Measurement of time-dependent $CP$ violation parameters in $B^{0} \to K_{S}^{0} \pi^{0} \gamma$ decays at Belle and Belle II

Using combined datasets from the Belle and Belle II experiments, this study presents the most precise measurements to date of time-dependent $CP$ violation parameters in $B^{0} \to K_{S}^{0} \pi^{0} \gamma$ decays, finding results consistent with Standard Model predictions across both $K^{*0}(892)$-dominated and non-resonant regions.

The Gist

The Big Picture: Catching a Ghost in the Machine

Imagine you are trying to watch a magic trick performed by two identical twins. One twin is the "good" version, and the other is the "evil" version. In the world of particle physics, these twins are B-mesons (specifically $B^0$ and $\bar{B}^0$). They are unstable particles that decay (fall apart) very quickly.

The scientists at the Belle and Belle II experiments (located in Japan) built massive, ultra-sensitive cameras to watch these twins decay. Their goal was to catch a specific, rare magic trick: a decay where a B-meson turns into a neutral Kaon ($K^0_S$), a neutral pion ($\pi^0$), and a flash of light (a photon, $\gamma$).

Why do they care? Because in our current understanding of the universe (the Standard Model), this specific trick should happen in a very predictable way. If the twins behave differently than expected, it means there is a "ghost" in the machine—some new, unknown force or particle messing with the rules.

The Setup: A High-Speed Dance

To study this, the researchers smash electrons and positrons (matter and antimatter) together at nearly the speed of light. This collision creates a heavy particle called the $\Upsilon(4S)$, which immediately splits into a pair of B-mesons.

Think of this like a synchronized dance:

1. The Twins: One B-meson is the "Signal" ($B_{sig}$) that performs the magic trick we want to watch. The other is the "Tag" ($B_{tag}$) that acts as a witness.
2. The Tag: The Tag twin decays into something easy to identify. This tells the scientists, "Hey, at this exact moment, the Signal twin was the 'good' version (or the 'evil' version)."
3. The Time Difference: Because the twins are moving, they don't decay at the exact same time. The scientists measure the tiny time gap ($\Delta t$) between the Tag's death and the Signal's death.

The Mystery: Left-Handed vs. Right-Handed

In the Standard Model, the photon (the flash of light) emitted during this decay is almost always left-handed (like a left-handed screw). It's very rare for it to be right-handed.

If the photon is strictly left-handed, the "good" and "evil" twins should decay at almost the same rate. The difference between them (called CP violation) should be tiny.

• The Goal: The scientists are looking for a "right-handed" photon. If they find one, it would mean the "good" and "evil" twins are behaving very differently, suggesting new physics (like Supersymmetry) is at play.

They measure two numbers to describe this difference:

• $S$ (The Mixing): How much the twins swap identities before decaying.
• $C$ (The Direct Difference): How much they prefer to decay as one type over the other immediately.

The Investigation: Two Different Neighborhoods

The researchers looked at the debris from the decay in two different "neighborhoods" based on the mass of the particles involved:

1. The $K^*$ Neighborhood (0.8 to 1.0 GeV): This is a busy, well-known area where a specific particle resonance ($K^*(892)$) dominates. It's like a crowded city square.
2. The Non-$K^*$ Neighborhood (1.0 to 1.8 GeV): This is a quieter, more chaotic area with no single dominant particle. It's like a scattered suburb.

They needed to check both because the rules might be different in the quiet suburb compared to the city square.

The Tools: Better Cameras and Smarter Algorithms

The paper highlights two major upgrades that made this study possible:

1. More Data: They combined data from the old Belle experiment (running 1999–2010) and the new Belle II experiment. This is like combining 772 million and 521 million photos to get a clearer picture.
2. Smarter AI: They used a new type of Artificial Intelligence called a Graph Neural Network (GNN). Imagine trying to figure out who is in a crowd photo. Old methods just looked at faces. This new AI looks at how everyone is connected, their movements, and their relationships to figure out exactly who is who. This helped them identify the "Tag" twin much more accurately.

The Results: The Twins Behave Themselves

After crunching the numbers, the scientists found:

• In the City Square ($K^*$ region): The difference between the twins was tiny. The numbers were $S = 0.09$ and $C = -0.09$.
• In the Suburb (Non-$K^*$ region): The difference was also small, though with slightly larger margins of error. The numbers were $S = -0.32$ and $C = -0.07$.

The Conclusion:
The "ghost" they were looking for wasn't there. The twins behaved exactly as the Standard Model predicted. The "right-handed" photon is still hiding, or at least, it's not showing up in this experiment.

However, this is a good result for science. It's like checking a bridge for cracks. Finding no cracks doesn't mean the bridge is boring; it means the bridge is safe and built exactly to the blueprints. These results are the most precise measurements ever made for this specific decay, improving upon previous attempts by about 24% to 31%.

Summary in One Sentence

Using a massive amount of data and a new AI system, the Belle and Belle II collaborations watched billions of particle "twins" decay and confirmed that they are behaving exactly as our current laws of physics predict, with no sign of mysterious new forces disrupting the process.

Technical Summary

Technical Summary: Measurement of Time-Dependent CP Violation Parameters in $B^0 \to K^0_S \pi^0 \gamma$ Decays

Problem and Motivation
The radiative decay $B^0 \to K^0_S \pi^0 \gamma$ serves as a sensitive probe for physics beyond the Standard Model (SM). This process occurs predominantly through $b \to s\gamma$ loop transitions, making it susceptible to contributions from heavy, virtual particles that could manifest at energy scales higher than those directly accessible in current colliders. In the SM, the emitted photon is predominantly left-handed due to the chiral structure of the weak interaction, leading to a suppression of the mixing-induced CP-violating parameter $S$ by a factor of $m_s/m_b$. Consequently, the SM predicts $S$ to be small (suppressed to the few-percent level), while the direct CP-violating parameter $C$ is expected to be negligible ($<1\%$). However, significant contributions from right-handed photons, potentially arising from new physics scenarios such as supersymmetry or extended Higgs sectors, could enhance $S$ to the order of 10%. Furthermore, theoretical predictions for non-resonant decays differ from resonant channels (dominated by $K^{*0}(892)$), motivating separate investigations. Previous measurements by Belle, BaBar, and Belle II were consistent with SM predictions but exhibited significant uncertainties.

Methodology
This study utilizes a combined dataset of approximately $772 \times 10^6$ and $521 \times 10^6$ $\Upsilon(4S) \to B\bar{B}$ decays collected by the Belle and Belle II experiments, respectively. The analysis employs a bottom-up reconstruction approach:

• Reconstruction: $K^0_S$ candidates are reconstructed from oppositely charged tracks, while $\pi^0$ and prompt photon candidates are identified via electromagnetic calorimeter (ECL) clusters. A mass-constrained fit improves $\pi^0$ momentum resolution.
• Event Selection: Multivariate classifiers (NeuroBayes for Belle, LightGBM/XGBoost for Belle II) are used to suppress backgrounds from misreconstructed $K^0_S$, $\Lambda^0$ decays, and continuum events. A Graph Neural Network (GNN)-based flavour tagger is implemented across both datasets to determine the flavour of the tagging $B$ meson ($B_{\text{tag}}$).
• Kinematic Variables: Signal candidates are identified using the beam-constrained mass ($M_{bc}$) and energy difference ($\Delta E$). To mitigate correlations caused by shower leakage in the ECL, a rescaling procedure is applied to the $\pi^0$ and photon energies before calculating $M_{bc}$.
• Fitting Strategy: The analysis performs a simultaneous extended unbinned maximum-likelihood fit to time-dependent (TD) and time-integrated (TI) datasets.
  • TD Fit: Uses events with well-reconstructed decay time differences ($\Delta t$) to extract both $S$ and $C$.
  • TI Fit: Uses events with poorly reconstructed $\Delta t$ to extract $C$ only, thereby reducing its uncertainty.
  • The fit model includes signal, $B\bar{B}$, and continuum ($q\bar{q}$) components, with Probability Density Functions (PDFs) validated using control channels such as $B^0 \to K^{*0} \gamma$ and $B^0 \to J/\psi K^0_S$.

Key Contributions

• Dataset Integration: The first combined analysis of Belle and Belle II data for this specific decay channel, leveraging the largest available statistics.
• Algorithmic Improvements: Implementation of an improved GNN-based flavour tagger for both experiments and an enhanced $K^0_S$ selection algorithm specifically for the Belle dataset.
• Methodological Refinement: The inclusion of events with poor time resolution in a time-integrated fit to constrain the direct CP violation parameter $C$, and the use of rescaled kinematic variables to reduce correlations between $M_{bc}$ and $\Delta E$.

Results
The analysis divides the data into two mass regions based on the invariant mass of the $K^0_S \pi^0$ system ($M_{K^0_S \pi^0}$):

1. Resonant Region ($K^{*0}(892)$ dominated): $M_{K^0_S \pi^0} \in [0.8, 1.0] \, \text{GeV}/c^2$.
2. Non-Resonant Region: $M_{K^0_S \pi^0} \in (1.0, 1.8] \, \text{GeV}/c^2$.

The measured CP violation parameters for the combined dataset are:

• Resonant Region:
  • $S = 0.09 \pm 0.16 \, (\text{stat}) \pm 0.02 \, (\text{syst})$
  • $C = -0.09 \pm 0.08 \, (\text{stat}) \pm 0.04 \, (\text{syst})$
• Non-Resonant Region:
  • $S = -0.32 \pm 0.33 \, (\text{stat}) \pm 0.09 \, (\text{syst})$
  • $C = -0.07 \pm 0.17 \, (\text{stat}) \pm 0.08 \, (\text{syst})$

Significance and Claims
The paper claims these results represent the most precise measurements to date for $B^0 \to K^0_S \pi^0 \gamma$ decays. Specifically, the combined results improve upon the precision of previous combinations of Belle and Belle II results by approximately 24% for $S$ and 31% for $C$ in the $K^{*0}(892)$ region. The authors state that these findings are consistent with Standard Model predictions and supersede earlier measurements for the same channel. The results do not indicate evidence for new physics contributions such as significant right-handed photon amplitudes, given the current precision and consistency with SM expectations.