Hadron spectroscopy is a field replete with frequent discoveries and surprises, and the theoretical complexities associated with understanding the strong interaction in the color confinement regime make the field even more fascinating. A very successful classification scheme for hadrons in terms of their valence quarks and antiquarks was independently proposed by Murray Gell-Mann [1] and George Zweig [2] in 1964. This classification, known as the quark model, essentially divides hadrons into two major families: mesons (quark-antiquark) and baryons (three-quarks). Theoretically, the basic theory of the strong interaction, quantum chromodynamics (QCD), allows for the existence of exotic hadrons beyond the conventional picture.

The first quarkonium-like state, the X ( 3872 ) , was discovered by the Belle collaboration in the decay B ± → K ± X ( 3872 ) → K ± ( π + π − J / ψ ) in 2003 [3]. It was subsequently confirmed by other experiments [4–6]. With the development of experimental techniques and the accumulation of data, a number of hadronic states beyond the conventional two-quark meson and three-quark baryon picture have been observed in the last 2 decades which are popular candidates for exotic hadrons [7]. By now, many approaches have been used to disentangle the nature of the numerous exotic hadrons discovered, but some difficulties remain [8, 9]. The study of exotic hadrons is also one of the most important topics in hadron physics.

The production yields of exotic states in high-energy collisions, which are expected to be strongly influenced by their internal structure, have received increasing attention [10–18]. The internal structure of exotic hadrons is still under debate. They are assumed to be loosely bound hadronic molecule, a compact tetraquark, or just a kinematic effect such as the triangle singularity, etc [8, 9]. The internal structure and interactions of compact multiquark states and hadronic molecular states have been extensively studied. The former are bound by the strong interaction directly, while the latter are bound by residual strong interaction [8, 9, 19].

The abundant number of quarks and antiquarks for both light and heavy flavors suggests that heavy-ion collisions provide an ideal environment for exotic hadron production, compared to electron-positron and proton-proton (or antiproton) collisions. The first evidence for the X ( 3872 ) production in relativistic heavy ion collisions was reported by the CMS Collaboration [16]. In this work, we think that the X ( 3872 ) may be a tetraquark, nuclear-like, or molecular state, and study their production using the dynamically constrained phase-space coalescence model (DCPC). We employ the parton and hadron cascade (PACIAE) model to simulate Pb-Pb collisions at s N N = 5.02 TeV in centralities of 0 − 10 % and 30 − 50 % . Using the DCPC, we then predict the yield and transverse momentum of the X ( 3872 ) .

The PACIAE model [20–22] is a parton and hadron cascade model based on PYTHIA [23]. It has been successfully used to describe particle multiplicity, transverse momentum, rapidity distributions, and other observables in high-energy collisions [17, 24–27]. The PACIAE Monte-Carlo (MC) simulation provides a complete description of one collision, which includes the partonic initialization stage, partonic rescattering stage, hadronization stage, and the hadronic rescattering stage. For nucleon-nucleon (NN) collisions, compared to PYTHIA, the partonic and hadronic rescattering are introduced before and after the hadronization, respectively. The initial-state free parton is produced by breaking the strings of quarks, antiquarks, and gluons formed in the Pb-Pb collision with the PACIAE model. The parton rescattering is further considered using the 2 → 2 leading-order (LO) perturbative QCD parton-parton cross sections [28]. The total and differential cross section in the evolution of the deconfined quark matter state are calculated using MC method. After the partonic freeze-out, the hadronization of the partonic matter is executed by the LUND string fragmentation [23] or the MC coalescence model [20]. Hadron rescattering is performed based on the two-body collision until the hadronic freeze-out.

The DCPC model was proposed to study production of the light nuclei in p p collisions [29]. In the DCPC model, based on the quantum statistical mechanics [30, 31], we can estimate the yield of a single particle in the six-dimension phase space by an integral Y 1 = ∫ E a ≤ H ≤ E b d q ⃗ d p ⃗ h 3 , (1) Here, E a , E b , and H denote the energy threshold and the energy function of the particle, respectively. The variables q ⃗ and p ⃗ correspond to the coordinates and momenta of the particle in the center-of-mass frame of the collision at the moment after hadronization. Furthermore, the yield of a cluster consisting of N particles is defined as following: Y N = ∫ ⋯ ∫ E a ≤ H ≤ E b d q ⃗ 1 d p ⃗ 1 ⋯ d q ⃗ N d p ⃗ N h 3 N . (2)

Therefore, the yield of an X ( 3872 ) consisting of D D ̄ * cluster in the DCPC model can be calculated by. Y X 3872 = ∫ … ∫ δ 12 d q ⃗ 1 d p ⃗ 1 d q ⃗ 2 d p ⃗ 2 h 6 , (3) δ 12 = 1 i f 1 ≡ D , 2 ≡ D ̄ * ; m X 3872 − Δ m ≤ m inv ≤ m X 3872 + Δ m ; q 12 ≤ R ; 0 o t h e r w i s e . (4)

where, m inv = E 1 + E 2 2 − p ⃗ 1 + p ⃗ 2 2 . (5)

The q 12 denote the relative distance between D and D ̄ * . The R represents the radius of the cluster (a free parameter). Obviously, the relative distance between D and D ̄ * ( q 12 ) in the compact picture is shorter than that in the nuclear or molecular picture. Consequently, the radius R of the compact state is also smaller. We assumed to that the X ( 3872 ) might exist in three different state: tetraquark, nuclear-like, or molecular state, each with a distinct radius. In our simulation, we distinguish these three structures of the X ( 3872 ) based on the value of R . According to the radius of deuteron and the result in Refs. [15, 19], the X ( 3872 ) is assumed to be a tetraquark state when R < 1.0 fm; a nuclear-like state when 1.0 < R < 1.74 fm; a molecular state, when 1.74 < R < 10.0 fm. The m X ( 3872 ) denotes the rest mass of X ( 3872 ) , and Δ m refers to its mass uncertainty. The E 1 , E 2 denote the energies of the two particles ( D and D ̄ * ), while p 1 ⃗ , p 2 ⃗ represent their respective momenta.

The DCPC model has been successfully applied to different collision systems at RHIC and LHC, including p − p [13, 17, 32–35], Cu-Cu [36, 37], Au-Au [24, 38–40], and Pb-Pb [41, 42] collisions. Especially, it has been successfully used to calculate the yields of the exotic states following transport model simulations [13, 17, 34, 35].

In this work, we produce the X ( 3872 ) and investigate its nature in Pb-Pb collisions at s N N = 5.02 TeV within the 0 − − 10 % and 30 − − 50 % centrality ranges using PACIAE + DCPC. The production involves a two-step process: first, simulating Pb-Pb collisions at s N N = 5.02 TeV to generate the multi-particle final states; then, combining the final states D 0 , D ̄ 0 , D 0 * and D ̄ 0 * to generate the tetraquark, nuclear-like and molecular states of the X ( 3872 ) using DCPC model.

In the production of final states particles with PACIAE, the impact parameter b is set to 0 − 4.89 , and 8.47 − 10.93 , according to Ref. [45], to simulate Pb-Pb collisions in the centrality ranges of 0 − − 10 % and 30 − − 50 % , respectively. The other model parameters are fixed at their default values given in the PYTHIA model, expect for the K factor and the parameters parj (1), parj (2), and parj (3). Here, the K factor is introduced to include the higher order and the nonperturbative corrections, parj (1) represents the suppression of diquark–antidiquark pair production relative to the quark–antiquark pair production, parj (2) denotes the suppression of strange quark pair production relative to up (down) quark pair production, parj (3) indicates the extra suppression for strange diquark production compared to the normal suppression of a strange quark. These parameters are determined by fitting to the ALICE data [43, 44] for D 0 , π , K , and p in Pb-Pb collisions at s N N = 5.02 TeV. The comparison of the yields for each final states between the simulation from the PACIAE model with determined parameters and the experimental measurements by ALICE collaboration is shown in Table 1, which are consistent with each other within uncertainties.

In this work, the X ( 3872 ) states are generated by combining the final state particles D 0 and D ̄ 0 * (or D ̄ 0 and D 0 * ) using the DCPC model, following the simulation of Pb-Pb collisions by the PACIAE model. First, we calculate the yield of the X ( 3872 ) in Pb-Pb collisions at s N N = 5.02 TeV, with parameter R varying from 1.0 fm to 10.0 fm, at a given mass uncertainty Δ m = 142 MeV/ c 2 (obtained from 2 m D < m inv < 2 m D ̄ * [18]). Depending on the value of R , the exotic state X ( 3872 ) can be classified into three structures: the tetraquark state for R < 1.0 fm, the nuclear-like state for 1.0 < R < 1.74 fm, and the molecular state for 1.74 < R < 10 fm. They are denoted as X T ( 3872 ) , X N ( 3872 ) and X M ( 3872 ) , respectively [15, 19]. Figure 1 present the distribution of the yield of these three different structures of the X ( 3872 ) as a function of the parameter R . From Figure 1, we can conclude that the yield of each structure of the X ( 3872 ) increase with parameter R at a given mass uncertainty Δ m = 142 MeV/ c 2 .

Then, we calculate the yields of three structures of the X ( 3872 ) in Pb-Pb collisions as parameter Δ m increases from 0.595 MeV (the half of the width of X ( 3872 ) ) to 142 MeV. The distribution of the yield of the X ( 3872 ) as a function of Δ m is shown in Figure 2. From Figure 2, we observe that the yields of X ( 3872 ) increase exponentially with increasing Δ m .

As a reasonable prediction, we can predict the yields of the X ( 3872 ) by assuming a mass uncertainty of Δ m = 142 MeV/ c 2 (obtained from 2 m D < m inv < 2 m D ̄ * [18]). The predicted yields of the X ( 3872 ) in Pb-Pb collision at s = 5.02 TeV within 0 − 10 % and 30 − 50 % centrality ranges are shown in Table 2. From these results, We observe that the yield is larger in central collisions. When comparing the yield in central Pb-Pb collision with pp collisions, we find that the yield in pp collision is lower.

Moreover, we calculate the transverse momentum distribution of the tetraquark, nuclear-like and molecular states the X ( 3872 ) . Figure 3 shows the transverse momentum p T distributions of these three different structures of the X ( 3872 ) in Pb-Pb collision at s N N = 5.02 TeV, for centralities of 0 − 10 % and 30 − 50 % . Obviously, the p T distributions of the X ( 3872 ) for the three different structures are similar to each other. From the p T distributions, we can find the yield of X ( 3872 ) increases with increasing p T in small p T range, and decreases with increasing p T in larger p T range. However, the molecular state X M ( 3872 ) exhibits a narrower p T distribution than the tetraquark state X T ( 3872 ) and nuclear-like state X N ( 3872 ) in the 0 − 10 % centrality range. In the 30 − 50 % centrality range, the p T differential yields of the compact and nuclear-like state of the X ( 3872 ) are smaller than that of the molecular state of the X ( 3872 ) , and their uncertainties are larger. The features of p T distributions may be used to distinguish X ( 3872 ) of different structure.

In Figure 3, we show the predicted pT-differential yields of the tetraquark, nuclear-like and molecular states of the X ( 3872 ) . We also analyze the pT-differential yield ratios for the X ( 3872 ) and D 0 , with the result shown in Figure 4.

From Figure 4, we observe that the yield ratio for the X ( 3872 ) and D 0 in the centrality ranges of 0 − 10 % is larger than that in the centrality ranges of 30 − 50 % . In 0 − 10 % centrality, the yield ratio for the molecular state of the X ( 3872 ) and D 0 is lower than that for the tetraquark and nuclear-like states of the X ( 3872 ) . However, in 30 − 50 % centrality, the yield ratio for the molecular state of the X ( 3872 ) and D 0 is higher than that for the tetraquark and nuclear-like states of the X ( 3872 ) .

In this paper, we study the production of the X ( 3872 ) in Pb-Pb collision at s N N = 5.02 TeV within the centrality ranges of 0 − 10 % and 30 − 50 % using the PACIAE + DCPC model. First, we investigate the dependence of the X ( 3872 ) production on the mass uncertainty Δ m and radius R . The results indicate that the yields of X ( 3872 ) increase with the increasing Δ m and R . We also predict the yield of the tetraquark, nuclear-like and molecular states of the X ( 3872 ) in Pb-Pb collision at s N N = 5.02 TeV for centralities of 0 − 10 % and 30 − 50 % , respectively. Subsequently, we examine the transverse momentum of these three different states of the X ( 3872 ) . We find that the p T distributions of the X ( 3872 ) for the three different structures are generally similar to each other. However, in the 0 − 10 % centrality range, the molecular state X M ( 3872 ) exhibits a narrower p T distribution than tetraquark state X T ( 3872 ) and nuclear-like state X N ( 3872 ) .