<p>In this work, we study the mass spectrum of the hidden-charm hybrid states with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>0</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>0</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>+</mo><mo>−</mo></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>2</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>2</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> via the quantum chromodynamics (QCD) sum rules in a consistent way. We calculate the vacuum condensates up to dimensions-6 by taking account of both the leading order and next-to-leading order contributions, and take the energy scale formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>μ</mi><mo>=</mo><msqrt><mrow><msubsup><mrow><mi>M</mi></mrow><mrow><mi>X</mi><mo>/</mo><mi>Y</mi><mo>/</mo><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mrow><mi mathvariant="double-struck">M</mi></mrow><mrow><mi>c</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></math> to choose the suitable energy scales of the QCD spectral densities, it is the first time to explore the energy scale dependence of the QCD sum rules for the hidden-charm hybrid states.</p>

China

Zhi-Gang Wang

Mass spectrum of the hidden-charm hybrid states via QCD sum rules

We introduce a relative $\textit{P}$-wave to construct the doubly charmed diquark $ (\widetilde{V}) $ vector. Therefore, scalar and tensor four-quark currents were constructed to investigate the decay widths of the fully charmed tetraquark states with $ J^{PC}=0^{++} $ , $ 1^{+-} $ and $ 2^{++} $ via quantum chromodynamics (QCD) sum rules. We observed that the total width of the ground state $ \widetilde{V}\overline{\widetilde{V}} $ -type scalar tetraquark state is compatible with that of the $ X(6552) $ within uncertainties, and the branching ratios are quite different from that of the first radial excitation of the $ A\bar{A} $ -type scalar tetraquark state. Other predictions can be verified in future experiments to shed light on the nature of the fully charmed tetraquark states.

Xiao-Song Yang

Strong decays of the fully charmed tetraquark states with explicit $\textit{P}$-waves via QCD sum rules * Supported by the National Natural Science Foundation (12175068), and the Fundamental Research Funds for the Central Universities

The production of heavy-quark (HQ) jets is a new area that addresses the mass effect of jet quenching in heavy-ion physics. This paper presents a theoretical study of HQ jet yield suppression in Pb+Pb collisions at the Large Hadron Collider (LHC) and focuses on the energy loss of HQ jets produced by different mechanisms. The $\textit{p+p}$ baseline is provided by the generator simulation of high-energy reactions of particles (SHERPA), and the jet-medium interactions are described by the SHELL transport model, which considers the elastic and inelastic partonic energy loss in the quark-gluon plasma (QGP). In $\textit{p+p}$ collisions, our numerical results indicate that the HQ jets from gluon splitting ( $ g \rightarrow Q $ -jet) are the dominant contribution at high $ p_T $ , displaying more dispersive structures than the HQ-initiated ( $ Q \rightarrow Q $ -jet). In nucleus-nucleus collisions, our calculations were consistent with the inclusive and b-jet $ R_{AA} $ recently measured by the ATLAS collaboration, revealing a remarkable manifestation of the mass effect of jet energy loss. As a result of the dispersive substructure, the $ g \rightarrow Q $ -jet loses more energy than the $ Q \rightarrow Q $ -jet in the QGP. Due to the significant contribution of $ g \rightarrow c $ -jet, the $ R_{AA} $ of $\textit{c}$-jet is comparable or even smaller than that of inclusive jet. To experimentally distinguish the $ g \rightarrow Q $ -jet and $ Q \rightarrow Q $ -jet, we propose event selection strategies based on their topological features and test their performances. By isolating the $ c \rightarrow c $ -jet, $ b \rightarrow b $ -jet, and the jets initiated by heavy quarks, we predicted that the order of their $ R_{AA} $ are in line with the mass hierarchy of energy loss. Future measurements on the $ R_{AA} $ of $ Q \rightarrow Q $ -jet and $ g \rightarrow Q $ -jet will provide a unique opportunity for testing the flavor/mass dependence of energy loss at the jet level.

Sa Wang

Shuang Li

Yao Li

Ben-Wei Zhang

Enke Wang

Probing the mass effect of heavy quark jets in high-energy nuclear collisions * Supported by the MajorProject of Basic and Applied Basic Research of Guangdong Province, China (2020B0301030008, 2023A1515011460), and the National Natural Science Foundation of China (11935007, 12035007, 12247127, 12375137). Sa Wang is supported by the Open Foundation of Key Laboratory of Quark and Lepton Physics (MOE) (QLPL2023P01) and the Talent Scientific Star-up Foundation of the China Three Gorges University, China (CTGU) (2024RCKJ013)

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.

Guo-Liang Yu

Zhen-Yu Li

Ze Zhou

Systematic analysis of the mass spectra of triply heavy baryons

Inspired by the first <math altimg='si1.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> observation at the LHCb Collaboration, the spectroscopic properties of the entire isoscalar and isovector <math altimg='si1.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> family are systematically investigated by means of multiple sorts of relativized and nonrelativistic diquark formalisms, which include the Godfrey-Isgur relativized diquark model, the modified Godfrey-Isgur relativized diquark model incorporating the color screening effects, the nonrelativistic diquark model with the Gaussian type hyperfine potential, and the nonrelativistic diquark model with the Yukawa type hyperfine potential. In terms of the 1 <italic>S</italic>-wave double-charm tetraquark state with <math altimg='si2.svg'><mi>I</mi><mo stretchy='false'>(</mo><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi></mrow></msup><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>0</mn><mo stretchy='false'>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>)</mo></math>, the predicted masses of most diquark-antidiquark scenarios are somewhat higher than the observed value of the <math altimg='si3.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub><msup><mrow><mo stretchy='false'>(</mo><mn>3875</mn><mo stretchy='false'>)</mo></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></math> structure. In light of the diquark-antidiquark configuration, this work unveils the mixing angles of the orbitally excited isovector <math altimg='si1.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> states and the magic mixing angles of the ideal heavy-light tetraquarks for the first time. As the advancement of the experimental detection capability, these phenomenological predictions will effectively boost the hunting for the prospective low-lying <math altimg='si1.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> states in the future.

Wen-Chao Dong

Hunting for the prospective <italic>T</italic> <inf><italic>cc</italic></inf> family based on the diquark-antidiquark configuration

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.

Bin Wu

Jie Lu

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 study, we use the optical theorem to calculate the next-to-leading order corrections to the QCD spectral densities directly in the QCD sum rules for the pseudoscalar and scalar B $_{ c }$ mesons. We use experimental data for guidance to perform an updated analysis. We obtain the masses and, in particular, decay constants, which are the fundamental input parameters in high energy physics. Ultimately, we obtain the pure leptonic decay widths, which can be compared with experimental data in the future.

  meson and its scalar cousin with QCD sum rules  

In this study, we chose the diquark-antidiquark type four-quark currents with an explicit P-wave between the diquark and antidiquark pairs to study the ground states and first radial excitations of the hidden-charm tetraquark states with quantum numbers   $J^{PC}=1^{--}$  . We also obtained the lowest vector tetraquark masses and made possible assignments of the existing   $J^{PC}=1^{--}$  states. There indeed exists a hidden-charm tetraquark state with   $J^{PC}=1^{--}$   at an energy of approximately 4.75 GeV as the first radial excitation that accounts for the BESIII data.