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A New Precise Measurement of the $e^+e^-\to\pi^+\pi^-(\gamma)$ Cross Section with the Initial-State Radiation Method at BaBar

Using the full 460 fb $^{-1}$ of BaBar data and a novel particle separation method based on angular distributions, this paper presents a precise measurement of the $e^+e^-\to\pi^+\pi^-(\gamma)$ cross section up to 1.4 GeV, yielding a refined calculation of the leading-order hadronic contribution to the muon $g-2$ anomaly.

Original authors: BaBar Collaboration

Published 2026-11-04

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CC BY 4.0

Original authors: BaBar Collaboration

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 are not just tiny, solid marbles; they are excitations in fields that constantly interact, creating a complex sea of virtual activity. One of the most precise tests of our understanding of this world involves the muon, a heavy cousin of the electron. Physicists have calculated how a muon should behave in a magnetic field based on the Standard Model, the theory that describes all known particles and forces. However, when scientists measure the muon's actual behavior, a tiny but persistent discrepancy appears. This difference, known as the magnetic anomaly, suggests that something is missing from our theoretical picture. The leading explanation for this gap is that the muon is interacting with a "cloud" of virtual particles popping in and out of existence, specifically a cloud made of hadrons, which are particles like protons and pions. To calculate the size of this cloud's effect, researchers need an incredibly precise map of how often electrons and positrons collide to produce pairs of pions. Without this map, the theoretical prediction remains incomplete, leaving the mystery of the muon's behavior unsolved.

A team of physicists working with the BABAR detector at the SLAC National Accelerator Laboratory has now provided a new, highly precise map of this crucial interaction. By analyzing a massive collection of data—equivalent to 460 inverse femtobarns of collisions—they have measured the rate at which electron-positron collisions produce pion pairs with a level of detail never before achieved. The researchers focused on a specific energy range, from the point where pions can first be created up to 1.4 billion electron volts. In this range, the production of pion pairs is dominated by a short-lived particle called the rho meson, which acts as a gateway for the hadronic cloud that influences the muon. The new measurement confirms the behavior of these particles with such clarity that it helps settle a long-standing debate among different experiments, some of which had reported conflicting results.

The method used to achieve this precision was clever and distinct from previous attempts. Instead of trying to identify every single particle by its specific signature, which can be difficult and error-prone, the team used a statistical trick based on how the particles fly apart. When an electron and a positron collide and emit a high-energy photon, the remaining energy creates a pair of particles. If the pair consists of muons, they fly apart in a pattern that is mathematically different from the pattern created by a pair of pions. By looking at the angles at which the particles emerge relative to the emitted photon, the researchers could statistically separate the pion events from the muon events, even without needing to know exactly what each particle was at the moment of detection. This approach allowed them to use a much wider range of particle speeds, including very slow ones that were previously difficult to study, thereby gathering far more data and reducing the statistical noise.

To ensure their results were not influenced by preconceived notions or external expectations, the team employed a "blind" analysis technique. For most of the study, the final numbers were hidden behind a series of secret offsets, known only to individual team members. This meant that while they were perfecting their methods and checking for errors, they could not see the final answer, preventing any unconscious bias from shaping their choices. Only after the entire analysis was complete and all internal checks were passed were the hidden numbers revealed. The result was a cross-section measurement, which is essentially a probability of the collision happening, that is significantly more precise than their previous work from 2009. The new data shows that the rate of pion production is consistent with the muon channel, validating the method and confirming that the theoretical tools used to describe these interactions are sound.

The implications of this measurement are immediate and profound for the calculation of the muon's magnetic anomaly. Using the new data, the team calculated the contribution of the pion pairs to the muon's magnetic moment. For the energy range between 0.3 and 1.4 billion electron volts, the contribution is found to be 525.9 with a statistical uncertainty of 3.5 and a systematic uncertainty of 3.8, measured in units of 10 to the power of negative 10. When combined with their older 2009 data, the value shifts slightly to 517.8, with even smaller uncertainties. These numbers are critical because they feed directly into the theoretical prediction of the muon's behavior. The new measurement helps to resolve discrepancies between different experimental approaches, particularly those that rely on different theoretical models for radiative corrections. By providing a robust, independent measurement that relies on the ratio of pion to muon events, the team has effectively canceled out many of the systematic errors that plagued earlier studies.

This work does not claim to solve the entire mystery of the muon's magnetic anomaly, but it provides a solid, high-precision foundation for the hadronic part of the calculation. The results stand in good agreement with other recent measurements from different experiments, suggesting that the path forward is clear. The precision achieved here, with total uncertainties well below one percent in the most important energy region, demonstrates that the Standard Model's description of these interactions is robust. As the community continues to refine the theoretical predictions, this new measurement serves as a definitive reference point, ensuring that the search for new physics beyond the Standard Model is built on the most accurate data possible. The clarity of the result, derived from a method that turns the angular distribution of particles into a powerful sorting tool, marks a significant step forward in our understanding of the fundamental forces that govern the universe.

Technical Summary: A New Precise Measurement of the $e^+e^- \to \pi^+\pi^-(\gamma)$ Cross Section with the Initial-State Radiation Method at BABAR

Problem and Motivation
Precise measurements of the $e^+e^- \to \text{hadrons}$ cross section are essential for evaluating dispersion integrals required to calculate the hadronic vacuum polarization (HVP) contribution to the muon magnetic moment anomaly, $a_\mu$ . The dominant contribution (approximately 73%) to the HVP comes from the $e^+e^- \to \pi^+\pi^-$ channel. However, the current theoretical prediction for $a_\mu$ is hindered by conflicting measurements in this channel, preventing a reliable combined average. Previous BABAR analyses and other experiments have raised issues regarding the treatment of radiative corrections, particularly in experiments relying on Next-to-Leading Order (NLO) generators like Phokhara. This study aims to provide an independent, high-precision measurement of the $e^+e^- \to \pi^+\pi^-(\gamma)$ cross section to clarify these discrepancies and improve the Standard Model prediction.

Methodology
The analysis utilizes the full BABAR dataset of 460 fb $^{-1}$ collected at the $\Upsilon(4S)$ resonance and 40 MeV below it. The measurement employs the Initial-State Radiation (ISR) method, where the cross section at a reduced center-of-mass energy $\sqrt{s'}$ is derived from the radiative process $e^+e^- \to X\gamma$ .

Key methodological features include:

Ratio Technique: The cross section is determined via the ratio of $\pi^+\pi^-$ to $\mu^+\mu^-$ ISR events. This approach cancels several systematic uncertainties, including the integrated luminosity, ISR photon detection efficiency, and vacuum polarization effects. The effective ISR luminosity is derived independently from the muon channel, which is compared against QED predictions.
Angular Separation ( $\pi\pi/\mu\mu$ ): Unlike the 2009 BABAR analysis, which relied on particle identification (PID) requiring tracks with momenta $>1$ GeV/ $c$ , this analysis statistically separates pions and muons using their distinct angular distributions in the two-particle center-of-mass frame ( $|\cos\theta^*|$ ). This allows the use of tracks with transverse momentum as low as $p_T > 0.1$ GeV/ $c$ , significantly increasing statistical precision, particularly for muons near threshold and at high masses.
Blind Analysis: To avoid bias, the analysis was conducted in a "blind" manner. Multiplicative offsets were applied to trigger and tracking efficiency corrections, as well as to the separation of $\pi\pi$ and $\mu\mu$ contributions. These offsets were only released after all internal consistency checks were completed.
Radiative Corrections and Simulation: The signal is simulated using the Phokhara generator at NLO. The analysis accounts for additional radiation (up to NNLO for ISR and FSR) and applies corrections for detector effects, including trigger efficiency, tracking efficiency, momentum calibration, and kinematic fit $\chi^2$ selection. Specific corrections are applied for pion-induced fake photons and secondary interactions, which are not well-modeled in standard simulations.
Unfolding: The measured mass spectra are unfolded to recover the true underlying distribution of $\sqrt{s'}$ , accounting for detector resolution and final-state radiation (FSR) migration effects using an iterative method with a response matrix derived from simulation.

Key Contributions and Improvements

New Separation Method: The shift from PID-based selection to angular distribution fitting allows for a much broader kinematic acceptance and reduces systematic uncertainties associated with PID efficiency.
Comprehensive Radiative Treatment: The analysis incorporates BABAR's previous findings on additional radiation, extending the measurement range to include NNLO contributions for both ISR and FSR, addressing limitations in the Phokhara generator.
Refined Efficiency Corrections: Detailed data-driven studies were performed for trigger, tracking, and kinematic fit efficiencies, including specific corrections for pion-specific effects like secondary interactions and fake photons.
Validation: The procedure is validated by comparing the unfolded $\mu^+\mu^-$ spectrum against QED predictions (via the Phokhara generator), showing consistency within uncertainties.

Results
The paper reports the bare cross section for $e^+e^- \to \pi^+\pi^-(\gamma)$ in the energy range from threshold to 1.4 GeV.

Cross Section: The measurement is presented with statistical and systematic uncertainties fully propagated. The total uncertainty in the central region ( $\sqrt{s'} \in [0.5, 1.0]$ GeV) is well below 0.3%.
HVP Contribution: The leading-order hadronic contribution to the muon magnetic anomaly ( $a_\mu^{\text{had, LO}}$ ) calculated using the new 2026 BABAR measurement in the range 0.3–1.4 GeV is $525.9 \pm 3.5_{\text{stat}} \pm 3.8_{\text{syst}}$ (in units of $10^{-10}$ ).
Combined Result: When combined with the 2009 BABAR measurement, the result is $517.8 \pm 2.4_{\text{stat}} \pm 3.3_{\text{syst}}$ in the same energy range.
Consistency: The new measurement shows fair agreement with the 2009 BABAR results, as well as with CMD-3, $\tau$ -based results, and lattice QCD calculations, though the paper notes that the situation regarding conflicting measurements in the $\pi^+\pi^-$ channel remains complex.

Significance
The paper claims that this new independent investigation is necessary to clarify the conflicting landscape of $e^+e^- \to \pi^+\pi^-$ cross-section measurements. By utilizing a complementary approach that avoids PID reliance and incorporates a more complete treatment of radiative corrections, the analysis provides a robust and precise determination of the cross section. The resulting HVP contribution helps refine the Standard Model prediction for the muon $g-2$ , a critical test for potential new physics. The author emphasizes that the blind analysis and rigorous cross-checks against QED ensure the reliability of the result, offering a solid foundation for future theoretical and experimental comparisons.

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