The forward-backward asymmetry induced $CP$ asymmetry in ${\overline{B}}^{0}\rightarrow K^{-}π^{+}π^{0}$ in phase space around the resonances ${\overline{K}}^{*}(892)^{0}$ and ${\overline{K}}^{*}_{0}(700)$

This paper investigates the significant CP violation induced by forward-backward asymmetry in the decay ${\overline{B}}^{0}\rightarrow K^{-}\pi^{+}\pi^{0}$ within the phase space dominated by $\overline{K}^{*}(892)^{0}$ and ${\overline{K}^{*}_{0}(700)}$ resonances, predicting asymmetries of up to 35% that are potentially observable by Belle and Belle-II collaborations.

The Gist

The Big Picture: A Cosmic Tug-of-War

Imagine the universe is a giant stage where tiny particles called B-mesons are actors. These actors are unstable and quickly fall apart (decay) into smaller particles. One specific actor, the $B^0$ meson, sometimes breaks apart into three children: a negative Kaon ($K^-$), a positive Pion ($\pi^+$), and a neutral Pion ($\pi^0$).

For decades, physicists have been hunting for a specific phenomenon called CP Violation. Think of this as a "cosmic rule-breaker." In a perfect world, if you filmed a particle decay and then played the movie backward (or swapped all particles for their "anti-particle" twins), the physics should look exactly the same. CP Violation is when the movie looks different when played backward. This difference is crucial because it helps explain why our universe is made of matter instead of being empty space where matter and antimatter cancelled each other out.

The Plot: Two Resonances Colliding

In this specific decay ($B^0 \to K^- \pi^+ \pi^0$), the three children don't just appear out of nowhere. They are born through two different "middlemen" or intermediate resonances:

1. $K^*(892)^0$: A heavy, spinning particle (like a spinning top).
2. $K^*_0(700)$: A lighter, non-spinning particle (like a smooth ball).

The paper argues that the "CP Violation" happens because these two middlemen are interfering with each other. Imagine two sound waves meeting in a room. Sometimes they boost each other (loud), and sometimes they cancel each other out (quiet). In the quantum world, these two particles are like waves crashing into each other, creating a complex pattern that reveals the "rule-breaking" CP violation.

The Detective Work: Forward vs. Backward

The authors introduce a clever way to spot this interference, called Forward-Backward Asymmetry (FBA).

• The Analogy: Imagine a crowd of people (the particles) leaving a concert. If the crowd is perfectly balanced, just as many people walk out the front door as the back door. This is "symmetry."
• The Twist: The paper suggests that because of the interference between the spinning top ($K^*$) and the smooth ball ($K^*_0$), the crowd gets pushed unevenly. More particles might fly "forward" than "backward" (or vice versa).
• The Discovery: The authors calculated that this imbalance can be quite large—up to 35% in certain conditions. This is a huge signal, much easier to spot than a tiny 1% difference.

The "Magic" Ingredient: The Phase Angle

The size of this effect depends on a hidden variable called the strong phase ($\delta$).

• The Analogy: Think of the two resonances as two drummers playing a beat. The "phase" is the timing of their drumsticks.
  • If they hit the drum at the exact same time, the sound is loud.
  • If one hits exactly when the other is lifting their stick, the sound is quiet.
  • The paper shows that depending on this timing (the phase), the "Forward-Backward" imbalance can flip signs or become massive.

The Conclusion: What This Means for Science

The paper claims that:

1. The Signal is Real: The interference between these two specific particles ($K^*(892)^0$ and $K^*_0(700)$) creates a strong, measurable "Forward-Backward" imbalance.
2. It's Detectable: This imbalance leads to a CP violation signal (called FB-CPA) that could be as high as 35%.
3. The Experimenters: The authors believe that current and future experiments, specifically the Belle and Belle-II collaborations (which are giant particle detectors in Japan), have enough data to see this effect soon.

In short: The paper provides a roadmap for how to find a massive "rule-breaking" signal in particle physics by looking at how particles fly forward or backward when two specific quantum "middlemen" interfere with each other. It's like finding a fingerprint on a crime scene that was previously invisible.

Technical Summary

Technical Summary: Forward-Backward Asymmetry Induced CP Asymmetry in $B^0 \to K^-\pi^+\pi^0$

Problem Statement
While CP violation (CPV) has been established in various meson systems, the Standard Model (SM) description via the CKM matrix is insufficient to explain the Baryon Asymmetry of the Universe. Recent observations of large regional CP asymmetries (CPAs) in multi-body decays of bottom and charmed hadrons suggest that interference between different intermediate resonances plays a critical role. Specifically, in three-body decays, the interference between resonances of different spins (e.g., S-wave and P-wave) generates a Forward-Backward Asymmetry (FBA). This FBA can induce a specific type of CP asymmetry, termed FB-CPA. Although the decay channel $B^0 \to K^-\pi^+\pi^0$ has been studied by BaBar and Belle, no CPV has been established due to low statistics. The authors investigate whether the interference between the scalar resonance $K^*_0(700)$ and the vector resonance $K^*(892)^0$ in this channel can generate significant, detectable FB-CPA.

Methodology
The authors analyze the decay amplitude of $B^0 \to K^-\pi^+\pi^0$ in the phase space region dominated by the intermediate resonances $K^*_0(700)$ and $K^*(892)^0$.

1. Amplitude Construction: The decay amplitude is parameterized as a sum of S-wave ($K^*_0$) and P-wave ($K^*$) components, incorporating Breit-Wigner factors for the resonances. The amplitudes are calculated using the naive factorization approach for the weak two-body decay processes $B^0 \to K^*(892)^0\pi^0$ and $B^0 \to K^*_0(700)\pi^0$.
2. Observables: The study defines and calculates the Forward-Backward Asymmetry ($A_{FB}$) and the CP asymmetry induced by FBA ($A_{FB}^{CP}$). The $A_{FB}$ arises from the interference between even- and odd-wave amplitudes. The $A_{FB}^{CP}$ is defined as the difference between the FBAs of the process and its CP-conjugate.
3. Parameter Variation: Numerical results are generated by varying the effective color number ($N_c^{eff} = 1, 2, 3$) to account for non-factorizable contributions and the relative strong phase ($\delta$) between the two resonance amplitudes. Input parameters, including Wilson coefficients, form factors, decay constants, and resonance masses/widths, are taken from established literature (e.g., Refs. [31–37]).
4. Analytical Decomposition: The interference term in the FBA is analytically decomposed into two parts, $\Delta$ and $\Sigma$, to understand the dependence on the strong phase $\delta$ and the invariant mass squared ($s$). This decomposition isolates the contributions that change sign across the resonance mass versus those that do not.

Key Results

1. Magnitude of FB-CPA: The analysis indicates that the interference between $K^*(892)^0$ and $K^*_0(700)$ can induce FB-CPAs as large as approximately 35% in the resonance region. This magnitude is potentially accessible to the Belle and Belle-II collaborations.
2. Sensitivity to Parameters:
   • Non-Factorizable Contributions: The FB-CPA is highly sensitive to $N_c^{eff}$. Significant non-zero FB-CPA is observed for $N_c^{eff} = 1$ and $2$, while the effect is negligibly small for $N_c^{eff} = 3$.
   • Strong Phase Dependence: The behavior of FB-CPA as a function of $s$ depends strongly on the relative strong phase $\delta$.
     • When $\delta \approx 0$ or $\pi$, the FB-CPA changes sign as $s$ crosses the mass squared of the vector resonance ($m_{K^*}^2$).
     • When $\delta \approx \pi/2$ or $3\pi/2$, the FB-CPA does not change sign across the resonance region.
3. Comparison with Data: Comparing the calculated CP-averaged FBA ($A_{FB}^{ave}$) with BaBar's experimental data (Ref. [27]) suggests that the strong phase $\delta$ is favored to lie in the range $[3\pi/2, 5\pi/3]$. Under these conditions (specifically $\delta \in [\pi/2, 2\pi/3]$ for $N_c^{eff}=1$), the FB-CPA in the combined resonance region can reach the ~35% level.
4. Correlation: A non-trivial correlation is identified between the FBA and the FB-CPA, which persists across parameter variations.

Significance and Claims
The paper claims to establish a theoretical framework for including $K^*(892)^0 - K^*_0(700)$ interference effects when studying CPV in multi-body decays. The authors assert that:

• The interference of these specific resonances provides a mechanism for potentially large, detectable CP violation in $B^0 \to K^-\pi^+\pi^0$, offering a target for future experimental measurements.
• The observed behavior explains similar regional CPV patterns seen in other decays, such as $\Lambda_b^0 \to pK^-\pi^+\pi^-$ and $B^0 \to p\bar{p}K^-\pi^+$.
• The analysis provides a model-independent correlation between FBA and FB-CPA that can be used to probe the underlying dynamics of CP violation.
• The study suggests that experimental collaborations should perform combined analyses of FBA, FB-CPA, and direct CP asymmetries in the interference regions of intermediate resonances to fully understand CPV mechanisms.

The authors note that their analysis assumes the strong phase $\delta$ is the dominant source of strong phases and that direct CP asymmetries in the intermediate two-body decays are negligible. They acknowledge that if large strong phases exist within the individual cascade amplitudes, the results would require modification, but the proposed framework remains applicable for a comprehensive combined analysis.