# The triply heavy tetraquark states

Xin-Zhen Weng<EMAIL_ADDRESS>Center of High Energy Physics, Peking
University, Beijing 100871, China  Wei-Zhen Deng<EMAIL_ADDRESS>School of
Physics, Peking University, Beijing 100871, China  Shi-Lin Zhu
<EMAIL_ADDRESS>Center of High Energy Physics, Peking University, Beijing
100871, China

###### Abstract

In the framework of an extended chromomagnetic model, we systematically study
the mass spectrum of the $S$-wave $qQ\bar{Q}\bar{Q}$ tetraquarks. Their mass
spectra are mainly determined by the color interaction. For the
$qc\bar{c}\bar{c}$, $qb\bar{c}\bar{c}$ and $qb\bar{b}\bar{b}$ tetraquarks, the
color interaction favors the color-sextet
$\ket{(qQ)^{6_{c}}(\bar{Q}\bar{Q})^{\bar{6}_{c}}}$ configuration over the
color-triplet $\ket{(qQ)^{\bar{3}_{c}}(\bar{Q}\bar{Q})^{3_{c}}}$ one. But for
the $qc\bar{b}\bar{b}$ tetraquarks, the color-triplet configuration is
favored. We find no stable states which lie below the thresholds of two
pseudoscalar mesons. The lowest axial-vector states with the
$qQ\bar{b}\bar{b}$ flavor configuration may be narrow. They lie just above the
thresholds of two pseudoscalar mesons, but cannot decay into these channels
because of the conservation of the angular momentum and parity.

##  I Introduction

Searching for the exotic states is an important and challenging topic in the
hadron physics. In the past two decades, we have witnessed tremendous
progresses in this area. In 2003, the first charmoniumlike state $X(3872)$ was
observed by the Belle Collaboration [1], and later confirmed by the CDF [2],
D0 [3], $BABAR$ [4], LHCb [5], CMS [6], and BESIII [7] Collaborations. Its
quantum number is $I^{G}J^{PC}=0^{+}1^{++}$ [8]. Since then, lots of
charmoniumlike and bottomoniumlike states are found. They are called $XYZ$
states. Many of them do not fit into the conventional charmonium or
bottomonium spectrum in the quark model. In particular, the charged
charmoniumlike and bottomoniumlike states like $Z_{c}(3900)$ [9, 10],
$Z_{c}(3885)$ [11, 12], $Z_{c}(4020)$ [13], $Z_{c}(4025)$ [14], $Z_{cs}(3985)$
[15], $Z_{cs}(4000)$ and $Z_{cs}$(4020) [16], $Z_{b}(10610)$ and
$Z_{b}(10650)$ [17] contain at least four quarks. They are candidates of
exotic structures like the compact tetraquark [18, 19], the molecule [20, 21,
22], the hybrid meson [23, 24], and so on. Interested readers may refer to
Refs. [25, 26, 27, 28, 29, 30, 31, 32, 33, 34] for more details.

Besides the charmoniumlike and bottomoniumlike states, various open heavy
flavor exotic states have been observed recently. In 2016, the D0
Collaboration reported the observation of the $X(5568)$ state in the
$B_{s}^{0}\pi^{\pm}$ channel [35]. It was very likely a $us\bar{d}\bar{b}$ or
$ds\bar{u}\bar{b}$ tetraquarks since it lies about 200 MeV below the
$\bar{B}K$ threshold [36]. However, the subsequent search of the LHCb
Collaboration did not confirm this state [37]. In 2020, the LHCb Collaboration
observed two resonances in the $D^{-}K^{+}$ channel with spin-$0$ and spin-$1$
[38, 39], whose minimal quark contents are $ud\bar{s}\bar{c}$. Recently, the
LHCb Collaboration observed an narrow state in the $D^{0}D^{0}\pi^{+}$ mass
spectrum just below the $D^{*+}D^{0}$ threshold [40, 41]. This is the first
doubly charmed tetraquark $T_{cc}^{+}$ observed in experiment. Moreover, in
2020, the LHCb collaboration observed a narrow structure and a wide structure
in the $J/\psi$-pair invariant mass spectrum in the range of 6.2–7.2 GeV,
which could be all-charm hadrons [42].

At this stage, it is natural to expect that the triply charmed tetraquark(s)
may be observed in the very near future. For the triply charmed system, a
molecule configuration requires the exchange of a $D$ meson. The corresponding
interaction range is $\sim 0.1~{}\mathrm{fm}$, which is a typical scale of the
compact tetraquark. From another point of view, the $D$-meson exchange
interaction is highly suppressed in the typical scale of the molecule
($\sim\mathrm{fm}$). In other words, the molecule interpretation is not
favored for the triply heavy system. If a $qc\bar{c}\bar{c}$ state is observed
in experiment, it is very likely a compact tetraquark state.

However, the theoretical study of this system is relatively scarce [43, 44,
18, 45, 46, 47, 48, 49, 50, 51]. In Ref. [45], Chen et al. found that some
axial-vector $cc\bar{b}\bar{q}$ states lie below the $B_{c}\bar{D}_{(s)}$
thresholds, which are stable against the strong decay. A lattice study
performed by Junnarkar et al. [47] gives similar results. They found that the
axial-vector $uc\bar{b}\bar{b}$ and $sc\bar{b}\bar{b}$ lie below their
thresholds by $6\pm 11~{}\mathrm{MeV}$ and $8\pm 3~{}\mathrm{MeV}$
respectively. However, a recent study by Lü et al. [51] predicted that the
corresponding states are about several hundred MeV higher. More theoretical
and experimental studies are needed to have a better understanding of these
systems.

For the triply heavy tetraquarks, the interaction is provided by the gluon
exchange and string confinement. The resulting interactions include the spin-
independent Coulomb-type interaction, linear confinement and spin-dependent
chromomagnetic interaction, tensor interaction and spin-orbit interaction.
When focusing on the $S$-wave states, the tensor and spin-orbit interactions
can be neglected. Furthermore, we can simplify the model by integrating out
spatial part of the interaction. Then the interaction becomes

$-\sum_{i<j}a_{ij}\bm{F}_{i}\cdot\bm{F}_{j}-\sum_{i<j}v_{ij}\bm{S}_{i}\cdot\bm{S}_{j}\bm{F}_{i}\cdot\bm{F}_{j}\,.$
(1)

This interaction gives good account of the $S$-wave mesons and baryons [52].
In this work, we use the this interaction to study the triply heavy
tetraquarks. The paper is organized as follows. In Sec. II, we introduce the
extended chromomagnetic model and construct the wave function bases for the
triply heavy tetraquarks. Then we discuss the numerical results in Sec. III.
We conclude in Sec. IV.

##  II The Extended Chromomagnetic Model

For the the $S$-wave tetraquark system, we consider the chromomagnetic model.
The Hamiltonian reads [53, 54, 55, 56, 57, 52, 58, 59, 60]

$H=\sum_{i}m_{i}+H_{\text{CE}}+H_{\text{CM}}$ (2)

where $m_{i}$ is the effective mass of $i$th constituent quark which consists
of the constituent quark mass, the kinetic energy, and so on, $H_{\text{CE}}$
is the colorelectric (CE) interaction which includes the color linear
confinement and Coulomb-type interaction [57, 52, 58, 59, 60]

$H_{\text{CE}}=-\sum_{i<j}a_{ij}\bm{F}_{i}\cdot\bm{F}_{j}\,,$ (3)

and $H_{\text{CM}}$ is the chromomagnetic (CM) interaction from the one-gluon-
exchange (OGE) [61, 62, 63, 18, 64, 45, 33]

$H_{\text{CM}}=-\sum_{i<j}v_{ij}\bm{S}_{i}\cdot\bm{S}_{j}\bm{F}_{i}\cdot\bm{F}_{j}\,.$
(4)

The coupling constants $a_{ij}$ and
$v_{ij}\propto\braket{\alpha_{s}(r)\delta^{3}(\bm{r})}/m_{i}m_{j}$ depend on
the spatial wave function and the constituent quark masses.
$\bm{S}_{i}=\bm{\sigma}_{i}/2$ and $\bm{F}_{i}={\bm{\lambda}}_{i}/2$ are the
quark spin and color operators. For the antiquark,

$\bm{S}_{\bar{q}}=-\bm{S}_{q}^{*}\,,\quad\bm{F}_{\bar{q}}=-\bm{F}_{q}^{*}\,.$
(5)

Since

$\displaystyle\sum_{i<j}\left(m_{i}+m_{j}\right)\bm{F}_{i}\cdot\bm{F}_{j}$
$\displaystyle={}$
$\displaystyle\left(\sum_{i}m_{i}\bm{F}_{i}\right)\cdot\left(\sum_{i}\bm{F}_{i}\right)-\frac{4}{3}\sum_{i}m_{i}\,,$
(6)

and the total color operator $\sum_{i}\bm{F}_{i}$ nullifies any color-singlet
physical state, we introduce the quark pair mass parameter

$m_{ij}=\left(m_{i}+m_{j}\right)+\frac{4}{3}a_{ij}\,,$ (7)

to combine the effective quark mass $m_{i}$ and the color interaction strength
$a_{ij}$. Then we can rewrite the model Hamiltonian as [52, 58, 59, 60]

$H=-\frac{3}{4}\sum_{i<j}m_{ij}V^{\text{C}}_{ij}-\sum_{i<j}v_{ij}V^{\text{CM}}_{ij}\,,$
(8)

where

$V^{\text{C}}_{ij}=\bm{F}_{i}\cdot\bm{F}_{j}\,,$ (9)

and

$V^{\text{CM}}_{ij}=\bm{S}_{i}\cdot\bm{S}_{j}\bm{F}_{i}\cdot\bm{F}_{j}$ (10)

are the color and CM interactions between quarks, respectively. Here, $m_{ij}$
and $v_{ij}$ are unknown parameters. In Ref. [52], we have used the
conventional mesons and baryons to fit them. Their values are listed in Table
1. In this work, we use the same set of parameters to estimate the masses of
the $S$-wave $qQ\bar{Q}\bar{Q}$ tetraquarks.

Table 1: Parameters of the $q\bar{q}$ and $qq$ pairs [52] (in units of MeV). Parameter | $m_{n\bar{n}}$ | $m_{n\bar{s}}$ | $m_{s\bar{s}}$ | $m_{n\bar{c}}$ | $m_{s\bar{c}}$ | $m_{c\bar{c}}$ | $m_{n\bar{b}}$ | $m_{s\bar{b}}$ | $m_{c\bar{b}}$ | $m_{b\bar{b}}$
---|---|---|---|---|---|---|---|---|---|---
Value | $615.95$ | $794.22$ | $936.40$ | $1973.22$ | $2076.14$ | $3068.53$ | $5313.35$ | $5403.25$ | $6322.27$ | $9444.97$
Parameter | $v_{n\bar{n}}$ | $v_{n\bar{s}}$ | $v_{s\bar{s}}$ | $v_{n\bar{c}}$ | $v_{s\bar{c}}$ | $v_{c\bar{c}}$ | $v_{n\bar{b}}$ | $v_{s\bar{b}}$ | $v_{c\bar{b}}$ | $v_{b\bar{b}}$
Value | $477.92$ | $298.57$ | $249.18$ | $106.01$ | $107.87$ | $85.12$ | $33.89$ | $36.43$ | $47.18$ | $45.98$
Parameter | $m_{nn}$ | $m_{ns}$ | $m_{ss}$ | $m_{nc}$ | $m_{sc}$ | $m_{cc}$ | $m_{nb}$ | $m_{sb}$ | $m_{cb}$ | $m_{b{b}}$
Value | $724.85$ | $906.65$ | $1049.36$ | $2079.96$ | $2183.68$ | $3171.51$ | $5412.25$ | $5494.80$ | $6416.07$ | $9529.57$
Parameter | $v_{n{n}}$ | $v_{n{s}}$ | $v_{ss}$ | $v_{n{c}}$ | $v_{s{c}}$ | $v_{c{c}}$ | $v_{n{b}}$ | $v_{s{b}}$ | $v_{c{b}}$ | $v_{b{b}}$
Value | $305.34$ | $212.75$ | $195.30$ | $62.81$ | $70.63$ | $56.75$ | $19.92$ | $8.47$ | $31.45$ | $30.65$

###  II.1 Wave function

Before calculating the tetraquark masses, we need to construct their wave
functions. In principle, the total wave function is a direct product of the
orbital, color, spin and flavor wave functions. In this work, we only consider
the $S$-wave states, the orbital wave function is always symmetric. In the
$qq{\otimes}\bar{q}\bar{q}$ configuration, we can construct the following
color-spin wave functions $\\{\alpha_{i}^{J}\\}$

  1. 1.

$J^{P}=0^{+}$:

$\displaystyle\alpha_{1}^{0}=\ket{\left(q_{1}q_{2}\right)_{1}^{6}\left(\bar{q}_{3}\bar{q}_{4}\right)_{1}^{\bar{6}}}_{0},$
$\displaystyle\alpha_{2}^{0}=\ket{\left(q_{1}q_{2}\right)_{0}^{6}\left(\bar{q}_{3}\bar{q}_{4}\right)_{0}^{\bar{6}}}_{0},$
$\displaystyle\alpha_{3}^{0}=\ket{\left(q_{1}q_{2}\right)_{1}^{\bar{3}}\left(\bar{q}_{3}\bar{q}_{4}\right)_{1}^{3}}_{0},$
$\displaystyle\alpha_{4}^{0}=\ket{\left(q_{1}q_{2}\right)_{0}^{\bar{3}}\left(\bar{q}_{3}\bar{q}_{4}\right)_{0}^{3}}_{0},$
(11)

  2. 2.

$J^{P}=1^{+}$:

$\displaystyle\alpha_{1}^{1}=\ket{\left(q_{1}q_{2}\right)_{1}^{6}\left(\bar{q}_{3}\bar{q}_{4}\right)_{1}^{\bar{6}}}_{1},$
$\displaystyle\alpha_{2}^{1}=\ket{\left(q_{1}q_{2}\right)_{1}^{6}\left(\bar{q}_{3}\bar{q}_{4}\right)_{0}^{\bar{6}}}_{1},$
$\displaystyle\alpha_{3}^{1}=\ket{\left(q_{1}q_{2}\right)_{0}^{6}\left(\bar{q}_{3}\bar{q}_{4}\right)_{1}^{\bar{6}}}_{1},$
$\displaystyle\alpha_{4}^{1}=\ket{\left(q_{1}q_{2}\right)_{1}^{\bar{3}}\left(\bar{q}_{3}\bar{q}_{4}\right)_{1}^{3}}_{1},$
$\displaystyle\alpha_{5}^{1}=\ket{\left(q_{1}q_{2}\right)_{1}^{\bar{3}}\left(\bar{q}_{3}\bar{q}_{4}\right)_{0}^{3}}_{1},$
$\displaystyle\alpha_{6}^{1}=\ket{\left(q_{1}q_{2}\right)_{0}^{\bar{3}}\left(\bar{q}_{3}\bar{q}_{4}\right)_{1}^{3}}_{1},$
(12)

  3. 3.

$J^{P}=2^{+}$:

$\displaystyle\alpha_{1}^{2}=\ket{\left(q_{1}q_{2}\right)_{1}^{6}\left(\bar{q}_{3}\bar{q}_{4}\right)_{1}^{\bar{6}}}_{2},$
$\displaystyle\alpha_{2}^{2}=\ket{\left(q_{1}q_{2}\right)_{1}^{\bar{3}}\left(\bar{q}_{3}\bar{q}_{4}\right)_{1}^{3}}_{2},$
(13)

where the superscript $3$, $\bar{3}$, $6$ or $\bar{6}$ denotes the color, and
the subscript $0$, $1$ or $2$ denotes the spin.

Next we consider the flavor wave functions. According to symmetric properties
of the two antiquarks, the triply heavy tetraquarks can be divided into two
categories. The $qQ\bar{c}\bar{b}$ tetraquarks are not constrained by the
Pauli principle, thus all the preceding color-spin wave functions are allowed.
On the other hand, the $qQ\bar{c}\bar{c}$ and $qQ\bar{b}\bar{b}$ tetraquarks
have symmetric flavor wave functions over the two antiquarks, thus the color-
spin wave functions must be asymmetric over the two antiquarks. More
precisely, the bases for the $qQ\bar{c}\bar{c}$ and $qQ\bar{b}\bar{b}$
tetraquarks are $\\{\alpha_{2}^{0},\alpha_{3}^{0}\\}$,
$\\{\alpha_{2}^{1},\alpha_{4}^{1},\alpha_{4}^{1}\\}$ and
$\\{\alpha_{2}^{2}\\}$.

Diagonalizing the Hamiltonian [Eq. (8)] in the corresponding bases, we can
obtain the masses and eigenvectors of the triply heavy tetraquarks.

##  III Numerical results

###  III.1 The $qc\bar{c}\bar{c}$ and $qb\bar{c}\bar{c}$ systems

---
(a) $nc\bar{c}\bar{c}$ states
(b) $sc\bar{c}\bar{c}$ states
Figure 1: Mass spectra of the $nc\bar{c}\bar{c}$ and $sc\bar{c}\bar{c}$
tetraquark states. The dotted lines indicate various meson-meson thresholds.
The masses are all in units of MeV.
---
(a) $nb\bar{c}\bar{c}$ states
(b) $sb\bar{c}\bar{c}$ states
Figure 2: Mass spectra of the $nb\bar{c}\bar{c}$ and $sb\bar{c}\bar{c}$ tetraquark states. The dotted lines indicate various meson-meson thresholds. Here the predicted mass $M_{B_{c}^{*}}=6338~{}\text{MeV}$ of Godfrey et al. [65] is used. The masses are all in units of MeV. Table 2: Masses and eigenvectors of the $nc\bar{c}\bar{c}$ and $sc\bar{c}\bar{c}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | Eigenvector
---|---|---|---
$nc\bar{c}\bar{c}$ | $0^{+}$ | $4936.7$ | $\\{0.817,0.576\\}$
|  | $5185.3$ | $\\{-0.576,0.817\\}$
| $1^{+}$ | $4968.1$ | $\\{0.915,-0.069,-0.397\\}$
|  | $5135.5$ | $\\{-0.071,-0.997,0.010\\}$
|  | $5154.0$ | $\\{-0.397,0.019,-0.918\\}$
| $2^{+}$ | $5198.4$ | $\\{1\\}$
$sc\bar{c}\bar{c}$ | $0^{+}$ | $5040.1$ | $\\{0.816,0.578\\}$
|  | $5290.6$ | $\\{-0.578,0.816\\}$
| $1^{+}$ | $5069.2$ | $\\{0.912,-0.074,-0.404\\}$
|  | $5239.9$ | $\\{-0.076,-0.997,0.011\\}$
|  | $5254.3$ | $\\{-0.404,0.020,-0.915\\}$
| $2^{+}$ | $5303.3$ | $\\{1\\}$
Table 3: The eigenvectors of the $nc\bar{c}\bar{c}$ tetraquarks in the $n\bar{c}{\otimes}c\bar{c}$ configuration. The masses are all in units of MeV. System | $J^{P}$ | Mass | $\bar{D}^{*}{J/\psi}$ | $\bar{D}^{*}{\eta_{c}}$ | $\bar{D}{J/\psi}$ | $\bar{D}{\eta_{c}}$
---|---|---|---|---|---|---
$nc\bar{c}\bar{c}$ | $0^{+}$ | $4936.7$ | $0.412$ |  |  | $0.622$
|  | $5185.3$ | $-0.643$ |  |  | $0.174$
| $1^{+}$ | $4968.1$ | $0.366$ | $0.460$ | $-0.516$ |
|  | $5135.5$ | $-0.037$ | $-0.439$ | $-0.375$ |
|  | $5154.0$ | $-0.604$ | $0.111$ | $-0.095$ |
| $2^{+}$ | $5198.4$ | $0.577$ |  |  |
Table 4: The eigenvectors of the $sc\bar{c}\bar{c}$ tetraquarks in the $s\bar{c}{\otimes}c\bar{c}$ configuration. The masses are all in units of MeV. System | $J^{P}$ | Mass | $\bar{D}_{s}^{*}{J/\psi}$ | $\bar{D}_{s}^{*}{\eta_{c}}$ | $\bar{D}_{s}{J/\psi}$ | $\bar{D}_{s}{\eta_{c}}$
---|---|---|---|---|---|---
$sc\bar{c}\bar{c}$ | $0^{+}$ | $5040.1$ | $0.410$ |  |  | $0.622$
|  | $5290.6$ | $-0.644$ |  |  | $0.172$
| $1^{+}$ | $5069.2$ | $0.361$ | $0.459$ | $-0.519$ |
|  | $5239.9$ | $-0.039$ | $-0.441$ | $-0.373$ |
|  | $5254.3$ | $-0.607$ | $0.107$ | $-0.091$ |
| $2^{+}$ | $5303.3$ | $0.577$ |  |  |
Table 5: The values of $k\cdot|c_{i}|^{2}$ for the $nc\bar{c}\bar{c}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | $\bar{D}^{*}{J/\psi}$ | $\bar{D}^{*}{\eta_{c}}$ | $\bar{D}{J/\psi}$ | $\bar{D}{\eta_{c}}$
---|---|---|---|---|---|---
$nc\bar{c}\bar{c}$ | $0^{+}$ | $4936.7$ | $\times$ |  |  | $172.8$
|  | $5185.3$ | $183.4$ |  |  | $27.0$
| $1^{+}$ | $4968.1$ | $\times$ | $\times$ | $25.9$ |
|  | $5135.5$ | $0.4$ | $114.1$ | $90.0$ |
|  | $5154.0$ | $125.7$ | $7.7$ | $6.1$ |
| $2^{+}$ | $5198.4$ | $159.5$ |  |  |
Table 6: The values of $k\cdot|c_{i}|^{2}$ for the $sc\bar{c}\bar{c}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | $\bar{D}_{s}^{*}{J/\psi}$ | $\bar{D}_{s}^{*}{\eta_{c}}$ | $\bar{D}_{s}{J/\psi}$ | $\bar{D}_{s}{\eta_{c}}$
---|---|---|---|---|---|---
$sc\bar{c}\bar{c}$ | $0^{+}$ | $5040.1$ | $\times$ |  |  | $178.2$
|  | $5290.6$ | $188.8$ |  |  | $27.0$
| $1^{+}$ | $5069.2$ | $\times$ | $\times$ | $26.4$ |
|  | $5239.9$ | $0.4$ | $117.3$ | $91.1$ |
|  | $5254.3$ | $124.4$ | $7.3$ | $5.6$ |
| $2^{+}$ | $5303.3$ | $163.1$ |  |  |
Table 7: The partial width ratios for the $nc\bar{c}\bar{c}$ tetraquarks. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | $\bar{D}^{*}{J/\psi}$ | $\bar{D}^{*}{\eta_{c}}$ | $\bar{D}{J/\psi}$ | $\bar{D}{\eta_{c}}$
---|---|---|---|---|---|---
$nc\bar{c}\bar{c}$ | $0^{+}$ | $4936.7$ | $\times$ |  |  | $1$
|  | $5185.3$ | $6.8$ |  |  | $1$
| $1^{+}$ | $4968.1$ | $\times$ | $\times$ | $1$ |
|  | $5135.5$ | $0.004$ | $1.3$ | $1$ |
|  | $5154.0$ | $20.7$ | $1.3$ | $1$ |
| $2^{+}$ | $5198.4$ | $1$ |  |  |
Table 8: The partial width ratios for the $sc\bar{c}\bar{c}$ tetraquarks. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | $\bar{D}_{s}^{*}{J/\psi}$ | $\bar{D}_{s}^{*}{\eta_{c}}$ | $\bar{D}_{s}{J/\psi}$ | $\bar{D}_{s}{\eta_{c}}$
---|---|---|---|---|---|---
$sc\bar{c}\bar{c}$ | $0^{+}$ | $5040.1$ | $\times$ |  |  | $1$
|  | $5290.6$ | $7.0$ |  |  | $1$
| $1^{+}$ | $5069.2$ | $\times$ | $\times$ | $1$ |
|  | $5239.9$ | $0.005$ | $1.3$ | $1$ |
|  | $5254.3$ | $22.1$ | $1.3$ | $1$ |
| $2^{+}$ | $5303.3$ | $1$ |  |  |
Table 9: Masses and eigenvectors of the $nb\bar{c}\bar{c}$ and $sb\bar{c}\bar{c}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | Eigenvector
---|---|---|---
$nb\bar{c}\bar{c}$ | $0^{+}$ | $8199.1$ | $\\{0.907,0.421\\}$
|  | $8444.8$ | $\\{-0.421,0.907\\}$
| $1^{+}$ | $8217.5$ | $\\{0.958,-0.151,-0.244\\}$
|  | $8430.0$ | $\\{0.207,0.953,0.222\\}$
|  | $8454.4$ | $\\{-0.199,0.263,-0.944\\}$
| $2^{+}$ | $8477.9$ | $\\{1\\}$
$sb\bar{c}\bar{c}$ | $0^{+}$ | $8300.5$ | $\\{0.895,0.445\\}$
|  | $8538.7$ | $\\{-0.445,0.895\\}$
| $1^{+}$ | $8324.8$ | $\\{0.952,-0.169,-0.256\\}$
|  | $8521.9$ | $\\{0.211,0.967,0.145\\}$
|  | $8553.3$ | $\\{-0.223,0.192,-0.956\\}$
| $2^{+}$ | $8569.1$ | $\\{1\\}$
Table 10: The eigenvectors of the $nb\bar{c}\bar{c}$ tetraquarks in the $n\bar{c}{\otimes}b\bar{c}$ configuration. The masses are all in units of MeV. System | $J^{P}$ | Mass | $\bar{D}^{*}\bar{B}_{c}^{*}$ | $\bar{D}^{*}\bar{B}_{c}$ | $\bar{D}\bar{B}_{c}^{*}$ | $\bar{D}\bar{B}_{c}$
---|---|---|---|---|---|---
$nb\bar{c}\bar{c}$ | $0^{+}$ | $8199.1$ | $0.520$ |  |  | $0.581$
|  | $8444.8$ | $-0.559$ |  |  | $0.282$
| $1^{+}$ | $8217.5$ | $0.454$ | $0.400$ | $-0.523$ |
|  | $8430.0$ | $0.210$ | $0.409$ | $0.369$ |
|  | $8454.4$ | $-0.500$ | $0.299$ | $-0.084$ |
| $2^{+}$ | $8477.9$ | $0.577$ |  |  |
Table 11: The eigenvectors of the $sb\bar{c}\bar{c}$ tetraquarks in the $s\bar{c}{\otimes}b\bar{c}$ configuration. The masses are all in units of MeV. System | $J^{P}$ | Mass | $\bar{D}_{s}^{*}\bar{B}_{c}^{*}$ | $\bar{D}_{s}^{*}\bar{B}_{c}$ | $\bar{D}_{s}\bar{B}_{c}^{*}$ | $\bar{D}_{s}\bar{B}_{c}$
---|---|---|---|---|---|---
$sb\bar{c}\bar{c}$ | $0^{+}$ | $8300.5$ | $0.505$ |  |  | $0.588$
|  | $8538.7$ | $-0.573$ |  |  | $0.266$
| $1^{+}$ | $8324.8$ | $0.445$ | $0.394$ | $-0.532$ |
|  | $8521.9$ | $0.181$ | $0.439$ | $0.350$ |
|  | $8553.3$ | $-0.519$ | $0.263$ | $-0.106$ |
| $2^{+}$ | $8569.1$ | $0.577$ |  |  |
Table 12: The values of $k\cdot|c_{i}|^{2}$ for the $nb\bar{c}\bar{c}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | $\bar{D}^{*}\bar{B}_{c}^{*}$ | $\bar{D}^{*}\bar{B}_{c}$ | $\bar{D}\bar{B}_{c}^{*}$ | $\bar{D}\bar{B}_{c}$
---|---|---|---|---|---|---
$nb\bar{c}\bar{c}$ | $0^{+}$ | $8199.1$ | $\times$ |  |  | $137.0$
|  | $8444.8$ | $172.6$ |  |  | $75.9$
| $1^{+}$ | $8217.5$ | $\times$ | $\times$ | $51.7$ |
|  | $8430.0$ | $22.4$ | $112.9$ | $111.6$ |
|  | $8454.4$ | $144.5$ | $65.2$ | $6.1$ |
| $2^{+}$ | $8477.9$ | $213.0$ |  |  |
Table 13: The values of $k\cdot|c_{i}|^{2}$ for the $sb\bar{c}\bar{c}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | $\bar{D}_{s}^{*}\bar{B}_{c}^{*}$ | $\bar{D}_{s}^{*}\bar{B}_{c}$ | $\bar{D}_{s}\bar{B}_{c}^{*}$ | $\bar{D}_{s}\bar{B}_{c}$
---|---|---|---|---|---|---
$sb\bar{c}\bar{c}$ | $0^{+}$ | $8300.5$ | $\times$ |  |  | $143.8$
|  | $8538.7$ | $175.2$ |  |  | $68.0$
| $1^{+}$ | $8324.8$ | $\times$ | $\times$ | $66.7$ |
|  | $8521.9$ | $15.7$ | $126.8$ | $100.4$ |
|  | $8553.3$ | $155.2$ | $50.7$ | $9.9$ |
| $2^{+}$ | $8569.1$ | $206.3$ |  |  |
Table 14: The partial width ratios for the $nb\bar{c}\bar{c}$ tetraquarks. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | $\bar{D}^{*}\bar{B}_{c}^{*}$ | $\bar{D}^{*}\bar{B}_{c}$ | $\bar{D}\bar{B}_{c}^{*}$ | $\bar{D}\bar{B}_{c}$
---|---|---|---|---|---|---
$nb\bar{c}\bar{c}$ | $0^{+}$ | $8199.1$ | $\times$ |  |  | $1$
|  | $8444.8$ | $2.3$ |  |  | $1$
| $1^{+}$ | $8217.5$ | $\times$ | $\times$ | $1$ |
|  | $8430.0$ | $0.2$ | $1.01$ | $1$ |
|  | $8454.4$ | $23.8$ | $10.7$ | $1$ |
| $2^{+}$ | $8477.9$ | $1$ |  |  |
Table 15: The partial width ratios for the $sb\bar{c}\bar{c}$ tetraquarks. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | $\bar{D}_{s}^{*}\bar{B}_{c}^{*}$ | $\bar{D}_{s}^{*}\bar{B}_{c}$ | $\bar{D}_{s}\bar{B}_{c}^{*}$ | $\bar{D}_{s}\bar{B}_{c}$
---|---|---|---|---|---|---
$sb\bar{c}\bar{c}$ | $0^{+}$ | $8300.5$ | $\times$ |  |  | $1$
|  | $8538.7$ | $2.6$ |  |  | $1$
| $1^{+}$ | $8324.8$ | $\times$ | $\times$ | $1$ |
|  | $8521.9$ | $0.2$ | $1.3$ | $1$ |
|  | $8553.3$ | $15.7$ | $5.1$ | $1$ |
| $2^{+}$ | $8569.1$ | $1$ |  |  |

First we consider the $nc\bar{c}\bar{c}$ and $sc\bar{c}\bar{c}$ tetraquarks.
Their masses and eigenvectors are listed in Tables 2. For simplicity, we will
use $T(qQ\bar{Q}\bar{Q},m,J^{P})$ to denote the $qQ\bar{Q}\bar{Q}$ tetraquarks
in the following. The isospin of the tetraquarks can be easily identified with
either $q=n$ or $q=s$. Here we assume that the $\mathrm{SU}(2)$ flavor
symmetry is exact and $n=\\{u,d\\}$. We plot their relative position in Fig 1,
along with the corresponding meson-meson thresholds. From the figure, we can
easily see that the quantum number of the ground states are $J^{P}=0^{+}$ in
both cases. They are $T(nc\bar{c}\bar{c},4936.7,0^{+})$ and
$T(sc\bar{c}\bar{c},5040.1,0^{+})$, respectively. On the other hand, the
highest mass states have quantum number $J^{P}=2^{+}$. Comparing to the
thresholds, we find that all states are above the meson-meson thresholds and
can decay through $S$-wave. They may all be _broad_ states [61]. The lowest
axial-vector states
$T(nc\bar{c}\bar{c},4968.1,1^{+})$/$T(sc\bar{c}\bar{c},5069.2,1^{+})$ lie just
above the $D_{(s)}J/\psi$ threshold. Thus they might be relatively narrow
compared to other states.

Besides the masses, the eigenvectors also provide important information of the
tetraquarks. The color configurations of the tetraquarks can be of
$\ket{(q_{1}q_{2})^{6_{c}}(\bar{q}_{3}\bar{q}_{4})^{\bar{6}_{c}}}$ or
$\ket{(q_{1}q_{2})^{\bar{3}_{c}}(\bar{q}_{3}\bar{q}_{4})^{3_{c}}}$. For
simplicity, we denote them as $6_{c}\otimes\bar{6}_{c}$ and
$\bar{3}_{c}\otimes 3_{c}$. From Table 2, we find that the ground states of
the $nc\bar{c}\bar{c}$ and $sc\bar{c}\bar{c}$ tetraquarks are both dominated
by the color-sextet components. The $T(nc\bar{c}\bar{c},4936.7,0^{+})$ has
$66.7\%$ of the $6_{c}\otimes\bar{6}_{c}$ component and the
$T(sc\bar{c}\bar{c},5040.1,0^{+})$ has $66.6\%$. This is very similar to the
fully charm tetraquarks [66, 67, 68, 59]. In the one-gluon-exchange (OGE)
model, the color interactions are attractive inside
$(q_{1}q_{2})^{\bar{3}_{c}}$/$(\bar{q}_{3}\bar{q}_{4})^{3_{c}}$ and repulsive
inside $(q_{1}q_{2})^{6_{c}}$/$(\bar{q}_{3}\bar{q}_{4})^{\bar{6}_{c}}$.
However, the attractions between $(q_{1}q_{2})^{6_{c}}$ and
$(\bar{q}_{3}\bar{q}_{4})^{\bar{6}_{c}}$ is much stronger than that between
$(q_{1}q_{2})^{\bar{3}_{c}}$ and $(\bar{q}_{3}\bar{q}_{4})^{3_{c}}$. In our
cases, the two competing effects result in the lower mass of the color-sextet
configurations. More precisely,

$\displaystyle\Braket{H_{\text{C}}\left(qc\bar{c}\bar{c}\right)}$
$\displaystyle={}$ $\displaystyle-\frac{3}{4}\left\langle
m_{qc}V_{12}^{\text{C}}+m_{cc}V_{34}^{\text{C}}+m_{q\bar{c}}\left(V_{13}^{\text{C}}+V_{14}^{\text{C}}\right)\right.$
$\displaystyle\qquad\left.+m_{c\bar{c}}\left(V_{23}^{\text{C}}+V_{24}^{\text{C}}\right)\right\rangle$
$\displaystyle={}$ $\displaystyle
m_{q\bar{c}}+m_{c\bar{c}}-\frac{3}{2}\delta{m}\Braket{V_{12}^{\text{C}}+V_{34}^{\text{C}}}$
$\displaystyle={}$ $\displaystyle
m_{q\bar{c}}+m_{c\bar{c}}+\delta{m}\begin{pmatrix}-1&0\\\ 0&+2\end{pmatrix}$
(14)

where $\delta{m}=(m_{qc}+m_{cc}-m_{q\bar{c}}-m_{c\bar{c}})/4$. Inserting the
parameters in Table 1, we have

$\delta{m}\left(nc\bar{c}\bar{c}\right)=52.43~{}\text{MeV}\,,$ (15)
$\delta{m}\left(sc\bar{c}\bar{c}\right)=52.63~{}\text{MeV}\,.$ (16)

Similar to the fully charm cases, the color interaction renders the
$6_{c}\otimes\bar{6}_{c}$ configuration more stable than the
$\bar{3}_{c}\otimes 3_{c}$ one. We also find that the color interaction does
not mix the two color configurations. This is because the two antiquarks are
identical [66, 60]. However, the two color configuration will mix due to the
CM interaction

$\displaystyle\Braket{H_{\text{CM}}\left(qc\bar{c}\bar{c}\right)}$
$\displaystyle={}$
$\displaystyle\begin{pmatrix}\frac{1}{4}\left(v_{qc}+v_{cc}\right)&-\frac{\sqrt{6}}{4}\left(v_{q\bar{c}}+v_{c\bar{c}}\right)\\\
-\frac{\sqrt{6}}{4}\left(v_{q\bar{c}}+v_{c\bar{c}}\right)&\frac{1}{6}\left(v_{qc}+v_{cc}\right)-\frac{1}{3}\left(v_{q\bar{c}}+v_{c\bar{c}}\right)\\\
\end{pmatrix}$ $\displaystyle\approx{}$
$\displaystyle\frac{v_{q\bar{c}}+v_{c\bar{c}}}{2}\begin{pmatrix}\frac{1}{3}&-\sqrt{\frac{3}{2}}\\\
-\sqrt{\frac{3}{2}}&-\frac{4}{9}\end{pmatrix}$ (17)

where we have used $v_{q_{1}q_{2}}/v_{q_{1}\bar{q}_{2}}\approx 2/3$ in the
last line [52]. Here, $(v_{n\bar{c}}+v_{c\bar{c}})/2=95.57~{}\text{MeV}$ and
$(v_{s\bar{c}}+v_{c\bar{c}})/2=96.50~{}\text{MeV}$. The diagonal part of the
CM interaction favors the $\bar{3}_{c}\otimes 3_{c}$ configuration by
approximately $75~{}\text{MeV}$. However, Eq.(III.1) indicates that the color
interaction disfavor the $\bar{3}_{c}\otimes 3_{c}$ configuration by
approximately $150~{}\text{MeV}$. Thus the ground states of the scalar
$qc\bar{c}\bar{c}$ tetraquarks are dominated by the $6_{c}\otimes\bar{6}_{c}$
component, while the higher scalar states mostly consist of
$\bar{3}_{c}\otimes 3_{c}$ component.

There are three bases for the axial-vector states, namely
$\ket{(qc)^{6}_{1}(\bar{c}\bar{c})^{\bar{6}}_{0}}$,
$\ket{(qc)^{\bar{3}}_{1}(\bar{c}\bar{c})^{3}_{1}}$ and
$\ket{(qc)^{\bar{3}}_{0}(\bar{c}\bar{c})^{3}_{1}}$. In these bases, we have

$\Braket{H_{\text{C}}\left(qc\bar{c}\bar{c}\right)}=m_{q\bar{c}}+m_{c\bar{c}}+\delta{m}\begin{pmatrix}-1&0&0\\\
0&+2&0\\\ 0&0&+2\end{pmatrix}$ (18)

The color interaction splits the three bases into two energy bands. The
$6_{c}\otimes\bar{6}_{c}$ configuration is more stable than the
$\bar{3}_{c}\otimes 3_{c}$. The CM interaction further splits the two bands
into the three bands structure in Fig. 1. As a consequence, the lightest
axial-vector states of both $nc\bar{c}\bar{c}$ and $sc\bar{c}\bar{c}$
tetraquarks have more than $80\%$ of the $6_{c}\otimes\bar{6}_{c}$ component.
Combining the scalar and axial-vector cases, we can conclude that the
tetraquark spectrum is dominantly determined by the color interaction, and the
CM interaction contributes to the finer structures.

The eigenvectors can also be used to study the decay properties of the
tetraquarks. We can transform the wave functions into the
$q\bar{q}{\otimes}q\bar{q}$ configuration. The corresponding color
configuration can be either of $\ket{(q\bar{q})^{1_{c}}(q\bar{q})^{1_{c}}}$ or
$\ket{(q\bar{q})^{8_{c}}(q\bar{q})^{8_{c}}}$. The former one can easily decay
into two $S$-wave mesons in $S$ wave (the so-called “Okubo-Zweig-Iizuka-
(OZI-)superallowed” decays), while the latter one can fall apart only through
gluon exchange. Following Refs. [61, 62, 69], we only consider the “OZI-
superallowed” decays. In Tables 3–4, we transform the eigenvectors of the
$qc\bar{c}\bar{c}$ tetraquarks into the $q\bar{c}{\otimes}c\bar{c}$
configuration. For simplicity, we only present the color-singlet components,
and we rewrite the bases as a direct product of two mesons. For each decay
channel, the decay width is proportional to the square of the coefficient
$c_{i}$ of the corresponding component in the eigenvectors, and also depends
on the phase space. For two body decay through $L$-wave, the partial decay
width reads [70, 58]

$\Gamma_{i}=\gamma_{i}\alpha\frac{k^{2L+1}}{m^{2L}}{\cdot}|c_{i}|^{2},$ (19)

where $\gamma_{i}$ is a quantity determined by the decay dynamics, $\alpha$ is
an effective coupling constant, $k$ is the momentum of the final states in the
rest frame of the initial state, and $m$ is the mass of the initial state. In
this work, the $(k/m)^{2}$ is always of $\mathcal{O}(10^{-2})$ or even
smaller. Thus we will only consider the $S$-wave decays. Next we consider the
$\gamma_{i}$. Generally, $\gamma_{i}$ depends on the spatial wave functions of
the initial tetraquark and final mesons, which are different for each decay
process. In the quark model, the spatial wave functions of the ground state
scalar and axial-vector meson are the same [71, 72]. Thus for each tetraquark,
we have

$\gamma_{M_{1}M_{2}}=\gamma_{M_{1}M_{2}^{*}}=\gamma_{M_{1}^{*}M_{2}}=\gamma_{M_{1}^{*}M_{2}^{*}}$
(20)

where $M_{i}$ and $M_{i}^{*}$ denote the pseudoscalar and axial-vector mesons.
With the eigenvectors obtained, we calculate the value of $k\cdot|c_{i}|^{2}$
and the relative widths for the $qc\bar{c}\bar{c}$ decays, as shown in Tables
5–8. Their ground states can easily decay into $\bar{D}_{(s)}J/\psi$, thus may
be _broad_ states [61]. The higher scalar states,
$T(nc\bar{c}\bar{c},5185.3,0^{+})$/$T(sc\bar{c}\bar{c},5290.6,0^{+})$, can
decay to $\bar{D}_{(s)}^{*}J/\psi$ and $\bar{D}_{(s)}J/\psi$ modes through
$S$-wave, with relative decay width ratios

$\frac{\Gamma\left[T(nc\bar{c}\bar{c},5185.3,0^{+}){\to}\bar{D}^{*}J/\psi\right]}{\Gamma\left[T(nc\bar{c}\bar{c},5185.3,0^{+}){\to}\bar{D}J/\psi\right]}\sim
6.8\,,$ (21)

and

$\frac{\Gamma\left[T(sc\bar{c}\bar{c},5290.6,0^{+}){\to}\bar{D}_{s}^{*}J/\psi\right]}{\Gamma\left[T(sc\bar{c}\bar{c},5290.6,0^{+}){\to}\bar{D}_{s}J/\psi\right]}\sim
7.0\,.$ (22)

The dominant decay modes are the $\bar{D}_{(s)}^{*}J/\psi$ final states. Next
we consider the axial-vector states. The $T(nc\bar{c}\bar{c},5135.3,1^{+})$
decay dominantly to $\bar{D}^{*}\eta_{c}$ and $\bar{D}J/\psi$

$\Gamma_{\bar{D}^{*}J/\psi}:\Gamma_{\bar{D}^{*}\eta_{c}}:\Gamma_{\bar{D}J/\psi}\sim
0.004:1.3:1\,.$ (23)

And for $T(nc\bar{c}\bar{c},5154.0,1^{+})$

$\Gamma_{\bar{D}^{*}J/\psi}:\Gamma_{\bar{D}^{*}\eta_{c}}:\Gamma_{\bar{D}J/\psi}\sim
20.7:1.3:1\,.$ (24)

Its dominant decay mode is $\bar{D}^{*}J/\psi$. The decay property of the
$sc\bar{c}\bar{c}$ tetraquarks is similar the the $nc\bar{c}\bar{c}$
tetraquarks.

Replacing the charm quark by a bottom quark, we get the $nb\bar{c}\bar{c}$ and
$sb\bar{c}\bar{c}$ tetraquarks. We list their masses and eigenvectors in Table
9 and plot their relative position in Fig. 2, along with the possible decay
channels. Since the $B_{c}^{*}$ meson has not been observed, we use the
Godfrey-Isgur (GI) model’s prediction $M_{B_{c}^{*}}=6338~{}\text{MeV}$ [65]
to estimate the meson-meson thresholds. The $qb\bar{c}\bar{c}$ tetraquarks are
very similar to the $qc\bar{c}\bar{c}$ tetraquarks. The ground states are both
scalar. They are $T(nb\bar{c}\bar{c},8199.1,0^{+})$ and
$T(sb\bar{c}\bar{c},8300.5,0^{+})$, whose color configurations are
$6_{c}\otimes\bar{6}_{c}$ as well ($>80\%$). The lightest axial-vector states,
$T(nb\bar{c}\bar{c},8217.5,1^{+})$ and $T(sb\bar{c}\bar{c},8324.8,1^{+})$ are
also dominated by $6_{c}\otimes\bar{6}_{c}$ component ($>90\%$). All higher
states mainly consist of $\bar{3}_{c}\otimes 3_{c}$ component.

The decay properties of the $qb\bar{c}\bar{c}$ tetraquarks can be found from
Tables 10–15. Similar to the $qc\bar{c}\bar{c}$ tetraquarks, the two lowest
states are just above their $S$-wave decay channels, while all other states
are above all meson-meson thresholds. The higher scalar states can decay into
$\bar{D}_{(s)}^{*}\bar{B}_{c}^{*}$ and $\bar{D}_{(s)}\bar{B}_{c}$ channels
with into comparable partial widths

$\frac{\Gamma\left[T(nb\bar{c}\bar{c},8444.8,0^{+}){\to}\bar{D}^{*}\bar{B}_{c}^{*}\right]}{\Gamma\left[T(nb\bar{c}\bar{c},8444.8,0^{+}){\to}\bar{D}\bar{B}_{c}\right]}\sim
2.3\,,$ (25)

and

$\frac{\Gamma\left[T(sb\bar{c}\bar{c},8538.7,0^{+}){\to}\bar{D}_{s}^{*}\bar{B}_{c}^{*}\right]}{\Gamma\left[T(sb\bar{c}\bar{c},8538.7,0^{+}){\to}\bar{D}_{s}\bar{B}_{c}\right]}\sim
2.6\,.$ (26)

For the highest axial-vector states, we have

$\Gamma_{\bar{D}^{*}\bar{B}_{c}^{*}}:\Gamma_{\bar{D}^{*}\bar{B}_{c}}:\Gamma_{\bar{D}\bar{B}_{c}^{*}}\sim
23.8:10.7:1$ (27)

for $T(nb\bar{c}\bar{c},8454.5,1^{+})$ and

$\Gamma_{\bar{D}_{s}^{*}\bar{B}_{c}^{*}}:\Gamma_{\bar{D}_{s}^{*}\bar{B}_{c}}:\Gamma_{\bar{D}_{s}\bar{B}_{c}^{*}}\sim
15.7:5.1:1$ (28)

for $T(sb\bar{c}\bar{c},8553.3,1^{+})$. The $\bar{D}_{(s)}^{*}\bar{B}_{c}^{*}$
and $\bar{D}_{(s)}^{*}\bar{B}_{c}$ modes are more important. However, for the
second higher axial-vector states of $qb\bar{c}\bar{c}$ tetraquarks, the
partial decay widths of $\bar{D}_{(s)}^{*}\bar{B}_{c}^{*}$,
$\bar{D}_{(s)}^{*}\bar{B}_{c}$ and $\bar{D}_{(s)}\bar{B}_{c}^{*}$ modes are
comparable, though the first mode is relatively smaller.

###  III.2 The $qb\bar{b}\bar{b}$ and $qc\bar{b}\bar{b}$ systems

---
(a) $nb\bar{b}\bar{b}$ states
(b) $sb\bar{b}\bar{b}$ states
Figure 3: Mass spectra of the $nb\bar{b}\bar{b}$ and $sb\bar{b}\bar{b}$
tetraquark states. The dotted lines indicate various meson-meson thresholds.
The masses are all in units of MeV.
---
(a) $nc\bar{b}\bar{b}$ states
(b) $sc\bar{b}\bar{b}$ states
Figure 4: Mass spectra of the $nc\bar{b}\bar{b}$ and $sc\bar{b}\bar{b}$ tetraquark states. The dotted lines indicate various meson-meson thresholds. Here the predicted mass $M_{B_{c}^{*}}=6338~{}\text{MeV}$ of Godfrey et al. [65] is used. The masses are all in units of MeV. Table 16: Masses and eigenvectors of the $nb\bar{b}\bar{b}$ and $sb\bar{b}\bar{b}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | Eigenvector
---|---|---|---
$nb\bar{b}\bar{b}$ | $0^{+}$ | $14706.1$ | $\\{0.932,0.362\\}$
|  | $14850.9$ | $\\{-0.362,0.932\\}$
| $1^{+}$ | $14712.1$ | $\\{0.977,0.049,-0.208\\}$
|  | $14845.1$ | $\\{-0.097,0.969,-0.228\\}$
|  | $14851.6$ | $\\{0.191,0.243,0.951\\}$
| $2^{+}$ | $14871.8$ | $\\{1\\}$
$sb\bar{b}\bar{b}$ | $0^{+}$ | $14793.1$ | $\\{0.924,0.382\\}$
|  | $14936.2$ | $\\{-0.382,0.924\\}$
| $1^{+}$ | $14804.5$ | $\\{0.976,0.041,-0.215\\}$
|  | $14929.2$ | $\\{0.058,-0.996,0.070\\}$
|  | $14943.7$ | $\\{-0.211,-0.081,-0.974\\}$
| $2^{+}$ | $14956.6$ | $\\{1\\}$
Table 17: Masses and eigenvectors of the $nc\bar{b}\bar{b}$ and $sc\bar{b}\bar{b}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | Eigenvector
---|---|---|---
$nc\bar{b}\bar{b}$ | $0^{+}$ | $11581.7$ | $\\{0.510,0.860\\}$
|  | $11694.9$ | $\\{0.860,-0.510\\}$
| $1^{+}$ | $11582.2$ | $\\{-0.425,-0.133,0.895\\}$
|  | $11624.7$ | $\\{-0.070,-0.981,-0.179\\}$
|  | $11658.6$ | $\\{-0.902,0.139,-0.408\\}$
| $2^{+}$ | $11651.7$ | $\\{1\\}$
$sc\bar{b}\bar{b}$ | $0^{+}$ | $11675.1$ | $\\{0.545,0.838\\}$
|  | $11787.2$ | $\\{0.838,-0.545\\}$
| $1^{+}$ | $11673.3$ | $\\{-0.465,-0.096,0.880\\}$
|  | $11722.2$ | $\\{-0.089,-0.984,-0.155\\}$
|  | $11746.3$ | $\\{0.881,-0.150,0.449\\}$
| $2^{+}$ | $11750.2$ | $\\{1\\}$
Table 18: The eigenvectors of the $nb\bar{b}\bar{b}$ tetraquarks in the $n\bar{b}{\otimes}b\bar{b}$ configuration. The masses are all in units of MeV. System | $J^{P}$ | Mass | ${B}^{*}{\Upsilon}$ | ${B}^{*}{\eta_{b}}$ | ${B}{\Upsilon}$ | ${B}{\eta_{b}}$
---|---|---|---|---|---|---
$nb\bar{b}\bar{b}$ | $0^{+}$ | $14706.1$ | $0.554$ |  |  | $0.562$
|  | $14850.9$ | $-0.525$ |  |  | $0.318$
| $1^{+}$ | $14712.1$ | $0.479$ | $0.479$ | $-0.439$ |
|  | $14845.1$ | $-0.149$ | $0.422$ | $0.369$ |
|  | $14851.6$ | $0.498$ | $-0.098$ | $0.296$ |
| $2^{+}$ | $14871.8$ | $0.577$ |  |  |
Table 19: The eigenvectors of the $sb\bar{b}\bar{b}$ tetraquarks in the $s\bar{b}{\otimes}b\bar{b}$ configuration. The masses are all in units of MeV. System | $J^{P}$ | Mass | ${B}_{s}^{*}{\Upsilon}$ | ${B}_{s}^{*}{\eta_{b}}$ | ${B}_{s}{\Upsilon}$ | ${B}_{s}{\eta_{b}}$
---|---|---|---|---|---|---
$sb\bar{b}\bar{b}$ | $0^{+}$ | $14793.1$ | $0.543$ |  |  | $0.568$
|  | $14936.2$ | $-0.537$ |  |  | $0.306$
| $1^{+}$ | $14804.5$ | $0.476$ | $0.477$ | $-0.444$ |
|  | $14929.2$ | $0.062$ | $-0.403$ | $-0.410$ |
|  | $14943.7$ | $-0.520$ | $0.162$ | $-0.228$ |
| $2^{+}$ | $14956.6$ | $0.577$ |  |  |
Table 20: The values of $k\cdot|c_{i}|^{2}$ for the $nb\bar{b}\bar{b}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | ${B}^{*}{\Upsilon}$ | ${B}^{*}{\eta_{b}}$ | ${B}{\Upsilon}$ | ${B}{\eta_{b}}$
---|---|---|---|---|---|---
$nb\bar{b}\bar{b}$ | $0^{+}$ | $14706.1$ | $\times$ |  |  | $136.4$
|  | $14850.9$ | $185.2$ |  |  | $109.7$
| $1^{+}$ | $14712.1$ | $\times$ | $\times$ | $\times$ |
|  | $14845.1$ | $14.2$ | $162.0$ | $115.6$ |
|  | $14851.6$ | $167.7$ | $8.9$ | $76.4$ |
| $2^{+}$ | $14871.8$ | $257.0$ |  |  |
Table 21: The values of $k\cdot|c_{i}|^{2}$ for the $sb\bar{b}\bar{b}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | ${B}_{s}^{*}{\Upsilon}$ | ${B}_{s}^{*}{\eta_{b}}$ | ${B}_{s}{\Upsilon}$ | ${B}_{s}{\eta_{b}}$
---|---|---|---|---|---|---
$sb\bar{b}\bar{b}$ | $0^{+}$ | $14793.1$ | $\times$ |  |  | $139.5$
|  | $14936.2$ | $186.2$ |  |  | $101.6$
| $1^{+}$ | $14804.5$ | $\times$ | $\times$ | $\times$ |
|  | $14929.2$ | $2.3$ | $144.9$ | $140.7$ |
|  | $14943.7$ | $185.0$ | $24.8$ | $46.6$ |
| $2^{+}$ | $14956.5$ | $249.2$ |  |  |
Table 22: The partial width ratios for the $nb\bar{b}\bar{b}$ tetraquarks. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | ${B}^{*}{\Upsilon}$ | ${B}^{*}{\eta_{b}}$ | ${B}{\Upsilon}$ | ${B}{\eta_{b}}$
---|---|---|---|---|---|---
$nb\bar{b}\bar{b}$ | $0^{+}$ | $14706.1$ | $\times$ |  |  | $1$
|  | $14850.9$ | $1.7$ |  |  | $1$
| $1^{+}$ | $14712.1$ | $\times$ | $\times$ | $\times$ |
|  | $14845.1$ | $0.1$ | $1.4$ | $1$ |
|  | $14851.6$ | $2.2$ | $0.1$ | $1$ |
| $2^{+}$ | $14871.8$ | $1$ |  |  |
Table 23: The partial width ratios for the $sb\bar{b}\bar{b}$ tetraquarks. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | ${B}^{*}{\Upsilon}$ | ${B}^{*}{\eta_{b}}$ | ${B}{\Upsilon}$ | ${B}{\eta_{b}}$
---|---|---|---|---|---|---
$sb\bar{b}\bar{b}$ | $0^{+}$ | $14793.1$ | $\times$ |  |  | $1$
|  | $14936.2$ | $1.8$ |  |  | $1$
| $1^{+}$ | $14804.5$ | $\times$ | $\times$ | $\times$ |
|  | $14929.2$ | $0.02$ | $1.03$ | $1$ |
|  | $14943.7$ | $3.97$ | $0.53$ | $1$ |
| $2^{+}$ | $14956.5$ | $1$ |  |  |
Table 24: The eigenvectors of the $nc\bar{b}\bar{b}$ tetraquarks in the $n\bar{b}{\otimes}c\bar{b}$ configuration. The masses are all in units of MeV. System | $J^{P}$ | Mass | ${B}^{*}{B}_{c}^{*}$ | ${B}^{*}{B}_{c}$ | ${B}{B}_{c}^{*}$ | ${B}{B}_{c}$
---|---|---|---|---|---|---
$nc\bar{b}\bar{b}$ | $0^{+}$ | $11581.7$ | $0.112$ |  |  | $0.638$
|  | $11694.9$ | $0.756$ |  |  | $0.096$
| $1^{+}$ | $11582.2$ | $0.120$ | $-0.486$ | $0.378$ |
|  | $11624.7$ | $-0.114$ | $-0.378$ | $-0.424$ |
|  | $11658.6$ | $-0.688$ | $-0.194$ | $0.307$ |
| $2^{+}$ | $11651.7$ | $0.577$ |  |  |
Table 25: The eigenvectors of the $sc\bar{b}\bar{b}$ tetraquarks in the $s\bar{b}{\otimes}c\bar{b}$ configuration. The masses are all in units of MeV. System | $J^{P}$ | Mass | ${B}_{s}^{*}{B}_{c}^{*}$ | ${B}_{s}^{*}{B}_{c}$ | ${B}_{s}{B}_{c}^{*}$ | ${B}_{s}{B}_{c}$
---|---|---|---|---|---|---
$sc\bar{b}\bar{b}$ | $0^{+}$ | $11675.1$ | $0.143$ |  |  | $0.642$
|  | $11787.2$ | $0.750$ |  |  | $0.070$
| $1^{+}$ | $11673.3$ | $0.091$ | $-0.483$ | $0.404$ |
|  | $11722.2$ | $-0.114$ | $-0.393$ | $-0.410$ |
|  | $11746.3$ | $0.692$ | $0.169$ | $-0.291$ |
| $2^{+}$ | $11750.2$ | $0.577$ |  |  |
Table 26: The values of $k\cdot|c_{i}|^{2}$ for the $nc\bar{b}\bar{b}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | ${B}^{*}{B}_{c}^{*}$ | ${B}^{*}{B}_{c}$ | ${B}{B}_{c}^{*}$ | ${B}{B}_{c}$
---|---|---|---|---|---|---
$nc\bar{b}\bar{b}$ | $0^{+}$ | $11581.7$ | $\times$ |  |  | $160.9$
|  | $11694.9$ | $246.9$ |  |  | $8.4$
| $1^{+}$ | $11582.2$ | $\times$ | $\times$ | $\times$ |
|  | $11624.7$ | $\times$ | $54.2$ | $36.7$ |
|  | $11658.6$ | $\times$ | $21.9$ | $46.0$ |
| $2^{+}$ | $11651.7$ | $\times$ |  |  |
Table 27: The values of $k\cdot|c_{i}|^{2}$ for the $sc\bar{b}\bar{b}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | ${B}_{s}^{*}{B}_{c}^{*}$ | ${B}_{s}^{*}{B}_{c}$ | ${B}_{s}{B}_{c}^{*}$ | ${B}_{s}{B}_{c}$
---|---|---|---|---|---|---
$sc\bar{b}\bar{b}$ | $0^{+}$ | $11675.1$ | $\times$ |  |  | $180.6$
|  | $11787.2$ | $250.2$ |  |  | $4.5$
| $1^{+}$ | $11673.3$ | $\times$ | $\times$ | $\times$ |
|  | $11722.2$ | $\times$ | $66.5$ | $53.6$ |
|  | $11746.3$ | $\times$ | $16.3$ | $41.7$ |
| $2^{+}$ | $11750.2$ | $\times$ |  |  |
Table 28: The partial width ratios for the $nc\bar{b}\bar{b}$ tetraquarks. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | ${B}^{*}{B}_{c}^{*}$ | ${B}^{*}{B}_{c}$ | ${B}{B}_{c}^{*}$ | ${B}{B}_{c}$
---|---|---|---|---|---|---
$nc\bar{b}\bar{b}$ | $0^{+}$ | $11581.7$ | $\times$ |  |  | $1$
|  | $11694.9$ | $29.5$ |  |  | $1$
| $1^{+}$ | $11582.2$ | $\times$ | $\times$ | $\times$ |
|  | $11624.7$ | $\times$ | $1.5$ | $1$ |
|  | $11658.6$ | $\times$ | $0.5$ | $1$ |
| $2^{+}$ | $11651.7$ | $\times$ |  |  |
Table 29: The partial width ratios for the $sc\bar{b}\bar{b}$ tetraquarks. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | ${B}_{s}^{*}{B}_{c}^{*}$ | ${B}_{s}^{*}{B}_{c}$ | ${B}_{s}{B}_{c}^{*}$ | ${B}_{s}{B}_{c}$
---|---|---|---|---|---|---
$sc\bar{b}\bar{b}$ | $0^{+}$ | $11675.1$ | $\times$ |  |  | $1$
|  | $11787.2$ | $55.7$ |  |  | $1$
| $1^{+}$ | $11673.3$ | $\times$ | $\times$ | $\times$ |
|  | $11722.2$ | $\times$ | $1.2$ | $1$ |
|  | $11746.3$ | $\times$ | $0.4$ | $1$ |
| $2^{+}$ | $11750.2$ | $\times$ |  |  |

Next we consider the $qb\bar{b}\bar{b}$ and $qc\bar{b}\bar{b}$ tetraquarks.
Their masses and eigenvectors are listed in Tables 16–17. By replacing the
three charm quark/antiquarks of the $qc\bar{c}\bar{c}$ tetraquarks into bottom
quark/antiquarks, we can obtain the $qb\bar{b}\bar{b}$ tetraquarks. Thus we
expect that the two systems share some similar properties. Specifically, the
quantum numbers of the lowest and highest $qb\bar{b}\bar{b}$ tetraquarks are
$J^{P}=0^{+}$ and $J^{P}=2^{+}$ respectively. Their ground states are
$T(nb\bar{b}\bar{b},14706.1,0^{+})$ and $T(sb\bar{b}\bar{b},14793.1,0^{+})$,
whose dominant color configurations are $6_{c}\otimes\bar{6}_{c}$ ($>85\%$).
The color triplet component is mostly in the higher scalar states. The axial-
vector states are similar. The $6_{c}\otimes\bar{6}_{c}$ component is mostly
in the lightest states ($>90\%$), and the $\bar{3}_{c}\otimes 3_{c}$
components are mostly in the higher states. Just like the $qc\bar{c}\bar{c}$
cases, the underlining reason is the color interaction

$\Braket{H_{\text{C}}\left(qb\bar{b}\bar{b}\right)}=m_{q\bar{b}}+m_{b\bar{b}}-\frac{3}{2}\delta{m}^{\prime}\Braket{V_{12}^{\text{C}}+V_{34}^{\text{C}}}$
(29)

where

$\delta{m}^{\prime}=\frac{m_{qb}+m_{bb}-m_{q\bar{b}}-m_{b\bar{b}}}{4}\sim
45~{}\text{MeV}\,.$ (30)

Note that the $\Braket{V_{12}^{\text{C}}+V_{34}^{\text{C}}}$ is diagonal in
the $6_{c}\otimes\bar{6}_{c}$ and $\bar{3}_{c}\otimes 3_{c}$ color bases, with
matrix elements $2/3$ and $-4/3$ respectively. Thus the
$6_{c}\otimes\bar{6}_{c}$ configuration is favored.

The $qc\bar{b}\bar{b}$ tetraquarks turn out to be quite different from the
$qb\bar{b}\bar{b}$ tetraquarks. The quantum number of the highest states
becomes $J^{P}=0^{+}$. For the $nc\bar{b}\bar{b}$ tetraquarks, the lightest
axial-vector state lies only $0.5~{}\text{MeV}$ above the lightest scalar
state. And the ground state of the $sc\bar{b}\bar{b}$ tetraquarks has quantum
number $J^{P}=1^{+}$. Moreover, the dominant color configuration of the lower
mass states becomes $\bar{3}_{c}\otimes 3_{c}$. For example, the
$T(nc\bar{b}\bar{b},11581.7,0^{+})$ has $74\%$ of $\bar{3}_{c}\otimes 3_{c}$
component, and the $T(sc\bar{b}\bar{b},11675.1,0^{+})$ has $70\%$. We again
resort to the color interaction

$\Braket{H_{\text{C}}\left(qc\bar{b}\bar{b}\right)}=m_{q\bar{b}}+m_{c\bar{b}}-\frac{3}{2}\delta{m}^{\prime\prime}\Braket{V_{12}^{\text{C}}+V_{34}^{\text{C}}}$
(31)

where

$\displaystyle\delta{m}^{\prime\prime}=\frac{m_{qc}+m_{bb}-m_{q\bar{b}}-m_{c\bar{b}}}{4}\,.$
(32)

Here $\delta{m}^{\prime\prime}=-6.5~{}\text{MeV}$ for $q=n$, and
$\delta{m}^{\prime\prime}=-3.1~{}\text{MeV}$ for $q=s$. The small negative
values indicate that the $\bar{3}_{c}\otimes 3_{c}$ configuration has slightly
lower mass. The CM interaction

$\displaystyle\Braket{H_{\text{CM}}\left(qc\bar{b}\bar{b}\right)}$
$\displaystyle={}$
$\displaystyle\begin{pmatrix}\frac{1}{4}\left(v_{qc}+v_{bb}\right)&-\frac{\sqrt{6}}{4}\left(v_{q\bar{b}}+v_{c\bar{b}}\right)\\\
-\frac{\sqrt{6}}{4}\left(v_{q\bar{b}}+v_{c\bar{b}}\right)&\frac{1}{6}\left(v_{qc}+v_{bb}\right)-\frac{1}{3}\left(v_{q\bar{b}}+v_{c\bar{b}}\right)\\\
\end{pmatrix}$ (33)

also favors the $\bar{3}_{c}\otimes 3_{c}$ configuration. Thus for both the
scalar and the axial-vector states, the lower mass states are dominated by the
$\bar{3}_{c}\otimes 3_{c}$ components, while the higher mass states have more
$6_{c}\otimes\bar{6}_{c}$ components.

Before concluding this section, we would like to compare the present results
with that of Refs. [59, 60]. It is interesting to note that the
$qb\bar{b}\bar{b}$ tetraquarks (as well as the $qc\bar{c}\bar{c}$ and
$qb\bar{c}\bar{c}$ tetraquarks) are very similar to the fully heavy
$cb\bar{b}\bar{b}$ tetraquarks, while the $qc\bar{b}\bar{b}$ tetraquarks are
more like the doubly heavy tetraquarks. This is quite natural since that the
triply heavy tetraquarks are intermediate states betwwen the doubly and fully
heavy tetraquarks. The reason is $m_{q}{\ll}m_{c}{\ll}m_{b}$. A detailed
dynamical study of the dependence of the spectrum and wave function with
respect to the quark masses would be very important for decoding the nature of
tetraquarks [73, 74].

Next we consider their decay properties. We transform the eigenvectors of the
$qb\bar{b}\bar{b}$ ($qc\bar{b}\bar{b}$) tetraquarks into the
$q\bar{b}{\otimes}b\bar{b}$ ($q\bar{b}{\otimes}c\bar{b}$) configuration, and
calculate the values of $k\cdot|c_{i}|^{2}$ and partial width ratios of all
possible decay channels. The corresponding results are listed in Tables 18–29.
We find six states which cannot decay into two mesons through $S$-wave. They
are $T(nb\bar{b}\bar{b},14712.1,1^{+})$, $T(sb\bar{b}\bar{b},14804.5,1^{+})$,
$T(nc\bar{b}\bar{b},11582.2,1^{+})$, $T(nc\bar{b}\bar{b},11651.7,2^{+})$,
$T(sc\bar{b}\bar{b},11673.3,1^{+})$ and $T(sc\bar{b}\bar{b},11750.2,2^{+})$.
Because of the conservation of the angular momentum and parity, their decays
into the thresholds below them must be of higher wave ($D$-wave for example),
which are highly suppressed. Thus we expect them to be narrow.

However, we need to bear in mind that our model is an oversimplified one. The
dynamical effects are not included explicitly, but rather concealed in the
interaction strengths $\\{m_{ij},v_{ij}\\}$. In the present work, we fit them
from the conventional mesons and baryons. But they may be altered in the
tetraquarks because of their wave function dependence. Nonetheless, even if
the masses of these states are pushed upward above their $S$-wave channels,
their phase spaces are still relatively small. Thus their decay widths are
still smaller compared to those of other states. A dynamical calculation may
help settle this issue. We hope the future experiments can search for these
states.

###  III.3 The $qc\bar{c}\bar{b}$ and $qb\bar{c}\bar{b}$ systems

---
(a) $nc\bar{c}\bar{b}$ states
(b) $sc\bar{c}\bar{b}$ states
Figure 5: Mass spectra of the $nc\bar{c}\bar{b}$ and $sc\bar{c}\bar{b}$
tetraquark states. The dotted lines indicate various meson-meson thresholds.
The scattering states are marked with a dagger ($\dagger$), along with the
proportion of their dominant components. Here the predicted mass
$M_{B_{c}^{*}}=6338~{}\text{MeV}$ of Godfrey et al. [65] is used. The masses
are all in units of MeV.
---
(a) $nb\bar{c}\bar{b}$ states
(b) $sb\bar{c}\bar{b}$ states
Figure 6: Mass spectra of the $nb\bar{c}\bar{b}$ and $sb\bar{c}\bar{b}$ tetraquark states. The dotted lines indicate various meson-meson thresholds. The scattering states are marked with a dagger ($\dagger$), along with the proportion of their dominant components. Here the predicted mass $M_{B_{c}^{*}}=6338~{}\text{MeV}$ of Godfrey et al. [65] is used. The masses are all in units of MeV. Table 30: Masses and eigenvectors of the $nc\bar{c}\bar{b}$ and $sc\bar{c}\bar{b}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | Eigenvector | Scattering state
---|---|---|---|---
$nc\bar{c}\bar{b}$ | $0^{+}$ | $8123.2$ | $\\{0.864,0.246,0.273,0.343\\}$ |  $\bar{D}{B}_{c}$ ($89.7\%$)
|  | $8281.2$ | $\\{-0.394,0.812,0.425,0.073\\}$ |
|  | $8388.0$ | $\\{-0.199,0.091,-0.502,0.837\\}$ |
|  | $8466.4$ | $\\{-0.240,-0.522,0.702,0.421\\}$ |
| $1^{+}$ | $8190.4$ | $\\{0.770,-0.430,0.259,0.229,-0.169,0.271\\}$ |  $\bar{D}{B}_{c}^{*}$ ($89.6\%$)
|  | $8268.0$ | $\\{-0.562,-0.380,0.609,-0.051,-0.317,0.256\\}$ |  $\bar{D}^{*}{B}_{c}$ ($83.6\%$)
|  | $8312.6$ | $\\{0.142,0.732,0.623,0.235,0.003,-0.033\\}$ |
|  | $8398.9$ | $\\{0.025,0.123,0.029,-0.379,0.530,0.747\\}$ |
|  | $8435.7$ | $\\{-0.264,-0.081,-0.159,0.861,0.315,0.242\\}$ |
|  | $8457.3$ | $\\{-0.030,0.336,-0.385,0.070,-0.700,0.493\\}$ |
| $2^{+}$ | $8335.1$ | $\\{0.950,0.313\\}$ |  $\bar{D}^{*}{B}_{c}^{*}$ ($91.4\%$)
|  | $8468.9$ | $\\{-0.313,0.950\\}$ |
$sc\bar{c}\bar{b}$ | $0^{+}$ | $8220.1$ | $\\{0.880,0.223,0.243,0.343\\}$ |  $\bar{D}_{s}{B}_{c}$ ($87.2\%$)
|  | $8375.2$ | $\\{0.351,-0.817,-0.456,-0.046\\}$ |
|  | $8488.1$ | $\\{0.238,-0.092,0.437,-0.862\\}$ |
|  | $8561.0$ | $\\{-0.215,-0.524,0.736,0.370\\}$ |
| $1^{+}$ | $8287.8$ | $\\{0.789,-0.431,0.240,0.204,-0.152,0.264\\}$ |  $\bar{D}_{s}{B}_{c}^{*}$ ($87.0\%$)
|  | $8368.4$ | $\\{-0.545,-0.397,0.623,-0.013,-0.309,0.249\\}$ |  $\bar{D}_{s}^{*}{B}_{c}$ ($80.5\%$)
|  | $8407.5$ | $\\{-0.168,-0.727,-0.622,-0.220,0.049,0.079\\}$ |
|  | $8498.8$ | $\\{0.006,0.183,0.043,-0.339,0.487,0.783\\}$ |
|  | $8531.2$ | $\\{0.227,0.033,0.164,-0.890,-0.266,-0.239\\}$ |
|  | $8551.1$ | $\\{-0.023,0.307,-0.373,0.045,-0.756,0.439\\}$ |
| $2^{+}$ | $8434.2$ | $\\{0.966,0.259\\}$ |  $\bar{D}_{s}^{*}{B}_{c}^{*}$ ($88.0\%$)
|  | $8566.3$ | $\\{-0.259,0.966\\}$ |
Table 31: Masses and eigenvectors of the $nb\bar{c}\bar{b}$ and $sb\bar{c}\bar{b}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | Eigenvector | Scattering state
---|---|---|---|---
$nb\bar{c}\bar{b}$ | $0^{+}$ | $11253.4$ | $\\{0.816,0.365,0.364,0.261\\}$ |  $\bar{D}{\eta}_{b}$ ($96.7\%$)
|  | $11438.6$ | $\\{0.463,-0.835,0.011,-0.296\\}$ |  $\bar{D}^{*}{\Upsilon}$ ($86.7\%$)
|  | $11673.1$ | $\\{0.160,-0.165,-0.688,0.688\\}$ |
|  | $11748.7$ | $\\{0.306,0.377,-0.628,-0.609\\}$ |
| $1^{+}$ | $11314.8$ | $\\{0.702,-0.408,0.397,0.298,-0.215,0.219\\}$ |  $\bar{D}{\Upsilon}$ ($96.2\%$)
|  | $11392.4$ | $\\{-0.609,-0.450,0.515,-0.218,-0.240,0.239\\}$ |  $\bar{D}^{*}{\eta}_{b}$ ($94.6\%$)
|  | $11446.6$ | $\\{-0.024,-0.688,-0.655,-0.028,-0.231,-0.205\\}$ |  $\bar{D}^{*}{\Upsilon}$ ($91.0\%$)
|  | $11697.5$ | $\\{0.080,-0.232,-0.184,-0.236,0.675,0.628\\}$ |
|  | $11719.1$ | $\\{0.353,0.101,0.028,-0.889,-0.267,-0.047\\}$ |
|  | $11744.9$ | $\\{-0.070,0.305,-0.337,0.130,-0.562,0.675\\}$ |
| $2^{+}$ | $11453.9$ | $\\{0.926,0.378\\}$ |  $\bar{D}^{*}{\Upsilon}$ ($94.9\%$)
|  | $11747.6$ | $\\{-0.378,0.926\\}$ |
$sb\bar{c}\bar{b}$ | $0^{+}$ | $11355.0$ | $\\{0.816,0.366,0.364,0.262\\}$ |  $\bar{D}_{s}{\eta}_{b}$ ($96.7\%$)
|  | $11539.4$ | $\\{0.468,-0.840,-0.008,-0.275\\}$ |  $\bar{D}_{s}^{*}{\Upsilon}$ ($84.7\%$)
|  | $11760.7$ | $\\{0.159,-0.124,-0.714,0.670\\}$ |
|  | $11836.7$ | $\\{0.300,0.381,-0.598,-0.638\\}$ |
| $1^{+}$ | $11416.4$ | $\\{0.698,-0.405,0.407,0.296,-0.219,0.218\\}$ |  $\bar{D}_{s}{\Upsilon}$ ($96.4\%$)
|  | $11496.1$ | $\\{-0.615,-0.437,0.522,-0.216,-0.238,0.235\\}$ |  $\bar{D}_{s}^{*}{\eta}_{b}$ ($94.3\%$)
|  | $11549.1$ | $\\{-0.029,-0.698,-0.651,-0.027,-0.229,-0.185\\}$ |  $\bar{D}_{s}^{*}{\Upsilon}$ ($89.9\%$)
|  | $11785.1$ | $\\{0.071,-0.248,-0.151,-0.206,0.717,0.595\\}$ |
|  | $11802.8$ | $\\{-0.354,-0.087,-0.019,0.899,0.239,0.024\\}$ |
|  | $11833.6$ | $\\{-0.066,0.296,-0.339,0.115,-0.520,0.713\\}$ |
| $2^{+}$ | $11558.4$ | $\\{0.926,0.378\\}$ |  $\bar{D}_{s}^{*}{\Upsilon}$ ($94.9\%$)
|  | $11832.7$ | $\\{-0.378,0.926\\}$ |
Table 32: The eigenvectors of the $nc\bar{c}\bar{b}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | $n\bar{c}{\otimes}c\bar{b}$ | $n\bar{b}{\otimes}c\bar{c}$
---|---|---|---|---
$\bar{D}^{*}{B}_{c}^{*}$ | $\bar{D}^{*}{B}_{c}$ | $\bar{D}{B}_{c}^{*}$ | $\bar{D}{B}_{c}$ | ${B}^{*}{J/\psi}$ | ${B}^{*}\eta_{c}$ | ${B}{J/\psi}$ | ${B}\eta_{c}$
$nc\bar{c}\bar{b}$ | $0^{+}$ | $8123.2$ | $-0.086$ |  |  | $0.947$ | $\\-0.277$ |  |  | $0.473$
|  | $8281.2$ | $0.649$ |  |  | $0.286$ | $-0.254$ |  |  | $-0.801$
|  | $8388.0$ | $0.709$ |  |  | $-0.113$ | $0.291$ |  |  | $0.314$
|  | $8466.4$ | $-0.263$ |  |  | $0.090$ | $0.880$ |  |  | $-0.187$
| $1^{+}$ | $8190.4$ | $-0.057$ | $0.129$ | $0.947$ |  | $0.219$ | $0.451$ | $0.252$ |
|  | $8268.0$ | $0.108$ | $-0.915$ | $0.224$ |  | $0.337$ | $-0.415$ | $-0.192$ |
|  | $8312.6$ | $0.770$ | $0.233$ | $0.123$ |  | $-0.048$ | $-0.576$ | $0.547$ |
|  | $8398.9$ | $0.610$ | $-0.165$ | $-0.116$ |  | $-0.143$ | $0.476$ | $-0.137$ |
|  | $8435.7$ | $0.089$ | $0.252$ | $0.146$ |  | $-0.015$ | $-0.245$ | $-0.762$ |
|  | $8457.3$ | $-0.113$ | $-0.039$ | $0.061$ |  | $-0.903$ | $-0.086$ | $-0.006$ |
| $2^{+}$ | $8335.1$ | $0.956$ |  |  |  | $0.595$ |  |  |
|  | $8468.9$ | $0.293$ |  |  |  | $-0.804$ |  |  |
Table 33: The eigenvectors of the $sc\bar{c}\bar{b}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | $s\bar{c}{\otimes}c\bar{b}$ | $s\bar{b}{\otimes}c\bar{c}$
---|---|---|---|---
$\bar{D}_{s}^{*}{B}_{c}^{*}$ | $\bar{D}_{s}^{*}{B}_{c}$ | $\bar{D}_{s}{B}_{c}^{*}$ | $\bar{D}_{s}{B}_{c}$ | ${B}_{s}^{*}{J/\psi}$ | ${B}_{s}^{*}\eta_{c}$ | ${B}_{s}{J/\psi}$ | ${B}_{s}\eta_{c}$
$sc\bar{c}\bar{b}$ | $0^{+}$ | $8220.1$ | $-0.100$ |  |  | $0.934$ | $-0.276$ |  |  | $0.508$
|  | $8375.2$ | $-0.612$ |  |  | $-0.326$ | $0.279$ |  |  | $0.796$
|  | $8488.1$ | $-0.720$ |  |  | $0.100$ | $-0.337$ |  |  | $-0.261$
|  | $8561.0$ | $-0.311$ |  |  | $0.109$ | $0.856$ |  |  | $-0.199$
| $1^{+}$ | $8287.8$ | $-0.064$ | $0.145$ | $0.933$ |  | $0.217$ | $0.483$ | $0.262$ |
|  | $8368.4$ | $0.106$ | $-0.897$ | $0.257$ |  | $0.361$ | $-0.419$ | $-0.200$ |
|  | $8407.5$ | $-0.726$ | $-0.238$ | $-0.135$ |  | $0.049$ | $0.581$ | $-0.595$ |
|  | $8498.8$ | $0.649$ | $-0.163$ | $-0.107$ |  | $-0.201$ | $0.416$ | $-0.133$ |
|  | $8531.2$ | $-0.092$ | $-0.294$ | $-0.171$ |  | $0.065$ | $0.268$ | $0.721$ |
|  | $8551.1$ | $-0.168$ | $-0.062$ | $0.072$ |  | $-0.880$ | $-0.096$ | $0.033$ |
| $2^{+}$ | $8434.2$ | $0.938$ |  |  |  | $0.639$ |  |  |
|  | $8566.3$ | $0.347$ |  |  |  | $-0.769$ |  |  |
Table 34: The eigenvectors of the $nb\bar{c}\bar{b}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | $n\bar{c}{\otimes}b\bar{b}$ | $n\bar{b}{\otimes}b\bar{c}$
---|---|---|---|---
$\bar{D}^{*}{\Upsilon}$ | $\bar{D}^{*}{\eta}_{b}$ | $\bar{D}{\Upsilon}$ | $\bar{D}{\eta}_{b}$ | ${B}^{*}\bar{B}_{c}^{*}$ | ${B}^{*}\bar{B}_{c}$ | ${B}\bar{B}_{c}^{*}$ | ${B}\bar{B}_{c}$
$nb\bar{c}\bar{b}$ | $0^{+}$ | $11253.4$ | $-0.050$ |  |  | $0.983$ | $-0.356$ |  |  | $0.322$
|  | $11438.6$ | $-0.931$ |  |  | $-0.094$ | $0.257$ |  |  | $0.577$
|  | $11673.1$ | $0.361$ |  |  | $-0.099$ | $0.197$ |  |  | $0.723$
|  | $11748.7$ | $0.019$ |  |  | $-0.119$ | $-0.877$ |  |  | $0.201$
| $1^{+}$ | $11314.8$ | $-0.005$ | $0.072$ | $0.981$ |  | $0.288$ | $0.289$ | $0.278$ |
|  | $11392.4$ | $0.037$ | $-0.972$ | $0.091$ |  | $0.361$ | $-0.289$ | $-0.236$ |
|  | $11446.6$ | $-0.954$ | $-0.047$ | $-0.005$ |  | $0.008$ | $0.420$ | $-0.425$ |
|  | $11697.5$ | $0.292$ | $-0.056$ | $-0.044$ |  | $0.047$ | $0.689$ | $-0.403$ |
|  | $11719.1$ | $-0.054$ | $-0.192$ | $-0.126$ |  | $-0.133$ | $0.424$ | $0.710$ |
|  | $11744.9$ | $0.028$ | $-0.083$ | $0.108$ |  | $-0.876$ | $-0.048$ | $-0.139$ |
| $2^{+}$ | $11453.9$ | $0.974$ |  |  |  | $0.538$ |  |  |
|  | $11747.6$ | $0.226$ |  |  |  | $-0.843$ |  |  |
Table 35: The eigenvectors of the $sb\bar{c}\bar{b}$ tetraquarks. The masses are all in units of MeV. System | $J^{P}$ | Mass | $s\bar{c}{\otimes}b\bar{b}$ | $s\bar{b}{\otimes}b\bar{c}$
---|---|---|---|---
$\bar{D}_{s}^{*}{\Upsilon}$ | $\bar{D}_{s}^{*}{\eta}_{b}$ | $\bar{D}_{s}{\Upsilon}$ | $\bar{D}_{s}{\eta}_{b}$ | ${B}_{s}^{*}\bar{B}_{c}^{*}$ | ${B}_{s}^{*}\bar{B}_{c}$ | ${B}_{s}\bar{B}_{c}^{*}$ | ${B}_{s}\bar{B}_{c}$
$sb\bar{c}\bar{b}$ | $0^{+}$ | $11355.0$ | $-0.048$ |  |  | $0.984$ | $-0.356$ |  |  | $0.321$
|  | $11539.4$ | $-0.920$ |  |  | $-0.095$ | $0.263$ |  |  | $0.598$
|  | $11760.7$ | $0.389$ |  |  | $-0.102$ | $0.152$ |  |  | $0.714$
|  | $11836.7$ | $0.001$ |  |  | $-0.115$ | $-0.884$ |  |  | $0.171$
| $1^{+}$ | $11416.4$ | $0.001$ | $0.066$ | $0.982$ |  | $0.290$ | $0.281$ | $0.283$ |
|  | $11496.1$ | $0.048$ | $-0.971$ | $0.085$ |  | $0.361$ | $-0.302$ | $-0.231$ |
|  | $11549.1$ | $-0.948$ | $-0.060$ | $0.005$ |  | $0.009$ | $0.426$ | $-0.437$ |
|  | $11785.1$ | $0.306$ | $-0.048$ | $-0.039$ |  | $0.106$ | $0.666$ | $-0.417$ |
|  | $11802.8$ | $0.046$ | $0.197$ | $0.129$ |  | $0.127$ | $-0.452$ | $-0.691$ |
|  | $11833.6$ | $0.054$ | $-0.088$ | $0.106$ |  | $-0.871$ | $-0.012$ | $-0.158$ |
| $2^{+}$ | $11558.4$ | $0.974$ |  |  |  | $0.538$ |  |  |
|  | $11832.7$ | $0.226$ |  |  |  | $-0.843$ |  |  |
Table 36: The values of $k\cdot|c_{i}|^{2}$ for the $nc\bar{c}\bar{b}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | $n\bar{c}{\otimes}c\bar{b}$ | $n\bar{b}{\otimes}c\bar{c}$
---|---|---|---|---
$\bar{D}^{*}{B}_{c}^{*}$ | $\bar{D}^{*}{B}_{c}$ | $\bar{D}{B}_{c}^{*}$ | $\bar{D}{B}_{c}$ | ${B}^{*}{J/\psi}$ | ${B}^{*}\eta_{c}$ | ${B}{J/\psi}$ | ${B}\eta_{c}$
$nc\bar{c}\bar{b}$ | $0^{+}$ | $8123.2$ | $\times$ |  |  | $\times$ | $\times$ |  |  | $\times$
|  | $8281.2$ | $\times$ |  |  | $52.3$ | $\times$ |  |  | $170.0$
|  | $8388.0$ | $179.3$ |  |  | $10.9$ | $\times$ |  |  | $68.6$
|  | $8466.4$ | $42.3$ |  |  | $8.0$ | $325.1$ |  |  | $31.0$
| $1^{+}$ | $8190.4$ | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ |
|  | $8268.0$ | $\times$ | $\times$ | $21.5$ |  | $\times$ | $\times$ | $\times$ |
|  | $8312.6$ | $\times$ | $16.1$ | $8.5$ |  | $\times$ | $43.5$ | $\times$ |
|  | $8398.9$ | $149.1$ | $16.2$ | $10.3$ |  | $\times$ | $133.8$ | $5.6$ |
|  | $8435.7$ | $4.1$ | $43.7$ | $17.8$ |  | $0.1$ | $42.1$ | $280.3$ |
|  | $8457.3$ | $7.5$ | $1.1$ | $3.2$ |  | $305.7$ | $5.6$ | $0.02$ |
| $2^{+}$ | $8335.1$ | $\times$ |  |  |  | $\times$ |  |  |
|  | $8468.9$ | $52.8$ |  |  |  | $278.7$ |  |  |
Table 37: The values of $k\cdot|c_{i}|^{2}$ for the $sc\bar{c}\bar{b}$ tetraquarks in the $s\bar{c}{\otimes}c\bar{b}$ configuration (in unit of MeV). System | $J^{P}$ | Mass | $s\bar{c}{\otimes}c\bar{b}$ | $s\bar{b}{\otimes}c\bar{c}$
---|---|---|---|---
$\bar{D}_{s}^{*}{B}_{c}^{*}$ | $\bar{D}_{s}^{*}{B}_{c}$ | $\bar{D}_{s}{B}_{c}^{*}$ | $\bar{D}_{s}{B}_{c}$ | ${B}_{s}^{*}{J/\psi}$ | ${B}_{s}^{*}\eta_{c}$ | ${B}_{s}{J/\psi}$ | ${B}_{s}\eta_{c}$
$sc\bar{c}\bar{b}$ | $0^{+}$ | $8220.1$ | $\times$ |  |  | $\times$ | $\times$ |  |  | $\times$
|  | $8375.2$ | $\times$ |  |  | $67.5$ | $\times$ |  |  | $196.4$
|  | $8488.1$ | $180.3$ |  |  | $8.8$ | $\times$ |  |  | $49.8$
|  | $8561.0$ | $57.6$ |  |  | $11.9$ | $321.4$ |  |  | $35.9$
| $1^{+}$ | $8287.8$ | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ |
|  | $8368.4$ | $\times$ | $\times$ | $28.7$ |  | $\times$ | $\times$ | $\times$ |
|  | $8407.5$ | $\times$ | $14.4$ | $10.2$ |  | $\times$ | $61.9$ | $\times$ |
|  | $8498.8$ | $165.9$ | $16.0$ | $8.8$ |  | $\times$ | $107.8$ | $6.5$ |
|  | $8531.2$ | $4.3$ | $58.9$ | $24.5$ |  | $1.2$ | $51.7$ | $268.0$ |
|  | $8551.1$ | $16.0$ | $2.8$ | $4.6$ |  | $303.7$ | $7.1$ | $0.7$ |
| $2^{+}$ | $8434.2$ | $\times$ |  |  |  | $\times$ |  |  |
|  | $8566.3$ | $73.5$ |  |  |  | $273.3$ |  |  |
Table 38: The partial width ratios for the $nc\bar{c}\bar{b}$ tetraquarks decay into $n\bar{c}{\otimes}c\bar{b}$ modes. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | $\bar{D}^{*}{B}_{c}^{*}$ | $\bar{D}^{*}{B}_{c}$ | $\bar{D}{B}_{c}^{*}$ | $\bar{D}{B}_{c}$
---|---|---|---|---|---|---
$nc\bar{c}\bar{b}$ | $0^{+}$ | $8123.2$ | $\times$ |  |  | $\times$
|  | $8281.2$ | $\times$ |  |  | $1$
|  | $8388.0$ | $16.4$ |  |  | $1$
|  | $8466.4$ | $5.3$ |  |  | $1$
| $1^{+}$ | $8190.4$ | $\times$ | $\times$ | $\times$ |
|  | $8268.0$ | $\times$ | $\times$ | $1$ |
|  | $8312.6$ | $\times$ | $1.9$ | $1$ |
|  | $8398.9$ | $14.5$ | $1.6$ | $1$ |
|  | $8435.7$ | $0.2$ | $2.5$ | $1$ |
|  | $8457.3$ | $2.3$ | $0.3$ | $1$ |
| $2^{+}$ | $8335.1$ | $\times$ |  |  |
|  | $8468.9$ | $1$ |  |  |
Table 39: The partial width ratios for the $nc\bar{c}\bar{b}$ tetraquarks decay into $n\bar{b}{\otimes}c\bar{c}$ modes. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | ${B}^{*}{J/\psi}$ | ${B}^{*}\eta_{c}$ | ${B}{J/\psi}$ | ${B}\eta_{c}$
---|---|---|---|---|---|---
$nc\bar{c}\bar{b}$ | $0^{+}$ | $8123.2$ | $\times$ |  |  | $\times$
|  | $8281.2$ | $\times$ |  |  | $1$
|  | $8388.0$ | $\times$ |  |  | $1$
|  | $8466.4$ | $10.5$ |  |  | $1$
| $1^{+}$ | $8190.4$ | $\times$ | $\times$ | $\times$ |
|  | $8268.0$ | $\times$ | $\times$ | $\times$ |
|  | $8312.6$ | $\times$ | $1$ | $\times$ |
|  | $8398.9$ | $\times$ | $1$ | $0.04$ |
|  | $8435.7$ | $0.001$ | $1$ | $6.7$ |
|  | $8457.3$ | $54.3$ | $1$ | $0.003$ |
| $2^{+}$ | $8335.1$ | $\times$ |  |  |
|  | $8468.9$ | $1$ |  |  |
Table 40: The partial width ratios for the $sc\bar{c}\bar{b}$ tetraquarks decay into $s\bar{c}{\otimes}c\bar{b}$ modes. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | $\bar{D}_{s}^{*}{B}_{c}^{*}$ | $\bar{D}_{s}^{*}{B}_{c}$ | $\bar{D}_{s}{B}_{c}^{*}$ | $\bar{D}_{s}{B}_{c}$
---|---|---|---|---|---|---
$sc\bar{c}\bar{b}$ | $0^{+}$ | $8220.1$ | $\times$ |  |  | $\times$
|  | $8375.2$ | $\times$ |  |  | $1$
|  | $8488.1$ | $20.5$ |  |  | $1$
|  | $8561.0$ | $4.8$ |  |  | $1$
| $1^{+}$ | $8287.8$ | $\times$ | $\times$ | $\times$ |
|  | $8368.4$ | $\times$ | $\times$ | $1$ |
|  | $8407.5$ | $\times$ | $1.4$ | $1$ |
|  | $8498.8$ | $18.9$ | $1.8$ | $1$ |
|  | $8531.2$ | $0.2$ | $2.4$ | $1$ |
|  | $8551.1$ | $3.5$ | $0.6$ | $1$ |
| $2^{+}$ | $8434.2$ | $\times$ |  |  |
|  | $8566.3$ | $1$ |  |  |
Table 41: The partial width ratios for the $sc\bar{c}\bar{b}$ tetraquarks decay into $s\bar{b}{\otimes}c\bar{c}$ modes. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | ${B}_{s}^{*}{J/\psi}$ | ${B}_{s}^{*}\eta_{c}$ | ${B}_{s}{J/\psi}$ | ${B}_{s}\eta_{c}$
---|---|---|---|---|---|---
$sc\bar{c}\bar{b}$ | $0^{+}$ | $8220.1$ | $\times$ |  |  | $\times$
|  | $8375.2$ | $\times$ |  |  | $1$
|  | $8488.1$ | $\times$ |  |  | $1$
|  | $8561.0$ | $8.9$ |  |  | $1$
| $1^{+}$ | $8287.8$ | $\times$ | $\times$ | $\times$ |
|  | $8368.4$ | $\times$ | $\times$ | $\times$ |
|  | $8407.5$ | $\times$ | $1$ | $\times$ |
|  | $8498.8$ | $\times$ | $1$ | $0.1$ |
|  | $8531.2$ | $0.02$ | $1$ | $5.2$ |
|  | $8551.1$ | $42.9$ | $1$ | $0.1$ |
| $2^{+}$ | $8434.2$ | $\times$ |  |  |
|  | $8566.3$ | $1$ |  |  |
Table 42: The values of $k\cdot|c_{i}|^{2}$ for the $nb\bar{c}\bar{b}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | $n\bar{c}{\otimes}b\bar{b}$ | $n\bar{b}{\otimes}b\bar{c}$
---|---|---|---|---
$\bar{D}^{*}{\Upsilon}$ | $\bar{D}^{*}{\eta}_{b}$ | $\bar{D}{\Upsilon}$ | $\bar{D}{\eta}_{b}$ | ${B}^{*}\bar{B}_{c}^{*}$ | ${B}^{*}\bar{B}_{c}$ | ${B}\bar{B}_{c}^{*}$ | ${B}\bar{B}_{c}$
$nb\bar{c}\bar{b}$ | $0^{+}$ | $11253.4$ | $\times$ |  |  | $\times$ | $\times$ |  |  | $\times$
|  | $11438.6$ | $\times$ |  |  | $6.5$ | $\times$ |  |  | $\times$
|  | $11673.1$ | $109.1$ |  |  | $11.5$ | $9.5$ |  |  | $432.4$
|  | $11748.7$ | $0.3$ |  |  | $18.2$ | $543.5$ |  |  | $42.7$
| $1^{+}$ | $11314.8$ | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ |
|  | $11392.4$ | $\times$ | $\times$ | $3.8$ |  | $\times$ | $\times$ | $\times$ |
|  | $11446.6$ | $\times$ | $0.8$ | $0.01$ |  | $\times$ | $\times$ | $\times$ |
|  | $11697.5$ | $75.4$ | $3.1$ | $2.1$ |  | $1.0$ | $357.0$ | $110.7$ |
|  | $11719.1$ | $2.7$ | $38.6$ | $18.0$ |  | $10.1$ | $149.3$ | $386.9$ |
|  | $11744.9$ | $0.8$ | $7.4$ | $13.7$ |  | $529.9$ | $2.1$ | $16.6$ |
| $2^{+}$ | $11453.9$ | $\times$ |  |  |  | $\times$ |  |  |
|  | $11747.6$ | $50.2$ |  |  |  | $499.3$ |  |  |
Table 43: The values of $k\cdot|c_{i}|^{2}$ for the $sb\bar{c}\bar{b}$ tetraquarks (in unit of MeV). System | $J^{P}$ | Mass | $s\bar{c}{\otimes}b\bar{b}$ | $s\bar{b}{\otimes}b\bar{c}$
---|---|---|---|---
$\bar{D}_{s}^{*}{\Upsilon}$ | $\bar{D}_{s}^{*}{\eta}_{b}$ | $\bar{D}_{s}{\Upsilon}$ | $\bar{D}_{s}{\eta}_{b}$ | ${B}_{s}^{*}\bar{B}_{c}^{*}$ | ${B}_{s}^{*}\bar{B}_{c}$ | ${B}_{s}\bar{B}_{c}^{*}$ | ${B}_{s}\bar{B}_{c}$
$sb\bar{c}\bar{b}$ | $0^{+}$ | $11355.0$ | $\times$ |  |  | $\times$ | $\times$ |  |  | $\times$
|  | $11539.4$ | $\times$ |  |  | $6.8$ | $\times$ |  |  | $\times$
|  | $11760.7$ | $123.5$ |  |  | $12.1$ | $4.8$ |  |  | $423.6$
|  | $11836.7$ | $0.001$ |  |  | $17.0$ | $545.9$ |  |  | $31.3$
| $1^{+}$ | $11416.4$ | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ |
|  | $11496.1$ | $\times$ | $\times$ | $3.4$ |  | $\times$ | $\times$ | $\times$ |
|  | $11549.1$ | $\times$ | $1.3$ | $0.01$ |  | $\times$ | $\times$ | $\times$ |
|  | $11785.1$ | $81.3$ | $2.2$ | $1.7$ |  | $4.9$ | $330.0$ | $118.7$ |
|  | $11802.8$ | $1.9$ | $39.9$ | $18.8$ |  | $8.7$ | $165.6$ | $360.9$ |
|  | $11833.6$ | $2.8$ | $8.3$ | $13.2$ |  | $519.6$ | $0.1$ | $21.7$ |
| $2^{+}$ | $11558.4$ | $\times$ |  |  |  | $\times$ |  |  |
|  | $11832.7$ | $49.4$ |  |  |  | $484.7$ |  |  |
Table 44: The partial width ratios for the $nb\bar{c}\bar{b}$ tetraquarks decay into $n\bar{c}{\otimes}b\bar{b}$ modes. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | $\bar{D}^{*}{\Upsilon}$ | $\bar{D}^{*}{\eta}_{b}$ | $\bar{D}{\Upsilon}$ | $\bar{D}{\eta}_{b}$
---|---|---|---|---|---|---
$nb\bar{c}\bar{b}$ | $0^{+}$ | $11253.4$ | $\times$ |  |  | $\times$
|  | $11438.6$ | $\times$ |  |  | $1$
|  | $11673.1$ | $9.5$ |  |  | $1$
|  | $11748.7$ | $0.02$ |  |  | $1$
| $1^{+}$ | $11314.8$ | $\times$ | $\times$ | $\times$ |
|  | $11392.4$ | $\times$ | $\times$ | $1$ |
|  | $11446.6$ | $\times$ | $59.0$ | $1$ |
|  | $11697.5$ | $35.6$ | $1.5$ | $1$ |
|  | $11719.1$ | $0.1$ | $2.1$ | $1$ |
|  | $11744.9$ | $0.1$ | $0.5$ | $1$ |
| $2^{+}$ | $11453.9$ | $\times$ |  |  |
|  | $11747.6$ | $1$ |  |  |
Table 45: The partial width ratios for the $nb\bar{c}\bar{b}$ tetraquarks decay into $n\bar{b}{\otimes}b\bar{c}$ modes. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | ${B}^{*}\bar{B}_{c}^{*}$ | ${B}^{*}\bar{B}_{c}$ | ${B}\bar{B}_{c}^{*}$ | ${B}\bar{B}_{c}$
---|---|---|---|---|---|---
$nb\bar{c}\bar{b}$ | $0^{+}$ | $11253.4$ | $\times$ |  |  | $\times$
|  | $11438.6$ | $\times$ |  |  | $\times$
|  | $11673.1$ | $0.02$ |  |  | $1$
|  | $11748.7$ | $12.7$ |  |  | $1$
| $1^{+}$ | $11314.8$ | $\times$ | $\times$ | $\times$ |
|  | $11392.4$ | $\times$ | $\times$ | $\times$ |
|  | $11446.6$ | $\times$ | $\times$ | $\times$ |
|  | $11697.5$ | $0.01$ | $3.2$ | $1$ |
|  | $11719.1$ | $0.03$ | $0.4$ | $1$ |
|  | $11744.9$ | $31.9$ | $0.1$ | $1$ |
| $2^{+}$ | $11453.9$ | $\times$ |  |  |
|  | $11747.6$ | $1$ |  |  |
Table 46: The partial width ratios for the $sb\bar{c}\bar{b}$ tetraquarks decay into $s\bar{c}{\otimes}b\bar{b}$ modes. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | $\bar{D}_{s}^{*}{\Upsilon}$ | $\bar{D}_{s}^{*}{\eta}_{b}$ | $\bar{D}_{s}{\Upsilon}$ | $\bar{D}_{s}{\eta}_{b}$
---|---|---|---|---|---|---
$sb\bar{c}\bar{b}$ | $0^{+}$ | $11355.0$ | $\times$ |  |  | $\times$
|  | $11539.4$ | $\times$ |  |  | $1$
|  | $11760.7$ | $10.2$ |  |  | $1$
|  | $11836.7$ | $0.00003$ |  |  | $1$
| $1^{+}$ | $11416.4$ | $\times$ | $\times$ | $\times$ |
|  | $11496.1$ | $\times$ | $\times$ | $1$ |
|  | $11549.1$ | $\times$ | $99.7$ | $1$ |
|  | $11785.1$ | $47.8$ | $1.3$ | $1$ |
|  | $11802.8$ | $0.1$ | $2.1$ | $1$ |
|  | $11833.6$ | $0.2$ | $0.6$ | $1$ |
| $2^{+}$ | $11558.4$ | $\times$ |  |  |
|  | $11832.7$ | $1$ |  |  |
Table 47: The partial width ratios for the $sb\bar{c}\bar{b}$ tetraquarks decay into $s\bar{b}{\otimes}b\bar{c}$ modes. For each state, we choose one mode as the reference channel, and the partial width ratios of the other channels are calculated relative to this channel. The masses are all in unit of MeV. System | $J^{P}$ | Mass | ${B}_{s}^{*}\bar{B}_{c}^{*}$ | ${B}_{s}^{*}\bar{B}_{c}$ | ${B}_{s}\bar{B}_{c}^{*}$ | ${B}_{s}\bar{B}_{c}$
---|---|---|---|---|---|---
$sb\bar{c}\bar{b}$ | $0^{+}$ | $11355.0$ | $\times$ |  |  | $\times$
|  | $11539.4$ | $\times$ |  |  | $\times$
|  | $11760.7$ | $0.01$ |  |  | $1$
|  | $11836.7$ | $17.4$ |  |  | $1$
| $1^{+}$ | $11416.4$ | $\times$ | $\times$ | $\times$ |
|  | $11496.1$ | $\times$ | $\times$ | $\times$ |
|  | $11549.1$ | $\times$ | $\times$ | $\times$ |
|  | $11785.1$ | $0.04$ | $2.8$ | $1$ |
|  | $11802.8$ | $0.02$ | $0.5$ | $1$ |
|  | $11833.6$ | $24.0$ | $0.01$ | $1$ |
| $2^{+}$ | $11558.4$ | $\times$ |  |  |
|  | $11832.7$ | $1$ |  |  |

Now we turn to the systems with two different heavy antiquarks. They are not
constrained by Pauli principle. The mass spectra and eigenvectors are listed
in Tables 30–31. We also transform their eigenvectors into the
$q\bar{c}{\otimes}Q\bar{b}$ and $q\bar{b}{\otimes}Q\bar{c}$ configurations, as
shown in Tables 32–35. From these tables, we see that some eigenstates couple
very strongly with two $S$-wave mesons. Taking $nb\bar{c}\bar{b}$ tetraquarks
as example, the lowest eigenstate

$T(nb\bar{c}\bar{b},11293.4,0^{+})=0.98341\bar{D}\eta_{b}+\cdots\,.$ (34)

This state couples almost completely ($96.7\%$) to the $\bar{D}\eta_{b}$
scattering state. Thus it may be very broad and hid in the continuum. Note
that this kind of state also exists in the calculation of the hidden charm
tetraquarks and pentaquarks, where the lower mass state couples strongly to a
heavy charmonium and a light hadron [18, 75, 58]. Moreover, the
$T(nb\bar{c}\bar{b},11314.8,1^{+})$ and $T(nb\bar{c}\bar{b},11392.4,1^{+})$
states couple very strongly to $\bar{D}\Upsilon$ and $\bar{D}^{*}\eta_{b}$
channels. Both the $T(nb\bar{c}\bar{b},11438.6,0^{+})$,
$T(nb\bar{c}\bar{b},11446.6,1^{+})$ and $T(nb\bar{c}\bar{b},11453.9,2^{+})$
couple very strongly to $\bar{D}^{*}\Upsilon$. They may be scattering states.
To draw a definite conclusion, dynamical studies like the complex scaling
methods [76, 77] are needed, which is beyond the present work. For clarity, we
add a fifth column in Tables 30–31 to indicate these scattering states.

We plot the relative position of the $qc\bar{c}\bar{b}$ and $qb\bar{c}\bar{b}$
tetraquarks in Figs. 5–6. For comparison, we also plot the possible scattering
states. They are marked with a dagger ($\dagger$), along with the proportion
of their dominant components. We can easily see that they all lie close to the
corresponding meson-meson thresholds.

After identifying the scattering states, the other states are genuine
tetraquarks. They all lie far above their $S$-wave decay channel(s). In
particular, the $qb\bar{c}\bar{b}$ tetraquarks lie above all possible
thresholds of two $S$-wave mesons. Thus they may be broad states. We also
study their decay properties, which can be found in Tables 36–47.

##  IV Conclusions

Experimentally, lots of $XYZ$ states have been found in this century. The
singly charmed $X(2900)$ and doubly charmed $T_{cc}^{+}$ were also observed
recently. For all these system, the exchange of a light meson may be important
since they are composed at least two light (anti-)quarks. It is difficult to
determine whether they are loosely bound molecular states or compact
tetraquark states. The recently discovered $X(6900)$’s are composed of four
charm quarks. They are very likely compact tetraquarks since their
interactions are mainly provided by gluon exchange. Similarly, if a triply
heavy exotic state is observed in experiment, it is very likely a compact
tetraquark.

In this work, we have systematically studied the triply heavy tetraquarks in
an extended chromomagnetic model, which includes both colorelectric and
chromomagnetic interactions. Our calculation suggests that the energy level is
mainly determined by the color interaction. For the $qc\bar{c}\bar{c}$,
$qb\bar{c}\bar{c}$ and $qb\bar{b}\bar{b}$ tetraquarks, the ground states are
dominated by the $6_{c}\otimes\bar{6}_{c}$ component, which is similar to the
fully heavy tetraquarks. However, the $qc\bar{b}\bar{b}$ tetraquarks are more
like the doubly heavy tetraquarks, where the $6_{c}\otimes\bar{6}_{c}$
component is mostly in the higher mass states, and the ground states are
dominated by $\bar{3}_{c}\otimes 3_{c}$ components. We find no stable state
which lie below the thresholds of two pseudoscalar mesons. The lowest axial-
vector states with $qQ\bar{b}\bar{b}$ flavor configuration might be narrow
because they lie just above the thresholds of two pseudoscalar mesons, but
they cannot decay into these channels because of the conservation of the
angular momentum and parity.

With the obtained wave functions, we also calculate the partial decay rates of
the tetraquarks. We hope that our studies can be of help for the future
experimental searches.

## Acknowledgments

This project was supported by the National Natural Science Foundation of China
(NSFC) under Grant No. 11975033 and No. 12070131001.

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