Spectroscopy of the hidden-charm $[qc][\bar q \bar c]$ and $[sc][\bar s \bar c]$ tetraquarks

We calculate the spectrum of $q\bar q c \bar c$ and $s\bar s c \bar c$ tetraquarks, where $q$, $s$ and $c$ stand for light ($u,d$), strange and charm quarks, respectively, in a relativized diquark model, characterized by one-gluon-exchange (OGE) plus confining potential. In the diquark model, a $q\bar q c \bar c$ ($s\bar s c \bar c$) tetraquark configuration is made up of a heavy-light diquark, $q c$ ($sc$), and anti-diquark, $\bar q \bar c$ ($\bar s \bar c$). According to our results, 13 charmonium-like observed states can be accommodated in the tetraquark picture, both in the hidden-charm ($q\bar q c \bar c$) and hidden-charm hidden-strange ($s\bar s c \bar c$) sectors.


I. INTRODUCTION
For a few decades after the formulation of the quark model, it was believed that baryons and mesons could be described as the bound states of three valence quarks and a constituent quark-antiquark pair, respectively. The classification of ground-state hadrons could be easily carried out in terms of group theoretical techniques and the quark model formalism, while resonances might be sorted by making use of effective potentials to describe the spatial excitations related to the inter-quark motion. For example, see Refs. [1][2][3][4].
Four quark states can in principle be bound by onegluon-exchange (OGE) forces. However, their possible emergence and stability is controversial because of the * naeem@itp.ac.cn † jak.ferretti@gmail.com ‡ santopinto@ge.infn.it lack of reliable and univocal experimental data. As a consequence, tetraquark model predictions are strongly model dependent and rely on the choice of a specific Hamiltonian, and also on the parameter fitting procedure. Despite of this, the tetraquark hypothesis is worth to be investigated.
It is worth noting that in the tetraquark hypothesis one obtains a four-quark spectrum which is richer than those generated by molecular models or the inclusion of dynamical/threshold effects in the quark model formalism. In particular, in molecular models one only expects to get bound states in the proximity of meson-meson decay thresholds; radial excitations cannot take place because of the smallness of meson-meson binding energies. On the contrary, if one includes threshold effects in the quark model formalism, one gets radial excitations, but exotic charged states of the type qqQQ, where Q is a heavy quark, are forbidden. A comparison between theoretical predictions for the spectrum and main decay modes of four-quark states and the existing experimental data may allow to distinguish between the previous hypotheses. The possible emergence of the fully-heavy QQQQ bound states may provide a strong indication in favor of the tetraquark one [22,[55][56][57][58][59].
It is also worth to remind that the heavy-light tetraquarks in a QQqq configuration are also of considerable interest. 1 It would be very interesting to test the possibility of a QQqq tetraquark that remains stable against strong decays, but unfortunately there is no experimental evidence yet. Theoretically, QQqq was first shown to be stable against strong decays by Lipkin [60] and Ader et al. [61] long ago. Very recently, bbqq was shown to be stable against strong decays but not its charm counterpart ccqq, nor the mixed (beauty+charm) bcqq state [62,63]. For the detailed discussions on the stability of different heavy-light tetraquarks, we refer to the following recent studies [64,65].
In this paper, we compute the spectrum of qqcc (q = u, d) and sscc tetraquarks. The calculations are performed within a diquark-antidiquark relativized model, characterized by a one-gluon-exchange potential. The effective degree of freedom of diquark describes two strongly correlated quarks, with no internal spatial excitation.
The tetraquark spectrum is obtained in a two-step process. First of all, the diquark masses are obtained by solving the Schrödinger equation with the relativized quark-quark potential [66]. In a second stage, the tetraquark spectrum is calculated by means of the relativized diquark-antidiquark potential [22]. Finally, by comparing our results to the data, we are able to provide some tentative assignments to XY Z-type states, including X(3872), Z c (3900), Z c (4020), Z c (4240), Z c (4430), Y (4008), Y (4260), Y (4360), Y (4630) and Y (4660) in the qqcc sector, plus X(4140), X(4500) and X(4700) in the sscc sector. The next step of our study of fully- [22] and doubly-heavy tetraquarks will be an analysis of the ground-state energies and dominant decay modes, including estimates of the total decay widths and production cross-sections.
The paper is organized as follows. In section II, we describe the relativized diquark model and calculation details, and enlist our model parameters. Section III is devoted to discussing the results, where we compare our mass predictions both for the qqcc and sscc tetraquarks with the experimental data and those of the previous theoretical studies. We compare our tentative tetraquark assignments for XY Z states with other theoretical interpretations (if available). Finally, we provide a short summary.

II. RELATIVIZED DIQUARK MODEL
In a diquark-antidiquark model, the effective degree of freedom of diquark, describing two strongly correlated quarks with no internal spatial excitations, is introduced. Tetraquark mesons are then interpreted as the bound states of a diquark, D, and an antidiquark,D.
The D −D relative motion is described in terms of a relative coordinate r rel (with conjugate momentum q rel ), thus neglecting the internal diquark (antidiquark) structure. As a result, one turns a four-body problem into a two-body one and gets a spectrum which is less rich than that of a four-body system. Something similar also happens in the baryon sector, where the spectrum of a quark-diquark system is characterized by a smaller number of states than that of a three quark one. For example, see Refs. [67][68][69][70][71][72].
As the tetraquark must be a color singlet, there are two possible diquark-antidiquark combinations: 1. diquark in3 c , antidiquark in 3 c 2. diquark in 6 c , antidiquark in6 c Diquarks (antidiquarks) are made up of two identical fermions and so they have to satisfy the Pauli principle, i.e. the diquark (antidiquark) total wave function, where ψ c , ψ sf and ψ sp are the color, spin-flavor and spatial wave functions, must be antisymmetric. Moreover, if for simplicity we neglect the diquarks' internal spatial excitations, their color-spin-flavor wave functions must be antisymmetric. This limits the possible representations to being only [71,73] color in3 c ; symmetric ψ sf (2a) and color in 6 c ; antisymmetric ψ sf .
In the study of qqcc and sscc tetraquarks, we consider diquarks (antidiquarks) of the cq-and cs-type, where q = u, d, with isospin I D = 1 2 or 0, respectively. Because of this, for qqcc states both I = 0 and I = 1 tetraquark isospin combinations are possible, while in the sscc case one necessarily has I = 0. We can determine the J P C quantum numbers of the tetraquarks by applying the restrictions for the diquark-antidiquark limit, i.e. L D = LD = 0 and color3 c ⊗ 3 c . This is because we expect that color-sextet diquarks, Eq. (2b), will be higher in energy than color-triplet ones or even that they will not be bound at all [18,73,74]. Thus, we are left with the (2a) diquark representation. The latter can be further decomposed in terms of the diquark spin and flavor content. As a result, we get a spin-0, flavor-antisymmetric representation, the scalar diquark, and a spin-1, flavor-symmetric representation, the axialvector diquark. With these restrictions, the parity of a tetraquark in the diquark-antidiquark limit is while the charge conjugation (obviously only for its eigenstates) is Here, L DD is the diquark-antidiquark relative orbital angular momentum, and S tot = S D + SD.

B. Relativized model Hamiltonian
We consider the following Hamiltonian where q 2 rel + m 2 D,D are the diquark (antidiquark) kinetic energies, with diquark (antidiquark) masses m D (mD), and V (r rel ) the OGE plus confining potential. The usual form for V (r rel ) is where λ a D,D are Gell-Mann color matrices, ∆E a constant, α s the strength of the color-Coulomb interaction, and β that of the linear confining potential.
The hyperfine interaction of Eq. (6) is an illegal operator in the Schrödinger equation; moreover, the Coulomblike potential should be regularized in the origin [13]. To overcome these difficulties, we follow the prescriptions of Refs. [66,67,75] and re-write Eq. (6) as where the Coulomb-like potential is given by [66,67] Here, Erf is the error function [76] and τ DDk [66,67] τ DDk = γ k σ DD with The values of the parameters α k and γ k (k = 1, 2, 3), σ 0 and s, extracted from Refs. [66,67], are given in Table I. The value of the qc scalar diquark mass, M s qc , is extracted from Ref. [16]. The values of the qs scalar and axial-vector diquark masses, M s sc and M av sc , are estimated by binding a sc (sc) pair via the OGE plus confining potential [22,66]. The only free parameters of our calculation are thus the strength of the linear confining interaction, β, the qc axial-vector diquark mass, M av qc , and ∆E (see Table I); they are fitted to the reproduction of the experimental data [77], as discussed in Sec. III A.

III. RESULTS AND DISCUSSIONS
A. qqcc tetraquark spectrum In Table III and Figs. 1 and 2, our theoretical predictions for the masses of qqcc (q = u, d) and sscc 0 ++ ,    Table I, are partly extracted from those of previous studies and partly fitted to the reproduction of the spectrum of suspected charmonium-like exotic states [48,77].

Xc and Zc states
It is worth noting that we are able to make some clear assignments, as in the case of X(3872), Z c (3900), Z c (4020) and Z c (4240). This is because the mass difference between the predicted and experimental masses is within the typical error of a quark model calculation, of the order of 30 − 50 MeV.
The X(3872), discovered by Belle in B ± → K ± π + π − J/ψ decays [10], was the first example of quarkonium-like candidate for a non-standard or exotic meson. This is a well established meson [77][78][79][80], with extremely peculiar features: its mass is 80 − 100 MeV below quark model predictions [66] and very close to the D 0D * 0 threshold, it is quite narrow (Γ < 1.2 MeV) and exhibits strong isospin violation in its decays.
In the present study, the X(3872) is interpreted as an S-wave scalar diquark, axial-vector antidiquark bound state with J P C = 1 ++ quantum numbers. This is the same interpretation as Ref. [16], where the authors calculated the spectrum of cqcq tetraquarks by means of an algebraic mass formula, giving the X(3872) mass as input, and Ref. [19], where the authors calculated the tetraquark spectrum in a relativistic diquark-antidiquark model with one-gluon exchange and long-range vector and scalar linear confining potentials. In the molecular model, the X(3872) is described as a D 0D * 0 meson-   Tables II and III. meson bound state [32][33][34][35][36][37][38][39].
The Z c (3900) is a charged charmonium-like meson, with 1 +− quantum numbers, observed at about the same time by BESIII [5] and Belle [6]. Its exotic quantum numbers and the value of its mass, about 12 MeV above the D 0 D * + threshold, is incompatible with both the charmonium and molecular model interpretations. In Ref. [24], the Z c (3900) was interpreted as a hadro-charmonium state, namely as a J/ψ embedded in an S-wave spin-less excitation of the light-quark matter with the quantum numbers of a pion, J P = 0 − . Our interpretation is the same as Refs. [16,19], namely as the C-odd partner of the X(3872) [16,Eq. (22)].
In 2014, LHCb confirmed the existence of the Z c (4430) in π ± ψ(2S) and, within the same dataset, also observed a lighter and wider structure named the Z c (4240) [81,82]. Further experimental confirmation of the Z c (4240) would be helpful. In our study, we interpret the Z c (4240) as a P -wave scalar diquark, axial-vector antidiquark bound state with J P C = 0 −− .
Finally, the Z c (4430) was the first established candidate for a charged charmonium-like meson. It was observed by Belle as a peak in the the invariant mass of the ψ(2S)π + system inB → ψ(2S)π + K [83]. In our study, we interpret the Z c (4430) as a D-wave scalar diquark, axial-vector antidiquark bound state with J P C = 1 +− quantum numbers. However, in this case the assignment is more dubious, because the experimental mass of the meson falls in the energy interval between the 2[(1, 0)1, 0]1 and 1[(1, 0)1, 2]1, J P C = 1 +− states of Table  III. It is worth noting that the Z c (4430) was interpreted as a 2S scalar diquark, axial-vector antidiquark bound state in Ref. [19]. Moreover, the Z c (3900) and Z c (4430) were assigned as the ground state and first radial excitation of the same tetraquark with J P = 1 + , and several strong decays were explored [84].

Yc states
There is a rich spectrum of charmonium-like J P C = 1 −− vector states, the so-called Y states. Below, we discuss our tetraquark model assignments.
Starting from Y (4008), the presence of a broad structure, with mass 4008 ± 40 +114 −28 MeV and width 226 ± 44 ± 87 MeV, was indicated by Belle in the measured π + π − J/ψ mass spectrum [85]. However, BaBar did not found the Y (4008) signal in the same e + e − → π + π − J/ψ process [86]. Future experiments will give a concluding answer about the Y (4008) existence. In our tetraquark model calculation, the Y (4008) is interpreted as a P -wave scalar diquark-antidiquark bound state.
Y (4260) was discovered by BaBar in e + e − → Y → π + π − J/ψ [83] and then confirmed by CLEO-c [87] and Belle [85]. We interpret it as a P -wave scalar diquark, axial-vector antidiquark bound state. In Ref. [19] it was described as a P -wave scalar diquark-antidiquark bound state, in Ref. [88] as the first orbital excitation of a diquark-antidiquark state cscs, but it was also interpreted as a hybrid charmonium in Refs. [89][90][91]. The authors of Refs. [26,92] also interpreted Y (4260) as DD 1 (2420) molecule with a binding energy of 29 MeV. Very recently, a possible molecular scenario was discussed in a coupled-channel analysis [93].
Finally, the Y (4630) was seen in e + e − → Y → Λ cΛc by Belle [99]. We interpret it as the Y (4260) radial excitation. In Ref. [100], the authors discussed the Y (4630) → Λ cΛc decay mode in the 3 P 0 model formalism, under the hypothesis that the Y (4630) is a 1 −− charmonium-like tetraquark. Because of its peculiar decay mode, Y (4630) was also described as a baryonium state, namely as a Λ cΛc bound state [101].

B. sscc tetraquark spectrum
There are charmonium-like mesons whose decay modes and production mechanisms suggest the presence of ss degrees of freedom in the tetraquark wave function. A typical example is the X(4140), observed in B → KY (4140), with Y (4140) → φJ/ψ, by CDF [102]. In addition to the X(4140), the CDF Collaboration found evidence of the X(4274) with approximate significance of 3.1σ [103]. The related peaks of J/ψφ mass structures around 4.3 GeV were also reported by LHCb, CMS, D0 and BaBar Collaborations [104][105][106][107], which may be the same state as the X(4274). Very recently, the X(4140) and X(4274) were confirmed by LHCb, which also found evidence of two more structures, the X(4500) and the X(4700) [108].
We interpret the X(4140) as the sscc counterpart of the X(3872); X(4500) and X(4700) as 0 ++ radial excitations of S-wave scalar diquark-antidiquark and axialvector diquark-antidiquark bound states, respectively. We cannot provide any assignment for the X(4274).
An investigation similar to ours was conducted in Ref. [109]. There, the authors studied sscc tetraquarks within the relativized quark model [66] and discussed possible assignments for X(4140), X(4274), X(4500) and X(4700). In the X(4140) case, their interpretation coincides with ours. They also obtain 0 ++ radial excitations of S-wave scalar diquark-antidiquark and axialvector diquark-antidiquark bound states characterized by similar energies: one of them can be assigned to X(4700). They could not accommodate the X(4274). Stancu calculated the sscc tetraquark spectrum within a simple quark model with chromomagnetic interaction [110]. She interpreted the X(4140) as the strange partner of the X(3872), but she could not accommodate the other sscc states, X(4274), X(4500) and X(4700). 2 Table I, is compared to the existing experimental data [77]. In the third column, we give the quantum numbers of the predicted tetraquark states: N stands for the radial quantum number, SD and SD are the spin of the diquark and antidiquark, respectively, coupled to the total spin of the meson, S; the latter is coupled to the orbital angular momentum, L, to get the total angular momentum of the tetraquark, J. For more details on the tetraquark basis, see App. A. molecular model description for the X(4140) as D * + s D * − s was proposed. By using QCD sum rules, the X(4140) and X(4274) were interpreted as S-wave ccss tetraquark states with opposite color structures [113], and, analogously, the X(4500) and X(4700) as the D-wave ccss tetraquark states with opposite color structures [112]. Maiani et al. suggested to accommodate X(4140), X(4274), X(4500) and X(4700) within two tetraquark multiplets. In particular, they suggested that the X(4500) and X(4700) are 2S cscs tetraquark states, the X(4140) the 1 ++ groundstate, and that the X(4274) may have 0 ++ or 2 ++ quantum numbers [114].

IV. SUMMARY
We calculated the spectrum of qqcc (q = u, d) and sscc tetraquarks in a relativized diquark model, characterized by one-gluon-exchange (OGE) plus confining potential [22]. According to our results, we were able to make some clear assignments, as in the case of X ( Table  III, with J P C = 1 +− . In the sscc sector, we could accommodate the X(4140), X(4500) and X(4700). We could not provide any assignment for the X(4274). A study of the main decay modes of XY Z-type exotics in the diquark model will be important to provide a more precise identification of tetraquark candidates.
Our relativized diquark-antidiquark model results are strongly model dependent. The possible sources of theoretical uncertainties lie in the choice of the effective Hamiltonian and model parameter fitting procedure, and also in the approximations introduced in the tetraquark wave function. The latter are strictly related to the possible ways of combinating the quark color representations to obtain a color singlet wave function for the tetraquark.
The next step of our study of fully-and doubly-heavy tetraquarks will be an analysis of the ground-state energies, dominant decay modes and production mechanisms, including estimates of total decay widths and production cross-sections. More precise experimental data for the exotic meson masses and properties and a detailed comparison between the calculated observables in the main interpretations (tetraquark, molecular model, hadro-quarkonium, and so on) may help to rule out one or more of these pictures.