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.

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

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.

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.

Zhen-Yu Li

Ze Zhou

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

The dineutron as a bound chargeless nucleus of two identical nucleons has been attracting attention for many decades. In addition to the study of the formation of a bound dineutron in the outgoing channel of specific nuclear reactions, cross-section estimates have been gaining interest as research targets. We used a traditional neutron activation technique to irradiate Lu samples followed by measurements of the induced activity with HPGe spectrometer in order to detect gamma-peaks of the 175Lu ( <italic>n</italic>, 2 <italic>n</italic>) 174 gLu reaction product when this reaction channel is not open for incident neutron energies. Based on two measurements of the instrumental gamma-ray spectra of Lu samples we reliably identified the presence of the 174 gLu isotope in the outgoing channel of the 175Lu ( <italic>n</italic>, 2 <italic>n</italic>) 174 gLu nuclear reaction. Also, the cross-section for this nuclear reaction was obtained and to be equal <math altimg='si1.svg'><mrow><mn>33</mn><mo>.</mo><msubsup><mn>5</mn><mrow><mo>&#8722;</mo><mn>6.7</mn></mrow><mrow><mo>+</mo><mn>7.0</mn></mrow></msubsup></mrow></math> mb. For the first time, the dineutron as a nuclearreaction product was indirectly detected via statistically significant observation of the induced activity of 174 gLu with a corresponding cross-section estimate.

Ukraine

Ihor Kadenko

Hungary

Barna Biró

Mihály Braun

András Fenyvesi

Kateryna Okopna

Nadiia Sakhno

Loránd Zakàny

Formation of bound dineutrons in the 175Lu ( <italic>n</italic>, 2 <italic>n</italic>) 174 gLu nuclear reaction and its cross-section

We revisit the proton emitter $^{149}$Lu utilizing the recently developed triaxial relativistic Hartree-Bogoliubov theory in continuum (TRHBc). By incorporating the microscopic nuclear structure properties from the TRHBc theory into the WKB approximation, we successfully reproduce the measured proton-emission half-life of $^{149}$Lu within experimental uncertainties. A triaxial ground state characterized by (<math><mi>β</mi><mo>=</mo><mn>0.17</mn><mo>,</mo><mi>γ</mi><mo>=</mo><msup><mrow><mn>31</mn></mrow><mrow><mo>∘</mo></mrow></msup></math>) has been clarified for $^{149}$Lu. The inclusion of triaxiality significantly changes nuclear density distributions and potentials, which results in enhanced binding of both the nuclear system and the proton-emitting orbital. As a result, a slightly extended half-life for the proton emission of $^{149}$Lu is achieved after considering triaxial deformation degrees of freedom. The examination of the dependence of the decay width on triaxial deformation parameters highlights the significance of microscopic description of nuclear deformation beyond the proton dripline.

Qi Lu

Kai-Yuan Zhang

Shi-Sheng Zhang

Triaxial shape of the one-proton emitter Lu

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

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

One-proton radioactivity in $^{149}$Lu, the latest identified proton emitter, is studied in the Wentzel-Kramers-Brillouin (WKB) approach with the proton-nucleus potential extracted from the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) for the first time. The predicted half-life turns out to be consistent with the experimental measurement within uncertainties and (almost) independent of the density functionals in the DRHBc theory. Such a microscopic self-consistent calculation reveals that $^{149}$Lu is oblately deformed with a quadrupole deformation <math><msub><mrow><mi>β</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≈</mo><mo>−</mo><mn>0.18</mn></math>, and rules out the possibility of a prolate quadrupole deformation suggested in the nonadiabatic quasiparticle model. We also check the validity of this approach in the description of $^{150,151}$Lu and their isomeric states. The deviations of the predicted half-lives from their experimental counterparts are mostly smaller than those of the theoretical studies without considering deformation effects. Furthermore, we predict $^{148}$Lu to be a more oblately deformed proton-emitter with a longer half-life than that of $^{149}$Lu, which can be checked in the future. Our studies show that the DRHBc + WKB approach provides a new alternative method to evaluate the half-lives of well-deformed proton emitters.