Measurement of the $e^+e^-\toπ^+π^-π^0$ cross section in the energy region from 0.56 to 1.1 GeV with the SND detector

Using data collected by the SND detector at the VEPP-2000 collider, this paper presents a precise measurement of the $e^+e^-\to\pi^+\pi^-\pi^0$ cross section in the 0.56–1.1 GeV energy range, yielding improved resonance parameters and a refined calculation of the leading-order hadronic contribution to the muon magnetic anomaly.

The Gist

Imagine you are trying to understand the rules of a very complex, invisible game played by the smallest particles in the universe. Physicists are like detectives trying to figure out how these particles interact, collide, and transform. One of the most important "clues" they are looking for is a tiny discrepancy in how a specific particle, the muon, spins. This discrepancy, called the "muon magnetic anomaly," could be the key to discovering new, hidden laws of physics that go beyond our current understanding.

To solve this mystery, the detectives need to know exactly how often certain particles collide and turn into others. Specifically, they need to measure how often an electron and a positron (a particle of antimatter) smash together to create three pions (a type of particle made of quarks).

This paper is the report from a team of detectives working at the SND detector in Novosibirsk, Russia. Here is what they did, explained simply:

1. The Collision Course: Building the "Particle Factory"

Think of the VEPP-2000 collider as a high-speed racetrack for electrons and positrons. The scientists accelerate these particles to nearly the speed of light and crash them into each other.

• The Goal: They wanted to see what happens when these particles collide at specific speeds (energies) between 0.56 and 1.1 billion electron-volts (GeV).
• The Result: Sometimes, the crash creates a "firework" of three pions ($\pi^+\pi^-\pi^0$).
• The Problem: In the past, different teams of detectives (like BABAR, Belle II, and earlier Russian experiments) had measured this "firework" rate, but their numbers didn't match. Some said it happened 8% more often than others. It was like three witnesses giving different stories about the same crime. The scientists needed a new, super-precise measurement to settle the argument.

2. The Detective Kit: The SND Detector

The SND detector is a giant, spherical camera that surrounds the crash site. It's like a high-tech onion with many layers, each designed to catch different clues:

• The Inner Layer (Tracking System): This is like a net that catches the paths of charged particles (the pions) as they fly out, recording exactly where they went.
• The Middle Layer (Calorimeter): This is a giant energy sponge. When particles hit it, it measures how much energy they had. It's crucial for spotting the "neutral pion" ($\pi^0$), which is invisible to the other layers but leaves a flash of light (photons) when it decays.
• The Outer Layer (Muon System): This catches the "ghosts" (muons) that pass right through the other layers.

3. The Investigation: Sorting the Noise from the Signal

The crash site is messy. For every "perfect" event where three pions are created, there are thousands of "fake" events caused by background noise (like stray particles from the beam or cosmic rays).

• The Filter: The scientists had to build a very strict filter. They looked for events with exactly two charged tracks and two flashes of light (from the neutral pion).
• The "Fit": They used a mathematical "fit" (like trying to force a square peg into a round hole, but in reverse) to see if the energy and momentum of the detected particles balanced out perfectly. If they did, it was likely a real signal. If not, it was background noise.
• The Challenge: The background noise changes depending on the energy of the crash. Near the "omega" resonance (a specific energy where particles love to form a temporary "omega" particle), the noise is different than near the "phi" resonance. The team had to use different rules for different energy zones to get a clean picture.

4. The Big Reveal: The Cross Section

The "cross section" is just a fancy word for "how likely this event is to happen."

• The team measured this probability with incredible precision. Their uncertainty was less than 1%.
• The Result: Their measurements settled the debate. They found that their data agrees well with the BABAR experiment but is about 7-8% lower than the Belle II experiment. This suggests that the earlier Belle II measurement might have been slightly off, or there are subtle differences in how the experiments were run.

5. Why Does This Matter? The Muon Mystery

Why do we care about counting pions?

• The Muon's Spin: The muon is a heavy cousin of the electron. It spins like a top. According to our current laws of physics (the Standard Model), we can predict exactly how fast it should spin.
• The Discrepancy: Real-world experiments show the muon spins slightly faster than predicted. This "anomaly" suggests that invisible, unknown particles are popping in and out of existence, nudging the muon.
• The Calculation: To calculate the prediction accurately, physicists need to know exactly how often electrons turn into hadrons (like pions). This new, precise measurement of the pion production rate allows them to recalculate the muon's expected spin.
• The Outcome: Using their new data, the calculated "hadronic contribution" to the muon anomaly is 45.95. This number is very close to the BABAR results but significantly different from the Belle II results. It brings us one step closer to understanding if there is "New Physics" hiding in the universe.

6. The Bonus: Mapping the "Resonance" Landscape

While counting the pions, the team also mapped out the "landscape" of the omega and phi particles.

• Think of these particles as musical notes. When the collision energy hits the exact "note" of the omega or phi particle, the production rate spikes (a resonance).
• By measuring the shape of these spikes, the team determined the exact mass (weight) and width (how long they last) of these particles with record-breaking accuracy. They found that the omega particle is slightly lighter and narrower than previously thought, and they improved the accuracy of how often a rho particle turns into three pions.

Summary

In short, this paper is a masterclass in precision. The SND team built a better map of how particles collide at specific energies. By resolving a conflict between previous measurements, they provided a cleaner, more accurate foundation for calculating the muon's magnetic anomaly. This helps the global physics community get closer to answering the ultimate question: Is there a hidden layer of reality beyond what we currently know?

Technical Summary

Here is a detailed technical summary of the paper "Measurement of the e+e−→π+π−π0 cross section in the energy region from 0.56 to 1.1 GeV with the SND detector."

1. Problem Statement and Motivation

The paper addresses the need for a high-precision measurement of the cross section for the process $e^+e^- \to \pi^+\pi^-\pi^0$ in the center-of-mass energy range of 0.56 to 1.1 GeV.

• Physics Context: This energy region is dominated by the $\omega(782)$ and $\phi(1020)$ vector meson resonances. Precise knowledge of this cross section is critical for calculating the leading-order hadronic contribution to the muon anomalous magnetic moment ($a_\mu$), a key parameter in testing the Standard Model.
• Experimental Discrepancies: Previous measurements by experiments such as CMD-2, SND (at VEPP-2M), BABAR, and Belle II show significant discrepancies. Notably, Belle II data near the $\omega$ resonance maximum is approximately 8% higher than BABAR data, with declared systematic uncertainties that do not fully explain the gap.
• Goal: To resolve these discrepancies by performing a new measurement with a systematic uncertainty of no worse than 1.5%, utilizing the upgraded VEPP-2000 collider and the SND detector.

2. Methodology

Experimental Setup

• Collider: Data was collected at the VEPP-2000 $e^+e^-$ collider in Novosibirsk.
• Detector: The Spherical Neutral Detector (SND), a non-magnetic detector featuring:
  • A tracking system (drift chamber + proportional chamber) with 94% solid angle coverage.
  • Aerogel Cherenkov counters for pion identification ($p < 436$ MeV/c).
  • A NaI(Tl) electromagnetic calorimeter (95% coverage) for photon detection.
  • A muon system.
• Data Sample: An integrated luminosity of 66 pb$^{-1}$ collected in 2018 across 102 energy points spanning 560–1100 MeV.

Event Selection and Analysis

The analysis focuses on events with two charged tracks (pions) and two photons (from $\pi^0$ decay). Due to varying background conditions across the energy range, the data was divided into five intervals (I–V), each with specific selection criteria:

1. Kinematic Fit: Events were fitted to the $\pi^+\pi^-\gamma\gamma$ hypothesis with energy-momentum conservation constraints.
2. Background Suppression:
   • $\chi^2_{3\pi}$: Cuts on the kinematic fit quality were tightened in the region between resonances (Interval II) to suppress background.
   • Angular Cuts: Cuts on the opening angle between pions ($\psi < 140^\circ$) and azimuthal difference ($\Delta\phi > 2.3^\circ$) were used to reject $e^+e^- \to \pi^+\pi^-(\gamma)$ and $e^+e^- \to e^+e^-\gamma$ backgrounds.
   • Energy Deposition: A cut on total calorimeter energy ($E_{EML}/E < 0.75$) suppressed $e^+e^- \to e^+e^-\gamma$ events in the low-energy region.
   • 4-Photon Veto: In the $\phi$ region, a kinematic fit to the $e^+e^- \to \pi^+\pi^-\pi^0\pi^0$ hypothesis was used to reject events with extra photons.
3. Signal Extraction: The number of signal events ($N_{3\pi}$) was determined by fitting the invariant mass spectrum of the two photons ($M_{\gamma\gamma}$) near the $\pi^0$ mass (70–200 MeV).
   • Signal Model: A sum of three Gaussian functions derived from Monte Carlo (MC) simulation, corrected for data-MC differences in line shape.
   • Background Model: Simulated backgrounds (QED processes, $K\bar{K}$, etc.) scaled by data-derived factors. A linear component was added to account for cosmic rays and simulation inaccuracies.

Corrections and Systematics

• Efficiency Corrections: Corrections were applied to account for differences between data and MC in photon loss, charged track reconstruction (especially at small polar angles), and specific selection cuts.
• Systematic Uncertainties: Key sources included:
  • Luminosity measurement (0.7%).
  • Beam energy measurement ($\Delta E_b/E_b \approx 6 \times 10^{-5}$).
  • Background subtraction and signal shape modeling.
  • Radiative corrections and energy spread effects.

3. Key Results

Cross Section Measurement

• The Born cross section for $e^+e^- \to \pi^+\pi^-\pi^0$ was measured with high precision.
• Systematic Uncertainty:
  • 0.9% at the $\omega$ resonance maximum ($E \approx 782.8$ MeV).
  • 1.2% at the $\phi$ resonance maximum ($E \approx 1019.1$ MeV).
• The results are presented in Tables III, IV, and V for 102 energy points.

Vector Meson Dominance (VMD) Fit

The measured cross section was fitted using a VMD model including $\rho$, $\omega$, $\phi$, and an excited $\omega'$ resonance. The fit yielded parameters with improved precision compared to current world averages (PDG):

• $\omega$ Meson:
  • $B(\omega \to e^+e^-)B(\omega \to 3\pi) = (6.648 \pm 0.022 \pm 0.058) \times 10^{-5}$ (PDG: $6.61 \pm 0.16$).
  • Mass $m_\omega = 782.703 \pm 0.011 \pm 0.047$ MeV.
  • Width $\Gamma_\omega = 8.598 \pm 0.023 \pm 0.004$ MeV.
• $\phi$ Meson:
  • $B(\phi \to e^+e^-)B(\phi \to 3\pi) = (4.154^{+0.102}_{-0.066} \pm 0.066) \times 10^{-5}$.
  • Mass and width agree with PDG but with slightly larger uncertainties than the $\omega$ parameters.
• $\rho$ Meson:
  • $B(\rho \to 3\pi) = (4.7 \pm 1.2 \pm 1.3) \times 10^{-5}$, an improvement over previous BABAR measurements.

Muon Anomalous Magnetic Moment ($a_\mu$)

Using the measured cross section (combined with previous SND data above 1.1 GeV), the leading-order hadronic contribution to the muon magnetic anomaly was calculated:

• Result: $a_\mu^{3\pi} (0.62 - 1.975 \text{ GeV}) = (45.95 \pm 0.06 \text{ (stat)} \pm 0.46 \text{ (syst)}) \times 10^{-10}$.
• Comparison:
  • Consistent with BABAR results and previous global fits.
  • 2.5$\sigma$ lower than the recent Belle II measurement ($48.91 \times 10^{-10}$).

4. Significance and Conclusion

• Resolution of Discrepancies: The SND measurement provides a high-precision benchmark that is in reasonable agreement with BABAR and CMD-2 data but significantly lower than Belle II data in the $\omega$ and $\phi$ regions. This suggests that the ~8% excess observed in Belle II data may not be fully explained by systematic errors within the stated uncertainties.
• Precision Physics: The measurement achieves the highest precision to date for this channel in the 0.56–1.1 GeV range, reducing the uncertainty on the hadronic contribution to $a_\mu$.
• Model Parameters: The extracted resonance parameters (masses, widths, and branching fractions) for the $\omega$ and $\rho$ mesons are more precise than current Particle Data Group (PDG) averages, refining our understanding of vector meson dynamics and isospin violation.
• Impact on Standard Model Tests: By providing a more precise value for $a_\mu^{3\pi}$, this work contributes to the ongoing effort to resolve the tension between the Standard Model prediction and the experimental measurement of the muon $g-2$.