fit 8 (P-pole, fixed $F_2$): $r_E = 0.815 \pm 0.011$ fm. This result is the most reliable and is consistent within uncertainties with the PRad experiment ($0.831 \pm 0.014$ fm).
  • Magnetic Radius ($r_M$): The extracted magnetic radii are generally consistent with the PDG average ($0.851 \pm 0.026$ fm), with Fit 8 yielding $0.831 \pm 0.004$ fm.

5. Significance

• Resolution of the Radius Puzzle: The results support the findings of the PRad experiment, suggesting that the proton charge radius is indeed smaller than the traditional PDG average derived from older elastic scattering data. This reinforces the hypothesis that the "radius puzzle" may stem from systematic differences in experimental methods or kinematic coverage.
• Complementary Tool: The study demonstrates that EP measurements, particularly in BH-dominated regions, are a viable and complementary method for determining nucleon structure. This is crucial for cross-validating results from elastic scattering and muonic hydrogen spectroscopy.
• Future Directions: The methodology provides a solid framework for future combined analyses of EP and elastic scattering data. It suggests that extending EP measurements to even smaller $|t|$ values could further refine the determination of the proton charge radius and potentially resolve remaining discrepancies in the field.
• Broader Application: The developed framework can be applied to other exclusive photon leptoproduction measurements (e.g., from Jefferson Lab Hall A) to extract FFs with improved precision.

In conclusion, the paper successfully demonstrates that by isolating the Bethe-Heitler contribution in DVCS/EP data, one can extract proton electromagnetic form factors that favor a smaller charge radius, aligning with recent high-precision measurements and offering a new perspective on the internal structure of the nucleon.