⚛️ phenomenology

NNLO initial-state corrections to $e^+\,e^-\to\mu^+\,\mu^-\,\gamma$

This paper presents next-to-next-to-leading-order QED initial-state corrections, including photonic and vacuum-polarisation effects, for the process $e^+e^-\to\mu^+\mu^-\gamma$ within the McMule framework, demonstrating that while total cross-section corrections are typically below the percent level, they can be locally significant and must not be neglected, especially at higher energies.

Original authors: Yizhou Fang, Sara Gündogdu, Sophie Kollatzsch, Marco Rocco, Adrian Signer, Yannick Ulrich

Published 2026-10-08

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CC BY 4.0

Original authors: Yizhou Fang, Sara Gündogdu, Sophie Kollatzsch, Marco Rocco, Adrian Signer, Yannick Ulrich

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 quiet hum of particle accelerators, scientists fire beams of electrons and their antimatter counterparts, positrons, into each other to study the fundamental building blocks of the universe. When these particles collide, they can vanish and reappear as different particles, such as muons, which are heavier cousins of the electron. Sometimes, this transformation is accompanied by the emission of a flash of light, a photon. To understand exactly how often these events happen and what they look like, physicists rely on complex mathematical models that predict the behavior of these collisions. These predictions are essential for experiments that measure the properties of matter with extreme precision, acting as a reference point against which real-world data is compared. If the theoretical prediction is slightly off, it could hide a new discovery or lead to a misinterpretation of the data. For decades, scientists have been refining these models, pushing them to higher levels of accuracy to match the incredible precision of modern detectors.

A team of researchers has now taken a significant step forward in this effort by calculating a specific type of correction for the collision process where an electron and a positron turn into a muon, an antimuon, and a photon. While previous calculations were good enough for many purposes, they missed subtle effects that become important when measurements need to be accurate to within a fraction of a percent. The researchers focused on the initial stage of the collision, where the electron and positron interact before creating the final particles. They found that by including these initial-state effects at a very high level of detail, along with corrections related to the vacuum itself, they could provide a much more reliable prediction. This work is part of a larger international effort to improve the theoretical tools used by major experiments operating at energies of a few billion electron volts, ensuring that the data collected from these machines can be interpreted with the utmost confidence.

The team used a sophisticated computer framework to perform these calculations, which involved simulating the complex interactions of particles and the light they emit. They looked at several different scenarios that mimic real experimental setups, ranging from lower energies around one billion electron volts to higher energies near ten billion electron volts. In these simulations, they found that the corrections they added were generally small, often less than one percent of the total result. However, in certain specific situations, particularly when looking at how the particles are distributed in space or when the collision energy is higher, these corrections became much more significant. The researchers discovered that ignoring these effects could lead to noticeable errors in the final prediction, especially when trying to distinguish between different types of particle interactions.

A key part of their work involved accounting for what happens in the empty space between particles, known as vacuum polarization. In quantum physics, the vacuum is not truly empty; it is filled with fleeting particles that can briefly appear and disappear, affecting how forces like electromagnetism behave. The team included the effects of these virtual particles in their calculations, finding that they contribute a sizeable amount to the overall result. In some cases, these vacuum effects were just as large as the corrections from the light particles themselves. They also examined how the mass of the electron influences the outcome. While they had to make a simplifying assumption about the electron's mass in one part of the calculation, they determined that this assumption introduced an error so small it would not affect the results for the scenarios they studied.

The researchers presented their findings for three distinct experimental setups, each with its own set of rules for which particles are detected. In the first scenario, which resembles an experiment called KLOE, they looked at collisions where a photon is emitted at a wide angle. They found that while a simpler approximation worked well in some regions, it failed to describe the full picture in others, particularly near a specific energy range where a known particle resonance occurs. In a second scenario, modeled after the BESIII experiment at higher energies, they observed that the corrections were crucial for accurately predicting the behavior of the particles across a wide range of energies. Here, the effects of the vacuum were again found to be significant, sometimes dominating the corrections in specific regions of the data. The third scenario, inspired by experiments at B factories operating at even higher energies, showed that the corrections remained important even at these elevated scales, with the vacuum effects contributing substantially to the final numbers.

Throughout their analysis, the team emphasized that these corrections are not just minor tweaks but necessary components for achieving the precision required by modern physics. They showed that relying on older, less complete calculations could lead to misunderstandings of the data, particularly when trying to isolate specific signals from background noise. By providing these new, more accurate predictions, the researchers have equipped experimentalists with better tools to test the Standard Model of particle physics. Their work serves as a building block for even more precise calculations in the future, helping to push the boundaries of what can be known about the fundamental forces of nature. The results are now available for use by the broader scientific community, ensuring that the next generation of experiments can proceed with a clearer and more accurate theoretical foundation.

Technical Summary: NNLO Initial-State Corrections to $e^+ e^- \to \mu^+ \mu^- \gamma$

Problem and Motivation
The paper addresses the need for improved theoretical predictions for low-energy electron-positron colliders operating at center-of-mass energies ( $\sqrt{s}$ ) of a few GeV. These predictions are critical inputs for precision observables, particularly for experiments such as Babar, KLOE, and BESIII. While the process $e^+ e^- \to \mu^+ \mu^- \gamma$ was previously available only at next-to-leading order (NLO) or via leading-order (LO) matched to parton showers, this work pushes the calculation to next-to-next-to-leading order (NNLO) in Quantum Electrodynamics (QED). Specifically, the authors focus on initial-state corrections (ISC) and vacuum-polarisation corrections (VPC), which are essential for achieving permille-level precision required by current and future experimental efforts like RadioMonteCarLow 2.

Methodology
The calculations are implemented within the Monte Carlo framework McMule, which utilizes the all-order FKS $\ell$ subtraction scheme and next-to-soft (NTS) stabilization techniques to handle infrared singularities in a fully differential manner.

Photonic Corrections: The authors compute NNLO photonic ISC, defined as contributions with charge counting $d\sigma^{(2)}_{\mu\mu\gamma}(q_e^8 q_\mu^2)$ $d σ_{μμ γ}^{(2)} (q_{e}^{8} q_{μ}^{2})$ . These include virtual-virtual, real-virtual, and real-real contributions.
- For the virtual-virtual component (two-loop amplitudes interfered with tree-level), the full electron mass dependence is not available. The authors utilize known massless two-loop amplitudes and apply a "massification" procedure to recover terms not polynomially suppressed by $m_e/Q$ . This introduces an error of order $(\alpha/\pi)^2 m_e^2/Q^2$ , which is deemed negligible for scenarios with tagged photons at large angles but restricts the applicability to collinear photon emissions.
- Real-virtual corrections are evaluated using OpenLoops with full mass dependence, employing the NTS stabilization technique to approximate matrix elements with soft photons.
- Real-real corrections are integrated over phase space using OpenLoops or in-house calculations.
Vacuum Polarisation (VP) Corrections: The authors include both leptonic and hadronic VP insertions.
- Non-factorising VPC: These involve VP insertions inside loop propagators (genuinely NNLO). The authors use the disperon QED method to express loop integrals involving VP through dispersion relations, allowing for the inclusion of generic VP functions (including hadronic VP inferred from experimental data) alongside leptonic loops.
- Factorising VPC: These appear in the photon line connecting the initial and final states. These are computed analytically, including the two-loop leptonic self-energy.
- Both types of VPC are included in a resummed form within the photon propagator.

Key Contributions

First NNLO ISC for $e^+ e^- \to \mu^+ \mu^- \gamma$ : This work provides the first ingredient towards full NNLO corrections for this generic radiative QED process, serving as a technical building block for other NNLO processes and ultimately for N3LO corrections to $e^+ e^- \to \mu^+ \mu^-$ .
Integration of VP Corrections: The paper updates the McMule framework to include a comprehensive set of vacuum-polarisation corrections, distinguishing between factorising and non-factorising contributions and treating them in a resummed manner.
Validation and Implementation: The results are part of the publicly available McMule version v.0.8.0, providing a tool for calculating arbitrary infrared-safe observables.

Results
The authors present results for three experimentally motivated scenarios: a KLOE-like large-angle scenario ( $\sqrt{s} \approx 1$ GeV), a BESIII-like scenario ( $\sqrt{s} \approx 4$ GeV), and a B-factory scenario ( $\sqrt{s} \approx 10$ GeV).

Total Cross Sections: For total cross sections, NNLO ISC corrections are typically below the percent level (often a few permille). However, VPCs are of similar magnitude and must not be neglected, particularly at higher energies.
Differential Distributions: While total cross-section corrections are small, local corrections in distributions can be significantly larger.
- KLOE-like: ISC effects alone are not a good approximation of the total corrections across the full phase space. The difference between full NLO and NLO ISC is significant (up to 20% in some regions). VPCs, dominated by factorisable diagrams, show sizeable impact, especially near the $\rho$ resonance.
- BESIII-like: Similar trends are observed, with large NLO corrections (10–20%). Non-factorisable VPCs become more important in regions of large cross section. Resonances such as $\phi$ and $J/\psi$ are visible.
- B Scenario: At 10 GeV, hadronic resonances (e.g., $\psi(2S)$ ) are pronounced. VPCs account for 30–50% of the NNLO correction depending on the invariant mass. The ISC approximation fails for extreme values of the invariant mass where the cross section is largest.
Comparison: The paper notes that in specific phase-space regions, the total NNLO prediction from McMule can differ from the NLO+PS prediction of BabaYaga@NLO by approximately 0.1%, highlighting the necessity of full fixed-order calculations.

Significance and Claims
The paper claims to advance the field of QED fixed-order predictions to a new level by providing the first NNLO initial-state corrections for $e^+ e^- \to \mu^+ \mu^- \gamma$ . It emphasizes that while ISC are often dominant due to collinear enhancement, they are insufficient to describe the full correction; a complete calculation including VPC is necessary for precision. The work serves as a foundational step for future N3LO calculations and improves the theoretical input for low-energy precision physics. The authors remain modest regarding the massification approximation, acknowledging that a full massive two-loop amplitude calculation would be beneficial to remove restrictions on collinear photon emission scenarios, a task left for future work.

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