Introduction

The study of baryon-antibaryon bound states dates back to the proposal by Fermi and Yang [1] to form pions from nucleon-antinucleon pairs. In the traditional one-boson-exchange theory of nucleon-nucleon interactions, it is shown that the nucleon-antinucleon system is more attractive than the nucleon-nucleon system owing to the strong ω-exchange [2]. Therefore, possible bound states and resonances of nucleon-antinucleon systems have been proposed for several years. An extensive and comprehensive review of the possible bound states of was provided in Ref. [3].

More recently, the BESIII Collaboration reported the observation of a new X(1880) state in the line shape of the 3(π⁺ π^-) invariant mass spectrum [4], which is considered as evidence for the existence of a proton-antiproton bound state. Many theoretical studies have been conducted to study the system and the properties of X(1880) [5-11]. In addition to the possible proton-antiproton state, there has also been significant progress in recent studies related to [12]. A narrow structure in the system near the mass threshold, named X(2085), is observed in the process with a statistical significance exceeding 20 σ. Its spin and parity are slightly favored to be JP = 1⁺ based on an amplitude analysis. Further theoretical results and discussions can be found in Refs. [13-15]. Building on the significant progress in studies of nucleon-antinucleon and nucleon- systems, it is natural to explore whether bound states or resonance states can be formed between nucleons and other hyperons or between nucleons and other antihyperons.

In recent years, progress in understanding the strange dibaryon pΩ has renewed interest in dibaryon systems. The STAR Collaboration measured the pΩ correlation functions in Au+Au collisions at the Relativistic Heavy-Ion Collider (RHIC) [16] and reported a positive scattering length for the pΩ interaction, which supports the hypothesis of a pΩ bound state. In addition, the ALICE Collaboration reported measurements of the p-Ω correlation in pp collisions at TeV at the Large Hadron Collider (LHC) [17]. Beyond the pΩ system, femtoscopic techniques and correlation function studies have made significant progress, both experimentally [18-24] and theoretically [25-57].

The S=-3, I=1/2, J=2 NΩ state was first predicted by J. T. Goldman et al. as a narrow resonance in a relativistic quark model [58]. M. Oka also proposed the existence of a quasi-bound state with I(JP)=1/2(2⁺) using a constituent quark model [59]. A lattice QCD study by the HAL QCD Collaboration reported that the pΩ state is a bound state at a pion mass of 875 MeV [60]. Later, the bound nature was also confirmed with nearly physical quark masses ( MeV and MeV) [61]. Using the interactions obtained from (2+1)-flavor lattice QCD simulations, K. Morita et al. studied the two-pair momentum correlation functions of the pΩ state in relativistic heavy-ion collisions to further investigate the existence of a pΩ bound state [62, 63]. This state has also been confirmed to be a bound state in the frameworks of the chromomagnetic model [64], QCD sum rules [65] and other quark models [66-68]. Additionally, studies on the production of pΩ and NN Ω systems can be found in Refs. [69-71].

By analogy to the nucleon-nucleon and nucleon-antinucleon systems, one may expect attractive interactions in both the pΩ and channels. If the pΩ state can be confirmed through further experimental measurements, we expect to observe an even stronger signal for the state in experiments. Furthermore, the nucleon-antinucleon state would annihilate rapidly in the ground state owing to the quark content of this system, making it challenging to provide a convincing theoretical confirmation of the nucleon-antinucleon bound state or resonance. In contrast, the state cannot annihilate into the vacuum, as the nucleon consists of three u(d) quarks and consists of three quarks. In this context, the state is expected to be relatively stable and may serve as an ideal system for studying baryon-antibaryon interactions. The copious production of anti-baryons in high-energy colliders offers excellent opportunities to study this type of spectrum. Clearly, the theoretical study of the system is both interesting and necessary, as it can provide valuable insights for the experimental search for baryon-antibaryon bound states.

In our previous work, we studied the pΩ interactions and correlation functions based on the quark delocalization color screening model (QDCSM) [72]. According to our calculations, the depletion of the pΩ correlation functions caused by the JP = 2⁺ bound state, which is not observed in the ALICE Collaboration’s measurements [17], can be explained by the contribution of the attractive JP = 1⁺ component in spin averaging. The QDCSM is a constituent quark model [73, 74] that introduces two key ingredients: first, quark delocalization, which accounts for orbital excitation by allowing quarks to delocalize from one cluster to another; second, the color screening factor, which modifies the confinement interaction between quarks in different clusters orbits. In the study of nucleon-nucleon and nucleon-hyperon interactions and the properties of the deuteron, the mechanism of quark delocalization and color screening plays a crucial role in generating intermediate-range attraction [75-77]. This model has also been used to investigate various dibaryon candidates, such as d^* [78], pΩ [67, 72], and others [79-83]. It has been extended to study baryon-antibaryon systems, including and [84, 85]. Extending it to the system is a natural progression. Therefore, we continue to investigate the system within the framework of the QDCSM. In this study, the system is studied from three aspects: energy spectrum, scattering processes, and correlation functions.

The remainder of this paper is organized as follows. A brief introduction to the QDCSM is provided in the next section. The correlation function and inverse scattering method are introduced in Sect. 2.2 and Sect. 2.3, respectively. Section 3 presents to the numerical results and discussion of the study. A summary is presented in the final section.

Theoretical formalism

Results and discussion

The S-wave systems with isospin I=1/2, spin parity JP=1^- and 2^- are investigated based on the QDCSM. To determine whether a bound state exists, a dynamic calculation was performed as the first step. The resonating group method (RGM) was employed to solve the bound-state problem. In this approach, the total wave function of the six-quark (three quarks and three antiquarks) system is constructed as:(25)where ϕp and are the internal wave functions of the proton and clusters, χ(R) represents the relative motion wave function, and is the antisymmetrization operator that accounts for quark exchange effects. The RGM equation is derived by projecting the Schrödinger equation onto the cluster basis, leading to a coupled integro-differential equation:(26)where H and N are Hamiltonian and normalization kernels, respectively. The relative wave function χ(R) is expanded using Gaussian basis functions, converting the integral equation into a generalized eigenvalue problem. Solving this equation provides the binding energies and wave functions of the states. This method has been widely and successfully applied to baryon-baryon interactions and multiquark systems [103-105].

The binding energies of the systems with JP = 1^- and JP = 2^-, denoted as E_B, are listed in Table 3. Here, represents the theoretical threshold and E^Theo represents the eigenvalue of the corresponding system. The calculation for the systems does not involve channel coupling because we limit our study to color-singlet sub-clusters consisting of three u/d quarks and three quarks.

From Table 3, we can see that both the JP=1^- and JP=2^- form bound states with binding energies of approximately 10 MeV and 9 MeV, respectively. By contrast, in our previous work on the pΩ systems, the single channel calculation showed that neither the JP=1⁺ nor JP=2⁺ pΩ is bound. After channel coupling, only the pΩ with JP=2⁺ forms a bound state with a binding energy of approximately 6 MeV. According to the results obtained with the second parameter set, the binding energies of the JP=1^- and JP=2^- states are approximately 18 MeV and 16 MeV, respectively. The corresponding binding energy of the pΩ state with JP=2⁺ after channel coupling is approximately 14 MeV. These numerical results indicate that it is more likely for the system rather than the pΩ system to form bound states in our calculations. Therefore, considering that the attractive pΩ interaction is implied in the experimental measurements of pΩ correlation functions [17], we look forward to the experimental progress on correlation functions in the future.

To further study the interaction between nucleon and , we calculated the scattering phase shifts of the systems. The calculation is based on the well-developed Kohn-Hulthen-Kato(KHK) variational method, and the details of this method can be found in Refs. [82, 104]. The low-energy scattering phase shifts of the systems with the two parameter sets are shown in Fig. 1. For the systems with both JP=1^- and JP=2^-, the scattering phase shifts approach 180° when MeV and rapidly decrease when E_c.m. increases, which indicates the existence of a bound state in these systems. This conclusion is consistent with the bound state calculation discussed earlier. Then, we can extract the scattering length a₀ and the effective range r_eff of the systems from the low-energy phase shifts obtained above using the expansion:(27)where k is the momentum of the relative motion with , μ is the reduced mass of the two baryons, and E_c.m. is the incident energy; δ is the low-energy scattering phase shift. The binding energy can be calculated using the following relation:(28)where α is the wave number, which can be obtained from the relation [106]:(29)Note that this is another way to calculate the binding energy; therefore, it is labeled . The scattering parameters of the systems, along with the binding energies obtained using the scattering parameters, are presented in Table 4.

Fig. 1Full-screen View

(Color online) Calculated phase shifts of the systems for JP = 1^- and 2^- with two parameter sets

From Table 4, our results show that the scattering lengths are positive for the systems with JP = 1^- and 2^-, which also confirms the existence of bound states. In addition, the binding energies of the two systems obtained using Eq. (28) is broadly consistent with the numerical results shown in Table 3, which were obtained by dynamic calculation. Additionally, in the method of obtaining the binding energies using scattering parameters, the binding energies of the two systems are also deeper than that of the pΩ system with JP=2⁺.

Moreover, by solving the inverse scattering problem, we can further study the behavior of the correlation functions based on the scattering process and the KP formula in Eq. (13). Before that, we can study the general properties of the correlation functions through an effective square well potential model. The correlation functions corresponding to different degrees of the square well potentials are presented in Fig. 2. The solid red lines represent the correlation functions influenced only by the repulsive Coulomb interaction. The dashed blue lines represent the correlation functions influenced only by the square well potentials. We introduce three square well potentials with a width r₀ = 2 fm: a weak attraction in panel (a) corresponding to V₀ = -10 MeV, a moderate attraction in panel (b) corresponding to V₀ = -28 MeV, and a relatively strong attraction in panel (c) corresponding to V₀ = -40 MeV. The dotted black lines represent the correlation functions influenced by both the Coulomb interaction and square well potentials through Eq. (19). Additionally, we adopt a source size parameter R = 0.95 fm in Eq. (14), which is the same value used in our previous pΩ correlation analysis [72]. This value was originally extracted by the ALICE Collaboration [17, 107]. It should be emphasized that the source size is determined by the specific collision system and experimental conditions in heavy-ion collisions.

Fig. 2Full-screen View

(Color online) Correlation functions for different square well potentials, with , where V₀ = -10 MeV in panel (a), V₀ = -28 MeV in panel (b), V₀ = -40 MeV in panel (c), and r₀ = 2 fm

In Fig. 2, panels (a) and (b), the correlation function affected only by the square potential is above unity in the low-energy region, which is due to the attractive interaction. The difference is that the weak attraction is not sufficient to form a bound state; therefore, the correlation function is always above unity, whereas the moderate attraction forms a shallow bound state. The existence of a bound state leads to the depletion of the correlation function; therefore, there exists a part below unity. After considering both the Coulomb interaction and square well potentials, which mainly dominate the low-energy region (0 MeV<k<25 MeV), the correlation function forms a peak-like structure. In panel (c), the correlation function remains below unity for a relatively strong attraction. A discussion of this phenomenon can be found in Refs. [39, 72]. After considering the Coulomb interaction, one can see that the correlation function in the low-energy region nearly coincides with the result obtained by considering only the Coulomb interaction. As the relative momentum k increases, the correlation function closely matches the result obtained by considering only a relatively strong attraction between the particles.

After replacing the square well potentials with the effective potentials obtained by solving the inverse scattering problem using the GLM method, which is briefly introduced in Sect. 2.3, we can study the correlation function of the system. Because the two parameter sets yielded similar results, the discussion of the correlation function is presented based on the first parameter set for conciseness. Both the effective potentials of the systems with JP = 1^- and 2^- are obtained. The total correlation function is a superposition of the correlation functions corresponding to the two quantum numbers according to Eq. (21). Because the system forms bound states for both quantum numbers and the interactions are similar, we omit the comparison of the correlation functions for the two quantum numbers here. The total correlation functions calculated for different values of the source size parameter R are shown in Fig. 3. In addition, recent studies on emission source properties can be found in Refs. [107-111].

Fig. 3Full-screen View

(Color online) correlation functions with different values of source size parameter R, where R = 1.0 fm in panel (a), R = 1.2 fm in panel (b), R = 1.5 fm in panel (c), R = 2.0 fm in panel (d), and R = 2.5 fm in panel (e) are summarized in panel (f)

In Fig. 3, panels (a)–(e), the dashed gray lines and the dotted orange lines represent the correlation functions considering only the Coulomb interaction and both the Coulomb interaction and strong interaction, respectively. In panel (f), the gray band represents the correlation functions influenced only by the Coulomb interaction with the size parameter R ranging from 1.0 to 2.5 fm, while the other lines summarize the correlation functions shown in panels (a)–(e). According to our results, the change in the size parameter R can greatly influence the correlation functions. An obvious feature is that as R increases, the peak-like structure caused by the different dominant regions of the Coulomb interaction and strong interaction gradually becomes less obvious and eventually disappears. Because two bound states are obtained in our calculation, it is very important to verify this conclusion using correlation functions. It can be seen that as the correlation function influenced only by the Coulomb interaction gradually approaches unity, the depletion caused by the bound states leads to the correlation function being below that of the Coulomb-only case.

While femtoscopic measurements in heavy-ion collisions provide important access to the interaction, such bound states may also be produced in other high-energy environments capable of forming multi-strange baryon-antibaryon pairs, such as pp and e⁺e^- collisions, as well as high-energy fixed-target experiments, although the production probability may vary across systems. If a bound state is formed, it can decay through the weak decay of the , leading to final states such as and , or through quark rearrangement into a three-meson final state, , which may provide complementary detection signatures beyond femtoscopy in future searches.

In recent years, experimental data on correlation functions have increased rapidly [18-24], providing unprecedented insights into hadron-hadron interactions across a variety of systems. Together with femtoscopic techniques, these studies open up new possibilities for extracting low-energy scattering parameters that are otherwise difficult to access. On the theoretical side, continuous progress in lattice QCD, effective field theory, and quark-model-based approaches has greatly enriched our understanding and offered valuable guidance for interpreting the experimental observations [25-57]. The synergy between experimental measurements and theoretical developments will not only deepen our knowledge of strong interactions, but also pave the way for future explorations of exotic hadronic states and nuclear physics [112-114]. In this context, it is well established that nucleon-nucleon and hyperon-nucleon interactions provide the basis for the formation of nuclei and hypernuclei [115-121], while antihyperon-antinucleon interactions give rise to anti-hypernuclei [122, 123]. An open question is whether antihyperon-nucleon interactions can also give rise to the formation of novel nuclear systems. As a continuation of the present study, we will further explore such systems based on the antihyperon-nucleon interactions obtained in this study.

Summary

In this study, we investigate the S-wave systems with isospin I=1/2, spin parity JP=1^- and 2^- in the framework of the QDCSM. The results show that the systems with both JP=1^- and 2⁺ form bound states, and the attraction between nucleon and is slightly stronger than that between nucleon and Ω, suggesting that the system has a higher likelihood of forming bound states than the pΩ system. The calculation of the low-energy scattering phase shifts and scattering parameters of the systems also supports the existence of bound states with JP=1^- and 2^-. In addition, considering that a nucleon is composed of three light quarks u(d) and of three strange quarks , the state cannot annihilate into the vacuum. In this context, the state is a special state that can provide useful information for the experimental search for baryon-antibaryon bound states.

By using the GLM method, we solve the inverse scattering problem and obtain the effective potentials. On this basis, the correlation functions are calculated, considering both the Coulomb interaction and spin averaging. We present correlation functions corresponding to different source size parameters R, which can be used for future comparisons with experimental measurements.

Understanding hadron-hadron interactions is an important issue in the study of hadron physics. The study of the interaction between baryons and antibaryons in this work is also an effective method for testing this mechanism. The femtoscopic correlation function has become an important method for exploring hadron-hadron interactions, and further theoretical and experimental investigations are essential for a deeper understanding of such baryon-antibaryon systems.