Exploring $T_{\psi\psi}$ tetraquark candidates in a coupled-channels formalism

Understanding the spectroscopy, structure and dynamics of exotic hadrons is one of the most challenging areas of contemporary physics research. In recent years, high-energy experiments have revealed a wealth of multiquark states that defy conventional explanations based on baryon ($qqq$) or meson ($q\bar{q}$) configurations. The seminal discovery of X(3872) by the Belle group Choi et al. (2003) marked a turning point, subsequently leading to the identification of several tetraquark states, including $Z_{c}(3900)$, $Z_{c}(4020)$ Ablikim et al. (2013a, b), $Z_{cs}(3985)^{-}$, $Z_{cs}(4220)^{+}$ Ablikim et al. (2021); Aaij et al. (2021), which exhibit charmonium-like properties, $Z_{b}(10610)$, $Z_{b}(10650)$ Bondar et al. (2012); Garmash et al. (2016), which resemble bottomonium states, or openly exotic states such as the $T_{cc}(3875)^{+}$ Aaij et al. (2022a, b) or the $T_{cs0}(2900)^{0}$, $T_{cs1}(2900)^{0}$ Aaij et al. (2020a) particles. The study of the properties and behaviour of these exotic hadrons promises to deepen our understanding of the fundamental interactions that govern the subatomic world, transcending the conventional quark compositions.

Recent breakthroughs have been made by the LHCb, CMS, and ATLAS Collaborations, as they have observed resonances in the di-$J/\psi$ and $J/\psi\psi^{\prime}$ ¹¹1For simplicity, in this work we will denote $\psi(2S)$ as $\psi^{\prime}$ and $\eta_{c}(2S)$ as $\eta_{c}^{\prime}$. invariant mass distributions Aaij et al. (2020b); Hayrapetyan et al. (2023); Zhang and Yi (2022); ATL (2022); Xu (2023) in proton-proton collision data at $\sqrt{s}=7$, $8$ and $13$ TeV. These resonances, with $cc\bar{c}\bar{c}$ minimum quark content, such as $T_{\psi\psi}(6200)$, $T_{\psi\psi}(6600)$, $T_{\psi\psi}(6900)$ and $T_{\psi\psi}(7200)$ ²²2In this work we will follow the naming convention of Ref. Gershon (2022). has sparked renewed interest in investigating fully charmed and beauty four-quark mesons. These experimental results provide a unique opportunity to test and refine our current understanding in this field.

The existence of heavy exotic mesons composed of two or four $c$ and $b$ quarks has intrigued researchers since the early stages of multiquark hadron studies Iwasaki (1976); Chao (1981); Ader et al. (1982); Badalian et al. (1987); Lloyd and Vary (2004); Berezhnoy et al. (2012); Karliner et al. (2017) and, since the experimental observation of $T_{\psi\psi}$ candidates a large number of theoretical studies have been devoted to explaining their properties, either as compact tetraquark states Park et al. (2019); Di et al. (2018); Lü et al. (2020); Karliner and Rosner (2020); Weng et al. (2021); Sonnenschein and Weissman (2021); Gordillo et al. (2020), diquark-antidiquark structures Debastiani and Navarra (2019); Chen et al. (2017); Wang et al. (2019); Bedolla et al. (2020); Giron and Lebed (2020); Jin et al. (2020); Deng et al. (2021); Faustov et al. (2021); Wang (2020); Mutuk (2021) and meson-meson molecules or coupled-channels effects Debastiani and Navarra (2019); Dong et al. (2021); Jin et al. (2020); Deng et al. (2021); Albuquerque et al. (2020); Guo and Oller (2021); Agaev et al. (2023); Niu et al. (2023).

Many of these exotic states, such as $X(3872)$, $Z_{b}(10610)$, $Z_{c}(3900)$, $P_{c}(4470)$ and others, tend to emerge close a two-hadron threshold. It is therefore tempting to infer a molecular nature for such kind of states. Similarly, many of the recent $T_{\psi\psi}$ states such as the $T_{\psi\psi}(6200)$, $T_{\psi\psi}(6600)$ or the $T_{\psi\psi}(6900)$, are close to many charmonium-charmonium thresholds such as the $J/\psi J/\psi$, $\eta_{c}\eta_{c}^{\prime}$ or the $J/\psi\psi^{\prime}$ threshold, respectively. Motivated by these observations, this study investigates the properties of the $T_{\psi\psi}$ candidates $T_{\psi\psi}(6200)$, $T_{\psi\psi}(6600)$, $T_{\psi\psi}(6700)$, $T_{\psi\psi}(6900)$ and $T_{\psi\psi}(7200)$ in a coupled-channels formalism based on a constituent quark model (CQM) Vijande et al. (2005); Segovia et al. (2013), which has been widely used in the heavy quark sector Segovia et al. (2008, 2016); Ortega et al. (2020) and extended to the study of other exotic states such as the $X(3872)$ Ortega et al. (2010, 2017, 2019, 2021, 2022), the $T_{cc}^{+}$ Ortega et al. (2023a) or the $T_{cs}$ and $T_{c\bar{s}}$ states Ortega et al. (2023b). The advantage of using an approach with a relatively long history is that all model parameters are already constrained by previous works. Consequently, from this point of view, we present a parameter-free calculation of the $T_{\psi\psi}$ states, extending our recent analysis of the similar $T_{cc}^{+}$ and $T_{cs}$ exotic candidates Ortega et al. (2023a, b).

The organization of the manuscript is as follows: After this introduction, section II provides a brief overview of the theoretical framework. Section III primarily focuses on the analysis and discussion of our theoretical findings. Lastly, in Sec. IV, we present a summary of our work and draw conclusions based on the obtained results.

In this work we will explore the $T_{\psi\psi}$ tetraquark candidates as meson-meson molecules. This system has many similar features as the recently discovered $T_{cc}^{+}$ tetraquark, with minimum quark content $cc\bar{u}\bar{d}$. Then, for the $T_{\psi\psi}$ we will follow the same formalism as in Ref. Ortega et al. (2023a), where the $T_{cc}^{+}$ was described as a $J^{P}=1^{+}$ $DD^{*}$ molecule. For this reason, in this section we will only briefly provide the most relevant theoretical aspects for the study of the $T_{\psi\psi}$ states.

The constituent quark model (CQM) employed in this work has been extensively detailed in the literature. For a full description, including expressions of all the potentials and the values of the model parameters, the reader is kindly referred to Ref. Vijande et al. (2005) and its update Ref. Segovia et al. (2008).

The main elements of our constituent quark model (CQM) encompass the constituent light quark masses and the exchanges involving Goldstone bosons, which arise as manifestations of the dynamical breaking of chiral symmetry in Quantum Chromodynamics (QCD). Additionally, the model incorporates the perturbative interaction of one-gluon exchange (OGE) and a non-perturbative confinement interaction Vijande et al. (2005); Segovia et al. (2013). However, it is worth noticing that, whereas the Goldstone boson exchanges are considered for two light quarks ($qq$), they are not allowed in the light-heavy ($qQ$) and heavy-heavy ($QQ$) configurations.³³3Here, we denote $q=\{u,d,s\}$ and $Q=\{c,b\}$. On the contrary, the most important contributions of the one-gluon exchange and confinement potentials are flavour-blind and are the only interactions relevant for this work, where all the quarks involved are beyond the chiral symmetry breaking scale.

Regarding the confinement interaction, while it has been proven that multi-gluon exchanges generate an attractive potential that rises linearly with the distance between infinitely heavy quarks Bali (2001), it is essential to consider the influence of sea quarks on the strong interaction dynamics. Sea quarks contribute to screening the rising potential at low momenta and eventually lead to the breaking of the quark-antiquark binding string Bali et al. (2005). To account for this behaviour, our CQM incorporates the following expression:

where $a_{c}$ and $\mu_{c}$ are model parameters. At short distances this potential exhibits a linear behavior with an effective confinement strength, $\sigma=-a_{c}\,\mu_{c}\,(\vec{\lambda}^{c}_{i}\cdot\vec{\lambda}^{c}_{j})$. However, it becomes constant at large distances, with a threshold defined by $\{\Delta-a_{c}\}(\vec{\lambda}^{c}_{i}\cdot\vec{\lambda}^{c}_{j})$.

Additionally, the model incorporates QCD perturbative effects mediated by the exchange of one gluon, derived from the vertex Lagrangian

Here, $\alpha_{s}$ represents an effective scale-dependent strong coupling constant, given by

where $\mu$ is the reduced mass of the $q\bar{q}$ pair and $\alpha_{0}$, $\mu_{0}$ and $\Lambda_{0}$ are parameters of the model Segovia et al. (2013).

The described CQM details the $qq$ ($q\bar{q}$) interaction at microscopic level and allows us to build the $c\bar{c}$ meson spectra Segovia et al. (2013, 2008), by solving the two-body Schrödinger equation through the use of the Gaussian Expansion Method Hiyama et al. (2003). This computational approach not only simplifies the evaluation of the necessary matrix elements but also ensures a satisfactory level of accuracy.

In order to describe the $c\bar{c}-c\bar{c}$ interaction from the underlying $qq$ dynamics we employ the Resonating Group Method Wheeler (1937). For that, we assume that the wave function of a system composed of two charmonium mesons $A$ and $B$ can be written as

where $\phi_{A(B)}$ is the wave functions of the $A(B)$ meson, $\chi_{L}$ the relative orbital wave function of the $AB$ pair, $\sigma_{ST}$ their spin-isospin wave function and $\xi_{c}$ their color wave function.

As we have two pair of identical quarks, we have to consider the full antisymmetric operator ${\cal A}$, so the wave function is completely antisymmetric. For the $c\bar{c}-c\bar{c}$ system, this operator can be written as ${\cal A}=(1-P_{c})(1-P_{\bar{c}})$, up to a normalization factor, where $P_{c}$ is the operator that exchanges $c$ quarks and $P_{\bar{c}}$ the operator that exchanges charm antiquarks between mesons. Following Ref. Ortega et al. (2023a), for identical mesons, the antisymmetrizer is reduced to $\Psi=(1-P_{\bar{c}})\left\{|\phi_{A}\phi_{B}\chi_{L}\sigma_{ST}\xi_{c}\rangle\right\}$, whereas for non-identical mesons, the wave functions is a combination of $AB$ and $BA$ configurations, given by

with $\mu=L+S-J_{A}-J_{B}$.

As the charmonium states are eigenstates of the $C$-parity operator, the $C$ parity of the $AB$ pair is defined as $(-1)^{L_{A}+S_{A}+L_{B}+S_{B}}$. Hence, it is equal to $C=1$ for $PP$ and $VV$ channels (where $P$ is a pseudoscalar meson and $V$ a vector meson) and $C=-1$ for $PV$ channels.

The interaction between $c\bar{c}-c\bar{c}$ mesons can be split into a direct term, with no quark exchange between clusters, and an exchange kernel, which incorporates them. The direct potential $V_{D}(\vec{P}^{\prime},\vec{P}_{i})$ can be written as

where $V_{ij}$ is the CQM potential between the quark $i$ and the quark $j$ of the mesons $A$ and $B$, respectively.

The exchange kernel $K_{E}$, that models the quark rearrangement between clusters, can be written as

which is a non-local and energy-dependent kernel, separated into a potential term $H_{E}$ plus a normalization term $N_{E}$. Here, $E_{T}$ denotes the total energy of the system and $\vec{P}_{i}$ is a continuous parameter. The exchange Hamiltonian and normalization can be written as

where ${\cal H}$ is the Hamiltonian at quark level.

The properties of the $T_{\psi\psi}$ tetraquark candidates, investigated here as meson-meson molecular systems, will be obtained as poles of the scattering matrix, given in non-relativistic kinematics as,

where $k_{\alpha}$ and $\mu_{\alpha}$ represents the on-shell momentum and reduced mass for channel $\alpha$, respectively. The $T$ matrix of the coupled-channels calculation is obtained from the Lippmann-Schwinger equation

where $\beta$ represents the set of quantum numbers necessary to determine a partial wave in the meson-meson channel, $V_{\beta}^{\beta^{\prime}}(p^{\prime},p)$ is the full RGM potential, sum of direct and exchange kernels, and $E_{\beta^{\prime\prime}}(q)$ is the energy for the momentum $q$ referred to the lower threshold.

The mass and the total width of resonances can be directly obtained from the complex energy of the poles, $\bar{E}=M_{r}-i\,\frac{\Gamma_{r}}{2}$. However, some caution should be taken in order to obtain the partial widths of the resonances to a specific final meson-meson channel. For that, we will follow Refs. Ortega et al. (2013); Grassi et al. (2001). In the neighborhood of a resonance, the $S$ matrix can be approximated as

where $g^{\beta}$ are the residues of the pole, which can be interpreted as the amplitude of the resonance to the final state. The partial width of the resonance to the final state $f$ can be defined as

where the integral is over the phase space of the final state with $\left(\sum_{n}p_{n}\right)^{2}=M_{r}^{2}$, with $M_{r}$ the mass of the resonance. In the case of a two meson decay, $\hat{\Gamma}_{\beta}$ can be written as

where ${k_{0}}_{\beta}$ is the relativistic onshell momentum of the final two meson state.

It is worth noticing that Eq. (13) does not guarantee that the sum of the partial widths must be equal to the total width. In fact, it is expected that $\sum_{f}\hat{\Gamma}_{f}\neq\Gamma_{r}$. To solve this problem we define the branching ratios as Grassi et al. (2001)

so the physical partial widths are given, as usual, by

with $\Gamma_{r}=-2\imaginary(\bar{E})$.

In this section we present the results of the coupled-channels calculation of the $c\bar{c}-c\bar{c}$ system in $J^{P}=0^{\pm},1^{\pm},2^{\pm}$. We have included the channels and partial waves shown in Table 1, which are the combination of the lowest lying $S$-wave charmonium resonances, that’s it: $J/\psi$, $\psi^{\prime}$, $\eta_{c}$ and $\eta_{c}^{\prime}$. We restrict ourselves to relative orbital momenta $L\leq 2$, since higher ones are expected to be negligible.

Direct interactions are only driven by gluon annihilation diagrams, which are rather small for charmonium. Confinement potential does not have direct interaction because we deal with a two-color-singlet system. Thus, the leading interaction is the exchange diagrams. This implies that their identification as pure molecules is questionable as we are not dealing with a residual direct interaction, but a short-range interaction that mixes quarks. Nevertheless, in this work we will denote the found states as molecules, in a broad sense of a resonant state of two colourless mesons, regardless of the binding mechanism.

Before presenting the results, it is worth mentioning that there is a theoretical uncertainty in the results as a consequence on the way the model parameters are adjusted to describe a certain number of hadron observables. Such fitting is done within a determinate range of agreement with the experiment, which is estimated to be around 10-20% for physical observables that help to fix the model parameters. This range of agreement will be taken as an estimate of the model uncertainty for the derived quantities and, in order to analyse its effect, we will estimate the error of the pole properties by varying the strength of the potentials by $\pm 10\%$.

The results of our calculations are shown in Table 2 (masses, widths and branching ratios). We find up to 29 poles in different $J^{P}$ sectors, that’s it: $2$ in $0^{-}$, $9$ in $0^{+}$, $5$ in $1^{-}$, $5$ in $1^{+}$, $2$ in $2^{-}$ and $6$ in $2^{+}$. Their masses range from $6.1$ to $7.6$ GeV and are quite broad. Due to Heavy Quark Spin Symmetry, the states are relatively degenerate between the $\{0^{-},1^{-},2^{-}\}$ and the $\{0^{+},1^{+},2^{+}\}$ sectors, but there are significant deviations due to the specific partial waves on each sector.

The most explored detection channels are $J/\psi J/\psi$ and $J/\psi\psi^{\prime}$. In Table 2 we can identify up to $13$ states with significant branching ratios to the $J/\psi J/\psi$ channel, and another $12$ states that can decay to the $J/\psi\psi^{\prime}$ channel. Among them, we can identify candidates for the experimental states $T_{\psi\psi}(6200)$, $T_{\psi\psi}(6600)$, $T_{\psi\psi}(6700)$, $T_{\psi\psi}(6900)$ and $T_{\psi\psi}(7200)$, which are described in more detail below.

Additionally, we have candidates that do not decay to the above channels. For example, the two $0^{--}$ and $2^{--}$ wide resonances with masses around $6740$ MeV/c² decay only to $\eta_{c}\psi^{\prime}$, while the two $0^{++}$ and $1^{+-}$ states with masses around $6100$ MeV/c² can only decay to $\eta_{c}\eta_{c}$ and $\eta_{c}J/\psi$, respectively. We also find a broad resonance in the $1^{--}$ sector with a mass of $6822_{-4}^{+3}$ MeV/c² and a width of $405_{-16}^{+18}$ MeV, which decays mostly to $\eta_{c}\psi^{\prime}$ and $\eta_{c}^{\prime}J/\psi$. Recently, Belle Collaboration searched for double-charmonium states in the $e^{+}e^{-}\to\eta_{c}J/\psi$ reaction and found no significant signal Yin et al. (2023). This is consistent with our results and points to $e^{+}e^{-}\to\eta_{c}\psi^{\prime}$ and $e^{+}e^{-}\to\eta_{c}^{\prime}J/\psi$ as more promising reactions.