Double-heavy tetraquark states with heavy diquark-antiquark symmetry

Recently, the LHCb Collaboration [1] observed the doubly charmed baryon $\Xi_{cc}^{++}$ in the $\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ mass distribution. Its mass was determined to be 3621.40±0.72(stat.)±0.27(syst.)±0.14($\Lambda_{c}^{+}$) MeV/c². This value is 100 MeV higher than the mass of $\Xi_{cc}^{+}$, which was determined in channels $\Lambda_{c}^{+}K^{-}\pi^{+}$ and $pD^+K^-$ by the SELEX Collaboration [2, 3] more than fifteen years ago. The doubly heavy baryons $\Xi_{cc}^{++}$ and $\Xi_{cc}^{+}$ have also been searched for in the FOCUS [4], BABAR [5], and Belle [6] detectors, with negative results. Thus far, the LHCb Collaboration has still not been able to confirm the $\Xi_{cc}^{+}$ baryon [7].

The confirmation of $\Xi_{cc}^{++}$ has important implications as it indicates that two identical charm quarks can exist in a hadronic state. The observation of this baryon has motivated several theoretical discussions regarding the possible double-charm tetraquark $T_{cc}$ and its partner states. We use $T_{cc}$ to specifically denote the lowest $cc\bar{u}\bar{d}$ tetraquark state with $I(J^P) = 0(1^+)$ in this article. Similarly, $T_{bb}$ represents the lowest $bb\bar{u}\bar{d}$ with $I(J^P) = 0(1^+)$. However, $T_{bc}$ represents the lowest $bc\bar{u}\bar{d}$ with $I(J^P) = 0(0^+)$. In the literature, various approaches have been applied to the double-heavy tetraquark structures $QQ\bar{q}\bar{q}$ ($Q = c,b$; $q = u,d,s$), including the chromomagnetic interaction (CMI) model [8-12], quark-level models [13-32], QCD sum rule method [33-40], lattice QCD simulation [41-48], and holographic model [49]. One may consult Ref. [50] for further discussions on such exotic states and related methods.

In Ref. [11], the $QQ\bar{q}\bar{q}$ states were systematically studied using a CMI model in which the color mixing effects between the $(QQ)_{\bar{3}_{c}}(\bar{q}\bar{q})_{3_{c}}$ and $(QQ)_{6_{c}}(\bar{q}\bar{q})_{\bar{6}_{c}}$ structures were considered, and the thresholds of the meson-meson channels were treated as reference scales to estimate the tetraquark masses. According to a series of studies on multiquark states that used the above model [11, 51-59], it appears that the method that uses thresholds usually yields underestimated masses [50]. A possible reason for the underestimation is that the color-electric contribution to the two heavy quarks in the tetraquarks was not explicitly considered [19, 60]. Considering the color-Coulomb interaction, the binding energy of two heavy quarks exhibits a positive correlation with their reduced mass. When the two heavy quarks are separated by a large distance, the $QQ\bar{q}\bar{q}$ state will form a mixed $(Q\bar{q})_{1c}(Q\bar{q})_{1c}$-$(Q\bar{q})_{8c}(Q\bar{q})_{8c}$ meson-meson type structure. In contrast, if the two heavy quarks move in a small spatial region because of the attraction, they may form a $\bar{3}_{c}$ substructure, and the tetraquark can be treated as a diquark-antidiquark state. In this case, it is not necessary for the distance between the two light antiquarks to be small owing to the considerable relativistic effect and small color-Coulomb potential. In this study, we aimed to perform a further investigation of double-heavy tetraquark systems, particularly the nonstrange double-charm system.

The heavy quark flavor-spin symmetry appears in limit $m_Q\to\infty$, and it is used extensively to study the properties of heavy quark hadrons. For states containing two heavy quarks, the heavy diquark-antiquark symmetry (HDAS) can also be considered [61-67]. According to this symmetry, the mass splittings between $QQq$ baryons and those between $\bar{Q}'q$ mesons can be related with the correspondence $QQ\leftrightarrow \bar{Q}'$. This consideration is based on the observations that i) the size of $QQ$ is small in the heavy quark limit, ii) the color representations of $QQ$ and $\bar{Q}'$ are both $\bar{3}_c$, and iii) the interaction between the light and heavy components is suppressed, even though $QQ$ and $\bar{Q}'$ have different spins. Similarly, one can relate the double-heavy tetraquarks $QQ\bar{q}\bar{q}$, with the color structure $(QQ)_{3_c}$, to the singly heavy antibaryons $\bar{Q}'\bar{q}\bar{q}$. According to the HDAS, for example, we can estimate the mass of a double-charm tetraquark with the relation $T_{cc\bar{n}\bar{n}}-\Xi_{cc}^{*} = \Sigma_{b}^{*}-\bar{B}^{*}$, where $n = u,d$, and the hadron symbol represents its mass. Obviously, the required unknown input is only the mass of $\Xi_{cc}^{*}$, which can be estimated with the experimental mass of $\Xi_{cc}^{++}$ in the CMI model. Although the quantum numbers of LHCb $\Xi_{cc}^{++}$ have not been measured, its mass is very close to the theoretical value of the ground state predicted by Karliner and Rosner in Ref. [68]. In the following calculations, we will use $m_{\Xi_{cc}}$ = 3621 MeV as the input to estimate the masses of the double-charm tetraquark states. Other double-heavy tetraquarks will also be systematically investigated. If the LHCb $\Xi_{cc}$ is actually the $\Xi_{cc}^*$ state with spin = 3/2, $m_{\Xi_{cc}^*}-m_{\Xi_{cc}}\approx 70$ MeV should be subtracted from the obtained masses of the relevant tetraquarks.

Unlike the conventional hadrons, the $QQ$ diquark in tetraquarks may also be in the color $6_c$ representation. In Ref. [9], the mass of $I(J^P) = 0(1^+)$ $T_{cc}$ with $6_c$ $cc$ was estimated. To include the mixing effects between the $(cc)_{\bar{3}_c}(\bar{n}\bar{n})_{3_c}$ and $(cc)_{6_c}(\bar{n}\bar{n})_{\bar{6}_c}$ configurations and estimate the masses of all the $(cc\bar{n}\bar{n})$ states, in the current study, we first need to identify the position of the $(cc)_{\bar{3}_c}(\bar{n}\bar{n})_{3_c}$ state determined with HDAS. Observing that the $J = 2$ tetraquark is the pure $(cc)_{\bar{3}_c}(\bar{n}\bar{n})_{3_c}$ state because of the constraint from the Pauli principle, we directly relate its mass to that of $\Xi_{cc}^*$. Thereafter, we determine the masses of the lower tetraquark states from the mass splittings within the CMI model. Other double-heavy tetraquark states will be studied similarly. This concept is contrary to the estimation strategy adopted in our recent works [11, 51-59, 69], wherein the multiquark masses were determined from lower mass scales.

The remainder of this paper is organized as follows. In Sec. II, we present the method and formalism of the study. Thereafter, we provide our analysis and numerical results for the $cc\bar{u}\bar{d}$ states in Sec. III and the predictions on their partners in Sec. IV. In Sec. V, we discuss the constraints on the masses of the involved heavy quark hadrons. Finally, Sec. VI presents our discussions and a summary.

For ground state hadrons, the mass splittings of different spin states with the same quark contents are mainly determined by the color-spin (color-magnetic) interaction in the quark model [70]:

Here, $i(j)$ represents the ith (jth) quark component of the tetraquark state, $\vec{\lambda}_{i}$ ($\vec{\lambda}_{j}$) is the vector containing the eight Gell-Mann matrices for the ith (jth) quark component, and $\vec{\sigma}_i$ ($\vec{\sigma}_j$) is the vector containing the three Pauli matrices for the ith (jth) quark component. It should be noted that $\vec{\lambda}_{i}$ ($\vec{\lambda}_{j}$) should be replaced with $-\vec{\lambda}_{i}^{*}$ ($-\vec{\lambda}_{j}^{*}$) if the quark component is an antiquark. The effective coupling parameters, $C_{ij}$, which actually depend on the systems, include effects from the spatial wave function and the constituent quark masses. Thus, the mass formula in the CMI model is

where the effective mass of the ith quark, $m_{i}$, includes the constituent quark mass and contributions from other terms, such as the color-Coulomb interaction and color confinement. In the following calculations, we will adopt the values of the parameters presented in Table 1, which are determined from the masses of conventional hadrons.

$C_{nn}=18.3$	$C_{ns}=12.0$	$C_{nc}=4.0$	$C_{nb}=1.3$	$C_{ss}=5.7$	$C_{sc}=4.4$	$C_{sb}=0.9$	$C_{cc}=3.2$	$C_{bb}=1.8$	$C_{cb}=2.0$
$C_{n\bar{n}}=29.9$	$C_{n\bar{s}}=18.7$	$C_{n\bar{c}}=6.6$	$C_{n\bar{b}}=2.1$	$C_{s\bar{s}}=9.3$	$C_{s\bar{c}}=6.7$	$C_{s\bar{b}}=2.3$	$C_{c\bar{c}}=5.3$	$C_{b\bar{b}}=2.9$	$C_{c\bar{b}}=3.3$

Table 1.  Coupling parameters (unit: MeV) extracted from conventional hadrons. The value of $C_{c\bar{b}}$ is estimated using the mass splitting in the Godfrey-Isgur model [71]. Approximations $C_{cc} = k C_{c\bar{c}}$, $C_{bb} = k C_{b\bar{b}}$, $C_{cb} = k C_{c\bar{b}}$, and $C_{s\bar{s}} = C_{ss}/k$ are adopted [72, 73], where $k\equiv C_{nn}/C_{n\bar{n}}\approx2/3$. The effective quark masses determined from the masses of the ground hadrons [74] are $m_n = 361.8$ MeV, $m_s = 542.4$ MeV, $m_c = 1724.1$ MeV, and $m_b = 5054.4$ MeV.

The CMI model can provide relatively reasonable predictions for the mass splittings for various hadronic systems, but it is not good enough to estimate hadron masses because the effective quark masses have large uncertainties. Ref. [11] presented two methods for estimating the double-heavy tetraquark masses: one employs the mass formula (2), and the other uses the modified formula

The first method, which uses the parameters in Table 1, provides theoretical upper limits for the masses. The differences between these upper limits and the "realistic" masses would be very large for heavy quark multiquark states. This can be observed, for example, in the results for conventional hadrons and the $cs\bar{c}\bar{s}$ tetraquark states [51]. A possible means of remedying the deviations is to include a color-electric term appropriately in the Hamiltonian of the CMI model [68, 75]. The second method yields more reasonable results than the first one, but it suffers from the problem of selecting the reference scale. For example, the threshold of the $J/\psi\phi$ channel leads to a lower mass $X(4140)$ than that of $D_s^+D_s^-$ does [51]. If the state has a mixed structure of $(c\bar{s})_{8_c}(\bar{c}s)_{8_c}$ and $(c\bar{s})_{1_c}(\bar{c}s)_{1_c}$, where the separation between $c$ and $\bar{s}$ is small and the distance between $c\bar{s}$ and $\bar{c}s$ is large, the choice to use $D_s^{(*)+}D_s^{(*)-}$ as a reference system to estimate the tetraquark masses is more natural, even though the resulting tetraquark mass is probably still lower than the measured one. In the $cc\bar{n}\bar{n}$ case, the threshold that may be used is only for the $D^{(*)}D^{(*)}$ channel. However, when $cc$ can be considered a diquark with a small spatial separation, using such a threshold as a reference scale appears to not be a good choice. The $T_{cc}$ mass (approximately 100 MeV below the $DD^*$ threshold) will probably also be underestimated.

In the second method, better choices for estimating the tetraquark masses than hadron-hadron thresholds should exist. While estimating the tetraquark masses of $Qq\bar{Q}\bar{q}$ and $q_1q_2\bar{q}_3\bar{q}_4$ in Refs. [56, 69], we attempted to relate the reference scales to the $X(4140)$ mass. The obtained masses were higher than those with the meson-meson thresholds. In the current study, we examine the results with the aid of the heavy diquark-antiquark symmetry. At present, it is not clear which choice yields more realistic results. Hopefully, future measurements regarding the predicted tetraquark states can provide an answer to this. In the following discussions, to compare the results, we will refer to the methods corresponding to these three reference choices as the threshold, $X(4140)$, and HDAS approaches.

A diquark is generally assumed to be a color-$\bar{3}$ correlated quark-quark subsystem of a bound or resonant state. It shares certain similar properties with an antiquark, and diquark-antiquark symmetry (DAS) may exist, even though the diquark and antiquark have different masses and spins, and their dynamics are not necessarily the same. This symmetry works for hadrons that contain a heavy quark with a mass that is much larger than $\Lambda_{\rm QCD}$. In this case, the heavy quark can be treated as a static color source, and many properties are independent of the quark mass. For example, if we treat the $\bar{n}\bar{n}$ antidiquark as a heavier light quark $n^\prime$, a $cc\bar{n}\bar{n}$ state will become a $ccn^\prime$ state, which resembles a baryon structure. Subsequently, the mass difference between the $ccn^\prime$ tetraquark and $ccn$ baryon is independent of the heavy quark mass in the heavy quark limit. This similarity based on DAS provides a method for estimating the masses of unknown states from those of known baryons. Likewise, as discussed by Savage and Wise in [64], if the $cc$ heavy diquark is treated as an antiquark $\bar{Q}^\prime$, the masses of various $\bar{Q}^\prime\bar{n}\bar{n}$ tetraquarks can be related to those of $Qnn$ baryons. As DAS is an approximate symmetry, whether or not the predictions based on it are correct needs to be tested experimentally. Before estimating the tetraquark masses, we provide further explanations of diquark-antiquark symmetry.

It is known that there are three types of light quarks ($u$, $d$, and $s$), which form the base representation of flavor $SU(3)$. As quarks have spin, the flavor-spin $SU(6)$ can conventionally be used to classify various quark states. If a light diquark is a stable object, it has been argued that a symmetry exists between the baryons and mesons (diquarks and antiquarks) [76-78]. Considering the color-$\bar{3}$ diquark and antiquark together, we obtain a flavor-spin 27-plet. Its $SU(3)\otimes SU(2)$ decomposition reads $27 = (6_{f},3_{s})+ (\bar{3}_f,1_{s})+(\bar{3}_{f},2_{s})$. The substructure $(6_{f},3_{s})$ represents an 18-plet with flavor-symmetric and spin-symmetric diquarks. Similarly, $(\bar{3}_{f},1_{s})$ represents a triplet with flavor-antisymmetric and spin-antisymmetric diquarks. The last substructure $(\bar{3}_{f},2_{s})$ represents the antiquark sextet. The decomposition indicates that the diquark and antiquark can be combined into the $SU(6/21)$ symmetry algebra. We present the group structures in Fig. 1.

Figure 1.  27-multiplet in $SU(6)$ group with diquark-antiquark symmetry. The first number in the parentheses denotes the flavor $SU(3)$ representation and the second the spin $SU(2)$ representation.

Lattice QCD simulations have indicated such a diquark-antiquark symmetry [79, 80]; the static quark-diquark potential is almost equal to the static quark-antiquark potential, and a quark-antiquark pair and a quark-diquark pair have similar wave functions. However, according to Ref. [61], the diquark-antiquark symmetry in a light quark sector is broken for at least three reasons: (1) a diquark and an antiquark have different masses, leading to kinematical differences; (2) the diquark and antiquark have different spin-dependent and velocity-dependent terms; and (3) the diquark is not a point particle, and its finite size must affect its interactions. Thus, the breaking effects for the diquark-antiquark symmetry between the light diquarks and the light antiquarks are significant.

In contrast, for hadrons containing one heavy quark, the aforementioned symmetry breaking effects will be largely suppressed [61]. According to the heavy quark effective theory (HQET) [81], the kinematic and spin-dependent terms are inversely proportional to the heavy quark mass and make small contributions to the hadron mass. The size of the diquark is not a major concern, because the constituent quark model is still successful in handling the properties of conventional hadrons, even though the constituent quark has a comparable size to the diquark [61]. Thus, the size of the diquark has no significant impact on the DAS. Therefore, in general, there is possibly a better diquark-antiquark symmetry for hadrons containing one heavy quark. In Ref. [82], Lichtenberg, Roncaglia, and Predazzi analyzed the relations for the masses of heavy quark hadrons using the Feynman-Hellmann theorem and semiempirical formulas. They obtained several mass sum rules for heavy quark hadrons. We have listed a selection of these as follows:

which are adopted in the following discussions. These also follow from the heavy quark flavor symmetry. These relations can be confirmed by observing that the values of $(l.h.s.-r.h.s)$ are $12.9$, $-0.6$, $-9.5$, and $-21.5$ MeV (${\Omega_b^*}-{\Omega_b} = 14.4$ MeV is used in the CMI model), respectively. The common feature of these four relations is that only the highest spins are involved. In fact, the final three relations also satisfy the diquark-antiquark symmetry for light quarks in which the diquark spin is 1. It is true that a better DAS exists for hadrons containing one heavy quark than for those without heavy quarks. Although more relations can be found in Ref. [82], it is not necessary to consider them in this work.

Our strategy for estimating the masses of double-heavy tetraquarks is to combine HDAS and the aforementioned four mass sum rules. To illustrate the concept, we temporarily focus only on Eq. (4). If we consider the heavy diquark-antiquark symmetry for the $cc$ diquark, we obtain

which can be used to estimate the mass of $\Omega_{cc}^{*}$using that of $\Xi_{cc}^{*}$. As a better heavy quark symmetry exists in bottom systems than in charmed systems, we use the masses of the bottom mesons. Here and in the following discussions, we only consider the highest spin hadrons while adopting the diquark-antiquark symmetry, because then, the possible contributions from other color or spin structures will be avoided. As explained in Sec. I, we assume that the spin of the LHCb $\Xi_{cc}$ is 1/2. We can evaluate the mass of $\Xi_{cc}^*$ with the CMI model $\Xi_{cc}^{*} = \Xi_{cc}+16C_{cn} = 3685$ MeV. Thereafter, the mass of $\Omega_{cc}^*$, namely $3776$ MeV, is obtained. By using the CMI model again, we can further obtain ${\Omega_{cc}} = 3706$ MeV.

At present, doubly heavy baryons other than $\Xi_{cc}$ have not been observed, and the accuracy of Eq. (8) cannot be verified. However, in theory, the reasonability of treating a $QQ$ diquark as a heavy antiquark $\bar{Q}^{\prime}$ can be argued. Because the heavy diquark has lower kinematic energy and less spin-dependent interaction, the heavy quark approximation works better than the single $Q$ case. Furthermore, the heavy diquark (with a light quark spectator) actually has better symmetry properties than the light diquark (with a heavy quark spectator) because a heavy diquark has small spatial separation, which means that its interaction with light quarks is not significantly affected by its size. In the following parts of this paper, we first focus on the double-charm tetraquark $cc\bar{n}\bar{n}$ states by treating them as systems that are composed of a small-size double-charm diquark and a light antidiquark. According to HDAS, the study of $cc\bar{n}\bar{n}$ states becomes that of heavy $Q^\prime nn$ "baryons." This approximate symmetry, together with the above mass sum rules, is evidently convenient for us to relate $cc\bar{n}\bar{n}$ to $\Xi_{cc}$. Following those on the $cc\bar{n}\bar{n}$ states, we perform similar studies on other double-heavy tetraquarks. Whether or not the adopted approximation is effective should be tested in future experiments.

Based on the symmetry consideration, we obtain a good flavor-spin supermultiplet [62] that contains both tetraquark mesons $cc\bar{q}\bar{q}$ and three-quark baryons $ccq$. This double-charm supermultiplet is classified into three types of states:

The members in Eq. (10) contain a light antidiquark with spin 1, and those in (11) have a light antidiquark with spin 0. We treat the $cc$ diquark as a heavy $\bar{3}_c$ "antiquark" $\bar{Q}^\prime$. Following the replacement, $cc\rightarrow \bar{Q}^\prime$,$\Xi_{cc}^{*}$ with spin 3/2 and the $cc\bar{n}\bar{n}$ with spin 2 are transformed into $\bar{M}_{Q^\prime}^{*} = (\bar{Q}^\prime n)^{J = 3/2}$ and $\bar{\Sigma}_{Q^\prime}^{*} = (\bar{Q}^\prime\bar{n}\bar{n})^{J = 2}$, respectively, i.e.,

Note that only the highest spins are involved, and the $cc$ diquark has unique quantum numbers $color = \bar{3}_c, \;spin = 1$. The forms on the right hand side remind us of the relation in Eq. (5), where the light diquark-antiquark symmetry is used. With that equation, one can naturally obtain

The obtained mass of $cc\bar{n}\bar{n}$ with $I = 1,J = 2$ is, thus, 4195 MeV (with $\Sigma_c^*$ and $D^*$) or 4194 MeV (with $\Sigma_b^*$ and $\bar{B}^{*}$). We select the latter value owing to the better heavy quark symmetry for the bottom hadrons. Subsequently, we have a good reference hadron $T_{cc\bar{n}\bar{n}}^{I = 1, J = 2}$ and can estimate the masses of the other $cc\bar{n}\bar{n}$ states by considering the CMI differences. When only $\bar{3}_c$ $cc$ is considered, the masses of the three other double-charm tetraquarks are $T_{cc\bar{n}\bar{n}}^{I = 1,J = 0} = 4087$ MeV, $T_{cc\bar{n}\bar{n}}^{I = 1,J = 1} = 4122$ MeV, and $T_{cc\bar{n}\bar{n}}^{I = 0,J = 1} = 3961$ MeV. It is obvious that these tetraquarks are all above the $DD^*$ threshold (3876 MeV).

Although the color structure of $\Xi_{cc}$ is unique, that of the exotic $cc\bar{n}\bar{n}$ states is not. The mixing or channel coupling effects from the $(cc)_{6_c}(\bar{n}\bar{n})_{\bar{6}_c}$ color structure may significantly change the tetraquark masses. When such contributions are considered, two more tetraquarks appear. We collect all the color-spin bases [11] for the $cc\bar{n}\bar{n}$ states, which are displayed in Table 2. With these wave functions, we finally obtain the numerical results listed in the fourth column of Table 3 and illustrated in Fig. 2(a). Moreover, we provide the masses estimated using Eq. (3) ($D^{(*)}D^{(*)}$ as the reference state) and those using Eq. (2) in the table. These can be viewed as the theoretical lower (fifth column) and upper (sixth column) limits, respectively, in the present framework. Comparing these three results from different considerations, it is found that the new masses fall within the range constrained by the lower and upper limits, and the values are slightly larger than the averages of the two limits. The masses of the lowest $1(0^{+})$ and $0(1^{+})$ states in Table 3 are evidently smaller than the above values without the $(cc)_{6_c}(\bar{n}\bar{n})_{\bar{6}_c}$ contributions. However, the $T_{cc}$ state, the mass of which in the HDAS approach is 155 MeV higher than the lower limit (3774 MeV), is still above the $DD^{*}$ threshold. The other $cc\bar{n}\bar{n}$ states are also above the respective fall-apart thresholds. Therefore, the HDAS approach results in an unstable $T_{cc}$, which is consistent with the conclusions obtained in Refs. [19, 20, 30, 31, 33, 39, 44, 45, 63, 83, 84].

States	$I(J^P)$		Bases
$(cc\bar{n}\bar{n})$	$1(2^{+})$	$[(cc)_{\bar{3} }^{1}(\bar{n}\bar{n})_{3}^{1}]^{2}$
	$1(1^{+})$	$[(cc)_{\bar{3}}^{1}(\bar{n}\bar{n})_{3}^{1}]^{1}$
	$1(0^{+})$	$[(cc)_{\bar{3}}^{1}(\bar{n}\bar{n})_{3}^{1}]^{0}$	$[(cc)_{6}^{0}(\bar{n}\bar{n})_{\bar{6}}^{0}]^{0}$
	$0(1^{+})$	$[(cc)_{\bar{3}}^{1}(\bar{n}\bar{n})_{3}^{0}]^{1}$	$[(cc)_{6}^{0}(\bar{n}\bar{n})_{\bar{6}}^{1}]^{1}$
$(cc\bar{n}\bar{s})$	$(2^{+})$	$[(cc)_{\bar{3}}^{1}(\bar{n}\bar{s})_{3}^{1}]^{2}$
	$(1^{+})$	$[(cc)_{\bar{3}}^{1}(\bar{n}\bar{s})_{3}^{1}]^{1}$	$[(cc)_{\bar{3}}^{1}(\bar{n}\bar{s})_{3}^{0}]^{1}$	$[(cc)_{6}^{0}(\bar{n}\bar{s})_{\bar{6}}^{1}]^{1}$
	$(0^{+})$	$[(cc)_{\bar{3}}^{1}(\bar{n}\bar{s})_{3}^{1}]^{0}$	$[(cc)_{6}^{0}(\bar{n}\bar{s})_{\bar{6}}^{0}]^{0}$
$(cc\bar{s}\bar{s})$	$(2^{+})$	$[(cc)_{\bar{3}}^{1}(\bar{s}\bar{s})_{3}^{1}]^{2}$
	$(1^{+})$	$[(cc)_{\bar{3}}^{1}(\bar{s}\bar{s})_{3}^{1}]^{1}$
	$(0^{+})$	$[(cc)_{\bar{3}}^{1}(\bar{s}\bar{s})_{3}^{1}]^{0}$	$[(cc)_{6}^{0}(\bar{s}\bar{s})_{\bar{6}}^{0}]^{0}$

Table 2.  Color-spin bases for $cc\bar{q}\bar{q}$ ($q = u,d,s$) states [11]. The superscripts indicate the spin, and the subscripts indicate the color representations.

$(cc\bar{n}\bar{n})$ $I(J^P)$	$\langle H_{CM}\rangle$	Eigenvalues	Mass (our) HDAS	Mass (low.) $D^{(*)}D^{(*)}$	Mass (up.) Eq. (2)
$1(2^{+})$	$\left(92.7\right)$	$\left(92.7\right)$	$\left(4194.4\right)$	$\left(4038.8\right)$	$\left(4264.5\right)$
$1(1^{+})$	$\left(22.3\right)$	$\left(22.3\right)$	$\left(4124.0\right)$	$\left(3968.4\right)$	$\left(4194.1\right)$
$1(0^{+})$	$\left(\begin{array}{cc}-12.9&129.3\\129.3&86.2\end{array}\right)$	$\left(\begin{array}{c}-101.9\\175.1\end{array}\right)$	$\left(\begin{array}{c}3999.8\\4276.8\end{array}\right)$	$\left(\begin{array}{c}3844.3\\4121.3\end{array}\right)$	$\left(\begin{array}{c}4069.9\\4346.9\end{array}\right)$
$0(1^{+})$	$\left(\begin{array}{cc}-137.7&-74.7\\-74.7&-11.4\end{array}\right)$	$\left(\begin{array}{c}-172.4\\23.2\end{array}\right)$	$\left(\begin{array}{c}3929.3\\4124.9\end{array}\right)$	$\left(\begin{array}{c}3773.8\\3969.4\end{array}\right)$	$\left(\begin{array}{c}3999.4\\4195.0\end{array}\right)$

Table 3.  Results for the $cc\bar{n}\bar{n}$ ($n = u,d$) states (unit: MeV). The second and third columns present the numerical values of the CMI matrices and their eigenvalues, respectively. The fifth and sixth columns list the masses estimated using Eq. (3) ($D^{(*)}D^{(*)}$ as the reference state) and Eq. (2) (parameters presented in Table 1), respectively. These can be viewed as the theoretical lower limits (low.) and upper limits (up.), respectively, for the tetraquark masses in the current framework. The fourth column displays our predictions with the heavy diquark-antiquark symmetry (HDAS) consideration.

Figure 2.  Relative positions of the double-charm tetraquark states (solid and dashed lines) and relevant meson-meson thresholds (dotted lines). The masses are given in MeV. In (a), the solid (dashed) lines denote the $I = 1$ ($I = 0$) states, and the almost degenerate masses of 4124 and 4125 MeV correspond to the isovector and isoscalar states, respectively.

Figure 3.  Relative positions for double-bottom tetraquark states (solid and dashed lines) and relevant meson-meson thresholds (dotted lines). The masses are given in MeV. In (a), the solid (dashed) lines denote the $I = 1$ ($I = 0$) states.

We have obtained the $cc\bar{n}\bar{n}$ spectrum with the mass of the LHCb $\Xi_{cc}$ state by considering the diquark-antiquark symmetry in the CMI model. It is natural to extend the study to other double-heavy tetraquark states, including $cc\bar{n}\bar{s}$, $cc\bar{s}\bar{s}$, $bb\bar{n}\bar{n}$, $bb\bar{n}\bar{s}$, $bb\bar{s}\bar{s}$, $bc\bar{n}\bar{n}$, $bc\bar{n}\bar{s}$, and $bc\bar{s}\bar{s}$. In Ref. [69], we estimated the masses of the $bc\bar{n}\bar{n}$ and $bc\bar{n}\bar{s}$ states by using $X(4140)$ as a reference system. It will be instructive to compare the results when using different reference states.

By following a similar procedure to that in Eq. (14), we can easily obtain two relations from Eqs. (6) and (7), respectively:

As the heavy quark symmetry breaking effects are larger for charmed systems than for bottom systems, we use the masses of $\Xi_b^*$ and $\bar{B}^{*}$ in the former relation. To obtain the latter relation, we employ Eq. (4) so that no $u$ or $d$ quark is involved. Actually, if the heavy meson is sufficiently heavy, the difference between the $u$, $d$, and $s$ cases can be neglected. Because the mass of $\Omega_b^*$ has not yet been measured yet, we opt to use the masses of $\Omega_c^*$ and $D_s^*$. With $\Xi_{cc}^* = 3685$ MeV and $\Omega_{cc}^* = 3776$ MeV, estimated using Eq. (8), we obtain $T_{cc\bar{n}\bar{s}}^{J = 2} = 4313$ MeV and $T_{cc\bar{s}\bar{s}}^{J = 2} = 4430$ MeV. These will be treated as reference scales to determine the masses of the other double-charm strange tetraquarks.

Before proceeding further, we investigate the masses of $T_{cc\bar{n}\bar{n}}^{J = 2}$, $T_{cc\bar{n}\bar{s}}^{J = 2}$, and $T_{cc\bar{s}\bar{s}}^{J = 2}$ in different approaches. The current estimation yields 177, 193, and 205 MeV for the mass distances measured from the $D^*D^*$, $D^*D_s^*$, and $D_s^*D_s^*$ thresholds, respectively, which are gradually increasing numbers. Those in Ref. [11] are gradually decreasing numbers, namely 23, 8, and -6 MeV, respectively. Therefore, no stable $J = 2$ tetraquarks are obtained in this study, whereas the states in Ref. [11] are around their fall-apart thresholds. This feature is an apparent difference between the HDAS and threshold approaches.

With the above reference states, $T_{cc\bar{n}\bar{s}}^{J = 2}$ and $T_{cc\bar{s}\bar{s}}^{J = 2}$, and the mass splittings in the CMI model, we can estimate the masses of the strange partners of the $cc\bar{n}\bar{n}$ states. The base structures for the calculation are presented in Table 2. We list the numerical results for all of the $(cc\bar{n}\bar{s})$ and $(cc\bar{s}\bar{s})$ states in Table 4, where we also display the theoretical lower and upper limits for the tetraquark masses. Comparing the values in the fourth, fifth, and six columns, it is obvious that our results with diquark-antiquark symmetry are slightly larger than the averages of the two limits, which is the same feature as in the $cc\bar{n}\bar{n}$ case. The relative positions for the $(cc\bar{n}\bar{s})$ and $(cc\bar{s}\bar{s})$ tetraquark states are illustrated in Fig. 2(b) and Fig. 2(c), respectively. According to the figure, similar to the $cc\bar{n}\bar{n}$case, all the obtained double-charm strange tetraquarks can decay through rearrangement mechanisms, and no such stable states exist. This observation is different from that in Ref. [11], where the lowest $1^+$ $cc\bar{n}\bar{s}$ is stable.

$(cc\bar{n}\bar{s})$ $J^{P}$	$\langle H_{CM}\rangle$	Eigenvalues	Mass (our) HDAS	Mass (low.) $D^{(*)}D_{s}^{(*)}$	Mass (up.) Eq. (2)
$2^{+}$	$\left(76.1\right)$	$\left(76.1\right)$	$\left(4313.3\right)$	$\left(4125.4\right)$	$\left(4428.5\right)$
$1^{+}$	$\left(\begin{array}{ccc}5.2&0.0&0.0\\0.0&-87.3&-75.2\\0.0&-75.2&-3.0\end{array}\right)$	$\left(\begin{array}{c}-131.4\\5.2\\41.1\end{array}\right)$	$\left(\begin{array}{c}4105.8\\4242.4\\4278.3\end{array}\right)$	$\left(\begin{array}{c}3917.9\\4054.5\\4090.4\end{array}\right)$	$\left(\begin{array}{c}4221.0\\4357.6\\4393.5\end{array}\right)$
$0^{+}$	$\left(\begin{array}{cc}-30.3&130.3\\130.3&61.0\end{array}\right)$	$\left(\begin{array}{c}-122.7\\153.4\end{array}\right)$	$\left(\begin{array}{c}4114.5\\4390.6\end{array}\right)$	$\left(\begin{array}{c}3926.6\\4202.7\end{array}\right)$	$\left(\begin{array}{c}4229.7\\4505.8\end{array}\right)$
($cc\bar{s}\bar{s}$) $J^{P}$	$\langle H_{CM}\rangle$	Eigenvalues	Mass(our) HDAS	Mass (low.) $D_{s}^{(*)}D_{s}^{(*)}$	Mass (up.) Eq. (2)
$2^{+}$	$\left(59.6\right)$	$\left(59.6\right)$	$\left(4429.9\right)$	$\left(4211.6\right)$	$\left(4592.6\right)$
$1^{+}$	$\left(-11.9\right)$	$\left(-11.9\right)$	$\left(4358.4\right)$	$\left(4140.1\right)$	$\left(4521.1\right)$
$0^{+}$	$\left(\begin{array}{cc}-47.6&131.3\\131.3&35.8\end{array}\right)$	$\left(\begin{array}{c}-143.7\\131.8\end{array}\right)$	$\left(\begin{array}{c}4226.6\\4502.1\end{array}\right)$	$\left(\begin{array}{c}4008.3\\4283.8\end{array}\right)$	$\left(\begin{array}{c}4389.3\\4664.8\end{array}\right)$

Table 4.  Results for $cc\bar{n}\bar{s}$ and $cc\bar{s}\bar{s}$ states (unit: MeV). The second and third columns provide the numerical values of the CMI matrices and their eigenvalues, respectively. The fifth and sixth columns list the masses estimated using Eq. (3) (with$D^{(*)}D_{s}^{(*)}/D_{s}^{(*)}D_{s}^{(*)}$ as the reference state) and Eq. (2) (parameters provided in Table 1), respectively. These can be viewed as the theoretical lower limits (low.) and upper limits (up.), respectively, for the tetraquark masses in the current framework. The fourth column displays our predictions with the heavy diquark-antiquark symmetry (HDAS) consideration.

According to the diquark-antiquark symmetry, no stable double-charm tetraquark states exist. In the bottom case, the attractive color-Coulomb interaction between the two heavy quarks may be sufficiently strong to aid in the formation of stable tetraquarks. Next, we investigate the $bb\bar{q}\bar{q}$ and $bc\bar{q}\bar{q}$ systems, where $q = u$, $d$, or $s$. First, we focus on the double-bottom tetraquarks, which have exactly the same group structure as the double-charm states. For the HDAS relations and wave function bases, we simply need to perform a simple substitution of $(bb)$ for $(cc)$ in Eqs. (14)-(16) and Table 2. However, considerable difficulty arises in applying the formulas, as the masses of $\Xi_{bb}^*$ and $\Omega_{bb}^*$ have not been measured. In this study, we need to select appropriate predictions for their values from various investigations.

In the literature, numerous analyses on the masses of $\Xi_{bb}$ and $\Xi_{bb}^*$ have been performed (see Table 1 of Ref. [85] for a collection). We list several of the results in Table 5, where the involved approaches include lattice QCD [86], chromomagnetic models [60, 68, 87], the relativistic quark model [88], the nonrelativistic quark model [89-91], the bag model [92], and the Bethe-Salpeter equation [93]. To select an appropriate value for the mass of $\Xi_{bb}$, we adopt the following criteria: 1) the baryon masses satisfy the light-flavor symmetry $\Omega_{bb}^{*}-\Omega_{bb} = \Xi_{bb}^{*}-\Xi_{bb}$ in the heavy quark limit; 2) the HDAS relation $\Omega_{bb}^{*}-\Xi_{bb}^{*} = \bar{B}_{s}^*-\bar{B}^*\approx 91$ MeV holds; and 3) the inequality $\Xi_{bb}- (\Xi_{bb})_{\rm CMI}- 2(\bar{B}-\bar{B}_{\rm CMI}) < \Xi_{cc}- (\Xi_{cc})_{\rm CMI}-2(D- D_{\rm CMI})$ for the mass of $\Xi_{bb}$ is required. The final criterion leading to $\Xi_{bb}<10327$ MeV means that the color-Coulomb contribution to the bottom diquark is larger than that of the charm case once the contributions from the effective quark masses and color-magnetic interactions have been subtracted. It can be confirmed that a similar inequality holds for heavy quarkona, $\eta_{c}-(\eta_{c})_{\rm CMI} - 2(D- D_{\rm CMI}) > \eta_{b}- (\eta_{b})_{\rm CMI}-2(\bar{B}-\bar{B}_{\rm CMI})$ (numerically, −882 MeV > −1181 MeV). The lattice results in Ref. [86] meet the first and third criteria but not the second because $\Omega_{bb}^{*}-\Xi_{bb}^{*}\approx 130\;{\rm{MeV}}>91$ MeV. In the chromomagnetic models [60, 68, 87], the results are compatible with all the criteria. In this case, we use the ground baryon mass $\Xi_{bb} = 10169$ MeV from Ref. [60], whereas the mass of $\Xi_{bb}^*$ is evaluated to be $\Xi_{bb}^* = \Xi_{bb}+16C_{bn} = 10190$ MeV. The mass of $\Omega_{bb}^*$ is, subsequently, further determined to be $\Omega_{bb}^{*} = \Xi_{bb}^*+\bar{B}_s^*-\bar{B}^* = 10280$ MeV and that of $\Omega_{bb}$ is $\Omega_{bb}^{*}- 16C_{bs} = 10266$ MeV.

	Ref. [86]	Ref. [60]	Ref. [88]	Ref. [89]	Ref. [90]	Ref. [91]	Refs. [68, 87]	Ref. [92]	Ref. [93]	Refs. [94, 95]
$\Xi_{bb}$	10143(30)(23)	10168.9±9.2	10202	10340	$10197_{-17}^{+10}$	10204	10162±12	10272	10090±10	10170±140
$\Xi_{bb}^{*}$	10178(30)(24)	10188±7.1	10237	10367	$10236 _{-17}^{+9}$	−	10184±12	−	10337	10220±150
$\Omega_{bb}$	10273(27)(20)	10259.0±15.5	10359	10454	$10260 _{-34}^{+14}$	10258	10208±18	10369	10180±5	10320±140
$\Omega_{bb}^{*}$	10308(27)(21)	10267.5±12.1	10389	10486	$10297 _{-28}^{+5}$	−	−	10429	−	10380±140
$\Xi_{bc}$	6943(33)(28)	6922.3±6.9	6933	7011	$6919 _{-7}^{+17}$	6932	6914±13	6838	6840±10	−
$\Xi_{bc}'$	6959(36)(28)	6947.9±6.9	6963	7047	$6948 _{-6}^{+17}$	−	6933±12	7028	−	−
$\Xi_{bc}^{*}$	6985(36)(28)	6973.2±5.5	6980	7074	$6986 _{-5}^{+14}$	−	6960±14	6986	−	−
$\Omega_{bc}$	6998(27)(20)	7010.7±9.3	7088	7136	$6986 _{-17}^{+27}$	6996	6968±19	6941	6945±5	−
$\Omega_{bc}'$	7032(28)(20)	7047.0±9.3	7116	7165	$7009 _{-15}^{+24}$	−	6984±19	7116	−	−
$\Omega_{bc}^{*}$	7059(28)(21)	7065.7±7.5	7130	7187	$7046 _{-9}^{+11}$	−	−	7077	−	−

Table 5.  Theoretical predictions for the masses of doubly heavy baryons (unit: MeV) in various approaches: lattice QCD [86], chromomagnetic models [60, 68, 87], relativistic quark model [88], nonrelativistic quark model [89-91], bag model [92], Bethe-Salpeter equation [93], and QCD sum rules [94, 95].

By repeating the procedure for studying the double-charm tetraquarks, we similarly obtain the masses of the highest-spin double-bottom tetraquark states:

These values are 49, 78, and 95 MeV (increasing numbers) higher than the $\bar{B}^*\bar{B}^*$, $\bar{B}^*\bar{B}_s^*$, and $\bar{B}_s^*\bar{B}_s^*$ thresholds, respectively. The mass distances from the corresponding thresholds in Ref. [11] are approximately 45, 29, and 13 MeV (decreasing numbers). With the newly obtained masses of the spin-2 tetraquarks, we can further estimate those of the lower double-bottom states in the CMI model. We list all the results in Table 6 and plot the relative positions for the $bb\bar{q}\bar{q}$ tetraquarks in Fig. 3.

$(bb\bar{n}\bar{n})$ $I(J^{P})$	$\langle H_{CM}\rangle$	Eigenvalues	Mass (our) HDAS	Mass (low.) $\bar{B}^{(*)}\bar{B}^{(*)}$	Mass (up.) Eq. (2)
$1(2^{+})$	$\left(\begin{array}{c}64.7\end{array}\right)$	$\left(\begin{array}{c}64.7\end{array}\right)$	$\left(\begin{array}{c}10698.7\end{array}\right)$	$\left(\begin{array}{c}10691.3\end{array}\right)$	$\left(\begin{array}{c}10897.1\end{array}\right)$
$1(1^{+})$	$\left(\begin{array}{c}42.3\end{array}\right)$	$\left(\begin{array}{c}42.3\end{array}\right)$	$\left(\begin{array}{c}10676.3\end{array}\right)$	$\left(\begin{array}{c}10668.9\end{array}\right)$	$\left(\begin{array}{c}10874.7\end{array}\right)$
$1(0^{+})$	$\left(\begin{array}{cc}31.1&41.2\\41.2&80.3\end{array}\right)$	$\left(\begin{array}{c}7.8\\103.7\end{array}\right)$	$\left(\begin{array}{c}10641.7\\10737.6\end{array}\right)$	$\left(\begin{array}{c}10634.3\\10730.2\end{array}\right)$	$\left(\begin{array}{c}10840.2\\10936.1\end{array}\right)$
$0(1^{+})$	$\left(\begin{array}{cc}-141.7&-23.8\\-23.8&-17.3\end{array}\right)$	$\left(\begin{array}{c}-146.1\\-12.9\end{array}\right)$	$\left(\begin{array}{c}10487.9\\10621.0\end{array}\right)$	$\left(\begin{array}{c}10480.5\\10613.6\end{array}\right)$	$\left(\begin{array}{c}10686.3\\10819.5\end{array}\right)$
$(bb\bar{n}\bar{s})$ $J^{P}$	$\langle H_{CM}\rangle$	Eigenvalues	Mass (our) HDAS	Mass (low.) $\bar{B}^{(*)}\bar{B}_{s}^{(*)}$	Mass (up.) Eq. (2)
$2^{+}$	$\left(\begin{array}{c}48.5\end{array}\right)$	$\left(\begin{array}{c}48.5\end{array}\right)$	$\left(\begin{array}{c}10817.6\end{array}\right)$	$\left(\begin{array}{c}10764.7\end{array}\right)$	$\left(\begin{array}{c}11061.5\end{array}\right)$
$1^{+}$	$\left(\begin{array}{ccc}25.0&0.0&0.0\\0.0&-91.3&-24.9\\0.0&-24.9&-8.9\end{array}\right)$	$\left(\begin{array}{c}-98.2\\-2.0\\25.0\end{array}\right)$	$\left(\begin{array}{c}10670.9\\10767.2\\10794.1\end{array}\right)$	$\left(\begin{array}{c}10618.0\\10714.2\\10741.2\end{array}\right)$	$\left(\begin{array}{c}10914.8\\11011.0\\11038.0\end{array}\right)$
$0^{+}$	$\left(\begin{array}{cc}13.3&43.1\\43.1&55.1\end{array}\right)$	$\left(\begin{array}{c}-13.7\\82.1\end{array}\right)$	$\left(\begin{array}{c}10755.4\\10851.2\end{array}\right)$	$\left(\begin{array}{c}10702.5\\10798.3\end{array}\right)$	$\left(\begin{array}{c}10999.3\\11095.1\end{array}\right)$
$(bb\bar{s}\bar{s})$ $J^{P}$	$\langle H_{CM}\rangle$	Eigenvalues	Mass (our) HDAS	Mass (low.) $\bar{B}_{s}^{(*)}\bar{B}_{s}^{(*)}$	Mass (up.) Eq. (2)
$2^{+}$	$\left(\begin{array}{c}32.2\end{array}\right)$	$\left(\begin{array}{c}32.2\end{array}\right)$	$\left(\begin{array}{c}10925.6\end{array}\right)$	$\left(\begin{array}{c}10838.1\end{array}\right)$	$\left(\begin{array}{c}11225.8\end{array}\right)$
$1^{+}$	$\left(\begin{array}{c}7.7\end{array}\right)$	$\left(\begin{array}{c}7.7\end{array}\right)$	$\left(\begin{array}{c}10901.0\end{array}\right)$	$\left(\begin{array}{c}10813.6\end{array}\right)$	$\left(\begin{array}{c}11201.3\end{array}\right)$
$0^{+}$	$\left(\begin{array}{cc}-4.6&45.1\\45.1&29.9\end{array}\right)$	$\left(\begin{array}{c}-35.6\\60.9\end{array}\right)$	$\left(\begin{array}{c}10857.7\\10954.3\end{array}\right)$	$\left(\begin{array}{c}10770.3\\10866.8\end{array}\right)$	$\left(\begin{array}{c}11158.0\\11254.5\end{array}\right)$

Table 6.  Results for the $bb\bar{q}\bar{q}$ ($q = u,d,s$) states (unit: MeV). The second and third columns provide the numerical values of the CMI matrices and their eigenvalues, respectively. The fifth and sixth columns list the masses estimated using Eq. (3) (with $\bar{B}^{(*)}\bar{B}^{(*)}$/$\bar{B}^{(*)}\bar{B}_{s}^{(*)}$/$\bar{B}_{s}^{(*)}\bar{B}_{s}^{(*)}$ as the reference state) and Eq. (2) (parameters provided in Table 1), respectively. These can be viewed as the theoretical lower limits (low.) and upper limits (up.), respectively, for the tetraquark masses in the current framework. The fourth column displays our predictions with the heavy diquark-antiquark symmetry (HDAS) consideration.

By comparing the current results with those in Ref. [11], a similar $bb\bar{n}\bar{n}$ spectrum can be found. According to Fig. 3(a), the $T_{bb}$ state with a mass of $10488$ MeV is approximately 116 MeV below the $\bar{B}\bar{B}^*$ threshold (10604 MeV), and it should be rather stable, but other $bb\bar{n}\bar{n}$ states are not. This observation is the same as that in Ref. [11]. For the $bb\bar{n}\bar{s}$ states, the masses in the current work are approximately 50 MeV higher than those in Ref. [11]. According to Fig. 3(b), only the lowest $T_{bb\bar{n}\bar{s}}^{J = 1}$, which is slightly ($\sim$20 MeV) below the $\bar{B}\bar{B}_s^*$ threshold, is possibly a stable tetraquark. This conclusion is similar to that of Ref. [11]. For the heaviest $bb\bar{s}\bar{s}$ states, the masses in the current work are approximately 80 MeV higher than those in Ref. [11], and no stable state is found in both approaches. Therefore, the HDAS and threshold approaches yield similar conclusions regarding the state stabilities for double-bottom tetraquarks.

It is interesting that the masses of the $bb\bar{n}\bar{n}$ states from the HDAS consideration coincide with those from the threshold approach. This coincidence may mean that the $(bb)(\bar{n}\bar{n})$ diquark-antidiquark structure and $(b\bar{n})(b\bar{n})$ molecule-like structure have similar effects on the mass spectrum. It probably also implies that the four quark components have an almost equal spatial distance. If this is true, it is also possible that more than one structure exists near the $\bar{B}\bar{B}^*$ threshold [27]. Of course, future experimental data are required to evaluate which approach provides more reasonable results and how large the effects from the mass uncertainty of $\Xi_{bb}$ would be.

HDAS relations similar to Eqs. (14), (15), and (16) can also be applied to bottom-charm tetraquark systems with the replacement $cc\to bc$. It should be noted that there are two bases for the highest-spin $bc\bar{n}\bar{n}$ and $bc\bar{n}\bar{s}$ tetraquarks. For the former case, the color-triplet (color-sextet) diquark exists only in the isovector (isoscalar) state. For the latter case, the two bases are nearly uncoupled, and the color-triplet diquark exists mainly in the higher state. The color-sextet contributions to the higher $T_{bc\bar{n}\bar{s}}^{J = 2}$ are not a concern.

Using the mass $\Xi_{bc} = 6922.3\pm6.9$ MeV from Ref. [60] and our CMI model, we obtain $\Xi_{bc}^* = 6974$ MeV [50] and $\Omega_{bc}^* = \Xi_{bc}^*+\bar{B}_s^*-\bar{B}^* = 7065$ MeV. Thereafter,

are obtained. These values are 150, 165, and 183 MeV (increasing numbers) higher than the $\bar{B}^*D^*$, $\bar{B}^*D_s^*$, and $\bar{B}_s^*D_s^*$ thresholds, respectively. In Ref. [11], such states are approximately 34, 5, and 1 MeV (decreasing numbers) higher than the corresponding thresholds. With the reference scales in Eq. (18), we obtain the numerical results for the $bc\bar{q}\bar{q}$ tetraquark states in the CMI model. These are listed in Table 7, and the spectra are plotted in Fig. 4.

$(bc\bar{n}\bar{n})$ $I(J^P)$	$\langle H_{CM}\rangle$	Eigenvalues	Mass (our) HDAS	Mass (low.) $\bar{B}^{(*)}D^{(*)}$	Mass (up.) Eq. (2)
$1(2^{+})$	$\left(\begin{array}{c}77.4\end{array}\right)$	$\left(\begin{array}{c}77.4\end{array}\right)$	$\left(\begin{array}{c}7483.2\end{array}\right)$	$\left(\begin{array}{c}7363.8\end{array}\right)$	$\left(\begin{array}{c}7579.5\end{array}\right)$
$1(1^{+})$	$\left(\begin{array}{ccc}31.0&17.0&36.0\\17.0&32.6&-49.2\\36.0&-49.2&70.5\end{array}\right)$	$\left(\begin{array}{c}-24.0\\47.9\\110.2\end{array}\right)$	$\left(\begin{array}{c}7381.9\\7453.7\\7516.0\end{array}\right)$	$\left(\begin{array}{c}7262.5\\7334.3\\7396.6\end{array}\right)$	$\left(\begin{array}{c}7478.1\\7550.0\\7612.3\end{array}\right)$
$1(0^{+})$	$\left(\begin{array}{cc}7.8&85.2\\85.2&81.3\end{array}\right)$	$\left(\begin{array}{c}-48.3\\137.4\end{array}\right)$	$\left(\begin{array}{c}7357.5\\7543.2\end{array}\right)$	$\left(\begin{array}{c}7238.1\\7423.8\end{array}\right)$	$\left(\begin{array}{c}7453.8\\7639.5\end{array}\right)$
$0(2^{+})$	$\left(\begin{array}{c}30.9\end{array}\right)$	$\left(\begin{array}{c}30.9\end{array}\right)$	$\left(\begin{array}{c}7436.7\end{array}\right)$	$\left(\begin{array}{c}7317.3\end{array}\right)$	$\left(\begin{array}{c}7533.0\end{array}\right)$
$0(1^{+})$	$\left(\begin{array}{ccc}-85.1&36.0&42.4\\36.0&-141.0&-49.2\\42.4&-49.2&-16.3\end{array}\right)$	$\left(\begin{array}{c}-182.7\\-70.2\\10.5\end{array}\right)$	$\left(\begin{array}{c}7223.1\\7335.6\\7416.3\end{array}\right)$	$\left(\begin{array}{c}7103.7\\7216.2\\7296.9\end{array}\right)$	$\left(\begin{array}{c}7319.4\\7431.9\\7512.6\end{array}\right)$
$0(0^{+})$	$\left(\begin{array}{cc}-143.1&85.2\\85.2&-162.6\end{array}\right)$	$\left(\begin{array}{c}-238.6\\-67.0\end{array}\right)$	$\left(\begin{array}{c}7167.2\\7338.8\end{array}\right)$	$\left(\begin{array}{c}7047.8\\7219.4\end{array}\right)$	$\left(\begin{array}{c}7263.5\\7435.1\end{array}\right)$
$(bc\bar{n}\bar{s})$ $J^{P}$	$\langle H_{CM}\rangle$	Eigenvalues	Mass (our) HDAS	Mass (low.) $\bar{B}^{(*)}D_{s}^{(*)}$	Mass (up.) Eq. (2)
$2^{+}$	$\left(\begin{array}{cc}61.0&0.3\\0.3&40.3\end{array}\right)$	$\left(\begin{array}{c}40.3\\61.0\end{array}\right)$	$\left(\begin{array}{c}7581.5\\7602.2\end{array}\right)$	$\left(\begin{array}{c}7429.8\\7450.5\end{array}\right)$	$\left(\begin{array}{c}7723.0\\7743.7\end{array}\right)$
$1^{+}$	$\left(\begin{array}{cccccc}13.8&-0.6&16.8&-0.3&35.6&-1.2\\-0.6&-90.6&-0.1&35.6&0.0&-50.1\\16.8&-0.1&15.8&-1.2&-50.1&0.0\\-0.3&35.6&-1.2&-77.7&-1.4&42.0\\35.6&0.0&-50.1&-1.4&45.3&-0.3\\-1.2&-50.1&0.0&42.0&-0.3&-7.9\end{array}\right)$	$\left(\begin{array}{c}-150.3\\-48.2\\-43.5\\21.9\\30.6\\88.1\end{array}\right)$	$\left(\begin{array}{c}7390.9\\7493.0\\7497.7\\7563.1\\7571.8\\7629.3\end{array}\right)$	$\left(\begin{array}{c}7239.2\\7341.4\\7346.1\\7411.5\\7420.2\\7477.7\\\end{array}\right)$	$\left(\begin{array}{c}7532.4\\7634.5\\7639.2\\7704.6\\7713.3\\7770.8\end{array}\right)$
$0^{+}$	$\left(\begin{array}{cccc}-9.8&0.2&-0.6&86.7\\0.2&-112.2&86.7&0.0\\-0.6&86.7&-136.7&0.6\\86.7&0.0&0.6&56.1\end{array}\right)$	$\left(\begin{array}{c}-212.0\\-69.6\\-36.8\\115.9\end{array}\right)$	$\left(\begin{array}{c}7329.2\\7471.6\\7504.3\\7657.1\end{array}\right)$	$\left(\begin{array}{c}7177.5\\7319.9\\7352.7\\7505.4\end{array}\right)$	$\left(\begin{array}{c}7470.7\\7613.1\\7645.9\\7798.6\end{array}\right)$
$(bc\bar{s}\bar{s})$ $J^{P}$	$\langle H_{CM}\rangle$	Eigenvalues	Mass (our) HDAS	Mass (low.) $\bar{B}_{s}^{(*)}D_{s}^{(*)}$	Mass (up.) Eq. (2)
$2^{+}$	$\left(\begin{array}{c}44.6\end{array}\right)$	$\left(\begin{array}{c}44.6\end{array}\right)$	$\left(\begin{array}{c}7710.1\end{array}\right)$	$\left(\begin{array}{c}7523.9\end{array}\right)$	$\left(\begin{array}{c}7907.9\end{array}\right)$
$1^{+}$	$\left(\begin{array}{ccc}-3.4&16.6&35.2\\16.6&-1.0&-50.9\\35.2&-50.9&20.1\end{array}\right)$	$\left(\begin{array}{c}-63.7\\13.0\\66.4\end{array}\right)$	$\left(\begin{array}{c}7601.9\\7678.5\\7731.9\end{array}\right)$	$\left(\begin{array}{c}7415.6\\7492.3\\7545.7\end{array}\right)$	$\left(\begin{array}{c}7799.6\\7876.3\\7929.7\end{array}\right)$
$0^{+}$	$\left(\begin{array}{cc}-27.4&88.2\\88.2&30.9\end{array}\right)$	$\left(\begin{array}{c}-91.1\\94.6\end{array}\right)$	$\left(\begin{array}{c}7574.4\\7760.1\end{array}\right)$	$\left(\begin{array}{c}7388.1\\7573.9\end{array}\right)$	$\left(\begin{array}{c}7772.2\\7957.9\end{array}\right)$

Table 7.  Results for the $bc\bar{q}\bar{q}$ ($q = u,d,s$) states (unit: MeV). The second and third columns provide the numerical values of the CMI matrices and their eigenvalues, respectively. The fifth and sixth columns list the masses estimated using Eq. (3) (with $\bar{B}^{(*)}D^{(*)}$/$\bar{B}^{(*)}D_{s}^{(*)}$/$\bar{B}_{s}^{(*)}D_{s}^{(*)}$ as the reference state) and Eq. (2) (parameters provided in Table 1), respectively. These can be viewed as the theoretical lower limits (low.) and upper limits (up.), respectively, for the tetraquark masses in the current framework. The fourth column displays our predictions with the heavy diquark-antiquark symmetry (HDAS) consideration.