<p>In this work, the soft rescattering parameters in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>±</mo></msup><mo stretchy="false">→</mo><msup><mi>π</mi><mo>±</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>±</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>±</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> decays with the light scalar meson <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></math> as the intermediate resonance are studied within the quantum chromodynamics (QCD) factorization. Considering the interference effect between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><msup><mo stretchy="false">)</mo><mn>0</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></math>, we utilize the experimentally more direct event yields for fitting and get the soft rescattering parameters <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msubsup><mi>ρ</mi><mi>k</mi><mrow><mi>S</mi><mi>P</mi></mrow></msubsup><mo stretchy="false">|</mo><mo>=</mo><mn>3.29</mn><mo>±</mo><mn>1.01</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msubsup><mi>ρ</mi><mi>k</mi><mrow><mi>P</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">|</mo><mo>=</mo><mn>2.33</mn><mo>±</mo><mn>0.73</mn></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>P</mi><mi>S</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>S</mi><mi>P</mi></math> decays (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math> denote pseudoscalar and scalar mesons, respectively), respectively. We also study the branching ratios and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> asymmetries in the decay modes involving other scalar mesons, including <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1450</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>K</mi><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1430</mn><mo stretchy="false">)</mo></math>, to test the rationality of the values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msubsup><mi>ρ</mi><mi>k</mi><mrow><mi>S</mi><mi>P</mi></mrow></msubsup><mo stretchy="false">|</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msubsup><mi>ρ</mi><mi>k</mi><mrow><mi>P</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">|</mo></math>. Meanwhile, the wealth of experimental data facilitate the examination of the forward-backward asymmetry induced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> asymmetries (FB-CPAs), and the localized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> asymmetries (LACPs). We investigate these asymmetries resulting from the interference between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><msup><mo stretchy="false">)</mo><mn>0</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>±</mo></msup><mo stretchy="false">→</mo><msup><mi>π</mi><mo>±</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>±</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>±</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> decays when the invariant mass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> locates in the low-energy region <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0.445</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>&lt;</mo><msub><mi>m</mi><mrow><mi>π</mi><mi>π</mi></mrow></msub><mo>&lt;</mo><mn>0.795</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. Our theoretical results of FB-CPAs and LACPs align with the experimental findings. We propose that the interference between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><msup><mo stretchy="false">)</mo><mn>0</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></math> can be extended to other beauty and charmed mesons decays.</p>

China

Jing-Juan Qi

Zhen-Yang Wang

Zhen-Hua Zhang

Ke-Wei Wei

Xin-Heng Guo

Probing the soft rescattering parameters in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math> decays involving a scalar meson with QCD factorization

A mechanism of generating <math><mrow><mi>C</mi><mi>P</mi></mrow></math> violation through the imaginary part of the interference of two amplitudes is proposed. This mechanism has shown clear evidence in decays such as <math><msup><mi>B</mi><mo>±</mo></msup><mo>→</mo><msup><mi>π</mi><mo>±</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math>. The proposed mechanism is helpful in searching for <math><mrow><mi>C</mi><mi>P</mi></mrow></math> violation in bottom and charmed baryon decay processes.

Jian-Yu Yang

<math><mi>C</mi><mi>P</mi></math> asymmetries corresponding to the imaginary parts of the interference terms in cascade decays of heavy hadrons

In this work, we systematically study the heavy baryonium and heavy dibaryon systems using the Bethe-Salpeter equation in the ladder and instantaneous approximations for the kernel. Our results indicate that all the heavy baryonium systems, specifically <math><msub><mi>Λ</mi><mi>Q</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>Q</mi></msub></math>, <math><msub><mi>Ξ</mi><mi>Q</mi></msub><msub><mover><mi>Ξ</mi><mo>¯</mo></mover><mi>Q</mi></msub></math>, <math><msub><mi>Σ</mi><mi>Q</mi></msub><msub><mover><mi>Σ</mi><mo>¯</mo></mover><mi>Q</mi></msub></math>, <math><msubsup><mi>Ξ</mi><mi>Q</mi><mo>′</mo></msubsup><msubsup><mover><mi>Ξ</mi><mo>¯</mo></mover><mi>Q</mi><mo>′</mo></msubsup></math>, and <math><msub><mi>Ω</mi><mi>Q</mi></msub><msub><mover><mi>Ω</mi><mo>¯</mo></mover><mi>Q</mi></msub></math> (<math><mrow><mi>Q</mi><mo>=</mo><mi>c</mi></mrow></math>, <math><mrow><mi>b</mi></mrow></math>), can form bound states. Among the heavy dibaryon systems, only the <math><msub><mi>Ξ</mi><mi>Q</mi></msub><msub><mi>Ξ</mi><mi>Q</mi></msub></math> system with <math><mi>I</mi><mo>=</mo><mn>0</mn></math> and the <math><msub><mi>Σ</mi><mi>Q</mi></msub><msub><mi>Σ</mi><mi>Q</mi></msub></math> systems with <math><mi>I</mi><mo>=</mo><mn>0</mn></math> and <math><mi>I</mi><mo>=</mo><mn>1</mn></math> can exist as bound states. Additionally, the <math><msub><mi>Σ</mi><mi>Q</mi></msub><msub><mover><mi>Σ</mi><mo>¯</mo></mover><mi>Q</mi></msub></math> system with <math><mi>I</mi><mo>=</mo><mn>2</mn></math> and the <math><msub><mi>Σ</mi><mi>Q</mi></msub><msub><mi>Σ</mi><mi>Q</mi></msub></math> system with <math><mi>I</mi><mo>=</mo><mn>1</mn></math> are not deeply bound.

Italy

Possible bound states of heavy baryonium and heavy dibaryon systems

In this work, we investigate possible bound states of the <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub></math> system in the Bethe-Salpeter formalism in the ladder and instantaneous approximations. By numerically solving the Bethe-Salpeter equation, we confirm the existence of <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub></math> bound states with quantum numbers <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>0</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> and <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math>. We further investigate the partial decay widths of the <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub></math> bound states into <math><mi>N</mi><mover><mi>N</mi><mo>¯</mo></mover></math>, <math><mi>D</mi><mover><mi>D</mi><mo>¯</mo></mover></math>, <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math>, <math><msup><mi>D</mi><mo>*</mo></msup><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math>, <math><mi>π</mi><mover><mi>π</mi><mo>¯</mo></mover></math>, and <math><mi>K</mi><mover><mi>K</mi><mo>¯</mo></mover></math>. Our results indicate that the decay width of the <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub></math> bound state with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> is much larger than that with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>0</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math>, and among their decay channels, the <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> final state is the main decay mode. We suggest experiments to search for the <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub></math> bound states in the <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> final state.

Shi-Ji Cao

Masses and decay widths of the <math><mi>S</mi></math>-wave <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub></math> bound states

In the framework of the QCD factorization approach, we study the localized CP violations of the <math><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mi>η</mi></math> and <math><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mi>η</mi></math> decays with and without the <math><msubsup><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>0</mn></mrow></msubsup><mo>(</mo><mn>980</mn><mo>)</mo><mo>−</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>980</mn><mo>)</mo></math> mixing mechanism, respectively, and find that the localized CP violations can be enhanced by this mixing effect when the invariant masses of the ππ, KK and πη pairs in the final states are located around 980 MeV in the aforementioned decays. We suggest that the <math><msubsup><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>0</mn></mrow></msubsup><mo>(</mo><mn>980</mn><mo>)</mo><mo>−</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>980</mn><mo>)</mo></math> mixing mechanism should be considered when studying the CP violation of the B or D meson decays theoretically and experimentally.

Study the Violation of the Quasi-two-body Decays of Meson With the And (980) Mixing

We introduce a set of observables representing angular distribution asymmetries, which can be viewed as a generalization of the forward-backward asymmetry of angular distributions, and can be used as an effective tool to search for     $$C\!P$$  <math> <mrow> <mi>C</mi> <mspace width="-0.166667em"></mspace> <mi>P</mi> </mrow> </math>   violation in three-body decays of bottom and charmed baryons. We propose to search for such     $$C\!P$$  <math> <mrow> <mi>C</mi> <mspace width="-0.166667em"></mspace> <mi>P</mi> </mrow> </math>   asymmetries (1) in decays with     $$\Lambda ^0$$  <math> <msup> <mi>Λ</mi> <mn>0</mn> </msup> </math>  ,     $$\Sigma ^\pm $$  <math> <msup> <mi>Σ</mi> <mo>±</mo> </msup> </math>  , or     $$\Lambda _c^+$$  <math> <msubsup> <mi>Λ</mi> <mi>c</mi> <mo>+</mo> </msubsup> </math>   involved, such as     $$\Lambda _b\rightarrow \Lambda ^0 D$$  <math> <mrow> <msub> <mi>Λ</mi> <mi>b</mi> </msub> <mo>→</mo> <msup> <mi>Λ</mi> <mn>0</mn> </msup> <mi>D</mi> </mrow> </math>   and     $$\Lambda _b\rightarrow \Lambda ^0\rho (770)^0$$  <math> <mrow> <msub> <mi>Λ</mi> <mi>b</mi> </msub> <mo>→</mo> <msup> <mi>Λ</mi> <mn>0</mn> </msup> <mi>ρ</mi> <msup> <mrow> <mo>(</mo> <mn>770</mn> <mo>)</mo> </mrow> <mn>0</mn> </msup> </mrow> </math>  ; and (2) in three-body decays of bottom baryons with opposite parity intermediate resonances involved. Typical examples include     $$\Xi _b^-\rightarrow p K^- K^-$$  <math> <mrow> <msubsup> <mi>Ξ</mi> <mi>b</mi> <mo>-</mo> </msubsup> <mo>→</mo> <mi>p</mi> <msup> <mi>K</mi> <mo>-</mo> </msup> <msup> <mi>K</mi> <mo>-</mo> </msup> </mrow> </math>  ,     $$\Lambda _b^0\rightarrow p K_S\pi ^-$$  <math> <mrow> <msubsup> <mi>Λ</mi> <mi>b</mi> <mn>0</mn> </msubsup> <mo>→</mo> <mi>p</mi> <msub> <mi>K</mi> <mi>S</mi> </msub> <msup> <mi>π</mi> <mo>-</mo> </msup> </mrow> </math>   and     $$\Lambda _b^0\rightarrow p \pi ^0\pi ^-$$  <math> <mrow> <msubsup> <mi>Λ</mi> <mi>b</mi> <mn>0</mn> </msubsup> <mo>→</mo> <mi>p</mi> <msup> <mi>π</mi> <mn>0</mn> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> </mrow> </math>  , in which the last decay channel is used as a toy model to illustrate the basic idea.

Analysis of angular distribution asymmetries and the associated     $$C\!P$$  <math> <mrow> <mi>C</mi> <mspace width="-0.166667em"></mspace> <mi>P</mi> </mrow> </math>   asymmetries in three-body decays of bottom baryons

In the framework of the QCD factorization approach, we study the localized   $ CP $   violations of the   $ CP $   decay with and without the   $ CP $   mixing mechanism and observe that the localized   $ CP $   violation can be enhanced by this mixing effect when the mass of the   $ CP $   pair is in the vicinity of the   $ CP $   resonance. The corresponding theoretical prediction results are   $ CP $   (   $ CP $   with the central value form) and   $ CP $   (   $ CP $   with the central value form), respectively. Meanwhile, we also calculate the branching fraction of the   $ CP $   decay, which is consistent with the experimental results. We suggest that the   $ CP $   mixing mechanism should be considered when theoretically and experimentally studying the   $ CP $   violation of the B or D meson decays.

Xuan Zhang

Chao Wang

Impact of Mixing on the Localized Violations of the Decay  

We interpret the  $$X_1(2900)$$  <math> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>2900</mn> <mo>)</mo> </mrow> </mrow> </math>   as an S-wave  $${\bar{D}}_1K$$  <math> <mrow> <msub> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mn>1</mn> </msub> <mi>K</mi> </mrow> </math>   molecular state in the Bethe–Salpeter equation approach with the ladder and instantaneous approximations for the kernel. By solving the Bethe–Salpeter equation numerically with the kernel containing one-particle-exchange diagrams and introducing three different form factors (monopole, dipole, and exponential form factors) in the verties, we find the bound state exists. We also study the decay width of the decay  $$X_1(2900)$$  <math> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>2900</mn> <mo>)</mo> </mrow> </mrow> </math>   to  $$D^-K^+$$  <math> <mrow> <msup> <mi>D</mi> <mo>-</mo> </msup> <msup> <mi>K</mi> <mo>+</mo> </msup> </mrow> </math>  .

Zhu-Feng Zhang

Studying the  $${\bar{D}}_1K$$  <math> <mrow> <msub> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mn>1</mn> </msub> <mi>K</mi> </mrow> </math>   molecule in the Bethe–Salpeter equation approach

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 the   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   and   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   pair invariant masses are   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   and   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  , with the pairs being dominated by the   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   and   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  , and   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   and   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   resonances, respectively. When dealing with the dynamical functions of these resonances,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   and   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   are modeled with the Bugg model, Gounaris-Sakurai function, Flatté formalism and LASS lineshape, respectively, while the others are described by the relativistic Breit-Wigner function. Adopting the end point divergence parameters   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   and   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  , our predicted results are   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   and   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$     $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  , based on the hypothetical   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   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   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   and all the individual four-body decays   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  , respectively. Our theoretical results for the two-body decays   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$     $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$     $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  ,   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$     $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$  , and   $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$   are consistent with the available experimental data, with the remaining predictions await testing in future high precision experiments.

Phenomenological studies on Decay  

In this work, we study the localized 
					 violation and the branching fraction of the four-body decay 
					 by employing a quasi-two-body QCD factorization approach. Considering the interference of 
					 and 
					 channels, we predict 
					 and 
					, respectively, which shows that the interference mechanism of these two channels can induce the localized 
					 violation to this four-body decay. Meanwhile, within the two quark model framework for the scalar mesons 
					 and 
					, we calculate the direct CP violations and branching fractions of the 
					 and 
					 decays, respectively. The corresponding results are 
					, 
					, 
					 and 
					, indicating that the 
					 violations of these two-body decays are both positive and the branching fractions quite different. These studies provide a new way to investigate the aforementioned four-body decay and can be helpful in clarifying the configuration of the structure of the light scalar meson.
				

Jing Xu

Studying the localized CP violation and the branching fraction of the 
					 decay Supported by National Natural Science Foundation of China (11575023, 11775024, 11947001, 11605150), the Fundamental Research Funds for the Provincial Universities of Zhejiang Province and Ningbo Natural Science Foundation (2019A610067)