Triply heavy tetraquark states: masses and other properties

In this work, we study masses and other static properties of triply heavy tetraquarks in the unified framework of the MIT bag which incorporates chromomagnetic interactions and enhanced binding energy. The masses, magnetic moments and charge radii of all strange and nonstrange (ground) states of triply heavy tetraquarks are computed, suggesting that all of triply heavy tetraquarks are above the respective two-meson thresholds. We also estimate relative decay widths of main decay channels of two-heavy mesons for these tetraquarks.


I. INTRODUCTION
Study of exotic multi-quark states and hadronic molecules has been an interesting topic and continuous attention since the observation of the charmoniumlike state X(3872) in 2003 whose quantum number was later shown to be J P = 1 ++ [1].Since then, a large number of XYZ states were discovered experimentally, such as the charmonium-like states Z c (3885) [2,3], the Z c (4020) [4], the Z c (4025) [3], the Z cs (3985) [5], the Z cs (4000) [6], the Z c (4200) [7] and the Z c (4430) [8], many of which are possible candidates of multiquark state or hadronic molecule [9,10].Some of the observed XYZ states, like the charged state Z c (3900) [11,12], are undoubtedly exotic.In 2020, the first fully charm tetraquark, the X(6900), has been observed by LHCb in the di-J/Ψ invariant mass spectrum [13].In 2021, the first doubly charmed tetraquark T ++ cc was discovered in experiment [14].If these states are confirmed to be exotic, they may be the cadidates of tetraquarks.Many theoretical discussions exist for the tetraquarks and multiquarks with different approaches [15][16][17][18][19][20][21][22][23] .For the review see Refs.([24, 25], [26][27][28], [29][30][31][32][33][34][35][36][37]) There exist several approaches devoted to triply heavy tetraquark states, such as AdS/QCD potential [38], the quark models [39][40][41] and the chromomagnetic interaction model[42, 43,46].In the discussions, the method of the MIT bag model is rarely employed to explore triply heavy tetraquarks though it is successful for light hadrons.Main reason may be that the conventional MIT bag model fails to apply to heavy hadrons where some extra binding energy, vanishing between light quarks (u/d and s), enters between heavy quarks(c and b) and between heavy and strange quarks [37] and are necessary to reconcile light flavor dynamics with heavy flavor dynamics within an unified bag model [45].When the chromomagnetic interaction added which has two different mass scales (one light mass scale and one heavy mass scale), the strong coupling α s (R) tends to favor running with the bag radii R in order to correctly describe mass splittings of hadrons [45].This makes it involving to explore the heavy hadrons like heavy tetraquarks in which two heavy quarks(or more) or heavy and strange quarks are involved using the bag picture like the MIT bag model.
In this work, we apply the MIT bag model [44,45] which incorporate the chromomagnetic interaction and enhanced binding energy between heavy flavors to calculate the masses, magnetic moments and charge radii for all ground (strange and nonstrange) states of triply heavy tetraquarks.The dynamical computation indicates that all of triply heavy tetraquarks are above the respective two-meson thresholds.By the way, we estimate the main decay widths of two-heavy mesons for these triply heavy tetraquarks.
This paper will be organized as follows.In Sect.II, we briefly review the MIT bag model that we use in this work.We present the color-spin wave functions for tetraquark states in details also in this section.In Sect.III, computation is given for the masses and other properties for the triply heavy tetraquarks.The paper ends with summary in Sect.IV.
The MIT bag model assumes hadron to be a spherical bag(with the radius R) with quarks confined in it.The mass formula which, includes the enhanced binding energy E B and chromomagnetic interactions(CMI) can be written as [44] The first three terms contain the kinetic energy of all quarks, the volume energy with bag constant B and the zero-point energy with coefficient Z 0 .The total binding energy E B = I<J B IJ stands for the sum of the enhanced binding energies B IJ between the I and J quarks.The last term M CMI = ∆H is the chromomagnetic interaction among quarks within bag, given by where λ i are the Gell-Mann matrices in color space, σ i are the Pauli matrices in spin space and C i j are the CMI parameters, given by Here, μi is reduced magnetic moment, F(x i , x j ) a rational function and α(R) the strong coupling, all of which are detailed in Appendix A, where two relations relations for F(x i , x j ) and α(R) of computing the CMI matrix elements is also given.The binding energy B IJ stems mainly from the short-range chromoelectric interaction between heavy quark I and/or antiquark J, which is greatly enhanced compared to that between light quarks.Since the latter is much smaller, it can be ignored.In the case that the quark (I) is heavy and the other quark (J) is strange, we assume B IJ to be nonvanishing, as assumed in Ref. [37,45].As such, for the color configuration 3c of the quark pair, there exist five binding energies : B cs , B cc , B bs , B bb and B bc .
For the other parameters like quark masses, zero-point energy Z 0 in Eqs. ( 1) and ( 2) except for B IJ , we apply the following input values [44,45].For the binding energy between quark pairs i j in the rep.3c , we employ the values from Ref. [45]: In computation, we solve the transcendental equation(A3) interactively to find x i depending on R. Here, the bag radii R varies with hadrons and can be solved numerically by Eq. (1) using variational method provided that the CMI matrix is evaluated for a given configuration of tetraquark wavefunction, which we address in the following.
B. The color-spin wavefunction.
In this subsection we present the color-spin wavefunctions of all triply heavy tetraquark states which are required to evaluate the CMI mass splitting due to the chromommagnetic interaction in Eq. (3).For simplicity, we express the flavor parts of the tetraquarks explicitly in terms of the quark flavors(n = u/d, s, c and b).For the colorless tetraquark T , its color wavefunction has two color structures 6 c ⊗ 6c or 3c ⊗ 3 c in the notation of the color representations 3 c ( 3c ) and 6 c ( 6c ), corresponding to the following color configurations respectively.For the spin part of tetraquark wavefunction, there are six configurations (spin bases in spin space) Combining the color and spin parts of wavefunctions, tetraquark has twelve wavefunctions in their ground(1S ) state, where the δ is defined to take flavor symmetry into account when writting tetraquark wavefunction.Firstly, δ S 12 = 0(δ A 12 = 0) if two quarks q 1 and q 2 are antisymmetric(symmetric) in flavor space; Secondly, δ A 34 = 0(δ S 34 = 0) if q 3 and q 4 are symmetric(antisymmetric) in flavor space; Thirdly, δ S 12 = δ A 12 = 1 and δ S 34 = δ A 34 = 1 in all other cases.For given color configurations one can write binding energy explicitly E B = I<J B IJ = I<J g IJ B IJ ( 3c ) with the help of the ratios g IJ of the color factor λ I • λ J for given quark pair q I q J relative to the color factor λ I • λ J 3c for the quark pair q I q J in the color antitriplet( 3c ), provided that the binding energy linearly scales with this factor.In the case of quark pairs in the color antitriplet 3c and singlet 1 c , this is known as the 1/2-rule the short-range interaction: B([qq ′ ] 3 c ) = (1/2)B([qq ′ ] 1 c ).For the tetraquark with color configuration φ T 1 = (q 1 q 2 ) 6 ( q3 q4 ) 6 , one can compute scaling ratios g IJ [45] for each pair q I q J in tetraquark (q 1 q 2 ) ( q3 q4 ) and find the scaled binding energy g IJ B IJ ( 3c ), giving rised to total binding energy, Similarly, for the tetraquark with color configuration φ T 2 = (q 1 q 2 ) 3( q3 q4 ) 3 , the total binding energy is found to be [45] where B i j = B i j ( 3c ) stands for the enhanced binding energy between quark i and j in 3c , which are nonvanishing only between heavy quarks or between heavy quarks and strange quarks.Here, the numeric prefactors in front of B i j are the color factor ratios g i j of the quark pair (i, j) in given configurations φ T 1,2 .In the case of the quark pair 12 in 6 c in the state φ T 1 , for instance, the color factor ratio is −1 : 2 relative to that in 3c , and that for the pair 13 the ratio is 5 : 4 relative to that in 3c .For the details of the color factors and their ratios [45], see Appendix B.
Notice that in the color representation 3c , the binding energy for the given flavor of the quark pairs i j can be obtained from Ref. [45].
For the color or spin matrices between quarks in given tetraquark with wavefunctions φ T 1,2 , one can employ the respective matrix formula (B1) and (B2) in Appendix B. For tetraquarks with wavefunction bases (φ T 1 , φ T 2 ), the color factor matrices are given explicitly in Appendix B.
Using the CG coefficients for the given tetraquark configurations φ T 1,2 , one can write the six spin wavefunctions explicitly.They are where the notation uparrow stands for the state of spin upward, and the downarrow for the state of spin downwards.Given the spin wavefunctions (14), one can then compute the spin matrices σ i • σ j of tetraquark states via the formulas (B2) in Appendix B. There are six spin matrices for the tetraquark in the subspace of {χ T 1−6 }, which are

C. Properties of triply heavy tetraquark states
In the framework of MIT bag model, the masses, magnetic moments and charge radii can all be calculated for the ground state hadrons, as shown in sect.2.A.Given the model parameters and formulas in section 2, one can then compute the masses of the triply heavy tetraquark states.We review in this subsection the related relations for other properties (magnetic moments and charge radii) of triply heavy tetraquark states.
Following the method in bag model[47], the contribution of a quark i or an antiquark i with electric charge Q i to the charge radii of hadron is Summing all of these contributions in Eq. ( 21) gives rise to the charge radii of hadron [47], Note that Eq. ( 22) also holds true when the system has the identical quark constituents so that chromomagnetic interaction may yield spin-color state mixing.
For the magnetic moment, the related formula is [45,47].
where g i = 2 and S iz is the third component of quark i and antiquark i.
We list in the Table I all magnetic moment formulas as a sum rule for the triply heavy tetraquarks.Given spin wavefunction χ T 1−6 , the corresponding magnetic moment of triply heavy tetraquark states can be calculated via Table I.In the case of spin mixing, the computation goes in the following way: The spin wavefunction I: Sum rule for magnetic moments of tetraquarks (q 1 q 2 ) ( q3 q4 ) and their spin-mixed systems.
).In addition, we present the way to compute the decay width Γ I of the tetraquark state [46], which is given by with γ I the prefactor involving deacy dynamics, α a coupling constant at decay vertex, p is the momentum of final hadrons(two mesons M 1 and M 2 ), m the mass of inital tetraquark state and c I eigenvector of the I-th decay channel.The factor γ I varies depending upon the initial tetraquark and the final decay state and it differs for each decay channel.Notice that the following prefactors are nearly same Given the model parameters and formulas detailed in section 2, one can compute the masses, magnetic moments and charge radii of all ground (strange and nonstrange) states of triply heavy tetraquarks.For this purpose, we proceed this computation for the tetraquarks with three heavy quarks(antiquarks) by grouping them into four classes, namely, the triply heavy tetraquarks nb Q Q and nc Q Q, the triply heavy tetraquarks sc Q Q, sb Q Q, the triply heavy tetraquarks nQ Q Q′ and the triply heavy tetraquarks sQ Q Q′, as to be addressed in the following subsections.We use the notation T (qQ ′′ Q Q′ , M, J P ) denote the triply heavy tetraquark with flavor content qQ ′′ Q Q′ , mass M in GeV and the spin-parity J P , for simplicity.

A. Triply heavy tetraquarks nb
We consider first the triply heavy tetraquarks nb Q Q and nc Q Q(Q = c, b) with identical flavor of two heavy antiquark.Owing to the chromomagnetic interaction (3), the color-spin states (φχ) of tetraquark nb Q Q or nc Q Q can mix among them to form a mixed state, a linear superposition of spin-color bases.For the heavy tetraquark, the J P = 2 + state consists of one state (φ 2 χ 1 ), with no mixing.The J P = 1 + state consists of three basis states (φ 2 χ 2 , φ 2 χ 5 , φ 1 χ 4 ) and can mix into a mixed state having the form of c 1 φ 2 χ 2 + c 2 φ 2 χ 5 + c 3 φ 1 χ 4 .Meanwhile, the J P = 0 + state, consisting of two basis states (φ 2 χ 3 , φ 1 χ 6 ), can mix into the mixed states with the form c 1 φ 2 χ 3 + c 2 φ 1 χ 6 .
First of all, we write the mass formula for the triply heavy tetraquarks nb Q Q and nc Q Q with J P = 2 + , 1 + , 0 + , which are obtained via diagonalization of the CMI for J P = 1 + and 0 + multiplets(mixed states).Then, one can minimize the diagonalized mass (1) using the variantional method to find the respective optimal bag radii R of the tetraquark and to solve x i depending on R via the transcendental equation(A3) interactively.This leads to the tetraquark masses (1) for each of the triply heavy tetraquark states, as shown in FIG. 1 and FIG. 2. Further, one can use Eqs.(22) and (24) to compute their charge radii and magnetic moments, and use Eq. ( 25) to estimate the relative decay widths of the tetraquark for the two-meson channel depicted in FIG. 1.In Table .II, we list the obtained mass(in GeV), magnetic moments(in µ N ) and charge radii (in f m) of the triply heavy tetraquarks nb Q Q and nc Q Q, which contain four states with given J P : nb bb , nbcc, nc bb , and nccc.For the tetraquark nb bb , the J P = 0 + state has mass of 15.335 GeV for the mixed configuration dominated relatively by 3c ⊗ 3 c and has the mass of 15.401 GeV for the mixed configuration dominated relatively by 6 c ⊗ 6c .The later (in 6 c ⊗ 6c ) are heavier than the former(in 3c ⊗ 3 c ) about 0.066 GeV.As depicted in FIG. 1, this state may decay to the final two-meson states of the B * Υ, BΥ, B * η b or Bη b .Our computation via Eq.(25) shows that the tetraquark T (nb bb , 15.401, J P = 0 + ) can predominantly decay to B * Υ, with the relative decay widths of Γ B * Υ : Γ Bη b ∼ 1.15 : 0.004.In addition, it turns out that tetraquark T (nb bb , 15.386, J P = 1 + ) can predominantly decay into B * Υ or B * η b , with relative decay widths of Γ B * Υ : Γ B * η b : Γ BΥ ∼ 0.52 : 0.56 : 0.001.
For the tetraquark state nbcc, the J P = 0 + state has mass of 8.757 GeV in the mixed state with configuration 6 c ⊗ 6c dominated relatively and has mass of 8.610 GeV with 3c ⊗ 3 c dominated relatively, with the later lighter than the former about 0.147 GeV.As indicated in FIG. 1 For the tetraquark nccc, the J P = 0 + state has mass of 5.334 GeV in the mixed state with configuration 3c ⊗ 3 c dominated relatively and has mass of 5.397 GeV with 6 c ⊗ 6c dominated relatively, with the later heavier than the former about 0.063 GeV.As depicted in FIG. 2, the 0 + state can decay to D * J/ψ, D * η c , DJ/ψ or Dη c .The dominant channel of the tetraquark T (nccc, 5.397, J P = 0 + ) is the D * J/ψ, while the tetraquark T (nccc, 5.367, J P = 1 + ) can predominantly decay to D * J/ψ or D * η c , with the relative width Γ D * J/ψ : Γ Dη c ∼ 0.36 : 0.04.The decay width ratio is Γ D * J/ψ : Γ D * η c : Γ DJ/ψ ∼ 0.12 : 0.14 : 0.07.
In Table .II, we also compare our calculated tetraquark masses to that in Ref.

B. Triply heavy tetraquarks sb
Due to chromomagnetic interaction, the triply heavy tetraquarks sb Q Q and sc Q Q can mix their spin-color bases to form mixed states, a linear combination of spin-color basis functions (φχ).A J P = 2 + state of tetraquark consists of a singlet φ 2 χ 1 TABLE II: Computed masses(in GeV), magnetic moments(in µ N ) and charge radii (in f m) of triply heavy tetraquarks nb Q Q. Bag radii R 0 is in GeV −1 .The first and second values charge radii (magnetic moment) correspond to n=u and n=d, respectively.while the J P = 1 + state consist of the mixed states c 1 φ 2 χ 2 + c 2 φ 2 χ 5 + c 3 φ 1 χ 4 in the subspace (φ 2 χ 2 , φ 2 χ 5 , φ 1 χ 4 ); the J P = 0 + state is composed of the mixed state c 1 φ 2 χ 3 + c 2 φ 1 χ 6 in the subspace (φ 2 χ 3 , c 2 φ 1 χ 6 ).Similarly, we employ MIT bag mode to compute the masses, bag radii, magnetic moments, and charge radii of the triply heavy tetraquarks sb Q Q and sc Q Q with the spin-color wavefunctions.The results are shown in Table .III.

State
The triply heavy tetraquarks sb Q Q and sc Q Q consitst in flavor context of the four states: the tetraquark sb bb , the sbcc, the sc bb , and the sccc.In the case of the sb bb , the J P = 0 + state has mass of 15.433This type of triply heavy tetraquarks consists of two classes of the tetraquark states: nbc b and ncc b.The state with J P = 2 + contains two configurations (φ 2 χ 1 , φ 1 χ 1 ) while the 1 + state, spaned by the basis states ( Treating the CMI as perturbation, one can calculate the mass and other properties, as listed in Table IV.Given the computed masses (shown in in FIG. 5  The triply heavy tetraquarks sQ Q Q consists of two types of tetraquark states: the sbc b and the scc b.For them, the 2 + state contains two states (φ 2 χ 1 , φ 1 χ 1 ) and the 1 + state forms six mixed states c  .V.

IV. SUMMARY
In this work, we employ the MIT bag model which incorporates chromomagnetic interactions and enhanced binding energy to systematically study the masses and other properties of all triply heavy tetraquarks (qQ 1 Q2 Q3 ) in their ground states.The variational analysis is applied to compute the masses, magnetic moments and charge radii of all strange and nonstrange (ground) states of triply heavy tetraquarks.Our computation indicates that all of these triply heavy tetraquark states are above their twomeson thresholds and can decay to their two-meson final states via strong interaction.We observe that the tetraquark states in configuration 6 c ⊗ 6c are usually heavier than that in 3c ⊗ 3 c about 50 − 150 MeV for the same quantum number J P , with exception for all J P = 1 + states with one multiplet in the middle of two others in the sense of the mass order.Furthermore, the mass splitting between them becomes smaller when more bottom quarks are involved.By the way, we estimate the relative decay widths of two-meson final states of the triply heavy tetraquarks and predict the primary decay channels for each of them.We hope that our prediction helps to find the triply heavy tetraquarks during experimental search in the future.There are some scattering states among the tetraquark states (sQ Q Q′, nQ Q Q′) which is based on the numerical evaluation of the I-th decay channel weight |c I | 2 .When |c I | 2 tends to be close to unity, the tetraquark state turns out to be a scattering state, for which the decay and width computation are considered.in which the bag radius R depends upon a dimensionless parameter x i = x i (mR) via a transcendental equation.
The matrix elements in the CMI (3) can be evaluated via the following relations where (n, m) and (x, y) refer to the indices of the color and spin states, respectively, i and j refer to quarks or antiquarks.Here, c in stand for the color bases (color wavefunctions) of the quark(antiquark) i, and χ ix stand for the spin bases of the quark(antiquark) i.
Based on formulas (B1) and the color wavefunctions in subsection 2.B, one can compute all color factors for the normal hadrons (mesons and baryons) addressed in this work.For instance, the color factors is for the quark-antiquark in meson with the wave function (φ M ), and the color factor is for quark pairs in the baryon with (φ B ).
For tetraquarks with two color configurations (φ B 1 , φ B 2 ) in the zero-order approximation, the color factors for them are which are 2 × 2 matrices in the subspace spanded by the base wavefunctions (φ T

FIG. 1 :
FIG. 1: Mass spectra(GeV) of the nb b b and nbcc tetraquark states(sold line), with meson-meson thresholds(GeV) also plotted in dotted lines.

FIG. 2 :
FIG.2: Mass spectra(GeV) of the nc bb and nccc tetraquark states in sold line, with meson-meson thresholds (GeV) plotted in dotted lines.

cFIG. 3 :
FIG. 3: Mass spectra(GeV) of the tetraquarks sb bb and sbcc plotted in sold line.The meson-meson thresholds (GeV) are also shown in dotted lines.
) of the tetraquark ncc b, they exceed the threshold and may decay to B * J/ψ, BJ/ψ, D * B * c , B * η c , D * Bc , Bη c , D * B * c or D Bc , as shown in FIG. 5. Notably, the analysis via eigenvector of the CMI matrix reveals that the state T (ncc b, 8.747, 2 + ) couples srongly to D * B * c , while the T (ncc b, 8.596, 1 + ) couples significantly to D * Bc .Furthermore, the state T (ncc b, 8.566, 0 + ) couples srongly to D Bc , a candidate of the scattering states.Regarding the tetraquark nbc b, the main decay channels contain B * B * c , B B * c , B * Bc , B Bc , D * Υ, D * η b , DΥ and Dη b , as depicted in FIG. 5. Notably, analysis via comuted masses and eigenvectors (shown in TableV), the state T (nbc b, 11.927, 2 + ) exhibit a pronounced coupling to D * Υ, while the T (nbc b, 11.809, 1 + ) states display a strong coupling to DΥ.Additionally, the T (nbc b, 11.898, 0 + ) states demonstrate a robust coupling to B * B * c , whereas the state T (nbc b, 11.797, 0 + ) s exhibit a strong coupling to Dη b , potentially indicating scattering states.

FIG. 4 :
FIG. 4: Mass spectra(GeV) of the tetraquarks sc bb and sccc plotted in sold line.The meson-meson thresholds (GeV) are shown in dotted lines.
The computed masses and other properties of the tetraquark states are listed in Table V.With their masses exceeding the two-meson thresholds, the triply heavy tetraquark scc b can decay to the channels B * s J/ψ, B s J/ψ, D * s B * c , B * s η c , D * s B c , B s ηc, Ds B * c and Ds Bc , as depicted in FIG. 6.The state sbc b can decay to B * s B * c , B * s Bc , B s Bc , B s B * c , D * s Υ, D * s ηb, Ds Υ or Ds η b .Both of the tetraquarks scc b and sbc b can be the candidates of the scattering states, as noted in Table

TABLE
GeV in the mixed state with configuration 3c ⊗ 3 c dominated relatively and has mass of 15.492 GeV with 6 c ⊗ 6c dominated relatively, with the later heavier than the former about 0.059 GeV.This state can decay into the final two-meson states of the B * s Υ, B s Υ, B * s η b or B s η b , as shown in FIG. 3. We find that relative decay width between the channals B * s Υ and B s η b is Γ B * s Υ : Γ B s η b ∼ 0.07 : 0.94.The state T (sb bb , 15.492, 0 + ) can dominantly decay into B s η b whereas the state T (sb bb , 15.476, 1 + ) can primarily decay into B * Regarding the tetraquark sbcc, J P = 0 + state has mass of 8.783 GeV in the mixed state with configuration 6 c ⊗ 6c dominated relatively and has mass of 8.839 GeV with 3c ⊗ 3 c dominated relatively, with the later heavier than the former about 0.056 GeV.As illustrated in FIG. 3, the tetraquark sbcc can potentially decay to D * s Υ or B * s η b , with computed relative decay width Γ B * s Υ : Γ B * s η b : Γ B s Υ ∼ 0.53 : 0.55 : 0.001.s B * c , D * s Bc , Ds B * c or Ds Bc , with the relative decay widths

TABLE III :
Computed masses(in GeV), magnetic moments and charge radii of triply heavy tetraquarks sb Q Q and sc Q Q. Bag radii R 0 is in GeV −1 .

TABLE IV :
Computed masses (in GeV) triply heavy tetraquarks nQ Q Q′, with bag radii R 0 in GeV −1 .The masses are compared to other calculations cited.The charge radii are 0.32 fm (n=u) or 0.55 fm (n=d) for the tetraquark nbc b;They are 0.62 fm(n=u) or 0.26 fm(n=d) for the ncc b.The Bag radii R 0 are 4.430 GeV −1 for nbc b and 4.700 GeV −1 for ncc b.The first and second values magnetic moment correspond to n=u and n=d, respectively.

TABLE V :
Computed masses (GeV) triply heavy tetraquarks sQ Q Q′ , with bag radii R 0 in GeV −1 .The masses are compared to the other calculations.(sbc b) state's charge radii respectively are 0.54(fm); (sbc b) state's Bag radii R 0 respectively are 4.490 is in GeV −1 .(scc b) state's charge radii respectively are 0.21(fm).(scc b) state's Bag radii R 0 respectively are 4.75 is in GeV −1 .