Production of $X_{cs\bar{c}\bar{s}}$ in heavy ion collisions

The yields of $X_{cs\bar{c}\bar{s}}$ with its two possible configurations, i.e., the hadronic molecular state and tetraquark state, for Pb-Pb collisions at $\sqrt{s_{NN}}=5.02~\rm{TeV}$ is studied. A volume effect is found from the centrality distribution of $X_{cs\bar{c}\bar{s}}$, which could help to distinguish the inner structure of $X_{cs\bar{c}\bar{s}}$. We also show the rapidity and the transverse momentum distributions of $X_{cs\bar{c}\bar{s}}$ production as well as its elliptic flow coefficient as a function of the transverse momentum.

In the molecular picture, a X cscs is formed by a strange-charmed meson D + s (D − s ) and a D * − s (D * + s ), while a X(3872) is formed by a charmed meson D 0 (D * 0 , * Mr.zhanghui@m.scnu.edu.cn D + , D − ) and aD 0 (D * 0 , D * − , D * + ). In the tetraquark picture, a X cscs is formed by a spin triplet diquark [cs] 1 (spin singlet diquark [cs] 0 ) and a spin singlet antidiquark [cs] 0 (spin triplet antidiquark [cs] 1 ), while a X(3872) is formed by a diquark [cq] 1 ([cq] 0 ) and a [cq] 0 ([cq] 1 ), q for u and d quarks. Although light quarks u and d in X(3872) are replaced with s quarks in X cscs , their inner structures may or may not be the same. This motivates the present study, in which we examine whether the approach we proposed in Ref. [75] can also be applied to the X cscs case and thus find a way to distinguish the two internal structures with heavy ion measurements. In this work, we try to distinguish the two aforementioned possible inner structures of X cscs , i.e., a loose hadronic molecule or a compact tetraquark, by studying its production in heavy ion collisions. The remainder of this paper is organized as follows. In section II, we introduce the generation mechanism of X cscs into the AMPT model corresponding to its two possible inner structures following the production of X(3872) described in Ref. [75]. In section III, we examine the production of X cscs as a function of centrality, transverse momentum, and rapidity. A volume effect is found, which can be a probe of the inner structure of X cscs . A summary and outlook are presented in section IV.

II. FRAMEWORK
In this study, we generate a total of one million minimum bias events for Pb-Pb collisions at √ s N N = 5.02 TeV by using the framework developed in Ref. [75]. We introduce the production mechanism to produce X cscs for its two possible configurations, i.e., the hadronic molecular configurations and the tetraquark configurations into the default version (v1.26t9b) of AMPT transport model [76]. Given that X cscs contains (anti-)charm quarks and (anti-)strange quarks, we need to generate a reasonable number of individual charm and strange quarks in the partonic phase. On top of the default version of AMPT, we modify the factor of K ( [77]) to enhance the initial c andc spectra because of a lack of some channels related to initial heavy quarks. The AMPT calculation gives a reasonable (order-of-magnitude) de-scription of the experimental data [78] for the total yield of D + + D + * in the low p T region (see upper panel of Fig. 1). For the strange quarks, an upper limit on the relative production of strange to non-strange quarks in AMPT is set to 0.6 because of the strangeness enhancement effect (see [79]), and our calculations also give a reasonable (order-of-magnitude) description of the experimental data [80] for the yield of D + s meson (see lower panel of Fig. 1). The main purpose of this work is to distinguish two inner structures of X cscs through their significantly different production rates. The difference of D and D + s mesons production between our calculation and experimental data should not influence the relative yield between two inner structures and thus cannot change the qualitative results.  [78]; Lower panel: the production of D + s from the ALICE Collaboration [80]. The bands reflect the uncertainty due to constituent composition as discussed around Eq. (1) that are obtained from varying the composition fraction by ±10%.
We use the same production mechanism developed in Ref. [75] for the hadronic molecule and tetraquark configurations of the X cscs . For the molecular picture, the charmed-strange mesons are collected after the hadronization process. Then, D + s (D − s ) and D − * s (D + * s ) are coalesced (similar to the hadronization process mentioned in [76]) to form the "molecule" X cscs according to the following conditions: the relative distance within the region [5fm, 7fm] and invariant mass within the region [2M D + s , 2M D + * s ]. For the tetraquark picture, the "tetra" X cscs is formed via two steps. (i) First, diquarks (cs) and diquarks (cs) are formed by matching a (anti-)charm quark with the nearest (in both position space and momentum space) (anti-)strange quark in the parton. (ii) Then, these (anti)diquarks are coalesced to form the X cscs according to the following conditions: the relative distance < 1fm and invariant mass within the region [2M [cs]1 , 2M [cs]0 ] (the spin triplet and singlet diquark masses are defined in Refs. [30,31]). Owing to a lack of spin information in the AMPT model for the formation of the charmed-strange mesons and (anti)diquarks, the relative yield ratios are estimated using the thermal model: where m A and m B represent the masses of hadrons A and B, respectively. Here, T freezeout 160 MeV is the freeze-out temperature. For the hadronic picture, A and B are the D + s and D + * s mesons, respectively. For the tetraquark picture, A and B are the spin triplet and singlet diquark, respectively. This estimate indicates a composition of 30%(70%) for D + s (D + * s ) and a composition of 35%(65%) for spin triplet(singlet) diquarks. We also vary the composition between 20%(80%) and 40%(60%) to show the uncertainty bands.

III. RESULTS AND DISCUSSIONS
Within this simulation framework, we use the Monte Carlo method to generate a total of one million minimum bias events for Pb-Pb collisions at √ s N N = 5.02 TeV.
The inclusive yield of X cscs is found to be approximately 42000 in the molecular picture and approximately 200 in the tetraquark picture. As a benchmark for comparison, we also estimate the yield of X(3872) within the same framework (see the production mechanism in Ref. [75], the yield should be multiplied a factor 1 4 owing to wavefunction normalization for both the molecular and tetraquark pictures). The inclusive yield of X(3872) is found to be approximately 171000 in the molecular picture and approximately 600 in the tetraquark picture. The yield of X cscs is approximately 1 4 of that of X(3872). Compared with the experimental data of X(3872) measured by the CMS collaboration for Pb-Pb collisions at √ s N N = 5.02 TeV [81], our finding suggests that an observable signal of X cscs could be measured in heavy ion collisions at the LHC energy.
One can also find the production in the molecular picture significantly exceeds that in the tetraquark picture, by a factor of 200 -a 2-order-of-magnitude difference. This result may be understood as follows: the c −c and s −s quarks must be pair produced in the initial conditions of heavy ion collisions and then expand and cool with the bulk flow; the molecule X cscs needs a large volume to be formed, while the tetraquark X cscs needs a compact volume to be formed; thus, the probability of the formation of hadron molecules is far higher than that for the tetraquark state.
We plot the X cscs production as a function of centrality in Pb-Pb collisions at √ s N N = 5.02 TeV for the hadronic molecular state and tetraquark state in Fig. 2. One can find the yield of the X cscs in the molecular picture is 2 orders of magnitude larger than that in the tetraquark picture. From the central collision region to the peripheral collision region, the production first increases then decreases for both the molecular state and the tetraquark state, and the slope of the decrease is far larger in the molecular state than in the tetraquark state. This results from a competing effect between the volume of the bulk system and the size of X cscs . For central collisions, the number of (anti-)charm and (anti-)strange quarks is large, the bulk volume is large, and its evolution time is long; thus, the (anti-)charm and (anti-)strange quarks separate sufficiently, which benefits the production of a large-size molecular state. For the peripheral collisions, both the number of (anti-)charm and (anti-)strange quarks and the size of the fireball are small; as such, the evolution time of the fireball is short, which benefits the production of small-sized tetraquark states. This size effect could help to explore the internal structure of X cscs through different collision systems, e.g., Pb-Pb, Au-Au, Xe-Xe, Cu-Cu, O-O, and d − A/p − A.
In Fig. 3, we present the rapidity and the transverse momentum distributions of X cscs . One can find that the distribution for both the hadronic molecular state and the tetraquark state is similar to that of the usual hadrons [82,83]. We also show the elliptic flow coefficient v 2 of X cscs as a function of the transverse momentum p T in Fig. 4. The elliptic flow is sensitive to the geometry of the initial fireball and the generation mechanism of X cscs .

IV. SUMMARY AND OUTLOOK
In this work, we studied the yields of X cscs for Pb-Pb collisions at √ s N N = 5.02 TeV by introducing the pro-  duction mechanism of two possible configurations, i.e., the hadronic molecular state and tetraquark state, into the AMPT model. We found that the production in the molecular picture exceeds in the tetraquark picture by two orders of magnitude. The centrality distribution of the yields of X cscs shows a strongly decreasing trend for the hadronic molecular state and a mild change for the tetraquark state. This system size dependence could be a good probe for the inner structure of X cscs . We also showed the rapidity and the transverse momentum distributions of X cscs production, as well as its elliptic flow coefficient, as a function of the transverse momentum, which can be tested in the future experimental measurements. In Ref. [80], a strangeness enhancement effect in heavy ion collisions was found by ALICE Collaboration, which could be evidence for quark-gluon plasma. We expect a similar effect to be found in the ratio of X cscs to X(3872), which will be studied in our future work.