Radiative decays of the spin-$2$ partner of $X(3872)$

The hadron spectroscopy of mesons and baryons with heavy quarks (charm and bottom) has been an important laboratory in the quest for understanding quantum chromodynamics (QCD) at the confinement scale due to the large bulk of experimental information accumulated over the last two decades. Many new hadronic states observed in the heavy sector seem to not fit into the predictions from the potential quark models such as the well-known Godfrey-Isgur quark model [1]. This fact has triggered a debate about the nature of those hadrons, and assumptions of different multiquark configurations beyond the conventional quark-antiquark/three quarks were put forward in order to explain their properties, such as mass, decay width, and the $J^{PC}$ quantum numbers, i.e., the total angular momentum $J$, parity $P$, and charge conjugation $C$ (see the reviews [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]).

Among these models, the molecular picture seems to be a natural one since the majority of the new hadrons are near some hadron-hadron threshold. In the charm sector, for instance, the $X(3872)$ state is just at the $D\bar{D}^{*}$ $(\bar{D}D^{*})$ threshold, and its properties are suitably described considering the $X(3872)$ as a $D\bar{D}^{*}$ molecular state (for a review focusing on the hadonic molecular model of the $X(3872)$, see Ref. [14]). In fact, the $X(3872)$ was the first among those new hadrons observed experimentally by the Belle Collaboration in 2003 [15], with $J^{PC}=1^{++}$ quantum numbers determined by the LHCb Collaboration a decade later [16]. It is, to date, the most well-studied state, and it is not a surprise that its experimental and theoretical information is used as inputs for predictions of new hadronic states in the heavy quark sector. In Ref. [17], the authors assumed the $X(3872)$ as a $D\bar{D}^{*}$ molecule and concluded that a $D^{*}\bar{D}^{*}$ state should exist as a consequence of the heavy-quark spin symmetry for the system under consideration. Specifically, using a contact-range (pionless) effective field theory, they claimed that the new state, from now on called $X_{2}$, is the spin-$2$ partner of the $X(3872)$, with a similar value for the binding energy and mass of about $4012$ MeV. Such a state was first predicted in Ref. [18] long ago with a mass $M=4015$ MeV and later in Refs. [19, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] using various phenomenological models.

Recently, the Belle collaboration reported a hint of an isoscalar structure with mass $M=(4014.3\pm 4.0\pm 1.5)$ MeV and width $\Gamma_{X_{2}}=(4\pm 11\pm 6)$ MeV, seen in the $\gamma\psi(2S)$ invariant mass distribution via a two-photon process [35]. The global significance is $2.8\sigma$. This new structure is located near the $D^{*}\bar{D}^{*}$ threshold which leads us to conclude that it is a promising candidate for the $D^{*}\bar{D}^{*}$ shallow bound state.¹¹1In Refs. [36, 37], this structure was assumed to be a $D^{*}\bar{D}^{*}$ molecule with $J^{PC}=0^{++}$. In addition, the mass value predicted in Refs. [17, 23] is in good agreement with the experimental one reported by Belle [35], and the measured width well matches the predicted one, of the order of a few MeV, in Ref. [27] despite that there is a sizeable uncertainty in the theoretical predictions [27, 28]. Thus, this narrow structure could be a hint for the $X_{2}$ state, supporting the theoretical predictions in Refs. [17, 18, 23].

Alternatively, by looking at the spectra of tetraquarks, there also exists spin partners of $1^{++}$ states that reproduce the $2^{++}$ quantum numbers of $X_{2}$ as well as its mass [38, 39, 40, 41]. Although the authors of these works were not particularly aiming at the $X_{2}$ state, it is still possible to assign the results to such a structure. On the other hand, a $2^{++}$ tensor state with a similar mass could also be described as a conventional $2P$ charmonium state [1, 42]; in this case, the $\chi_{c2}(3930)$ [43] would be an exotic meson.

One way to disentangle those different multiquark configurations from the molecular point of view is to check the mass splitting between the $2^{++}$ and $1^{++}$ states. In Refs. [17, 23], the corresponding mass splitting is approximately equal to that between the vector and pseudoscalar charmed mesons, that is

with $m_{D}(m_{D^{*}})$ the pseudoscalar (vector) charmed meson mass. On the other hand, within the tetraquark approach, for instance, in Ref. [38] the $2^{++}-1^{++}$ is about $80$ MeV, which is smaller than the difference given in Eq. (1). A similar conclusion is found for the difference between the first radially excited charmonia $2^{++}$ and $1^{++}$ by looking at the results for both the Godfrey-Isgur quark model [1] and the one using a screened potential [42], which are about $30$ MeV and $40$ MeV, respectively.

Even though only the Belle experiment has reported a signal relevant to the spin-$2$ partner of the $X(3872)$, such a structure can also be searched for in other ongoing and future experiments, e.g., BESIII and its upgrade, LHCb, and PANDA. Belle II also has plans to search for the $X_{2}$ state soon. In line with the current and upcoming experiments that will provide more information about such a structure, it is crucial to extend the theoretical studies surveying the $X_{2}$ system. In other words, we should further explore the $2^{++}$ tensor state to help discriminate the various multiquark models used to describe the $X_{2}$ structure.

The decays of a $2^{++}$ tensor structure have been studied in Refs. [44, 45]. In particular, considering that state as the first radial excitation of the $P$-wave $\chi_{c2}$ ($2^{3}P_{2}$) charmonium, the quark model adopted in Ref. [44, 45] provides width estimates for the $X_{2}$ decay to charmed mesons around tens of MeV. Moreover, the hadronic decays of the $D^{*}\bar{D}^{*}$ $S$-wave hadronic molecule, into $D\bar{D}$ and $D\bar{D}^{*}$ meson pairs were estimated to be of the order of a few MeV in Ref. [27] and can be as large as 50 MeV in Ref. [28].

Furthermore, the $X_{2}\to\gamma D\bar{D}^{*}$ decay width was also calculated in Ref. [27] to be of the keV order. In contrast to the hadronic decays that, according to Ref. [27], has a strong dependence on the ultraviolet (UV) form factors and therefore are sensitive to the short-distance details, radiative decays into $\gamma D\bar{D}^{*}$ are more sensitive to the long-distance structure of the resonance. Thus, as argued in Ref. [27], one can extract valuable information about the $X_{2}$ wave function, as well as about $D\bar{D}^{*}$ interactions, by surveying such decays. It is not difficult to understand this feature. In the $(D^{*}\bar{D}^{*})\to\gamma DD^{*}$ process, the final state receives leading contribution from the one-body transition $D^{*}\to D\gamma$, which has no direct relation to the two-body interaction accounting for the short-distance part of the $X_{2}$ state. Therefore, the long-range structure of $X_{2}$ that determines its coupling to $D^{*}\bar{D}^{*}$ has an essential role in its radiative decay into $\gamma D\bar{D}^{*}$.

Additional interesting decay modes of the $X_{2}$ which have not been explored before include the radiative decays into $\gamma\psi(2S)$ and $\gamma J/\psi$ channels. Such a study may help to discriminate the $X_{2}$ nature from the $c\bar{c}$ meson $\chi_{c2}(2P)$ possibility. As discussed in, e.g., Ref. [46], the radiative decay matrix element is proportional to the overlap between the wave functions corresponding to the initial and final states. Specifically, for transitions between two charmonia, that overlap is influenced by the position of the nodes of the wave function. Hence, the one-node wave function for the $\psi(2S)$ state has an overlap with the $\chi_{c2}(2P)$ larger than that for the $J/\psi$ one, which is nodeless, such that the following ratio

should be much larger than the one if the initial particle is the $\chi_{c2}(2P)$ charmonium. Table 1 shows some results for $R_{X_{2}}$ obtained in different quark models [44, 45]. Therefore, to confront these results, we evaluate the $X_{2}$ radiative decays into $\gamma\psi(2S)$ and $\gamma J/\psi$, assuming that the $X_{2}$ resonance is a $D^{*}\bar{D}^{*}$ molecular partner of the $X(3872)$ state, predicted in Refs. [17, 23] according to heavy quark spin symmetry (HQSS).

In order to calculate those radiative decay widths, we employ the couplings of the $\psi$ mesons to charmed mesons respecting HQSS and the magnetic and electric couplings of charmed mesons and a photon. The nonrelativistic effective field theory is applied to depict the coupling of the $X_{2}$ to $D^{*}\bar{D}^{*}$, which is related to the binding energy of $X_{2}$. After calculating the radiative decay widths for $X_{2}\to\psi\gamma$ with $\psi=J/\psi,\psi(2S)$, the upper limit of the ratio $R_{X_{2}}$ is predicted with the use of the experimental result reported by the BESIII collaboration $R_{X(3872)}<0.59$ at $90\%$ confidence level [47], and the results for the radiative decays of $X(3872)$ in the hadronic molecular picture [46]. Taking into account the signal yield of $X_{2}$ in the $\gamma\psi(2S)$ invariant mass distribution measured by Belle [35], and the predicted upper bound of $R_{X_{2}}$, we also estimate the lower limit of the signal yield of $X_{2}$ in the $\gamma J/\psi$ mode via the two-photon process at Belle, with $J/\psi$ reconstructed in lepton-antilepton pairs.

The structure of the paper is as follows. In Section II, we discuss the interaction Lagrangians and the main parameters used as well as the relevant Feynman diagrams contributing to the decay process $X_{2}\to\gamma\psi$. The radiative decay ratio $R_{X_{2}}$ and the signal yield of $X_{2}$ in the $\gamma J/\psi$ mode are predicted in Section III. A summary is given in Section IV. Finally, in Appendix A, we provide an update for the results corresponding to the radiative decay widths for $X(3872)\to\gamma\psi$ discussed in Ref. [46].