⚛️ phenomenology

QED corrections to $e^+e^-\to\pi^+\pi^-(\gamma)$ with amplitude-level resummation

This paper presents a comprehensive calculation of QED corrections to the $e^+e^-\to\pi^+\pi^-(\gamma)$ process implemented in two independent Monte Carlo generators (KKMC and SHERPA) using amplitude-level resummation and higher-order loop corrections, demonstrating excellent agreement between the codes and providing precise estimates of theoretical uncertainties across various energy scales.

Original author: Stéphane Delorme, Charles Earnshaw, Frank Krauss, Alan Price, Nora Salone, Andrzej Siódmok

Published 2026-11-06

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CC BY 4.0

Original author: Stéphane Delorme, Charles Earnshaw, Frank Krauss, Alan Price, Nora Salone, Andrzej Siódmok

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, where particles collide and vanish in flashes of energy, scientists are trying to solve a decades-old mystery regarding the magnetic personality of the muon. The muon is a heavy cousin of the electron, and its magnetic field is slightly stronger than standard physics predicts. For years, this tiny discrepancy hinted at the existence of new, undiscovered particles. However, a recent shift in how scientists calculate the background noise of the universe has muddied the waters. The noise in question comes from the "hadronic vacuum polarization," a quantum effect where the vacuum itself briefly sprouts pairs of particles that interact with the muon. To understand this noise, researchers must measure how electrons and positrons annihilate to create pairs of pions, the lightest particles made of quarks. The challenge is that these collisions are rarely clean; they are often accompanied by the emission of light particles called photons, which can carry away energy and distort the measurement. To extract the true signal, physicists must strip away these radiative effects with extreme precision, a task that requires simulating the complex dance of particles and light with mathematical rigor.

A team of physicists has now taken a significant step forward in this effort by creating two independent, high-precision computer programs to simulate the process of an electron and a positron colliding to produce a pair of pions, with or without an extra photon. The researchers, working with the KKMC and SHERPA software tools, developed a method to calculate the corrections caused by the emission of light to a level of detail never before achieved for this specific process. They treated the pions not as simple points, but as objects with an internal structure that changes how they interact with light, a property known as a form factor. By combining a technique that sums up the effects of countless soft, low-energy photons with exact calculations for the emission of one or two hard, high-energy photons, they created a simulation that captures the full complexity of the event. The two programs, built on completely different mathematical foundations, were run side-by-side to verify each other's results, acting as a powerful cross-check to ensure no hidden errors had slipped into the calculations.

The results of this dual-simulation effort are remarkably consistent. When the researchers compared the total number of events predicted by both programs under specific experimental conditions, the numbers agreed to within a fraction of a percent. In the most precise scenarios, the difference between the two independent calculations was less than four hundredths of one percent. This level of agreement gives the scientific community high confidence that the underlying physics is being modeled correctly. The team also mapped out how the energy and angles of the resulting particles are distributed, finding that the two programs produced identical patterns across the board. This consistency is crucial because it means that the theoretical uncertainty—the "fuzziness" in the prediction caused by missing higher-order effects—is now well-defined and can be quantified. The researchers introduced a new way to estimate these missing pieces, creating a safety margin that accounts for effects that are too complex to calculate directly but are known to exist.

The study also explored how different assumptions about the internal structure of the pion affect the outcome. By testing a specific model where the pion's interaction with light is treated as a sum of various resonance states, the team found that this approach significantly reduces a specific type of imbalance in the data known as forward-backward asymmetry. This finding helps resolve a tension between different experimental measurements, particularly those from the CMD-3 experiment, which had previously reported higher cross-sections than other groups. The new calculations show that when the internal structure of the pion is properly accounted for in the loops of the calculation, the theoretical prediction shifts to align better with the experimental data. This adjustment is vital for refining the value of the hadronic vacuum polarization, which is the dominant source of uncertainty in the muon magnetic moment calculation.

In scenarios where a high-energy photon is detected, the researchers found that the corrections to the standard predictions were substantial, sometimes reaching nearly ten percent. This highlights the importance of including these higher-order effects, as ignoring them would lead to a significant misinterpretation of the experimental data. The team demonstrated that their new method handles these large corrections effectively, providing a reliable tool for future experiments at facilities like BESIII and the proposed Future Circular Collider. By offering a precise, cross-validated calculation that includes the full complexity of photon emission and the pion's internal structure, this work provides a solid foundation for the next generation of measurements. The goal is to determine whether the remaining discrepancy in the muon's magnetic moment is indeed a sign of new physics or simply a result of imperfect calculations. With these refined tools, the path to answering that question has become clearer, allowing scientists to peer deeper into the fundamental laws that govern the universe.

Technical Summary: QED Corrections to $e^+e^- \to \pi^+\pi^-(\gamma)$ with Amplitude-Level Resummation

Problem and Motivation
Precise theoretical predictions for the process $e^+e^- \to \pi^+\pi^-(\gamma)$ are critical for determining the leading-order hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment, $a_\mu$ . While recent lattice QCD calculations have shifted the Standard Model prediction, tensions remain between lattice results and data-driven estimates based on $e^+e^-$ cross-sections. Furthermore, discrepancies exist among experimental datasets (e.g., CMD-3 vs. KLOE and BaBar). Extracting the HVP contribution requires high-precision radiative corrections, particularly for experiments utilizing radiative return or energy scans. Existing Monte Carlo (MC) generators often rely on fixed-order calculations or parton showers that may not fully capture higher-order effects or coherent interference at the required precision.

Methodology
This work presents a new calculation of QED radiative corrections for $e^+e^- \to \pi^+\pi^-(\gamma)$ implemented independently in two MC event generators: KKMC and SHERPA. The calculation is based on the Yennie-Frautschi-Suura (YFS) formalism, which performs all-order resummation of soft photons.

Theoretical Framework: Pions are treated as point-like scalar particles within scalar QED (sQED), dressed with the pion electromagnetic form factor $F_\pi(q^2)$ . The calculation utilizes the STRONG2020 parametrization for $F_\pi$ .
Resummation and Amplitudes: The approach combines all-order soft-photon exponentiation with exact tree-level amplitudes for one- and two-photon emissions.
- KKMC: Implements the Coherent Exclusive Exponentiation (CEEX) method.
- SHERPA: Implements the newly developed ACRAIC method.
- Both generators include exact one-loop virtual corrections (initial-state, final-state, and initial-final interference boxes) and exact one-loop corrections to one-photon emission.
Form Factor Treatments: Two treatments of the form factor in loops are considered:
1. Factorised approach ( $F \times$ sQED): The form factor is applied to the s-channel photon vertex.
2. Generalised Vector Meson Dominance (GVMD): The form factor is attached to every photon coupling to pions, including loop photons, effectively treating loop photons as having complex masses.
Uncertainty Estimation: Instead of relying on the spread between generators or the size of the last included correction, the authors employ theory nuisance parameters. These parameters estimate missing higher-order terms (e.g., two-loop boxes, genuine two-loop pion vertices, and higher-order photonic corrections) by rescaling identified missing contributions. A specific "radiative-return parameter" is introduced for scenarios where first-order corrections are of order one.

Key Contributions

Dual Implementation: The first amplitude-level implementation of these corrections in two independent generators (KKMC and SHERPA), providing a robust cross-check.
Higher-Order Precision: Inclusion of exact one- and two-photon tree amplitudes, complete one-loop corrections (including initial-final boxes), and photonic two-loop vertices.
New Uncertainty Framework: Application of theory nuisance parameters to YFS resummation to systematically estimate missing higher orders, including a specialized rule for radiative return.
Form Factor Modeling: Detailed comparison of factorised and GVMD approaches, particularly regarding their impact on charge asymmetry.

Results
The authors evaluated the generators across five fiducial scenarios corresponding to experimental setups: CMD, KLOE-I, KLOE-II, BESIII, and a high-energy "B" scenario.

Agreement between Generators: At first order ( $O(\alpha)$ ), the fiducial cross sections of KKMC and SHERPA agree to $\lesssim 0.04\%$ , and differential distributions agree within statistical precision. At second order ( $O(\alpha^2 L)$ ), the agreement remains within statistical uncertainties.
Uncertainty Bands:
- CMD (Elastic): The second-order uncertainty band is $\sim 0.005\%$ .
- KLOE-I: $\sim 0.1\%$ .
- KLOE-II: $\sim 0.02\%$ .
- BESIII and B (Tagged Photon): $\sim 0.4\%$ and $\sim 0.9\%$ respectively, dominated by the radiative-return parameter.
Form Factor Impact: The GVMD treatment reduces the forward-backward charge asymmetry for CMD-3 kinematics by up to $\sim 1\%$ near the $\rho$ resonance compared to the factorised approach. The two generators agree on this shift within statistical uncertainties.
Comparison with BabaYaga@NLO:
- In the CMD, KLOE-I, and KLOE-II scenarios, the $O(\alpha^2 L)$ fiducial cross sections agree with BabaYaga@NLO within $0.01\%$ to $0.1\%$ .
- In the polar angle distributions for KLOE-I, both generators follow the shape of BabaYaga@NLO, differing by only a few tenths of a percent, supporting the BabaYaga@NLO prediction over PHOKHARA in this region.
- For tagged-photon scenarios (BESIII, B), the $O(\alpha)$ predictions for $\pi^+\pi^-$ (without explicit photon) lie significantly above BabaYaga@NLO in regions requiring a second hard photon. However, the explicit $\pi^+\pi^-\gamma$ calculation at NLO agrees with BabaYaga@NLO to $\sim 0.1\%$ .

Significance and Claims
The paper claims to provide a rigorous, amplitude-level treatment of QED corrections for $e^+e^- \to \pi^+\pi^-(\gamma)$ that goes beyond fixed-order predictions by including all-order soft-photon resummation matched to exact higher-order matrix elements. The primary significance lies in the high level of agreement between two independent implementations, validating the theoretical framework. The authors assert that their theory nuisance parameter method offers a systematic way to estimate missing higher orders, which is crucial for reducing theoretical uncertainties in the extraction of $a_\mu$ and $\alpha(s)$ . The work establishes a baseline precision (sub-percent level for most scenarios) against which future experimental data and theoretical improvements (such as two-loop boxes and genuine two-loop vertices) can be measured. The authors note that the limiting factor for further precision is now the calculation of the specific higher-order amplitudes, which can be straightforwardly implemented into the existing YFS framework.

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