Measurement of the $CP$ asymmetry in $D^0\to\pi^+\pi^-\pi^0$ decays at Belle II

Using a dataset of 428 fb$^{-1}$ collected by the Belle II experiment, researchers measured the time- and phase-space-integrated $CP$ asymmetry in $D^0\to\pi^+\pi^-\pi^0$ decays to be $(0.29\pm0.27\pm0.13)\%$, which represents the most precise result to date and is consistent with $CP$ conservation.

The Gist

The Big Picture: A Cosmic Coin Flip

Imagine you have a magical coin. In our everyday world, if you flip a coin a million times, you expect roughly 50% heads and 50% tails. This is the rule of symmetry.

In the subatomic world, there are particles called D0 mesons. These are like tiny, unstable coins that can decay (break apart) into other particles. Sometimes, a D0 meson breaks into a specific set of particles: two pions and a neutral pion (think of them as two red marbles and one blue marble).

The big question physicists ask is: Does nature play fair?

• If a D0 meson breaks into "Red-Red-Blue," does its antimatter twin (the anti-D0) break into "Anti-Red-Anti-Red-Anti-Blue" at the exact same rate?
• If the rates are slightly different, it's called CP Violation. It's like if the universe had a tiny, invisible bias that made the coin land on heads 50.0003% of the time instead of 50%.

Finding this bias is huge. It helps us understand why the universe is made of matter (us) instead of being empty space where matter and antimatter canceled each other out after the Big Bang.

The Detective Work: Belle II and the "Tag"

The scientists in this paper are the Belle II Collaboration. They are using a giant, high-tech camera (the Belle II detector) located in Japan to watch these cosmic coin flips happen. They collected data from 2019 to 2022, which is like watching 428 billion billion collisions.

The Problem:
When a D0 meson is created, it's a mystery. You don't know if it's a "matter" coin or an "antimatter" coin until it decays. To solve this, the scientists look at how the D0 was born. It usually comes from a slightly heavier particle called a D+*.

• The D*+ decays into a D0 and a tiny, low-energy pion (a "tag pion").
• The Analogy: Imagine the D*+ is a parent giving birth to twins. One twin is the D0 (the mystery), and the other is the tag pion. If the tag pion is positively charged, you know the D0 is a "matter" coin. If it's negative, the D0 is an "antimatter" coin. The tag pion acts like a name tag on the mystery box.

The Goal:
They counted how many "matter" D0s turned into the Red-Red-Blue combo versus how many "antimatter" D0s did the same. They calculated the difference (the asymmetry).

The "Dirty Laundry" Problem: Cleaning the Data

Here is the tricky part. The universe might be fair, but the machine isn't.

• The detector is made of metal, silicon, and magnets.
• Positive particles might get stuck in the metal slightly more often than negative ones.
• The way the computer reconstructs the data might be slightly biased.

If you just count the results, you might think you found a "cosmic bias," but you actually just found a "machine bias."

The Solution: The Control Group
To fix this, the scientists used a "Control Group" (like a placebo group in a medical trial).

1. The Signal: The rare, tricky decay (D0 → π+π−π0).
2. The Control: A very common, well-understood decay (D0 → K−π+). We know this one should be perfectly fair (symmetric) because of how the Standard Model works.

They measured the "machine bias" using the Control Group. Then, they subtracted that bias from the Signal Group. It's like weighing yourself on a scale that you know is off by 2 pounds. You weigh yourself, see 152 lbs, subtract the 2-pound error, and know you actually weigh 150.

The Results: The Verdict

After counting millions of decays, correcting for the machine's quirks, and doing some very complex math to account for the angle at which the particles fly out, here is what they found:

The Asymmetry is 0.29% ± 0.27% (statistical) ± 0.13% (systematic).

What does that mean?

• The number is very close to zero.
• The "error bars" (the uncertainty) are larger than the number itself.
• The Conclusion: The result is consistent with zero. The universe is playing fair in this specific case. There is no evidence of CP violation in this particular decay channel.

Why is this paper important if the answer is "nothing"?

1. Precision: This is the most precise measurement ever made for this specific decay. They improved the precision by 34% compared to the previous best record (held by the BABAR experiment), even though they didn't collect that much more data. They did it by being smarter about how they selected and analyzed the data.
2. Ruling Out New Physics: If they had found a big asymmetry, it would have been a Nobel Prize-winning discovery of "New Physics" (physics beyond our current understanding). By finding nothing, they are putting strict limits on what "New Physics" can look like. It's like searching for a ghost in a house; finding nothing tells you exactly where the ghost isn't.
3. The Journey: The paper details a masterclass in how to handle massive amounts of data, correct for subtle machine errors, and use statistical weighting to ensure a fair comparison.

Summary Analogy

Imagine you are trying to see if a specific brand of dice is loaded (biased).

• You roll the dice 428 billion times.
• You notice that the table you are rolling on is slightly tilted, which makes the dice roll one way more often.
• You use a second set of dice (the control group) that you know are fair to measure exactly how much the table is tilted.
• You subtract the table's tilt from your results.
• Result: The dice are perfectly fair.

This paper says, "We rolled the dice more times and more carefully than anyone before, and after correcting for the table, the dice are still fair." It's a victory for precision and a reminder that the universe is stubbornly symmetrical in this specific corner.

Technical Summary

1. Problem and Motivation

• Physics Context: Searches for Charge-Parity (CP) violation in the charm sector are crucial for testing the Standard Model (SM) and probing physics beyond it. In the SM, CP violation in charm transitions is predicted to be extremely small ($\mathcal{O}(10^{-4} \text{--} 10^{-3})$) due to the suppression of third-generation quark contributions.
• Current Status: While the LHCb Collaboration observed CP violation in 2019 (in $D^0 \to K^-K^+$ and $D^0 \to \pi^-\pi^+$), the origin remains debated between new physics and unaccounted non-perturbative QCD effects.
• Specific Goal: This paper aims to measure the time- and phase-space-integrated CP asymmetry ($A_{CP}$) in the singly Cabibbo-suppressed decay $D^0 \to \pi^+\pi^-\pi^0$. Improving the precision of this measurement is essential to constrain theoretical models and search for deviations from the SM.

2. Methodology

Data and Detector

• Dataset: The analysis uses $e^+e^- \to c\bar{c}$ events collected by the Belle II experiment at the SuperKEKB collider between 2019 and 2022.
• Integrated Luminosity: $428 \text{ fb}^{-1}$.
• Flavor Tagging: The flavor of the neutral $D^0$ meson at production is determined by reconstructing the decay chain $D^{*+} \to D^0 \pi^+_{\text{tag}}$. The charge of the low-momentum "tag pion" ($\pi^+_{\text{tag}}$) identifies the initial flavor ($D^0$ or $\bar{D}^0$).

Event Selection and Reconstruction

• Signal Sample: $D^0 \to \pi^+\pi^-\pi^0$ candidates are reconstructed using charged tracks and photon clusters in the electromagnetic calorimeter.
  • $\pi^0$ candidates are formed from photon pairs with invariant mass between 116 and 150 MeV/$c^2$.
  • Particle Identification (PID) via Neural Networks (NNPID) is used to reject kaon misidentification.
  • A kinematic fit constrains the $D^{*+}$ vertex to the interaction region.
  • Cuts are applied to reject $B$-meson decays and combinatorial background (e.g., requiring $D^{*+}$ momentum $> 1.06 \times$ the kinematic limit for $B$ decays).
• Control Samples: To correct for experimental asymmetries (reconstruction and production), two control samples of the Cabibbo-favored decay $D^0 \to K^-\pi^+$ are used:
  1. Tagged Sample: $D^0 \to K^-\pi^+$ with a reconstructed $D^{*+} \to D^0 \pi^+_{\text{tag}}$.
  2. Untagged Sample: $D^0 \to K^-\pi^+$ without the tag pion.

Correction for Asymmetries

The raw asymmetry ($A_{\text{raw}}$) contains contributions from the physical CP asymmetry ($A_{CP}$), production asymmetry ($A_{\text{prod}}$), and detection efficiency asymmetries ($A_{\epsilon}$).

• Tag Pion Correction ($A_{\pi_{\text{tag}}}^{\epsilon}$): Calculated by subtracting the raw asymmetry of the untagged sample from the tagged sample: $A_{\pi_{\text{tag}}}^{\epsilon} = A_{\text{raw}}^{\text{tagged}} - A_{\text{raw}}^{\text{untagged}}$.
• Kinematic Weighting: To ensure the control samples match the signal sample's kinematic distributions (which affect detection efficiency), per-candidate weights are applied. This matches the $p_T$ and $\cos\theta$ distributions of the tag pion, kaon, and pions between the signal and control samples.
• Production Asymmetry Cancellation: The production asymmetry is an odd function of $\cos\theta_{CM}$ (the polar angle in the center-of-mass frame). The data is divided into eight bins of $\cos\theta_{CM}$. The CP asymmetry is calculated as the average of positive and negative bins to cancel the odd $A_{\text{prod}}$ contribution.

Statistical Analysis

• Fitting: Unbinned extended maximum-likelihood fits are performed on the invariant mass ($M$) and mass difference ($\Delta M = M_{D^*} - M_D$) distributions.
• PDFs: Signal components are modeled using Johnson's SU distributions; backgrounds use Crystal Ball, threshold-like, and Chebyshev polynomial functions.
• Blinding: The raw asymmetry values were shifted by an undisclosed offset during the analysis to prevent bias.

3. Key Contributions

• Precision Improvement: This measurement achieves a statistical precision significantly higher than previous results, despite only a ~10% increase in integrated luminosity compared to the previous best measurement (BABAR). This is attributed to novel candidate selection strategies and improved detector performance.
• Comprehensive Systematic Control: The analysis rigorously addresses systematic uncertainties, including:
  • $D^0$ reconstruction asymmetry (verified via MC).
  • Fixed fit parameters from simulation.
  • Fit biases (evaluated via pseudoexperiments).
  • Residual PDF mismodeling (evaluated via bootstrap subsamples).
  • Kinematic weighting uncertainties.
  • Wrong-sign $D^0 \to K^+\pi^-$ contamination in control samples.
• Validation of CP Conservation: The result provides the most stringent test to date for CP conservation in this specific three-body charm decay channel.

4. Results

The measured time-integrated CP asymmetry is:
$$A_{CP}(D^0 \to \pi^+\pi^-\pi^0) = (0.29 \pm 0.27_{\text{stat}} \pm 0.13_{\text{syst}})\%$$

• Statistical Uncertainty: $0.27\%$ (includes uncertainty from the control samples).
• Systematic Uncertainty: $0.13\%$.
• Total Uncertainty: $\approx 0.30\%$.
• Comparison: The result is consistent with CP conservation (zero asymmetry) and is compatible with previous measurements by BABAR, Belle, and LHCb.
• Precision Gain: The result is 34% more precise than the previous world-best measurement from BABAR.

5. Significance

• Standard Model Test: The result confirms the SM prediction that CP violation in charm decays is extremely small, placing tight constraints on the magnitude of potential New Physics contributions.
• Methodological Benchmark: The analysis demonstrates the high precision capabilities of the Belle II experiment and validates the use of kinematic weighting and control samples to suppress experimental biases in CP asymmetry measurements.
• Future Prospects: This level of precision sets a new benchmark for future charm physics studies, helping to guide theoretical efforts to understand the interplay between perturbative and non-perturbative QCD effects in charm decays.