1. Introduction
Recently, the LHCb collaboration [1] reported the important observation of a doubly charmed tetraquark containing two charm quarks, an anti-u and an anti-d quark, using the LHCb-experiment data at CERN, which manifests itself as a narrow peak in the mass spectrum of D0D0π+ mesons just below the D*+D0 mass threshold. This invites quantitative study of the mass spectroscopy of the multiquark hadrons and raises the issue of if there are more (strongly) stable doubly heavy tetraquark TQQ against the two-meson decay. Most mass computations of the compact tetraquark Tcc [2–7] predict masses around 3.9 − 4.1 GeV, above the D*+D0 mass threshold (3876 MeV).
For doubly charmed hadrons, one crucial experimental input is the strength of the interaction between two charm quarks, which can be provided by the doubly charmed baryon ${{\rm{\Xi }}}_{{cc}}^{++}={cuu}$ discovered by the LHCb Collaboration at CERN. The updated mass of the doubly charmed baryon ${{\rm{\Xi }}}_{{cc}}^{++}$ is 3621.55 ± 0.53 MeV [8]. This value is consistent with several predictions, including the computed values of 3620 MeV [9] and 3627 ± 12 MeV [10].
Interestingly, a narrow structure X(6900) around 6.9 GeV, the candidate for all-charm tetraquark, was observed in 2020 in the J/ψ-pair mass spectrum (above five standard deviations) by LHCb collaboration at CERN [11], with the Breit-Wigner mass m[X(6900)] = 6905 ± 11 ± 7 MeV and natural width Γ[X(6900)] = 80 ± 19 ± 33 MeV. This observation is confirmed recently by CMS experiment [12] in the di-J/ψ mass data with the resonance mass 6927 ± 9 ± 5 MeV. The recent ATLAS experiment at CERN also finds a resonance in the di-J/ψ channel [13], with the mass $6.87\pm {0.03}_{-0.01}^{+0.06}$ GeV and the width $0.12\pm {0.04}_{-0.01}^{+0.03}$ GeV, which is also consistent with X(6900) reported by the LHCb. Some other broad structures at lower masses and around 6.9 GeV are also seen in these experiments, whose natures are under study.
Given the mass of the ${{\rm{\Xi }}}_{{cc}}^{++}(3620)$ [8], a simple sum rule $M({cc}\bar{q}\bar{q})$ = 2M(ccu)-$M({cc}\bar{c}\bar{c})/2$ predicts the mass of the tetraquark ${T}_{{cc}\bar{q}\bar{q}}$ to be around 3790.6 ± 10.06 MeV, which is slightly below the D*+D0 mass threshold. Here, we used the mass $M({cc}\bar{c}\bar{c})$ = 6905 ± 18 MeV, taken from the LHCb measured data of the fully charmed tetraquark X(6900), around 6.9 GeV just above twice the J/$\Psi$ mass [11].
We use Regge relation methods in [14, 16] and earlier works [18], which are validated by many successful predictions of the excited heavy baryons [14, 16], to analyze low-lying mass spectra of the doubly heavy (DH) tetraquark. Masses of the DH tetraquark in their ground state and 1P wave are computed. In the analysis, the measured data of the ${{\rm{\Xi }}}_{{cc}}^{++}$ [8] and other mass estimates of the ground-state of doubly heavy baryons [9] is compatible with this data and are used to extract masses of heavy diquark MQQ (Q = c, b) and other trajectory parameters [18].
2. Method for excited spectrum
We write the mass of the doubly heavy (DH) tetraquarks TQQ as the sum of two parts: $M=\bar{M}+{\rm{\Delta }}M$, where $\bar{M}$ is the spin-independent part and ΔM is the spin-dependent mass. In the picture of heavy-diquark-light-antidiquark($\bar{q}\bar{q}$), the diquark QQ consisting of two heavy quarks is quite heavy compared to light antidiquark $\bar{q}\bar{q}$ so the heavy-light limit applies to the DH tetraquarks ${QQ}\bar{q}\bar{q}$. One can then derive, by analogy with the Regge relation in Ref. [14], a linear Regge relation from the quantum chromodynamics (QCD) string model for the DH tetraquark TQQ by viewing it to be a system of a massive QCD string with the diquark QQ(D) at one end and $\bar{q}\bar{q}$($\bar{D}$) at the other.
2.1. Spin-independent mass
We consider systems of a DH tetraquark TQQ(=${QQ}\bar{q}\bar{q}$) consisting of an S-wave heavy diquark QQ and an S-wave light antidiquark $\bar{q}\bar{q}$, which are in relative states of the orbital angular momentum L = 0 or L = 1 between two diquarks. We also assume the heavy diquark QQ is in colour antitriplet (${\bar{3}}_{c}$) and $\bar{q}\bar{q}$ is in colour triplet (3c). Thus, the DH tetraquark we consider is that with colour structure $({\bar{3}}_{c}\times {3}_{c})$. To describe TQQ, especially its excited states, we employ a linear Regge relation for the tetraquark TQQ, which is rooted in the general scattering theory of strong interaction and based on the observed spectrum of established hadrons [17]. The relation takes its form from [14]:1 ), $\bar{M}={M}_{{QQ}}+{m}_{{qq}}+{({M}_{{QQ}}{v}_{{QQ}})}^{2}/{M}_{{QQ}}$ where ${v}_{{QQ}}^{2}=1-{({m}_{\mathrm{bare}{QQ}}/{M}_{{QQ}})}^{2}$. It agrees with mass expansion at the heavy quark limit of DH hadrons.
$\begin{eqnarray}{\bar{M}}_{L}={M}_{{QQ}}+\sqrt{\pi {aL}+{\left[{m}_{{qq}}+{M}_{{QQ}}-{m}_{\mathrm{bare}\ \mathrm{QQ}}^{2}/{M}_{{QQ}}\right]}^{2}},\end{eqnarray}$
where MQQ and mqq are the effective masses of the heavy and light diquarks, respectively, and a stands for the tension of the QCD string connecting the diquark QQ at one end and the antidiquark $\bar{q}\bar{q}$ at the other. Here, ${m}_{\mathrm{bare}{QQ}}$ is the bare mass of QQ, given approximately by the sum of the bare masses of each quark Q: ${m}_{\mathrm{bare}{QQ}}$ $=\ {m}_{\mathrm{bare}Q}+{m}_{\mathrm{bare}Q}$, where ${m}_{{bare},Q}$ of the quarks Q(=c, b) are ${m}_{{bare},c}=1.275\,\mathrm{GeV}$ and ${m}_{{bare},b}=4.18$ GeV [15]. Numerically, one has $\begin{eqnarray}\begin{array}{l}{m}_{\mathrm{bare}{cc}}=2.55\,\mathrm{GeV},{m}_{\mathrm{bare}{bb}}=8.36\,\mathrm{GeV},\\ {m}_{\mathrm{bare}{bc}}=5.455\,\mathrm{GeV}.\end{array}\end{eqnarray}$
Applying to the S wave, one sees the reduction of the Regge relation (| 1. | (1) 1S wave. The relative orbital angular momentum L = 0 with respect to QQ. In this case, the Regge relation ( $\begin{eqnarray}\begin{array}{rcl}\bar{M}(1S) & = & {M}_{{QQ}}+{m}_{{qq}}+\displaystyle \frac{{k}_{{QQ}}^{2}}{{M}_{{QQ}}},\\ {k}_{{QQ}}^{2} & \equiv & {M}_{{QQ}}^{2}-{m}_{\mathrm{bare}{QQ}}^{2}.\end{array}\end{eqnarray}$ Putting masses in table 1 and equations ( |
| 2. | (2) 1P wave. As examined in [14], the Regge slope πa of heavy baryons, in the sense of quantum averaging in hadrons, is nearly independent of the spin of light diquarks (=0 or 1), but relies on the mass MQQ of heavy quarks. One then expects for tetraquark TQQ that its effective value of the tension a varies only with the heavy content of the flavor combinations QQ. |
Table 2. Mean (spin-averaged) mass(in MeV) of the doubly heavy tetraquarks TQQ in ground states, [] and {} stands for S[] = 0 and S{} = 1. |
| System | State | QQ = cc | QQ = bb | QQ = bc |
|---|---|---|---|---|
| IJP | Mass | |||
| $\{{QQ}\}[\bar{u}\bar{d}]$ | 01+ | 3997 | 10530 | 7270 |
| $[{QQ}][\bar{u}\bar{d}]$ | 00+ | – | – | 7268 |
| $[{QQ}]\{\bar{u}\bar{d}\}$ | 11+ | – | – | 7478 |
| $\{{QQ}\}\{\bar{u}\bar{d}\}$ | 10+, 11+, 12+ | 4207 | 10740 | 7480 |
| $\{{QQ}\}[\bar{u}\bar{s}]$ | $\tfrac{1}{2}{1}^{+}$ | 4180 | 10713 | 7453 |
| $[{QQ}][\bar{u}\bar{s}]$ | $\tfrac{1}{2}{0}^{+}$ | – | – | 7451 |
| $[{QQ}]\{\bar{u}\bar{s}\}$ | $\tfrac{1}{2}{1}^{+}$ | – | – | 7605 |
| $\{{QQ}\}\{\bar{u}\bar{s}\}$ | $\tfrac{1}{2}{0}^{+}$  , $\tfrac{1}{2}{1}^{+}$, $\tfrac{1}{2}{2}^{+}$ | 4334 | 10867 | 7607 |
| $\{{QQ}\}\{\bar{s}\bar{s}\}$ | 00+, 01+, 02+ | 4450 | 10987 | 7725 |
| $[{QQ}]\{\bar{s}\bar{s}\}$ | 01+ | – | – | 7724 |
Let us consider first the tetraquark ${T}_{{QQ}}=({QQ})(\bar{u}\bar{d})$ and scale its tension to the Λc,b( = Qud, Q = c, b) baryons which share the heavy-light structure similar to singly-heavy mesons. We assume, for simplicity, a power-law of the mass scaling for two string tensions of the doubly heavy tetraquark $({QQ})(\bar{u}\bar{d})$ and Λc,b:
$\begin{eqnarray}\displaystyle \frac{{a}_{{{\rm{\Lambda }}}_{c}}}{{a}_{{{\rm{\Lambda }}}_{b}}}={\left(\displaystyle \frac{{M}_{c}}{{M}_{b}}\right)}^{{P}_{1}},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{a}_{{{\rm{\Lambda }}}_{c}}}{{a}_{({QQ})(\bar{u}\bar{d})}}={\left(\displaystyle \frac{{M}_{c}}{{M}_{{QQ}}}\right)}^{{P}_{1}},\end{eqnarray}$
Here, the relevant string tensions are evaluated previously in [14], as listed in table 3.Table 3. The string tension coefficient a (GeV)2 of Λc/Λb,Ξc/Ξb,Ωc/Ωb baryons and the effective mass of component quark (GeV) of Mc and Mb. |
| Parameters | Mc | Mb | mu | ${a}_{{{\rm{\Lambda }}}_{c}}$ | ${a}_{{{\rm{\Lambda }}}_{b}}$ | ${a}_{{{\rm{\Xi }}}_{c}}$ | ${a}_{{{\rm{\Xi }}}_{b}}$ | ${a}_{{{\rm{\Omega }}}_{c}}$ | ${a}_{{{\rm{\Omega }}}_{b}}$ |
|---|---|---|---|---|---|---|---|---|---|
| The value | 1.44 | 4.48 | 0.23 | 0.212 | 0.246 | 0.255 | 0.307 | 0.316 | 0.318 |
The same procedure applies to the strange ${T}_{{QQ}}=({QQ})(\bar{u}\bar{s})$ and the associated Ξc,b(=c(us), b(us)) baryons, the ${T}_{{QQ}}=({QQ})(\bar{s}\bar{s})$ and the Ωc,b(=c(ss), b(ss)) baryons, for which the mass scaling, corresponding to equations (4 ) and (5 ), has the same form
$\begin{eqnarray}\displaystyle \frac{{a}_{{{\rm{\Xi }}}_{c}}}{{a}_{{{\rm{\Xi }}}_{b}}}={\left(\displaystyle \frac{{M}_{c}}{{M}_{b}}\right)}^{{P}_{2}},\displaystyle \frac{{a}_{{{\rm{\Xi }}}_{c}}}{{a}_{({QQ})(\bar{u}\bar{s})}}={\left(\displaystyle \frac{{M}_{c}}{{M}_{{QQ}}}\right)}^{{P}_{2}},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{a}_{{{\rm{\Omega }}}_{c}}}{{a}_{{{\rm{\Omega }}}_{b}}}={\left(\displaystyle \frac{{M}_{c}}{{M}_{b}}\right)}^{{P}_{3}},\displaystyle \frac{{a}_{{{\rm{\Omega }}}_{c}}}{{a}_{({QQ})(\bar{s}\bar{s})}}={\left(\displaystyle \frac{{M}_{c}}{{M}_{{QQ}}}\right)}^{{P}_{3}}.\end{eqnarray}$
Putting the parameters in table 3 into equations (4 )–(7 ), one obtains the all power parameter P and ensuing effective values of the string tension9 ) and table 1 to equation (1 ), one can obtain the mean (spin-averaged) masses of the TQQ system $({QQ})(\bar{q}\bar{q})$ in P-wave(L = 1). The results are
$\begin{eqnarray}\begin{array}{c}{P}_{1}=0.1311\ \ ,{P}_{2}=0.1635\ \ ,{P}_{3}=0.0056\ \ ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\left\{\begin{array}{c}{a}_{({cc})(\bar{u}\bar{d})}=0.2320\ \ {\mathrm{GeV}}^{2},{a}_{({bb})(\bar{u}\bar{d})}=0.2692\ \ {\mathrm{GeV}}^{2},{a}_{({bc})(\bar{u}\bar{d})}=0.2320\ \ {\mathrm{GeV}}^{2},\\ {a}_{({cc})(\bar{u}\bar{s})}=0.2854\ \ {\mathrm{GeV}}^{2},{a}_{({bb})(\bar{u}\bar{s})}=0.3436\ \ {\mathrm{GeV}}^{2},{a}_{({bc})(\bar{u}\bar{s})}=0.3211\ \ {\mathrm{GeV}}^{2},\\ {a}_{({cc})(\bar{s}\bar{s})}=0.3172\ \ {\mathrm{GeV}}^{2},{a}_{({bb})(\bar{s}\bar{s})}=0.3192\ \ {\mathrm{GeV}}^{2},{a}_{({bc})(\bar{s}\bar{s})}=0.3185\ \ {\mathrm{GeV}}^{2},\end{array}\right\}\end{eqnarray}$
Applying the data in equation ( $\begin{eqnarray}\left\{\begin{array}{c}{\bar{M}}_{({cc}[\bar{u}\bar{d}])}=4.283\ \ \mathrm{GeV},{\bar{M}}_{({bb}[\bar{u}\bar{d}])}=10.774\ \ \mathrm{GeV},{\bar{M}}_{({bc}[\bar{u}\bar{d}])}=7.535\ \ \mathrm{GeV},\\ {\bar{M}}_{({cc}\{\bar{u}\bar{d}\})}=4.455\ \ \mathrm{GeV},{\bar{M}}_{({bb}\{\bar{u}\bar{d}\})}=10.959\ \ \mathrm{GeV},{\bar{M}}_{({bc}\{\bar{u}\bar{d}\})}=7.714\ \ \mathrm{GeV},\\ {\bar{M}}_{({cc}[\bar{u}\bar{s}])}=4.485\ \ \mathrm{GeV},{\bar{M}}_{({bb}[\bar{u}\bar{s}])}=10.992\ \ \mathrm{GeV},{\bar{M}}_{({bc}[\bar{u}\bar{s}])}=7.748\ \ \mathrm{GeV},\\ {\bar{M}}_{({cc}\{\bar{u}\bar{s}\})}=4.613\ \ \mathrm{GeV},{\bar{M}}_{({bb}\{\bar{u}\bar{s}\})}=11.127\ \ \mathrm{GeV},{\bar{M}}_{({bc}\{\bar{u}\bar{s}\})}=7.879\ \ \mathrm{GeV},\\ {\bar{M}}_{({cc}\{\bar{s}\bar{s}\})}=4.741\ \ \mathrm{GeV},{\bar{M}}_{({bb}\{\bar{s}\bar{s}\})}=11.216\ \ \mathrm{GeV},{\bar{M}}_{({bc}\{\bar{s}\bar{s}\})}=7.981\ \ \mathrm{GeV}.\end{array}\right\}\end{eqnarray}$
where QQ = cc, bb, bc stand for the axial diquark {QQ} with spin S{QQ}=1.2.2. Spin-dependent mass
For mass splitting ΔM = ⟨HSD⟩ due to spin interaction between heavy and light diquarks with spins SQQ and Sqq, we consider the spin-dependent Hamiltonian [16, 30]
$\begin{eqnarray}\begin{array}{rcl}{H}^{{SD}} & = & {a}_{1}{\bf{L}}\cdot {{\bf{S}}}_{{qq}}+{a}_{2}{\bf{L}}\cdot {{\bf{S}}}_{{QQ}}+{{bS}}_{12}+c{{\bf{S}}}_{{qq}}\cdot {{\bf{S}}}_{{QQ}},\\ {S}_{12} & = & 3{{\bf{S}}}_{{qq}}\cdot \hat{{\bf{r}}}{{\bf{S}}}_{{QQ}}\cdot \hat{{\bf{r}}}-{{\bf{S}}}_{{qq}}\cdot {{\bf{S}}}_{{QQ}},\end{array}\end{eqnarray}$
where the first two terms are spin-orbit forces, the third is a tensor force, and the last describes hyperfine splitting. We discuss the following states of the DH tetraquark:| 1. | (1) 1S wave. For the 1S wave, L = 0, the spin-interaction Hamiltonian is simply $\begin{eqnarray}{H}^{{SD}}=c{{\bf{S}}}_{{QQ}}\cdot {{\bf{S}}}_{{qq}},\end{eqnarray}$ in which SQQ · Sqq has the eigenvalues −2, −1, 1, 0 when SQQ = 1. One has the mass formula then, $\begin{eqnarray}\begin{array}{l}M({QQ}\bar{q}\bar{q},1S)=\bar{M}({QQ}\bar{q}\bar{q})+c({QQ}\bar{q}\bar{q})\\ {diag}\{-2,-1,1,0\}.\end{array}\end{eqnarray}$ Based on the similarity between TQQ and heavy mesons $Q\bar{q}$(we choose D meson typically), one has a relation of mass scaling for the spin coupling c, $\begin{eqnarray}c\left(\{{QQ}\}\left(\bar{q}\bar{q}\right)\right)=\displaystyle \frac{{M}_{c}}{{M}_{{QQ}}}\cdot \displaystyle \frac{{m}_{q}}{{m}_{{qq}}}\cdot c{\left(D\right)}_{1S}.\end{eqnarray}$ in which c(D)1S = 140.6 MeV, mq = 230 MeV and other masses are given in table 1 and table 3. |
| 2. | (2) 1P wave. In the heavy-diquark(D)-light-antidiquark($\bar{{\rm{D}}}$) picture, the total spin of the TQQ is denoted by ${S}_{{tot}}={S}_{D}+{S}_{\bar{D}}$, which takes value Stot = 2, 1, 0 when ${S}_{D}={S}_{\bar{D}}=1$ and Stot = 1 when SD = 1 and ${S}_{\bar{D}}=0$. In the scheme of LS coupling, coupling Stot = 2 to L = 1 gives the states with the total angular momentums J = 3, 2, 1, while coupling Stot = 1 to L = 1 leads to the states with J = 2, 1, 0; coupling Stot = 0 to L = 1 leads to the states with J = 1. Normally, one uses the LS basis ${}^{2{S}_{{tot}}+1}{P}_{J}$ ={3P0,1P1,3P1,5P1,3P2,5P2,5P3} to label these multiplets in P-wave. We consider two cases for DH tetraquarks: |
| a | (a) ${S}_{D}={S}_{\bar{D}}=1$. There are seven states, all of which are negative parity. In the LS basis, the three J = 1 states and two J = 2 states are unmixed unless a1 = a2. Otherwise, they are the respective eigenstates of a 3 × 3 and 2 × 2 matrices ΔMJ representing HSD with J = 1 and J = 2. The matrices of mass shifts are then (see $\begin{eqnarray}{\rm{\Delta }}{M}_{J=0}=-{a}_{1}-{a}_{2}-2b-c,\end{eqnarray}$ $\begin{eqnarray}\begin{array}{l}{\rm{\Delta }}{M}_{J=1}=\left[\begin{array}{ccc}0 & \displaystyle \frac{2}{\sqrt{3}}\left({a}_{1}-{a}_{2}\right) & 0\\ \displaystyle \frac{2}{\sqrt{3}}\left({a}_{1}-{a}_{2}\right) & \displaystyle \frac{1}{2}\left({a}_{1}+{a}_{2}\right) & \displaystyle \frac{\sqrt{5}}{6}\left({a}_{1}-{a}_{2}\right)\\ 0 & \displaystyle \frac{\sqrt{5}}{6}\left({a}_{1}-{a}_{2}\right) & -\displaystyle \frac{3}{2}\left({a}_{1}+{a}_{2}\right)\end{array}\right],\\ \quad +b\left[\begin{array}{ccc}0 & 0 & \displaystyle \frac{32}{15\sqrt{5}}\\ 0 & 1 & 0\\ \displaystyle \frac{32}{15\sqrt{5}} & 0 & -\displaystyle \frac{7}{5}\end{array}\right]+c\left[\begin{array}{ccc}-2 & 0 & \displaystyle \frac{1}{6\sqrt{5}}\\ 0 & -1 & 0\\ \displaystyle \frac{1}{6\sqrt{5}} & 0 & 1\end{array}\right],\end{array}\end{eqnarray}$ $\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{M}_{J=2} & = & \left[\begin{array}{cc}\displaystyle \frac{1}{2}\left({a}_{2}+{a}_{1}\right) & \displaystyle \frac{\sqrt{3}}{2}\left({a}_{1}-{a}_{2}\right)\\ \displaystyle \frac{\sqrt{3}}{2}\left({a}_{1}-{a}_{2}\right) & \displaystyle \frac{1}{2}\left({a}_{2}+{a}_{1}\right)\end{array}\right]\\ & & +b\left[\begin{array}{cc}-\displaystyle \frac{1}{5} & 0\\ 0 & \displaystyle \frac{7}{5}\end{array}\right]+c\left[\begin{array}{cc}-1 & 0\\ 0 & 1\end{array}\right],\end{array}\end{eqnarray}$ $\begin{eqnarray}{\rm{\Delta }}{M}_{J=3}={a}_{1}+{a}_{2}-\displaystyle \frac{2}{5}b+c,\end{eqnarray}$ |
In the heavy quark limit, the terms involving SQQ in equation (13 ) behave as 1/MQQ and are suppressed. Due to heavy quark spin symmetry(SQQ conserved), the total angular momentum of the light quark j = L + Sqq = J − SQQ is conserved and forms a set of the conserved operators {J, j}, together with the total angular momentum J of the DH tetraquark. So instead of the LS coupling, one can use the jj coupling scheme in which the light-antidiquark spin Sqq couples to L to form total angular momentum j of the light quark and then to the conserved spin SQQ of the heavy-diquark. So, for the tetraquarks TQQ, we label the P-wave multiplets in terms of the basis ∣J, j⟩ of the jj coupling (appendix ), in which the diquark QQ is infinitely heavy (MQQ → ∞ ) and L · Sqq becomes diagonal. As such, one can employ the jj coupling to find the formula for mass splitting ΔM by diagonalizing L · Sqq and treating other interactions in equation (11 ) proportional to a2, b and c perturbatively. The results for ΔM(J, j) are listed in table 4. Given the matrices in table 4, one can use the lowest perturbation theory to find the mass splitting ΔM(J, j) of the DH tetraquarks in the P-wave. The result is
$\begin{eqnarray}{\rm{\Delta }}M(0,0)=-{a}_{1}-{a}_{2}-2b-c,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}M(1,0)=-2{a}_{2}+\displaystyle \frac{4}{135}b+\displaystyle \frac{1}{27}c,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}M(1,1)=-{a}_{1}-\displaystyle \frac{1}{2}{a}_{2}-\displaystyle \frac{47}{45}b-\displaystyle \frac{5}{9}c,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}M(1,2)={a}_{1}-\displaystyle \frac{3}{2}{a}_{2}+\displaystyle \frac{83}{135}b-\displaystyle \frac{40}{27}c,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}M(2,1)=-{a}_{1}+\displaystyle \frac{1}{2}{a}_{2}+b+\displaystyle \frac{1}{2}c,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}M(2,2)={a}_{1}-\displaystyle \frac{1}{2}{a}_{2}+\displaystyle \frac{1}{5}b-\displaystyle \frac{1}{2}c,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}M(3,2)={a}_{1}+{a}_{2}-\displaystyle \frac{2}{5}b+c,\end{eqnarray}$
which express the TQQ mass splitting in terms of four parameters (a1, a2, b, c) of the spin couplings. Adding the spin-dependent $\bar{M}=\bar{M}(1P)$, one obtains the respective mass formula $M(J,j)=\bar{M}(1P)+{\rm{\Delta }}M(J,j)$ for the TQQ in P-wave. Here, one can check that the spin-weighted sum of these mass shifts is zero: $\begin{eqnarray}\displaystyle \sum _{J}(2J+1){\rm{\Delta }}M(J,j)=0.\end{eqnarray}$
Table 4. The matrix elements of the mass splitting operators in the P-wave TQQ state in the jj coupling. |
| (J, j) | $\left\langle {\bf{L}}\cdot {{\bf{S}}}_{{qq}}\right\rangle $ | $\left\langle {\bf{L}}\cdot {{\bf{S}}}_{{QQ}}\right\rangle $ | $\left\langle {S}_{12}\right\rangle $ | $\left\langle {{\bf{S}}}_{{qq}}\cdot {{\bf{S}}}_{{QQ}}\right\rangle $ |
|---|---|---|---|---|
| (0, 0) | −1 | −1 | −2 | −1 |
| | ||||
| (1, 0) | −2 | 0 | 4/135 | 1/27 |
| (1, 1) | −1 | −1/2 | −47/45 | −5/9 |
| (1, 2) | 1 | −2/3 | 83/135 | −40/27 |
| | ||||
| (2, 1) | −1 | 1/2 | 1 | 1/2 |
| (2, 2) | 1 | −1/2 | 1/5 | −1/2 |
| | ||||
| (3, 2) | 1 | 1 | −2/5 | 1 |
To evaluate spin coupling parameters (a1, a2, b, c) in equation (19 ) through equation (25 ), we use the following mass scaling,31 ).31 ) and tables 3 and 1 into equations (27 )–(30 ), one obtains the spin couplings for the doubly heavy tetraquarks TQQ, with the results listed collectively in table 5.
$\begin{eqnarray}{a}_{1}[{QQ}(\bar{q}\bar{q})]=\displaystyle \frac{{M}_{c}}{{M}_{{QQ}}}\cdot \displaystyle \frac{{m}_{s}}{{m}_{({qq})}}\cdot {a}_{1}\left({D}_{s}\right),\end{eqnarray}$
$\begin{eqnarray}{a}_{2}[{QQ}(\bar{q}\bar{q})]=\displaystyle \frac{{M}_{c}}{{M}_{{QQ}}}\cdot \displaystyle \frac{{m}_{s}}{1+{m}_{({qq})/{M}_{c}}}\cdot {a}_{2}\left({D}_{s}\right),\end{eqnarray}$
$\begin{eqnarray}b[{QQ}(\bar{q}\bar{q})]=\displaystyle \frac{{M}_{c}}{{M}_{{QQ}}}\cdot \displaystyle \frac{{m}_{s}}{1+{m}_{({qq})/{m}_{s}}}\cdot b\left({D}_{s}\right),\end{eqnarray}$
$\begin{eqnarray}c[{QQ}(\bar{q}\bar{q})]=\displaystyle \frac{{M}_{c}}{{M}_{{QQ}}}\cdot \displaystyle \frac{{m}_{{ss}}}{{m}_{({qq})}}\cdot c\left({{\rm{\Omega }}}_{c}\right),\end{eqnarray}$
which apply successfully to the singly-heavy baryons [14, 16]. For this, we list the parameters of spin couplings for the Ds and Ωc in [14, 16, 29] and the effective masses of mss and ms in [14, 29] collectively in equation ( $\begin{eqnarray}\left\{\begin{array}{c}{a}_{1}({D}_{s})=89.36\ \ \mathrm{MeV},{a}_{2}({D}_{s})=40.7\ \ \mathrm{MeV},b({D}_{s})=65.6\ \ \mathrm{MeV},\\ c({{\rm{\Omega }}}_{c})=4.04\ \ \mathrm{MeV},{m}_{s}=328\ \ \mathrm{MeV},{m}_{{ss}}=991\ \ \mathrm{MeV},\end{array}\right\}\end{eqnarray}$
Putting the data in equation (Table 5. Spin-coupling parameters (in MeV) in the spin interaction ( |
| TQQ | ${cc}\{\bar{u}\bar{d}\}$ | ${bb}\{\bar{u}\bar{d}\}$ | ${bc}\{\bar{u}\bar{d}\}$ | ${cc}\{\bar{u}\bar{s}\}$ | ${bb}\{\bar{u}\bar{s}\}$ | ${bc}\{\bar{u}\bar{s}\}$ | ${cc}\{\bar{s}\bar{s}\}$ | ${bb}\{\bar{s}\bar{s}\}$ | ${bc}\{\bar{s}\bar{s}\}$ |
|---|---|---|---|---|---|---|---|---|---|
| a1 | 19.77 | 6.35 | 9.61 | 16.89 | 5.43 | 8.21 | 14.86 | 4.78 | 7.23 |
| a2 | 13.48 | 4.33 | 6.56 | 12.74 | 4.09 | 6.20 | 12.12 | 3.89 | 5.89 |
| b | 10.08 | 3.42 | 4.90 | 9.01 | 2.90 | 4.38 | 8.20 | 2.63 | 3.99 |
| c | 2.70 | 0.87 | 1.31 | 2.31 | 0.74 | 1.12 | 2.03 | 0.65 | 0.99 |
| a | (b) SD = 1,${S}_{\bar{D}}=0$. There are three states with J = 0, 1, 2, corresponding to mass shifts of the spin interaction as follows $\begin{eqnarray}{\rm{\Delta }}{M}_{J=0}=-2{a}_{2},\end{eqnarray}$ $\begin{eqnarray}{\rm{\Delta }}{M}_{J=1}=-{a}_{2},\end{eqnarray}$ $\begin{eqnarray}{\rm{\Delta }}{M}_{J=2}={a}_{2},\end{eqnarray}$ for which ∑J(2J + 1)ΔMJ = 0 ( $\begin{eqnarray}\left\{\begin{array}{c}{a}_{2}({cc}[\bar{u}\bar{d}])=14.91\ \ \mathrm{MeV},{a}_{2}({bb}[\bar{u}\bar{d}])=4.79\ \ \mathrm{MeV},{a}_{2}({bc}[\bar{u}\bar{d}])=7.25\ \ \mathrm{MeV},\\ {a}_{2}({cc}[\bar{u}\bar{s}])=13.65\ \ \mathrm{MeV},{a}_{2}({bb}[\bar{u}\bar{s}])=4.39\ \ \mathrm{MeV},{a}_{2}({bc}[\bar{u}\bar{s}])=6.64\ \ \mathrm{MeV},\end{array}\right\}\end{eqnarray}$ |
3. Numerical results and discussions
Based on section 2.2 , one can calculate spin-dependent mass ΔM corresponding to the spin-dependent interaction. Adding these masses to the spin-averaged masses discussed in section 2.1 , one can find the mass $M={\bar{M}}_{L}+{\rm{\Delta }}M$ of a given DH tetraquark. In tables 6–10, we list the mass results for the double heavy tetraquarks in 1S and 1P waves.
Table 6. Ground-state masses M of the cc tetraquarks, the lowest threshold T for its decaying into two D($c\bar{q}$) mesons and Δ = M − T. All values are given in MeV. |
| System | IJP | M | T | Δ | [31] | [32] | [3] | [33] | [4] | [25] |
|---|---|---|---|---|---|---|---|---|---|---|
| $({cc})(\bar{u}\bar{d})$ | 01+ | 3997 | 3871 | 126 | 3935 | 3947 | 3978 | |||
| 10+ | 4163 | 3729 | 434 | 4056 | 4111 | 4146 | ||||
| 11+ | 4185 | 3871 | 314 | 4079 | 4133 | 4167 | 4117 | 4201 | 4268 | |
| 12+ | 4229 | 4041 | 188 | 4118 | 4177 | 4210 | 4179 | 4271 | 4318 | |
| | ||||||||||
| $({cc})(\bar{u}\bar{s})$ | $\tfrac{1}{2}{1}^{+}$ | 4180 | 3975 | 205 | 4143 | 4124 | 4156 | |||
| $\tfrac{1}{2}{0}^{+}$ | 4297 | 3833 | 464 | 4221 | 4232 | |||||
| $\tfrac{1}{2}{1}^{+}$ | 4315 | 3975 | 340 | 4239 | 4254 | 4314 | 4363 | 4394 | ||
| $\tfrac{1}{2}{2}^{+}$ | 4352 | 4119 | 233 | 4271 | 4298 | 4305 | 4434 | 4440 | ||
| | ||||||||||
| $({cc})(\bar{s}\bar{s})$ | 00+ | 4420 | 3936 | 484 | 4359 | 4352 | ||||
| 01+ | 4436 | 4080 | 356 | 4375 | 4374 | 4382 | 4526 | 4493 | ||
| 02+ | 4470 | 4224 | 246 | 4402 | 4418 | 4433 | 4597 | 4536 | ||
Table 7. Ground-state masses M of the bb tetraquarks, the lowest threshold T for its decaying into two B($b\bar{q}$) mesons and Δ = M − T. All values are in MeV. |
| System | IJP | M | T | Δ | [31] | [32] | [3] | [33] | [4] | [25] |
|---|---|---|---|---|---|---|---|---|---|---|
| $({bb})(\bar{u}\bar{d})$ | 01+ | 10530 | 10604 | −74 | 10502 | 10471 | 10482 | |||
| 10+ | 10726 | 10588 | 168 | 10648 | 10664 | 10674 | ||||
| 11+ | 10733 | 10604 | 129 | 10657 | 10671 | 10681 | 10854 | 10875 | 10779 | |
| 12+ | 10747 | 10650 | 97 | 10673 | 10685 | 10695 | 10878 | 10897 | 10799 | |
| | ||||||||||
| $({bb})(\bar{u}\bar{s})$ | $\tfrac{1}{2}{1}^{+}$ | 10713 | 10693 | 20 | 10706 | 10644 | 10643 | |||
| $\tfrac{1}{2}{0}^{+}$ | 10855 | 10649 | 208 | 10802 | 10781 | |||||
| $\tfrac{1}{2}{1}^{+}$ | 10861 | 10693 | 168 | 10809 | 10788 | 10974 | 11010 | 10897 | ||
| $\tfrac{1}{2}{2}^{+}$ | 10873 | 10742 | 131 | 10823 | 10802 | 10997 | 11060 | 10915 | ||
| | ||||||||||
| $({bb})(\bar{s}\bar{s})$ | 00+ | 10976 | 10739 | 237 | 10932 | 10898 | ||||
| 01+ | 10981 | 10786 | 195 | 10939 | 10905 | 11099 | 11199 | 10986 | ||
| 02+ | 10991 | 10833 | 158 | 10950 | 10919 | 11119 | 11224 | 11004 | ||
It is seen from tables 6–8 that our mass predictions of the DH tetraquarks are compatible with most other calculations cited. In the nonstrange sector, our predictions for the P wave masses are higher than the ground-state mass about 300 MeV for the cc tetraquarks, about 300–400 MeV for the bb tetraquarks and about 220 MeV for the bc tetraquarks. In the strange sector, our predictions for the P wave masses are higher than the ground-state mass about 300 MeV for the cc tetraquarks, about 270 MeV for the bb tetraquarks and about 300 MeV for the bc tetraquarks. It turns out that the monotonous suppression of the S-P wave splitting of tetraquark masses by heavy diquark mass MQQ breaks when light diquark becomes more strange.
Table 8. Ground-state masses M of bc tetraquarks, lowest threshold T for its decaying into B and D mesons and Δ = M − T. All values are in MeV. |
| System | IJP | M | T | Δ | [31] | [32] | [3] | [33] | [4] | [25] |
|---|---|---|---|---|---|---|---|---|---|---|
| $({bc})(\bar{u}\bar{d})$ | 00+ | 7268 | 7144 | 154 | 7239 | 7248 | 7229 | |||
| 01+ | 7269 | 7190 | 79 | 7246 | 7289 | 7272 | ||||
| 11+ | 7478 | 7190 | 288 | 7403 | 7455 | 7439 | ||||
| 10+ | 7458 | 7114 | 344 | 7383 | 7460 | 7461 | ||||
| 11+ | 7469 | 7190 | 279 | 7396 | 7474 | 7472 | ||||
| 12+ | 7490 | 7332 | 158 | 7422 | 7503 | 7493 | 7586 | 7531 | 7582 | |
| | ||||||||||
| $({bc})(\bar{u}\bar{s})$ | $\tfrac{1}{2}{0}^{+}$ | 7451 | 7232 | 219 | 7444 | 7422 | 7406 | |||
| $\tfrac{1}{2}{1}^{+}$ | 7452 | 7277 | 175 | 7451 | 7456 | 7445 | ||||
| $\tfrac{1}{2}{1}^{+}$ | 7605 | 7277 | 328 | 7555 | 7573 | |||||
| $\tfrac{1}{2}{0}^{+}$ | 7558 | 7232 | 356 | 7540 | 7578 | |||||
| $\tfrac{1}{2}{1}^{+}$ | 7579 | 7277 | 320 | 7552 | 7592 | |||||
| $\tfrac{1}{2}{2}^{+}$ | 7616 | 7420 | 196 | 7572 | 7621 | 7705 | ||||
| | ||||||||||
| $({bc})(\bar{s}\bar{s})$ | 01+ | 7710 | 7336 | 374 | 7673 | 4696 | ||||
| 01+ | 7724 | 7381 | 343 | 7684 | 7691 | |||||
| 10+ | 7717 | 7381 | 336 | 7683 | 7710 | |||||
| 11+ | 7733 | 7525 | 208 | 7701 | 7739 | 7779 | 7908 | 7798 | ||
Regarding our calculations on the spectra of DH tetraquarks, we make the following remarks:
| i | (i) The ground state masses of the DH tetraquarks increase mostly with the quantum number JP of the states listed, as shown in table 6 through table 8. A few exceptions are the IJP = 01+ and 10+ states of the nonstrange tetraquarks $({cc})(\bar{u}\bar{d})$ and $({bb})(\bar{u}\bar{d})$, the IJP = (1/2)0+ and (1/2)1+ states of the strange bc-tetraquarks $({bc})(\bar{u}\bar{s})$ and the IJP = 01+ and 10+ states of the double-strange bc-tetraquarks $({bc})(\bar{s}\bar{s})$. |
| ii | (ii) The state number of the excited DH tetraquarks is far less than that allowed by the four-body quark systems. As such, some P-wave excited states have not been predicted in tables 9 and 10, which are associated with the internal excitations of the light or heavy diquarks. |
| iii | (iii) Generally, a multiquark state may form a mixture of all possible colour components, including the colour singlets and hidden colour configurations. For example, the ${T}_{{cc}}^{+}$ state $| {\bar{3}}_{c},{3}_{c}\rangle $ (with colour structure ${\bar{3}}_{c}\times {3}_{c}$) with IJP = 01+ can in principle mix with the tetraquark $| {6}_{c},{\bar{6}}_{c}\rangle $ (colour structure ${6}_{c}\times {\bar{6}}_{c}$) with IJP = 01+ to form the two states ∣1c, 1c⟩ and ∣8c, 8c⟩: $\begin{eqnarray}| {1}_{c},{1}_{c}\rangle =-\displaystyle \frac{1}{\sqrt{3}}| {\bar{3}}_{c},{3}_{c}\rangle +\sqrt{\displaystyle \frac{2}{3}}| {6}_{c},{\bar{6}}_{c}\rangle ,\end{eqnarray}$ $\begin{eqnarray}| {8}_{c},{8}_{c}\rangle =\sqrt{\displaystyle \frac{2}{3}}| {\bar{3}}_{c},{3}_{c}\rangle +\displaystyle \frac{1}{\sqrt{3}}| {6}_{c},{\bar{6}}_{c}\rangle .\end{eqnarray}$ |
Table 9. P wave masses (MeV) of the DH tetraquarks TQQ with configuration ${QQ}\{\bar{q}\bar{q}\}$. |
| JP | ${cc}\{\bar{u}\bar{d}\}$ | ${bb}\{\bar{u}\bar{d}\}$ | ${bc}\{\bar{u}\bar{d}\}$ | ${cc}\{\bar{u}\bar{s}\}$ | ${bb}\{\bar{u}\bar{s}\}$ | ${bc}\{\bar{u}\bar{s}\}$ | ${cc}\{\bar{s}\bar{s}\}$ | ${bb}\{\bar{s}\bar{s}\}$ | ${bc}\{\bar{s}\bar{s}\}$ |
|---|---|---|---|---|---|---|---|---|---|
| 0− | 4429 | 10941 | 7687 | 4563 | 11110 | 7855 | 4696 | 11201 | 7959 |
| 1− | 4446 | 10946 | 7695 | 4579 | 11116 | 7863 | 4711 | 11206 | 7966 |
| 1− | 4447 | 10946 | 7696 | 4579 | 11116 | 7863 | 4710 | 11206 | 7966 |
| 1− | 4487 | 10959 | 7715 | 4613 | 11127 | 7879 | 4739 | 11215 | 7980 |
| 2− | 4484 | 10958 | 7714 | 4612 | 11126 | 7879 | 4741 | 11216 | 7981 |
| 2− | 4499 | 10963 | 7721 | 4624 | 11130 | 7884 | 4751 | 11219 | 7985 |
| 3− | 4517 | 10969 | 7730 | 4641 | 11136 | 7893 | 4766 | 11224 | 7993 |
Table 10. P wave masses (MeV) of the DH tetraquark with configurations ${QQ}[\bar{u}\bar{d}]$ and ${QQ}[\bar{u}\bar{s}]$. |
| JP | Mass (MeV) | |||||
|---|---|---|---|---|---|---|
| ${cc}[\bar{u}\bar{d}]$ | ${bb}[\bar{u}\bar{d}]$ | ${bc}[\bar{u}\bar{d}]$ | ${cc}[\bar{u}\bar{s}]$ | ${bb}[\bar{u}\bar{s}]$ | ${bc}[\bar{u}\bar{s}]$ | |
| 0− | 4253 | 10764 | 7520 | 4458 | 10983 | 7735 |
| 1− | 4268 | 10769 | 7528 | 4472 | 10988 | 7741 |
| 2− | 4298 | 10779 | 7542 | 4499 | 10996 | 7754 |
Another possibility is that the ${T}_{{cc}}^{+}$ state is a mixture of the two colour singlets (molecule states) DD* and D*D*, and two colour octets ${D}_{8}{D}_{8}^{* }$ and ${D}_{8}^{* }{D}_{8}^{* }$. These possible mixture effects as well as the coupling with the molecule channels of the DD* and D*D* may shift the ground-state mass prediction 3997 MeV of the state $| {\bar{3}}_{c},{3}_{c}\rangle $ (table 6) down to be near the measured mass (about 3874.7 MeV).
4. Summary and remarks
In this work, we utilize linear Regge relation, which is derived from the QCD string model, and mass scaling with respect to the D(Ds) mesons to study low-lying excited mass spectra of the doubly heavy tetraquark in the heavy diquark-antidiquark picture. For the spin-independent masses of the DH tetraquarks, the measured data of the Ξcc baryon and other compatible estimates of ground-state masses of doubly heavy baryons are used to extract the trajectory parameters, including the heavy diquark MQQ (Q = c, b). For the spin-dependent masses, we employ the D/Ds meson masses in relation to mass scaling to determine the hyperfine mass splitting. We find that except for the ground states of doubly bottom tetraquarks Tbb(${bb}\bar{u}\bar{d}$ 1S with IJP = 01+), all DH tetraquarks are not stable against strong decays, provided that the DH tetraquarks are in the pure state of compact exotic hadrons.
The heavy quark pair in diquark QQ in ${\bar{3}}_{c}$ may (when MQ is very large) stay close to each other to form a compact core due to the strong Coulomb interaction, with the light quarks moving around the QQ -core [4], while it is also possible (when MQ is comparable with 1/ΛQCD ≈ 3 GeV−1 ) that Q attracts $\bar{q}$ to bind into a colourless clustering (in 1c) and other pair of ${Q}^{{\prime} }$ and ${\bar{q}}^{{\prime} }$ binds into another colourless clustering (in 1c), giving a molecular system $(Q\bar{q})({Q}^{{\prime} }{\bar{q}}^{{\prime} })$. In the former case, DH tetraquarks mimic the helium-like QCD-atom, while in the latter case, DH tetraquarks resemble the hydrogen-like QCD molecules. In the real world with finite heavy-quark mass, especially in the case of the cc tetraquark state, a DH tetraquark may form exotic states of a compact tetraquark, or form an exotic molecular state consisting of two heavy mesons, or a mixture of both structures [34].
We stress that our picture of an antiquark−heavy-diquark relies on the diquark size ⟨R⟩ compared to the scale 1/ΛQCD, as two heavy quarks Q and ${Q}^{{\prime} }$ in colour antitriplet (${\bar{3}}_{c}$) in DH tetraquark may not act as a compact colour source when ⟨R⟩ ≫ 1/ΛQCD. The explicit size of the diquark depends on two-quark dynamics [35–37] and the mixing of the molecule components in DH tetraquarks is of interest and remains to be explored.
Acknowledgments
D J is supported by the National Natural Science Foundation of China under No. 12165017. Y S thanks Wen-Xuan Zhang for many discussions.
Appendix
For a system composed of a heavy-diquark QQ and a light antidiquark $\bar{q}\bar{q}$, the matrix elements of L · Sqq, L · SQQ, $\widehat{{\bf{B}}}$ and SQQ · Sqq may be evaluated by explicit construction of the tetraquark states with a given J3 as linear combinations of states $\left|{S}_{{QQ}3},{S}_{{qq}3},{L}_{3}\right\rangle $, where ${S}_{{qq}3}+{S}_{{QQ}3}+{L}_{3}={S}_{\bar{D}3}\,+{S}_{D3}+{L}_{3}={J}_{3}$. Due to the rotation invariance of the matrix elements, it suffices to use a single J3 for each and, one can use
$\begin{eqnarray}{\bf{L}}\cdot {{\bf{S}}}_{i}=\displaystyle \frac{1}{2}\left[{L}_{+}{S}_{i-}+{L}_{-}{S}_{i+}\right]+{L}_{3}{S}_{i3},\end{eqnarray}$
to find their elements by applying L · Si on the third components of angular momenta, where $i=D,\bar{D}$. In the $\left|J,{J}_{3}\right\rangle $ representation in the LS coupling, these components are given by the following basis states(when ${S}_{D}={S}_{\bar{D}}=1$ ) $\begin{eqnarray}\begin{array}{l}\left|{}^{3}{P}_{0},J=0\right\rangle =\displaystyle \frac{1}{\sqrt{6}}| 1,-1,0\rangle \\ \quad +\displaystyle \frac{1}{\sqrt{6}}| 0,1,-1\rangle -\displaystyle \frac{1}{\sqrt{6}}| -1,1,0\rangle \\ \quad -\displaystyle \frac{1}{\sqrt{6}}| 1,0,-1\rangle -\displaystyle \frac{1}{\sqrt{6}}| 0,-1,1\rangle +\displaystyle \frac{1}{\sqrt{6}}| -1,0,1\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left|{}^{1}{P}_{1},J=1\right\rangle =\displaystyle \frac{1}{\sqrt{3}}| 1,-1,1\rangle \\ \quad -\displaystyle \frac{1}{\sqrt{3}}| 0,0,1\rangle +\displaystyle \frac{1}{\sqrt{3}}| -1,1,0\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left|{}^{3}{P}_{1},J=1\right\rangle =\displaystyle \frac{1}{2}| 1,-1,1\rangle \\ \quad -\displaystyle \frac{1}{2}| -1,1,1\rangle -\displaystyle \frac{1}{2}| 1,0,0\rangle +\displaystyle \frac{1}{2}| 0,1,0\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left|{}^{5}{P}_{1},J=1\right\rangle =\displaystyle \frac{1}{2\sqrt{15}}| 1,-1,1\rangle \\ \quad +\displaystyle \frac{1}{\sqrt{15}}| 0,0,1\rangle +\displaystyle \frac{1}{2\sqrt{15}}| -1,1,1\rangle \\ \quad -\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{3}{5}}| 1,0,0\rangle -\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{3}{5}}| 0,1,0\rangle \\ \quad +\sqrt{\displaystyle \frac{3}{5}}| 1,1,-1\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\left|{}^{3}{P}_{2},J=2\right\rangle =\displaystyle \frac{1}{\sqrt{2}}| 1,0,-1\rangle +\displaystyle \frac{1}{\sqrt{2}}| 0,1,1\rangle ,\end{eqnarray}$
$\begin{eqnarray}\left|{}^{5}{P}_{2},J=2\right\rangle =\displaystyle \frac{1}{\sqrt{6}}| 1,0,1\rangle +\displaystyle \frac{1}{\sqrt{6}}| 0,1,1\rangle -\sqrt{\displaystyle \frac{2}{3}}| 1,1,0\rangle ,\end{eqnarray}$
$\begin{eqnarray}\left|{}^{5}{P}_{3},J=3\right\rangle =| 1,1,1\rangle ,\end{eqnarray}$
In the state subspace of J = 1, using the basis $\left[{}^{1}{P}_{1},{}^{1}{P}_{1}\right]$, $\left[{}^{1}{P}_{1},{}^{3}{P}_{1}\right]$, $\left[{}^{1}{P}_{1},{}^{5}{P}_{1}\right]$ and $\left[{}^{3}{P}_{1},{}^{3}{P}_{1}\right]$, $\left[{}^{3}{P}_{1},{}^{5}{P}_{1}\right]$, $\left[{}^{5}{P}_{1},{}^{5}{P}_{1}\right]$, one can compute the matrix elements of L · Si, B and ${{\bf{S}}}_{\bar{D}}\cdot {{\bf{S}}}_{D}$,
$\begin{eqnarray}\begin{array}{rcl}{\left\langle {\bf{L}}\cdot {{\bf{S}}}_{\bar{D}}\right\rangle }_{J=1} & = & \left[\begin{array}{ccc}0 & \displaystyle \frac{2}{\sqrt{3}} & 0\\ \displaystyle \frac{2}{\sqrt{3}} & -\displaystyle \frac{1}{2} & \displaystyle \frac{\sqrt{15}}{6}\\ 0 & \displaystyle \frac{\sqrt{15}}{6} & -\displaystyle \frac{3}{2}\end{array}\right],\\ {\left\langle {\bf{L}}\cdot {{\bf{S}}}_{D}\right\rangle }_{J=1} & = & \left[\begin{array}{ccc}0 & -\displaystyle \frac{2}{\sqrt{3}} & 0\\ -\displaystyle \frac{2}{\sqrt{3}} & -\displaystyle \frac{1}{2} & -\displaystyle \frac{\sqrt{15}}{6}\\ 0 & -\displaystyle \frac{\sqrt{15}}{6} & -\displaystyle \frac{3}{2}\end{array}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\left\langle {{\bf{S}}}_{\bar{D}}\cdot {{\bf{S}}}_{D}\right\rangle }_{J=1} & = & \left[\begin{array}{ccc}-2 & 0 & \displaystyle \frac{1}{6\sqrt{5}}\\ 0 & -1 & 0\\ \displaystyle \frac{1}{6\sqrt{5}} & 0 & 1\end{array}\right],\\ \langle {\bf{B}}{\rangle }_{J=1} & = & \left[\begin{array}{ccc}0 & 0 & \displaystyle \frac{32}{15\sqrt{5}}\\ 0 & 1 & 0\\ \displaystyle \frac{32}{15\sqrt{5}} & 0 & -\displaystyle \frac{7}{5}\end{array}\right],\end{array}\end{eqnarray}$
In the subspace of J = 2, one finds $\begin{eqnarray}\begin{array}{rcl}{\left\langle {\bf{L}}\cdot {{\bf{S}}}_{\bar{D}}\right\rangle }_{J=2} & = & \left[\begin{array}{cc}\displaystyle \frac{1}{2} & \displaystyle \frac{\sqrt{3}}{2}\\ \displaystyle \frac{\sqrt{3}}{2} & -\displaystyle \frac{1}{2}\end{array}\right],\\ {\left\langle {\bf{L}}\cdot {{\bf{S}}}_{D}\right\rangle }_{J=2} & = & \left[\begin{array}{cc}\displaystyle \frac{1}{2} & -\displaystyle \frac{\sqrt{3}}{2}\\ -\displaystyle \frac{\sqrt{3}}{2} & -\displaystyle \frac{1}{2}\end{array}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\left\langle {{\bf{S}}}_{\bar{D}}\cdot {{\bf{S}}}_{D}\right\rangle }_{J=2} & = & \left[\begin{array}{cc}-1 & 0\\ 0 & 1\end{array}\right],\\ \langle {\bf{B}}{\rangle }_{J=2} & = & \left[\begin{array}{cc}-\displaystyle \frac{1}{5} & 0\\ 0 & \displaystyle \frac{7}{5}\end{array}\right],\end{array}\end{eqnarray}$
In the subspace of J = 0, the matrix elements are $\begin{eqnarray}\begin{array}{rcl}\langle {\bf{L}}\cdot {{\bf{S}}}_{}{\rangle }_{J=0} & = & -1,\langle {\bf{L}}\cdot {{\bf{S}}}_{\bar{D}}{\rangle }_{J=0}=-1,\\ {\left\langle {{\bf{S}}}_{\bar{D}}\cdot {{\bf{S}}}_{D}\right\rangle }_{J=0} & = & -1,\quad \langle {\bf{B}}{\rangle }_{J=0}=-2,\end{array}\end{eqnarray}$
In the subspace of J = 3, the results are $\begin{eqnarray}\begin{array}{rcl}\langle {\bf{L}}\cdot {{\bf{S}}}_{}{\rangle }_{J=3} & = & 1,\langle {\bf{L}}\cdot {{\bf{S}}}_{\bar{D}}{\rangle }_{J=3}=1,\\ {\left\langle {{\bf{S}}}_{\bar{D}}\cdot {{\bf{S}}}_{D}\right\rangle }_{J=3} & = & 1,\quad \langle {\bf{B}}{\rangle }_{J=3}=-\displaystyle \frac{2}{5},\end{array}\end{eqnarray}$
Given the above matrices, one can solve eigenvalues λ of L · Sqq and the corresponding eigenvectors for a given J, and for that J one can write the hadron states $\left|J,j\right\rangle $ in the jj coupling. Then, these hadron states can be expressed to be the linear combinations of LS bases $\left|{}^{2S+1}{P}_{J}\right\rangle $:
$\begin{eqnarray}\lambda =+1:| J=0,j=0\rangle =\left|1{}^{3}{P}_{0}\right\rangle ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\lambda =-2:| J & = & 1,j=0\rangle =\displaystyle \frac{1}{3}\left|{1}^{1}{P}_{1}\right\rangle \\ & & -\displaystyle \frac{1}{\sqrt{3}}\left|{1}^{3}{P}_{1}\right\rangle +\displaystyle \frac{\sqrt{5}}{3}\left|{1}^{1}{P}_{1}\right\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\lambda & = & -1:| J=1,j=1\rangle =\displaystyle \frac{1}{\sqrt{3}}\left|{1}^{1}{P}_{1}\right\rangle -\displaystyle \frac{1}{2}\left|{1}^{3}{P}_{1}\right\rangle \\ & & -\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{5}{3}}\left|{1}^{1}{P}_{1}\right\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\lambda & = & 1:| J=1,j=2\rangle =\displaystyle \frac{\sqrt{5}}{3}\left|{1}^{1}{P}_{1}\right\rangle \\ & & -\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{5}{3}}\left|{1}^{3}{P}_{1}\right\rangle +\displaystyle \frac{1}{6}\left|{1}^{1}{P}_{1}\right\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\lambda =-1:| J=2,j=1\rangle =\displaystyle \frac{1}{2}\left|{1}^{3}{P}_{2}\right\rangle -\displaystyle \frac{\sqrt{3}}{2}\left|{1}^{5}{P}_{2}\right\rangle ,\end{eqnarray}$
$\begin{eqnarray}\lambda =1:| J=2,j=2\rangle =\displaystyle \frac{\sqrt{3}}{2}\left|{1}^{3}{P}_{2}\right\rangle -\displaystyle \frac{1}{2}\left|{1}^{5}{P}_{2}\right\rangle ,\end{eqnarray}$
$\begin{eqnarray}\lambda =+1:| J=3,j=2\rangle =\left|{1}^{5}{P}_{3}\right\rangle .\end{eqnarray}$
This gives the required baryon states in the heavy-diquark limit, by which the diagonal matrix elements of L · SQQ, $\widehat{{\bf{B}}}$ and SQQ · Sqq can be obtained.In the $\left|J,{J}_{3}\right\rangle $ representation in the LS coupling, these components are given by the following basis states(for SD = 1, ${S}_{\bar{D}}=0$)
$\begin{eqnarray}\begin{array}{l}\left|{}^{3}{P}_{0},J=0\right\rangle =\displaystyle \frac{1}{\sqrt{3}}| 0,1,-1\rangle \\ +\displaystyle \frac{1}{\sqrt{3}}| 0,-1,1\rangle -\displaystyle \frac{1}{\sqrt{3}}| 0,0,0\rangle ,\\ \left|{}^{3}{P}_{0},J=1\right\rangle =\displaystyle \frac{1}{\sqrt{2}}| 0,1,0\rangle -\displaystyle \frac{1}{\sqrt{2}}| 0,0,1\rangle ,\\ \left|{}^{3}{P}_{2},J=2\right\rangle =| 0,1,1\rangle ,\end{array}\end{eqnarray}$
In the following, we write only the nonzero matrix elements for the spin–orbit coupling operators with J = 0, 1: $\begin{eqnarray}\langle {\bf{L}}\cdot {{\bf{S}}}_{\bar{D}}{\rangle }_{J=0}=-2(J=0)\end{eqnarray}$
$\begin{eqnarray}\langle {\bf{L}}\cdot {{\bf{S}}}_{\bar{D}}{\rangle }_{J=1}=-1(J=1)\end{eqnarray}$
$\begin{eqnarray}\langle {\bf{L}}\cdot {{\bf{S}}}_{\bar{D}}{\rangle }_{J=2}=1,(J=1).\end{eqnarray}$
with the results collected in table 4.