Search for Possible Stable Structures in the Tccq¯s¯ System

In this work, we investigate the resonance structures in the system from both three-quark and five-quark perspectives within the framework of the chiral quark model. An accurate few-body computational approach, the Gaussian expansion method, is employed to construct the orbital wave functions of multiquark states. To reduce the model dependence on parameters, we fit two sets of parameters to check the stability of the results. The calculations show that our results remain stable despite changes in the parameters. In the three-quark calculations, two states are obtained with energies around 1.8 GeV, which are good candidates for the experimentally observed and . In the five-quark configuration, several stable resonance states are identified, including , , and . These resonance states survive the channel-coupling calculations under the complex-scaling framework and manifest as stable structures. Our results support the existence of a two-pole structure for the system, predominantly composed of and configurations, analogous to the well-known - ( - ) system. On the other hand, although the energy of the configuration is close to that of and , the obtained width is not consistent with the experimental values. This suggests that the state needs to mix with three-quark components to better explain the experimental and states. According to our decay width calculations, the predicted two resonance states are primarily composed of and , with their main decay channel being . Therefore, we encourage experimental groups to search for the predicted two-pole structure of the system in the invariant mass spectrum of .

Inspired by the recent observation of a very narrow state, called , by the LHCb collaboration, the possible bound states and low-lying resonance states of the doubly-heavy tetraquark states ( ) and ( ) are searched in the framework of a chiral quark model with an accurate few-body method, the Gaussian expansion method. The real scaling method is also applied to identify the genuine resonance states. In the calculation, the meson–meson structure, diquark–antidiquark structure, and their coupling are all considered. The numerical results show: (i) For and , only states are bound in different quark structures. The binding energy varies from a few MeV for the meson–meson structure to over 100 MeV for the diquark–antidiquark structure. For example, for , in the meson–meson structure, there exists a weakly bound molecule state around 3841.4 MeV, 1.8 MeV below the , which may be a good candidate for the observed state by LHCb; however in the diquark–antidiquark structure, a deeper bound state with mass 3700.9 MeV is obtained. When considering the structure mixing, the energy of system decreases to 3660.7 MeV and the shallow bound state disappears. (ii) Besides bound states, several resonance states for with are proposed.

In this work, we systematically study N(1440), N(1535), and Λ(1405) in both the quenched three-quark and five-quark frameworks using the Gaussian expansion method within the chiral quark model. Our calculations show that N(1535) can be reproduced as a three-quark state [N(1P)], while N(1440) and Λ(1405) cannot be accommodated as the three-quark candidates, [N(2S) and Λ(1P)], respectively. In the five-quark framework, we find that the ΛK state for N(1535) cannot form a bound state, while in the NK¯ channel there will Λ(1405) form a shallow bound state. Based on the complex-scaling method, we performed complete coupled-channel calculations and obtained six resonance states with energies ranging from 1.8 to 2.2 GeV, in addition to one bound state located around Σπ channel. However, neither molecular candidates in ΛK channel for N(1535) nor NK¯ for Λ(1405) are included in these states. This is because the strong coupling between NK¯ and Σπ will make the NK¯ unbound, while the weak coupling between ΛK and ΣK cannot help form a stable structure around ΛK threshold. Thus, under the quenched quark model, our results support N(1535) as a three-quark state, while N(1440) is neither a three-quark nor a five-quark state. In addition, we find that although Λ(1405) can be primarily a five-quark state, it requires a mixture of three- and five-quark components for stability. In the future, an exploration on the mixing effects between bare baryons with these relevant two-body hadronic channel components will be carried out to further test our conclusions.

Stimulated by the newly observed charged hidden-charm state ${Z}_{cs}(3985{)}^{\ensuremath{-}}$ by BESIII Collaboration, ${Z}_{cs}(4000{)}^{+}$, ${Z}_{cs}(4220{)}^{+}$ and the excited ${B}_{s}^{0}$ states by LHCb Collaboration, a full calculation including masses, decay widths, and the inner structures of the states has emerged in the chiral quark model. For ${Z}_{cs}$ states, we assign quantum numbers $I({J}^{P})=\frac{1}{2}({1}^{+})$ and quark composition $c\overline{c}s\overline{u}$ according to the experiment. For ${B}_{s}^{0}$ states, systematically, investigations are performed with $I({J}^{P})=0({0}^{+}),0({1}^{+}),0({2}^{+})$ in both two-body $b\overline{s}$ and four-body $b\overline{s}q\overline{q}$ ($q=u$ or $d$) systems. Each tetraquark calculation takes all structures including meson-meson, diquark-antidiquark, and all possible color configurations into account. Among the numerical techniques to solve the two-body and four-body Schr\"odinger equation, the spatial wave functions are expanded in series of Gaussian basis functions for high precision, which is the way Gaussian expansion method (GEM) so called. Our results indicate that the low-lying states of the four-quark system are all higher than the corresponding thresholds either for $c\overline{c}s\overline{u}$ or for $b\overline{s}q\overline{q}$ systems. With the help of the real scaling method, we found two molecular resonance states with masses of 4023 and 4042 MeV for the $c\overline{c}s\overline{u}$ system. The state $c\overline{c}s\overline{u}(4042)$ has a close mass and decay width with the recent observed state ${Z}_{cs}(3985{)}^{\ensuremath{-}}$. For the $b\overline{s}q\overline{q}$ system with $J=0$, some resonance states are also found. The newly observed excited ${B}_{s}^{0}$ states can be accommodated in the chiral quark model as $2S$ or $1D$ states, and the mixing with four-quark states also needs to be considered.

Since the discovery of $T_{cc}$ by LHCb, there has been considerable interest in $T_{cc}$ and its heavy-flavor partners. However, the study of its strange partner $T_{ss}$ has been largely overlooked. Within the framework of the chiral quark model, we conducted a systematic study of the bound states of $T_{ss}$ utilizing the Gaussian Expansion Method. Considering all physical channels with $01^{+}$, including molecular and diquark structures. Our calculations revealed that upon considering the coupling between diquarks and molecular states, we identified a deep bound state with a bounding energy of 60 MeV, primarily composed of $K K^{*}$. Using the $^3P_0$ model, we calculated the decay width of $K^{*}$ within the $KK^{*}$ bound state, which is approximated as the decay width of the bound state in the $T_{ss}$ system. The results indicate that due to the effect of binding energy, the decay width of $K^{*}$ in $KK^{*}$ is approximately $3$ MeV smaller than that of $K^{*}$ in vacuum. Additionally, resonance state calculations were performed. Utilizing the real-scaling method, we searched for possible resonance states in the $T_{ss}$ sysytem. Due to the strong attraction in the $[K^{*}]_8[K^{*}]_8$ configuration, four resonance states were found in the vicinity of $2.2$-$2.8$ GeV, predominantly featuring hidden-color structures, and their decay widths are all less than $10$ MeV. We strongly recommend experimental efforts to search for the resonance states in the $T_{ss}$ system predicted by our calculations.

This work presents the first prediction of tetralepton resonant states containing muons, extending beyond the simplest tetralepton system, dipositronium (Ps2). With the rapid advancements in experimental facilities, the production and study of these intriguing states may be within reach. We perform a comprehensive analysis of S-wave trilepton and tetralepton systems within the framework of a QED Coulomb potential. We employ the Gaussian expansion method to solve the three- or four-body Schrödinger equation and utilize the complex scaling method to identify resonant states. We uncover a series of bound and resonant states in the trilepton systems e+e+e−, μ+μ+μ−, e+e+μ−, and μ+μ+e−, as well as the tetralepton systems e+e+e−e−, μ+μ+μ−μ−, and μ+μ+e−e−. The energies of these states range from −30 eV to −1 eV below the total mass of three or four leptons, with their widths varying from less than 0.01 eV to approximately 0.07 eV. Additionally, we calculate the spin configurations and root mean square radii of these states, providing insight into their spatial structures. No bound or resonant states are found in the trilepton e+μ+e−, μ+e+μ− systems, nor in the tetralepton μ+e+μ−e− system. A comparison with fully heavy tetraquark systems reveals that the additional color degree of freedom in QCD results in the absence of low-energy bound and resonant states. However, this extra degree of freedom allows for a broader range of JPC quantum numbers to produce resonant states, highlighting the rich complexity of QCD systems. Published by the American Physical Society 2025

We report a theoretical calculation for the bound and S-wave resonance states of the positronic copper atom (e+Cu). A positron is a positively charged particle; therefore, a positronic atom has an attractive correlation between the positron and electron. A Gaussian expansion method is adopted to directly describe this correlation as well as the strong repulsive interaction with the nucleus. The correlation between the positron and electron is much more important than that between electrons in an analogous system of Cu−, although the formation of a positronium (Ps) in e+Cu is not expressed in the ground state structure explicitly. Resonance states are calculated with a complex scaling method and identified above the first excited state of the copper atom. Resonance states below Ps (n = 2) + Cu+ classified to a dipole series show agreement with a simple analytical law. Comparison of the resonance energies and widths of e+Cu with those of e+K, of which the potential energy of the host atom resembles that of e+Cu, reveals that the positions of the resonance for the e+Cu dipole series deviate equally from those of e+K.

We review our calculation method, Gaussian expansion method (GEM), and its applications to various few-body (3- to 5-body) systems such as 1) few-nucleon systems, 2) few-body structure of hypernuclei, 3) clustering structure of light nuclei and unstable nuclei, 4) exotic atoms/molecules, 5) cold atoms, 6) nuclear astrophysics and 7) structure of exotic hadrons. Showing examples in our published papers, we explain i) high accuracy of GEM calculations and its reason, ii) wide applicability of GEM and iii) successful predictions by GEM calculations before measurements. GEM was proposed 30 years ago and has been applied to a variety of subjects. To solve few-body Schroedinger equations accurately, use is made of the Rayleigh-Ritz variational method for bound states, the complex-scaling method for resonant states and the Kohn-type variational principle to S-matrix for scattering states. The total wave function is expanded in terms of few-body Gaussian basis functions spanned over all the sets of rearrangement Jacobi coordinates. Gaussians with ranges in geometric progression work very well both for short-range and long-range behavior of the few-body wave functions. Use of Gaussians with complex ranges gives much more accurate solution when the wave function has many oscillations.

In the last two decades, a large number of exotic hadron states have been observed in experiments, which arouses great attention in the hadron physics community. In this short review, we briefly summarize progresses of our group on several exotic hadron states. Two approaches, quenched and unquenched quark models, are adopted in the calculation. The channel coupling effects, the multiquark states couple to open channels and the quark-antiquark states couple to meson-meson states, are emphasized. X(3872) is showed to be a mixture state of and in the unquenched quark model. X(2900) can be explained as a resonance state with the quantum numbers in both the quark delocalization color screening model and the chiral quark model. The reported state X(6900) can be explained as a compact resonance state with in both two quark models, and several fully heavy tetraquark states are predicated. The possible hidden-charm pentaquarks are systematically investigated in QDCSM, and seven resonance states are obtained in the corresponding baryon-meson scattering process, among which the with , with and are consistent with the experimental report of P c (4312), P c (4440), and P c (4457), respectively. More experimental data on exotic hadron states are vital for understanding exotic hadron states in quark models.

Inspired by the newly observed $T_{cc}^+$ state, we systematically investigate the $S$-wave triple-charm molecular states composed of $D^*D^*D$ and $D^*D^*D^*$. We employ the one-boson-exchange model to derive the interactions between $D(D^*)$ and $D^*$ and solve the three-body Schr\"odinger equations with the Gaussian expansion method. The $S$-$D$ mixing and coupled channel effects are carefully assessed in our study. Our results show that the $I(J^P)=\frac{1}{2}(0^-,1^-,2^-)$ $D^*D^*D$ and $I(J^P)=\frac{1}{2}(0^-,1^-,2^-,3^-)$ $D^*D^*D^*$ systems could form bound states, which can be viewed as three-body hadronic molecules. We present not only the binding energies of the three-body bound states, but also the root-mean-square radii of $D $-$D^*$ and $D^*$-$D^*$, which further corroborate the molecular nature of these states. These predictions could be tested in the future at LHC or HL-LHC.

We make a systematic calculation of the spectra and hadronic decays of the ${D}_{s}$ system in a coupled channel framework, where the unquenched effects are induced by the $^{3}{P}_{0}$ model. In the calculation, the wave functions are obtained by using a nonrelativistic potential model and are handled precisely with Gaussian expansion method. Even though the fitting mainly focuses on the spectrum, our model agrees well with the experiments on both the spectra and the hadronic decays, suggesting that the coupled channel effect could result in a reasonable and coherent description of the ${D}_{s}$ mesons. Based on the calculation, we give a detailed analysis on various aspects of the excited states, especially ${D}_{s0}^{*}(2317),\phantom{\rule{0ex}{0ex}}{D}_{s1}(2460),{D}_{s1}(2536),{D}_{s1}^{*}(2860),{D}_{s0}(2590)$, and ${D}_{sJ}^{*}(3040)$. We also predict that ${D}_{s0}^{*}({2}^{3}{P}_{0})$ should be a ${D}^{*}{K}^{*}$ dominant molecule with mass 2894 MeV, which is only 5 MeV below the ${D}^{*}{K}^{*}$ threshold.

Inspired by the experimental discoveries of $T_{cc}$, $\Sigma_c(2800)$, and $\Lambda_c(2940)$ and the theoretical picture where they are $DD^*$, $DN$, and $D^*N$ molecular candidates, we investigate the double charm heptaquark system of $DD^*N$. We employ the one-boson-exchange model to deduce the pairwise $D$-$D^*$, $D$-$N$, and $D^*$-$N$ potentials and then study the $DD^*N$ system with the Gaussian expansion method. We find two good hadronic molecular candidates with $I(J^P)=\frac{1}{2}(\frac{1}{2}^-)$ and $\frac{1}{2}(\frac{3}{2}^-)$ $DD^*N$ with only $s$-wave pairwise interactions. The conclusion remains unchanged even taking into account the $S$-$D$ mixing and coupled channel effects. In addition to providing the binding energies, we also calculate the root-mean-square radii of the $DD^*N$ system, which further support the molecular nature of the predicted states. They can be searched for at the upcoming LHC run 3 and run 4.

An analytical expression for the beam propagation factor (M 2 factor) of truncated Gaussian beams was derived by using the complex Gaussian functions expansion method. The reasonability of the approximation of complex Gaussian functions expansion method is studied, and a comparison of this method with the generalised truncated second-order moments method and the asymptotic analysis method is also made. In general, an easy analytical expression for the M 2 factor of truncated laser beams can be derived by using the complex Gaussian functions expansion method. The M 2 factor obtained by using the complex Gaussian functions expansion method is more consistent with that in practice than that obtained by using two other methods. The analytical results obtained by using the complex Gaussian functions expansion method can be reduced to that for the non-truncated case when the truncation parameter is sufficiently large. Therefore, the complex Gaussian functions expansion method is a suitable approximation method for studying the M 2 factor of truncated laser beams.

Motivated by the recently obtained Hadrons to Atomic nuclei from Lattice QCD potentials for the N − ϕ , N − J / ψ , and N − η c interactions, we investigate the structure of the exotic nuclei Be ϕ 8 , Be J / ψ 8 , and Be η c 8 as α + α + M three-body systems ( M denotes the meson). The bound and resonant states are calculated consistently using the Gaussian expansion method, with resonances identified via the complex scaling method. For the α ϕ and α -charmonium interactions, a folding potential is constructed based on the Hadrons to Atomic nuclei from Lattice (HAL) QCD potentials and fitted to a Woods-Saxon form. We find that the ϕ meson exhibits a strong “gluelike” effect, binding the 0 1 + , 2 1 + , and 4 1 + resonant states of Be 8 into stable states and significantly reducing the α − α distance. In contrast, the interactions of J / ψ and η c with the nucleus are weaker, forming only shallow bound states with the 0 1 + state of Be 8 and even increasing the α − α separation. Notably, our analysis predicts weakly bound α − J / ψ states in the S 4 3 / 2 and S 2 1 / 2 channels, a result not reported in prior studies, which suggests that Be J / ψ 8 may not be a Borromean nucleus. The sensitivity of the Be ϕ 8 ( 4 1 + ) state—transitioning from bound to resonant depending on the α -particle radius—highlights the subtle dynamics at play. These results provide a systematic theoretical comparison of how different vector mesons modify nuclear clustering, offering critical predictions for future experimental searches of such exotic hadron-nucleus systems.

In the framework of color flux-tube model with a four-body confinement potential, the lowest charged tetraquark states $[Qq][\bar{Q}'\bar{q}']~(Q=c,b,q=u,d,s)$ are studied by using the variational method, Gaussian expansion method. The results indicate that some compact resonance states can be formed, the states can not decay into two color singlet mesons $Q\bar{q}'$ and $\bar{Q}'q$ through the breakdown and recombination of color flux tubes but into $Q\bar{Q}'$ and $q\bar{q}'$. The four-body confinement potential is an crucial dynamical mechanism for the formation of states, The decay mechanism is similar to that of compound nucleus and therefore the states should be called "color confined, multi-quark resonance" states. The newly observed charged states $Z_c(3900)$ and $Z_c(4025)/Z_c(4020)$ can be accommodated in the color flux-tube model and can be interpreted as the $S$-wave tetraquark states $[cu][{\bar{c}\bar{d}}]$ with quantum numbers $I=1$ and $J=1$ and 2, respectively.