$X(3872)$ production in high energy heavy ion collisions

We have determined the production cross sections of the $X(3872)$ state in the reactions $\overline{D}D\to \pi X$, ${\overline{D}}^{*}D\to \pi X$, and ${\overline{D}}^{*}{D}^{*}\to \pi X$, information which is useful for studies of the $X(3872)$ meson abundance in heavy ion collisions. We construct a formalism considering $X$ as a molecular bound state of ${\overline{D}}^0{D}^{*0}-\mathrm{c}.\mathrm{c}.$, $D^{-}{D}^{*+}-\mathrm{c}.\mathrm{c}.$, and $D_s^{-}{D}_s^{*+}-\mathrm{c}.\mathrm{c}.$ To obtain the amplitudes related to these processes we have made use of effective field Lagrangians. The evaluation of the cross section of the processes involving $D^{*}$ meson(s) requires the calculation of an anomalous vertex, $X{\overline{D}}^{*}{D}^{*}$, which has been obtained by considering triangular loops motivated by the molecular nature of $X(3872)$. Proceeding in this way, we have evaluated the cross section for the reaction ${\overline{D}}^{*}D\to \pi X$, and find that the diagrams involving the $X{\overline{D}}^{*}{D}^{*}$ vertex give a large contribution. Encouraged by this finding we estimate the $X{\overline{D}}^{*}{D}^{*}$ coupling, which turns out to be $1.95\pm 0.22$. We then use it to obtain the cross section for the reaction ${\overline{D}}^{*}{D}^{*}\to \pi X$ and find that, in this case too, the $X{\overline{D}}^{*}{D}^{*}$ vertex is relevant. We also discuss the role of the charged components of $X$ in the determination of the production cross sections.

Figure 1

Diagrams contributing to the process $\overline{D}D\to \pi X$ (top) and ${\overline{D}}^{*}D\to \pi X$ (bottom). The diagram containing a filled box is calculated by summing the set of diagrams shown in Fig. 2, as explained in the text.

Figure 2

Diagrams considered for the determination of the filled box shown in Fig. 1. The hadrons $P_X$ and $V_X$ represent the pseudoscalars and vectors coupling to the state $X$, while $P$ and $V$ are any pseudoscalar and vector mesons which can be exchanged conserving different quantum numbers. For a list of the different exchanged hadrons considered here see Tables 5 and 6 of the Appendix pp1.

Figure 3

Cross section for the reaction $\overline{D}D\to \pi X$ considering only the neutral components of $X$ (dashed line) and adding the charged components (solid line).

Figure 4

Cross section for the reaction ${\overline{D}}^{*}D\to \pi X$. The solid line has the same meaning as in Fig. 3, and we have shown it for the purpose of comparison. The dashed line represents the result for the cross section of the process ${\overline{D}}^{*}D\to \pi X$ considering only the $t$-channel diagram in Fig. 1. The shaded region is the result obtained with both $t$- and $u$-channel diagrams of Fig. 1 considering cutoffs in the range 700–1000 MeV.

Figure 5

Cross section for the reaction ${\overline{D}}^{*}D\to \pi X$. The dark color shaded region has the same meaning as the shaded region in Fig. 4. The light color shaded region represents the result for the cross section when considering the Lagrangian in Eq. (11) to determine the $X{\overline{D}}^{*}{D}^{*}$ vertex with the value of the coupling given in Eq. (31).

Figure 6

Different diagrams contributing to the reaction ${\overline{D}}^{*}{D}^{*}\to \pi X$.

Figure 7

Cross section for the reaction ${\overline{D}}^{*}{D}^{*}\to \pi X$. The solid line represents the cross section without the contribution from the diagrams in Fig. 6 and 6, which contain the vertex $X{\overline{D}}^{*}{D}^{*}$. The shaded region represents the result for the cross section when including the contribution of all the diagrams in Fig. 6, with the vertex $X{\overline{D}}^{*}{D}^{*}$ obtained using the Lagrangian in Eq. (11) with the value of the coupling given in Eq. (31).

Figure 8

Cross sections for the different reactions studied. The dashed line, dark shaded region, and light shaded region represent the total cross section for the $\overline{D}D\to \pi X$, ${\overline{D}}^{*}D\to \pi X$, and ${\overline{D}}^{*}{D}^{*}\to \pi X$ reactions, respectively.

Figure 9

Cross sections for the different reactions studied using form factors. The dashed line, dark shaded region, and light shaded region correspond to the total cross section for the $\overline{D}D\to \pi X$, ${\overline{D}}^{*}D\to \pi X$, and ${\overline{D}}^{*}{D}^{*}\to \pi X$ reactions, respectively.