$X(3872)$ production and absorption in a hot hadron gas

We calculate the time evolution of the $X(3872)$ abundance in the hot hadron gas produced in the late stage of heavy ion collisions. We use effective field Lagrangians to obtain the production and dissociation cross sections of $X(3872)$. In this evaluation we include diagrams involving the anomalous couplings $\pi D^*\bar{D}^*$ and $X \bar{D}^{\ast} D^{\ast}$ and also the couplings of the $X(3872)$ with charged $D$ and $D^*$ mesons. With these new terms the $X(3872)$ interaction cross sections are much larger than those found in previous works. Using these cross sections as input in rate equations, we conclude that during the expansion and cooling of the hadronic gas, the number of $X(3872)$, originally produced at the end of the mixed QGP/hadron gas phase, is reduced by a factor of 4.

The structure of the new charmonium states has been subject of an intense debate. In the case of X(3872), calculations using constituent quark models give masses for possible charmonium states with J P C = 1 ++ quantum numbers, which are much bigger than the observed X(3872) mass: 2 3 P 1 (3990) and 3 3 P 1 (4290) [15]. These results, together with the observed isospin violation in their hadronic decays, motivated the conjecture that these objects are multiquark states, such as mesonic molecules or tetraquarks. Indeed, the vicinity of the X mass to theDD * threshold inspired the proposal that the X(3872) could be a molecularDD * bound state with a small binding energy [16,17]. Another well known interpretation of the X(3872) is that it can be a tetraquark state resulting from the binding of a diquark and an antidiquark [18,19]. There are other proposals as well [2][3][4]. One successful approach in describing experimental data is to treat the X as an admixture of two and four-quark states [20].
Until now it has not been possible to determine the strucute of the X, since the existing data on masses and decay widths can be explained by quite different models. However this situation can change, as we address the production of exotic charmonium in hadronic reactions, i.e. proton-proton, proton-nucleus and nucleus-nucleus collisions. Hadronic collisions seem to be a promising testing ground for ideas about the structure of the new states. It has been shown [3,21] that it is extremely difficult to produce meson molecules in p p collisions. In the molecular approach the estimated cross section for X(3872) production is two orders of magnitude smaller than the measured one. On the other hand, in Ref. [22] a simple model was proposed to compute the X production cross section in p p collisions in the tetraquark approach. Predictions were made for the next run of the LHC.
As pointed out in Refs. [23,24], high energy heavy ion collisions offer an interesting scenario to study the production of multiquark states. In these processes, a quite large number of heavy quarks are expected to be produced, reaching as much as 20 cc pairs per unit rapidity in Pb + Pb collisions at the LHC. Moreover, the formation of quark gluon plasma (QGP) may enhance the production of exotic charmonium states, since the charm quarks are free to move over a large volume and they may coalesce to form bound states at the end of the QGP phase or, more precisely, at the end of the mixed phase, since the QGP needs some time to hadronize. One of the main conclusions of Refs. [23,24] was that, if the production mechanism is coalescence, then the production yield of these exotic hadrons at the moment of their formation strongly reflects their internal structure. In particular it was shown that in the coalescence model the production yield of the X(3872), at RHIC or LHC energies, is almost 20 times bigger for a molecular structure than for a tetraquark configuration.
After being produced at the end of the quark gluon plasma phase, the X(3872) interacts with other hadrons during the expansion of the hadronic matter. Therefore, the X(3872) can be destroyed in collisions with the comoving light mesons, such as X + π →D + D, X + π →D * + D * , but it can also be produced through the inverse reactions, such as D +D → X + π,D * + D * → π + X. We expect these cross sections to depend on the spatial configuration of the X(3872). Charm tetraquarks in a diquark -antidiquark configuration (cq) − (cq) have a typical radius comparable to (or smaller than) the radius of the charmonium groundstates, i.e. r 4q ≃ rc c ≃ 0.3 − 0.4 fm. Charm meson molecules are bound by meson exchange and hence r mol ≃ 1/m π ≃ 1.5 fm. In fact, the calculations of Ref. [25] show that r mol ≃ 2.0 − 3.0 fm. Molecules are thus much bigger than tetraquarks and their absorption cross sections may also be much bigger. On the other hand, when these states are produced from D −D * fusion in a hadron gas, what matters is the overlap between the initial and final state configurations. Assuming that the radius of the D and D * mesons is r D ≃ 0.6 fm [26], the initial D + D * state has a larger spatial overlap with a molecule than with a tetraquark and, therefore, the production of molecules is favored. Hence from geometrical arguments we expect that in a hot hadronic environment molecules are easier to produce and also easier to destroy than tetraquarks. Of course geometrical estimates of cross sections are more reliable if we apply them to high energy collisions. Here the typical collision energies are of the order of the temperature T ≃ 100 − 180 MeV and are probably not high enough. Nevertheless, at a qualitative level, they can be useful as guidance.
In Ref. [27] the interactions of the X in a hadronic medium were studied in the framework of SU (4) effective Lagrangians. The authors computed the corresponding production and absorption cross sections, finding that the absorption cross section is two orders of magnitude larger than the production one. The effective Lagrangians include the X particle as a fundamental degree of freedom and the theory is unable to distinguish between molecular and tetraquark configurations. Presumably this information might be included in the form factors introduced in the vertices. The authors find that it is much easier to destroy the X than to create it. In particular, for the largest thermally averaged cross sections they find: < σv > πX→D * D * ≃ 30 < σv > D * D * →πX . In spite of this difference, the authors of Ref. [27] arrived at the intriguing conclusion that the number of X's stays approximately constant during the hadronic phase. In Ref. [27] the coupling of the X(3872) with charged charm mesons (such as D − D * + ) was neglected and only neutral mesons were considered (such as D 0D0 * ). Moreover, the terms with anomalous couplings were not included in the calculations. In Ref. [28] we showed that the inclusion of the couplings of the X(3872) to charged D's and D * 's and those of the anomalous vertices, πD * D * and XD * D * , increases the cross sections by more than one order of magnitude. Similar results were also observed in the case of J/ψπ cross section [29]. These anomalous vertices also give rise to new reaction channels, namely,D + D * → π + X and π + X →D + D * . Thus it is important to evaluate the changes that the above mentioned contributions can produce in the X abundance (and in its time evolution) in reactions as those considered in Ref. [27]. This is the subject of this work.
The formalism used in Refs. [27] and [28] was originally developed to study the interaction of charmonium states (specially the J/ψ) with light mesons in a hot hadron gas many years ago [29,30]. The conclusions obtained in the past can help us now, giving some baseline for comparison. For example, if it is true that the X has a large cc component, we may expect that the corresponding production and absorption cross sections are comparable to the J/ψ ones. If, alternatively, they turn out to be much larger, this could be an indication of a strong multiquark and possibly molecular component.
The paper is organized as follows. In the next section we discuss the cross sections averaged over the thermal distributions. In Sec.III we investigate the time evolution of the X(3872) abundance by solving the kinetic equation based on the phenomenological model of Ref. [27]. Finally in Sec.IV we present our conclusions.

II. CROSS SECTIONS AVERAGED OVER THE THERMAL DISTRIBUTION
In this section we calculate the cross sections averaged over the thermal distributions for the processesDD → πX, D * D → πX andD * D * → πX, and for the inverse reactions. This information is the input to the study of the time evolution of the X(3872) abundance in hot hadronic matter. In Fig. 1 we show the different diagrams contributing to each process. In Ref. [27] it was shown that the contribution from the reactions involving the ρ meson is very small compared to the reactions with pions and thus we neglect the former in what follows. The cross sections for the processes shown in Fig. 1 were obtained in Ref. [28]. Using effective Lagrangians based on SU(4), the coupling of the X(3872) toD * D * was estimated through the evaluation of triangular loops and an effective Lagrangian was proposed to describe this vertex. As a result, it was found that the contributions coming from the coupling of the X(3872) to charged D's and D * 's and from the anomalous vertices play an important role in determining the cross sections. The coupling constant of the XD * D * vertex was found to be g XD * D * = 12.5 ± 1.0. For more details about the calculations, we refer the reader to Ref. [28]. In the present manuscript, we follow Ref. [28] and obtain the cross sections of the processes in Fig. 1 using a form factor of the type where Λ = 2.0 GeV is the cutoff and q the three-momentum transfer in the center of mass frame. Following Refs. [27,31], the thermally averaged cross section for a process ab → cd can be calculated using the expression where f a and f b are Bose-Einstein distributions, σ ab→cd are the cross sections computed in [28], v ab represents the relative velocity of the two interacting particles (a and b), α i = m i /T , with m i being the mass of particle i and T the temperature, z 0 = max(α a + α b , α c + α d ), and K 1 and In Fig. 2a we show the thermally averaged cross section for the processDD → πX(3872), considering only the coupling of the X(3872) to the neutral statesD 0 D * 0 (solid line) and adding the coupling to the charged components (dashed line). As can be seen, the thermally averaged cross section increases by a factor of about 2.5 when we include the charged D's and D * 's.
In Figs. 2b and 2c we show the thermally averaged cross sections for the processesD * D → πX(3872) and D * D * → πX(3872), considering only the coupling of the X to neutral D's and D * 's (dashed line), including couplings to charged D's and D * 's (dotted line) and finally adding also the contribution from the anomalous vertices (shadded region). As can be seen, the contribution from the anomalous vertices produces an enhancement of the thermally averaged cross sections by a factor of 100 − 150.
To close this section we show in Fig. 3a the total thermally averaged cross sections for the processes involving the production of the X(3872) state, i.e.,DD → πX,D * D → πX andD * D * → πX reactions, while in Fig. 3b we show the inverse processes, i.e., the dissociation of X(3872) through the reactions πX →DD, πX →D * D, πX →D * D * , respectively. For the latter cases, we use the principle of detailed balance to determine the corresponding cross sections. Figure 3 should be compared with the Fig. 3 of Ref. [27]. Our cross sections are a factor 100 larger in reactions with D * D * in the initial or final state This can be attributed mostly to the inclusion of the anomalous terms. Moreover, our cross sections are a factor 10 larger in the case of DD mesons in the initial or final state. The difference comes from the inclusion of the coupling of the X to charged charged D's and D * 's, which was not considered in Ref. [27].
On the other hand, in both works, the absorption cross sections are more than fifty times larger than the production ones.
In the computation of the time evolution of the X abundance, we will need to know how the temperature changes with time and this is highly model dependent. Fortunately, as one can see in Fig. 3, the dependence of < σv > on the temperature is relatively weak.

III. TIME EVOLUTION OF THE X(3872) ABUNDANCE
Following Ref. [27] we study the yield of X(3872) in central Au-Au collisions at √ s N N = 200 GeV. By using the thermally averaged cross sections obtained in the previous section, we can now analyze the time evolution of the X(3872) abundance in hadronic matter, which depends on the densities and abundances of the particles involved in the processes of Fig. 1, as well as the cross sections associated with these reactions (and the corresponding inverse reactions), Figs. 3a and 3b. The momentum-integrated evolution equation has the form [27,[31][32][33] where N X (τ ), N c ′ (τ ), n c (τ ) and n π (τ ) are the abundances of X(3872), of charmed mesons of type c ′ , of charmed mesons of type c and of pions at proper time τ , respectively. The term R QGP (τ ) in Eq. (3) represents the X production from the quark-gluon plasma in the mixed phase, since the hadronization of the QGP takes a finite time, and it is given by [27,33]: where the times τ C = 5.0 fm/c and τ H = 7.5 fm/c determine the beginning and the end of the mixed phase respectively. The constant N 0 X corresponds to the total number of X(3872) produced from quark-gluon plasma. To solve Eq. (3) we assume that the pions and charmed mesons in the reactions contributing to the abundance of X are in equilibrium. Therefore N c ′ (τ ), n c (τ ) and n π (τ ) can be written as [27,[31][32][33] where γ i and g i are the fugacity factor and the spin degeneracy of particle i respectively. As can be seen in Eq. (5), the time dependence in Eq. (3) enters through the parametrization of the temperature T (τ ) and volume V (τ ) profiles suitable to describe the dynamics of the hot hadron gas after the end of the quark-gluon plasma phase. Following Refs. [27,32,33], we assume the τ dependence of T and V to be given by [27,32,33] These expressions are based on the boost invariant picture of Bjorken [34] with an accelerated transverse expansion. In the above equation R C = 8.0 fm denotes the final size of the quark-gluon plasma, while v C = 0.4 c and a C = 0.02 c 2 /fm are its transverse flow velocity and transverse acceleration at τ C respectively. The critical temperature of the quark gluon plasma to hadronic matter transition is T C = 175 MeV; T H = T C = 175 MeV is the temperature of the hadronic matter at the end of the mixed phase. The freeze-out takes place at the freeze-out time τ F = 17.3 fm/c, when the temperature drops to T F = 125 MeV.
To solve Eq. (3), we assume that the total number of charm quarks in charm hadrons is conserved during the production and dissociation reactions, and that the total number of charm quark pairs produced at the initial stage of the collisions at RHIC is 3, yielding the charm quark fugacity factor γ C ≈ 6.4 in Eq. (5) [24,27]. In the case of pions, we follow Ref. [33] and work with the assumption that their total number at freeze-out is 926, which fixes the value of γ π appearing in Eq. (5) to be ∼ 1.725 . In Ref. [27] the authors studied the yields obtained for the X(3872) abundance within two different approaches: the statistical and the coalescence models. In the statistical model, hadrons are produced in thermal and chemical equilibrium. This model does not contain any information related to the internal structure of the X(3872) and, for this reason we do not consider it in this work. In the case of the coalescence model, the determination of the abundance of a certain hadron is based on the overlap of the density matrix of the constituents in an emission source with the Wigner function of the produced particle. This model contains information on the internal structure of the considered hadron, such as angular momentum, multiplicity of quarks, etc. According to Ref. [24], the number of X(3872) produced at the end of the mixed phase, assuming that the X(3872) is a tetraquark state with J P C = 1 ++ , is given by: In order to determine the time evolution of the X(3872) abundance we solve Eq. (3) starting at the end of the mixed phase, i.e. at τ H = 7.5 fm/c, and assuming that the X(3872) is a tetraquark, formed according to the prescription of the coalescence model. The initial condition is given by Eq. (7). We use this initial abundance to integrate Eq. (3) and we show the result in Fig. 4. In the figure the solid line represents the result obtained using the same approximations as those made in Ref. [27]. Our curve is slightly different from that of Ref. [27] because we did not include the contribution of the ρ mesons, as discussed earlier. The dashed line shows the result when we include the couplings of the X(3872) to charged D's and D * 's. The light-shaded band shows the results obtained with the further inclusion of the diagrams containing the anomalous vertices. The band reflects the uncertainty in the XD * D * coupling constant, which is g XD * D * = 12.5 ± 1.0 [28]. As can be seen, without the inclusion of the anomalous coupling terms, the abundance of X remains basically constant. This is because the magnitude of the thermally averaged cross sections for the X production and absorption reactions obtained within this approximation is so small that the second term in the right hand side of Eq. (3) is completely negligible compared to the first term. When including the coupling of the X to charged D's and D * 's we basically do not find any important change for the time evolution of the X abundance, since, as can be seen in Fig. 2, the thermally averaged cross sections do not change drastically in this case. On the other hand, the inclusion of the anomalous coupling terms, depicted in Figs. 1c, 1d, 1f, 1g and 1h, modifies the behavior of the X(3872) yield, producing a fast decrease of the X abundance with time. This is the main result of this work. We emphasize that the X(3872) abundance, whose time evolution was studied above, is the only one which comes from the QGP and is what could be observed if the X(3872) is a tetraquark state. However, if the X(3872) is a molecular state, it will be formed by hadron coalescence at the end of the hadronic phase. According to Ref. [27], at this time the average number of X's, considering it as a DD * molecule, is which is about 80 times larger than the contribution for a tetraquark state at the end of the hadronic phase (see Fig. 4). We can then conclude that the QGP contribution for the X(3872) production (as a tetraquark state and from quark coalescence) after being suppressed during the hadronic phase, becomes insignificant at the end of the hadronic phase.

IV. CONCLUDING REMARKS
In this work we have studied the time evolution of the X(3872) abundance in heavy ion collisions. If the X(3872) is a tetraquark state it will be produced at the mixed phase by quark coalescence. After being produced at the end of the quark gluon plasma phase, the X(3872) interacts with other comoving hadrons during the expansion of the hadronic matter. Therefore, the X(3872) can be destroyed in collisions with the comoving light mesons, such as X + π →D + D, X + π →D * + D * but it can also be produced through the inverse reactions, such as D +D → X + π, D * + D * → π + X. In this work we have considered the contributions of anomalous vertices, πD * D * and XD * D * , and the contributions from charged D and D * mesons to the X(3872) production and dissociation cross sections. These vertices, apart from enhancing the cross sections associated with theD * D * channel, give rise to additional production/absorption mechanisms of X, which are found to be relevant.
The cross sections, averaged over the thermal distribution, have been used to analyze the time evolution of the X(3872) abundance in hadronic matter. We have found that the abundance of a tetraquark X drops from N X(4q) ≈ 4.0 × 10 −5 at the begining of the hadronic phase [27] to N X(4q) ∼ 1.0 × 10 −5 at the end of the hadronic phase.
On the other hand, if the X(3872) is a molecular state it will be produced by hadron coalescence at the end of the hadronic phase. According to Ref. [27], at this time the average number of X's, considering it as a DD * molecule, is N X(mol) ≈ 7.8 × 10 −4 , which is about 80 times larger than N X(4q) .
As expected, the results show that the X multiplicity in relativistic ion collisions depends on the structure of X(3872). Our main conclusion is that the contribution from the anomalous vertices play an important role in determining the time evolution of the X(3872) abundance and they lead to strong suppression of this state during the hadronic phase. Therefore, within the uncertainties of our calculation we can say that if the X(3872) were observed in a heavy ion collision it must have been produced at the end of the hadronic phase and, hence, it must be a molecular state.