Spectroscopic behavior of fully heavy tetraquarks

Stimulated by the observation of the X(6900) from LHCb in 2020 and the recent results from CMS and ATLAS in the di-J/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi $$\end{document} invariant mass spectrum, in this work we systematically study all possible configurations for the ground states of fully heavy tetraquark in the constituent quark model. By our calculation, we present their spectroscopic behaviors including binding energy, lowest meson–meson thresholds, specific wave function, magnetic moment, transition magnetic moment, radiative decay width, rearrangement strong width ratio, internal mass contributions, relative lengths between (anti)quarks, and the spatial distribution of four valence (anti)quarks. We cannot find a stable S-wave state for the fully heavy tetraquark system. We hope that our results will be valuable to further experimental exploration of fully heavy tetraquark states.

Briefly reviewing the status of heavy flavor tetraquark states, we must mention the fully heavy tetraquark with the QQ Q Q configuration, which has attracted the attention of both theorists and experimentalists.Chao et al. suggested that the peculiar resonance-like structures of R(e + e − → hadrons) for √ s = 6 − 7 GeV may be due to the production of the predicted P-wave (cc)-(cc) states in the energy range of 6.4 − 6.8 GeV, which could dominantly decay into charmed mesons [73].The calculation of the fully heavy tetraquark was then carried out using the potential model [74,75] and the MIT bag model with the Born-Oppenheimer approximation [76].This system has also been studied in a non-relativistic potential model, where no QQ Q Q bound state can be found [77].However, Lloyd et al. adopted a parameterized non-relativistic Hamiltonian to study such system [78], where they found several closely-lying bound states with a large oscillator basis.Later, Karliner et al. estimated the masses of the fully heavy tetraquark states by a simple quark model, and obtained M(X cccc ) = 6192±25 MeV and M(X bb bb) = 18826 ± 25 MeV for the fully charmed and fully bottom tetraquarks with the J P C = 0 ++ quantum number, respectively [79].Anwar et al. have calculated the ground-state energy of the bb bb bound state in a nonrelativistic effective field theory with one-gluonexchange (OGE) color Coulomb interaction, and the ground state bb bb tetraquark mass is predicted to be (18.72 ± 0.02) GeV [80].In Ref. [81], Bai et al. presented a calculation of the bb bb tetraquark ground-state energy using a diffusion Monte Carlo method to solve the non-relativistic many-body system.Debastiani et al.
extended the updated Cornell model to study the fully charmed tetraquark in a diquark-antidiquark configuration [82].Chen et al. used a moment QCD sum rule method to give the existence of the exotic states cccc and bb bb in the compact diquark-antidiquark configuration, where they suggested to search for them in the J/ψJ/ψ and η c (1S)η 1S channels [83].
With the accumulation of experimental data, many collaborations have tried to search for it.The CMS Collaboration reported the first observation of the Υ(1S) pair production in pp collisions, where there is evidence for a structure around 18.4 GeV with a global significance of 3.6 σ exists in the four-lepton channel, which is probably a fully-bottom tetraquark state [84].However, this structure was not confirmed by the later CMS analysis [85].Subsequently, the LHCb Collaboration studied the Υ(1S) µ + µ − invariant mass distribution to search for a possible bb bb exotic meson, but they did not see any significant excess in the range 17.5 − 20.0 GeV [86].By 2020, the LHCb Collaboration declared a narrow resonance X(6900) in the di-J/ψ mass spectrum with a significance of more than 5σ [87].In addition, a broad structure ranging from 6.2 to 6.8 GeV and an underlying peak near 7.3 GeV were also reported at the same time [87].Recently, the ATLAS and CMS collaborations published their measurements on the di-J/ψ invariant mass spectrum.Here, they not only confirmed the existence of the X(6900), but also found some new peaks [88][89][90].There have been extensive discussions about the observed X(6900) from different approaches and with different assignments [91][92][93][94][95][96][97][98][99][100][101][102][103][104][105][106][107][108].
The problem of the stability of the fully heavy tetraquark state has been debated for a long time.Debastiani et al. found that the lowest S-wave cccc tetraquarks may be below the dicharmonium thresholds in their updated Cornell model [109].The 1 + bb bc state is thought to be a narrow state in the extended chromomagnetic model [110].However, many other studies have suggested that the ground state of fully heavy tetraquarks is above the di-meson threshold.Wang et al. also calculated the fully-heavy tetraquark state in two nonrelativistic quark models with different OGE Coulomb, linear confinement and hyperfine potentials [111].Based on the numerical calculations, they suggested that the ground states should be located about 300 − 450 MeV above the lowest scattering states, indicating that there is no bound tetraquark state.The lattice nonrelativistic QCD method was applied to study the lowest energy eigenstate of the bb bb system, and no state was found below the lowest bottomonium-pair threshold [112].Furthermore, Richard et al. also claimed that the fully heavy configuration QQ Q Q is not stable if one adopts a standard quark model and treats the four-body problem appropriately [113].Jin et al. studied full-charm and full-bottom tetraquarks using the quark delocalization color screening model and the chiral quark model, respectively, and the results within the quantum numbers J P = 0 + , 1 + , and 2 + show that the bound state exists in both models [114].Frankly speaking, theorists have not come to an agreement on the stability of the fully heavy tetraquark state.
Facing the present status of fully heavy tetraquark, in this work we adopt the variational method to systematically study the fully heavy tetraquark states, where the mass spectrum of the fully heavy tetraquark is given in the framework of nonrelativistic quark model associated with a potential containing Coulomb, linear, and hyperfine terms.The constructed total wave functions involved in these discussed systems satisfy the requirement of the Pauli principle.We should emphasize that we can also reproduce the masses of these conventional hadrons with the same parameters, which is a test of our adopted framework.With this preparation, we calculate the binding energies, the lowest meson-meson thresholds, and the rearrangement strong width ratio, and study the stability of the fully heavy tetraquark states against the decay into two meson states.Furthermore, we discuss whether the discussed tetraquarks have a compact configuration based on the eigenvalue of the hyperfine potential matrix.According to specific wave functions, we could obtain the magnetic moments, transition magnetic moments, and radiative decay widths, which may reflect their electromagnetic properties and internal structures.We also give the the size of the tetraquarks, the relative distances between (anti)quarks, and the spatial distribution of the four valence (anti)quarks for each state.Through the present systematic work, we can test whether compact bound fully heavy tetraquarks exist within the given Hamiltonian.
This paper is organized as follows.After the introduction, we present the Hamiltonian of the constituent quark model and list the corresponding parameters in Section II.Then we give the spatial function with a simple Gaussian form and construct the flavor, color, and spin wave functions of the fully heavy tetraquark states (see Section III).In Section IV, we show the numerical results obtained by the variational method and further calculate their magnetic moment, transition magnetic moment, radiative decay width, rearrangement strong width ratio, the internal mass contributions and relative lengths between (anti) quarks.In addition, a comparison of our results with those of other theoretical groups is made in Section V. Finally, the paper ends with a short summary in Section VI.

II. HAMILTONIAN
We choose a nonrelativistic Hamiltonian for the fully heavy tetraquark system, which is written as, Here, m i is the (anti)quark mass, λ c i is the SU (3) color operator for the i-th quark, and for the antiquark, λ c i is replaced by −λ c * i .The internal quark potentials V Con ij and V SS ij have the following forms: where r ij = |r i − r j | is the distance between the i-th (anti)quark and the j-th (anti)quark, and the σ i is the SU (2) spin operator for the i-th quark.As for the r 0ij and κ ′ , we have The corresponding parameters appearing in Eqs.(2-3) are shown in Table I.Here, κ and κ ′ are the couplings of the Coulomb and hyperfine potentials, respectively, and they are proportional to the running coupling constant α s (r) of QCD.The Coulomb and hyperfine interactions can be deduced from the one-gluon-exchange model.1/a 2 0 represents the strength of the linear potential.r 0ij is the Gaussian-smearing parameter.Furthermore, we introduce κ 0 and γ in κ ′ to better describe the interaction between different quark pairs [115].

III. WAVE FUNCTIONS
Here, we focus on the ground fully heavy tetraquark states.We present the flavor, spatial, and color-spin parts of the total wave function for fully-heavy tetraquark system.In order to consider the constraint by the Pauli principle, we use a diquark-antidiquark picture to analyze this tetraquark system.

A. Flavor Part
First we discuss the flavor part.Here, we list all the possible flavor combinations for the fully-heavy

System
Flavor combinations tetraquark system in Table II.
In Table II, the three flavor combinations in the first row are purely neutral particles and the C parity is a "good" quantum number.For the other six states in the second row, each state has a charge conjugation antipartner, and their masses, internal mass contributions, relative distances between (anti)quarks are absolutely identical, so we only need to discuss one of the pair.
Furthermore, the cccc, bb bb , and cc bb states have the two pairs of (anti)quarks which are identical, but only the first two quarks in the ccc b and bb bc states are identical.

B. Spatial Part
In this part, we construct the wave function for the spatial part in a simple Gaussian form.We denote the fully heavy tetraquark state as the Q(1)Q( 2) Q(3) Q(4) configuration, and choose the Jacobian coordinate system as follows: x 2 = 1/2(r 3 − r 4 ), Here, we set the Jacobi coordinates with the following conditions: Based on this, we construct the spatial wave functions of the QQ Q Q states in a single Gaussian form.The spatial wave function can satisfy the required symmetry property: where C 11 , C 22 , and C 33 are the variational parameters.
It is also useful to introduce the center of mass frame so that the kinetic term in the Hamiltonian of Eq. ( 1) can be reduced appropriately for our calculations.The kinetic term, denoted by T c , is as follows where different states have different reduced masses m ′ i , which are listed in Table III.
In the color space, the color wave functions can be analyzed using the SU(3) group theory, where the direct product of the diquark and antidiquark components reads Based on this, we get two types of color-singlet states: In the spin space, the allowed wave functions are in the diquark-antidiquark picture: In the notation , the spin1, spin2, and spin3 represent the spin of the diquark, the spin of the antidiquark, and the total spin of the tetraquark state, respectively.Since the flavor part and spatial parts are chosen to be fully symmetric for the (anti)diquark, the color-spin part of the total wave function should be fully antisymmetric.Combining the flavor part, we show all possible color-spin part satisfying the Pauli principle with J P C in Table IV.
In addition, it is convenient to consider the strong decay properties, we use the meson-meson configuration to represent color-singlet and spin wave functions, again.The color wave functions in the meson-meson configuration can be derived from the following direct product: (10) Based on Eq. ( 10), they can be expressed as Similarly, the spin wave functions in the meson-meson configuration read as TABLE IV.The allowed color-spin parts for each flavor configuration. Type IV. NUMERICAL ANALYSIS

Mass spectrum, internal contribution, and spatial size
In this subsection, we check the consistency between the experimental masses and the obtained masses of traditional hadrons using the variational method based on the Hamiltonian of Eq. ( 1) and the parameters in Table I.We show the results in Table VI and note that our values are relatively reliable since the deviations for most states are less than 20 MeV.
In addition, we have systematically constructed the total wave function satisfied by the Pauli principle in the previous section.The corresponding total wave function could be expanded as follows: To study the mass of the fully heavy tetraquarks with the variational method, we calculate the Schrödinger equation H|Ψ α ⟩ = E α |Ψ α ⟩, diagonalize the corresponding matrix, and then determine the ground state masses for the fully heavy tetraquarks.According to the corresponding variational parameters, we also give the internal mass contributions, including the quark mass part, the kinetic energy part, the confinement potential part, and the hyperfine potential part.For comparison, we also show the lowest meson-meson thresholds for the tetraquarks with different quantum numbers and their internal contributions.This is how we define the binding energy: where M tetraquark , M meson1 , and M meson2 are the masses of the tetraquark and the two mesons at the lowest threshold allowed in the rearrangement decay of the tetraquark, respectively.To facilitate the discussion in the next subsection, we also define the V C , which is the sum of the Coulomb potential and the linear potential.
Here, it is also useful to investigate the spatial size of the tetraquarks, which is strongly related to the magnitude of the various kinetic energies and the potential energies between the quarks.It is also important to understand the relative lengths between the quarks in the tetraquarks and their lowest thresholds, and the relative distance between the heavier quarks is generally shorter than that between the lighter quarks [61].This tendency is also maintained in each tetraquark state according to the corresponding tables.

Magnetic moments, transition magnetic moments, and radiative decay widths
The magnetic moment of hadrons is a physical quantity that reflects their internal structures [121].The total magnetic moment ⃗ µ total of a compound system contains the spin magnetic moment ⃗ µ spin and the orbital magnetic moment ⃗ µ orbital from all of its constituent quarks.For ground hadron states, their contribution of the orbital magnetic moment ⃗ µ orbital is zero, and so we only concentrate on the spin magnetic moment ⃗ µ spin .The explicit expression for the spin magnetic moment ⃗ µ spin is written as where Q ef f i and M ef f i are the effective charge and effective mass of the i-th constituent quark, respectively.The ⃗ σ i denotes the Pauli's spin matrix of the i-th constituent quark.According to Ref. [122], the effective charge of the quark is affected by other quarks in the inner hadron.We now assume that the effective charge is linearly dependent on the charge of the shielding quarks.So the effective charge Q ef f i is defined as where Q i is the bare charge of the i-th constituent quark, the α ij is a corrected parameter that reflects how much the charge of other quarks affects the charge of the i-th quark.To simplify the calculation, we also set α ij always equal to 0.033 according to the Ref. [122].The effective quark masses M ef f i contain the contributions from both the bare quark mass terms and the interaction terms in the chromomagnetic model, and their values are taken from Ref. [123].
To obtain the magnetic moment of the discussed hadron, we calculate the z-component of the magnetic moment operator μz sandwiched by the corresponding total wave function Ψ α (Eq.[9]).Now, only the spin part of the total wave function is involved.The total spin wave functions of the discussed hadrons are written as Based on this, we can quantitatively obtain the magnetic moment of the discussed hadron .. where µ tr is the cross-term representing the transition moment, and C 1 , C 2 are the eigenvectors of the given mixing state [124].Similarly, the transition magnetic moments between the hadrons can be obtained as µ According to Eq. ( 18), the numerical values for the magnetic moments of the traditional hadrons have been listed in Table VI.Here, µ N = e/2m N is the nuclear magnetic moment with m N = 938 MeV as the nuclear mass, which is the unit of the magnetic moment.For comparison, we also show the experimental values and other theoretical results from Refs.[121,122,[124][125][126][127].Because of the µ Q = −µ Q, the magnetic moment of all of the J P = 0 + ground mesons and tetraquarks and the ground states with certain C-parity is 0.
where the J i and J f are the total angular momentums of the initial and final hadrons, respectively.The M i and M f in Eq. ( 18) represent initial and final hadron masses, respectively.

Relative decay widths of tetraquarks
In addition to radiative decay, we also consider the rearrangement strong decay properties for fully heavy tetraquarks.Based on Eqs.(10)(11)(12), the color wave function also falls into two categories: the colorsinglet 1 ⟩ which can easily decay into two S-wave mesons, and the color-octet 8 ⟩ which can only fall apart by the gluon exchange.Thus we transform the total wave functions Ψ α into the new configuration, Among the decay behaviors of the tetraquarks, one decay mode is that the quarks simply fall apart into the final decay channels without quark pair creations or annihilations, which is donated as " Okubo-Zweig-Iizuka (OZI)-superallowd" decays.In this part, we will only focus this type of decay channels.For two body decay by L-wave, the partial decay width reads as [72,110,[138][139][140]: where α is an effective coupling constant, c i is the overlap corresponding exactly to C ′α ij of Eq. ( 19), m is the mass of the initial state, k is the momentum of the final state in the rest frame of the initial state.For the decays of the S-wave tetraquarks, (k/m) −2 is of order O(10 −2 ) or even smaller, so all higher-wave decays are suppressed.So we only need to consider the S-wave decays.As for γ i , it is determined by the spatial wave functions of the initial and final states, which are different for each decay process.In the quark model in the heavy quark limit, the spatial wave functions of the ground S-wave pseudoscalar and the vector meson are the same.The relations of γ i for fully heavy tetraquarks are given in Table V.Based on this, the branching fraction is proportional to the square of the coefficient of the corresponding component in the eigenvectors, and the strong decay phase space, i.e., k • |c i | 2 , for each decay mode.From the value of k • |c i | 2 , one can roughly estimate the ratios of the relative decay widths between different decay processes of different initial tetraquarks.
In the following subsections, we concretely discuss all possible configurations for fully heavy tetraquarks.

A. cccc and bb bb states
First we investigate the cccc and bb bb systems.There are two J P C = 0 ++ states, one J P C = 1 +− state, and one J P C = 2 ++ state according to Table IV.We show the masses of the ground states, the variational parameters, the internal mass contributions, the relative lengths between the quarks, their lowest meson-meson thresholds, the specific wave function, the magnetic moments, the transition magnetic moments, the radiative decay widths, and the rearrangement strong width ratios in Tables VII-IX, respectively.

States
Here, we take the J P C = 0 ++ bb bb ground state as an example, and others have similar discussions according to Tables VII-IX.We now analyze the numerical results obtained from the variational method.For the J P C = 0 ++ bb bb ground state, its mass is 19240.0MeV and the corresponding binding energy B T is +461.9MeV.Its variational parameters are given as C 11 = 7.7 fm −2 , C 22 = 7.7 fm −2 , and C 33 = 11.4 fm −2 , giving roughly the inverse ratios of the size for the diquark, the antidiquark, and between the center of the diquark and the antidiquark, respectively.We naturally find that the C 11 is equal to C 22 , so the distance of (b − b) would be equal to that of ( b − b), and the reason is that the bb bb system is a neutral system.
The total wave function in the diquark-antidiquark configuration is given by The meson-meson configuration is connected to the diquark-antidiquark configuration by a linear transformation.We then obtain the total wave function in the meson-meson configuration: According to Eq. ( 22), we are sure that the overlaps c i of η b η b and ΥΥ are 0.560 and 0.558, respectively.Then, based on Eq. ( 20), the rearrangement strong width ratios are i.e., both the ΥΥ and η b η b are dominant decay channels for the T b 2b2 (19240.0,ΥΥ) state., and µ the(2) are theoretical masses and magnetic moments for Eq.(1), Eq. ( 15), and Refs.[121,122,124], respectively.Mexp and µexp are the observed values of masses and magnetic moments.The masses and errors are in units of MeV.The magnetic moment is in units of the nuclear magnetic moment µN .The variational parameter is in units of fm −2 .As for the magnetic moments of the cccc and bb bb ground states, their values are all 0, because the same quark and antiquark have exactly opposite magnetic moments, which cancel each other out.
We also discuss the transition magnetic moment of the T b 2b2 (19303.9, 1 +− ) → T b 2b2 (19240.0,0 ++ )γ process.We construct their flavor ⊗ spin wave functions as And then, the transition magnetic momentum of the T b 2b2 (19303.9, 1 +− ) → T b 2b2 (19240.0,0 ++ )γ process can be given by the z-component of the magnetic moment operator μz sandwiched by the flavor-spin wave functions of the T b 2b2 (19303.9, 1 +− ) and T b 2b2 (19240.0,0 ++ ).So, the corresponding transition magnetic momentum is As for the transition magnetic moment of the T b 2b2 (19327.9, 2 ++ ) → T b 2b2 (19240.0,0 ++ )γ process, its value is 0 due to the C parity conservation restriction.Furthermore, according to Eq. ( 18) and Eq. ( 25), we also obtain the corresponding radiative decay widths

Relative distances and symmetry
Here, we concentrate on the the relative distances between the (anti)quarks in tetraquarks.Looking at the relative distances in Table IX, we find that the relative distances of (1,2) and (3,4) pairs are the same, and other relative distances are the same in all the cccc and bb bb states This is due to the permutation symmetry for the ground state wave function in each tetraquark [65].For the c 1 c 2 c3 c4 and b 1 b 2 b3 b4 states, they need to satisfy the Pauli principle for identical particles are as follows: where the operator A ij means exchanging the coordinate of Q i ( Qi ) and Q j ( Qj ).Meanwhile, they are pure neutral particles with definite C-parity, so the permutation symmetries for total wave functions are as follows: where A 12−34 means that the coordinates of the diquark and the antidiquark are exchanged.
Based on this, the relationship of the relative distances for all the c 1 c 2 c3 c4 and b 1 b 2 b3 b4 states can be obtained as follows: and Obviously, our theoretical derivations are in perfect agreement with the calculated results in Table IX.We can also prove three Jacobi coordinates, , are orthogonal to each other for all the cccc and bb bb states: and According to the relative distances in Table IX and the relationship of Eqs.(28)(29)(30)(31)(32)(33)(34), we can well describe the relative positions of the four valence quarks for all the cccc and bb bb states.Meanwhile, using the relative distances between (anti)quarks and the orthogonal relation, one can also determine the relative distance of ( 12) − (34), which is consistent with our results in Table IX.We can also give the relative position of R c and the spherical radius of the tetraquarks.Here, we define R c to be the geometric center of the four quarks (the center of the sphere).Based on these results, we show the spatial distribution of the four valence quarks for the J P C = 0 ++ bb bb ground state in Fig. 1.
In quark model, a compact tetraquark state has no color-singlet substruture, while a hadronic molecule is a loosely bound state which contains several color-singlet hadrons.According to Table IX, we easily find the relative distances of (1,2), (1,3), (1,4), (2,3), (2,4), and (3,4) quark pairs are all 0.227 or 0.204 fm.Meanwhile, the radius of the state is only 0.130 fm.Thus, in this state, all the distances between the quark pairs are roughly the same order of magnitude apart.If it is a molecular configuration, the distances between two quarks and two antiquarks should be much greater than the distances in the compact multiquark scheme.And the radius of molecular configuration can reach several femtometers.So, our calculations are consistent with the compact tetraquark expectations.

The internal contribution
Let us now turn our discussion to the internal mass contribution for the J P C = 0 ++ bb bb ground state.
First, for the kinetic energy, this bb bb state has 814.0 MeV, which can be understood as the sum of three pair, the total kinetic energies in this bb bb state are still smaller than that in the η b η b state.However, this does not lead the ground J P C = 0 ++ bb bb state to a stable state due to the confinement potential part.
As for the confinement potential part, the contributions from V C for the J P C = 0 ++ bb bb ground state in Table IX are all attractive.Thus, this state has a large positive binding energy.However, it is still above the meson-meson threshold because the V C (b b) in η b is very attractive.As for the other internal contributions, the quark contents of this state are the same as the corresponding rearrangement decay threshold.Moreover, the mass contribution from the hyperfine potential term is negligible compared to the contributions from other terms.

Comparison with two models of chromomagnetic interaction
Now, we compare the numerical values for the cccc and bb bb systems between the constituent quark model and two CMI models [110,123] in Tables VII-IX.The comparisons of the values for cccc and bb bb states, which are in the last three columns of Tables VII-IX, can be summarized in the following important conclusions.
First, we find that there is no stable state below the lowest heavy quarkonium pair thresholds in any of the three different models.In all three models, we consider two possible color configurations, the color-sextet According to the extended chromomagnetic model [110], the ground state is always dominated by the color-sextet configuration.This view is consistent with the specific wave function of the ground state in Eq. (21) given by the constituent quark model.
In contrast, the obtained masses from the constituent quark model are systematically larger than those from the extended CMI model [110] according to Tables VII-IX.Meanwhile, the obtained masses from the the CMI model [123] are obviously larger than those of the constituent quark model.Their mass differences are mainly due to the effective quark masses as given in the last three columns of Tables VII-IX.The effective quark masses are the sum of the quark mass, the relevant kinetic term, and all the relevant interaction terms in the constituent quark model, which indeed seems to approximately reproduce the effective quark mass from two CMI model [110,123].We compare the subtotal values of c and c quark part in Table VIII.The c effective quark mass in constituent quark model is 3225 MeV, which is about 100 MeV larger than that of the extended CMI model in the J P C = 0 ++ cccc state.Correspondingly, we also find that the c effective quark mass in the CMI model [123] is 3450 MeV and about 200 MeV larger than that of constituent quark model.The effective quark masses in extended CMI model depend on the parameters of the traditional hadron.However, the effective quark masses should be different depending on whether they are inside a meson, a baryon, or a tetraquark.The effective quark masses trend to be large when they are inside configurations with larger constituents in the extended CMI model, as can be seen from the comparisons of the last three columns in Tables VII-IX.Moreover, we notice the similar situation also occurs in the udcc state in Table X of Ref. [61].

B. cc bb state
Here, we will concentrate on the cc bb system.Similar to the cccc and bb bb systems, the cc bb system is also satisfied with fully antisymmetric for diquarks and antiquarks.There are two J P = 0 + states, one J P = 1 + state, and one J P = 2 + state in the cc bb system.We show the masses of the ground states, the variational parameters, the internal mass contributions, the relative lengths between the quarks, their lowest meson-meson thresholds, the specific wave function, the magnetic moments, the transition magnetic moments, the radiative decay widths, and the rearrangement strong width ratios in Tables XI and XII.
First, we take the J P = 0 + cc bb ground state as an example to discuss its properties with the variational method.A similar situation occurs in the other two quantum numbers according to Tables XI, and XII.The mass of the lowest J P = 0 + cc bb state is 12920.0MeV, and the corresponding binding energy B T is +344.2MeV according to Table XI.Thus, the state is obviously higher than the corresponding rearrangement meson-meson thresholds.The wave function is given by Here, we see that the mass contribution of ground state mainly comes from the The meson-meson configuration is connected to the diquark-antidiquark configuration by a linear transformation.Then, we obtain the total wave function in the meson-meson configuration: According to Eq. ( 36), we are sure that the overlaps c i of B c B c and B * c B * c are 0.095 and 0.589, respectively.Then, based on Eq. ( 20), the rearrangement strong width TABLE VII.The masses, binding energies, variational parameters, the internal contribution, total wave functions, magnetic moments, transition magnetic moments, radiative decay widths, rearrangement strong width ratios, and the relative lengths between the quarks for the J P C = 0 ++ , 1 +− cccc states and their lowest meson-meson thresholds.Here, (i, j) denotes the contribution of the i-th and j-th quarks.The number is given as i=1 and 2 for the quarks, 3 and 4 for the antiquark.The masses and corresponding contributions are given in units of MeV, and the relative lengths (variational parameters) are in units of fm (fm −2 ).Meanwhile, we give a comparison with the other two CMI models [110,123], to further secure the effective quark mass.The magnetic moments: The transition magnetic moments: -28.4 -28.2The magnetic moments:   The magnetic moments: µ T b 2b2 (19303.9, +V SS (23) + V SS (24) 11.9The magnetic moments: µ T b 2b2 (19303.9, The transition magnetic moments: -15.3 -15.5 Total contribution 19303.918696.420211.6 ratios is i.e., B c B c is the dominant rearrangement decay channel for the T c 2b2 (12920.0,0 + ) state.
As for the magnetic moment of the J P = 0 + cc bb ground state, its value is 0, while the magnetic moment of all J P = 0 + tetraquark states is 0. As for the J P = 1 + cc bb state, we construct its flavor ⊗ spin wave functions as So the corresponding transition magnetic momentum is We also discuss the transition magnetic moment of the T c 2b2 (12939.9, 1 + ) → T c 2b2 (12920.0,0 + )γ process.We still construct their flavor ⊗ spin wave functions as And then, the transition magnetic momentum of the T c 2b2 (12939.9, 1 + ) → T c 2b2 (12920.0,0 + )γ process can be described by the z-component of the magnetic moment operator μz sandwiched by the flavor-spin wave functions of the T c 2b2 (12939.9, 1 + ) and T c 2b2 (12920.0,0 + ).So the corresponding transition magnetic momentum is Further, according to Eq. ( 18) and Eq. ( 41), we also obtain the radiative decay widths Γ T c 2b2 (12960.9,2+ )→T c 2b2 (12939.9,1+ )γ = 3.6 keV.(42) Finally, we turn to the internal contribution for the cc bb ground state.For the kinetic energy part, the J P = 0 + cc bb state receives 835.9 MeV, which is smaller than that of the meson-meson threshold B c B c .The potential part of this state is much smaller than that of the lowest meson-meson threshold.Furthermore, we find that all the V C for this state are attractive.However, compared to the V C of B c B c , these attractive values seem to trivial.This is because the length between c − b in tetraquarks is longer than that in B c according to Table XI and XII.In summary, we tend to think that these cc bb states are unstable compact states.
C. ccc b and bb bc states Here, we discuss the ccc b and bb bc systems.For these two systems, they only need to satisfy the antisymmetry for the diquark.Thus, compared to the above three systems, the ccc b and bb bc systems have more allowed states.There are two J P = 0 + states, three J P = 1 + states, one J P = 2 + state in the ccc b and bb bc systems.We calculate the masses of the ground states, the corresponding variational parameters, the various internal contributions, the relative lengths between the quarks, and their lowest meson-meson thresholds, their lowest meson-meson thresholds, specific wave functions, magnetic moments, transition magnetic moments, radiative decay widths, and rearrangement strong width ratios in Tables XIII, XIV, and XV, respectively.We now analyse the numerical results of the J P = 1 + ground bb bc state obtained from the variational method according to Table XIV Here, we notice that the mass contribution of ground state mainly comes from the |(Q 1 Q 2 ) 6 0 ( Q3 Q4 ) 6 1 ⟩ 1 component, and the other two components are negligible.Then we transform Eq. ( 43) to the meson-meson configuration via a linear transformation, and the corresponding wave function is given as: Furthermore, we can sure that its rearrangement strong width ratios is: And its radiative decay widths are: Let us now focus on the internal contributions for this state and the relative lengths between the quarks.For the kinetic energy part, the state gets 876.1 MeV, which is obviously smaller than that of the lowest meson-meson threshold B c η b .The actual kinetic energy of the b − b (b − c) in the J P = 1 + bb bc state is smaller than that in the η b (B * c ) meson.The reason for this can be seen in Table XIV.The size of this pair is larger in the J P = 1 + bb bc state than in the meson: the distance (3,4) is 0.245 fm in this tetraquark while it is 0.148 fm in η b .
Here, let us turn our discussion to the potential parts.The potential part of this state is much smaller than that of its lowest meson-meson threshold.Although the V C between quark and antiquark are attractive, the V C in the diquark and antiquark are repulsive.However, relative to the η b and B c mesons, the V C in the tetraquark are less attractive.Therefore, they still have a relatively large positive binding energy in this state.

D. cbc b state
Finally, we investigate the cbc b system.Similar to the cccc and bb bb systems, the cbc b system is also a pure neutral system and has a certain C-parity.Thus the corresponding magnetic moment is 0µ N for all the ground cbc b states.Moreover, the Pauli principle does not impose any constraints on the wave functions of the cbc b system.Thus, compared to other discussed tetraquark systems, the cbc b system has more allowed states.There are four J P C = 0 ++ states, four J P C = 1 +− states, two J P C = 1 ++ states, two J P C = 2 ++ states in the cbc b system.
Here, we now analyze the numerical results about the cbc b system obtained from the variational method.Here, we take the J P C = 0 ++ cbc b ground state as an example to discuss specifically, and others would have similar discussions.The mass of the lowest J P C = 0 ++ cbc b state is 12759.3MeV, and the corresponding binding energy B T is +371.8MeV.Thus, the state obviously has a larger mass than the lowest rearrangement meson-meson decay channel η b η c , and it should be an unstable compact tetraquark state.Its variational parameters are given as C 11 = 11.9 fm −2 , C 22 = 11.9 fm −2 , and C 33 = 22.9 fm −2 .Since this state is a pure neutral state, we naturally notice that the value of C 11 is equal to C 22 , which means that the distance of (b − b) is equal to ( b − b).Our results also reflect these properties according to Table XVI.The corresponding wave function is given as: Based on Eq. ( 46), we find that its mass contribution to the ground state mainly comes from the 6⊗ 6 component, the corresponding 3⊗ 3 component being negligible.Then we transform Eq. ( 46) to cc−b b and c b−bc configurations via a linear transformation, and the corresponding two wave functions are given as: Further, we can certain its rearrangement strong width ratios.
where both the B * c Bc and B c B * c channels are the dominant decay modes for the T cbc b(12796.9, 1 +− ) tetraquark state.
Γ T cbc b (12796.9,1+− )→η b J/ψ : Γ T cbc b (12796.9,1+− )→Υηc = 1 : 18.4. ( The dominant decay channel is the Υη c final states in the cc − b b decay mode.We also calculate the transition magnetic moments for this state: Let us now turn our discussion to the internal contribution for the J P C = 1 +− cbc b ground state.For the kinetic energy part, the state obtains 858.5 MeV, which is smaller than the 1001.2MeV of the lowest meson-meson threshold B c η b according to Table XVI.As for the potential part, although the V C between quark and antiquark are attractive, the V C in the diquark and antiquark are repulsive.However, relative to the lowest meson-meson threshold B c η b , the total V C is not attractive than the B c η b , which leads to this state having a relatively larger mass. We also notice that the V C (1, 3), V C (2, 3), V C (1, 4), and V C (2, 4) are absolutely the same, and meanwhile the distances of (1,3), (1,4), (2,3), and (2,4) are also the same.These actually reflect Obviously, it is unreasonable that the distance of cc is exactly the same as that of the c b and b b.According to Sec IV of Ref. [65], we only consider single Gaussian form which the l 1 = l 2 = l 3 = 0 in spatial part of the total wave function is not sufficient.These lead to the cbc b state, which is far away from the real structures in nature.We have reason enough to believe that the ⟨Ψ tot |(R 1,2 • R 3,4 )|Ψ tot ⟩ should not be zero.Meanwhile, considering other spatial basis would reduce the corresponding to the binding energy B T [65].But these corrections would be powerless against the higher binding energy B T of the ground J P C = 1 +− cbc b.In conclusion, we tend to think that the J P C = 1 +− cbc b ground state should be an unstable compact state.

V. COMPARISON WITH OTHER WORK
The mass spectra have been studied with different approaches such as different nonrelativistic constituent quark models, different chromomagnetic models, the relativistic quark models, the nonrelativistic chiral quark model, the diquark models, the diffusion Monte Carlo calculation, and the QCD sum rule.In addition, these fully heavy tetraquark systems have been discussed with different color structures such as the 8 Q Q ⊗ 8 Q Q configuration, the diquark-antiquark configuration (3⊗ 3 and the 6 ⊗ 6) and the couplings between the above color configurations.For comparison, we briefly list our results and other theoretical results in Table X.
Compared to other systems, there is the most extensive discussion about the cccc system.So we will concentrate on the cccc system, but other systems can be discussed in a similar way.After comparing our results with other researches, we can see that the most theoretical masses of cccc in ground states lie in a wide range of 6.0 − 6.8 GeV in Table X.Our results are 6.38, 6.45, and 6.48 GeV for the 0 ++ , 1 +− , and 2 ++ cccc ground states, respectively.These three ground states are expected to be broad because they can all decay to charmonium pairs: η c η c , η c J/ψ, or J/ψJ/ψ through the quark (antiquark) rearrangements.Therefore, these types of decays are favored both dynamically and kinematically.According to Table X, we can conclude that the obtained masses of the ground states are obviously smaller than the X(6900) observed by the LHCb collaboration.The observed X(6900) is less likely to be the ground compact tetraquark state and could be a first or second radial excited cccc state.
Although we all use a similar Hamiltonian expression as in the nonrelativistic constituent quark model [111,[141][142][143], the spatial wave function is mostly expanded in the Gaussian basis according to Ref. [144], while we treat the spatial function as a Gaussian function, which is convenient for use in further variational methods to handle calculations in the four-body problem.Our results for the cccc system are roughly compatible with other nonrelativistic constituent quark models, although different papers have chosen different potential forms.
It is also interesting to note that relatively larger results are also given by the QCD sum rules [145], the Monte Carlo method [146], the diquark model [147], and the chiral quark model [148].However, the results given by the QCD sum rules [145] are about 1 GeV below those of the constituent quark models for the bb bb system.In contrast, our results are obviously larger than the chromomagnetic models [79,107,110,123], and the diquark models [100,149], where these models usually neglect the kinematic term and explicitly include confining potential contributions or adopt a diquark picture.

VI. SUMMARY
The discovery of exotic structures in the di-J/ψ invariant mass spectrum from the LHCb, CMS, and ATLAS collaborations gives us strong confidence to investigate the fully heavy tetraquark system.Thus, we use the variational method to systematically calculate the masses of all possible configurations for fully heavy tetraquarks within the framework of the constituent quark model.Meanwhile, we also give the corresponding internal mass contributions, the relative lengths between (anti)quarks, their lowest meson-meson thresholds, the specific wave function, the magnetic moments, the transition magnetic moments, the radiative decay widths, the rearrangement strong width ratios, and the comparisons with the two different CMI models.
To obtain the above results, we need to construct the total wave functions of the tetraquark states, including the flavor part, the color part, the spin part, and the spatial part, which is chosen to be a simple Gaussian form.Here, we first estimate the theoretical values of traditional hadrons, which are used to compare the experimental values to prove the reliability of this model.Before the discussing the numerical analysis, we analyze the stability condition by using only the color-spin interaction.Then, we obtain the specific numerical values and show them in corresponding Tables and the spatial distribution of valence quarks for the J P C = 0 ++ bb bb ground state in Fig. 1.
For the cccc and bb bb systems, there are two pure neutral systems with definite C-parity.There are only two J P C = 0 ++ states, one J P C = 1 +− state, and one J P C = 2 ++ state, due to the Pauli principle.We also find that these states with different quantum numbers are all above the lowest thresholds, and have larger masses.Since these states are pure neutral particles, the corresponding magnetic moments are all 0 for the ground cccc and bb bb states.Meanwhile, of course, the variational parameters C 11 and C 22 are the same, so the distances of the diquark and antidiquark are also the same.Moreover, the distances between quark and antiquark are all the same according to the symmetry analysis of Eqs.(30)(31).Furthermore, three Jacobi coordinates are orthogonal to each other according to Eqs. (32)(33)(34).Based on this, we take the J P C = 0 ++ bb bb ground state as an example to show the spatial distribution of four valence quarks.As for the internal contribution, although the kinetic energy part is smaller than that of the η b η b state, the V C in η b is much more attractive relative to the J P C = 0 ++ bb bb ground state, which is the main reason why this state has a larger mass than the meson-meson threshold.Similar situations also occur in other systems.Similar to the cccc and bb bb systems, the cc bb system has the same number of the allowed ground states.According to the specific function, their mass contribution mainly comes from the 3 ⊗ 3 component within the diquark-antiquark configuration.Furthermore, we get the relevant the values of the magnetic moments, the transition magnetic moments, and the radiative decay widths.We also obtain the rearrangement strong width ratios within the meson-meson configuration.
As for the ccc b and bb bc systems, there are more allowed states due to fewer symmetry restrictions.Considering only the hyperfine potential, we can expect to have a compact stable state for J P = 1 + bb bc configuration.However, since the V C of the tetraquark are less attractive than the corresponding mesons, this state still has a mass larger than the meson-meson threshold.
In the cbc b system, these states are also pure neutral particles, and we naturally obtain that their variational parameters C 11 and C 22 are the same.There is no constraint from the Pauli principle, so there are four J P C = 0 ++ states, four J P C = 1 +− states, two Then we compare our results with other theoretical work.Our results are roughly compatible with other nonrelativistic constituent quark models, although different papers have chosen different potential forms.Meanwhile, it is also interesting to that similar mass ranges are given by the QCD sum rules, the Monte Carlo method, and the chiral quark model.This shows that our results are quite reasonable.
In summary, our theoretical calculations show that the masses of the cccc ground states are around 6.45 GeV, which is obviously lower than 6.9 GeV.Thus, the experimentally observed X(6900) state does not seem to be a ground cccc tetraquark state, but could be a radially or orbitally excited state.We also find that these lowest states all have a large positive binding energy B T .In other words, all these states are found to have masses above the corresponding two meson decay thresholds via the quark rearrangement.Hence, we conclude that there is no compact bound ground fully heavy tetraquark state which is stable against the strong decay into two mesons within the constituent quark model.Finally, we hope that more relevant experimental analyses will be able to focus on this system in the near future.

VIII. APPENDIX
In this appendix, we show the masses, binding energies, variational parameters, internal contribution, total wave functions, magnetic moments, transition magnetic moments, radiative decay widths, rearrangement strong width ratios, and the relative lengths between the quarks for the cc bb , ccc b, bb bc, and cbc b states with different J P (C) quantum numbers and their lowest meson-meson thresholds.

RR ′ =0. 126 Radius=0. 130 FIG. 1 .
FIG.1.Relative positions for four valence quarks and Rc in the J P C = 0 ++ bb bb ground state.Meanwhile, we label the relative distances of R b,b , R b, b, Rb , b, R ′ , and the radius (units: fm).
internal kinetic energies: kinetic energies of two pairs of the b− b, and the (b b)−(b b) pair.Accordingly, the sum of the internal kinetic energies of the η b η b state only comes from the two pairs of the b − b.Therefore, this bb bb state has an additional kinetic energy needed to bring the η b η b into a compact configuration.The actual kinetic energies of two pairs of the b − b in the the J P C = 0 ++ bb bb ground state are smaller than those in the η b η b state.This is because, as can be seen in Table IX, the distance of b − b is larger in the tetraquark state than in the meson: the distance of b − b is 0.204 fm in this bb bb state while it is 0.148 fm in η b .Meanwhile, we find that even if we consider the additional kinetic energy between the (b b)−(b b)
states.All of the cbc b states have larger masses relative to the lowest thresholds.Moreover, they all have two different rearrangement strong decay modes: cc − b b and c b − bc.
This work is supported by the China National Funds for Distinguished Young Scientists under Grant No. 11825503, National Key Research and Development Program of China under Contract No. 2020YFA0406400, the 111 Project under Grant No. B20063, the National Natural Science Foundation of China under Grant No. 12247101, and the project for top-notch innovative talents of Gansu province.Z.W.L. would like to thank the support from the National Natural Science Foundation of China under Grants No. 12175091, and 11965016, and CAS Interdisciplinary Innovation Team.

TABLE I .
Parameters of the Hamiltonian.

TABLE II .
All possible flavor combinations for the fully-heavy tetraquark system.

TABLE III .
The reduced mass m ′ i in different states.

TABLE V .
The approximate relation about γi for the QQ Q Q system.

TABLE VI .
(1)ses and magnetic moments of some ground hadrons obtained from the theoretical calculations.M result , µ results , µ bag , µ the(1)

TABLE VIII .
The masses, binding energies, variational parameters, the internal contribution, total wave functions, magnetic moments, transition magnetic moments, radiative decay widths, rearrangement strong width ratios, and the relative lengths between quarks for the J P C = 2 ++ cccc and bb bb states and their lowest meson-meson thresholds.The notation is the same as that in TableVII.

TABLE IX .
The masses, binding energies, variational parameters, the internal contribution, total wave functions, magnetic moments, transition magnetic moments, radiative decay widths, rearrangement strong width ratios, and the relative lengths between quarks for the J P C = 0 ++ , 1 +− bb bb states and their lowest meson-meson thresholds.The notation is the same as that in TableVII.
. Other states would have similar discussions from Tables XIII-XV.The mass of the lowest J P = 1 + bb bc state is 16043.2MeV, and the corresponding binding energy B T is +303.7 MeV.Thus, the state is obviously above the lowest rearrangement meson-meson decay channel B * c η b , and it is an unstable tetraquark state.Its variational parameters are given as C 11 = 12.4fm −2 , C 22 = 21.0fm−2 , and C 33 = 28.9fm−2 .The corresponding wave function is given by

TABLE X .
Comparison of the results of different methods for the QQ Q Q tetraquark states. .

TABLE XII .
The masses, binding energies, variational parameters, the internal contribution, total wave functions, magnetic moments, transition magnetic moments, radiative decay widths, rearrangement strong width ratios, and the relative lengths between quarks for the J P = 2 + cc bb state and its lowest meson-meson threshold.The notation is the same as that in TableVII.

TABLE XIV .
The masses, binding energy, variational parameters, the internal contribution, total wave functions, magnetic moments, transition magnetic moments, radiative decay widths, rearrangement strong width ratios, and the relative lengths between quarks for the J P = 0 + , 1 + bb bc states and their lowest meson-meson thresholds.The notation is the same as that in TableVII.

TABLE XV .
The masses, binding energy, variational parameters, the internal contribution, total wave functions, magnetic moments, transition magnetic moments, radiative decay widths, rearrangement strong width ratios, and the relative lengths between quarks for the J P = 2 + ccc b and bb bc states and their lowest meson-meson thresholds.The notation is the same as that in TableVII.

TABLE XVI .
The masses, binding energies, variational parameters, the internal contribution, total wave functions, magnetic moments, transition magnetic moments, radiative decay widths, rearrangement strong width ratios, and the relative lengths between quarks for the the J P C = 0 ++ , 2 ++ cbc b states and their lowest meson-meson thresholds.The notation is the same as that in TableVII. .0

TABLE XVII .
The masses, binding energies, variational parameters, the internal contribution, total wave functions, magnetic moments, transition magnetic moments, radiative decay widths, rearrangement strong width ratios, and the relative lengths between quarks for the the J P C = 1 +− , 1 ++ cbc b states and their lowest meson-meson thresholds.The notation it the same as that in TableVII.