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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 460 fb $^{-1}$ of BaBar initial-state radiation data, this paper presents a new precise measurement of the $e^+e^-\to\pi^+\pi^-(\gamma)$ cross section from threshold to 1.4 GeV that improves upon previous analyses by statistically separating pion and muon contributions, ultimately yielding an updated leading-order hadronic vacuum polarization contribution to the muon magnetic anomaly of $525.9\pm 3.5_\mathrm{stat}\pm3.8_\mathrm{syst}$ in units of $10^{-10}$ .

Original author: BaBar Collaboration

Published 2026-11-04

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CC BY 4.0

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

In the subatomic world, particles do not simply bounce off one another; they briefly vanish and reappear as a swirling cloud of energy before settling back into matter. Physicists call this fleeting transformation "vacuum polarization." It is a subtle effect where the empty space between particles is actually teeming with virtual particles that pop in and out of existence, momentarily altering how forces behave. One of the most precise tests of our understanding of the universe involves a tiny particle called the muon, which is essentially a heavy cousin of the electron. Scientists have measured how a muon wobbles as it spins in a magnetic field, a property known as its magnetic anomaly. Theoretical predictions for this wobble rely heavily on calculating how much the muon interacts with that seething vacuum of virtual particles. However, a gap has emerged between what theory predicts and what experiments observe, leaving physicists unsure if the discrepancy points to new, undiscovered physics or simply to an error in how the vacuum's influence is calculated.

To resolve this, researchers must measure the strength of the interaction between electrons and positrons—the antimatter counterparts of electrons—as they transform into pairs of pions, which are the lightest particles made of quarks. This specific transformation is the dominant source of the vacuum's influence on the muon's wobble. For years, different experiments have produced slightly different measurements of this process, creating a confusing picture that prevents a definitive answer. The question is not just about a number; it is about whether our fundamental map of reality is complete or if it needs a new landmark.

A team of physicists working with the BABAR detector at the SLAC National Accelerator Laboratory has now provided a fresh, highly precise look at this process. They analyzed a massive collection of data, equivalent to 460 trillion electron-positron collisions, gathered over many years. Instead of trying to identify every single particle by its specific "fingerprint" as it passes through the detector, the researchers used a clever statistical trick. They looked at the angles at which the particles flew apart after the collision. Because pions and muons have different masses and spins, they scatter at different angles in a predictable way. By studying the distribution of these angles across millions of events, the team could mathematically separate the pion signals from the muon signals without needing to rely on traditional particle identification methods. This approach allowed them to use a much wider range of particle speeds, including those moving more slowly, which previous measurements had to discard.

The researchers also employed a technique called initial-state radiation. In this method, one of the colliding particles emits a high-energy photon before the actual crash, effectively lowering the energy of the collision. This allows the experiment to scan a continuous range of energies from the lowest possible threshold up to 1.4 billion electron volts, all using the same dataset collected at a single, fixed energy setting. By comparing the rate of pion production to the rate of muon production, many of the experimental uncertainties canceled out, leaving a very clean measurement of the pion cross section, which is a measure of how likely the transformation is to occur.

The result is a new, precise map of how often electrons and positrons turn into pion pairs across this energy range. The team found that in the most important energy region, where the interaction is strongest, their new measurement is about two percent higher than a previous BABAR result from 2009. While the two results are statistically compatible within their margins of error, the new data offers a more inclusive and independent view. When the researchers combined their new findings with the 2009 data, they calculated the total contribution of this pion channel to the muon's magnetic anomaly. The combined value is 517.8 with a small margin of error, while the new measurement alone stands at 525.9.

This new measurement does not immediately solve the mystery of the muon's wobble, but it clarifies the landscape. The result sits comfortably between some of the older measurements and aligns well with recent calculations based on the behavior of quarks and gluons inside atomic nuclei. However, it remains in tension with other specific experiments that reported significantly lower values. By providing a robust, independent check that avoids the limitations of earlier techniques, this work helps physicists narrow down where the true value lies. The path forward now relies on combining these diverse experimental efforts to see if the gap between theory and observation can be closed, or if it truly signals a crack in the Standard Model of physics.

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 Statement
The leading-order hadronic vacuum polarization (HVP) contribution is a critical, limiting ingredient in the Standard Model prediction of the muon magnetic anomaly ( $g-2$ ). Data-driven evaluations rely on dispersion integrals over measured $e^+e^- \to \text{hadrons}$ cross sections, where the $e^+e^- \to \pi^+\pi^-$ channel provides the dominant contribution. However, the current experimental landscape is unsettled: precise measurements in this channel are not mutually consistent, preventing straightforward combination and hindering progress. Independent measurements utilizing different experimental strategies are required to clarify these discrepancies and reduce the uncertainty assigned to the hadronic contribution.

Methodology
This work presents a new measurement of the $e^+e^- \to \pi^+\pi^-(\gamma)$ cross section (including final-state radiation, FSR) from threshold to 1.4 GeV. The analysis utilizes the full BABAR data set collected at the PEP-II asymmetric-energy $e^+e^-$ storage rings, corresponding to an integrated luminosity of 460 fb $^{-1}$ , primarily collected near the $\Upsilon(4S)$ resonance.

The measurement employs the Initial-State Radiation (ISR) method. In this approach, the emission of a hard photon from the incoming $e^+e^-$ state lowers the effective collision energy ( $\sqrt{s'}$ ), allowing the full energy range to be scanned with a single data sample.

Key methodological features include:

Statistical Separation: Unlike the previous 2009 BABAR analysis which relied on particle identification (PID) to separate pions and muons, this analysis statistically separates the $\pi^+\pi^-$ and $\mu^+\mu^-$ contributions using their distinct angular distributions in the two-particle center-of-mass frame. Pion pairs follow a $\sin^2 \theta^*$ distribution, while muon pairs follow a $1 + \cos^2 \theta^*$ distribution at higher masses. This allows for a less restrictive track-momentum selection, retaining momentum-asymmetric topologies that populate the large- $|\cos \theta^*|$ region.
Ratio Method: The bare cross section is derived from the ratio of the pion to muon ISR spectra ( $\pi^+\pi^-(\gamma)/\mu^+\mu^-(\gamma)$ ). This approach cancels or strongly reduces common systematic uncertainties, including the absolute $e^+e^-$ luminosity, ISR-photon reconstruction efficiency, and vacuum polarization effects.
Inclusive Strategy: Following recent findings regarding discrepancies in next-to-leading-order Phokhara Monte Carlo simulations, this analysis adopts an inclusive treatment of extra photons. Events with additional ISR or FSR photons are retained if compatible with the signal topology, as the impact of dominant additional ISR largely cancels in the pion-to-muon ratio.
Unfolding and Validation: The measured spectra are unfolded from reconstructed mass to true final-state energy ( $\sqrt{s'}$ ) using matrices derived from simulation and validated with closure tests. The method is validated by comparing the unfolded dimuon mass spectrum against the QED prediction, yielding a ratio $R_{\mu\mu} = 0.9920 \pm 0.0030_{\text{stat}} \pm 0.0031_{\text{syst}}$ , consistent with a constant over the full mass range with a precision of 0.69%.

Key Contributions

Independence: By utilizing statistical separation based on angular distributions rather than PID, this measurement is essentially independent of the previous BABAR result, which was limited by PID performance and used only half the available data.
Precision: The analysis achieves statistical and systematic precisions of 0.67% and 0.72%, respectively, over the full 0.3–1.4 GeV range. In the dominant 0.5–1.0 GeV region, precisions reach 0.57% (stat) and 0.56% (syst).
Complementarity: The new measurement is statistically and systematically complementary to the 2009 BABAR result, particularly in the central $\rho$ -resonance region where the dispersive kernel makes the measurement most critical for HVP calculations.

Results
The paper reports the following leading-order HVP contributions ( $a_\mu^{\pi\pi}$ ) in units of $10^{-10}$ :

New 2026 Measurement (0.3–1.4 GeV): $525.9 \pm 3.5_{\text{stat}} \pm 3.8_{\text{syst}}$ .
Combined 2009 + 2026 BABAR Measurement: $517.8 \pm 2.4_{\text{stat}} \pm 3.3_{\text{syst}}$ .

The new cross section is approximately 2% higher than the 2009 BABAR result in the $\rho$ -resonance region. Local significance scans indicate that differences between the two BABAR measurements are typically below 2 standard deviations in the central region, with small statistical correlations and conservative systematic correlation scenarios altering local significances by at most 0.2 standard deviations.

In the common energy range of 0.6–0.8832 GeV, the 2026 result is within two standard deviations of most previous results, including the 2009 BABAR ( $-1.4\sigma$ ) and CMD-3 ( $+0.9\sigma$ ) measurements. However, it remains inconsistent with KLOE ( $-4.2\sigma$ ) and BESIII ( $-3.0\sigma$ ). The result is consistent with $\tau$ -based estimates and recent lattice QCD calculations.

Significance
The paper claims that this measurement provides a complementary and essentially independent input to the data-driven HVP determination. By resolving the pion and muon components statistically, the analysis overcomes the dominant particle-identification limitations of the earlier BABAR measurement. The result contributes to the ongoing effort to clarify the experimental situation regarding the $e^+e^- \to \pi^+\pi^-$ channel, which remains a bottleneck in reducing the uncertainty of the Standard Model prediction for the muon magnetic anomaly. The author emphasizes that independent measurements with different strategies are essential for progress in this field.

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