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 ${\overline{K}}^{*}_{0}(700)$

TL;DR

This work investigates CP violation arising from the interference between the resonances $\overline{K}^{*}(892)^0$ and $\overline{K}^{*}_{0}(700)$ in the three-body decay $\overline{B}^0\to K^-\pi^+\pi^0$, focusing on the forward-backward asymmetry (FBA) and the CP asymmetry induced by FBA (FB-CPA). Using a naive factorization framework, the authors express the decay amplitude as a sum of two cascade amplitudes with Breit–Wigner propagators and a relative strong phase $\delta$, i.e. $\mathcal{M}=\mathcal{A}_{K_0^*}+\mathcal{A}_{K^{*0}}e^{i\delta}\cos\theta$, with $\mathcal{A}_X=\tilde{\mathcal{A}}_X/(s-m_X^2+i m_X\Gamma_X)$. They show that $A^{FB}$ is driven by the real part of the interference between even and odd partial waves, and that $A^{FB}_{CP}$ can reach up to roughly $35\%$ around the $K^{*}(892)^0$ region for certain $N_c^{\text{eff}}$ and $\delta$ values, while $N_c^{\text{eff}}$ (non-factorizable effects) and the strong phase critically shape the CP-violating signal. The results imply a robust, testable framework for FB-CPA in multi-body decays and suggest experimental prospects for Belle and Belle-II, with potential applicability to related decays like $\Lambda_b^0$ channels.

Abstract

The interference between amplitudes corresponding to different intermediate resonances plays an important role in generating large CP asymmetries in phase space in multi-body decays of bottom and charmed mesons. In this paper, we study the CP violation in the decay channel ${\overline{B}}^{0}\rightarrow K^{-}π^{+}π^{0}$ in phase space region where the intermediate resonances $\overline{K}^{*}(892)^{0}$ and ${\overline{K}^{*}_{0}(700)}$ dominate. The Forward-Backward Asymmetry (FBA) and the CP asymmetry induced by FBA (FB-CPA), which are closely related to the interference effects between the two aforementioned resonances, are especially investigated. The non-trivial correlation between FBA and FB-CPA is analyzed. The analysis indicates that the FB-CPAs around the resonance $\overline{K}^{*}(892)^{0}$ can be as large as about 35\%, which can be potentially accessible by Belle and Belle-II collaborations in the near future.

Figures (5)

• Figure 1: The definition of $\theta$ in the $B^{0}\rightarrow K^{+}\pi^{-}\pi^{0}$ decay channel.
• Figure 2: The $s$-dependence of $A^{FB}$, $\overline{A^{FB}}$, $A^{FB}_{CP}$, and $A^{FB}_{\text{ave}}$ of the decay channel $B^{0}\rightarrow K^{+}\pi^{-}\pi^{0}$ for $\delta=0$, $\pi/2$, $\pi$, and $3\pi/2$, respectively, and for $N_{c}^{\text{eff}}=1$. The range of $s$ is taken from 0.4 GeV/$c^{2}$ to 1.2 GeV/$c^{2}$. The dotted, dashed, dash-dotted and solid lines represent $A^{FB}$, $\overline{A^{FB}}$, $A^{FB}_{\text{ave}}$, and $A^{FB}_{CP}$, respectively. The shadowed region indicate the location of the vector resonances $K^{*}(892)^{0}$ ( $(m_{K^{*}(892)^{0}}-\Gamma_{K^{*}(892)^{0}})^{2}<s<(m_{K^{*}(892)^{0}}+\Gamma_{K^{*}(892)^{0}})^{2}$).
• Figure 3: The same as Fig. \ref{['FIG.Nc1']} but for $N_{c}^{\text{eff}}=2$.
• Figure 4: The same as Figs. \ref{['FIG.Nc1']} and \ref{['FIG.Nc2']} but for $N_{c}^{\text{eff}}=3$.
• Figure 5: The FB-CPAs of three different regions in phase space, $L$ ($m_P-\Gamma_P<s<m_P$), $R$ ($m_P<s<m_P+ \Gamma_P$), and $L+R$ ($m_P-\Gamma_P<s<m_P+\Gamma_P$), which are respectively denoted as $A^{FB}_{CP}(L)$, $A^{FB}_{CP}(R)$, and $A^{FB}_{CP}(L+R)$ in the maintext and are represented by dotted, dashed, and solid lines, respectively, as functions of the strong phase $\delta$ for $N_c^{\text{eff}}=1$. The shadowed area indicates the range of $\delta$ preferred by the data in Ref. BaBar:2007hmp.