Elastic lepton-proton two-photon exchange scattering: An exact HB$χ$PT analysis including hadronic effects at NNLO

This paper presents an exact analytical evaluation of the two-photon exchange correction to elastic lepton-proton scattering at low energies using heavy-baryon chiral perturbation theory up to NNLO, revealing non-vanishing proton structure effects and demonstrating good perturbative convergence for the kinematic regime relevant to the MUSE experiment.

The Gist

Imagine you are trying to measure the size of a tiny, bouncy ball (a proton) by throwing other tiny balls (electrons or muons) at it. You want to know exactly how the ball bounces back. In the world of physics, this is called "scattering."

For a long time, scientists used a simple rulebook to predict how these balls would bounce. They assumed the interaction was like a game of billiards: one ball hits another, and that's it. This is called "one-photon exchange."

However, in recent years, experiments have shown that the real world is messier than billiards. Sometimes, the balls don't just exchange one "messenger" (a photon); they exchange two messengers at the same time. This is called Two-Photon Exchange (TPE). This extra exchange changes the bounce slightly, and if you ignore it, your measurements of the proton's size and shape are wrong.

This paper is a new, ultra-precise calculation of exactly how much this "two-messenger" exchange changes the bounce, specifically for the low-energy experiments being planned by the MUSE collaboration.

Here is the breakdown of what the authors did, using simple analogies:

1. The Old Way vs. The New Way

• The Old Way (Soft-Photon Approximation): Previous calculations were like trying to predict a storm by only looking at the gentle breeze. Scientists assumed the "messengers" (photons) exchanged were very soft and low-energy. They used a shortcut called the "Soft-Photon Approximation" (SPA). It's like saying, "The wind is so light, we can ignore the gusts."
• The New Way (Exact Analysis): This paper says, "Wait, sometimes the wind is a hurricane!" The authors decided to stop using shortcuts. They calculated the interaction exactly, accounting for every possible way the two photons could be exchanged, even if they are "hard" (high energy) and wild. They used a sophisticated mathematical framework called Heavy-Baryon Chiral Perturbation Theory (HBχPT), which is like a highly detailed map of the proton's internal structure.

2. The "Recoil" Problem

Imagine the proton isn't a giant, immovable boulder, but a heavy bowling ball. When a tiny marble (the electron) hits it, the bowling ball wobbles. This wobble is called recoil.

• In the past, scientists mostly ignored the wobble or approximated it.
• This paper calculates the wobble with extreme precision, going up to a level of detail called NNLO (Next-to-Next-to-Leading Order). Think of this as measuring the wobble not just in inches, but in microns. They found that these tiny wobbles, when combined with the two-photon exchange, create small but important corrections to the final result.

3. The Proton's "Internal Structure"

The proton isn't a solid, featureless marble; it's a fuzzy cloud of quarks and gluons.

• The Discovery: When the authors did their exact calculation, they found that the proton's internal "fuzziness" (its structure) actually leaves a fingerprint on the two-photon exchange.
• The Surprise: In the old "shortcut" methods (SPA), these structural fingerprints seemed to disappear or cancel out completely. But in the new, exact calculation, they do not vanish. They remain as a small, measurable effect. It's like realizing that the texture of the bowling ball actually changes how the marble bounces, even if the ball is heavy.

4. Did the Math Work? (Convergence)

When you do complex math like this, you often worry that adding more layers of detail will make the answer explode into nonsense.

• The Good News: The authors found that their math is stable. The first layer of correction (NLO) was large, but the next layer (NNLO) was small.
• The Metaphor: Imagine you are climbing a ladder. The first rung is big. The second rung is smaller. The third rung is tiny. This tells us the ladder is stable, and we can trust the result. The "perturbative expansion" (the method of adding corrections one by one) is working well.

5. Electrons vs. Muons

The MUSE experiment will use two types of particles: electrons and muons (muons are like heavier, "cousin" electrons).

• Electrons: The math for electrons involves a lot of big numbers that cancel each other out perfectly. It's like a tug-of-war where both teams are pulling hard, but the net result is small.
• Muons: For muons, the forces don't cancel out as much; they add up.
• The Result: Despite these different internal mechanics, the final "bounce" (the total correction) ends up being roughly the same size for both particles. This is a crucial finding because it helps scientists understand why previous experiments using only electrons might have seen different results than those using muons.

Summary of the Conclusion

The authors conclude that:

1. Shortcuts are dangerous: The old "Soft-Photon" method missed significant physics, especially regarding the proton's internal structure and the "hard" exchanges of photons.
2. The new math is solid: By doing the full, exact calculation, they confirmed that the corrections are small enough to be trusted, meaning the theory is converging nicely.
3. Structure matters: The proton's internal shape (its radius and magnetic moment) plays a real role in these interactions, even at this level of precision.

In short, this paper provides a much more accurate "rulebook" for the MUSE experiment, ensuring that when they measure the proton, they aren't being misled by the complex dance of two-photon exchanges. They have removed the guesswork and replaced it with a precise, exact calculation.

Technical Summary

Problem Statement
The precise determination of the proton's electromagnetic form factors and charge radius relies on elastic lepton-proton scattering experiments. Recent precision measurements, such as those by the MUSE collaboration, have highlighted discrepancies between polarized and unpolarized form factor extractions and between scattering and atomic spectroscopy results (the proton radius puzzle). A dominant source of systematic uncertainty at the sub-percent level is the two-photon exchange (TPE) radiative correction to the one-photon exchange (OPE) process. While previous theoretical analyses within Heavy-Baryon Chiral Perturbation Theory (HB$\chi$PT) have addressed TPE effects up to Next-to-Leading Order (NLO), they often relied on the Soft-Photon Approximation (SPA). The SPA simplifies loop integrals but fails to capture kinematic regions where both exchanged photons are hard (far off-shell), potentially missing significant contributions. Furthermore, standard HB$\chi$PT power counting at NLO treats the proton as a point-like particle, neglecting finite-size hadronic structure effects (such as pion-loop contributions and form factors) that enter at Next-to-Next-to-Leading Order (NNLO). This paper addresses the need for an exact, model-independent analytical evaluation of TPE corrections up to NNLO accuracy, specifically targeting the low-momentum transfer regime ($Q^2 < 0.1 \text{ GeV}^2$) relevant to the MUSE experiment.

Methodology
The authors employ the framework of SU(2) Heavy-Baryon Chiral Perturbation Theory (HB$\chi$PT), which expands observables in powers of small external momenta and inverse proton mass ($1/M$). The analysis is conducted in the laboratory frame where the target proton is at rest.

• Exact Analytical Evaluation: Unlike previous works that utilized the SPA to simplify 4-point loop integrals, this study evaluates all loop integrals analytically without approximations. This includes the full kinematic region of the TPE box and crossed-box diagrams.
• Order of Accuracy: The calculation extends to NNLO, retaining terms up to $O(\alpha/M^2)$ in the recoil expansion. This includes:
  1. Kinematical NNLO Corrections: Arising from the interference of LO and NLO TPE amplitudes with OPE amplitudes, and from the recoil expansion of kinematical pre-factors (e.g., outgoing lepton energy $E'$ and velocity $\beta'$).
  2. Dynamical NNLO Corrections: Arising from genuine NNLO Feynman diagrams ($O(e^4/M^2)$). These include "reducible" two-loop diagrams where pion-loops factorize at the proton-photon vertex, effectively dressing the vertices with the proton's Dirac ($F_1^p$) and Pauli ($F_2^p$) form factors.
• Renormalization and Divergences: Infrared (IR) divergences are isolated using dimensional regularization and shown to cancel against corresponding bremsstrahlung contributions. Ultraviolet (UV) divergences from pion-loops are renormalized via local counterterms and low-energy constants (LECs), specifically the proton's anomalous magnetic moment ($\kappa_p$) and mean-square charge radius ($r_p$).
• Diagrammatic Scope: The analysis considers the dominant elastic intermediate state. It includes LO ($O(\alpha^2)$), NLO ($O(\alpha^2/M)$), and NNLO ($O(\alpha^2/M^2)$) diagrams. Notably, it omits "irreducible" two-loop TPE diagrams involving non-factorizable pion-loops, as their analytical evaluation is significantly more complex and their numerical contribution is expected to be suppressed.

Key Contributions

1. Exact Analytical Formulas: The paper provides the first exact analytical expressions for the fractional TPE corrections to the elastic differential cross section up to $O(\alpha/M^2)$, including all kinematically suppressed recoil terms and dynamical hadronic structure effects.
2. Proton Structure at NNLO: The authors demonstrate that at NNLO, structure-dependent effects (mediated by the proton's charge radius and anomalous magnetic moment) do not vanish. In contrast to SPA-based analyses where these effects cancel or are pushed to higher orders, the exact treatment reveals non-vanishing residual proton structure effects of $O(\alpha/M^2)$.
3. Re-evaluation of NLO Contributions: The study revisits NLO contributions, showing that diagrams (e) through (h) add constructively and yield sizable positive corrections, particularly for electron scattering. This contrasts with the LO diagrams (a) and (b), which exhibit substantial cancellations.
4. Comparison with SPA: The work systematically compares exact results with previous SPA-based calculations (e.g., Refs. [56, 60]), highlighting significant discrepancies. The SPA is shown to underestimate contributions, particularly those arising from hard photon exchanges and specific interference terms involving NLO OPE amplitudes.

Results

• Perturbative Convergence: The numerical results indicate that the chiral expansion exhibits reasonably good perturbative convergence. While NLO corrections are numerically sizable, the net NNLO corrections (both kinematical and dynamical) are found to be small, approximately 0.5% for electron-proton and 0.8% for muon-proton scattering relative to the LO Born contribution in the MUSE kinematic range.
• Lepton Mass Dependence:
  • For electron scattering, NLO corrections are negative at very low $Q^2$ and turn positive at higher momentum transfers. The total TPE correction reaches approximately 7.5% at the highest beam momenta.
  • For muon scattering, NLO corrections remain positive and grow monotonically. The total correction is approximately 4.4%.
  • The net NNLO corrections are negative and much smaller in magnitude than lower-order terms.
• Structure Effects: The dynamical NNLO contributions, which depend on the proton rms charge radius ($r_p$), are uniformly negative. The paper quantifies the sensitivity of these results to the proton radius puzzle by varying $r_p$ between values derived from CREMA (muonic hydrogen) and MAMI (electron scattering) experiments.
• SPA vs. Exact: The exact evaluation reveals substantial cancellations among various contributions that are not manifest in SPA results. Specifically, the SPA fails to capture the non-vanishing structure-dependent terms at NNLO and underestimates the magnitude of NLO corrections arising from diagrams (e)–(h).

Significance and Claims
The paper claims that an SPA-based analysis of TPE suffers from serious shortcomings due to the improper treatment of kinematics in loop integrations, particularly missing contributions from regions where both exchanged photons are hard. By providing an exact analytical evaluation, the authors establish a more robust baseline for interpreting low-energy lepton-proton scattering data.

• The work validates the efficacy of the HB$\chi$PT expansion scheme, showing that despite large NLO terms, the series converges well at NNLO.
• It highlights that previous SPA analyses missed significant structure-dependent effects at $O(\alpha/M^2)$, which are now shown to be non-negligible and essential for sub-percent precision.
• The authors conclude that while their calculation is a significant step forward, it is not yet a fully systematic treatment, as it neglects irreducible two-loop diagrams and inelastic intermediate states (such as the $\Delta(1232)$ resonance), which may become relevant at higher energies. They emphasize that their results provide a critical theoretical input for the ongoing MUSE experiment to resolve discrepancies in proton structure measurements.