<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> violation in hadronic multibody decays has been extensively studied, and the experimental breakthrough in the heavy baryon sector was made recently. Partial-wave <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> asymmetries (PWCPAs) in multibody decays of heavy hadrons, which although provide us with more interference information, suffer from the normalization problem, as is pointed out in this paper. We propose a novel solution to this problem. We introduce a set of extra factors to rescale the PWCPAs to the proper sizes. Instead of determining the set of factors according to the normalization requirement, we demand that all the PWCPAs have the same statistical errors. In this way, we obtain a set of quasi-normalized PWCPAs, in the sense that they are close to the ideal normalized ones. As an application, we perform an analysis of PWCPAs in the decay channel <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>. We focus on the phase space region where the invariant mass of the <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> pair varies around the vector resonance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>ρ</mi><mn>0</mn></msup><mo stretchy="false">(</mo><mn>1450</mn><mo stretchy="false">)</mo></math>. Based on the data of the LHCb Collaboration, the interference patterns among the resonances <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>ρ</mi><mn>0</mn></msup><mo stretchy="false">(</mo><mn>1450</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo></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>1500</mn><mo stretchy="false">)</mo></math> and their contributions to the quasinormalized PWCPAs are analyzed. The analysis indicates that the quasi-normalized PWCPAs can avoid potential misleading or distorted results comparing with some other alternatively defined ones.

China

Jing-Juan Qi

Zhen-Yang Wang

Zhen-Hua Zhang

Xin-Heng Guo

Normalization of partial wave <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> asymmetries in three-body decays of heavy hadrons

In this work, we investigate possible bound states in the $ D_s\bar{D}_s $ system using the Bethe-Salpeter (BS) formalism within both the ladder and instantaneous approximations. By numerically solving the Bethe-Salpeter equation with a kernel that incorporates contributions from $\textit{ϕ}$ and $ J/\psi $ meson exchanges, we confirm the existence of a loosely bound state. Furthermore, we explore the partial decay widths of the $ D_s\bar{D}_s $ bound state into the $ D\bar{D} $ , $ \eta_c\eta $ , and $ J/\psi\omega $ channels, and observe that these widths are sensitive to the model parameter $\textit{α}$. Notably, we find the dominant decay channel for the $ D_s\bar{D}_s $ bound state to be $ D\bar{D} $ .

Zhu-Feng Zhang

The properties of the $\textit{S}$-wave D$_{s}$ D $_{s}$ bound state * Supported by the National Natural Science Foundation of China (12405115, 12275024, 12105149)

Within the diquark-antidiquark framework, this study investigates the masses of ground-state tetraquarks composed of heavy charm (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi></math>) and bottom (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi></math>) quarks and antiquarks, using the Bethe-Salpeter formalism. We establish the Bethe-Salpeter equations for the fully heavy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mi>c</mi><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> tetraquarks to the leading order in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>/</mo><msub><mi>m</mi><mi>Q</mi></msub></math> expansion. These equations are subsequently solved numerically under the covariant instantaneous approximation with kernels containing scalar confinement and one-gluon exchange terms. Our results show that the spectra of all possible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>-wave tetraquark states are above the corresponding two lowest meson decay thresholds via the quark rearrangement. This implies that the ground <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mi>c</mi><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> tetraquark states should be broad.

Spectra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mi>c</mi><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> tetraquark states from a diquark-antidiquark perspective

In the perturbative QCD framework, we use the quasi-two-body method to study the CP violations and branching ratios in the $${\bar{B}}_{s} \rightarrow [\phi (\rho ^{0},\omega )\rightarrow K^{+}K^{-}][K^{*0}\rightarrow K^{+}\pi ^{-}]$$ and $${\bar{B}}_{s} \rightarrow [\rho ^{0}(\omega ,\phi )\rightarrow (\pi ^{+}\pi ^{-})][K^{*0}\rightarrow K^{+}\pi ^{-}]$$ decay processes. Owing to the interference effects of $$\rho ^{0}$$ , $$\omega $$ , and $$\phi $$ , new strong phases associated with vector mesons will be generated, resulting in relatively large CP violation within the interference region. Moreover, we provide numerical comparisons of CP violation from resonance effects and non-resonance contributions. In order to provide better theoretical predictions for future experiments, we integrate CP violation over invariant mass to obtain the regional CP violation value. Furthermore, we have calculated the polarization fractions and branching ratios for different intermediate states in the four-body decay process, which may be influenced by vector interference effects. Additionally, this study may provide valuable reference for the future exploration of the LHC experiment.

Jiao Wu

Na Wang

Gang Lü

Study of $$K\pi $$ final states from four-body decay of $$\bar{B}_{s}$$ meson under perturbative QCD

Within the framework of the perturbative QCD approach utilizing $k_T$ factorization, we investigated the CP violation and branching ratios in the decay processes of $B_{c}^{+}\to D_{(s)} ^{+}V(V\rightarrow\pi^{+}\pi^{-})$ and $B_{c}^{+}\to D_{(s)}^{+}V(V\rightarrow K^{+}K^{-})$ , where V denotes the three vector mesons $\rho^0$ , $\textit{ω}$, and $\textit{ϕ}$. During the $V\to \pi^+\pi^-$ and $V\to K^+K^-$ decay processes, we incorporated the $\rho^{0}-\omega-\phi$ mixing mechanism to describe the amplitudes of these quasi-two-body decays. Within the interference region of the three vector particles, we observed distinct changes in both CP violations and branching ratios. Furthermore, our study presents evidence for local CP violations and branching ratios that warrants further experimental investigation.

Kun Shuai Ye

Jian Chai

CP violations and branching ratios for $\textit{B}$ $\textit{c}$ $^{+}$→ $\textit{D}$ $_{($\textit{s}$)}$ $^{+}$ $\textit{π}$ $^{+}$ $\textit{π}$ $^{−}$($\textit{K}$ $^{+}$ $\textit{K}$ $^{−}$) from interference of the vector mesons in Perturbative QCD * Supported by Natural Science Foundation of Henan (252300420319), National Natural Science Foundation of China (12275024).

We analyze the recent data from the BESIII collaboration on the $ X(3872) $ state in the $ J/\psi\pi^+\pi^- $ and $ D^0\bar{D}^0\pi^0 $ decay channels. The quantum number and mass of the $ X(3872) $ state allow us to exploit the universal feature of the very near-threshold $ D\bar D^* $ scattering in the $\textit{S}$ wave. The analysis of $ J/\psi\pi^+\pi^- $ and $ D^0\bar{D}^0\pi^0 $ data separately, as well as the combined analysis of these data together, all support the conclusion that $ X(3872) $ is an extremely weakly bound charm meson molecule.

Xian-Wei Kang

Jin-Zhe Zhang

Nature of $\textit{X}$(3872) from recent BESIII data: considering the universal feature of an $\textit{S}$-wave threshold resonance * Supported by the National Natural Science Foundation of China (12275023)

The direct CP asymmetry in quasi-two-body decays of $ B \rightarrow (V\rightarrow \pi^{+}\pi^{-})P $ is investigated using the perturbative QCD (PQCD) method, where $\textit{P}$ represents a pseudoscalar meson, and $\textit{V}$ refers to $\textit{ρ}$, $\textit{ω,}$ and $\textit{ϕ}$ mesons. We present the amplitude of the quasi-two-body decay process and investigate the effects of mixed resonances involving $ \rho^{0}-\omega $ , $ \rho^{0}-\phi $ , and $ \omega-\phi $ while considering the impact of isospin symmetry breaking. We observe significant CP asymmetry when the invariant mass of the $ \pi^{+}\pi^{-} $ pair is within the resonance ranges of $\textit{ρ}$, $\textit{ω,}$ and $\textit{ϕ}$ mesons. Consequently, we quantify the regional CP asymmetry in these resonance regions. A significant difference is observed when comparing the results obtained with and without the interference of the three vector mesons and isospin conservation. The CP asymmetry results obtained from the three-body decay process, without interference owing to isospin conservation by the PQCD method, are in agreement with the newly updated data acquired by the LHCb experiment.

Wan-Ying Yao

Hai-Feng Ou

CP asymmetry from the effect of isospin symmetry breaking during $\textit{B-}$meson decay * Supported by the Natural Science Foundation of Henan Province, China(252300420319) and the National Natural Science Foundation of China (12275024)

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.

Ke-Wei Wei

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

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

<math><mi>C</mi><mi>P</mi></math> violation in baryon decay processes is still undiscovered to date. We present a general analysis of the decay-angular distributions and the corresponding <math><mi>C</mi><mi>P</mi></math> asymmetries in cascade decays of the type <math><mi>H</mi><mo>→</mo><mi>R</mi><mo>(</mo><mo>→</mo><mi>a</mi><mi>b</mi><mo>)</mo><mi>c</mi></math>, where <math><mi>H</mi></math> is a heavy hadron that decays through weak interactions <math><mi>H</mi><mo>→</mo><mi>R</mi><mi>c</mi></math> and the resonance <math><mi>R</mi></math> decays strongly via <math><mi>R</mi><mo>→</mo><mi>a</mi><mi>b</mi></math>. Based on the analysis, we propose to search for <math><mi>C</mi><mi>P</mi></math> violation in the decay-angular distributions in the cascade decay processes <math><mi>B</mi><mo>→</mo><mi>B</mi><mi>M</mi></math>, with <math><mi>B</mi></math> or <math><mi>M</mi></math> subsequently decaying through strong interactions, where <math><mi>B</mi></math> is the mother baryon, <math><mi>B</mi></math> and <math><mi>M</mi></math> are the daughter baryon and meson, respectively, and <math><mi>M</mi></math> has to be spin nonzero. We also present some typical decay channels in which the search for such kinds of <math><mi>C</mi><mi>P</mi></math> asymmetries can be performed.