Sensitivity of the photon-induced processes to the proton radius

This paper investigates the sensitivity of exclusive dilepton production in proton-proton collisions to the proton radius, finding that while current ATLAS and CMS data within a dipole form factor model suggest an effective radius of $1.002 \pm 0.038$ fm, this result is not yet sufficient to definitively resolve the proton radius puzzle.

The Gist

Imagine the proton as a tiny, fuzzy cloud of energy rather than a solid marble. For decades, scientists have been trying to measure the exact size of this cloud, but they've hit a snag: when they measure it using electrons, they get one size, but when they use muons (a heavier cousin of the electron), they get a noticeably smaller size. This disagreement is known as the "Proton Radius Puzzle."

This paper is like a new detective story where the authors try to solve this puzzle using a completely different crime scene: high-energy collisions at the Large Hadron Collider (LHC).

Here is the breakdown of their investigation, using simple analogies:

1. The Experiment: A "Ghostly" Collision

Usually, when protons smash into each other at the LHC, they shatter into a million pieces. But sometimes, they barely graze each other. Imagine two fast-moving cars passing each other on a highway so close that their windshields shake, but they don't crash.

In this "grazing" scenario, the protons don't break apart. Instead, they exchange invisible packets of energy called photons (light particles). These photons collide with each other and briefly turn into a pair of particles (like a muon and an anti-muon), which then fly away, leaving the original protons intact.

The authors studied these "ghostly" collisions to see if the size of the proton cloud changes how often these events happen.

2. The Tool: The "Fuzzy Lens"

To understand the proton's size, the scientists used a mathematical model called a dipole form factor. Think of this as a "fuzzy lens" through which we view the proton.

• The Conventional Lens: For a long time, scientists used a standard setting for this lens (a specific number called $\Lambda^2 = 0.71$).
• The Puzzle Lenses: The authors swapped this standard lens for two new settings based on the two conflicting measurements from the "electron vs. muon" puzzle:
  • The "Big Proton" Lens: Based on electron measurements (radius $\approx 0.875$ fm).
  • The "Small Proton" Lens: Based on muon measurements (radius $\approx 0.841$ fm).

3. The Discovery: Where to Look

The authors found that the size of the proton doesn't matter equally everywhere.

• The "Backyard" Analogy: If you look at the proton from far away (low energy), the "Big" and "Small" lenses look almost identical. The difference is too small to see.
• The "Front Yard" Analogy: However, if you zoom in very close (high energy, large mass, or extreme angles), the difference becomes obvious. The "Big Proton" lens blocks more of the view than the "Small Proton" lens.

They discovered that the sensitivity to the proton's size is greatest when:

• The created particle pair is very heavy (high invariant mass).
• The particles fly off at very sharp angles (forward or backward).

4. The "Traffic Jam" (Absorptive Corrections)

In the real world, protons aren't just passing through each other; they sometimes have a "traffic jam" (soft interactions) that ruins the clean collision. The authors had to account for this using a "survival factor."

• The Result: This traffic jam mostly happens when protons get very close (small impact parameter). Since the proton size matters most when they are close, this traffic jam actually dampens the difference between the "Big" and "Small" lenses.
• The Takeaway: Even with the traffic jam, the difference between the two sizes is still visible, though slightly smaller.

5. The Verdict: Fitting the Data

The team took their theoretical predictions and compared them to real data collected by the ATLAS and CMS experiments at the LHC.

• The Problem: The standard "Conventional Lens" (the one everyone usually uses) predicted too many collisions compared to what was actually seen.
• The Fit: When they adjusted the lens to fit the data perfectly, the math suggested a proton radius of $1.002$ fm.
  • This is actually larger than both the "Big" and "Small" puzzle values.
  • The "Big" and "Small" values (0.875 and 0.841) didn't fit the LHC data as well as this new, larger value did.

6. The Conclusion: A Clue, Not a Solution

The authors are careful not to claim they have solved the puzzle.

• What they proved: The LHC data is indeed sensitive to the proton's size. Changing the size parameter changes the predictions, and the data can "feel" that difference.
• What they didn't prove: They cannot yet say definitively which size is correct. In fact, the data seems to prefer a size that is different from both the electron and muon measurements.
• The Caveat: The fact that the data prefers a weird, larger size suggests that their theoretical model might be missing something (perhaps how the magnetic fields of the proton interact, or how the "traffic jam" is modeled).

In summary: The paper shows that high-energy proton collisions are a new, sensitive way to measure the proton's size. While the current data doesn't solve the "Proton Radius Puzzle" yet, it proves that this method works and that the standard way of calculating these collisions might need a tune-up.

Technical Summary

Technical Summary: Sensitivity of Photon-Induced Processes to the Proton Radius

Problem Statement
The proton charge radius remains a subject of significant debate, known as the "proton radius puzzle," due to the discrepancy between values extracted from electron scattering/atomic spectroscopy ($r_p \approx 0.88$ fm) and muonic hydrogen spectroscopy ($r_p \approx 0.84$ fm). While high-energy proton-proton ($pp$) collisions offer a complementary avenue to probe the proton's electromagnetic structure via exclusive dilepton production ($pp \to pp\ell^+\ell^-$), the sensitivity of these processes to the proton radius parameter has not been systematically quantified within a rigorous theoretical framework. Specifically, it is unclear how variations in the proton radius, modeled through the dipole form factor cutoff parameter $\Lambda^2$, manifest in observables measured at the Large Hadron Collider (LHC), and whether current data can constrain this parameter.

Methodology
The authors employ two complementary theoretical approaches to calculate the exclusive dilepton production cross section:

1. Momentum-Space Approach: A $2 \to 4$ process treatment ($pp \to pp\ell^+\ell^-$) where the proton electromagnetic structure is incorporated at the amplitude level via proton-photon vertices. This method retains full event kinematics and allows for a fully differential treatment without relying on the Equivalent Photon Approximation (EPA).
2. Impact-Parameter Space Approach: Formulated within the EPA, this method treats the reaction as a convolution of photon fluxes with the elementary $\gamma\gamma \to \ell^+\ell^-$ subprocess. Crucially, this framework naturally incorporates the proton's electromagnetic structure (via form factors) into the photon flux $n(\omega, b)$ and allows for the inclusion of absorptive corrections (rescattering effects) via a survival factor $S^2_{\gamma\gamma}(b)$.

The study compares three specific scenarios for the dipole cutoff parameter $\Lambda^2$, which relates to the proton radius $r_p$ via the slope relation $r_p^2 = 12/(\Lambda^2 (\hbar c)^2)$:

• Conventional (C): $\Lambda^2_C = 0.71$ GeV$^2$ ($r_p \approx 0.81$ fm).
• Small Proton (S): $\Lambda^2_S = 0.6587$ GeV$^2$ ($r_p = 0.84087$ fm, from muonic hydrogen).
• Large Proton (L): $\Lambda^2_L = 0.6081$ GeV$^2$ ($r_p = 0.8751$ fm, from electron scattering).

The analysis utilizes the full Sachs form factor treatment (including both electric $G_E$ and magnetic $G_M$ contributions) to ensure quantitative reliability, contrasting this with electric-only approximations. Theoretical predictions are confronted with exclusive dilepton data from ATLAS and CMS at $\sqrt{s} = 7$ and $13$ TeV.

Key Contributions and Results

• Kinematic Sensitivity: The study demonstrates that sensitivity to the proton radius is not uniform across phase space. It is minimal in the low-$Q^2$, large impact parameter regime but becomes pronounced in kinematic regions probing larger photon virtualities. Specifically, deviations between the $\Lambda^2$ scenarios are enhanced at:
  • Large dilepton invariant masses ($M_{\ell\ell}$).
  • Forward and backward rapidities (away from midrapidity).
  • Large transverse momenta of the outgoing protons.
• Impact of Absorptive Corrections: While rescattering effects (survival factor) suppress the cross section, particularly in the small impact parameter region where radius sensitivity is highest, they do not eliminate the dependence on $\Lambda^2$. The relative differences between scenarios persist after corrections, though they are slightly reduced (e.g., the difference between the conventional and "Large" scenarios drops from $\sim 8.3\%$ to $\sim 6.9\%$ in the integrated $\tau^+\tau^-$ cross section).
• Comparison with Data:
  • The full Sachs form factor treatment (including magnetic contributions) yields cross sections that are generally higher than electric-only approximations. Interestingly, the electric-only results lie closer to the central experimental values, suggesting that current fiducial data cannot yet isolate the radius dependence from other normalization effects.
  • A global $\chi^2$ fit of the theoretical model to ATLAS and CMS data (treating $\Lambda^2$ as a free parameter) yields a minimum at $\Lambda^2 = 0.465 \pm 0.056$ GeV$^2$, corresponding to an effective radius $r_p = 1.002 \pm 0.038$ fm.
  • The fit quality ($\chi^2/\text{ndf} \approx 0.75$) indicates satisfactory agreement, but the best-fit $\Lambda^2$ value lies outside the physically reasonable range of the three benchmark scenarios (S, L, and C).

Significance and Claims
The paper asserts that exclusive dilepton production at the LHC exhibits non-trivial sensitivity to the proton radius scale. The authors conclude that the conventional dipole parameter ($\Lambda^2 = 0.71$ GeV$^2$) is not a numerically neutral choice; using it introduces systematic shifts in cross-section predictions compared to values motivated by modern radius determinations.

However, the authors are modest regarding the extraction of a definitive proton radius. They explicitly state that the current result ($r_p \approx 1.00$ fm) should not be interpreted as a precision measurement or a solution to the proton radius puzzle. Instead, it serves as an indicator of the radius scale preferred by the current theoretical implementation when confronted with existing LHC data. The discrepancy between the best-fit value and the benchmark scenarios suggests "unknown problems in the theoretical description of the photon-induced processes."

The study highlights that transforming this observable into a precision tool requires:

1. Thorough checks of theoretical systematics (modeling of magnetic contributions, survival factors, and form factor parametrizations).
2. Future measurements with larger datasets (e.g., full Run 2 and Run 3 luminosity) to extend kinematic coverage to higher invariant masses and forward rapidities.
3. The use of forward proton detectors to measure proton transverse-momentum distributions.

Ultimately, the work establishes that while the LHC is already sensitive to radius-scale variations, a precise determination of $r_p$ via this channel awaits a consistent treatment of associated theoretical uncertainties and higher-statistics data.