Phenomenological studies on the $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^+\pi^-$ decay

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Jing-Juan Qi, Zhen-Yang Wang, Zhu-Feng Zhang and Xin-Heng Guo. Phenomenological studies on the $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^+\pi^-$ decay[J]. Chinese Physics C. doi: 10.1088/1674-1137/abeb06
Jing-Juan Qi, Zhen-Yang Wang, Zhu-Feng Zhang and Xin-Heng Guo. Phenomenological studies on the $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^+\pi^-$ decay[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abeb06 shu
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Phenomenological studies on the $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^+\pi^-$ decay

    Corresponding author: Zhen-Yang Wang, wangzhenyang@nbu.edu.cn
    Corresponding author: Xin-Heng Guo, xhguo@bnu.edu.cn
  • 1. Junior College, Zhejiang Wanli University, Zhejiang 315101, China
  • 2. Physics Department, Ningbo University, Zhejiang 315211, China
  • 3. College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China

Abstract: Within the quasi-two-body decay model, we study the localized $CP$ violation and branching fraction of the four-body decay $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$ when $K^-\pi^+$ and $\pi^-\pi^+$ pair invariant masses are $0.35<m_{K^-\pi^+}<2.04 \; \mathrm{GeV}$ and $0<m_{\pi^-\pi^+}<1.06\; \mathrm{GeV}$, with the pairs being dominated by the $\bar{K}^*_0(700)^0$, $\bar{K}^*(892)^0$, $\bar{K}^*(1410)^0$, $\bar{K}^*_0(1430)$ and $\bar{K}^*(1680)^0$, and $f_0(500)$, $\rho^0(770)$, $\omega(782)$ and $f_0(980)$ resonances, respectively. When dealing with the dynamical functions of these resonances, $f_0(500)$, $\rho^0(770)$, $f_0(980)$ and $\bar{K}^*_0(1430)$ are modeled with the Bugg model, Gounaris-Sakurai function, Flatté formalism and LASS lineshape, respectively, while others are described by the relativistic Breit-Wigner function. Adopting the end point divergence parameters $\rho_A\in[0,0.5]$ and $\phi_A\in[0,2\pi]$, our predicted results are $\mathcal{A_{CP}}(\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-)\in[-0.365,0.447]$ and $\mathcal{B}(\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-)\in [6.11,185.32]\times10^{-8}$ based on the hypothetical $q\bar{q}$ structures for the scalar mesons in the QCD factorization approach. Meanwhile, we calculate the $CP$ violating asymmetries and branching fractions of the two-body decays $\bar{B}^0\rightarrow SV(VS)$ and all the individual four-body decays $\bar{B}^0\rightarrow SV(VS) \rightarrow K^-\pi^+\pi^-\pi^+$, respectively. Our theoretical results for the two-body decays $\bar{B}^0\rightarrow \bar{K}^*(892)^0$$f_0(980)$, $\bar{B}^0\rightarrow \bar{K}^*_0(1430)^0$$\omega(782)$, $\bar{B}^0\rightarrow \bar{K}^*(892)^0f_0(980)$, $\bar{B}^0\rightarrow \bar{K}^*_0(1430)^0\rho$, and $\bar{B}^0\rightarrow\bar{K}^*_0(1430)^0\omega$ are consistent with the available experimental data, with the remaining predictions awaiting the tests in future examinations with high precision.

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    I.   INTRODUCTION
    • Four-body decays of heavy mesons are hard to be investigated because of their complicated phase spaces and relatively smaller branching fractions. This leads to much less research in four-body dacays than that in two- and three- body decays [1-11]. We have discussed localized $ CP $ violation and the branching fraction of the four-body decays $ \bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+ $ in Ref. [12]. We have focused on the $ \pi\pi $ and $ K\pi $ invariant masses are near the masses of $ f_0(500) $ and $ K_0^*(700) $ mesons. The more resonance states, the more abundant physical mechanisms can be provided to us. Now we will further expand the area of our research to predict the study the $ CP $ violation and the branching fraction in the $ \bar{B}^0 $ four-body decays, which include more contributions from more different resonances. Specifically, the invariant mass of the $ K^-\pi^+ $ pair lies in the range $ 0.35<m_{K^-\pi^+}<2.04 \; \mathrm{GeV} $ which is dominated by the $ \bar{K}^*_0(700)^0 $, $ \bar{K}^*(892)^0 $, $ \bar{K}^*(1410)^0 $, $ \bar{K}^*_0(1430) $ and $ \bar{K}^*(1680)^0 $ resonances, and that of the $ \pi^-\pi^+ $ pair is in the range $ 0<m_{\pi^-\pi^+}<1.06\; \mathrm{GeV} $ which includes the $ f_0(500) $, $ \rho^0(770) $, $ \omega(782) $ and $ f_0(980) $ resonances. Meanwhile, studying the multibody decays can provide rich information for their intermediate resonances especially for the unclear compositions of scalar mesons. The basic structure of the scalar meson is not well established because it is very difficult to identify experimentally [13, 14]. In $ B\rightarrow f_0(980)K $ channel, B decay into a scalar meson was first observed and updated in [15], and also confirmed by BABAR later [16]. In Refs. [17, 18], there are two typical scenarios for scalar mesons based on their mass spectra and strong or electromagnetic decays. In Scenario 1 (S1), the light scalar mesons (such as $ f_0(500) $, $ \bar{K}^*_0(700)^0 $, $ f_0(980) $ and $ a_0(980) $ mesons) are regarded as the lowest-lying $ q\bar{q} $ states, and some others (their masses near 1.5 $ \mathrm{GeV} $ including $ a_0(1450) $, $ K^*_0(1430) $, $ f_0(1370) $ and $ f_0(1500) $ [19-21]) are treated as the first corresponding excited states. In Scenario 2 (S2),the heavier nonet mesons are regarded as the ground states of $ q\bar{q} $, while those lighter nonet ones are not regular mesons and might be four-quark states. To further improve our understanding of QCD mechanism and the quark confinement, it is necessary for us to study the structural composition of scalar mesons and related content.

      In 2019, LHCb collaboration study the $ B^0\rightarrow \rho(770)^0 K^*(892)^0 $ decay within an quasi-two-body decay mode $ B^0\rightarrow (\pi^+\pi^-)(K^+\pi^-) $ [22]. In our work, we will adopt this mechanism to study the four-body decay $ \bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+ $, i.e. $ \bar{B}^0\rightarrow \bar{\kappa}\rho\rightarrow K^-\pi^+\pi^-\pi^+ $, $ \bar{B}^0\rightarrow \bar{\kappa}\omega\rightarrow K^-\pi^+\pi^-\pi^+ $, $ \bar{B}^0\rightarrow\bar{K}^*(892)^0\sigma\rightarrow K^-\pi^+\pi^-\pi^+ $, $ \bar{B}^0\rightarrow \bar{K}^*(892)^0 f_0(980)\!\rightarrow \!K^-\pi^+\pi^-\pi^+ $, $ \bar{B}^0\!\rightarrow \!\bar{B}^0\rightarrow\bar{K}^*(1410)^0\sigma\!\rightarrow \!K^-\pi^+\pi^-\pi^+ $, $ \bar{B}^0\rightarrow \bar{K}^*(1410)^0f_0(980)\rightarrow K^-\pi^+\pi^-\pi^+ $, $ \bar{B}^0\rightarrow \bar{K}^*_0(1430)^0\rho\rightarrow K^-\pi^+\pi^-\pi^+ $,$ \bar{B}^0\rightarrow \bar{K}^*_0(1430)^0\omega\rightarrow K^-\pi^+\pi^-\pi^+ $, $ \bar{B}^0\rightarrow \bar{K}^* (1680)^0\sigma\rightarrow K^-\pi^+\pi^-\pi^+ $ and $ \bar{B}^0\rightarrow \bar{K}^* (1680)^0f_0(980)\rightarrow K^-\pi^+\pi^-\pi^+ $, where the scalar mesons will be treated in S1 as mentioned above. We can then calculate the localized $ CP $ violations and branching fractions of the four-body decay $ \bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+ $. Besides we can also calculate the $ CP $ violations and branching fractions of the two-body decays $ \bar{B}^0\rightarrow SV(VS) $ and all the individual four-body decays $ \bar{B}^0\rightarrow SV(VS) \rightarrow K^-\pi^+\pi^-\pi^+ $, respectively. In fact, with the great development of the large hadron collider beauty (LHCb) and Belle-II experiments, more and more decay modes involving one or two scalar states in the B and D meson decays are expected to be measured with the high precision in the future.

      The remainder of this paper is organized as follows. Our theoretical framework are presented in Sect. II. In Sect. III, we give our numerical results. And we summarize our work in Sect IV. Appendix A collects the explicit formulas for all four-body decay amplitudes. The dynamical functions for the corresponding resonances are summarized in Appendix B. We also consider the $ f_0(500)- f_0 (980) $ mixing in Appendix C. Related theoretical parameters are listed in Appendix D.

    II.   THEORETICAL FRAMEWORK

      A.   B decay in QCD factorization approach

    • Under the framework of the QCD factorization approach [4, 23], the effective Hamiltonian matrix elements can be written as

      $ \langle{M_1M_2}|\mathcal{H}_{\rm eff}|B\rangle = \sum_{p = u,c}\lambda_{p}^{(D)}\langle{M_1M_2}|\mathcal{T}_A^p+\mathcal{T}_B^p|B\rangle, $

      (1)

      where $ \mathcal{H}_{\rm eff} $ is the effective weak Hamiltonian, $ \lambda_p^{(D)} = V_{pb}V_{pD}^* $, $ V_{pb} $ and $ V_{pD} $ are the CKM matrix elements, $ \mathcal{T}_A^p $ and $ \mathcal{T}_B^p $ describe the contributions from non-annihilation and annihilation amplitudes, respectively, they can be expressed in terms of $ a_i^p $ and $ b_i^p $.

      Generally, $ a_i^p $ includes the contributions from naive factorization, vertex correction, penguin amplitude and spectator scattering, which have the following expressions [4]

      $ \begin{aligned}[b] a_i^p{(M_1M_2)} =& {\left(c_i+\frac{c_{i\pm1}}{N_c}\right)}N_i{(M_2)}+\frac{c_{i\pm1}}{N_c}\frac{C_F\alpha_s}{4\pi}\\&\times{\bigg[V_i{(M_2)}+\frac{4\pi^2}{N_c}H_i{(M_1M_2)}\bigg]}+P_i^p{(M_2)}, \end{aligned} $

      (2)

      where $ c_i $ are the Wilson coefficients, $ N_i{(M_2)} $ is leading-order coefficient, $ V_i{(M_2)} $, $ H_i{(M_1M_2)} $ and $ P_i^p{(M_1M_2)} $ are one-loop vertex corrections, hard spectator interactions with a hard gluon exchange between the emitted meson and the spectator quark of the B meson and penguin contractions, respectively. $ C_F = {(N_c^2-1)}/{2N_c} $ with $ N_c = 3 $ [4].

      The weak annihilation contributions can be expressed in terms of $ b_i $ and $ b_{i,EW} $, which are

      $ \begin{aligned}[b] b_1 =& \frac{C_F}{N_c^2}c_1A_1^i, \quad b_2 = \frac{C_F}{N_c^2}c_2A_1^i, \\ b_3^p =& \frac{C_F}{N_c^2}\bigg[c_3A_1^i+c_5(A_3^i+A_3^f)+N_cc_6A_3^f \bigg],\\ b_4^p =& \frac{C_F}{N_c^2}\bigg[c_4A_1^i+c_6A_2^i \bigg], \\ b_{3,EW}^p =& \frac{C_F}{N_c^2}\bigg[c_9A_1^i+C_7(A_3^i+A_3^f)+N_cc_8A_3^f \bigg],\\ b_{4,EW}^p = &\frac{C_F}{N_c^2}\bigg[c_{10}A_1^i+c_8A_2^i \bigg], \end{aligned} $

      (3)

      where the subscripts 1, 2, 3 of $ A_n^{i,f}(n = 1,2,3) $ stand for the annihilation amplitudes induced from $ (V-A)(V-A) $, $ (V-A)(V+A) $, and $ (S-P)(S+P) $ operators, respectively, the superscripts i and f refer to gluon emission from the initial- and final-state quarks, respectively. The explicit expressions for $ A_n^{i,f} $ can be found in Ref. [24].

      In the expressions for the spectator and annihilation corrections, there are end-point divergences $ X = \int_0^1 {\rm d}x/(1-x) $, which can be parametrized as [17]

      $ X_{H,A} = (1+\rho_{H,A} {\rm e}^{{\rm i}\phi_{H,A}})\ln\frac{m_B}{\Lambda_h}, $

      (4)

      with $ \Lambda_h $ being a typical scale of order 500 $ \mathrm{MeV} $, $ \rho_{A,H} $ an unknown real parameter and $ \phi_{A,H} $ the free strong phase in the range $ [0,2\pi] $.

    • B.   Four-body decay amplitudes

    • For the four-body decay $ \bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+ $, we consider the two-body cascade decays mode $ \bar{B}^0\rightarrow[K^-\pi^+]_{S/V} [\pi^-\pi^+]_{V/S}\rightarrow K^-\pi^+\pi^-\pi^+ $. Within the QCDF framework in Ref. [4], we can deduce the two-body weak decay amplitudes of $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^-\pi^+]_{V/S} $, which are

      $ \begin{aligned}[b] \mathcal{M}(\bar{B}^0\rightarrow \bar{K}^{*0}_{0i}\rho ) =& {\rm i}G_F\sum_{p = u,c}\lambda_p^{(s)}\bigg\{\bigg[\delta_{pu}\alpha_2(\bar{K}^{*0}_{0i}\rho) +\frac{3}{2}\alpha_{3,EW}^p(\bar{K}^{*0}_{0i}\rho)\bigg] f_\rho m_\rho \varepsilon_\rho^*\cdot p_BF_1^{\bar{B}^0 \bar{K}^{*0}_{0i}}(m_{\rho}^2)\\& +\bigg[\alpha_4^p(\rho\bar{K}^{*0}_{0i})-\frac{1}{2}\alpha_{4,EW}^p(\rho\bar{K}^{*0}_{0i})\bigg] \bar{f}_{\bar{K}^{*0}_{0i}}m_\rho \varepsilon_\rho^*\cdot p_BA_0^{\bar{B}^0\rho}(m_{\bar{K}^{*0}_{0i}}^2)\\& +\bigg[\frac{1}{2}b_3^p(\rho\bar{K}^{*0}_{0i})-\frac{1}{4}b_{3,EW}^p(\rho\bar{K}^{*0}_{0i})\bigg] f_{\bar{B}^0}f_\rho \bar{f}_{\bar{K}^{*0}_{0i}}\bigg\},\end{aligned} $

      (5)

      $ \begin{aligned}[b] \mathcal{M}(\bar{B}^0\rightarrow \bar{K}^{*0}_{0i}\omega) =& {\rm i}G_F\sum_{p = u,c}\lambda_p^{(s)}\bigg\{\bigg[\delta_{pu}\alpha_2(\bar{K}^{*0}_{0i}\omega)+2\alpha_3^p(\bar{K}^{*0}_{0i}\omega) +\frac{1}{2}\alpha_{3,EW}^p(\bar{K}^{*0}_{0i}\omega)\bigg] f_\omega m_\omega \varepsilon_\omega^*\cdot p_BF_1^{\bar{B}^0 \bar{K}^{*0}_{0i}}(m_{\omega}^2)\\& +\bigg[\frac{1}{2}\alpha_{4,EW}^p(\omega\bar{K}^{*0}_{0i})-\alpha_4^p(\omega\bar{K}^{*0}_{0i})\bigg] \bar{f}_{\bar{K}^{*0}_{0i}}m_\omega \varepsilon_\omega^*\cdot p_BA_0^{\bar{B}^0\omega}(m_{\bar{K}^{*0}_{0i}}^2)\\& +\bigg[\frac{1}{4}b_{3,EW}^p(\omega\bar{K}^{*0}_{0i})-\frac{1}{2}b_3^p(\omega\bar{K}^{*0}_{0i})\bigg] f_{\bar{B}^0}f_\rho \bar{f}_{\bar{K}^{*0}_{0i}}\bigg\},\end{aligned} $

      (6)

      with $ \bar{K}^{*0}_{0i} = \bar{K}^*_0(700)^0,\;\bar{K}^*_0(1430)^0 $ corresponding to $ i = 1,\;2 $, respectively, and

      $ \begin{aligned}[b] \mathcal{M}(\bar{B}^0\rightarrow \bar{K}^{*0}_if_{0j}) =& -iG_F\sum_{p = u,c}\lambda_p^{(s)}\bigg\{\bigg[\delta_{pu}\alpha_2(\bar{K}^{*0}_if_{0j})+2\alpha_3^p(\bar{K}^{*0}_if_{0j})+\frac{1}{2}\alpha_{3,EW}^p(\bar{K}^{*0}_if_{0j})\bigg]\\& \times\bar{f}_{f_{0j}^n}m_{\bar{K}^{*0}_i} \varepsilon_{\bar{K}^{*0}_i}^*\cdot p_BA_0^{\bar{B}^0 \bar{K}^{*0}_i}(m_{f_{0j}}^2)+\bigg[\sqrt{2}\alpha_3^p(\bar{K}^{*0}_if_{0j})+\sqrt{2}\alpha_4^p(\bar{K}^{*0}_if_{0j})\\& -\frac{1}{\sqrt{2}}\alpha_{3,EW}^p(\bar{K}^{*0}_if_{0j})-\frac{1}{\sqrt{2}}\alpha_{4,EW}^p(\bar{K}^{*0}_if_{0j})\bigg]\bar{f}_{f_{0j}^s}m_{\bar{K}^{*0}_i} \varepsilon_{\bar{K}^{*0}_i}^*\cdot p_BA_0^{\bar{B}^0\bar{K}^{*0}_i}(m_{f_{0j}}^2)\\& +\bigg[\frac{1}{2}\alpha_{4,EW}^p(f_{0j}\bar{K}^{*0}_i)-\alpha_4^p(f_{0j}\bar{K}^{*0}_i)\bigg]f_{\bar{K}^{*0}_i}m_{\bar{K}^{*0}_i} \varepsilon_{\bar{K}^{*0}_i}^*\cdot p_B F_1^{\bar{B}^0f_{0j}}(m_{\bar{K}^{*0}_i}^2)+\bigg[\frac{1}{\sqrt{2}}b_3^p(\bar{K}^{*0}_if_{0j})\\& -\frac{1}{2\sqrt{2}}b_{3,EW}^p(\bar{K}^{*0}_if_{0j})\bigg]f_{\bar{B}^0}f_{\bar{K}^{*0}_i}\bar{f}_{f_{0j}}^s+\bigg[\frac{1}{2}b_3^p(f_{0j}\bar{K}^{*0}_i)-\frac{1}{4}b_{3,EW}^p(f_{0j}\bar{K}^{*0}_i)\bigg]f_{\bar{B}^0}f_{\bar{K}^{*0}_i}\bar{f}_{f_{0j}}^n\bigg\},\end{aligned} $

      (7)

      with $ \bar{K}^{*0}_i = \bar{K}^*(892)^0,\;\bar{K}^*(1410)^0,\;\bar{K}^*(1680)^0 $ corresponding to $ i = 1,\;2,\;3 $, respectively, $ f_{0j} = f_0(500) $, $ f_0(980) $ when $ j = 1,2 $, respectively. In Eqs. (5)-(7), $ F_1^{\bar{B}^0 \rightarrow S}(m_V^2) $ and $ A_0^{\bar{B}^0\rightarrow V}(m_S^2) $ are the form factors for $ \bar{B}^0 $ to scalar and vector meson transitions, respectively, $ f_V $, $ \bar{f}_S $, and $ f_{\bar{B}^0} $ are decay constants of vector, scalar, and $ \bar{B}^0 $ mesons, respectively, $ \bar{f}_{f_{0j}}^s $ and $ \bar{f}_{f_{0j}}^n $ are decay constants of $ f_{0j} $ mesons coming from the up and strange quark components, respectively.

      In the framework of the two two-body decays, the four-body decay can be factorized into three pieces as the following:

      $ \mathcal{M}(\bar{B}^0\rightarrow [K^-\pi^+]_S[\pi^-\pi^+]_V\rightarrow K^-\pi^+\pi^-\pi^+) = \frac{\langle SV|\mathcal{H}_{\rm eff}|\bar{B}^0\rangle \langle K^-\pi^+|\mathcal{H}_{S K^-\pi^+}|S\rangle \langle \pi^-\pi^+|\mathcal{H}_{V \pi^-\pi^+}|V\rangle}{s_{S}s_{V}}, $

      (8)

      and

      $ \mathcal{M}(\bar{B}^0\rightarrow [K^-\pi^+]_{V}[\pi^-\pi^+]_{S}\rightarrow K^-\pi^+\pi^-\pi^+) = \frac{\langle VS|\mathcal{H}_{\rm eff}|\bar{B}^0\rangle \langle K^-\pi^+|\mathcal{H}_{V K^-\pi^+}|V\rangle \langle \pi^-\pi^+|\mathcal{H}_{S\pi^-\pi^+}|S\rangle}{s_Vs_S}, $

      (9)

      where $ \mathcal{H}_{\rm eff} $ is the effective weak Hamiltonian, $ \langle M_1M_2|\mathcal{H}_s|V\rangle = g_{VM_1M_2}(p_{M_1}-p_{M_2})\cdot\epsilon_V $ and $ \langle M_1M_2|\mathcal{H}_s|S\rangle = g_{SM_1M_2} $, $ g_{VM_1M_2} $ and $ g_{SM_1M_2} $ are the strong coupling constants of the corresponding vector and scalar mesons decays, $ s_{S/V} $ are the reciprocal of dynamical functions $ T_{S/V} $ for the corresponding resonances. The specific kinds and expressions of $ T_{S/V} $ are given in the fifth column of Table 1 and Appendix C, respectively.

      Decay modeBABARPDG [25][18]This work
      $\bar{\kappa}$$\rho$$-10.66\pm3.14$
      $\bar{\kappa}$$\omega$$17.43\pm6.53$
      $\bar{K}^*(892)^0$$\sigma$$25.57\pm10.42$
      $\bar{K}^*(892)^0$$f_0(980)$$7\pm10\pm2$$7\pm10$$9.31\pm1.04$
      $\bar{K}^*(1410)^0$$\sigma$$0.43\pm0.13$
      $\bar{K}^*(1410)^0$$f_0(980)$$-2.01\pm0.19$
      $\bar{K}^*_0(1430)^0$$\rho$$0.54^{+0.45+0.02+3.76}_{-0.46-0.02-1.80}$$6.03\pm0.97$
      $\bar{K}^*_0(1430)^0$$\omega$$-7\pm9\pm2$$0.03^{+0.37+0.01+0.29}_{-0.35-0.01-3.00}$$-9.53\pm3.88$
      $\bar{K}^*(1680)^0$$\sigma$$3.03\pm0.77$
      $\bar{K}^*(1680)^0$$f_0(980)$$-2.76\pm0.20$

      Table 1.  Direct $ CP $ violations (in units of $ 10^{-2} $) of the two-body decays $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} $. The experimental branching fractions are taken from Ref. [34]. The theoretical errors come from the uncertainties of the form factors, decay constants, Gegenbauer moments and divergence parameters.

      When considering the contributions from $ \bar{B}^0\rightarrow [K^-\pi^+]_S[\pi^-\pi^+]_V\!\rightarrow \!K^-\pi^+\pi^-\pi^+ $ and $ \bar{B}^0\!\rightarrow\! [K^-\pi^+]_{V}[\pi^-\pi^+]_{S}\!\rightarrow K^-\pi^+\pi^-\pi^+ $ channels as listed in Eqs. (8) and (9), the total decay amplitude of the $ \bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^- $ decay can be written as

      $ \begin{aligned}[b] \mathcal{M} =&\mathcal{M}(\bar{B}^0\rightarrow [K^-\pi^+]_S[\pi^-\pi^+]_V\rightarrow K^-\pi^+\pi^-\pi^+ )\\&+\mathcal{M}(\bar{B}^0\rightarrow [K^-\pi^+]_{V}[\pi^-\pi^+]_{S}\rightarrow K^-\pi^+\pi^-\pi^+). \end{aligned}$

      (10)
    • C.   Kinematics of the four-body decay and localized CP violation

    • One can use the five variables $ s_{\pi\pi} $, $ s_{K\pi} $, $ \phi $, $ \theta_\pi $ and $ \theta_K $ to describe the kinematics of the four-body decay $ \bar{B}^0\rightarrow K^-(p_1)\pi^+(p_2)\pi^-(p_3)\pi^+(p_4) $ [26-29], $ s_{\pi\pi} $ and $ s_{K\pi} $ are the invariant mass squared of the $ \pi\pi $ system and $ K\pi $ system, respectively, $ \theta_\pi $ is the angle between the $ \pi\pi $ and $ K\pi $ planes, $ \theta_\pi $ (or $ \theta_K) $ is the angle of the $ \pi^+ $ (or $ K^- $) in the $ \pi\pi $ (or $ K\pi $) center-of-mass system with respect to the $ \pi\pi $ (or $ K\pi $) line of flight in the $ \bar{B}^0 $ rest frame. Their specific physical ranges can be found in Refs. [12, 26-29] for detail.

      For the convenience of presentation and calculation, it is more convenient to replace the individual momenta $ p_1 $, $ p_2 $, $ p_3 $, $ p_4 $ with the following kinematic variables

      $ \begin{aligned}[b] & P = p_1+p_2,\quad Q = p_1-p_2,\\& L = p_3+p_4,\quad N = p_3-p_4.\end{aligned} $

      (11)

      Using the above formula, we can get

      $ \begin{aligned}[b] P^2 =& s_{K\pi},\quad Q^2 = 2(p_K^2+p_\pi^2)-s_{K\pi},\quad L^2 = s_{\pi\pi},\\ P\cdot L =& \frac{1}{2}(m_{\bar{B}^0}^2-s_{K\pi}-s_{\pi\pi}),\quad P\cdot N = X\cos\theta_\pi,\\ L\cdot Q =& \sigma(s_{K\pi})X\cos\theta_K,\end{aligned} $

      (12)

      where

      $ \sigma(s_{K\pi}) = \sqrt{1-(m_K^2+m_\pi^2)/s_{K\pi}}, $

      (13)

      with the decay amplitude, one can get the decay rate of the four-body decay [32],

      $ {\rm d}^5\Gamma = \frac{1}{4(4\pi)^6m_{\bar{B}^0}^3}\sigma(s_{\pi\pi})X(s_{\pi\pi},s_{K\pi})\sum_{\mathrm{spins}}|\mathcal{M}|^2{\rm d}\Omega, $

      (14)

      where $ \sigma(s_{\pi\pi}) = \sqrt{1-4m_\pi^2/s_{\pi\pi}} $, $ \Omega $ represents the phase space with $ {\rm d}\Omega = {\rm d}s_{\pi\pi}{\rm d}s_{K\pi}{\rm dcos}\theta_\pi {\rm dcos}\theta_K{\rm d}\phi $.

      The differential $ CP $ asymmetry parameter and the localized integrated $ CP $ asymmetry take the following forms

      $ \mathcal{A_{CP}} = \frac{|\mathcal{M}|^2-|\bar{\mathcal{M}}|^2}{|\mathcal{M}|^2+|\bar{\mathcal{M}}|^2}, $

      (15)

      and

      $ \mathcal{A^\mathrm{\Omega}_{CP}} = \frac{\int {\rm d}\Omega(|\mathcal{M}|^2-|\bar{\mathcal{M}}|^2)}{\int {\rm d}\Omega(|\mathcal{M}|^2+|\bar{\mathcal{M}}|^2)}, $

      (16)

      respectively.

    III.   NUMERICAL RESULTS
    • When dealing with the scalar mesons, we adopt Scenario 1 in Ref. [17], in which those with masses below or near 1 GeV ($ \sigma $, $ f_0(980) $, $ \kappa $) and near 1.5 $ \mathrm{GeV} $ ($ K^*_0(1430) $) are suggested as the lowest-lying $ q\bar{q} $ states and the first excited state, respectively. As for the decay constants of $ f_{0j} $ mesons, we consider the $ f_0(500)-f_0(980) $ mixing with the mixing angle $ |\varphi_m| = 17^0 $ (see Appendix A for details). As for the decay constants and Gegenbauer moments of the $ \bar{K}^*(1410)^0 $ and the $ \bar{K}^*(1680)^0 $ mesons, we assume they have the same central values as that of $ \bar{K}^*(892)^0 $ and assign their uncertainties to be $ \pm0.1 $ [33]. With the QCDF approach, we have obtained the amplitudes of the two-body decays $ \bar{B}^0\rightarrow SV $ and $ \bar{B}^0\rightarrow VS $, which are listed in Eqs. (5)-(7). Generally, the end-point divergence parameter $ \rho_A $ is constrained in the range of $ [0,1] $ and $ \phi_A $ is treated as a free strong phase. The experimental data of B two-body decays can provide important information to restrict the ranges of these two parameters. In fact, compared with the $ B\rightarrow PV/VP/PP $ decays, there are much less experimental data for the $ B \rightarrow VS/PS $ and $ B\rightarrow SV/SP $ decays, so the values of $ \rho_A $ and $ \phi_A $ have not been determined well in these decays. Thus we adopt $ \rho_{A,H}<0.5 $ and $ 0\leqslant\phi_{A,H}\leqslant 2\pi $ as in Refs. [17, 24]. With more accumulation of experimental data, both of them could be defined in small regions in the future.

      Substituting Eqs. (5)-(7) into Eq. (15), we obtain the $ CP $ violating asymmetries of the two-body decays $ \bar{B}^0\rightarrow SV $ and $ \bar{B}^0\rightarrow VS $ with the parameters given in Table 1 and Appendix F, which are listed in Table 2. From Table 2, one can see our theoretical results for the $ CP $ asymmetries of $ \bar{B}^0\rightarrow \bar{K}^*(892)^0 f_0(980) $ and $ \bar{B}^0\rightarrow \bar{K}^*_0(1430)^0 $$ \omega $ are consistent with the data from BABAR Collaboration. However, the predicted central values of the $ CP $ asymmetries of the $ \bar{B}^0\rightarrow \bar{K}^*_0(1430)^0 \rho $ and $ \bar{B}^0\rightarrow \bar{K}^*_0(1430)^0\omega $ are larger than those in Ref. [18]. The main difference between our work and Ref. [18] is the structure of the $ \bar{K}^*_0(1430)^0 $ meson, which is explored in S1 in our work and S2 in Ref. [18], respectively. Besides, we predict the $ CP $ asymmetries of some other channel decays. We find the signs of the $ CP $ asymmetries are negative in $ \bar{B}^0\rightarrow \bar{\kappa}\rho $, $ \bar{B}^0\rightarrow \bar{K}^*(1410)^0 f_0(980) $ and $ \bar{B}^0\rightarrow \bar{K}^*(1680)^0 f_0(980) $ decays, with the first one being one order larger than the other two. For the positive values of the $ CP $ asymmetries in our work, those for the $ \bar{B}^0\rightarrow\bar{\kappa}\omega $ and $ \bar{B}^0\rightarrow\bar{K}^*(892)^0\sigma $ decays are also one order larger than the others. Moreover, we also calculate the branching fractions of the two-body decays $ \bar{B}^0\rightarrow SV $ and $ \bar{B}^0\rightarrow VS $ which are listed in Table 3. As can be seen from Table 3, our results are consistent with the available experimental data for the $ \bar{B}^0\rightarrow \bar{K}^*(892)^0f_0(980) $, $ \bar{B}^0\rightarrow\bar{K}^*_0(1430)^0\rho $ and $ \bar{B}^0\rightarrow\bar{K}^*_0(1430)^0\omega $ decays. Meanwhile, we find the magnitudes of the branching fractions are of order $ 10^{-5} $ for $ \bar{B}^0\rightarrow\bar{K}^*(892)^0f_0(980) $, $ \bar{B}^0\rightarrow\bar{K}^*(1410)^0\sigma $ and $ \bar{B}^0\rightarrow \bar{K}^*(1410)^0f_0(980) $, but of $ 10^{-6} $ for $ \bar{B}^0\rightarrow\bar{\kappa}\rho $, $ \bar{B}^0\rightarrow\bar{\kappa}\omega $, $ \bar{B}^0\rightarrow\bar{K}^*_0(1430)^0\rho $ and $ \bar{B}^0\rightarrow\bar{K}^*_0(1430)^0\omega $. We note that the predicted branching fraction of $ \bar{B}^0\rightarrow\bar{K}^*(892)^0 $$ \sigma $ is the smallest one with the order of $ 10^{-7} $.

      Decay modeBABARBelleLHCb [24]PDG [25]QCDF [18]pQCD [35, 36]This work
      $\bar{\kappa}$$\rho$$1.35\pm0.47$
      $\bar{\kappa}$$\omega$$3.87\pm1.65$
      $\bar{K}^*(892)^0$$\sigma$$0.11\pm0.04$
      $\bar{K}^*(892)^0$$f_0(980)$$11.4\pm1.4$$<4.4$$7.8^{+4.2}_{-3.6}$$9.1^{+1.0+1.0+5.3}_{-0.4-0.5-0.7}$$11.2\sim13.7$$9.48\pm2.88$
      $\bar{K}^*(1410)^0$$\sigma$$25.41\pm9.13$
      $\bar{K}^*(1410)^0$$f_0(980)$$14.39\pm4.22$
      $\bar{K}^*_0(1430)^0$$\rho$$27\pm4\pm2\pm3$$10.0^{+2.4+0.5+12.1}_{-2.0-0.4-3.1}$$27.0\pm6.0$$4.1^{+1.1+0.2+2.6}_{-1.0-0.2-0.1}$$4.8^{+1.1+1.0+0.3}_{-0.0-1.0-0.3}$$8.13\pm2.03$
      $\bar{K}^*_0(1430)^0$$\omega$$6.4^{+1.4+0.3+4.0}_{-1.2-0.2-0.9}$$16.0\pm3.4$$9.3^{+2.7+0.3+3.9}_{-2.2-0.3-1.3}$$9.3^{+2.1+3.6+1.2}_{-2.0-2.9-1.0}$$5.02\pm1.06$
      $\bar{K}^*(1680)^0$$\sigma$$27.64\pm8.59$
      $\bar{K}^*(1680)^0$$f_0(980)$$21.76\pm8.33$

      Table 2.  Branching fractions (in units of $ 10^{-6} $) of the two-body decays $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} $. We have used $ \mathcal{B}(f_0(980)\rightarrow \pi^+\pi^-) = 0.5 $ to obtain the experimental branching fractions for $ f_0(980)V $. The theoretical errors come from the uncertainties of the form factors, decay constants Gegenbauer moments and divergence parameters.

      As mentioned in the abstract, for different intermediate resonance states, we use different models to deal with their dynamical functions which are listed in Table 1 and Appendix D in detail, where $ \sigma $, $ \rho^0(770) $, $ f_0(980) $ and $ \bar{K}^*_0(1430) $ are modeled with the Bugg model [37], Gounaris-Sakurai function [38], Flatté formalism [39] and LASS lineshape [40-42], respectively, while others are described by the relativistic Breit-Wigner function [43]. Inserting Eqs. (A1)-(A3) into Eqs. (16) and (14), we can directly obtain the $ CP $ asymmetries and branching fractions of all the individual four-body decay channel $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^+\pi^- $ by integrating the phase space of Eq. (14), respectively, both of which are summarized in Table 4. From this table, we can conclude that the range of these $ CP $ asymmetries and branching fractions are about $ [-7.03, 24.33]\times10^{-2} $ and $ [0.11, 27.3]\times 10^{-6} $, respectively. Considering the contributions from all four-body decays listed in Table 4, we can obtain the localized integrated $ CP $ asymmetries and branching fractions of the $ \bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^- $ decay by integrating the phase space. Our results are in the ranges $ \mathcal{A_{CP}}(\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-) = [-0.365,0.447] $ and $ \mathcal{B}(\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-) = [6.11,185.32]\times10^{-8} $ when the invariant masses of $ K^-\pi^+ $ and $ \pi^-\pi^+ $ are in the ranges $ 0.35<m_{K^-\pi^+}<2.04 \, \mathrm{GeV} $ and $ 0<m_{\pi^-\pi^+}<1.06\; \mathrm{GeV} $, where the $ K\pi $ channel is dominated by the $ \kappa $, $ \bar{K}^*(892)^0 $, $ \bar{K}^*(1410)^0 $, $ \bar{K}^*_0(1430) $ and $ \bar{K}^*(1680)^0 $ resonances, and the $ \pi\pi $ channel is dominated by the $ \sigma $, $ \rho^0(770) $, $ \omega(782) $ and $ f_0(980) $ resonances, respectively, and the range of $ \rho_A $ and $ \phi_A $ are taken as $ [0,0.5] $ and $ [0,2\pi] $, respectively. Both of them are expected to be tested in the near future experiments.

      Decay mode$CP$ asymmetriesBranching fractions
      $\bar{\kappa}$$\rho$$(\rightarrow K^-\pi^+\pi^+\pi^-)$$-10.03\pm5.01$$1.46\pm0.51$
      $\bar{\kappa}$$\omega$ $(\rightarrow K^-\pi^+\pi^+\pi^-)$$18.34\pm5.17$$4.10\pm0.63$
      $\bar{K}^*(892)^0$$\sigma$ $(\rightarrow K^-\pi^+\pi^+\pi^-)$$24.33\pm9.01$$0.11\pm0.05$
      $\bar{K}^*(892)^0$$f_0(980)$ $(\rightarrow K^-\pi^+\pi^+\pi^-)$$-3.85\pm1.01$$9.22\pm4.15$
      $\bar{K}^*(1410)^0$$\sigma$ $(\rightarrow K^-\pi^+\pi^+\pi^-)$$0.41\pm0.53$$21.18\pm6.32$
      $\bar{K}^*(1410)^0$$f_0(980)$$(\rightarrow K^-\pi^+\pi^+\pi^-)$$-2.38\pm0.49$$16.01\pm4.04$
      $\bar{K}^*_0(1430)^0$$\rho$ $(\rightarrow K^-\pi^+\pi^+\pi^-)$$-7.03\pm2.47$$2.03\pm0.41$
      $\bar{K}^*_0(1430)^0$$\omega$ $(\rightarrow K^-\pi^+\pi^+\pi^-)$$10.39\pm3.42$$2.55\pm0.87$
      $\bar{K}^*(1680)^0$$\sigma$ $(\rightarrow K^-\pi^+\pi^+\pi^-)$$8.05\pm3.01$$27.30\pm7.05$
      $\bar{K}^*(1680)^0$$f_0(980)$$(\rightarrow K^-\pi^+\pi^+\pi^-)$$-5.03\pm0.62$$19.89\pm4.01$

      Table 3.  Direct $ CP $ violations (in units of $ 10^{-2} $) and branching fractions (in units of $ 10^{-6} $) of the four-body decays $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^+\pi^- $. The theoretical errors come from the uncertainties of the form factors, decay constants, Gegenbauer moments and divergence parameters.

      ResonanceMass/$\mathrm{MeV}$Width/$\mathrm{MeV}$$J^P$Model
      $\sigma$$475\pm75$$550\pm150$$0^+$BUGG
      $\rho$$775.26\pm0.25$$149.1\pm0.8$$1^-$GS
      $\omega$$782.65\pm0.12$$8.49\pm0.08$$1^-$RBW
      $f_0(980)$$990\pm20$$65\pm45$$0^+$FLATT$\acute{\mathrm{E}}$
      $\bar{\kappa}$$824\pm30$$478\pm50$$0^+$RBW
      $\bar{K}^*(892)^0$$895.5\pm0.20$$47.3\pm0.5$$1^-$RBW
      $\bar{K}^*(1410)^0$$1421\pm9$$236\pm18$$1^-$RBW
      $\bar{K}^*_0(1430)^0$$1425\pm50$$270\pm80$$0^+$LASS
      $\bar{K}^*(1680)^0$$1718\pm18$$322\pm110$$1^-$RBW

      Table 4.  The masses, widths and decay models of the intermediate resonances [25].

    IV.   SUMMARY
    • In this work, we have revisited the four-body decay $ \bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+ $ in the framework of the two two-body decays. We consider more contributions from more different resonances. Meanwhile, we update the model when dealing with the dynamical function for the $ \rho $ resonance. The most important thing is that we have added the relevant calculations to further test the rationality of the two-quark model for scalar mesons in the two-body decay of the $ \bar{B}^0 $ meson. In this analysis, we first calculate the direct $ CP $ violating asymmetries and branching fractions of the two-body decays $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} $ within the QCDF approach which are listed in Table 2 and Table 3, respectively. From these two tables, we can see that our theoretical results are consistent with the available experimental data including the $ CP $ asymmetries of the $ \bar{B}^0\rightarrow \bar{K}^*(892)^0 $$ f_0(980) $ and $ \bar{B}^0\rightarrow \bar{K}^*_0(1430)^0 $$ \omega $ decays and the branching fractions of the $ \bar{B}^0\rightarrow \bar{K}^*(892)^0f_0(980) $, $ \bar{B}^0\rightarrow\bar{K}^*_0(1430)^0\rho $ and $ \bar{B}^0\rightarrow\bar{K}^*_0(1430)^0\omega $ decays. Because of different structures of the $ \bar{K}^*_0(1430)^0 $ meson, our predicted central values of the $ CP $ asymmetries are larger than those in Ref. [18] for the $ \bar{B}^0\rightarrow \bar{K}^*_0(1430)^0 \rho $ and $ \bar{B}^0\rightarrow \bar{K}^*_0(1430)^0\omega $ decays. It is found that the signs of the $ CP $ asymmetries are negative for the $ \bar{B}^0\rightarrow \bar{\kappa}\rho $, $ \bar{B}^0\rightarrow \bar{K}^*(1410)^0 f_0(980) $ and $ \bar{B}^0\rightarrow \bar{K}^*(1680)^0 f_0(980) $ decays and are positive for others decays. The magnitudes of branching fractions for our considered two-body decays $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} $ are of orders $ 10^{-7}\sim10^{-5} $. Then, under the assumption of the quasi-two-body decay mode, we regard $ \bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+ $ decay as happening through $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S}\rightarrow K^-\pi^+\pi^-\pi^+ $ and calculate the direct $ CP $ asymmetries and branching fractions of all the individual four-body decay channel $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^+\pi^- $. The range of them are about $ [-7.03, 24.33]\times10^{-2} $ and $ [0.11, 27.3]\times10^{-6} $, respectively. Finally, considering the contributions from all these decay channels decays, we obtain the localized integrated $ CP $ asymmetries and the branching fraction of $ \bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+ $ when $ 0.35<m_{K^-\pi^+}<2.04 \, \mathrm{GeV} $ and $ 0<m_{\pi^-\pi^+}<1.06\; \mathrm{GeV} $, which are dominated by the $ \bar{K}^*_0(700)^0 $, $ \bar{K}^*(892)^0 $, $ \bar{K}^*(1410)^0 $, $ \bar{K}^*_0(1430) $ and $ \bar{K}^*(1680)^0 $, and $ f_0(500) $, $ \rho^0(770) $ , $ \omega(782) $ and $ f_0(980) $ resonances, respectively. The predicted results are $ \mathcal{A_{CP}}(\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-) = [-0.365,0.447] $ and $ \mathcal{B}(\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-) = [6.11,185.32]\times10^{-8} $. In our analysis, the errors come from the uncertainties of the form factors, decay constants, Gegenbauer moments and divergence parameters. These theoretical predictions await the test in the future examinations with high precision. If our predictions are confirmed, the viewpoint that scalars have the $ q\bar{q} $ composition may be supported. However, to exclude other possible structures, more investigations will be needed due to uncertainties from both theory and experiments.

    APPENDIX A: FOUR-BODY DECAY AMPLITUDES
    • Considering the related weak and strong decays, one can obtain the four-body decay amplitudes of the $ \bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^+\pi^- $ channels as the following:

      $\tag{A1} \begin{aligned}[b] \mathcal{M}(\bar{B}^0\rightarrow \bar{K}^{*0}_{0i}\rho\rightarrow K^-\pi^+\pi^+\pi^- ) =& \frac{{\rm i}G_Fg_{\bar{K}^{*0}_{0i}K\pi}g_{\rho\pi\pi}}{S_{\bar{K}^{*0}_{0i}}S_\rho }\bigg[(P\cdot N)+(L\cdot N)+\frac{1}{m_{\rho}^2}(L\cdot P+L^2)(L\cdot N)\bigg]\\ &\times\sum_{p = u,c}\lambda_p^{(s)}\bigg\{\bigg[\delta_{pu}\alpha_2(\bar{K}^{*0}_{0i}\rho) +\frac{3}{2}\alpha_{3,EW}^p(\bar{K}^{*0}_{0i}\rho)\bigg] f_\rho m_{\bar{B}^0}p_cF_1^{\bar{B}^0 \bar{K}^{*0}_{0i}}(m_{\rho}^2)\\& +\bigg[\alpha_4^p(\rho\bar{K}^{*0}_{0i})-\frac{1}{2}\alpha_{4,EW}^p(\rho\bar{K}^{*0}_{0i})\bigg] \bar{f}_{\bar{K}^{*0}_{0i}}m_{\bar{B}^0}p_cA_0^{\bar{B}^0\rho}(m_{\bar{K}^{*0}_{0i}}^2)\\& +\bigg[\frac{1}{2}b_3^p(\rho\bar{K}^{*0}_{0i})-\frac{1}{4}b_{3,EW}^p(\rho\bar{K}^{*0}_{0i})\bigg] \frac{f_{\bar{B}^0}f_\rho \bar{f}_{\bar{K}^{*0}_{0i}}m_{\bar{B}^0}p_c}{m_\rho}\bigg\},\end{aligned} $

      $\tag{A2} \begin{aligned}[b] \mathcal{M}(\bar{B}^0\rightarrow \bar{K}^{*0}_{0i}\omega\rightarrow K^-\pi^+\pi^+\pi^- ) =& \frac{{\rm i}G_Fg_{\bar{K}^{*0}_{0i}K\pi}g_{\omega\pi\pi}}{S_{\bar{K}^{*0}_{0i}}S_\omega}\bigg[(P\cdot N)+(L\cdot N)+\frac{1}{m_{\omega}^2}(L\cdot P+L^2)(L\cdot N)\bigg]\\& \times\sum_{p = u,c}\lambda_p^{(s)}\bigg\{\bigg[\delta_{pu}\alpha_2(\bar{K}^{*0}_{0i}\omega)+2\alpha_3^p(\bar{K}^{*0}_{0i}\omega) +\frac{1}{2}\alpha_{3,EW}^p(\bar{K}^{*0}_{0i}\omega)\bigg]\\& \times f_\omega m_{\bar{B}^0}p_cF_1^{\bar{B}^0 \bar{K}^{*0}_{0i}}(m_{\omega}^2) +\bigg[\frac{1}{2}\alpha_{4,EW}^p(\omega\bar{K}^{*0}_{0i})-\alpha_4^p(\omega\bar{K}^{*0}_{0i})\bigg]\bar{f}_{\bar{K}^{*0}_{0i}}m_{\bar{B}^0}p_c\\& \times A_0^{\bar{B}^0\omega}(m_{\bar{K}^{*0}_{0i}}^2) +\bigg[\frac{1}{4}b_{3,EW}^p(\omega\bar{K}^{*0}_{0i})-\frac{1}{2}b_3^p(\omega\bar{K}^{*0}_{0i})\bigg] \frac{f_{\bar{B}^0}f_\rho \bar{f}_{\bar{K}^{*0}_{0i}}m_{\bar{B}^0}p_c}{m_\omega}\bigg\},\end{aligned} $

      and

      $\tag{B3} \begin{aligned}[b] \mathcal{M}(\bar{B}^0\rightarrow \bar{K}^{*0}_if_{0j}\rightarrow K^-\pi^+\pi^+\pi^-) = &-\frac{{\rm i}G_Fg_{\bar{K}^{*0}_iK\pi}g_{f_{0j}\pi\pi}}{S_{\bar{K}^{*0}_i}S_{f_{0j}}} \bigg[-(P\cdot Q)-(L\cdot Q)+\frac{1}{{m_{\bar{K}^{*0}_i}}^2}(P^2+P\cdot L)(P\cdot Q)]\bigg]\\&\times \sum_{p = u,c}\lambda_p^{(s)}\bigg\{\bigg[\delta_{pu}\alpha_2(\bar{K}^{*0}_if_{0j})+2\alpha_3^p(\bar{K}^{*0}_if_{0j}) +\frac{1}{2}\alpha_{3,EW}^p(\bar{K}^{*0}_if_{0j})\bigg]\\& \times \bar{f}_{f_{0j}^n} m_{\bar{B}^0}p_cA_0^{\bar{B}^0 \bar{K}^{*0}_i}(m_{f_{0j}}^2)+\bigg[\sqrt{2}\alpha_3^p(\bar{K}^{*0}_if_{0j})+\sqrt{2}\alpha_4^p(\bar{K}^{*0}_if_{0j})\\ &-\frac{1}{\sqrt{2}}\alpha_{3,EW}^p(\bar{K}^{*0}_i\sigma)-\frac{1}{\sqrt{2}}\alpha_{4,EW}^p(\bar{K}^{*0}_i\sigma)\bigg]\bar{f}_{\sigma^s} m_{\bar{B}^0}p_cA_0^{\bar{B}^0\bar{K}^{*0}_i}(m_{f_{0j}}^2)\\& +\bigg[\frac{1}{2}\alpha_{4,EW}^p(f_{0j}\bar{K}^{*0}_i)-\alpha_4^p(f_{0j}\bar{K}^{*0}_i)\bigg]f_{\bar{K}^{*0}_i}m_{\bar{B}^0}p_c F_1^{\bar{B}^0f_{0j}}(m_{\bar{K}^{*0}_i}^2)\\& +\bigg[\frac{1}{\sqrt{2}}b_3^p(\bar{K}^{*0}_if_{0j})-\frac{1}{2\sqrt{2}}b_{3,EW}^p(\bar{K}^{*0}_if_{0j})\bigg] \frac{f_{\bar{B}^0}f_{\bar{K}^{*0}_i}\bar{f}_{f_{0j}}^s m_{\bar{B}^0}p_c}{m_{\bar{K}^{*0}_i}}\\& +\bigg[\frac{1}{2}b_3^p(f_{0j}\bar{K}^{*0}_i)-\frac{1}{4}b_{3,EW}^p(f_{0j}\bar{K}^{*0}_i)\bigg]\frac{f_{\bar{B}^0}f_{\bar{K}^{*0}_i}\bar{f}_{f_{0j}}^nm_{\bar{B}^0}p_c}{m_{\bar{K}^{*0}_i}}\bigg\}.\end{aligned} $

    APPENDIX B: DYNAMICAL FUNCTIONS FOR THE CORRESPONDING RESONANCES

      B.1.   BUGG MODEL

    • We adopt the Bugg model [37] to parameter the $ \sigma $ resonance

      $\tag{B1} T_R(m_{\pi\pi}) = 1/[M^2-s_{\pi\pi}-g_1^2(s_{\pi\pi})\frac{s_{\pi\pi}-s_A}{M^2-s_A}z(s_{\pi\pi})-{\rm i}M\Gamma_{\mathrm{tot}}(s_{\pi\pi})], $

      where $ z(s_{\pi\pi}) = j_1(s_{\pi\pi})-j_1(M^2) $ with $j_1(s_{\pi\pi}) = \dfrac{1}{\pi}\bigg[2+ \rho_1 \times \ln\bigg(\dfrac{1-\rho_1}{1+\rho_1}\bigg)\bigg]$, $ \Gamma_{\mathrm{tot}}(s_{\pi\pi}) = \displaystyle\sum\limits _{i = 1}^4 \Gamma_i(s_{\pi\pi}) $ with

      $\tag{B2} \begin{aligned}[b] M\Gamma_1(s_{\pi\pi}) =& g_1^2(s_{\pi\pi})\frac{s_{\pi\pi}-s_A}{M^2-s_A}\rho_1(s_{\pi\pi}),\\ M\Gamma_2(s_{\pi\pi}) =& 0.6g_1^2(s_{\pi\pi})(s_{\pi\pi}/M^2)\mathrm{exp}(-\alpha|s_{\pi\pi}-4m_K^2|)\rho_2(s_{\pi\pi}),\\ M\Gamma_3(s_{\pi\pi}) =& 0.2g_1^2(s_{\pi\pi})(s_{\pi\pi}/M^2)\mathrm{exp}(-\alpha|s_{\pi\pi}-4m_\eta^2|)\rho_3(s_{\pi\pi}),\\ M\Gamma_4(s_{\pi\pi}) =& Mg_4\rho_{4\pi}(s_{\pi\pi})/\rho_{4\pi}(M^2), \end{aligned}$

      and

      $\tag{B3} \begin{aligned}[b] g_1^2(s_{\pi\pi}) =& M(b_1+b_2s)\mathrm{exp}[-(s_{\pi\pi}-M^2)/A],\\ \rho_{4\pi}(s_{\pi\pi}) =& 1.0/[1+\mathrm{exp}(7.082-2.845s_{\pi\pi})].\end{aligned}$

      In the above two formulas, the relevant parameters are specifically fixed as $ M = 0.953\;\mathrm{GeV} $, $ g_{4\pi} = 0.011\; \mathrm{GeV} $, $ s_A = 0.14m_\pi^2 $, $ A = 2.426\; \mathrm{GeV}^2 $, $ b_1 = 1.302 \;\mathrm{GeV}^2 $, $ b^2 = 0.340 $ in Ref. [37]. The phase-space factors parameters $ \rho_1 $, $ \rho_2 $ and $ \rho_3 $ have the following forms

      $ \tag{B4} \rho_i(s_{\pi\pi}) = \sqrt{1-4\frac{m_i^2}{s_{\pi\pi}}}, $

      with $ m_1 = m_\pi $, $ m_2 = m_K $ and $ m_3 = m_\eta $.

    • B.2.   THE GOUNARIS-SAKURAI FUNCTION

    • In the framework of the Gounaris-Sakurai model which includes an analytic dispersive term, the propagator of the $ \rho^0(770) $ resonance can be expressed as [38]

      $ \tag{B5} T_R(m_{\pi\pi}) = \frac{1+D\Gamma_0/m_0}{m_0^2-s_{\pi\pi}+f(m_{\pi\pi})-{\rm i}m_0\Gamma(m_{\pi\pi})}, $

      where $ m_0 $ and $ \Gamma_0 $ are the the mass and decay width of the $ \rho^0(770) $ meson, respectively, $ f(m_{\pi\pi}) $ is given by

      $ \tag{B6} f(m_{\pi\pi}) = \Gamma_0\frac{m_0^2}{q_0^3}\left[q^2\left[h(m_{\pi\pi})-h(m_0)\right]+(m_0^2-m_{\pi\pi}^2)q^2_0\frac{\mathrm{d}h}{\mathrm{d}m_{\pi\pi}^2}\bigg|_{m_0}\right], $

      where $ q_0 $ is the value of $ q = |\vec{q}| $ when the mass of the $ \pi\pi $ pair satisfy $ m_{\pi\pi} = m_{\rho^0(770)} $, with

      $ \tag{B7} h(m_{\pi\pi}) = \frac{2}{\pi}\frac{q}{m_{\pi\pi}}\log\bigg(\frac{m_{\pi\pi}+2q}{2m_\pi}\bigg),$

      $\tag{B8} \frac{\mathrm{\rm d}h}{\mathrm{\rm d}m_{\pi\pi}^2}\bigg|_{m_0} = h(m_0)\left[(8q_0^2)^{-1}-(2m_0^2)^{-1}\right]+(2\pi m_0^2)^{-1}. $

      In Eq. (B5), the concrete form of the constant parameter D is

      $ \tag{B9} D = \frac{3}{\pi}\frac{m_\pi^2}{q_0^2}\log\bigg(\frac{m_0+2q_0}{2m_\pi}\bigg)+\frac{m_0}{2\pi q_0}-\frac{m_\pi^2 m_0}{\pi q_0^3}. $

    • B.3.   FLATTÉ MODEL

    • In Refs. [39, 44], when study the $ f_0(980) $ resonance, we can use the Flatté model to dealing with it, which has the following form

      $\tag{B10} T_R(m_{\pi\pi}) = \frac{1}{m_R^2-s_{\pi\pi}-{\rm i}m_R(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}F_{KK}^2\rho_{KK})}, $

      where $ m_R $ is the mass of the $ f_0(980) $ meson, $ g_{\pi\pi} $ (or $ g_{KK} $ )is the coupling constants of the $ f_0(980) $ resonance decay to $ \pi^+\pi^- $ (or $ K^+K^- $) pair. Within the Lorentz-invariant phase spaces, the phase-space $ \rho $ factors are given by

      $\tag{B11} \begin{aligned}[b] \rho_{\pi\pi} =& \frac{2}{3}\sqrt{1-\frac{4m_{\pi^\pm}^2}{s_{\pi\pi}}}+\frac{1}{3}\sqrt{1-\frac{4m_{\pi^0}^2}{s_{\pi\pi}}},\\ \rho_{KK} =& \frac{1}{2}\sqrt{1-\frac{4m_{K^\pm}^2}{s_{\pi\pi}}}+\frac{1}{2}\sqrt{1-\frac{4m_{K^0}^2}{s_{\pi\pi}}}.\end{aligned} $

      Compared to the normal Flatté function, a form factor $ F_{KK} = \mathrm{exp}(-\alpha k^2) $ in Eq. (B10) is introduced above the $ KK $ threshold and serves to reduce the $ \rho_{KK} $ factor as $ s_{\pi\pi} $ increases, where k is momentum of each kaon in the $ KK $ rest frame, and $ \alpha = (2.0\pm0.25)\;\mathrm{GeV}^{-2} $ [44]. This parametrization slightly decreases the $ f_0(980) $ width above the $ KK $ threshold. The parameter $ \alpha $ is fixed to be $ 2.0 \;\mathrm{GeV}^{-2} $, which is not very sensitive to the fit.

    • B.4.   LASS MODEL

    • Generally, LASS model can describe the low mass of the $ K^+\pi^- $ resonance, which has been used in theories and experiments widely [40-42] and has been written as

      $ \tag{B12} \begin{aligned}[b] T(m_{K\pi}) =& \frac{m_{K\pi}}{|\vec{q}|\cot\delta_B-{\rm i}|\vec{q}|}\\&+{\rm e}^{2{\rm i}\delta_B}\dfrac{m_0\Gamma_0\frac{m_0}{|q_0|}}{m_0^2-s_{K\pi}^2-{\rm i}m_0\Gamma_0\dfrac{|\vec{q}|}{m_{K\pi}}\dfrac{m_0}{|q_0|}},\end{aligned}$

      where $ m_0 $ and $ \Gamma_0 $ are the mass and width of the $ K_0^*(1430) $ state, respectively, $ |\vec{q_0}| $ is the value of $ |\vec{q}| $ when $ m_{K\pi} = m_{K_0^*(1430)} $, $ |\vec{q}| $ is the momentum vector of the resonance decay product measured in the resonance rest frame, $ \cot\delta_B $ has two terms which is $ \cot\delta_B = \dfrac{1}{a|\vec{q}|}+\dfrac{1}{2}r|\vec{q}| $, with $ a = (3.1\pm1.0)\,\mathrm{GeV}^{-1} $ and $ r = (7.0\pm2.3)\,\mathrm{GeV}^{-1} $ are the scattering length and effective range [42], respectively.

    • B.5.   RELATIVISTIC BREIT-WIGNER

    • We adopt the relativistic Breit-Wigner function to describe the distributions of the $ \bar{K}^*_0(700)^0 $, $ \bar{K}^*(892)^0 $, $ \bar{K}^*(1410)^0 $ and $ \bar{K}^*(1680)^0 $ resonances [43],

      $\tag{B13} T_R(m_{K\pi}) = \frac{1}{M_R^2-s_{K\pi}-{\rm i}M_R\Gamma_{K\pi}} \quad\quad\quad(R = \bar{\kappa},\bar{K}^*), $

      with

      $\tag{B14} \Gamma_{K\pi} = \Gamma_0^R\bigg(\frac{p_{K\pi}}{p_R}\bigg)^{2J+1}\bigg(\frac{M_R}{m_{K\pi}}\bigg)F^2_R, $

      where $ M_R $ and $ \Gamma_0^R $ are the mass and width, respectively, $ m_{K\pi} $ is the invariant mass of the $ K\pi $ pair, $ p_{K\pi}(p_R) $ is the momentum of either daughter in the $ K\pi $ (or R) rest frame, $ F_R $ is the Blatt-Weisskopf centrifugal barrier factor [45], which are listed in Table B1 and depend on a single parameter $ R_r $ which can be adopted $ R_r = 1.5\;\mathrm{GeV}^{-1} $ [46]

      Spin $ F_R $
      0 1
      1 $\dfrac{\sqrt{1+(R_r p_R)^2} }{\sqrt{1+(R_r p_{AB})^2} }$

      Table B1.  Summary of the Blatt-Weisskopf penetration form factors.

    APPENDIX C: $ f_0(500)-f_0(980) $ MIXING
    • Analogous to the $ \eta-\eta' $ mixing, using a $ 2 \times 2 $ rotation matrix, the $ f_0(500)-f_0(980) $ mixing can be parameterized as

      $\tag{C1} \left( \begin{array}{cc} f_0(980)\\ f_0(500)\\ \end{array} \right) = \left( \begin{array}{cc} \cos\varphi_m& \sin\varphi_m \\ -\sin\varphi_m& \cos\varphi_m \end{array} \right ) \left( \begin{array}{cc} f_s\\ f_q\\ \end{array} \right ), $

      where $ f_s\equiv s\bar{s} $ and $ f_q\equiv \dfrac{u\bar{u}+d\bar{d}}{\sqrt{2}} $, $ \varphi_m $ is the mixing angle which has been summarized in the Refs. [18, 47]. However, based on the measurement by the LHCb Collaboration the range of $ \varphi_m $ is $ |\varphi_m|<31^0 $ [48]. In our calculation, we adopt $ |\varphi_m| = 17^0 $ [18].

    APPENDIX D: THEORETICAL INPUT PARAMETERS
    • The predictions obtained in the QCDF approach depend on many input parameters. The values of the Wolfenstein parameters are taken from Ref. [49]: $ \bar{\rho} = 0.117\pm0.021 $, $ \bar{\eta} = 0.353\pm0.013 $.

      For the masses used in $ \bar{B}^0 $ decays, we use the following values except those listed in Table 1 (in $ \mathrm{GeV} $) [49]:

      $\tag{D1} \begin{aligned}[b]& m_u = m_d = 0.0035,\quad m_s = 0.119, \quad m_b = 4.2,\\& m_{\pi^\pm} = 0.14,\quad m_{K^-} = 0.494,\quad m_{\bar{B}^0} = 5.28,\end{aligned} $

      while for the widths we shall use (in units of $ \mathrm{GeV} $) [49]

      $\tag{D2} \begin{aligned}[b]& \Gamma_{\rho\rightarrow\pi\pi} = 0.149,\quad\Gamma_{\omega\rightarrow\pi\pi} = 0.00013,\quad\Gamma_{\sigma\rightarrow\pi\pi} = 0.3,\\& \Gamma_{f_0(980)\rightarrow \pi\pi} = 0.33,\quad \Gamma_{\bar{K}^*(892)^0\rightarrow K\pi} = 0.0487,\\&\Gamma_{\bar{K}^*(1410)^0\rightarrow K\pi} = 0.015,\quad \Gamma_{\bar{K}^*(1680)^0\rightarrow K\pi} = 0.10,\\& \Gamma_{K^*_0(1430)\rightarrow K\pi} = 0.251.\end{aligned} $

      The Wilson coefficients used in our calculations are taken from Refs. [50-53]:

      $ \tag{D3}\begin{aligned}[b]&c_1 = -0.3125, \quad c_2 = 1.1502, \quad c_3 = 0.0174,\\& c_4 = -0.0373,\quad c_5 = 0.0104,\quad c_6 = -0.0459,\\& c_7 = -1.050\times10^{-5},\quad c_8 = 3.839\times10^{-4}, \\& c_9 = -0.0101,\quad c_{10} = 1.959\times10^{-3}. \end{aligned} $

      The following relevant decay constants (in $ \mathrm{GeV} $) are used [17, 54, 55]:

      $\tag{D4} \begin{aligned}[b]& f_{\pi^\pm} = 0.131,\quad f_{\bar{B}^0} = 0.21\pm0.02, \quad f_{K^-} = 0.156\pm0.007, \\& \bar{f}^s_{\sigma} = -0.21\pm0.093,\quad \bar{f}_{\sigma}^u = 0.4829\pm0.076,\\& \bar{f}_{\bar{\kappa}} = 0.34\pm0.02,\quad f_{\rho} = 0.216\pm0.003,\\& f_{\rho}^\perp = 0.165\pm0.009,\quad f_{\omega} = 0.187\pm0.005,\\& f_{\omega}^\perp = 0.151\pm0.009,\quad f_{\bar{K}^*(892)^0} = 0.22\pm0.005,\\&f_{\bar{K}^*(892)^0}^\perp = 0.185\pm0.010,\quad \bar{f}_{\bar{K}^*_0(1430)^0} = -0.300\pm0.030. \\& \bar{f}^s_{f_0(980)} = 0.325\pm0.016,\quad \bar{f}_{f_0(980)}^u = 0.1013\pm0.005.\end{aligned} $

      As for the form factors, we use [17, 33, 55, 56]:

      $\tag{D5} \begin{aligned}[b]& F_0^{\bar{B}^0\rightarrow K}(0) = 0.35\pm0.04,\quad F_0^{\bar{B}^0\rightarrow \sigma}(0) = 0.45\pm0.15,\quad F^{\bar{B}^0\rightarrow\kappa}(0) = 0.3\pm0.1,\\ &A_0^{\bar{B}^0\rightarrow \bar{K}^*(892)^0}(0) = 0.374\pm0.034, \quad F_0^{\bar{B}^0\rightarrow \pi}(0) = 0.25\pm0.03, \quad F_0^{\bar{B}^0\rightarrow \bar{K}^*_0(1430)^0}(0) = 0.21,\\& A_0^{\bar{B}^0\rightarrow \bar{K}^*(1410)^0}(0) = 0.26\pm0.0275, \quad A_0^{\bar{B}^0\rightarrow \bar{K}^*(1680)^0}(0) = 0.2154\pm0.0281 \quad A_0^{\bar{B}^0\rightarrow \rho}(0) = 0.303\pm0.029,\end{aligned} $

      The values of Gegenbauer moments at $ \mu = 1\; \mathrm{GeV} $ are taken from [17, 54, 55],

      $ \tag{D6}\begin{aligned}[b]& \alpha_1^\rho = 0,\quad \alpha_2^\rho = 0.15\pm0.07, \quad \alpha_{1,\perp}^\rho = 0,\quad \alpha_{2,\perp}^\rho = 0.14\pm0.06, \quad \alpha_1^\omega = 0,\quad \alpha_2^\omega = 0.15\pm0.07, \quad \alpha_{1,\perp}^\omega = 0,\quad \alpha_{2,\perp}^\omega = 0.14\pm0.06, \\& \alpha_1^{\bar{K}^*(892)^0} = 0.03\pm0.02,\quad \alpha_{1,\perp}^{\bar{K}^*(892)^0} = 0.04\pm0.03,\quad \alpha_2^{\bar{K}^*(892)^0} = 0.11\pm0.09,\quad \alpha_{2,\perp}^{\bar{K}^*(892)^0} = 0.10\pm0.08,\\& B_{1,\sigma}^u = -0.42\pm0.074,\quad B_{3,\sigma}^u = -0.58\pm0.23,\quad B_{1,\sigma}^s = -0.35\pm0.061,\quad B_{3,\sigma}^s = -0.43\pm0.18,\\& B_{1,f_0(980)}^u = -0.92\pm0.08,\quad B_{3,f_0(980)}^u = -0.74\pm0.064,\quad B_{1,f_0(980)}^s = -1\pm0.05,\quad B_{3,f_0(980)}^s = -0.8\pm0.04,\\& B_{1,\bar{\kappa}} = -0.92\pm0.11,\quad B_{3,\bar{\kappa}} = 0.15\pm0.09,\quad B_{1,\bar{K}^*_0(1430)^0} = 0.58\pm0.07,\quad B_{3,\bar{K}^*_0(1430)^0} = -1.20\pm0.08.\end{aligned} $

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