<p>In this article, the vector, axial vector, and tensor form factors of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub><mo stretchy="false">→</mo><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>=</mo><mn>0</mn></math>, 1, 2) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub><mo stretchy="false">→</mo><msub><mi>h</mi><mi>c</mi></msub></math> are analyzed within the framework of three-point QCD sum rules. With the calculated vector and axial vector form factors, we directly study the decay widths and branching ratios of semileptonic decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>−</mo></msubsup><mo stretchy="false">→</mo><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub><mi>l</mi><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mi>l</mi></msub><mo>,</mo><msub><mi>h</mi><mi>c</mi></msub><mi>l</mi><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mi>l</mi></msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mi>l</mi><mo>=</mo><mi>e</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>μ</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>τ</mi></math>) and analyze the nonleptonic decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>−</mo></msubsup><mo stretchy="false">→</mo><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub><msup><mi>π</mi><mo>−</mo></msup><mo>,</mo><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub><msup><mi>K</mi><mo>−</mo></msup><mo>,</mo><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub><msup><mi>ρ</mi><mo>−</mo></msup><mo>,</mo><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub><msup><mi>K</mi><mrow><mo>*</mo><mo>−</mo></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>−</mo></msubsup><mo stretchy="false">→</mo><msub><mi>h</mi><mi>c</mi></msub><msup><mi>π</mi><mo>−</mo></msup><mo>,</mo><msub><mi>h</mi><mi>c</mi></msub><msup><mi>K</mi><mo>−</mo></msup><mo>,</mo><msub><mi>h</mi><mi>c</mi></msub><msup><mi>ρ</mi><mo>−</mo></msup><mo>,</mo><msub><mi>h</mi><mi>c</mi></msub><msup><mi>K</mi><mrow><mo>*</mo><mo>−</mo></mrow></msup></math> by using the naive factorization approach. These results can provide more information to understand the properties of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub></math> meson and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math>-wave charmonia and to study the heavy quark dynamics.</p>

China

Jie Lu

Dian-Yong Chen

Guo-Liang Yu

Zhi-Gang Wang

Bin Wu

Systematic analysis of the form factors of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math>-wave charmonia and corresponding weak decays

The mass spectra, root mean square (r.m.s.) radii and radial density distributions of $$\Omega _{ccb}$$ and $$\Omega _{bbc}$$ baryons are firstly analyzed in the present work. The calculations are carried out in the frame work of relativized quark model, where the baryon is regarded as a real three-quark system. Our results show that the excited energy of charmed-bottom triply baryons are always associated with heavier quark. This means the lowest state of $$\Omega _{ccb}$$ baryon is dominated by the $$\lambda $$ -mode, however, the dominant orbital excitation for $$\Omega _{bbc}$$ baryon is $$\rho $$ -mode. In addition, the influence of configuration mixing on mass spectrum, which is induced by different angular momentum assignments, is also analyzed. It shows that energy of the lowest state will be further lowered by this mixing effect. According to this conclusion, we systematically analyze the mass spectra of the ground and excited states( $$1S\sim 4S$$ , $$1P\sim 4P$$ , $$1D\sim 4D$$ , $$1F\sim 4F$$ and $$1G\sim 4G$$ ) of $$\Omega _{ccb}$$ , $$\Omega _{bbc}$$ , $$\Omega _{ccc}$$ and $$\Omega _{bbb}$$ baryons. Finally, with the predicated mass spectra, the Regge trajectories of these heavy baryons in the ($\textit{J}$, $$M^{2}$$ ) plane are constructed.

Zhen-Yu Li

Ze Zhou

Systematic analysis of the mass spectra of triply heavy baryons

We study the transverse momentum ( $p_T$ ) differential anisotropic flow and flow fluctuation in Pb+Pb collisions at $\sqrt{s_{_{NN}}}$ =5.02 TeV at the LHC. A (3+1)-dimensional CLVisc hydrodynamics framework with fluctuating TRENTO (or AMPT) initial conditions is utilized to simulate the space-time evolution of the quark-gluon plasma (QGP) medium. The effects of shear viscosity and the sub-nucleon structure on anisotropic flow and flow fluctuation are analyzed. Our result shows that shear viscosity tends to suppress both flow coefficients ( ${v_2\{2}\}$ , ${v_2\{4\}}$ , and ${\langle v_2\rangle}$ ) and flow fluctuation ( ${\sigma_{v_2}}$ ) owing to its smearing effect on local density fluctuation. The flow coefficients appear to be insensitive to the sub-nucleon structure, whereas the flow fluctuation ${\sigma_{v_2}}$ tends to be suppressed by the sub-nucleon structure in central collisions but enhanced in peripheral collisions. After taking into account the sub-nucleon structure effect, our numerical result can qualitatively describe the relative flow fluctuations ( ${v_2\{4\}/v_2\{2\}}$ , $F({v_2})$ ) measured by the ALICE Collaboration at the LHC. We further investigate the effects of shear viscosity, sub-nucleon structure, and initial condition model on the flow angle and flow magnitude decorrelations ( ${A_2^f}$ , ${M_2^f}$ ) using the four-particle correlation method. We find that the flow decorrelation effect is typically stronger in central collisions than in peripheral collisions. The flow angle decorrelation is found to be insensitive to the shear viscosity and sub-nucleon structure, whereas the flow magnitude decorrelation shows quite different behaviors when using the TRENTO and AMPT initial condition models. Our study sheds light on the anisotropic flow, transport properties, and initial structure of the QGP created in high-energy nuclear collisions.

Germany

Jie Zhu

Canada

Xiang-Yu Wu

Guang-You Qin

Anisotropic flow, flow fluctuation, and flow decorrelation in relativistic heavy-ion collisions: the roles of sub-nucleon structure and shear viscosity * Supported in part by the Natural Science Foundation of China (12225503, 11890710, 11890711, 11935007). Zhu Jie is supported in part by the China Scholarship Council (202306770009). Wu Xiang-Yu is supported in part by the National Science Foundation within the framework of the JETSCAPE collaboration (OAC-2004571) and in part by the Natural Sciences and Engineering Research Council of Canada. Some of the calculations were performed in the Nuclear Science Computing Center at Central China Normal University (NSC$^{3}$), Wuhan, Hubei, China.

In the framework of the relativized quark model, the calculation of spin-orbit interactions is improved by considering the contribution from the light-quark cluster in a singly heavy baryon. It modifies the energy level splitting of the orbital excitation significantly and causes the emergence of fine structures for $$\Sigma _{Q}$$ , $$\Xi '_{Q}$$ and $$\Omega _{Q}$$ baryons. Based on this improvement, we systematically analyze the fine structures and retest the heavy quark dominance mechanism. This mechanism is found to be violated in the 1 -wave states of the $$\Sigma _{c}$$ , $$\Xi '_{c}$$ and $$\Omega _{c}$$ baryons although it remains effective overall, which may help to understand the nature of the heavy quarks and strong interactions. With the predicted fine structures, we make the precise assignments of those observed heavy baryons which once could not be accurately explained due to their close mass values. The method used in this work is instructive and applicable for the study of more complex exotic hadrons, such as the heavy tetraquarks and pentaquarks.

Jian-Zhong Gu

Heavy-quark dominance and fine structure of excited heavy baryons $$\Sigma _{Q}$$ , $$\Xi '_{Q}$$ and $$\Omega _{Q}$$

The form factors of <math><msub><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>→</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></math> and <math><msub><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi></math> are analyzed in the framework of three-point QCD sum rules. In the calculations, the contributions of the vacuum condensate terms <math><mo>〈</mo><msubsup><mrow><mi>g</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>G</mi><mi>G</mi><mo>〉</mo></math> and <math><mo>〈</mo><msubsup><mrow><mi>g</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mi>G</mi><mi>G</mi><mi>G</mi><mi>f</mi><mo>〉</mo></math> are considered. In addition, the decay widths and branching ratios of several decay channels are predicted by using the calculated form factors. These decay processes include the nonleptonic decays <math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>→</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><msup><mrow><mi>K</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msup></math>, <math><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mrow><mi>K</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msup></math>, and the semileptonic decays <math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>→</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><mi>l</mi><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></math>, <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>l</mi><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></math>. These results about the form factors and decay properties of <math><msub><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow></msub></math> meson provide useful information for us to study the heavy-quark dynamics and find new physics beyond Standard Model.

Systematic analysis of the form factors of → , / And Corresponding Weak Decays

Inspired by the great progress in the observations of charmonium-like states in recent years, we perform a systematic analysis about the ground states and the first radially excited states of  $$qc\bar{q}\bar{c}$$  <math> <mrow> <mi>q</mi> <mi>c</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   (q = u/d and s) tetraquark systems. Their mass spectra, root mean square (r.m.s.) radii and radial density distributions are predicted within the framework of relativized quark model. By comparing with experimental data, some potential candidates for hidden-charm tetraquark states are suggested. For  $$qc\bar{q}\bar{c}$$  <math> <mrow> <mi>q</mi> <mi>c</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   (q = u/d) system, if  $$Z_{c}(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>   is supposed to be a compact tetraquark state with  $$J^{PC}=1^{+-}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>1</mn> <mrow> <mo>+</mo> <mo>-</mo> </mrow> </msup> </mrow> </math>  , Z(4430) can be interpreted as the first radially excited states of  $$Z_{c}(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>  . Another broad structure  $$Z_{c}(4200)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>4200</mn> <mo>)</mo> </mrow> </mrow> </math>   can also be explained as a partner of  $$Z_{c}(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>  , and it arise from a higher state with  $$J^{PC}=1^{+-}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>1</mn> <mrow> <mo>+</mo> <mo>-</mo> </mrow> </msup> </mrow> </math>  . In addition, theoretical predictions indicate that the possible assignments for X(3930), X(4050) and X(4250) are low lying  $$0^{++}$$  <math> <msup> <mn>0</mn> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msup> </math>   tetraquark states. As for the  $$sc\bar{s}\bar{c}$$  <math> <mrow> <mi>s</mi> <mi>c</mi> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   system, X(4140) and X(4274) structures can be interpreted as this type of tetraquark states with  $$J^{PC}=1^{++}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>1</mn> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msup> </mrow> </math>  , and X(4350) can be described as a  $$sc\bar{s}\bar{c}$$  <math> <mrow> <mi>s</mi> <mi>c</mi> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   tetraquark with  $$J^{PC}=0^{++}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>0</mn> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msup> </mrow> </math>  . With regard to  $$qc\bar{s}\bar{c}$$  <math> <mrow> <mi>q</mi> <mi>c</mi> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   (q = u/d) system, we find two potential candidates for this type of tetraquark, which are  $$Z_{cs}(4000)$$  <math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>cs</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>4000</mn> <mo>)</mo> </mrow> </mrow> </math>   and  $$Z_{cs}(4220)$$  <math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>cs</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>4220</mn> <mo>)</mo> </mrow> </mrow> </math>   structures. The measured masses of these two structures are in agreement with theoretical predictions for the  $$1^{+}$$  <math> <msup> <mn>1</mn> <mo>+</mo> </msup> </math>   state.

The ground states of hidden-charm tetraquarks and their radial excitations

In this article, we analyze the electromagnetic form factors of the vector heavy-light mesons to the pseudoscalar heavy-light mesons in the framework of three-point QCD sum rules, where the contributions of vacuum condensate terms <math><mo>〈</mo><mover><mrow><mi>q</mi></mrow><mo>‾</mo></mover><mi>q</mi><mo>〉</mo></math>, <math><mo>〈</mo><mover><mrow><mi>q</mi></mrow><mo>‾</mo></mover><msub><mrow><mi>g</mi></mrow><mrow><mi>s</mi></mrow></msub><mi>σ</mi><mi>G</mi><mi>q</mi><mo>〉</mo></math>, <math><mo>〈</mo><msubsup><mrow><mi>g</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>〉</mo></math>, <math><mo>〈</mo><msup><mrow><mi>f</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>G</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>〉</mo></math> and <math><mo>〈</mo><mover><mrow><mi>q</mi></mrow><mo>‾</mo></mover><mi>q</mi><mo>〉</mo><mo>〈</mo><msubsup><mrow><mi>g</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>〉</mo></math> are considered. With these results, we also obtain the radiative decay widths of the vector heavy-light mesons and then compare our results with those of other collaboration's. The final results about the radiative decay widths are <math><mi>Γ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo><mn>0</mn></mrow></msup><mo>→</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><mi>γ</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mn>1.74</mn></mrow><mrow><mo>−</mo><mn>0.37</mn></mrow><mrow><mo>+</mo><mn>0.40</mn></mrow></msubsup></math> keV, <math><mi>Γ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo><mo>+</mo></mrow></msup><mo>→</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>+</mo></mrow></msup><mi>γ</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mn>0.17</mn></mrow><mrow><mo>−</mo><mn>0.07</mn></mrow><mrow><mo>+</mo><mn>0.08</mn></mrow></msubsup></math> keV, <math><mi>Γ</mi><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>→</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow></msub><mi>γ</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mn>0.029</mn></mrow><mrow><mo>−</mo><mn>0.008</mn></mrow><mrow><mo>+</mo><mn>0.009</mn></mrow></msubsup></math> keV, <math><mi>Γ</mi><mo>(</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>⁎</mo><mn>0</mn></mrow></msup><mo>→</mo><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><mi>γ</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mn>0.018</mn></mrow><mrow><mo>−</mo><mn>0.005</mn></mrow><mrow><mo>+</mo><mn>0.006</mn></mrow></msubsup></math> keV, <math><mi>Γ</mi><mo>(</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>⁎</mo><mo>+</mo></mrow></msup><mo>→</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>+</mo></mrow></msup><mi>γ</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mn>0.015</mn></mrow><mrow><mo>−</mo><mn>0.007</mn></mrow><mrow><mo>+</mo><mn>0.007</mn></mrow></msubsup></math> keV and <math><mi>Γ</mi><mo>(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>→</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow></msub><mi>γ</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mn>0.016</mn></mrow><mrow><mo>−</mo><mn>0.005</mn></mrow><mrow><mo>+</mo><mn>0.003</mn></mrow></msubsup></math> keV.

Analysis of the electromagnetic form factors and the radiative decays of the vector heavy-light mesons

A mechanism of the heavy quark dominance in the orbital excitation is proposed in this paper which is testified to be reasonable for singly and doubly heavy baryons. In the relativistic quark model, an analysis of the Hamiltonian figures out the mechanism that the excitation mode with lower energy levels is always associated with the heavy quark(s), and the splitting of the energy levels is suppressed by the heavy quark(s). So, the heavy quarks dominate the orbital excitation of singly and doubly heavy baryons. Furthermore, a physical understanding of this mechanism is given in a semi-classical way. Accordingly, the predicted mass spectra of singly and doubly heavy baryons confirm the rationality of this mechanism. In addition, an interesting consequence of this mechanism is that a heavy-light meson is more likely to be produced in the strong decay of the high-orbital excited states, which is supported by experiments. This mechanism is rooted in the breakdown of the mass symmetry. Therefore, it may be also valid for other multi-quark systems, such as the tetraquarks Qqqq and QQqq, or the pentaquarks Qqqqq and QQqqq.

Heavy quark dominance in orbital excitation of singly and doubly heavy baryons

In this study, the strong coupling constants of vertices   $ BB\Upsilon $  ,   $ BB\Upsilon $  ,   $ BB\Upsilon $  ,   $ BB\Upsilon $   , and   $ BB\Upsilon $   are analyzed in the framework of QCD sum rules. All possible off-shell cases and the contributions of vacuum condensate terms including   $ BB\Upsilon $  ,   $ BB\Upsilon $  ,   $ BB\Upsilon $  ,   $ BB\Upsilon $   , and   $ BB\Upsilon $   are considered. The momentum dependent strong coupling constants are first calculated and then fitted into the analytical function   $ BB\Upsilon $  , which is extrapolated to time-like regions to obtain the final values of strong coupling constants. The final results are   $ BB\Upsilon $  ,   $ BB\Upsilon $   GeV  $ BB\Upsilon $  ,   $ BB\Upsilon $  ,   $ BB\Upsilon $   , and   $ BB\Upsilon $   GeV  $ BB\Upsilon $  . These strong coupling constants are important input parameters that reflect the dynamic properties of the interactions among the mesons and quarkonia.

Strong vertices of bottom mesons And * and Bottomonia ,   

In this work, we analyze the strong vertices  $$\Sigma _{c}\Delta D^{*}$$  <math> <mrow> <msub> <mi>Σ</mi> <mi>c</mi> </msub> <mi>Δ</mi> <msup> <mi>D</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </math>   and  $$\Sigma _{b}\Delta B^{*}$$  <math> <mrow> <msub> <mi>Σ</mi> <mi>b</mi> </msub> <mi>Δ</mi> <msup> <mi>B</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </math>   using the three-point QCD sum rules under the tensor structures  $$i\epsilon ^{\rho \tau \alpha \beta }p_{\alpha }p_{\beta }$$  <math> <mrow> <mi>i</mi> <msup> <mi>ϵ</mi> <mrow> <mi>ρ</mi> <mi>τ</mi> <mi>α</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>p</mi> <mi>α</mi> </msub> <msub> <mi>p</mi> <mi>β</mi> </msub> </mrow> </math>  ,  $$p^{\rho }p'^{\tau }$$  <math> <mrow> <msup> <mi>p</mi> <mi>ρ</mi> </msup> <msup> <mi>p</mi> <mrow> <mo>′</mo> <mi>τ</mi> </mrow> </msup> </mrow> </math>   and  $$p^{\rho }p^{\tau }$$  <math> <mrow> <msup> <mi>p</mi> <mi>ρ</mi> </msup> <msup> <mi>p</mi> <mi>τ</mi> </msup> </mrow> </math>  . We firstly calculate the momentum dependent strong coupling constants  $$g(Q^{2})$$  <math> <mrow> <mi>g</mi> <mo>(</mo> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </math>   by considering contributions of the perturbative part and the condensate terms  $$\langle {\overline{q}}q\rangle $$  <math> <mrow> <mo>⟨</mo> <mover> <mi>q</mi> <mo>¯</mo> </mover> <mi>q</mi> <mo>⟩</mo> </mrow> </math>  ,  $$\langle g_{s}^{2}GG \rangle $$  <math> <mrow> <mo>⟨</mo> <msubsup> <mi>g</mi> <mrow> <mi>s</mi> </mrow> <mn>2</mn> </msubsup> <mi>G</mi> <mi>G</mi> <mo>⟩</mo> </mrow> </math>  ,  $$\langle {\overline{q}}g_{s}\sigma Gq\rangle $$  <math> <mrow> <mo>⟨</mo> <mover> <mi>q</mi> <mo>¯</mo> </mover> <msub> <mi>g</mi> <mi>s</mi> </msub> <mi>σ</mi> <mi>G</mi> <mi>q</mi> <mo>⟩</mo> </mrow> </math>   and  $$\langle {\overline{q}}q\rangle ^{2}$$  <math> <msup> <mrow> <mo>⟨</mo> <mover> <mi>q</mi> <mo>¯</mo> </mover> <mi>q</mi> <mo>⟩</mo> </mrow> <mn>2</mn> </msup> </math>  . By fitting these coupling constants into analytical functions and extrapolating them into time-like regions, we then obtain the on-shell values of strong coupling constants for these vertices. The results are  $$g_{1\Sigma _{c}\Delta D^{*}}=5.13^{+0.39}_{-0.49}\,\hbox {GeV}^{-1}$$  <math> <mrow> <msub> <mi>g</mi> <mrow> <mn>1</mn> <msub> <mi>Σ</mi> <mi>c</mi> </msub> <mi>Δ</mi> <msup> <mi>D</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>.</mo> <msubsup> <mn>13</mn> <mrow> <mo>-</mo> <mn>0.49</mn> </mrow> <mrow> <mo>+</mo> <mn>0.39</mn> </mrow> </msubsup> <mspace width="0.166667em"></mspace> <msup> <mtext>GeV</mtext> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math>  ,  $$g_{2\Sigma _{c}\Delta D^{*}}=-3.03^{+0.27}_{-0.35}\,\hbox {GeV}^{-2}$$  <math> <mrow> <msub> <mi>g</mi> <mrow> <mn>2</mn> <msub> <mi>Σ</mi> <mi>c</mi> </msub> <mi>Δ</mi> <msup> <mi>D</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mn>3</mn> <mo>.</mo> <msubsup> <mn>03</mn> <mrow> <mo>-</mo> <mn>0.35</mn> </mrow> <mrow> <mo>+</mo> <mn>0.27</mn> </mrow> </msubsup> <mspace width="0.166667em"></mspace> <msup> <mtext>GeV</mtext> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math>  ,  $$g_{3\Sigma _{c}\Delta D^{*}}=17.64^{+1.51}_{-1.95}\,\hbox {GeV}^{-2}$$  <math> <mrow> <msub> <mi>g</mi> <mrow> <mn>3</mn> <msub> <mi>Σ</mi> <mi>c</mi> </msub> <mi>Δ</mi> <msup> <mi>D</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mn>17</mn> <mo>.</mo> <msubsup> <mn>64</mn> <mrow> <mo>-</mo> <mn>1.95</mn> </mrow> <mrow> <mo>+</mo> <mn>1.51</mn> </mrow> </msubsup> <mspace width="0.166667em"></mspace> <msup> <mtext>GeV</mtext> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math>  ,  $$g_{1\Sigma _{b}\Delta B^{*}}=20.97^{+2.15}_{-2.39}\,\hbox {GeV}^{-1}$$  <math> <mrow> <msub> <mi>g</mi> <mrow> <mn>1</mn> <msub> <mi>Σ</mi> <mi>b</mi> </msub> <mi>Δ</mi> <msup> <mi>B</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mn>20</mn> <mo>.</mo> <msubsup> <mn>97</mn> <mrow> <mo>-</mo> <mn>2.39</mn> </mrow> <mrow> <mo>+</mo> <mn>2.15</mn> </mrow> </msubsup> <mspace width="0.166667em"></mspace> <msup> <mtext>GeV</mtext> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math>  ,  $$g_{2\Sigma _{b}\Delta B^{*}}=-11.42^{+1.17}_{-1.28}\,\hbox {GeV}^{-2}$$  <math> <mrow> <msub> <mi>g</mi> <mrow> <mn>2</mn> <msub> <mi>Σ</mi> <mi>b</mi> </msub> <mi>Δ</mi> <msup> <mi>B</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mn>11</mn> <mo>.</mo> <msubsup> <mn>42</mn> <mrow> <mo>-</mo> <mn>1.28</mn> </mrow> <mrow> <mo>+</mo> <mn>1.17</mn> </mrow> </msubsup> <mspace width="0.166667em"></mspace> <msup> <mtext>GeV</mtext> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math>   and  $$g_{3\Sigma _{b}\Delta B^{*}}=24.87^{+2.57}_{-2.82}\,\hbox {GeV}^{-2}$$  <math> <mrow> <msub> <mi>g</mi> <mrow> <mn>3</mn> <msub> <mi>Σ</mi> <mi>b</mi> </msub> <mi>Δ</mi> <msup> <mi>B</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mn>24</mn> <mo>.</mo> <msubsup> <mn>87</mn> <mrow> <mo>-</mo> <mn>2.82</mn> </mrow> <mrow> <mo>+</mo> <mn>2.57</mn> </mrow> </msubsup> <mspace width="0.166667em"></mspace> <msup> <mtext>GeV</mtext> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math>  . These strong coupling constants are important parameters which can help us to understand the strong decay behaviors of hadrons.

Analysis of the strong vertices of  $$\Sigma _{c}\Delta D^{*}$$  <math> <mrow> <msub> <mi>Σ</mi> <mi>c</mi> </msub> <mi>Δ</mi> <msup> <mi>D</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </math>   and  $$\Sigma _{b}\Delta B^{*}$$  <math> <mrow> <msub> <mi>Σ</mi> <mi>b</mi> </msub> <mi>Δ</mi> <msup> <mi>B</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> </mrow> </math>   in QCD sum rules