arXiv subjects

Guo-Liang Yu

Publications and source records attributed to Guo-Liang Yu.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of the two-body strong decays of the hidden-charm pentaquark states in QCD sum rules

In the present work, we study the two-body strong decays of the hidden-charm pentaquark states with the quark content $uudc\bar c$ and the quantum numbers $I(J^P)=\frac{1}{2}(\frac{1}{2}^-)$ in the framework of the three-point QCD sum rules. The initial pentaquark states are described by four local diquark-diquark-antiquark type interpolating currents with definite isospin. We construct the three-point correlation functions for the decay channels $P_c\to \eta_c p$, $J/\psi p$, $\Lambda_c\bar D$, $\Lambda_c\bar D^{*}$ and $\Sigma_c\bar D$, and derive the corresponding QCD sum rules for the strong coupling constants. At the hadron side, the correlation functions are expressed in terms of the hadron masses, pole residues, decay constants and strong coupling constants. At the QCD side, they are calculated by carrying out the operator product expansion with the full quark propagators, where the vacuum condensates up to dimension 10 are taken into account. After matching the two representations and performing the double Borel transformations, we extract the strong coupling constants from the selected Lorentz structures. With the obtained coupling constants, we evaluate the partial decay widths and discuss the possible assignments of the corresponding pentaquark states. The numerical results indicate that two of the compact hidden-charm pentaquark states can be related to the $P_c(4312)$ and $P_c(4457)$, respectively, while the other two lower-mass states may be regarded as possible hidden-charm pentaquark candidates to be searched for in future experiments. The present results may be useful for identifying the hidden-charm pentaquark states in future experiments.

hep-ph↗

arXiv2026

Low-lying singly heavy baryon states based on the rigorous calculation with the relativized quark model

In this work, the low-lying $\mathbf{6}_{F}$ singly heavy baryon states with positive-parity are studied in detail in the framework of the relativized quark model by using the improved calculation scheme which has successfully explained the fine structure of the low-lying negative-parity singly heavy baryons. The complete mass spectra of all the singly heavy baryon families obtained in the same framework and calculation scheme are systematically analyzed. The baryon states marked with (mass)$J^{P}$ are obtained by considering the mixing effect rigorously. It is found that the mixing effect in the singly heavy baryons depends on the flavor symmetry of the two light quarks and the baryon parity. The results show that the high-precision calculation can reproduce most of the data perfectly, and the statistical error between the calculated masses and the experimental data is only 6.96 MeV. This confirms the reliability of the improved calculation scheme. The rigorous calculation achieved by the two-step GEM enables us to analyze the detailed behavior of the various strong interaction components within the baryons with a high-precision and discover the truth of the ``soft QCD''. The large amount of data obtained in this work serves as the reliable references for related experimental and theoretical researches.

hep-ph↗

arXiv2026

The semileptonic decays of $\mathcal{B}_{Q_{1}Q_{2}}(\frac{1}{2}^{+})\rightarrow\mathcal{B}_{Q_{1}}^{*}(\frac{3}{2}^{+})$ in QCD sum rules

In the framework of QCD sum rules, we systematically analyze the weak transition process $\mathcal{B}_{Q_{1}Q_{2}}(\frac{1}{2}^{+})\rightarrow\mathcal{B}_{Q_{1}}^{*}(\frac{3}{2}^{+})$. When doing the operator product expansion in the QCD side, we consider the contributions of perturbative part and vacuum condensate terms up to dimension 6. In the phenomenological side, we eliminate the interferences of the low spin states and negative parity states by employing 16 different dirac structures. As an application, these form factors are finally used to analyze the semileptonic decays of $\mathcal{B}_{Q_{1}Q_{2}}(\frac{1}{2}^{+})\rightarrow\mathcal{B}_{Q_{1}}^{*}(\frac{3}{2}^{+})l\nu$, where these decays are driven by the transition processes $c\rightarrow d/s+l^{+}+\nu_{l}$ and $b\rightarrow u+l^{-}+\overline{\nu}_{l}$. The predicted physical quantities include not only the partial widths, ratios of $\Gamma_{L}/\Gamma_{T}$ and the branching fractions, but also some observables such as the forward-backward asymmetry parameter $A_{FB}^{l}$ of lepton, the $P_z^{F}$ component of the polarization vector for daughter baryon and the longitudinal polarization of the lepton $P_z^{l}$. We hope all of these theoretical predictions about the weak decays will be helpful for studying the properties of doubly heavy baryons in experiments in the future.

hep-ph↗

arXiv2026

Spin-dependent interactions and fine structure in the negative-parity singly heavy baryons

In order to meet the high-precision measurement of the current baryon spectroscopy, for the first time, we rigorously analyze the spin-dependent interactions and the fine structure of the negative-parity singly heavy baryons in the relativized quark model, which was previously unfeasible in the three-quark system. This gains access to the exploration of the strong interactions dominated by the non-perturbative QCD, and reveals how the various forms of strong interactions in a baryon compete with each other, affect the evolution of the energy levels, cause the energy level splitting and contribute to the mixing effect responsible for the formation of the fine structures. It is shown that the rigorous calculation can perfectly reproduce the data, the averaged deviation between the calculated and experimental energy levels is less than 5 MeV for the negative-parity singly heavy baryons. Therefore, the theoretical precision has reached the experimental high precision. This confirms the reliability of the calculation and also helps make reasonable assignments for the observed negative-parity baryons. The large amount of data obtained by the rigorous calculations contains a wealth of interaction information and is helpful for both of the theoretical and experimental studies. The key to the rigorous calculation in this work is the proposal of a new method, namely the two-step Gaussian expansion method. This new method not only overcomes the long-standing unresolved problem in the relativized quark model, but also provides an effective approach for the high-precision calculation of other few-body systems such as the compact tetraquarks and pentaquarks, especially for the treatment of spin-orbit interactions and tensor interactions which actually appear ubiquitously in all of quantum many-body systems.

hep-ph↗

arXiv2026

The $\Lambda_c\Sigma_c$ dibaryon and $\bar{\Lambda}_c\Sigma_c\pm \bar{\Sigma}_c\Lambda_c$ baryonium states via the QCD sum rules

In the present work, we study the $\Lambda_c\Sigma_c$ dibaryon and $\bar{\Lambda}_c\Sigma_c\pm \bar{\Sigma}_c\Lambda_c$ baryonium states via the QCD sum rules. We construct four (eight) currents with definite $J^P$ ($J^{PC}$) to interpolate the dibaryon (baryonium) states and obtain twelve QCD sum rules. For the dibaryon states, the state with the $J^P=1^+$ lies below the $\Lambda_c\Sigma_c$ threshold, and is a molecule candidate. For the baryonium states, the states with the $J^P=0^{-\pm}$ and $1^{-\pm}$ lie below the $\bar{\Lambda}_c\Sigma_c$ threshold, and are four molecule candidates. The present predictions can be confronted to the experimental data in the future and make contributions to the exotic spectroscopy.

hep-ph↗

arXiv2026

Analysis of the semileptonic decays $\Sigma_b\to\Sigma_cl\bar{\nu}_l$, $\Xi'_b\to\Xi'_cl\bar{\nu}_l$ and $\Omega_b\to\Omega_cl\bar{\nu}_l$ in QCD sum rules

In this article, the electroweak transition form factors of $\Sigma_b\to\Sigma_c$, $\Xi'_b\to\Xi'_c$ and $\Omega_b\to\Omega_c$ are analyzed within the framework of three-point QCD sum rules. In phenomenological side, all possible couplings of interpolating current to hadronic states are considered, and the Dirac structure dependence on the form factors is systematically eliminated. In QCD side, our calculation incorporates both the perturbative part and the contributions from vacuum condensates up to dimension 8. This systematic inclusion of higher-dimensional terms accounts for a broader set of Feynman diagrams, thereby enhancing the comprehensiveness and reliability of the operator product expansion. Using the obtained form factors, we study the partial widths of semileptonic decays $\Sigma_b\to\Sigma_cl\bar{\nu}_l$, $\Xi'_b\to\Xi'_cl\bar{\nu}_l$ and $\Omega_b\to\Omega_cl\bar{\nu}_l$ ($l=e$, $\mu$ and $\tau$). The results indicate that these decay widths approximately satisfy SU(3) flavor symmetry. Next, we calculate the branching ratios for the decay process $\Omega_b\to\Omega_cl\bar{\nu}_l$ and compare them with the results from other collaborations. Furthermore, the lepton universality ratios and some asymmetry parameters of these decay processes are also analyzed, which provide information for the study of new physics. We hope that these results will serve as a useful reference for future theoretical and experimental studies of weak decays involving heavy flavor baryons.

hep-ph↗

arXiv2026

Analysis of the semileptonic decays of $\Xi_{cc}$ and $\Omega_{cc}$ baryons in QCD sum rules

We firstly carry out a systematic analysis on the spin $\frac{1}{2}^{+}\rightarrow\frac{3}{2}^{+}$ weak transition process in the framework of three-point QCD sum rules, where the initial and final states are doubly and singly charmed baryons. In the phenomenological side, all possible couplings of interpolating current to hadronic states are considered. In doing operator production expansion at QCD side, the contributions of the perturbative part, vacuum condensate terms of $\langle{\bar qq}\rangle$, $\langle g_{s}^{2}GG\rangle$, $\langle \bar q g_{s}\sigma Gq\rangle$ and $g_{s}^{2}\langle{\bar qq}\rangle^{2}$ are all considered. After the form factors in space-like region ($Q^2>0$) are obtained, the numerical results are extrapolated into time-like region ($Q^2<0$) by a fitting function. Using the predicted form factors, we finally analyze the semileptonic decays of $\Xi_{cc}^{++}\rightarrow \Sigma_{c}^{*+}l^{+}\nu_{l}$, $\Xi_{cc}^{++}\rightarrow \Xi_{c}^{\prime*+}l^{+}\nu_{l}$, $\Omega_{cc}^{+}\rightarrow\Xi_{c}^{\prime*0}l^{+}\nu_{l}$ and $\Omega_{cc}^{+}\rightarrow \Omega_{c}^{*0}l^{+}\nu_{l}$ with $l=e,\mu$. The predictions in this work can deepen our understanding of the dynamics in the decay processes of doubly heavy baryons and provide useful information to explore the possibility of new physics in heavy baryonic decay channels.

hep-ph↗

arXiv2025

Analysis of the strong vertices of hadronic molecules $DK$, $D^K$, $DK^$ and their bottom analogs

In this work, we analyze the strong vertices of hadronic molecules $DK$, $D^*K$, $DK^*$ and their bottom analogs within the framework of three-point QCD sum rules. The coupling between interpolating currents and low spin particles is considered in the phenomenological side, and the vacuum condensates $\left\langle \bar qq \right\rangle ,\left\langle g_s^2GG \right\rangle ,\left\langle \bar qg_s\sigma Gq \right\rangle ,\left\langle g_s^3GGG \right\rangle ,{\left\langle \bar qq \right\rangle ^2}$ are included in the QCD side. As an application of strong coupling constants, we also obtain the partial decay widths of these states, where $\Gamma_{T_{DK}\rightarrow D_s\pi}=10.3_{-2.9}^{+3.0}\mathrm{KeV}$, $\Gamma_{T_{D^*K}\rightarrow D_s^*\pi}=24.8_{-5.4}^{+5.5}\mathrm{KeV}$, $\Gamma_{T_{BK}\rightarrow BK}=101_{-21}^{+36}\mathrm{MeV}$ and $\Gamma_{T_{B^*K}\rightarrow B^*K}=142_{-25}^{+52}\mathrm{MeV}$. It is shown that the results of $\Gamma_{T_{DK}\rightarrow D_s\pi}$ and $\Gamma_{T_{D^*K}\rightarrow D_s^*\pi}$ are compatible with the experimental data of $D_{s0}^*(2317)$ and $D_{s1}(2460)$.

hep-ph↗

arXiv2025

Systematic analysis of the transition form factors of $B_c$ to $D$-wave charmonia and corresponding semileptonic decays

In this article, the vector, axial vector and tensor form factors of $B_c$ to $D$-wave charmonia $\psi_J$ ($J=1,2,3$) and $\eta_{c2}$ are analyzed within the framework of three-point QCD sum rules. With the estimated vector and axial vector form factors, we study the decay widths and branching ratios of semileptonic decays $B_c\to\psi_Jl\bar{\nu_l}$ and $B_c\to\eta_{c2}l\bar{\nu_l}$. Our results indicate that the semileptonic decay branching ratios of $B_c$ to $D$-wave charmonia decreases as the total angler momentum of the final state $D$-wave charmonia increases. For decay processes $B_c\to \psi_1l\bar{\nu}_l$, $B_c\to \psi_2[\eta_{c2}]l\bar{\nu}_l$ and $B_c\to \psi_3l\bar{\nu}_l$, the branching ratios can reach the order $10^{-3}$, $10^{-4}$ and $10^{-5}$, respectively, which can provide a valuable reference for future experimental measurements for $B_c$ meson by LHCb collaboration. Furthermore, these results also can provide useful information to study the properties of $B_c$ meson and $D$-wave charmonia.

hep-ph↗

arXiv2025

Analysis of the semileptonic decays $\Lambda_b\to\Lambda_cl\bar{\nu}_l$ and $\Xi_b\to\Xi_cl\bar{\nu}_l$ in QCD sum rules

In this article, the electroweak transition form factors of $\Lambda_b\to\Lambda_c$ and $\Xi_b\to\Xi_c$ are analyzed within the framework of three-point QCD sum rules. In phenomenological side, all possible couplings of interpolating current to hadronic states are considered. In QCD side, the perturbative part and the contributions of vacuum condensates up to dimension 8 are also included. With the estimated form factors, we study the decay widths and branching ratios of semileptonic decays $\Lambda_b\to\Lambda_cl\bar{\nu}_l$ and $\Xi_b\to\Xi_cl\bar{\nu}_l$ ($l=e,\mu$ and $\tau$). Our results for the branching ratios of $\Lambda_b\to\Lambda_cl\bar{\nu}_l$ are comparable with experimental data and the results from other collaborations. In addition, our prediction for the branching ratio of $\Xi_b\to\Xi_cl\bar{\nu}_l$ can provide a valuable reference for future experimental measurements.

hep-ph↗

arXiv2025

Analysis of the hadronic molecules $DK$, $D^K$, $DK^$ and their bottom analogs with QCD sum rules

In this work, we construct the color-singlet-color-singlet type currents to study the masses and pole residues of charm-strange tetraquark states and their bottom analogs with $J^P$ = $0^+$ and $1^+$ by using two-point QCD sum rules, where the vacuum condensates are considered up to dimension 12. The predicted masses for $DK$, $D^*K$ and $DK^*$ molecular states are $2.322_{ - 0.072}^{ + 0.066}$ GeV, $2.457_{ - 0.068}^{ + 0.064}$ GeV and $2.538_{ - 0.062}^{ + 0.059}$ GeV. These results are consistent well with the experimental data of $D_{s0}(2317)$, $D_{s1}(2460)$ and ${D}_{s1}(2536)$, respectively. The theoretical results for $BK$ and $B^*K$ molecular states are $5.970_{ - 0.064}^{ + 0.061}$ GeV and $6.050_{ - 0.064}^{ + 0.062}$ GeV which are all higher than their own thresholds. Finally, the mass of hadronic molecule $BK^*$ is predicted to be $6.158_{ - 0.063}^{ + 0.061}$ GeV. This value is lower than the threshold of $BK^*$, which implies that it may be a bound hadronic molecular state.

hep-ph↗

arXiv2025

Analysis of the strong vertices $\Lambda_cD^{()}N^(1535)$ and $\Lambda_bB^{()}N^(1535)$ in QCD sum rules

In this article, we firstly analyze the mass and pole residue of negative parity nucleon $N^*(1535)$ within the two-point QCD sum rules. Basing on these results, we continuously study the strong coupling constants of vertices $\Lambda_cDN^*$, $\Lambda_cD^*N^*$, $\Lambda_bBN^*$ and $\Lambda_bB^*N^*$ in the framework of three-point QCD sum rules. At hadron side, all possible couplings of interpolating current to hadronic states are considered. At QCD side, the contributions of vacuum condensate terms $\langle\bar{q}q\rangle$, $\langle g_s^2GG\rangle$, $\langle\bar{q} g_s\sigma Gq\rangle$, $\langle\bar{q}q\rangle^2$ and $g_s^2\langle\bar{q}q\rangle^2$ are also considered. By setting the four momentum of $D^{(*)}[B^{(*)}]$ mesons off-shell, the strong coupling constants in deep space-like regions ($Q^2=-q^2\gg\Lambda_{QCD}^2$) are obtained. Then, the momentum dependent coupling constants in space-like regions are fitted into analytical function $G(Q^2)$ and are extrapolated into time-like regions ($Q^2<0$). Finally, the on-shell values of strong coupling constants are obtained by taking $Q^{2}=-m_{D^{(*)}[B^{(*)}]}^2$. The results are $G_{\Lambda_cDN^*}(Q^2=-m_D^2)=4.06^{+0.96}_{-0.75}$, $f_{\Lambda_cD^*N^*}(Q^2=-m_{D^*}^2)=3.73^{+0.68}_{-0.16}$, $g_{\Lambda_cD^*N^*}(Q^2=-m_{D^*}^2)=9.22^{+3.16}_{-0.36}$, $G_{\Lambda_bBN^*}(Q^2=-m_B^2)=9.11^{+1.54}_{-1.61}$, $f_{\Lambda_bB^*N^*}(Q^2=-m_{B^*}^2)=8.55^{+2.69}_{-2.21}$ and $g_{\Lambda_bB^*N^*}(Q^2=-m_{B^*}^2)=-0.25^{+0.16}_{-0.01}$.

hep-ph↗

arXiv2025

Mass spectra of doubly heavy baryons in the relativized quark model with heavy-quark dominance

In the framework of the relativized quark model, the mass spectra of the doubly heavy baryons are rigorously calculated in the three-quark system under the heavy-quark dominance mechanism, by using the Gaussian expansion method and the infinitesimally-shifted Gaussian basis functions. With the obtained mass spectra of all doubly heavy baryon families, the contribution of each Hamiltonian term to the energy levels is analyzed. It is found that the spin splitting is mainly determined by the spin-dependent interactions associated with the light quark. Moreover, it is shown that the spin splitting evolves regularly with the mass of heavy quarks by the evolution of the spectral structure, which is consistent with the heavy quark symmetry. Meanwhile, the orbital excitation is dominated by the $\rho$-mode, which is different from that of the singly heavy baryons. At last, our analysis indicates that the $\Xi_{cc}^{+}(3520)$ state should not exist truly and the $\Xi_{cc}^{++}(3621)$ should be the true ground state with $J^{P}$ = $\frac{1}{2}^{+}$. It is recommended to design the corresponding experiments to search for the $\Xi_{cc}^{*}$ in the energy range from 3694 to 3714 MeV.

hep-ph↗

arXiv2025

Mass spectra of singly heavy baryons in the relativized quark model with heavy-quark dominance

The rigorous calculation of the spin-orbit terms in the three-quark system is realized based on the Gaussian expansion method and the infinitesimally-shifted Gaussian basis functions in the frame work of the relativized quark model, by ignoring the mixing between different excited states. Then, the complete mass spectra of the singly heavy baryons are obtained rigorously, under the mechanism of the heavy-quark dominance. On these bases, the systematical analyses are carried out for the reliability and predictive power of the model, the fine structure of the singly heavy baryon spectra, the assignments of the excited baryons, and some important topics about the heavy baryon spectroscopy such as the missing states, the `spin-orbit puzzle', the clustering effect, etc. The result confirms that under the heavy-quark dominance mechanism, the relativized quark model can describe the excitation spectra and the fine structures of the singly heavy baryons correctly and precisely.

hep-ph↗

arXiv2025

Systematic analysis of the form factors of $B_{c}$ to $P$-wave charmonia and corresponding weak decays

In this article, the vector, axial vector and tensor form factors of $B_{c}\to \chi_{cJ}$ ($J=0,1,2$) and $B_{c}\to h_{c}$ 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 $B_{c}^{-}\to \chi_{cJ}l \bar{\nu}_l, h_{c}l \bar{\nu}_l$ $(l=e, \mu$ and $\tau)$ and analyze the nonleptonic decays $B_{c}^{-}\to \chi_{cJ}\pi^{-}, \chi_{cJ}K^{-}, \chi_{cJ}\rho^{-}, \chi_{cJ}K^{*-}$, $B_{c}^{-}\to h_{c}\pi^{-}, h_{c}K^{-}, h_{c}\rho^{-}, h_{c}K^{*-}$ by using the naive factorization approach (NFA). These results can provide more information to understand the properties of $B_{c}$ meson and $P$-wave charmonia and to study the heavy quark dynamics.

hep-ph↗

arXiv2025

Systematic analysis of the mass spectra of triply heavy baryons

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($1S\sim4S$, $1P\sim4P$, $1D\sim4D$, $1F\sim4F$ and $1G\sim4G$) 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 ($J$,$M^{2}$) plane are constructed.

hep-ph↗

arXiv2024

Analysis of the form factors of $B_c\rightarrow D^{()}$, $D_{s}^{()}$ and their nonleptonic decays

This article is devoted to calculating the form factors of $B_c \to D^{*}$, $B_c \to D$, $B_c \to D_s^{*}$ and $B_c \to D_s$ transitions in the framework of three-point QCD sum rules. At the QCD side, the contributions of $\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma Gq\rangle$, $\langle g_{s}^{2}G^{2}\rangle$, $\langle f^{3}G^{3}\rangle$ and $\langle\overline{q}q\rangle \langle g_{s}^{2}G^{2}\rangle$ are taken into account. With the obtained form factors, the decay widths and branching ratios of several two-body nonleptonic decay processes $B_c \to \eta_c D^{*}$, $\eta_c D$, $ J/\psi D^{*}$, $ J/\psi D$, $\eta_c D_s^{*}$, $\eta_c D_s$, $J/\psi D_s^{*}$ and $J/\psi D_s$ are predicted. These results about the form factors and decay properties of $B_c$ meson provide useful information for us to study the heavy-quark dynamics.

hep-ph↗

arXiv2024

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

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 $qc\bar{q}\bar{c}$ ($q$=$u/d$ 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 $qc\bar{q}\bar{c}$ ($q$=$u/d$) system, if $Z_{c}(3900)$ is supposed to be a compact tetraquark state with $J^{PC}=1^{+-}$, $Z(4430)$ can be interpreted as the first radially excited states of $Z_{c}(3900)$. Another broad structure $Z_{c}(4200)$ can also be explained as a partner of $Z_{c}(3900)$, and it arise from a higher state with $J^{PC}=1^{+-}$. In addition, theoretical predictions indicate that the possible assignments for $X(3930)$, $X(4050)$ and $X(4250)$ are low lying $0^{++}$ tetraquark states. As for the $sc\bar{s}\bar{c}$ system, $X(4140)$ and $X(4274)$ structures can be interpreted as this type of tetraquark states with $J^{PC}=1^{++}$, and $X(4350)$ can be described as a $sc\bar{s}\bar{c}$ tetraquark with $J^{PC}=0^{++}$. With regard to $qc\bar{s}\bar{c}$ ($q$=$u/d$) system, we find two potential candidates for this type of tetraquark, which are $Z_{cs}(4000)$ and $Z_{cs}(4220)$ structures. The measured masses of these two structures are in agreement with theoretical predictions for the $1^{+}$ state.

hep-ph↗