$X_J(2900)$ states in a hot hadronic medium

In this work we investigate the hadronic effects on the $X_{J=0,1} (2900)$ states in heavy-ion collisions. We make use of Effective Lagrangians to estimate the cross sections and their thermal averages of the processes $X_J \pi \to \bar{D}^{*} K , K^{*} \bar{D} $, as well as those of the corresponding inverse processes, considering also the possibility of different isospin assignments ($I=0,1$). We complete the analysis by solving the rate equation to follow the time evolution of the $X_J (2900)$ multiplicities and determine how they are affected by the considered reactions during the expansion of the hadronic matter. We also perform a comparison of the $X_J (2900)$ abundances considering them as hadronic molecular states ($J=0$ as a $S$-wave and $J=1$ as a $P$-wave) and tetraquark states at kinetic freeze-out.


I. INTRODUCTION
quark-gluon plasma (QGP), heavy quarks coalesce to form bound states and possibily exotic states at the end of the QGP phase. After that, the multiquark states interact with other hadrons during the hadronic phase, and can be destroyed in collisions with the comoving light mesons, but can also be produced through the inverse processes [22][23][24][25][26][27]. So, their final multiplicities depend on the interaction cross sections, which in principle depend on the spatial configuration of the quarks. Thus, the evaluation of the interactions of X 0,1 (2900) states with hadron matter and the measurement of their abundances might be useful to determine the structure of these states. For example, Refs. [23,25] suggested that the X(3872) multiplicity at the end of the QGP phase is reduced due to the interactions with the hadron gas, and if it was observed in HICs, it would be most likely a molecular state, since this structure would be dominantly produced at the end of hadron phase by hadron coalescence mechanism.
Motivated by the discussion above, in the present work we intend to analyze the hadronic effects on the X J (2900) states in HICs. We make use of Effective Lagrangians to calculate the cross sections and their thermal averages of the processes X J π →D * K, K * D , as well as those of the corresponding inverse processes. Considering also the possibility of different isospin assignments (I = 0, 1), we analyze dependence of the magnitude of the cross sections and their thermal averages with the quantum numbers. We then solve the rate equation to follow the time evolution of the X J (2900) multiplicities and determine how they are affected by the considered reactions during the expansion of the hadronic matter. We also perform a comparison of the X J (2900) abundances considering them as hadronic molecular states (J = 0 as a S-wave and J = 1 as a P -wave) and tetraquark states at kinetic freeze-out.
The paper is organized as follows. In Section II we describe the effective formalism and calculate the cross sections of X J (2900) − π absorption and production and their thermal averages. With these results, in Section III we solve the rate equation and follow the time evolution of the X J (2900) abundances. Finally, in Section IV we present some concluding remarks.

A. The effective formalism
We start by analyzing the interactions of X J (2900) with the surrounding hadronic medium composed of the lightest pseudoscalar meson (π). We expect that the reactions involving pions provide the main contributions to the X J in hadronic matter, due to their large multiplicity with respect to other light hadrons [22][23][24][25][26][27]. In particular, we focus on the reactions X J π →D * K and X J π → K * D , as well as the inverse processes. In Fig. 1 we show the lowest-order Born diagrams contributing to each process, without the specification of the particle charges. To calculate the respective cross sections, we make use of the effective hadron Lagrangian framework. Accordingly, we follow Refs. [11,22] and employ the effective Lagrangians involving π, K, D, K * and D * mesons, where τ are the Pauli matrices in the isospin space; π denotes the pion isospin triplet; and D ( * ) = (D ( * )0 , D ( * )+ ) and K ( * ) = (K ( * )+ , D ( * )0 ) T represent the isospin doublets for the pseudoscalar (vector) D ( * ) and K ( * ) mesons, respectively. The coupling constants are determined from decay widths of D * and K * , having the following values [22,23]: g πDD * = 6.3 and g πKK * = 3.25.
The couplings including the X J (2900) states have been built in order to yield the transition matrix elements X J →D * 0 K * 0 , D * − K * + [11], In the expressions above, X 0 and X 1 denote respectively the X 0 (2866) and X 1 (2904) states; this notation notation will be used henceforth. Since there is no experimental information yet available on their isospin (I), we consider here the different isospin assignments: I = 0, 1.
In this sense, C (I) Y labels the isospin factor according to the value of I and the respective channel Y =D * 0 K * 0 , D * − K * + . The values of the coupling constants are chosen according to the analysis based on the compositeness condition performed in Ref. [11], in which for a cutoff α = (1.0 ± 0.1) GeV we have g X 0D * K * = 3.82 −0.16 +0.11 GeV and g X 1D * K * = 7.84 −0.30 +1.00 GeV.
, without specification of the charges of the particles.
With the effective Lagrangians introduced above, the amplitudes of the processes shown in Fig. 1 can be calculated; the are given by where τ  Y of the processes described in Eq. (3) by considering the charges Q 1f and Q 2f for each of the two particles in final state [24].
for X 0 and X 1 , respectively.
We determine the isospin coefficients τ Y of the processes reported in Eq. (3) by considering the charges Q 1f and Q 2f for each of the two particles in final state [24], whose combination must yield the total charge Q = Q 1f + Q 2f = 0, +1, −1. Thus, there are four possible charge configurations (Q 1f , Q 2f ) for each reaction in Eq. (3). In Table I

B. Cross sections
We define the isospin-spin-averaged cross section in the center of mass (CM) frame for the processes in Eq. (3) as where r = a, b designates the reactions according to Eq. (3); √ s is the CM energy; | p i | and | p f | denote the three-momenta of initial and final particles in the CM frame, respectively; the symbol S,I stands for the sum over the spins and isospins of the particles in the initial and final state, weighted by the isospin and spin degeneracy factors of the two particles forming the initial state for the reaction r, i.e. [24] S,I where g 1i,r = (2I 1i,r + 1)(2I 2i,r + 1) and g 2i,r = (2S 1i,r + 1)(2S 2i,r + 1) are the degeneracy factors of the particles in the initial state, and S,I Thus, the contributions for the isospin-spin-averaged cross section are distinguished by the possible charge configurations (Q 1f , Q 2f ), according to Table I.
Also, we can evaluate the cross sections related to the inverse processes, where the X J is produced, using the detailed balance relation.
Another feature is that when evaluating the cross sections, to prevent the artificial increase of the amplitudes with the energy we make use of a Gaussian form factor defined as [11]: where p E is the Euclidean Jacobi momentum. The size parameter α is taken according to the choice stated in previous subsection [32].
In Fig. 2 the X J π absorption and production cross sections for the processes involving the particlesD * , K in the final or initial states are plotted as a function of the CM energy √ s. The absorption cross sections are exothermic and become infinite near the threshold.
Within the range 3.2 ≤ √ s ≤ 4.2 GeV, they are found to be ∼ 1 × 10 −2 − 1 × 10 −1 mb, with the magnitudes between the situations involving X J with same spin but different isospins being distinguishable because of the degeneracy factor. But if we compare the magnitudes of reactions with the same isospin, it can be noticed that the cross sections for X 1 are bigger than those for X 0 by a factor about 2 − 3. In the case of the production cross sections, they are endothermic and there is no distinction between the processes with different isospins.
Their magnitudes are ∼ 2 × 10 −3 − 3 × 10 −2 mb, within the range 3. suggesting that the production cross sections are smaller than the absorption ones due to kinematic effects.
The cross sections for the processes involving the particles K * ,D in the final or initial states are plotted in Fig. 3. In general, the results are similar to the processes analyzed previously, except to the fact that the difference between the cross sections for X 1 production become bigger than those for X 0 by a factor about 20 − 25. Thus, we can infer that the spin-1 state can be formed and absorbed by light mesons more easily than the spin-0 state.

C. Cross sections averaged over the thermal distribution
Now use the results reported above to compute the thermally averaged cross sections for the X J production and absorption reactions. To this end, let us introduce the cross section averaged over the thermal distribution for a reaction involving an initial two-particle state going into two final particles ab → cd. It is given by [23,25,29] where v ab represents the relative velocity of the two initial interacting particles a and b; the The main point here is that the thermally averaged cross sections for X J π annihilation and production reactions have different magnitudes, and this might play an important role in the time evolution of the X J multiplicity, to be analyzed in the next section.

III. TIME EVOLUTION OF X J ABUNDANCE
Now we study the time evolution of the X J abundance in hadronic matter, using the thermally averaged cross sections estimated in the previous section. We focus on the influence of X J − π interactions on the abundance of X J during the hadronic stage of heavy ion collisions. The momentum-integrated evolution equation for the abundances of particles included in the processes previously discussed reads [22,23,[25][26][27][29][30][31] where n i (τ ) are N i (τ ) denote the density and the abundances of involved mesons in hadronic matter at proper time τ . Since the lifetime of the X J states is less than that of the hadronic stage presumed in this work (of the order of 10 fm/c), thus the decay of X J and its regeneration from the daughter particlesD and K are included in the last two lines of the rate equation. We adopt a similar approach to Ref. [30]: the scattering cross section for the X J state production fromD and K mesons is given by the spin-averaged relativistic Breit-Wigner cross section, where g X J , gD and g K are the degeneracy of X J ,D and K mesons, respectively; p cm is the momentum in CM frame; Γ X J →DK is the total decay width for the reaction X J →DK, which is supposed to be effectively √ s-dependent in the form with M Γ ∝ g X JD K {1, ǫ(p X ) · (pD − p K )} for J = 0 and J = 1, respectively; the value of constant g X JD K is determined from the experimental value of Γ X J →DK ( √ s) with the system in the rest frame of the X J : Γ X 0 = 57.2 MeV, Γ X 1 = 110.3 MeV. It can be noticed that the cross section for the X J state production in Eq. (10) are about 2.5-8 mbarn for J = 0, and about 6.5-20 mbarn for J = 1, depending on the isospin. Its average over the thermal distribution, σD K→X J vD K , can be determined from Eq. (8). The thermally averaged decay width of X J is given by [30] Γ In the obtention of solutions of Eq. (9), we assume that the pions, charm and strange mesons in the reactions contributing to the X J abundance are in equilibrium. Therefore, the density n i (τ ) can be written as [20,23,25,29] where γ i and g i are the fugacity factor and the degeneracy factor of the particle, respectively.
In this sense, the multiplicity N i (τ ) is obtained by multiplying the density n i (τ ) by the volume V (τ ). The time dependence of n i (τ ) is inserted through the parametrization of the temperature T (τ ) and volume V (τ ) used to model the dynamics of relativistic heavy ion collisions after the end of the QGP phase. The hydrodynamical expansion and cooling of the hadron gas are based on the boost invariant Bjorken picture with an accelerated transverse expansion [20,23,30]. Accordingly, the τ dependence of V (τ ) and T are given by where R C and τ C label the final transverse and longitudinal sizes of the QGP; v C and a C are its transverse flow velocity and transverse acceleration at τ C ; T C is the critical temperature for the QGP to hadronic matter transition; T H is the temperature of the hadronic matter at the end of the mixed phase, occurring at the time τ H ; and the kinetic freeze-out temperature T F leads to a freeze-out time τ F [20,23,30].
The evolution of X J multiplicity is evaluated with the hadron gas formed in central P b − P b collisions at √ s N N = 5 TeV at the LHC. The parameters which we use as input in Eq. (14) are listed in Table 3.1 of Ref. [20], and are summarized in Table II for completeness.
We also suppose that the total number of charm quarks, N c , in charm hadrons is conserved during the processes, i .e. n c (τ ) × V (τ ) = N c . This implies that the charm quark fugacity factor γ c in Eq. (13) is time-dependent in order to keep N c constant. The total numbers of pions and strange mesons at freeze-out were also based on Ref. [20]. Considering that the pions and strange mesons might be out of chemical equilibrium in the later part of the hadronic evolution, they also have time dependent fugacities.
We study the evolution of yields obtained for the X J abundance in different approaches: the statistical and the coalescence models. In the statistical model, hadrons are produced in  [20], given by Eq. (14).
v C (c) a C (c 2 /fm) R C (fm) 0.5 0.09 11  [20,23,30], the number of X J 's produced is given by : where g j and N j are the degeneracy and number of the j-th constituent of the X J and σ i = (µ i ω) −1/2 ; ω is the oscillator frequency, assuming an harmonic oscillator prescription for the hadron internal structure; µ the reduced mass, given by ; and l i is the angular momentum of the wave function associated with the relative coordinate, being 0 for an S-wave, and 1 for a P -wave constituent. If we consider the X J as a tetraquark state produced via quark coalescence mechanism from the QGP phase at the critical temperature T c when the volume is V C , then for J = 0 we have a relative S-wave and for J = 1 a P -wave. On the other hand, for the yields of X J as a weakly bound hadronic molecule from the coalescence of hadrons, they are evaluated at the kinetic freeze-out temperature T F and volume V F . On this regard, J = 0 should be a S-wave hadronic molecule and J = 1 a Pwave, as speculated in Ref. [11]. In Table III we give the yields in these different approaches.
The comparison between the values of N 0(Stat.) indicates that the number of X J 's calculated with the statistical model is greater than the four-quark state (formed by quark coalescence) by about one order of magnitude. Also, for tetraquark coalescence the case with J = 1 has a smaller initial yield than the state with J = 0.
In the present context we evaluate the time evolution of the X J abundance by solving Eq. (9), with initial conditions given within statistical and tetraquark coalescence models.
We emphasize that in the case of molecular states, since they are dominantly formed by hadron coalescence at the end of the hadronic phase, we use the yields shown in last column of Table III just for comparison with the results obtained from the time evolution of initial yields in statistical and tetraquark coalescence models up to kinetic freeze-out, i.e. N X J (τ F ). The X J yields in central P b − P b collisions at √ s N N = 5 TeV at the LHC using statistical and molecular/four-quark coalescence models, according to Eqs. (13) and (15). The oscillator frequency for tetraquark states produced via quark coalescence mechanism is ω = 588 MeV [20,22].
In the case of molecular states formed by hadron coalescence at the end of the hadronic phase, we have used ω = 6B = 217 MeV and 108 MeV for a S-wave and a P -wave, respectively (B being the binding energy). it can be seen that the results are strongly dependent of the initial yields. With the initial conditions calculated from statistical model, the abundances suffer a reduction which is more pronounced for the state with J = 1. On the other hand, when we use the initial conditions at the end of the mixed phase from four-quark coalescence model, we see that the X J multiplicities experiment an increasing by a factor about 1.5 for J = 0 and 3 for J = 1. It is also worth mentioning that the modifications in the abundances are mostly due to the terms associated to the spontaneous decay/regeneration of X J (i.e. the terms in the last two lines of Eq. (9)). To illustrate this issue, in Fig. (7) we plot the time evolution of the X J abundance using four-quark coalescence model to fix the initial conditions, but taking Γ X 0 ,X 1 = 0 MeV. Within these conditions, there is a relative equilibrium between production and absorption reactions, resulting in a number of X J 's throughout the hadron gas phase nearly constant in the case J = 0; for J = 1 is suggested an increasing of the multiplicity by a factor about 25% and 10% in the cases of I = 0 and I = 1, respectively. We also show in Fig. 8 the evolution of the ratio of the X 1 abundance to the sum of the X 0 and X 1 abundances. In the context of initial conditions with the statistical model, the ratio monotonically decreases from 70% to 56%. Using the tetraquark coalescence model, however, the ratio undergoes an abrupt growth from 41% to 71%, and a further reduction up to 58% in the end. Finally, we stress that the outputs reported above are based on the evolution of the X J multiplicities which come from the quark-gluon plasma and might be modified due to hadronic effects via the considered processes in previous sections. In this scenario, with the initial yields calculated in tetraquark coalescence model (X 0 and X 1 as S-wave and P -wave tetraquark states), we have obtained estimations for the case of reminiscent X J abundances.
As stated before, another possible formation mechanism of X J is the hadron coalescence, dominant at the end of the hadronic phase. According to Table III and the results from Fig. 7 (taking Γ X 0 ,X 1 = 0 MeV), the yield of hadronic molecular state for J = 0 is about 3 times smaller and for J = 1 is 6 times greater than the contribution from the tetraquark state at kinetic freeze-out. When the spontaneous decay/regeneration of X J is taken into account (Fig. 6), these ratios are about 4 times smaller for J = 0 and 2.5 greater for J = 1.
Hence, remarking that the calculated cross sections does not account for the size of the hadrons, our findings suggest that at the end of the hadronic phase the production of X 0 as a hadronic molecular state is reduced with respect to tetraquark state, while for the case of X 1 the most prominent production comes from the hadronic molecular state.

IV. CONCLUDING REMARKS
We have investigated in this work the hadronic effects on the recently observed X J=0,1 (2900) states in heavy ion collisions. We have made use of Effective Lagrangians to calculate the cross sections and their thermal averages of the processes X J π →D * K, K * D , as well as those of the corresponding inverse processes. Considering also the possibility of different isospin assignments (I = 0, 1), we have found that the magnitude of the cross sections and their thermal averages depend on the quantum numbers, since the energy dependence is different for J = 0, 1. In general the cross sections for X 1 are bigger than those for X 0 , and those involving X J π production are smaller than the absorption ones due to kinematic effects. As a consequence the thermally averaged cross sections for X J π annihilation and production reactions also have different magnitudes.
Taking the thermally averaged cross sections as inputs, we have solved the rate equation to determine the time evolution of the X J (2900) multiplicities. We have found that the X J abundance is also strongly affected by the quantum numbers I, J during the expansion of the hadronic matter. Considering the X J as a tetraquark state produced via quark coalescence mechanism from the QGP phase, in which X 0 is a relative S-wave and X 1 a P -wave, then when we neglect the X J spontaneous decays/regenerations their multiplicities are not significantly affected by the interactions with the pions; and hence the number of X J 's would remain essentially unchanged during the hadron gas phase in the case of J = 0.
But the inclusion of these effects (present in the last two lines of Eq. (9) ) leads to an increasing by a factor about 1.5 for J = 0 and 3 for J = 1 at kinetic freeze-out. Besides, concerning the evolution of the ratio of the X 1 abundance to the sum of the X 0 and X 1 abundances, it undergoes a growth from 41% at the end of mixed phase to 58% in the end of hadronic phase.
When we compare the multiplicity of X J (2900) as hadronic molecular states (J = 0 as a S-wave and J = 1 as a P -wave) and tetraquark states at kinetic freeze-out, the production of X 0 as a hadronic molecular state is smaller than the tetraquark state, while for the case of X 1 the most prominent production comes from the hadronic molecular state. therefore, as pointed out in analyses performed in the scenario of other exotic states [23][24][25]31], we believe that the evaluation of the X J (2900) abundance in relativistic heavy ion collisions might shed some light on the discrimination of its structure, although the ratio between these two approaches obtained in the present work is not large as in the case of X(3872) discussed in [23,25]. In our case, if a vertex detector is able to cumulate a number of charmed mesons by about 10 3 , a few X 0 (2900) and X 1 (2900) are expected to be yielded if they are S-wave tetraquark state and P -wave hadronic molecular state for J = 0 and J = 1, respectively.
In this sense, In the near future we intend to refine this phenomenological description of the hadronic medium, with inclusion of other processes, and perform improvements in the hydrodynamical model, to have a more compelling perspective on the role of the hot hadronic medium in the evolution of these states yielded at the end of the QGP phase.