Method to study $CP$ violation in $B_s^0\to K_S K^\pm \pi^\mp$ decays

This paper proposes and demonstrates the feasibility of a novel, simultaneous tagged decay-time-dependent Dalitz-plot analysis method for $B_s^0\to K_S K^\pm \pi^\mp$ decays to precisely measure the weak phase $\phi_s^{\rm eff}$ and search for new sources of $CP$ violation, a technique implemented in the Laura++ package and validated through pseudoexperiments.

The Gist

Imagine the universe is a giant, complex dance floor where tiny particles called B-mesons are the dancers. Physicists want to understand the rules of this dance to see if they match the "Standard Model" (the current rulebook of physics) or if there are secret new moves (New Physics) that we haven't discovered yet.

This paper is about a specific, tricky dance move: the $B^0_s$ meson decaying (dancing apart) into three other particles ($K^0_S$, $K$, and $\pi$).

Here is the breakdown of what the authors did, using simple analogies:

1. The Mystery: "Left-Handed" vs. "Right-Handed" Dancers

In the world of particles, there is a concept called CP violation. Think of it like this: If you film a particle decaying and then play the film backward (or look at its mirror image), does it look exactly the same?

• Standard Model: Usually, yes. The dance looks the same forward and backward.
• The Goal: The authors are looking for a dance where the forward version looks different from the backward version. Finding this difference is the "smoking gun" for new physics.

2. The Challenge: A Two-Stage Dance Routine

Usually, physicists study one type of dance at a time. But this specific decay is special because it can happen in two different ways (two final states) that are mirror images of each other.

• The Problem: To catch the "CP violation," you can't just watch one dancer. You have to watch both dancers simultaneously and compare their moves frame-by-frame over time.
• The Analogy: Imagine trying to spot a subtle difference between a left-handed and a right-handed tennis player. If you only watch the lefty, you can't tell if they are doing something weird. You have to watch both players on the court at the same time, comparing every swing, every step, and how long they stay on the court.

3. The Method: The "Dalitz Plot" Map

The authors propose a new way to analyze this data called a Dalitz-plot analysis.

• The Map: Imagine a map of a city where every point represents a different way the particles could fly apart.
• The Time Factor: This isn't just a static map; it's a movie. The authors are creating a method to watch how the dancers move across this map over time.
• The Tag: To make this work, they need to know which dancer started as the "left-handed" version and which started as the "right-handed" version. This is called "flavor tagging." It's like putting a red hat on one dancer and a blue hat on the other at the start of the race.

4. The Experiment: The "Fake" Dance Floor

Since they don't have enough real data from the LHCb experiment yet to do this complex analysis, they built a simulation (called "pseudoexperiments").

• The Simulation: They created a computer program that generated 500 "fake" datasets, pretending to be the real data LHCb will collect in the future (specifically from Runs 1, 2, and 3).
• The Test: They fed these fake datasets into their new analysis method to see if the method could successfully find the hidden "CP violation" signals they had planted in the code.

5. The Results: It Works!

The paper claims that their new method is feasible.

• Success: When they ran their "fake" experiments, the method successfully recovered the hidden parameters. It could tell the difference between the two dance styles and measure the "weak phase difference" (the angle of the CP violation) with good precision.
• Precision: They found that with the data LHCb is currently collecting (Runs 1–3), they can measure this angle very accurately. If they wait for even more data in the future (Runs 4–6), the precision will get even better.
• The Tool: They have already built this method into a software package called Laura++, which other scientists can use.

6. Why It Matters

• New Physics: If the real data (when it arrives) shows a different result than the Standard Model predicts, it means there is new physics hiding in the shadows.
• A Blueprint: This paper doesn't just study one decay; it provides a blueprint (a recipe) for how to study any complex particle decay that has two mirror-image outcomes.

Summary

Think of this paper as a training manual for a very difficult detective game. The authors invented a new way to compare two mirror-image particle dances over time. They tested their method on a computer simulation and proved it works. Now, they are ready to apply this method to the real data coming from the Large Hadron Collider to see if the universe has any secret, rule-breaking moves left to discover.

Technical Summary

Technical Summary: Method to study CP violation in $B^0_s \to K^0_S K^\pm \pi^\mp$ decays

Problem and Motivation
The study of CP violation in charmless $B$-meson decays is essential for testing the Standard Model (SM) and probing new physics (NP). The decay channel $B^0_s \to K^0_S K^\pm \pi^\mp$ is of particular interest. In the SM, when an intermediate $K^{*0}$ resonance is involved, the decay is dominated by a single $t$-loop penguin diagram, implying no direct CP violation and no mixing-induced CP violation (as the weak phase difference vanishes). However, contributions from higher-order $c$- and $u$-loop diagrams or NP could modify this expectation. Conversely, when $K^{*\pm}$ resonances are involved, tree-level diagrams contribute, potentially allowing for CP violation and sensitivity to the CKM angle $\gamma$.

To access the full set of CP-violating observables in these decays, a flavor-tagged, decay-time-dependent Dalitz-plot analysis is required. While such analyses have been performed for single self-conjugate final states (e.g., $B^0 \to K^0_S \pi^+ \pi^-$) and for four-body decays, a simultaneous analysis of the two charge-conjugate final states ($K^0_S K^+ \pi^-$ and $K^0_S K^- \pi^+$) with decay-time dependence has never been performed. Current LHCb data from Runs 1 and 2 lack the statistics for a flavor-tagged analysis, but the Run 3 dataset (and future runs) is expected to provide sufficient yield.

Methodology
The authors propose a formalism and a method to perform a simultaneous, flavor-tagged, decay-time-dependent Dalitz-plot analysis for the two final states $f \equiv K^0_S K^+ \pi^-$ and $\bar{f} \equiv K^0_S K^- \pi^+$.

1. Formalism: The differential decay widths for initially produced $B^0_s$ and $\bar{B}^0_s$ mesons decaying to $f$ and $\bar{f}$ are expressed as functions of decay time $t$ and phase-space variables $\Phi_3$. These equations incorporate $B^0_s$-$\bar{B}^0_s$ mixing parameters ($\Delta m_s$, $\Delta \Gamma_s$, $q/p$) and decay amplitudes ($A_f, \bar{A}_f, A_{\bar{f}}, \bar{A}_{\bar{f}}$). CP violation manifests as differences in amplitude magnitudes (direct CPV) and in the interference between mixing and decay (mixing-induced CPV), characterized by an effective weak phase difference $\phi^{\text{eff}}_s$.
2. Amplitude Model: The decay amplitudes are constructed using the isobar formalism, summing contributions from intermediate resonances. For the feasibility study, a simplified model is adopted including only the $K^*(892)$ resonances ($K^{*0}$, $K^{*+}$, $K^{*-}$). Complex coefficients ($c_i$) are decomposed into CP-conserving parts ($c$) and CP-violating parts ($\Delta c$).
3. Pseudoexperiment Setup: The method is implemented in the Laura++ Dalitz-plot analysis package. Pseudoexperiments are generated based on the LHCb Run 1–3 dataset (approx. 130,000 total yield, ~6,500 effective tagged yield).
4. Fitting Strategy: An unbinned maximum-likelihood fit is performed on ensembles of 500 pseudoexperiments per scenario. To resolve fit ambiguities, specific constraints are applied:
   • The reference amplitude coefficient $c_-$ is fixed to unity.
   • The imaginary part of $\Delta c_-$ is set to zero.
   • A constraint is applied to the relative phase between $A_{\bar{f}}$ and $\bar{A}_{\bar{f}}$.
   • Solutions for $\phi^{\text{eff}}_s$ are restricted to within $\pm \pi/3$ of the input value to resolve the $\pi$-ambiguity.
   • 100 fits with random starting points are performed per pseudoexperiment to ensure convergence to the global minimum.

Key Contributions

• Novel Analysis Framework: This paper outlines, for the first time, the method to perform a simultaneous, flavor-tagged, decay-time-dependent Dalitz-plot analysis on two charge-conjugate final states.
• Implementation: The method is implemented in the Laura++ package, establishing a baseline framework for this class of measurements.
• Sensitivity Study: The authors evaluate the sensitivity to the effective weak phase $\phi^{\text{eff}}_s$ and CP-violating phases in the amplitude coefficients across various model configurations (varying magnitudes, phases, and introducing CP violation in $K^*$ amplitudes).

Results

• Feasibility: The pseudoexperiments demonstrate that input parameters, including $\phi^{\text{eff}}_s$ and amplitude coefficients, can be determined reliably.
• Precision on $\phi^{\text{eff}}_s$: For the LHCb Run 1–3 dataset, a statistical precision of approximately 26 mrad on $\phi^{\text{eff}}_s$ is achievable. This is comparable to current LHCb/CMS measurements using $B^0_s \to J/\psi \phi$ (22 mrad) and significantly better than $B^0_s \to \phi \phi$ (75 mrad).
• Precision on Amplitude Phases: Sensitivity to the phases of the CP-violating amplitude components ($\arg(\Delta c_0)$, $\arg(\Delta \bar{c}_0)$) is lower, with uncertainties around 200 mrad. This is attributed to the limited phase-space overlap between the reference charged $K^*$ amplitudes and the neutral $K^*$ amplitudes.
• Future Projections: Scaling the statistics to expected LHCb Run 4 (60 fb$^{-1}$) and Run 6 (300 fb$^{-1}$) datasets, the projected uncertainties on $\phi^{\text{eff}}_s$ improve to 17 mrad and 8.6 mrad, respectively, following inverse square-root scaling with luminosity.
• Model Dependence: The precision on $\phi^{\text{eff}}_s$ is shown to depend on the underlying amplitude model. In the quasi-two-body limit, sensitivity decreases when strong phase differences approach $\pm \pi/2$. However, the authors note that a more realistic model including additional resonances (e.g., $K^*_0(1430)$) would likely reduce this dependence.

Significance and Claims
The paper claims that the proposed method is feasible and provides a path to measuring CP violation in $B^0_s \to K^0_S K^\pm \pi^\mp$ decays with the LHCb Run 3 dataset. The study establishes a necessary baseline for future data analyses, which are expected to yield further improvements in precision with Run 4 and beyond. The authors emphasize that while the current study uses a simplified model and neglects experimental effects (background, resolution, tagging imperfections), the demonstrated sensitivity suggests that an interesting level of precision is achievable. The method is also noted to be extensible to other multibody decays with multiple final states. The counterpart $B^0 \to K^0_S K^\pm \pi^\mp$ decay is highlighted as a future target where U-spin symmetry could relate amplitudes between $B^0_s$ and $B^0$ systems to control hadronic uncertainties.