Measurement of time-dependent $CP$ asymmetries in $B^0 \to K_{\rm S}^0 \: π^{+} π^{-} γ$ decays at Belle and Belle II

Using combined data from the Belle and Belle II experiments, this paper reports the first measurement of time-dependent $CP$ asymmetry parameters ($C$, $S$, $S^+$, and $S^-$) in $B^0 \to K_{\rm S}^0 \pi^+ \pi^- \gamma$ decays, yielding results consistent with zero within uncertainties.

The Gist

The Big Picture: A Cosmic Dance of Twins

Imagine two identical twins born at the exact same moment, dancing in a perfect, synchronized routine. In the world of particle physics, these "twins" are a pair of particles called B mesons (specifically a $B^0$ and its antiparticle, $\bar{B}^0$). They are created together in a high-energy collision at the SuperKEKB and KEKB colliders in Japan.

Because they are born together in a "quantum entangled" state, they are linked. If one twin decides to change its identity (a process called "flavor oscillation") at a specific moment, the other twin instantly knows about it.

The scientists in this paper (the Belle and Belle II collaborations) are acting like high-speed photographers trying to capture a very specific, rare dance move performed by these twins. They are looking for a specific decay:

• The Star of the Show: A $B^0$ meson decaying into a photon (a particle of light), a neutral kaon ($K^0_S$), and two pions ($\pi^+\pi^-$).
• The Goal: To see if the "dance" of the particles follows the rules of the Standard Model (the current rulebook of physics) or if there is a glitch that hints at "New Physics" (rules we haven't discovered yet).

The Mystery: Left-Handed vs. Right-Handed Light

In the Standard Model, when a $B^0$ meson decays and emits a photon, that photon is almost always "left-handed" (it spins in a specific direction). A "right-handed" photon is so rare it's like finding a needle in a haystack.

However, if there are unknown forces or particles (Physics Beyond the Standard Model), they might make the "right-handed" photon appear more often. The scientists are looking for a subtle asymmetry in the timing of the decay to see if this "right-handed" influence is sneaking in.

The Experiment: A Race Against Time

To catch this rare event, the scientists used two massive "cameras" (detectors):

1. Belle: An older camera that ran from 1999 to 2010.
2. Belle II: A newer, sharper camera that started in 2019.

They collected a massive amount of data, equivalent to 1,076 "inverse femtobarns" (a unit of collision data). To put that in perspective, they watched billions of particle collisions to find just a few hundred of the specific "dance moves" they were interested in.

The Challenge:
The $B^0$ meson decays incredibly fast. To measure the time difference between the two twins dancing, the scientists had to reconstruct the "story" of the event:

• The Signal ($B_{sig}$): The twin they are studying.
• The Tag ($B_{tag}$): The other twin. By figuring out what the "tag" twin decayed into, they can deduce what the "signal" twin was doing at the very start.

The Measurement: The "CP Asymmetry"

The scientists measured something called CP Asymmetry. Think of this as checking if the universe treats matter and antimatter exactly the same way.

• If the universe is perfectly fair, the "dance" should look the same whether you watch it forward or backward in time.
• If there is an asymmetry, it means the universe has a slight preference, which could explain why our universe is made of matter instead of being empty.

They measured four specific numbers (parameters) to describe this asymmetry:

1. $C$ and $S$: The main scores for the asymmetry.
2. $S_+$ and $S_-$: New, more detailed scores. The scientists split their data into two halves based on how the particles were moving (like splitting a dance floor into a "left" and "right" side) to get a more granular view of the physics.

The Results: What Did They Find?

After crunching the numbers from both the old and new cameras, here is what they found:

• The Scores: They measured the asymmetry parameters to be roughly:

  • $C \approx -0.17$
  • $S \approx -0.29$
  • $S_+ \approx -0.57$
  • $S_- \approx 0.31$
    (Note: These numbers have "error bars" because measuring subatomic particles is like trying to weigh a feather in a hurricane.)

• The Verdict:

  • The results are consistent with the Standard Model. The "dance" looks mostly like what the rulebook predicted.
  • However, the measurements for the new parameters ($S_-$) are slightly "tense" (about 2 standard deviations away from zero). This isn't a definitive proof of new physics yet, but it's a hint that keeps the scientists interested.
  • The biggest achievement is precision. By combining data from both experiments, they cut the uncertainty in half compared to previous measurements. This makes the "ruler" they are using to measure the universe much sharper.

Why Does This Matter?

This paper doesn't claim to have found a new particle or a new force. Instead, it has tightened the net.

Imagine you are trying to find a specific type of fish in a huge ocean. Previous studies cast a wide net and caught a few fish, but the net had big holes. This study used a finer mesh net. They didn't find a "monster fish" (New Physics) yet, but they proved that if the monster fish is there, it must be very small or very shy.

By measuring these parameters with such high precision, they have placed strict limits on how much "right-handed" light can exist in these decays. This helps theorists rule out certain ideas about what might be hiding beyond our current understanding of the universe.

Summary in a Nutshell

The Belle and Belle II teams took a massive snapshot of billions of particle collisions to watch a rare, fleeting dance between matter and antimatter. They measured the timing of this dance with unprecedented precision. The dance mostly follows the known rules of physics, but the measurements are now so precise that they can spot even the tiniest deviations, helping scientists narrow down where the secrets of the universe might be hiding.

Technical Summary

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

Problem and Motivation
Flavor-changing neutral currents, specifically $b \to s\gamma$ transitions, serve as sensitive probes for physics beyond the Standard Model (SM). In the SM, the emitted photon in these transitions is predominantly left-handed, with right-handed contributions suppressed by a factor proportional to $m_s^2/m_b^2$. Consequently, the mixing-induced CP violation in decays to CP eigenstates ($B^0 \to f_{CP}\gamma$) is predicted to be very small. Enhanced right-handed currents, predicted by various Beyond-the-Standard-Model (BSM) scenarios, could significantly increase the mixing-induced CP asymmetry. This paper presents a measurement of these asymmetries in the decay $B^0 \to K^0_S \pi^+ \pi^- \gamma$, a channel where the intermediate resonance $K_{res}$ decays to $K^0_S \rho^0$ (a CP eigenstate) but also includes contributions from non-CP eigenstates involving strange meson resonances.

Methodology
The analysis utilizes the combined datasets from the Belle and Belle II experiments collected at the $\Upsilon(4S)$ resonance. The total integrated luminosity comprises $711 \text{ fb}^{-1}$ from Belle and $365 \text{ fb}^{-1}$ from Belle II.

• Candidate Selection: Signal candidates are reconstructed by identifying a photon ($E_\gamma > 1.4/1.5 \text{ GeV}$), a $K^0_S$ (decaying to $\pi^+\pi^-$), and a prompt pion pair forming a $\rho^0$ candidate. The $K_{res}$ invariant mass is constrained to $[0.9, 1.8] \text{ GeV}/c^2$, and the $B^0$ candidate is required to satisfy beam-energy-constrained mass ($M_{bc}$) and energy difference ($\Delta E$) criteria. A multivariate Boosted Decision Tree (BDT) is employed to suppress continuum background ($e^+e^- \to q\bar{q}$).
• Flavor Tagging and Time Dependence: The flavor of the signal $B^0$ meson ($B_{sig}$) at the time of decay is inferred from the decay products of the companion $B$ meson ($B_{tag}$) using flavor-tagging algorithms. For Belle, a standard multivariate algorithm is used; for Belle II, a Graph Neural Network (GFlat) is employed. The proper time difference ($\Delta t$) between the two decays is reconstructed using the vertex positions along the boost direction.
• Fit Strategy: An unbinned maximum likelihood fit is performed on the observables $M_{bc}$, $\Delta E$, and $\Delta t$. The fit model includes the signal component and three background sources: continuum events, misreconstructed $B\bar{B}$ decays, and self-cross-feed (SCF) where a pion from the tag side is misidentified as a signal pion.
• CP Observables: The analysis extracts the standard mixing-induced ($S$) and direct ($C$) CP violation parameters. Additionally, two new observables, $S_+$ and $S_-$, are measured. These are defined by splitting the phase space into two halves based on the inequality $m^2(K^0_S \pi^+) \gtrless m^2(K^0_S \pi^-)$, allowing for the separation of CP-even and CP-odd contributions within the $K_{res}$ system.

Key Contributions

1. Combined Measurement: This work provides the first combined measurement of time-dependent CP asymmetries in $B^0 \to K^0_S \pi^+ \pi^- \gamma$ using the full Belle dataset and the Belle II dataset collected up to 2022.
2. New Observables: The paper reports the first measurement of the CP observables $S_+$ and $S_-$, which are proposed to constrain photon polarization and Wilson coefficients ($C_7, C'_7$) when combined with amplitude analysis parameters from the isospin partner mode.
3. Validation: The analysis strategy is validated using simulated Monte Carlo samples and the control mode $B^+ \to K^+ \pi^+ \pi^- \gamma$, confirming the absence of significant biases in the extraction of CP parameters.

Results
The measured values for the combined datasets (statistical uncertainty first, systematic second) are:

• $C = -0.17 \pm 0.09 \pm 0.04$
• $S = -0.29 \pm 0.11 \pm 0.05$
• $S_+ = -0.57 \pm 0.23 \pm 0.10$
• $S_- = 0.31 \pm 0.24 \pm 0.05$

The results for $S$, $C$, and $S_+$ are consistent with the Standard Model prediction within $1\sigma$ (Belle) and $2.3\sigma$ (Belle II). The value of $S_-$ shows a slight tension ($2.2\sigma$) but remains compatible with the SM prediction of zero. The correlations between the parameters are accounted for in the combination.

Significance
The authors state that this combined measurement improves the uncertainties on the CP observables $S$ and $C$ by at least a factor of two compared to previous measurements by the Belle and BaBar collaborations. The result supersedes the effective mixing-induced CP asymmetry ($S_{eff}$) presented in earlier works. Furthermore, the measurement of $S_+$ and $S_-$ provides new constraints for new physics models that enhance right-handed currents, offering a more complete picture of the photon polarization in $b \to s\gamma$ transitions. The paper does not claim a discovery of new physics but establishes tighter constraints on the SM and potential BSM contributions.