⚛️ phenomenology

Radiative pion-pair production in disperon QED

This paper presents a next-to-leading order calculation of radiative pion-pair production ( $ee\to\pi\pi\gamma$ ) using the disperon QED method to incorporate experimental data via the pion vector form factor, with results implemented in the McMule Monte Carlo framework to assess theoretical uncertainties across various experimental scenarios.

Original authors: Yizhou Fang, Sophie Kollatzsch, Adrian Signer, Max Zoller

Published 2026-09-02

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CC BY 4.0

Original authors: Yizhou Fang, Sophie Kollatzsch, Adrian Signer, Max Zoller

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, particles do not always behave like solid billiard balls; sometimes they act more like ripples in a pond, interacting with invisible forces that shape the universe. One of the most persistent mysteries in modern physics involves the muon, a heavy cousin of the electron. Scientists have long tried to predict how this particle wobbles in a magnetic field, a property known as its magnetic moment. When they compare their theoretical predictions to actual measurements, a troubling gap appears. The numbers do not match. This discrepancy suggests that our current understanding of the forces holding matter together is incomplete. To solve this puzzle, researchers must look at how muons interact with the "hadronic" world—the realm of particles made of quarks, such as protons and pions. A major source of this interaction comes from the creation of pairs of pions, the lightest particles made of quarks, in high-energy collisions. To understand the muon's behavior, physicists need to measure these pion pair creations with extreme precision, down to a fraction of a percent.

The challenge lies in the complexity of the collision itself. When an electron and a positron smash together, they can produce a pair of pions and a flash of light, or a photon. To predict exactly how often this happens, scientists use computer simulations that account for every possible way the particles can interact. However, pions are not simple point-like objects; they have an internal structure that makes the math incredibly difficult. The researchers in this study, working with a sophisticated computer framework called McMule, tackled the problem of calculating these collisions at a very high level of detail. They focused on a specific type of calculation where they could reliably include the messy, internal structure of the pion without making unproven guesses about how it behaves at every step. By doing so, they were able to separate the parts of the calculation they could trust from the parts that remain too uncertain to model accurately.

The team's work centers on a method they call "disperon QED," a technique designed to handle the complicated data about pion structure within complex computer loops. Imagine trying to calculate the path of a ball rolling through a forest where the trees move; you need a way to describe the trees' movement without getting lost in the details of every single leaf. In this case, the "trees" are the internal parts of the pion, and the "ball" is the photon. The researchers found that they could reliably calculate the effects where the pion's structure acts like a simple, static object during the collision. However, they also identified specific scenarios where the pion's internal parts interact in more complex, dynamic ways that their current tools cannot yet describe with certainty. They labeled these difficult parts as "beyond" their reliable scope. By isolating the trustworthy calculations from the uncertain ones, they created a clearer picture of what the computer models can and cannot tell us.

When they applied these refined calculations to different experimental setups, the results varied significantly depending on the energy of the collision. For experiments similar to those conducted by the KLOE detector, which operates at an energy of 1.02 GeV, the new calculations showed that the "trustworthy" parts of the theory change the predicted outcome by about 15 percent. This is a massive shift in the world of particle physics, where even tiny adjustments matter. More importantly, they found that the "beyond" parts—the uncertain, complex interactions—contributed a small but noticeable amount, reaching about 0.1 percent in the large-angle KLOE scenario. While this might sound small, it is significant because the goal of these experiments is to reach a precision of 0.1 percent. This means that for the KLOE-like experiments, the current theoretical models are just barely good enough, and the uncertainty from the uncalculated parts is starting to limit how precisely scientists can test the muon mystery.

In contrast, when the researchers looked at experiments with much higher energies, such as those resembling the BESIII detector at 4 GeV or the B factories at 10 GeV, the story changed. At these higher energies, the complex, uncertain parts of the calculation became almost invisible, contributing less than a tiny fraction of a percent. The dominant effects were the simpler, well-understood interactions. This suggests that for high-energy experiments, the current theoretical tools are sufficient, but for the lower-energy experiments that are currently the most sensitive to the muon puzzle, the theory is hitting a wall. The researchers emphasize that their work does not solve the muon mystery, but it clarifies where the uncertainty lies. They have shown that for the most critical low-energy experiments, the theoretical prediction is limited not by the simple parts of the math, but by the complex, uncalculated interactions of the pion's internal structure.

The study concludes that while the team has successfully mapped out the reliable territory of the calculation, the "beyond" territory remains a frontier. They have provided a way to estimate the size of the unknown errors, which is a crucial step for experimentalists who need to know how much trust to place in their data. By separating the known from the unknown, the researchers have given the scientific community a better tool to interpret the next generation of experiments. The path forward involves refining the description of those complex pion interactions, perhaps by treating them as intermediate states of heavier particles, but for now, the focus is on understanding the limits of what can be calculated. The work serves as a vital checkpoint, ensuring that as we push toward the 0.1 percent precision needed to solve the muon anomaly, we are not misled by gaps in our theoretical understanding.

Technical Summary: Radiative Pion-Pair Production in Disperon QED

Problem Statement
The calculation of the process $e^+ e^- \to \pi^+ \pi^- \gamma$ at next-to-leading order (NLO) is critical for extracting the hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment ( $a_\mu$ ). While initial-state radiation (ISR) corrections are well-understood, the treatment of final-state radiation (FSR) and mixed corrections involving the composite structure of pions presents significant challenges. Specifically, the interaction vertex $\gamma^* \to \pi\pi$ requires a non-perturbative vector form factor (VFF), $F_\pi^V$ . Extending this to higher-order processes involving Compton tensors (e.g., $\gamma^*\gamma^* \to \pi\pi$ and $\gamma^*\gamma^*\gamma \to \pi\pi$ ) introduces conceptual and technical difficulties. Standard scalar QED (sQED) fails to capture the necessary non-perturbative dynamics, while Generalized Vector Meson Dominance (GVMD) sacrifices analyticity. The core problem addressed is how to incorporate data-driven VFFs into loop integrals for Monte Carlo simulations without compromising theoretical consistency or numerical stability, and determining which subsets of NLO corrections can be reliably approximated using these methods.

Methodology
The authors implement a calculation within the Monte Carlo framework McMule, utilizing the disperon QED method. The approach involves:

Form-Factor Scalar QED (FsQED): The authors adopt a prescription where pion-photon interactions are described by sQED blocks equipped with a pion VFF. Crucially, they distinguish between a "technical" definition (inserting VFFs into loop integrals) and the "original" definition (including pole terms of Compton tensors).
Charge Counting and Classification: To address conceptual limitations, the NLO corrections are decomposed based on charge counting ( $q_e$ $q_{e}$ for electrons, $q_\pi$ $q_{π}$ for pions) into:
- Initial-State Corrections (ISC): Purely electron-line corrections.
- Mixed Corrections: Interactions between initial and final states, subdivided into $q_e^5 q_\pi^3$ , $q_e^4 q_\pi^4$ , and $q_e^3 q_\pi^5$ .
- Final-State Corrections (FSC): Purely pion-line corrections.
  The authors argue that only the ISC and the $q_e^5 q_\pi^3$ subset of mixed corrections can be reliably calculated using FsQED, as the latter requires only the $\gamma^*\gamma^* \to \pi\pi$ Compton tensor (pole terms). Contributions involving radiative corrections to Compton tensors (e.g., in the $q_e^4 q_\pi^4$ sector) or the $\gamma^*\gamma^*\gamma \to \pi\pi$ tensor are deemed physically unjustified within the current FsQED framework.
Disperon QED Implementation: To handle the VFFs inside loop integrals, the authors apply a once-subtracted dispersion relation to the VFF. This replaces the loop-momentum-dependent VFF with an integral over a dispersive parameter $s_1$ involving massive "disperon" propagators. This splits diagrams into photon-photon (pp), photon-disperon (pd), and disperon-disperon (dd) topologies.
Technical Handling:
- Automation: Amplitudes are generated using OpenLoops with a dedicated disperon QED model to manage the tedious proliferation of diagrams.
- Numerical Stability: For large dispersive parameters, the amplitude is expanded using "disperon effective theory" to avoid numerical instabilities in the infinite integration range.
- Threshold Singularities: Singularities arising when the dispersive parameter matches the kinematic threshold ( $s_1 = s_{th}$ ) are handled via analytic subtraction of a counterterm (CT) derived from the singular part of the diagram.

Key Contributions

Reliable Subset Definition: The paper defines a physically motivated subset of NLO corrections for $e^+ e^- \to \pi^+ \pi^- \gamma$ that can be reliably computed: the complete ISC and the $q_e^5 q_\pi^3$ subset of mixed corrections.
Estimation of Uncertainty: For the more complex $q_e^4 q_\pi^4$ mixed corrections (involving $\gamma^*\gamma^*\gamma \to \pi\pi$ ), the authors introduce a "hadronic counting" parameter $\lambda_{hadr}$ and selection rules to estimate their impact. These are labeled as "beyond" contributions.
Implementation in McMule: The full NLO calculation (ISC + reliable mixed + estimated "beyond" contributions) is implemented in McMule, making the code and results publicly available.
Phenomenological Analysis: The authors present differential cross-section results for three experimental scenarios: KLOE-like (large and small angle), BESIII-like, and B-factory-like, analyzing the relative impact of various correction terms.

Results

KLOE-like Large-Angle Scenario ( $\sqrt{s} = 1.02$ GeV):
- NLO ISC corrections reach 15–20% relative to the Leading Order (LO) result.
- The reliable mixed corrections ( $q_e^5 q_\pi^3$ ) modify the distribution shape and reach up to ~15% relative to LO.
- The "beyond" contributions ( $q_e^4 q_\pi^4$ ) are small but non-negligible, reaching the 0.1% level relative to the NLO ISC. This limits the theoretical precision in this specific scenario to the sub-percent level.
- The choice of VFF (e.g., from Ref [45] vs. Ref [6]) significantly affects the absolute cross-section (~4% difference) but has negligible impact on the shape of distributions or relative higher-order corrections.
KLOE-like Small-Angle Scenario:
- Corrections are significantly smaller (NLO ISC ~3%, mixed ~1%) compared to the large-angle scenario.
- "Beyond" contributions are negligible.
BESIII-like ( $\sqrt{s} = 4$ GeV) and B-factory-like ( $\sqrt{s} = 10$ GeV) Scenarios:
- Mixed corrections and FSC become increasingly suppressed due to the high-energy behavior of the pion VFF ( $F_\pi^V(q^2) \to 0$ ).
- ISC corrections remain dominant (enhanced by collinear logarithms), reaching ~5% relative to LO.
- "Beyond" contributions are negligible in these high-energy regimes.

Significance and Claims
The paper claims to provide a physics-motivated subset of corrections that can be computed reliably, distinguishing between what is theoretically sound within the FsQED framework and what requires further non-perturbative development. By explicitly separating the calculable $q_e^5 q_\pi^3$ terms from the estimated "beyond" terms, the authors offer a method to quantify the theory uncertainty associated with $e^+ e^- \to \pi^+ \pi^- \gamma$ .

The authors modestly state that while their "best" prediction includes the reliable subset and an estimate of the "beyond" terms, a full calculation of the $q_e^4 q_\pi^4$ sector (including radiative corrections to Compton tensors) is currently not feasible with a physically justified description. Therefore, the comparison between their result and a hypothetical full calculation serves as a lower estimate of the theory uncertainty. The work is presented as a necessary step toward the 0.1% precision required for upcoming analyses like KLOE-NXT, highlighting that for large-angle KLOE-like scenarios, the "beyond" contributions are the limiting factor for theoretical precision. Future work is identified as extending ISC to NNLO and developing more sophisticated descriptions for the $\gamma^*\gamma^*\gamma \to \pi\pi$ Compton tensor.

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