# Properties of $Q^{5}q$ dibaryons

Xin-Zhen Weng<EMAIL_ADDRESS>School of Physics and Astronomy, Tel
Aviv University, Tel Aviv 6997801, Israel

###### Abstract

We investigate heavy flavor dibaryons with five heavy quarks $Q$
($Q=\\{c,b\\}$) and one light quark $q$ ($q=\\{u,d,s\\}$), namely the $Q^{5}q$
dibaryons. In the framework of an extended chromomagnetic model, we
systematically study the mass spectrum of these dibaryons. We find no stable
state below the corresponding baryon-baryon thresholds. In addition to the
analysis of the masses, we also study their two body decay properties by
estimating the relative width ratios of the decay channels. We hope our study
will be of help for future experiments.

##  I Introduction

In the last few decades, lots of exotic states which cannot be explained as
the conventional mesons or baryons were observed in experiment. In 2003, the
Belle Collaboration observed the $X(3872)$ state in the exclusive
$B^{\pm}{\to}K^{\pm}\pi^{+}\pi^{-}J/\psi$ decays [1]. It was later confirmed
by the CDF [2], D0 [3], BABAR [4], LHCb [5], CMS [6], and BESIII [7]
Collaborations. Its quantum numbers are $I^{G}J^{PC}=0^{+}1^{++}$ [8]. Since
then, lots of charmoniumlike and bottomoniumlike $XYZ$ states have been
observed in experiments [8]. In 2020, A narrow structure, $X(6900)$, matching
the lineshape of a resonance and a broad structure next to the di-$J/\psi$
mass threshold were found by the LHCb collaboration in the $J/\psi$-pair
invariant mass spectrum [9]. These structures were later confirmed by the
ATLAS collaboration [10]. Recently, the CMS collaboration confirmed the
$X(6900)$ and observed two additional states with mass $6552~{}\text{MeV}$ and
$7287~{}\text{MeV}$ [11]. These states could be the $cc\bar{c}\bar{c}$
tetraquarks. Besides the hidden flavor states, many open heavy flavor states
were observed as well. In 2020, the LHCb Collaboration observed two resonances
in the $D^{-}K^{+}$ channel [12, 13]. They are the $X_{0/1}(2900)$ states
which consist of only one heavy charm quark. In the next year, a narrow
doubly-charmed tetraquark $T_{cc}^{+}$ was observed in the $D^{0}D^{0}\pi^{+}$
mass spectrum, just below the $D^{*+}D^{0}$ threshold [14, 15]. There are also
great experimental progresses in the pentaquark sector. In 2015, the LHCb
Collaboration observed the $P_{c}(4380)$ and $P_{c}(4450)$ states in the
${J/\psi}p$ mass spectrum [16]. An later study found that the
$P_{c}(4450)^{+}$ is actually composed of two narrow resonances,
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$. And they also found a lower mass
state $P_{c}(4312)^{+}$ [17]. In 2021, the LHCb collaboration found an
additional state $P_{c}(4337)$ in the same channel [18]. These states have a
minimal quark content of $uudc\bar{c}$. Moreover, the LHCb collaboration also
observed their strange partners, $P_{cs}(4338)$ [19] and $P_{cs}(4459)$ [20],
in the $J/\psi\Lambda$ channel. More details of the experimental as well as
theoretical progresses can be found in review Refs. [21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34] and references therein.

The study of the six-quark systems has also been pursued for very long time.
In 1932, Urey, Brickwedde and Murphy discovered the deuteron in the atomic
spectrum of hydrogen gas in a discharge tube [35, 36]. It is a very loosely
bound $p^{+}n^{0}$ molecule with a binding energy of only $2.2~{}\text{MeV}$.
Its quantum numbers are $IJ^{P}=01^{+}$. Another dibaryon candidate is the
$d^{*}(2380)$. In Ref. [37], the CELSIUS/WASA Collaboration studied the
double-pionic fusion reaction $pn{\rightarrow}d\pi^{0}\pi^{0}$. They observed
a structure with with the mass and width of the $s$-channel resonance being
$M\approx 2.36~{}\text{GeV}$ and $\Gamma\approx 80~{}\text{MeV}$. This
resonance was later confirmed by experiments performed by the WASA-at-COSY
Collaboration [37, 38, 39, 40, 41, 42]. Its quantum numbers are
$IJ^{P}=03^{+}$. This state was extensively explored theoretically [43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. For example, such a system was
studied by Dyson and Xuong in as early as 1964, with an impressive prediction
of its mass around $2350~{}\text{MeV}$ [43]. In Ref. [44], Kamae and Fujita
studied the $\Delta\Delta$ system by a non-relativistic one-boson-exchange
(OBE) potential model and found that the two $\Delta$ isobars are bound by
about $100~{}\text{MeV}$. Besides the light systems, dibaryons with heavy
flavor(s) have also been extensively studied in various models, such as the
potential quark model [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
70, 71, 72], the OBE potential model [73, 74, 75, 76, 77, 78, 79, 80], the QCD
sum rules [81, 82], the effective field theory [83, 84, 85], the lattice QCD
[86, 87, 88, 89], and the dispersion relation technique [90, 91]. More details
can be found in the recent review Ref. [92] and references therein.

For most of these systems, the exchange of a light mesons may be important. If
so, it is difficult to determine whether they are loosely bound molecular
states or compact dibaryons. However, for the $Q^{5}q$ and $Q^{6}$ systems, a
molecular configuration requires exchange of a heavy meson, which is highly
suppressed in the typical scale of the molecule ($\sim\mathrm{fm}$). Thus the
molecular interpretation is not favored. If these dibaryons are observed in
experiment, they are very likely compact states.

Table 1: Parameters of the $qq$ pairs for baryons [93] (in units of MeV). Parameter | $m_{nn}^{b}$ | $m_{ns}^{b}$ | $m_{ss}^{b}$ | $m_{nc}^{b}$ | $m_{sc}^{b}$ | $m_{cc}^{b}$ | $m_{nb}^{b}$ | $m_{sb}^{b}$ | $m_{cb}^{b}$ | $m_{b{b}}^{b}$
---|---|---|---|---|---|---|---|---|---|---
Value | $724.85$ | $906.65$ | $1049.36$ | $2079.96$ | $2183.68$ | $3171.51$ | $5412.25$ | $5494.80$ | $6416.07$ | $9529.57$
Parameter | $v_{n{n}}^{b}$ | $v_{n{s}}^{b}$ | $v_{ss}^{b}$ | $v_{n{c}}^{b}$ | $v_{s{c}}^{b}$ | $v_{c{c}}^{b}$ | $v_{n{b}}^{b}$ | $v_{s{b}}^{b}$ | $v_{c{b}}^{b}$ | $v_{b{b}}^{b}$
Value | $305.34$ | $212.75$ | $195.30$ | $62.81$ | $70.63$ | $56.75$ | $19.92$ | $8.47$ | $31.45$ | $30.65$

For these compact systems, the interaction is provided by gluon exchange and
string confinement. The interactions include spin-independent Coulomb-type
interaction and linear confinement, plus higher order terms such as spin-spin
chromomagnetic interaction, tensor interaction, and spin-orbit interaction. In
this work, we only consider the $S$-wave states, thus the tensor and spin-
orbit interactions can be neglected. Then we can use the extended
chromomagnetic model [94, 93, 30]. In this model, the hadron masses receive
contributions from effective quark masses, color interaction and
chromomagnetic interaction. This simplified model provides good account of all
$S$-wave mesons and baryons [93]. In Ref. [69], we have used this model to
investigate the fully heavy dibaryons. Here, we estimate the mass spectrum and
decay property of the $Q^{5}q$ dibaryons within the same model.

The paper is organized as follows. In Sec. II, we introduce the extended
chromomagnetic model and present the wave function bases of the $Q^{5}q$
dibaryons. In Sec. III, we calculate and discuss the numerical results. We
conclude in Sec. IV.

##  II The Extended Chromomagnetic Model

In the quark model, the Hamiltonian of the $S$-wave hadron reads [95, 94]

$H=\sum_{i}m_{i}+H_{\text{CE}}+H_{\text{CM}}\,,$ (1)

where $m_{i}$ is the effective mass of $i$th quark. $H_{\text{CE}}$ is the
colorelectric (CE) interaction

$H_{\text{CE}}=-\sum_{i<j}a_{ij}\bm{F}_{i}\cdot\bm{F}_{j}\,,$ (2)

and $H_{\text{CM}}$ is the chromomagnetic (CM) interaction

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

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

Since

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

and the total color operator $\sum_{i}\bm{F}_{i}$ nullifies any color-singlet
physical state, we can rewrite the Hamiltonian as [93]

$H=-\frac{3}{4}\sum_{i<j}m_{ij}V^{\text{C}}_{ij}-\sum_{i<j}v_{ij}V^{\text{CM}}_{ij}\,,$
(5)

by introducing the quark pair mass parameter

$m_{ij}=\left(m_{i}+m_{j}\right)+\frac{4}{3}a_{ij}\,,$ (6)

where $V^{\text{C}}_{ij}\equiv\bm{F}_{i}\cdot\bm{F}_{j}$ and
$V^{\text{CM}}_{ij}\equiv\bm{S}_{i}\cdot\bm{S}_{j}\bm{F}_{i}\cdot\bm{F}_{j}$
are the color and CM interactions between quarks. The model parameters
$m_{ij}$ and $v_{ij}$ can be fitted from conventional mesons and baryons.
Their values are listed in Table 1. In this work, we use these parameters to
estimate the masses of the $S$-wave $Q^{5}q$ dibaryons.

To obtain the mass spectra of the dibaryons, we need to construct the wave
functions. In principle, the total wave function is a direct product of the
spatial, flavor, color and spin wave functions. In Table 2, we construct the
color and spin wave functions of the dibaryons in the
$\left\\{\left[\left(q_{1}q_{2}{\otimes}q_{3}\right){\otimes}q_{4}\right]{\otimes}q_{5}\right\\}{\otimes}q_{6}$
and
$\left[\left(q_{1}q_{2}{\otimes}q_{3}\right){\otimes}q_{4}q_{5}\right]{\otimes}q_{6}$
configurations. In this work, we consider the ground state and assume that the
spatial wave function is totally symmetric. Then we need to construct the
totally anti-symmetric $\text{flavor}\otimes\text{color}\otimes\text{spin}$
wave functions. The total wave functions of the $Q^{5}q$ dibaryons are listed
in Tables 3. More details can be found in Ref. [69]. Diagonalizing the
Hamiltonian in these bases, we can obtain the masses and eigenvectors of the
$Q^{5}q$ dibaryons.

Table 2: The dibaryon color and spin wave functions in the $\left\\{\left[\left(q_{1}q_{2}{\otimes}q_{3}\right){\otimes}q_{4}\right]{\otimes}q_{5}\right\\}{\otimes}q_{6}$ and $\left[\left(q_{1}q_{2}{\otimes}q_{3}\right){\otimes}q_{4}q_{5}\right]{\otimes}q_{6}$ configurations. The superscripts $\\{1,3,\bar{3},6,8\\}$ are color representations, and the subscripts $\\{0,1/2,1,3/2,2,5/2,3\\}$ are spins. |  | $\left\\{\left[\left(q_{1}q_{2}{\otimes}q_{3}\right){\otimes}q_{4}\right]{\otimes}q_{5}\right\\}{\otimes}q_{6}$ | $\left[\left(q_{1}q_{2}{\otimes}q_{3}\right){\otimes}q_{4}q_{5}\right]{\otimes}q_{6}$
---|---|---|---
Color |  | $\phi_{\alpha 1}=\ket{(\\{[(q_{1}q_{2})^{6}q_{3}]^{8}q_{4}\\}^{\bar{6}}q_{5})^{\bar{3}}q_{6}}^{1}$ | $\phi_{\beta 1}=\ket{\\{[(q_{1}q_{2})^{6}q_{3}]^{8}(q_{4}q_{5})^{6}\\}^{\bar{3}}q_{6}}^{1}$
|  | $\phi_{\alpha 2}=\ket{(\\{[(q_{1}q_{2})^{\bar{3}}q_{3}]^{8}q_{4}\\}^{\bar{6}}q_{5})^{\bar{3}}q_{6}}^{1}$ | $\phi_{\beta 2}=\ket{\\{[(q_{1}q_{2})^{\bar{3}}q_{3}]^{8}(q_{4}q_{5})^{6}\\}^{\bar{3}}q_{6}}^{1}$
|  | $\phi_{\alpha 3}=\ket{(\\{[(q_{1}q_{2})^{6}q_{3}]^{8}q_{4}\\}^{3}q_{5})^{\bar{3}}q_{6}}^{1}$ | $\phi_{\beta 3}=\ket{\\{[(q_{1}q_{2})^{6}q_{3}]^{8}(q_{4}q_{5})^{\bar{3}}\\}^{\bar{3}}q_{6}}^{1}$
|  | $\phi_{\alpha 4}=\ket{(\\{[(q_{1}q_{2})^{\bar{3}}q_{3}]^{8}q_{4}\\}^{3}q_{5})^{\bar{3}}q_{6}}^{1}$ | $\phi_{\beta 4}=\ket{\\{[(q_{1}q_{2})^{\bar{3}}q_{3}]^{8}(q_{4}q_{5})^{\bar{3}}\\}^{\bar{3}}q_{6}}^{1}$
|  | $\phi_{\alpha 5}=\ket{(\\{[(q_{1}q_{2})^{\bar{3}}q_{3}]^{1}q_{4}\\}^{3}q_{5})^{\bar{3}}q_{6}}^{1}$ | $\phi_{\beta 5}=\ket{\\{[(q_{1}q_{2})^{\bar{3}}q_{3}]^{1}(q_{4}q_{5})^{\bar{3}}\\}^{\bar{3}}q_{6}}^{1}$
Spin | $J=0$ | $\chi^{0}_{\alpha 1}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}q_{4}\\}_{1}q_{5})_{1/2}q_{6}}_{0}$ | $\chi^{0}_{\beta 1}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}(q_{4}q_{5})_{1}\\}_{1/2}q_{6}}_{0}$
|  | $\chi^{0}_{\alpha 2}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}q_{4}\\}_{1}q_{5})_{1/2}q_{6}}_{0}$ | $\chi^{0}_{\beta 2}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}(q_{4}q_{5})_{1}\\}_{1/2}q_{6}}_{0}$
|  | $\chi^{0}_{\alpha 3}=\ket{(\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}q_{4}\\}_{1}q_{5})_{1/2}q_{6}}_{0}$ | $\chi^{0}_{\beta 3}=\ket{\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}(q_{4}q_{5})_{1}\\}_{1/2}q_{6}}_{0}$
|  | $\chi^{0}_{\alpha 4}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}q_{4}\\}_{0}q_{5})_{1/2}q_{6}}_{0}$ | $\chi^{0}_{\beta 4}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}(q_{4}q_{5})_{0}\\}_{1/2}q_{6}}_{0}$
|  | $\chi^{0}_{\alpha 5}=\ket{(\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}q_{4}\\}_{0}q_{5})_{1/2}q_{6}}_{0}$ | $\chi^{0}_{\beta 5}=\ket{\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}(q_{4}q_{5})_{0}\\}_{1/2}q_{6}}_{0}$
| $J=1$ | $\chi^{1}_{\alpha 1}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}q_{4}\\}_{2}q_{5})_{3/2}q_{6}}_{1}$ | $\chi^{1}_{\beta 1}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}(q_{4}q_{5})_{1}\\}_{3/2}q_{6}}_{1}$
|  | $\chi^{1}_{\alpha 2}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}q_{4}\\}_{1}q_{5})_{3/2}q_{6}}_{1}$ | $\chi^{1}_{\beta 2}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}(q_{4}q_{5})_{0}\\}_{3/2}q_{6}}_{1}$
|  | $\chi^{1}_{\alpha 3}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}q_{4}\\}_{1}q_{5})_{3/2}q_{6}}_{1}$ | $\chi^{1}_{\beta 3}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}(q_{4}q_{5})_{1}\\}_{3/2}q_{6}}_{1}$
|  | $\chi^{1}_{\alpha 4}=\ket{(\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}q_{4}\\}_{1}q_{5})_{3/2}q_{6}}_{1}$ | $\chi^{1}_{\beta 4}=\ket{\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}(q_{4}q_{5})_{1}\\}_{3/2}q_{6}}_{1}$
|  | $\chi^{1}_{\alpha 5}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}q_{4}\\}_{1}q_{5})_{1/2}q_{6}}_{1}$ | $\chi^{1}_{\beta 5}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}(q_{4}q_{5})_{1}\\}_{1/2}q_{6}}_{1}$
|  | $\chi^{1}_{\alpha 6}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}q_{4}\\}_{1}q_{5})_{1/2}q_{6}}_{1}$ | $\chi^{1}_{\beta 6}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}(q_{4}q_{5})_{1}\\}_{1/2}q_{6}}_{1}$
|  | $\chi^{1}_{\alpha 7}=\ket{(\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}q_{4}\\}_{1}q_{5})_{1/2}q_{6}}_{1}$ | $\chi^{1}_{\beta 7}=\ket{\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}(q_{4}q_{5})_{1}\\}_{1/2}q_{6}}_{1}$
|  | $\chi^{1}_{\alpha 8}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}q_{4}\\}_{0}q_{5})_{1/2}q_{6}}_{1}$ | $\chi^{1}_{\beta 8}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}(q_{4}q_{5})_{0}\\}_{1/2}q_{6}}_{1}$
|  | $\chi^{1}_{\alpha 9}=\ket{(\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}q_{4}\\}_{0}q_{5})_{1/2}q_{6}}_{1}$ | $\chi^{1}_{\beta 9}=\ket{\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}(q_{4}q_{5})_{0}\\}_{1/2}q_{6}}_{1}$
| $J=2$ | $\chi^{2}_{\alpha 1}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}q_{4}\\}_{2}q_{5})_{5/2}q_{6}}_{2}$ | $\chi^{2}_{\beta 1}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}(q_{4}q_{5})_{1}\\}_{5/2}q_{6}}_{2}$
|  | $\chi^{2}_{\alpha 2}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}q_{4}\\}_{2}q_{5})_{3/2}q_{6}}_{2}$ | $\chi^{2}_{\beta 2}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}(q_{4}q_{5})_{1}\\}_{3/2}q_{6}}_{2}$
|  | $\chi^{2}_{\alpha 3}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}q_{4}\\}_{1}q_{5})_{3/2}q_{6}}_{2}$ | $\chi^{2}_{\beta 3}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}(q_{4}q_{5})_{0}\\}_{3/2}q_{6}}_{2}$
|  | $\chi^{2}_{\alpha 4}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}q_{4}\\}_{1}q_{5})_{3/2}q_{6}}_{2}$ | $\chi^{2}_{\beta 4}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{1/2}(q_{4}q_{5})_{1}\\}_{3/2}q_{6}}_{2}$
|  | $\chi^{2}_{\alpha 5}=\ket{(\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}q_{4}\\}_{1}q_{5})_{3/2}q_{6}}_{2}$ | $\chi^{2}_{\beta 5}=\ket{\\{[(q_{1}q_{2})_{0}q_{3}]_{1/2}(q_{4}q_{5})_{1}\\}_{3/2}q_{6}}_{2}$
| $J=3$ | $\chi^{3}_{\alpha 1}=\ket{(\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}q_{4}\\}_{2}q_{5})_{5/2}q_{6}}_{3}$ | $\chi^{3}_{\beta 1}=\ket{\\{[(q_{1}q_{2})_{1}q_{3}]_{3/2}(q_{4}q_{5})_{1}\\}_{5/2}q_{6}}_{3}$
Table 3: The total wave functions of $Q^{5}q$ dibaryons. The color ($\phi$) and spin ($\chi$) wave functions are defined in Table 2. | $\varphi_{\text{A}}=\\{c^{5}q,b^{5}q\\}$
---|---
$J=0$ | $\Psi_{\text{A}1}^{0}=\frac{1}{\sqrt{5}}\varphi_{\text{A}}\left(\phi_{\alpha 1}\chi^{0}_{\alpha 5}-\phi_{\alpha 2}\chi^{0}_{\alpha 4}-\phi_{\alpha 3}\chi^{0}_{\alpha 3}+\phi_{\alpha 4}\chi^{0}_{\alpha 2}-\phi_{\alpha 5}\chi^{0}_{\alpha 1}\right)$
$J=1$ | $\Psi_{\text{A}1}^{1}=\frac{1}{\sqrt{5}}\varphi_{\text{A}}\left(\phi_{\alpha 1}\chi^{1}_{\alpha 9}-\phi_{\alpha 2}\chi^{1}_{\alpha 8}-\phi_{\alpha 3}\chi^{1}_{\alpha 7}+\phi_{\alpha 4}\chi^{1}_{\alpha 6}-\phi_{\alpha 5}\chi^{1}_{\alpha 5}\right)$
| $\varphi_{\text{B}}=\\{c^{4}bq,b^{4}cq\\}$ | $\varphi_{\text{C}}=\\{c^{3}b^{2}q,b^{3}c^{2}q\\}$
$J=0$ | $\Psi_{\text{B}1}^{0}=\frac{1}{\sqrt{2}}\varphi_{\text{B}}\left(\phi_{\alpha 1}\chi^{0}_{\alpha 5}-\phi_{\alpha 2}\chi^{0}_{\alpha 4}\right)$ | $\Psi_{\text{C}1}^{0}=\frac{1}{\sqrt{2}}\varphi_{\text{C}}\left(\phi_{\beta 1}\chi^{0}_{\beta 5}-\phi_{\beta 2}\chi^{0}_{\beta 4}\right)$
| $\Psi_{\text{B}2}^{0}=\frac{1}{\sqrt{3}}\varphi_{\text{B}}\left(\phi_{\alpha 3}\chi^{0}_{\alpha 3}-\phi_{\alpha 4}\chi^{0}_{\alpha 2}+\phi_{\alpha 5}\chi^{0}_{\alpha 1}\right)$ | $\Psi_{\text{C}2}^{0}=\frac{1}{\sqrt{2}}\varphi_{\text{C}}\left(\phi_{\beta 3}\chi^{0}_{\beta 3}-\phi_{\beta 4}\chi^{0}_{\beta 2}\right)$
|  | $\Psi_{\text{C}3}^{0}=\varphi_{\text{C}}\phi_{\beta 5}\chi^{0}_{\beta 1}$
$J=1$ | $\Psi_{\text{B}1}^{1}=\frac{1}{\sqrt{2}}\varphi_{\text{B}}\left(\phi_{\alpha 1}\chi^{1}_{\alpha 9}-\phi_{\alpha 2}\chi^{1}_{\alpha 8}\right)$ | $\Psi_{\text{C}1}^{1}=\frac{1}{\sqrt{2}}\varphi_{\text{C}}\left(\phi_{\beta 1}\chi^{1}_{\beta 9}-\phi_{\beta 2}\chi^{1}_{\beta 8}\right)$
| $\Psi_{\text{B}2}^{1}=\frac{1}{\sqrt{3}}\varphi_{\text{B}}\left(\phi_{\alpha 3}\chi^{1}_{\alpha 4}-\phi_{\alpha 4}\chi^{1}_{\alpha 3}+\phi_{\alpha 5}\chi^{1}_{\alpha 2}\right)$ | $\Psi_{\text{C}2}^{1}=\frac{1}{\sqrt{2}}\varphi_{\text{C}}\left(\phi_{\beta 3}\chi^{1}_{\beta 4}-\phi_{\beta 4}\chi^{1}_{\beta 3}\right)$
| $\Psi_{\text{B}3}^{1}=\frac{1}{\sqrt{3}}\varphi_{\text{B}}\left(\phi_{\alpha 3}\chi^{1}_{\alpha 7}-\phi_{\alpha 4}\chi^{1}_{\alpha 6}+\phi_{\alpha 5}\chi^{1}_{\alpha 5}\right)$ | $\Psi_{\text{C}3}^{1}=\frac{1}{\sqrt{2}}\varphi_{\text{C}}\left(\phi_{\beta 3}\chi^{1}_{\beta 7}-\phi_{\beta 4}\chi^{1}_{\beta 6}\right)$
|  | $\Psi_{\text{C}4}^{1}=\varphi_{\text{C}}\phi_{\beta 5}\chi^{1}_{\beta 1}$
|  | $\Psi_{\text{C}5}^{1}=\varphi_{\text{C}}\phi_{\beta 5}\chi^{1}_{\beta 5}$
$J=2$: | $\Psi_{\text{B}1}^{2}=\frac{1}{\sqrt{3}}\varphi_{\text{B}}\left(\phi_{\alpha 3}\chi^{2}_{\alpha 5}-\phi_{\alpha 4}\chi^{2}_{\alpha 4}+\phi_{\alpha 5}\chi^{2}_{\alpha 3}\right)$ | $\Psi_{\text{C}1}^{2}=\frac{1}{\sqrt{2}}\varphi_{\text{C}}\left(\phi_{\beta 3}\chi^{2}_{\beta 5}-\phi_{\beta 4}\chi^{2}_{\beta 4}\right)$
|  | $\Psi_{\text{C}2}^{2}=\varphi_{\text{C}}\phi_{\beta 5}\chi^{2}_{\beta 1}$
|  | $\Psi_{\text{C}3}^{2}=\varphi_{\text{C}}\phi_{\beta 5}\chi^{2}_{\beta 2}$
$J=3$ |  | $\Psi_{\text{C}1}^{3}=\varphi_{\text{C}}\phi_{\beta 5}\chi^{3}_{\beta 1}$

##  III Numerical results

Table 4: Masses and eigenvectors of the $cccccq$, $bbbbbq$, $ccccbq$ and $bbbbcq$ dibaryons. The masses are all in units of MeV. System | $J^{P}$ | Mass | Eigenvector | System | $J^{P}$ | Mass | Eigenvector
---|---|---|---|---|---|---|---
$cccccn$ | $0^{+}$ | $8599$ | $\\{1\\}$ | $bbbbbn$ | $0^{+}$ | $24553$ | $\\{1\\}$
| $1^{+}$ | $8516$ | $\\{1\\}$ |  | $1^{+}$ | $24526$ | $\\{1\\}$
$cccccs$ | $0^{+}$ | $8711$ | $\\{1\\}$ | $bbbbbs$ | $0^{+}$ | $24624$ | $\\{1\\}$
| $1^{+}$ | $8617$ | $\\{1\\}$ |  | $1^{+}$ | $24612$ | $\\{1\\}$
$ccccbn$ | $0^{+}$ | $11693$ | $\\{0.900,0.437\\}$ | $bbbbcn$ | $0^{+}$ | $21306$ | $\\{0.192,0.981\\}$
|  | $11840$ | $\\{-0.437,0.900\\}$ |  |  | $21473$ | $\\{0.981,-0.192\\}$
| $1^{+}$ | $11698$ | $\\{0.902,-0.405,0.150\\}$ |  | $1^{+}$ | $21251$ | $\\{0.092,-0.836,0.540\\}$
|  | $11776$ | $\\{0.307,0.359,-0.882\\}$ |  |  | $21303$ | $\\{0.045,0.546,0.837\\}$
|  | $11806$ | $\\{0.303,0.841,0.448\\}$ |  |  | $21448$ | $\\{0.995,0.052,-0.088\\}$
| $2^{+}$ | $11765$ | $\\{1\\}$ |  | $2^{+}$ | $21286$ | $\\{1\\}$
$ccccbs$ | $0^{+}$ | $11793$ | $\\{0.875,0.484\\}$ | $bbbbcs$ | $0^{+}$ | $21398$ | $\\{0.164,0.987\\}$
|  | $11941$ | $\\{-0.484,0.875\\}$ |  |  | $21550$ | $\\{0.987,-0.164\\}$
| $1^{+}$ | $11802$ | $\\{-0.872,0.467,-0.150\\}$ |  | $1^{+}$ | $21341$ | $\\{0.072,-0.859,0.506\\}$
|  | $11871$ | $\\{-0.266,-0.192,0.945\\}$ |  |  | $21395$ | $\\{0.098,0.511,0.854\\}$
|  | $11907$ | $\\{0.412,0.863,0.291\\}$ |  |  | $21524$ | $\\{0.993,0.012,-0.121\\}$
| $2^{+}$ | $11855$ | $\\{1\\}$ |  | $2^{+}$ | $21383$ | $\\{1\\}$
Table 5: Masses and eigenvectors of the $cccbbq$ dibaryons. The masses are all in units of MeV. System | $J^{P}$ | Mass | Eigenvector | Scattering state
---|---|---|---|---
$cccbbn$ | $0^{+}$ | $14887$ | $\\{0.087,0.925,0.369\\}$ |
|  | $14981$ | $\\{-0.287,-0.331,0.899\\}$ |  $\Omega_{ccc}^{*}\Xi_{bb}^{*}$ ($80.8\%$)
|  | $15093$ | $\\{0.954,-0.183,0.237\\}$ |
| $1^{+}$ | $14833$ | $\\{0.066,0.898,0.418,0.116,0.020\\}$ |
|  | $14884$ | $\\{0.089,-0.358,0.857,-0.357,-0.038\\}$ |
|  | $14953$ | $\\{-0.258,0.090,-0.075,-0.426,0.859\\}$ |  $\Omega_{ccc}^{*}\Xi_{bb}$ ($79.3\%$)
|  | $14984$ | $\\{0.133,0.234,-0.277,-0.823,-0.416\\}$ |
|  | $15039$ | $\\{-0.950,0.038,0.091,-0.026,-0.294\\}$ |
| $2^{+}$ | $14872$ | $\\{0.948,-0.314,0.048\\}$ |
|  | $14962$ | $\\{-0.274,-0.730,0.626\\}$ |  $\Omega_{ccc}^{*}\Xi_{bb}$ ($87.0\%$)
|  | $14976$ | $\\{0.161,0.607,0.778\\}$ |  $\Omega_{ccc}^{*}\Xi_{bb}^{*}$ ($93.6\%$)
| $3^{+}$ | $14974$ | $\\{1\\}$ |  $\Omega_{ccc}^{*}\Xi_{bb}^{*}$ ($100\%$)
$cccbbs$ | $0^{+}$ | $14986$ | $\\{0.044,0.861,0.507\\}$ |
|  | $15069$ | $\\{-0.272,-0.478,0.835\\}$ |
|  | $15187$ | $\\{0.961,-0.175,0.214\\}$ |
| $1^{+}$ | $14934$ | $\\{-0.051,-0.902,-0.403,-0.142,-0.033\\}$ |
|  | $14984$ | $\\{0.123,-0.302,0.826,-0.449,-0.096\\}$ |
|  | $15041$ | $\\{-0.263,0.038,-0.015,-0.319,0.910\\}$ |
|  | $15072$ | $\\{0.016,0.307,-0.372,-0.822,-0.303\\}$ |
|  | $15127$ | $\\{0.955,-0.004,-0.126,-0.024,0.266\\}$ |
| $2^{+}$ | $14976$ | $\\{0.911,-0.411,0.033\\}$ |
|  | $15051$ | $\\{-0.135,-0.222,0.966\\}$ |
|  | $15061$ | $\\{-0.389,-0.884,-0.258\\}$ |
| $3^{+}$ | $15053$ | $\\{1\\}$ |  $\Omega_{ccc}^{*}\Omega_{bb}^{*}$ ($100\%$)
Table 6: Masses and eigenvectors of the $bbbccq$ dibaryons. The masses are all in units of MeV. System | $J^{P}$ | Mass | Eigenvector | Scattering state
---|---|---|---|---
$bbbccn$ | $0^{+}$ | $18003$ | $\\{-0.109,0.047,0.993\\}$ |  $\Omega_{bbb}^{*}\Xi_{cc}^{*}$ ($98.6\%$)
|  | $18167$ | $\\{-0.894,-0.441,-0.077\\}$ |
|  | $18280$ | $\\{0.434,-0.896,0.090\\}$ |
| $1^{+}$ | $17941$ | $\\{0.068,0.043,-0.022,0.737,-0.671\\}$ |  $\Omega_{bbb}^{*}\Xi_{cc}$ ($99.3\%$)
|  | $18004$ | $\\{0.092,-0.048,0.009,-0.672,-0.733\\}$ |  $\Omega_{bbb}^{*}\Xi_{cc}^{*}$ ($98.9\%$)
|  | $18161$ | $\\{0.916,0.346,0.178,-0.007,0.101\\}$ |
|  | $18233$ | $\\{0.385,-0.849,-0.355,0.051,0.053\\}$ |
|  | $18250$ | $\\{0.028,0.394,-0.917,-0.046,0.008\\}$ |
| $2^{+}$ | $17942$ | $\\{0.041,-0.893,0.449\\}$ |  $\Omega_{bbb}^{*}\Xi_{cc}$ ($99.8\%$)
|  | $18006$ | $\\{0.017,0.450,0.893\\}$ |  $\Omega_{bbb}^{*}\Xi_{cc}^{*}$ ($99.97\%$)
|  | $18230$ | $\\{0.999,0.029,-0.034\\}$ |
| $3^{+}$ | $18006$ | $\\{1\\}$ |  $\Omega_{bbb}^{*}\Xi_{cc}^{*}$ ($100\%$)
$bbbccs$ | $0^{+}$ | $18109$ | $\\{-0.126,0.024,0.992\\}$ |  $\Omega_{bbb}^{*}\Omega_{cc}^{*}$ ($98.3\%$)
|  | $18251$ | $\\{-0.867,-0.489,-0.098\\}$ |
|  | $18362$ | $\\{0.482,-0.872,0.083\\}$ |
| $1^{+}$ | $18040$ | $\\{0.069,0.041,-0.018,0.735,-0.672\\}$ |  $\Omega_{bbb}^{*}\Omega_{cc}$ ($99.3\%$)
|  | $18110$ | $\\{0.096,-0.038,-0.008,-0.675,-0.730\\}$ |  $\Omega_{bbb}^{*}\Omega_{cc}^{*}$ ($98.9\%$)
|  | $18256$ | $\\{0.880,0.426,0.184,-0.010,0.101\\}$ |
|  | $18323$ | $\\{0.448,-0.880,-0.137,0.055,0.056\\}$ |
|  | $18328$ | $\\{0.101,0.204,-0.973,-0.018,0.030\\}$ |
| $2^{+}$ | $18041$ | $\\{0.030,-0.893,0.450\\}$ |  $\Omega_{bbb}^{*}\Omega_{cc}$ ($99.9\%$)
|  | $18112$ | $\\{0.039,0.451,0.892\\}$ |  $\Omega_{bbb}^{*}\Omega_{cc}^{*}$ ($99.8\%$)
|  | $18308$ | $\\{0.999,0.010,-0.049\\}$ |
| $3^{+}$ | $18112$ | $\\{1\\}$ |  $\Omega_{bbb}^{*}\Omega_{cc}^{*}$ ($100\%$)

###  III.1 The $cccccq$ and $bbbbbq$ systems

The masses and eigenvectors of the $Q^{5}q$ dibaryons are listed in Tables
4–6. In this work, we assume that the $\mathrm{SU}(2)$ flavor symmetry and
denote $n=\\{u,d\\}$. In the following, we will use $D(QQQQQq,m,J^{P})$ to
denote the $QQQQQq$ dibaryon state. The isospin is not explicitly shown since
it can be easily identified with either $q=n$ or $q=s$.

First we consider the $cccccq$ dibaryons. In Fig. 1(a–b), We plot the relative
position of these dibaryons, along with the corresponding baryon-baryon
thresholds which they can decay into through quark rearrangement. From the
figure, we can easily see that all states are above thresholds. They may all
be _broad_ states since they can decay into two baryons through $S$-wave [96].
We find that the quantum number of the ground states are $J^{P}=1^{+}$ in both
cases. They are the $D(cccccn,8516,1^{+})$ and the $D(cccccs,8617,1^{+})$
states, respectively. Unlike the conventional mesons and baryons, the $cccccq$
scalar states, which have lower spin, are heavier then the axial-vector states
by almost 100 MeV. Similar to the fully heavy $c^{5}b$ and $b^{5}c$ dibaryons,
the underlying reason of this mass reversal is that the Pauli principle
imposes large restriction over the wave functions [69]. The colorelectric
interaction

$\Braket{H_{\text{C}}\left(cccccq\right)}=2m_{cc}+m_{cq}\,,$ (7)

is identical for the two states. By introducing the
$\mathrm{SU}(6)_{cs}=\mathrm{SU}(3)_{c}\otimes\mathrm{SU}(2)_{s}$ group, the
chromomagnetic interaction can be written as [97, 69]

$\Braket{H_{\text{CM}}\left(cccccq\right)}=2v_{cc}+v_{cq}\left[1+\frac{S\left(S+1\right)}{12}-\frac{\Braket{\text{C}_{6}\left(cccccq\right)}}{32}\right]\,,$
(8)

where $\text{C}_{6}$ is the $\mathrm{SU}(6)_{cs}$ Casimir operator. The spin
term does contribute larger mass for the higher spin state, but the
$\text{C}_{6}$ term has opposite effect. In the $\mathrm{SU}(6)_{cs}$ group,
the Pauli principle requires that the five charm quarks be anti-symmetric

$\Yvcentermath
1\young(c,c,c,c,c)_{cs}\sim\young(cc,cc,c)_{c}\otimes\young(ccc,cc)_{s}\,.$
(9)

Adding the light quark, we have two possible $\text{SU}(6)_{cs}$
representations. Namely a $J=0$ representation

$\Yvcentermath
1\young(c,c,c,c,c,q)_{cs}\sim\young(cc,cc,cq)_{c}\otimes\young(ccc,ccq)_{s}\,,$
(10)

and a $J=1$ representation

$\Yvcentermath
1\young(cq,c,c,c,c)_{cs}\sim\young(cc,cc,cq)_{c}\otimes\young(cccq,cc)_{s}\,,$
(11)

where we have require the states to be of color-singlet. The former
representation ($J=0$) is a $\text{SU}(6)_{cs}$ singlet with
$\braket{\text{C}_{6}}=0$, while the latter one ($J=1$) is a 35-plet with
$\braket{\text{C}_{6}}=48$ [97]. Thus we have

$\Braket{H_{\text{CM}}\left(cccccq\right)}=\left\\{\begin{split}&2v_{cc}+v_{cq}~{}\text{
for }~{}J=0\,,\\\ &2v_{cc}-\frac{v_{cq}}{3}~{}\text{ for
}~{}J=1\,.\end{split}\right.$ (12)

The effect of $\text{C}_{6}$ term surpasses that of the spin term. Thus the
chromomagnetic interaction favors the axial-vector state over the scalar
state.

The spectra of their bottom partners, $bbbbbq$ dibaryons, share similar
pattern. As shown in Fig. 1(c–d), the ground states are the
$D(bbbbbn,24526,1^{+})$ and the $D(bbbbbs,24612,1^{+})$, respectively. While
the excitations are the $D(bbbbbn,24553,0^{+})$ and the
$D(bbbbbs,24624,0^{+})$ states, respectively. The corresponding mass
splittings are about $20~{}\text{MeV}$, which is much smaller than that of
their charm partners. The reason is that the splitting is inversely
proportional to the quark masses [see Eq. (12)]

$\Delta{m}\left(Q^{5}q\right){\propto}v_{Qq}{\propto}1/m_{Q}m_{q}\,.$ (13)

The bottom quark is much heavier than the charm quark, thus the mass splitting
of the $bbbbbq$ dibaryons is smaller than that of the $cccccq$ dibaryons.

Besides the masses, the eigenvectors can also be used to investigate the decay
properties of the dibaryons. We can transform the wave function into the
$q^{3}{\otimes}q^{3}$ configuration. The $q^{3}$ component can be either of
color-singlet or of color-octet. The
$\ket{(qqq)^{1_{c}}{\otimes}(qqq)^{1_{c}}}$ dibaryon can easily dissociate
into two baryons in relative $S$-wave. We call this kind of decay the “Okubo-
Zweig-Iizuka- (OZI-)superallowed” decay. On the other hand, the
$\ket{(qqq)^{8_{c}}{\otimes}(qqq)^{8_{c}}}$ dibaryon cannot fall apart without
the gluon exchange. For simplicity, we follow Refs. [96, 97, 98, 69] and only
consider the “OZI-superallowed” decays in this work. In Table 7, we transform
the eigenvectors of the $cccccq$ ($bbbbbq$) dibaryons into the
$ccc{\otimes}ccq$ ($bbb{\otimes}bbq$) configuration. For simplicity, we only
present the color-singlet components, and rewrite the bases as a direct
product of two baryons

$\Psi=\sum_{i}c_{i}\Ket{\psi_{i}\left(B{\otimes}B\right)}+\cdots\,,$ (14)

where $c_{i}$ represents the overlap between the dibaryon and a particular
baryon $\times$ baryon channel. For each decay channel, the decay width is
proportional to the square of the coefficient $c_{i}$ of the corresponding
component in the eigenvector, and also depends on the phase space. For two
body $L$-wave decay, the partial decay width reads [99, 100]

$\Gamma_{i}=\gamma_{i}\alpha\frac{k^{2L+1}}{m^{2L}}{\cdot}|c_{i}|^{2},$ (15)

where $\gamma_{i}$ is a quantity related to the decay dynamics, $\alpha$ is an
effective coupling constant, $k$ is the momentum of the final baryons in the
rest frame of the initial state, $m$ is the dibaryon mass, and $L$ is the
relative partial wave between the two baryons. For the decay processes studied
in this work, $(k/m)^{2}$’s are all of $\mathcal{O}(10^{-2})$ or even smaller.
The higher wave decays are all suppressed. Thus we only consider the $S$-wave
decays. Generally, $\gamma_{i}$ depends on the spatial wave functions of the
initial and final states, which are different for each decay process. In the
quark model, the spatial wave functions of the ground state $1/2^{+}$ and
$3/2^{+}$ baryons are similar. Thus for _each_ dibaryon state, we have

$\gamma_{B_{1}B_{2}}\approx\gamma_{B_{1}B_{2}^{*}}\approx\gamma_{B_{1}^{*}B_{2}}\approx\gamma_{B_{1}^{*}B_{2}^{*}}\,,$
(16)

where $B_{i}$ and $B_{i}^{*}$ are the $1/2^{+}$ and $3/2^{+}$ baryons with the
same flavor contents. In Table 8, we calculate the values of
$k\cdot|c_{i}|^{2}$ for the $cccccq$ and $bbbbbq$ dibaryons. Then we compare
their relative widths in Table 9.

For the $cccccq$ and $bbbbbq$ dibaryons, the scalar states only have one
$S$-wave decay channel consists of two $3/2^{+}$ baryons. They are broad
states since their phase spaces are quite large (see Fig. 1). The axial-vector
states have two $S$-wave decay channels. From Table 9, we have

$\frac{\Gamma\left[D(cccccn,8516,1^{+}){\to}\Omega_{ccc}^{*}\Xi_{cc}\right]}{\Gamma\left[D(cccccn,8516,1^{+}){\to}\Omega_{ccc}^{*}\Xi_{cc}^{*}\right]}\sim
1.3\,,$ (17)
$\frac{\Gamma\left[D(cccccs,8617,1^{+}){\to}\Omega_{ccc}^{*}\Omega_{cc}\right]}{\Gamma\left[D(cccccs,8617,1^{+}){\to}\Omega_{ccc}^{*}\Omega_{cc}^{*}\right]}\sim
1.5\,,$ (18)
$\frac{\Gamma\left[D(bbbbbn,24526,1^{+}){\to}\Omega_{bbb}^{*}\Xi_{bb}\right]}{\Gamma\left[D(bbbbbn,24526,1^{+}){\to}\Omega_{bbb}^{*}\Xi_{bb}^{*}\right]}\sim
1.0\,,$ (19)

and

$\frac{\Gamma\left[D(bbbbbs,24612,1^{+}){\to}\Omega_{bbb}^{*}\Omega_{bb}\right]}{\Gamma\left[D(bbbbbs,24612,1^{+}){\to}\Omega_{bbb}^{*}\Omega_{bb}^{*}\right]}\sim
0.9\,.$ (20)

For axial-vector states, the partial decay widths of the two channels are
compatible.

###  III.2 The $ccccbq$ and $bbbbcq$ systems

Now we turn to the $ccccbq$ and $bbbbcq$ dibaryons. We list their masses and
eigenvectors in Table 4 and plot their relative position in Fig. 2–3, along
with the possible decay channels. Similar to the $cccccq$ and $bbbbbq$
dibaryon cases, all $ccccbq$ dibaryon states are above baryon-baryon
thresholds. The ground states are both scalar states. They are
$D(ccccbn,11693,0^{+})$ and $D(ccccbs,11793,0^{+})$, respectively. The
heaviest states are also scalar states. Namely $D(ccccbn,11840,0^{+})$ and
$D(ccccbs,11941,0^{+})$. The mass splittings of the $ccccbn$ and $ccccbs$
dibaryons are $247~{}\text{MeV}$ and $248~{}\text{MeV}$, respectively.

Besides the masses, the eigenvectors also provide important information of the
dibaryons. As shown in Tables 2–3, there are two scalar bases, three axial-
vector bases and one tensor base. Their color configurations can be of
$\ket{(c_{1}c_{2}c_{3}c_{4})^{\bar{6}_{c}}(b_{5}q_{6})^{6_{c}}}$ or
$\ket{(c_{1}c_{2}c_{3}c_{4})^{3_{c}}(b_{5}q_{6})^{\bar{3}_{c}}}$. For
simplicity, we denote them as $\bar{6}_{c}{\otimes}6_{c}$ and
$3_{c}{\otimes}\bar{3}_{c}$. From Table 4, we find that the ground states of
the $ccccbn$ and $ccccbs$ dibaryons are both dominated by the color-sextet
components. More precisely, the $D(ccccbn,11693,0^{+})$ has $80.9\%$ of the
$\bar{6}_{c}{\otimes}6_{c}$ component and the $D(ccccbs,11793,0^{+})$ has
$76.6\%$. Moreover, the lowest axial-vector states are also dominated by the
color-sextet components. The $D(ccccbn,11698,1^{+})$ and
$D(ccccbs,11802,1^{+})$ possess $81.4\%$ and $76.0\%$ of the
$\bar{6}_{c}{\otimes}6_{c}$ component, respectively. On the other hand, all
higher states are dominated by the color-triplet components. The underlying
reason is the color interaction. More precisely, we have

$\Braket{H_{\text{C}}\left(ccccbq\right)}=2m_{cc}+m_{bq}-\frac{3}{8}\delta{m}\cdot\mathrm{C}_{3}\left(cccc\right)$
(21)

where $\text{C}_{3}$ is the $\mathrm{SU}(3)_{c}$ Casimir operator and

$\delta{m}=\frac{m_{cc}+m_{bq}-m_{cb}-m_{cq}}{4}\,.$ (22)

Inserting the parameters in Table 1, we have

$\left\\{\begin{split}\delta{m}\left(ccccbn\right)=21.93~{}\text{MeV}\,,\\\
\delta{m}\left(ccccbs\right)=16.64~{}\text{MeV}\,.\end{split}\right.$ (23)

Note that the $\mathrm{C}_{3}(cccc)$ is diagonal in the
$\bar{6}_{c}{\otimes}6_{c}$ and $3_{c}{\otimes}\bar{3}_{c}$ color bases, with
matrix elements $40/3$ and $16/3$, respectively. Thus

$\Braket{H_{\text{C}}\left(ccccbq\right)}=\left\\{\begin{split}2m_{cc}+m_{bq}-5\delta{m}~{}\text{for}~{}\bar{6}_{c}{\otimes}6_{c}\,,\\\
2m_{cc}+m_{bq}-2\delta{m}~{}\text{for}~{}3_{c}{\otimes}\bar{3}_{c}\,.\end{split}\right.$
(24)

The color interaction favors the $\bar{6}_{c}{\otimes}6_{c}$ configuration.
Then we obtain the two-band structure in the mass spectrum. The CM interaction
is suppressed by $1/m_{Q}$ and thus cannot alter the order. Alas, the CM
interaction mixes the states and gives small shift of the bands.

Unlike the $ccccbq$ dibaryons, the quantum numbers of the $bbbbcq$ ground
states become $J^{P}=1^{+}$. All $bbbbcq$ dibaryons are above their baryon-
baryon threshold. However, some of them may be relatively narrow because they
are not too much higher than their $S$-wave decay channels (see Fig. 3). The
$D(bbbbcn,21251,1^{+})$ can decay to the $\Omega_{bbb}^{*}\Xi_{cb}$ mode, but
their mass difference is only $19~{}\text{MeV}$. The $D(bbbbcn,21306,0^{+})$
has three decay channels. It can only decay into the
$\Omega_{bbb}^{*}\Xi_{cb}$ and $\Omega_{bbb}^{*}\Xi_{cb}^{\prime}$ channels by
$D$-wave due to the conservation of the angular momentum. According to Eq.
(15), these channels are highly suppressed. On the other hand, it can also
decay into the $\Omega_{bbb}^{*}\Xi_{cb}^{*}$ channel through $S$-wave, but
their mass difference is only $23~{}\text{MeV}$. The $bbbbcs$ dibaryons are
similar, the $D(bbbbcs,21341,1^{+})$/$D(bbbbcs,21398,0^{+})$ is only
$21/23~{}\text{MeV}$ above the
$\Omega_{bbb}^{*}\Omega_{cb}$/$\Omega_{bbb}^{*}\Omega_{cb}^{*}$ channel. Due
to their small phases, these stats are expected to be relatively narrow.

Another difference of the $ccccbq$ and $bbbbcq$ systems is that the dominant
color configuration of the lower mass $bbbbcq$ dibaryon states becomes
$3_{c}{\otimes}\bar{3}_{c}$. For instance, the $D(bbbbcn,21251,1^{+})$ has
$99.2\%$ of $3_{c}{\otimes}\bar{3}_{c}$ component, and the
$D(bbbbcs,21341,1^{+})$ has $99.5\%$. We again resort to the color interaction

$\Braket{H_{\text{C}}\left(bbbbcq\right)}=2m_{bb}+m_{cq}-\frac{3}{8}\delta{m}^{\prime}\cdot\mathrm{C}_{3}\left(bbbb\right)$
(25)

where

$\delta{m}^{\prime}=\frac{m_{bb}+m_{cq}-m_{cb}-m_{bq}}{4}\,.$ (26)

The main difference between $ccccbq$ and $bbbbcq$ dibaryons is that the
$\delta{m}$’s are positive, while the $\delta{m}^{\prime}$’s are negative

$\left\\{\begin{split}\delta{m}^{\prime}\left(bbbbcn\right)=-54.70~{}\text{MeV}\,,\\\
\delta{m}^{\prime}\left(bbbbcs\right)=-49.40~{}\text{MeV}\,.\end{split}\right.$
(27)

Thus the lower mass states are dominated by the $3_{c}{\otimes}\bar{3}_{c}$
components, while the higher one have more $\bar{6}_{c}{\otimes}6_{c}$
components.

Before concluding this section, we would like to compare the present results
with that of Refs. [101, 102, 103]. It is interesting to note that $ccccbq$
dibaryons are very similar to the triply heavy $\bar{c}\bar{c}bq$ tetraquarks
and the fully heavy tetraquarks like $\bar{b}\bar{b}bc$. On the other hand,
the $bbbbcq$ dibaryons are more like the triply heavy $\bar{b}\bar{b}cq$
tetraquarks and the doubly heavy tetraquarks. The underlying reason is the
similarity between the $cccc$/$bbbb$ cluster in the dibaryons and the
$\bar{c}\bar{c}$/$\bar{b}\bar{b}$ cluster in the tetraquarks. For example,
their possible color representations are both $\\{\bar{6}_{c},3_{c}\\}$, while
their possible spins are both $\\{0,1\\}$ as well. In fact, the similarity can
also be inferred from the color interaction (taking $\bar{c}\bar{c}bq$ as an
example)

$\Braket{H_{\text{C}}\left(\bar{c}\bar{c}bq\right)}=m_{cc}+m_{bq}-\frac{3}{8}\delta\tilde{m}\cdot\mathrm{C}_{3}\left(\bar{c}\bar{c}\right)$
(28)

where

$\delta\tilde{m}=\frac{m_{cc}+m_{bq}-m_{\bar{c}b}-m_{\bar{c}q}}{4}\,.$ (29)

A more 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 dibaryons.

Next we consider their decay properties. We transform the eigenvectors of the
$ccccbq$ ($bbbbcq$) dibaryons into the $ccc{\otimes}cbq$ ($bbb{\otimes}cbq$)
and $ccb{\otimes}ccq$ ($bbc{\otimes}bbq$) configurations. For all possible
channels, we calculate the values of $k{\cdot}|c_{i}|^{2}$ and the ratios of
the partial decay widths. Note that for a dibaryon state decays into two
channels through two different quark rearrangements, the $\gamma_{i}$’s, which
depend on the wave functions of final states, may not equal (or approximately
equal). Thus we compare only the decay width ratios between channels through
same quark rearrangement type. the corresponding results are listed in Table
10–12. It is easy to see that the decay properties of the $ccccbq$ and the
$bbbbcq$ dibaryons are quite different. Taking the higher scalar states as
example. For the $ccccbq$ dibaryons, we have

$\frac{\Gamma\left[D(ccccbn,11840,0^{+}){\to}\Omega_{ccb}^{*}\Xi_{cc}^{*}\right]}{\Gamma\left[D(ccccbn,11840,0^{+}){\to}\Omega_{ccb}\Xi_{cc}\right]}\sim
11.0\,,$ (30)

and

$\frac{\Gamma\left[D(ccccbs,11941,0^{+}){\to}\Omega_{ccb}^{*}\Omega_{cc}^{*}\right]}{\Gamma\left[D(ccccbs,11941,0^{+}){\to}\Omega_{ccb}\Omega_{cc}\right]}\sim
19.5\,.$ (31)

The former modes always dominate. But for the $bbbbcq$ dibaryons, we have

$\frac{\Gamma\left[D(bbbbcn,21473,0^{+}){\to}\Omega_{bbc}^{*}\Xi_{bb}^{*}\right]}{\Gamma\left[D(bbbbcn,21473,0^{+}){\to}\Omega_{bbc}\Xi_{bb}\right]}\sim
2.6\,,$ (32)

and

$\frac{\Gamma\left[D(bbbbcs,21550,0^{+}){\to}\Omega_{bbc}^{*}\Omega_{bb}^{*}\right]}{\Gamma\left[D(bbbbcs,21550,0^{+}){\to}\Omega_{bbc}\Omega_{bb}\right]}\sim
2.5\,.$ (33)

The two modes are compatible. As discussed above, the wave functions of the
$ccccbq$ and the $bbbbcq$ dibaryons are quite different, which results in the
difference of their decay properties.

###  III.3 The $cccbbq$ and $bbbccq$ systems

The eigenvalues and eigenvectors of the $cccbbq$ system are listed in Table 5.
We transform their eigenvectors into the $ccc{\otimes}bbq$, $ccb{\otimes}cbq$
and $ccq{\otimes}bbc$ configurations, as shown in Table 13. From these tables,
we see that some eigenstates couple very strongly with two $S$-wave baryons.
For example, the second scalar state

$D\left(cccbbn,14981,0^{+}\right)=0.899\Omega_{ccc}^{*}\Xi_{bb}^{*}+\cdots\,.$
(34)

This state couples strongly $(80.8\%)$ to the $\Omega_{ccc}^{*}\Xi_{bb}^{*}$
scattering state. It might be a broad state which belongs to the continuum.
Various calculations also found existence of such kind of states in
tetraquarks [104, 105, 101, 103, 106], pentaquarks [100] and dibaryons [68,
69]. Besides the scalar case, the $cccbbn$ states of $14984~{}\text{MeV}$
(with $J^{P}=1^{+}$), $14976~{}\text{MeV}$ (with $J^{P}=2^{+}$) and
$14974~{}\text{MeV}$ (with $J^{P}=3^{+}$) also couple strongly to the
$\Omega_{ccc}^{*}$ and $\Xi_{bb}^{*}$ baryons, while the states of
$14953~{}\text{MeV}$ (with $J^{P}=1^{+}$) and $14962~{}\text{MeV}$ (with
$J^{P}=2^{+}$) couple very strongly to the $\Omega_{ccc}^{*}{\otimes}\Xi_{bb}$
channel. They may be scattering states. On the other hand, the $cccbbs$ state
of $15053~{}\text{MeV}$ with $J^{P}=3^{+}$ is a
$\Omega_{ccc}^{*}{\otimes}\Omega_{bb}$ scattering state. Among them,
$D(cccbbn,14984,1^{+})$ state couples relatively weaker ($73.8\%$) to the
$\Omega_{ccc}^{*}{\otimes}\Xi_{bb}^{*}$ channel, thus we cannot rule out the
possibility that it is a dibaryons. To draw a definite conclusion, dynamical
studies like the complex scaling methods [107] are needed, which is beyond the
present work. For clarity, we indicate the scattering states in the last
column in Table 5. Going to the $bbbccq$ dibaryons, we also find various
scattering states. They are listed in the fifth column of Table 6.

We plot the relative position of the $cccbbq$ and $bbbccq$ dibaryons in Figs.
4–5. 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
$QQQ$ baryon-$QQ^{\prime}q$ baryon thresholds. Interestingly, we find that the
scattering states of the $cccbbq$ systems lie in the middle of the spectra,
whit those of the $bbbccq$ systems lie in the bottom of the spectra. The
reason is as follows. From Tables 2–3, we see that the $Q^{3}Q^{\prime 2}q$
dibaryons has three possible color bases

$\displaystyle
8_{c}{\otimes}6_{c}{\otimes}3_{c}:\left\\{\left[\left(QQQ\right)^{8_{c}}{\otimes}\left(Q^{\prime}Q^{\prime}\right)^{6_{c}}\right]^{\bar{3}_{c}}{\otimes}q^{3_{c}}\right\\}^{1_{c}}\,,$
(35) $\displaystyle
8_{c}{\otimes}\bar{3}_{c}{\otimes}3_{c}:\left\\{\left[\left(QQQ\right)^{8_{c}}{\otimes}\left(Q^{\prime}Q^{\prime}\right)^{\bar{3}_{c}}\right]^{\bar{3}_{c}}{\otimes}q^{3_{c}}\right\\}^{1_{c}}\,,$
(36) $\displaystyle
1_{c}{\otimes}\bar{3}_{c}{\otimes}3_{c}:\left\\{\left[\left(QQQ\right)^{1_{c}}{\otimes}\left(Q^{\prime}Q^{\prime}\right)^{\bar{3}_{c}}\right]^{\bar{3}_{c}}{\otimes}q^{3_{c}}\right\\}^{1_{c}}\,.$
(37)

Note that the last one is the dominant component of the scattering states. For
these systems, the color interaction reads

$\displaystyle\Braket{H_{\text{C}}\left(Q^{3}Q^{\prime 2}q\right)}$
$\displaystyle={}$ $\displaystyle
M-\frac{3}{8}\delta{m}_{\alpha}\cdot\mathrm{C}_{3}\left(QQQ\right)-\frac{3}{8}\delta{m}_{\beta}\cdot\mathrm{C}_{3}\left(Q^{\prime}Q^{\prime}\right)\,,$
(38)

where

$M=\frac{3m_{QQ}+2m_{Q^{\prime}Q^{\prime}}+m_{qQ}+m_{qQ^{\prime}}-m_{QQ^{\prime}}}{2}\,,$
(39)
$\delta{m}_{\alpha}=\frac{m_{QQ}+m_{qQ^{\prime}}-m_{qQ}-m_{QQ^{\prime}}}{4}\,,$
(40)

and

$\delta{m}_{\beta}=\frac{m_{Q^{\prime}Q^{\prime}}+m_{qQ}-m_{qQ^{\prime}}-m_{QQ^{\prime}}}{4}\,.$
(41)

The color interaction is diagonal over the three bases. For the $cccbbq$
dibaryons, $\delta{m}_{\alpha}\sim 20~{}\text{MeV}$ and
$\delta{m}_{\beta}\sim-50~{}\text{MeV}$. Thus the $\mathrm{C}_{3}(ccc)$ term
favors the bases with $(ccc)^{8_{c}}$ cluster, while the $\mathrm{C}_{3}(bb)$
term favors the bases with $(bb)^{\bar{3}_{c}}$ cluster. Since
$|\delta{m}_{\alpha}|<|\delta{m}_{\beta}|$, the $\mathrm{C}_{3}(bb)$ term
dominates the mass splitting while the $\mathrm{C}_{3}(ccc)$ term give smaller
splittings afterward. In summary, we have

$8_{c}{\otimes}\bar{3}_{c}{\otimes}3_{c}<1_{c}{\otimes}\bar{3}_{c}{\otimes}3_{c}\ll
8_{c}{\otimes}6_{c}{\otimes}3_{c}\,.$ (42)

Thus the scattering states of the $cccbbq$ systems lie in the middle of the
spectra. For the $bbbccq$ dibaryons, $\delta{m}_{\alpha}\sim-50~{}\text{MeV}$
and $\delta{m}_{\beta}\sim 20~{}\text{MeV}$. With similar argument, we have

$1_{c}{\otimes}\bar{3}_{c}{\otimes}3_{c}\ll
8_{c}{\otimes}6_{c}{\otimes}3_{c}<8_{c}{\otimes}\bar{3}_{c}{\otimes}3_{c}\,.$
(43)

Thus the scattering states of the $bbbccq$ systems lie in the bottom of the
spectra. When including the CM interaction, these color bases will be mixed.
Since the splittings between $1_{c}{\otimes}\bar{3}_{c}{\otimes}3_{c}$ base
and other bases are quite large in the $bbbccq$ system, the corresponding
mixing effects become small. Thus the proportions of $bbbccq$ scattering
states’ dominant components are always very large ($\sim 100\%$). On the
contrary, the splitting between the $1_{c}{\otimes}\bar{3}_{c}{\otimes}3_{c}$
and $8_{c}{\otimes}\bar{3}_{c}{\otimes}3_{c}$ bases is quite small in the
$cccbbq$ system, the corresponding mixing effect becomes pretty important. So
the proportions of $cccbbq$ scattering states’ dominant components become
significantly smaller (see Table 5).

After identifying the scattering states, the other states are genuine
dibaryons. They all lie far above their $S$-wave decay channel(s). Thus they
may be broad states. We also study their decay properties, which can be found
in Tables 13–18.

##  IV Conclusions

In this work, we have systematically studied the mass spectrum of dibaryons
with five heavy quarks, in the framework of an extended chromomagnetic model,
which includes both the colorelectric and chromomagnetic interactions. To do
this, we first constructed the flavor $\otimes$ color $\otimes$ spin wave
functions of the $Q^{5}q$ dibaryons to satisfy the $\mathrm{SU}(2)_{f}$
isospin symmetry and the Pauli principle. Then we systematically calculated
the interaction Hamiltonian under these bases and obtain the corresponding
masses. The numerical results suggest that all states are above the
corresponding baryon-baryon thresholds. Most of them are expected to be broad
states. However, the $D(bbbbcn,21306,0^{+})$, $D(bbbbcn,21251,1^{+})$,
$D(bbbbcs,21398,0^{+})$ and $D(bbbbcs,21341,1^{+})$ both have only one
$S$-wave decay channel and they lie above the corresponding channels by only
about $20~{}\text{MeV}$. Due to the small phases, these states may be
relatively narrow. In addition to computing the masses, we also use the wave
functions to investigate the decay properties of the dibaryons. We hope that
our studies can be of help for future experimental searches.

## Acknowledgments

X.Z.W is grateful to Professor Marek Karliner and Professor Shi-Lin Zhu for
helpful comments and discussions. This project was supported by the NSFC-ISF
under Grant No. 3423/19.

|  |
---|---|---
(a) $cccccn$ states |  | (b) $cccccs$ states
|  |
(c) $bbbbbn$ states |  | (d) $bbbbbs$ states
Figure 1: Masses of the $cccccq$ and $bbbbbq$ dibaryons. The dotted lines indicate various baryon-baryon thresholds, where the baryon masses are calculated in the same model [93]. The masses are all in units of MeV. Table 7: The eigenvectors of the $cccccq$ and $bbbbbq$ dibaryon states. The masses are all in units of MeV. System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Xi_{cc}^{*}$ | $\Omega_{ccc}^{*}\Xi_{cc}$ | System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Xi_{bb}^{*}$ | $\Omega_{bbb}^{*}\Xi_{bb}$
---|---|---|---|---|---|---|---|---|---
$cccccn$ | $0^{+}$ | $8599$ | $-0.447$ |  | $bbbbbn$ | $0^{+}$ | $24553$ | $-0.447$ |
| $1^{+}$ | $8516$ | $-0.333$ | $0.298$ |  | $1^{+}$ | $24526$ | $-0.333$ | $0.298$
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Omega_{cc}^{*}$ | $\Omega_{ccc}^{*}\Omega_{cc}$ | System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Omega_{bb}^{*}$ | $\Omega_{bbb}^{*}\Omega_{bb}$
$cccccs$ | $0^{+}$ | $8711$ | $-0.447$ |  | $bbbbbs$ | $0^{+}$ | $24624$ | $-0.447$ |
| $1^{+}$ | $8617$ | $-0.333$ | $0.298$ |  | $1^{+}$ | $24612$ | $-0.333$ | $0.298$
Table 8: The values of $k\cdot|c_{i}|^{2}$ for the $cccccq$ and $bbbbbq$ dibaryon states (in units of MeV). System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Xi_{cc}^{*}$ | $\Omega_{ccc}^{*}\Xi_{cc}$ | System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Xi_{bb}^{*}$ | $\Omega_{bbb}^{*}\Xi_{bb}$
---|---|---|---|---|---|---|---|---|---
$cccccn$ | $0^{+}$ | $8599$ | $140.6$ |  | $bbbbbn$ | $0^{+}$ | $24553$ | $160.7$ |
| $1^{+}$ | $8516$ | $41.8$ | $56.3$ |  | $1^{+}$ | $24526$ | $63.7$ | $66.8$
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Omega_{cc}^{*}$ | $\Omega_{ccc}^{*}\Omega_{cc}$ | System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Omega_{bb}^{*}$ | $\Omega_{bbb}^{*}\Omega_{bb}$
$cccccs$ | $0^{+}$ | $8711$ | $144.8$ |  | $bbbbbs$ | $0^{+}$ | $24624$ | $149.2$ |
| $1^{+}$ | $8617$ | $38.7$ | $57.5$ |  | $1^{+}$ | $24612$ | $72.1$ | $64.3$
Table 9: The partial width ratios for the $cccccq$ and $bbbbbq$ dibaryon states. 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 | $\Omega_{ccc}^{*}\Xi_{cc}^{*}$ | $\Omega_{ccc}^{*}\Xi_{cc}$ | System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Xi_{bb}^{*}$ | $\Omega_{bbb}^{*}\Xi_{bb}$
---|---|---|---|---|---|---|---|---|---
$cccccn$ | $0^{+}$ | $8599$ | $1$ |  | $bbbbbn$ | $0^{+}$ | $24553$ | $1$ |
| $1^{+}$ | $8516$ | $1$ | $1.3$ |  | $1^{+}$ | $24526$ | $1$ | $1.0$
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Omega_{cc}^{*}$ | $\Omega_{ccc}^{*}\Omega_{cc}$ | System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Omega_{bb}^{*}$ | $\Omega_{bbb}^{*}\Omega_{bb}$
$cccccs$ | $0^{+}$ | $8711$ | $1$ |  | $bbbbbs$ | $0^{+}$ | $24624$ | $1$ |
| $1^{+}$ | $8617$ | $1$ | $1.5$ |  | $1^{+}$ | $24612$ | $1$ | $0.9$
---
(a) $ccccbn$ states
(b) $ccccbs$ states
Figure 2: Masses of the $ccccbq$ dibaryons. The dotted lines indicate various
baryon-baryon thresholds, where the baryon masses are calculated in the same
model [93]. The masses are all in units of MeV.
---
(a) $bbbbcn$ states
(b) $bbbbcs$ states
Figure 3: Masses of the $bbbbcq$ dibaryons. The dotted lines indicate various baryon-baryon thresholds, where the baryon masses are calculated in the same model [93]. The masses are all in units of MeV. Table 10: The eigenvectors of the $ccccbq$ and $bbbbcq$ dibaryon states. The masses are all in units of MeV. |  |  | $ccc{\otimes}cbn$ | $ccb{\otimes}ccn$
---|---|---|---|---
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Xi_{cb}^{*}$ | $\Omega_{ccc}^{*}\Xi_{cb}$ | $\Omega_{ccc}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{ccb}^{*}\Xi_{cc}^{*}$ | $\Omega_{ccb}^{*}\Xi_{cc}$ | $\Omega_{ccb}\Xi_{cc}^{*}$ | $\Omega_{ccb}\Xi_{cc}$
$ccccbn$ | $0^{+}$ | $11693$ | $0.252$ |  |  | $-0.340$ |  |  | $-0.419$
|  | $11840$ | $0.519$ |  |  | $0.379$ |  |  | $-0.099$
| $1^{+}$ | $11698$ | $-0.059$ | $-0.068$ | $0.233$ | $-0.213$ | $0.356$ | $-0.304$ | $0.166$
|  | $11776$ | $-0.270$ | $0.289$ | $-0.382$ | $-0.307$ | $0.128$ | $0.113$ | $-0.092$
|  | $11806$ | $-0.449$ | $0.091$ | $0.305$ | $0.213$ | $0.153$ | $0.248$ | $0.034$
| $2^{+}$ | $11765$ | $0.408$ | $-0.377$ | $-0.156$ | $-0.272$ | $0.136$ | $-0.136$ |
|  |  | $ccc{\otimes}cbs$ | $ccb{\otimes}ccs$
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Omega_{cb}^{*}$ | $\Omega_{ccc}^{*}\Omega_{cb}$ | $\Omega_{ccc}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{ccb}^{*}\Omega_{cc}^{*}$ | $\Omega_{ccb}^{*}\Omega_{cc}$ | $\Omega_{ccb}\Omega_{cc}^{*}$ | $\Omega_{ccb}\Omega_{cc}$
$ccccbs$ | $0^{+}$ | $11793$ | $0.279$ |  |  | $-0.319$ |  |  | $-0.423$
|  | $11941$ | $0.505$ |  |  | $0.397$ |  |  | $-0.077$
| $1^{+}$ | $11802$ | $0.077$ | $0.057$ | $-0.266$ | $0.190$ | $-0.360$ | $0.291$ | $-0.170$
|  | $11871$ | $0.348$ | $-0.281$ | $0.331$ | $0.268$ | $-0.161$ | $-0.150$ | $0.081$
|  | $11907$ | $0.388$ | $-0.061$ | $-0.350$ | $-0.278$ | $-0.104$ | $-0.243$ | $-0.039$
| $2^{+}$ | $11855$ | $0.408$ | $-0.385$ | $-0.135$ | $-0.272$ | $0.136$ | $-0.136$ |
|  |  | $bbb{\otimes}cbn$ | $bbc{\otimes}bbn$
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Xi_{cb}^{*}$ | $\Omega_{bbb}^{*}\Xi_{cb}$ | $\Omega_{bbb}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{bbc}^{*}\Xi_{bb}^{*}$ | $\Omega_{bbc}^{*}\Xi_{bb}$ | $\Omega_{bbc}\Xi_{bb}^{*}$ | $\Omega_{bbc}\Xi_{bb}$
$bbbbcn$ | $0^{+}$ | $21306$ | $-0.567$ |  |  | $0.098$ |  |  | $-0.331$
|  | $21473$ | $0.111$ |  |  | $-0.500$ |  |  | $-0.275$
| $1^{+}$ | $21251$ | $0.022$ | $0.571$ | $-0.066$ | $0.215$ | $0.149$ | $-0.130$ | $0.167$
|  | $21303$ | $-0.526$ | $-0.009$ | $-0.236$ | $-0.007$ | $-0.217$ | $-0.254$ | $0.011$
|  | $21448$ | $0.022$ | $-0.052$ | $0.017$ | $-0.373$ | $0.312$ | $-0.292$ | $0.096$
| $2^{+}$ | $21286$ | $-0.408$ | $-0.053$ | $-0.405$ | $-0.272$ | $0.136$ | $-0.136$ |
|  |  | $bbb{\otimes}cbs$ | $bbc{\otimes}bbs$
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Omega_{cb}^{*}$ | $\Omega_{bbb}^{*}\Omega_{cb}$ | $\Omega_{bbb}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{bbc}^{*}\Omega_{bb}^{*}$ | $\Omega_{bbc}^{*}\Omega_{bb}$ | $\Omega_{bbc}\Omega_{bb}^{*}$ | $\Omega_{bbc}\Omega_{bb}$
$bbbbcs$ | $0^{+}$ | $21398$ | $-0.570$ |  |  | $0.113$ |  |  | $-0.323$
|  | $21550$ | $0.094$ |  |  | $-0.497$ |  |  | $-0.284$
| $1^{+}$ | $21341$ | $0.043$ | $0.567$ | $-0.088$ | $0.222$ | $0.153$ | $-0.114$ | $0.164$
|  | $21395$ | $-0.523$ | $-0.001$ | $-0.238$ | $-0.016$ | $-0.195$ | $-0.273$ | $0.023$
|  | $21524$ | $0.048$ | $-0.041$ | $0.030$ | $-0.369$ | $0.325$ | $-0.281$ | $0.098$
| $2^{+}$ | $21383$ | $-0.408$ | $-0.076$ | $-0.401$ | $-0.272$ | $0.136$ | $-0.136$ |
Table 11: The values of $k\cdot|c_{i}|^{2}$ for the $ccccbq$ and $bbbbcq$ dibaryon states (in units of MeV). |  |  | $ccc{\otimes}cbn$ | $ccb{\otimes}ccn$
---|---|---|---|---
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Xi_{cb}^{*}$ | $\Omega_{ccc}^{*}\Xi_{cb}$ | $\Omega_{ccc}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{ccb}^{*}\Xi_{cc}^{*}$ | $\Omega_{ccb}^{*}\Xi_{cc}$ | $\Omega_{ccb}\Xi_{cc}^{*}$ | $\Omega_{ccb}\Xi_{cc}$
$ccccbn$ | $0^{+}$ | $11693$ | $\times$ |  |  | $\times$ |  |  | $103.8$
|  | $11840$ | $183.9$ |  |  | $113.6$ |  |  | $10.3$
| $1^{+}$ | $11698$ | $\times$ | $\times$ | $\times$ | $\times$ | $58.5$ | $21.8$ | $16.7$
|  | $11776$ | $22.6$ | $51.7$ | $71.2$ | $51.1$ | $12.8$ | $8.7$ | $7.4$
|  | $11806$ | $104.2$ | $6.2$ | $59.8$ | $30.4$ | $20.4$ | $48.1$ | $1.1$
| $2^{+}$ | $11765$ | $31.8$ | $81.1$ | $10.3$ | $36.3$ | $13.8$ | $11.7$ |
|  |  | $ccc{\otimes}cbs$ | $ccb{\otimes}ccs$
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Omega_{cb}^{*}$ | $\Omega_{ccc}^{*}\Omega_{cb}$ | $\Omega_{ccc}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{ccb}^{*}\Omega_{cc}^{*}$ | $\Omega_{ccb}^{*}\Omega_{cc}$ | $\Omega_{ccb}\Omega_{cc}^{*}$ | $\Omega_{ccb}\Omega_{cc}$
$ccccbs$ | $0^{+}$ | $11793$ | $\times$ |  |  | $\times$ |  |  | $108.0$
|  | $11941$ | $182.8$ |  |  | $122.4$ |  |  | $6.3$
| $1^{+}$ | $11802$ | $\times$ | $0.6$ | $\times$ | $\times$ | $64.8$ | $18.7$ | $18.6$
|  | $11871$ | $41.1$ | $51.6$ | $51.7$ | $35.5$ | $20.2$ | $14.4$ | $5.7$
|  | $11907$ | $84.9$ | $3.0$ | $79.8$ | $50.7$ | $9.5$ | $45.6$ | $1.5$
| $2^{+}$ | $11855$ | $24.6$ | $86.0$ | $6.5$ | $29.6$ | $13.4$ | $10.5$ |
|  |  | $bbb{\otimes}cbn$ | $bbc{\otimes}bbn$
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Xi_{cb}^{*}$ | $\Omega_{bbb}^{*}\Xi_{cb}$ | $\Omega_{bbb}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{bbc}^{*}\Xi_{bb}^{*}$ | $\Omega_{bbc}^{*}\Xi_{bb}$ | $\Omega_{bbc}\Xi_{bb}^{*}$ | $\Omega_{bbc}\Xi_{bb}$
$bbbbcn$ | $0^{+}$ | $21306$ | $148.7$ |  |  | $\times$ |  |  | $\times$
|  | $21473$ | $16.4$ |  |  | $241.4$ |  |  | $92.0$
| $1^{+}$ | $21251$ | $\times$ | $137.5$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $21303$ | $120.7$ | $0.1$ | $36.4$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $21448$ | $0.6$ | $3.9$ | $0.4$ | $113.4$ | $91.3$ | $85.6$ | $10.1$
| $2^{+}$ | $21286$ | $29.4$ | $2.0$ | $84.7$ | $\times$ | $\times$ | $\times$ |
|  |  | $bbb{\otimes}cbs$ | $bbc{\otimes}bbs$
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Omega_{cb}^{*}$ | $\Omega_{bbb}^{*}\Omega_{cb}$ | $\Omega_{bbb}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{bbc}^{*}\Omega_{bb}^{*}$ | $\Omega_{bbc}^{*}\Omega_{bb}$ | $\Omega_{bbc}\Omega_{bb}^{*}$ | $\Omega_{bbc}\Omega_{bb}$
$bbbbcs$ | $0^{+}$ | $21398$ | $151.5$ |  |  | $\times$ |  |  | $\times$
|  | $21550$ | $11.5$ |  |  | $236.7$ |  |  | $93.9$
| $1^{+}$ | $21341$ | $\times$ | $140.5$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $21395$ | $118.2$ | $\sim 0$ | $34.1$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $21524$ | $2.8$ | $2.3$ | $1.1$ | $109.4$ | $90.7$ | $78.6$ | $9.9$
| $2^{+}$ | $21383$ | $43.3$ | $4.4$ | $79.5$ | $\times$ | $\times$ | $\times$ |
Table 12: The partial width ratios for the $ccccbq$ and $bbbbcq$ dibaryon states. 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. For a dibaryon state decays into two channels through two different quark rearrangements, the $\gamma_{i}$’s, which depend on the wave functions of final states, may not equal (or approximately equal). Thus we compare only the decay width ratios between channels through same quark rearrangement type. The masses are all in unit of MeV. |  |  | $ccc{\otimes}cbn$ | $ccb{\otimes}ccn$
---|---|---|---|---
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Xi_{cb}^{*}$ | $\Omega_{ccc}^{*}\Xi_{cb}$ | $\Omega_{ccc}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{ccb}^{*}\Xi_{cc}^{*}$ | $\Omega_{ccb}^{*}\Xi_{cc}$ | $\Omega_{ccb}\Xi_{cc}^{*}$ | $\Omega_{ccb}\Xi_{cc}$
$ccccbn$ | $0^{+}$ | $11693$ | $\times$ |  |  | $\times$ |  |  | $1$
|  | $11840$ | $1$ |  |  | $11.0$ |  |  | $1$
| $1^{+}$ | $11698$ | $\times$ | $\times$ | $\times$ | $\times$ | $3.5$ | $1.3$ | $1$
|  | $11776$ | $1$ | $2.3$ | $3.1$ | $6.9$ | $1.7$ | $1.2$ | $1$
|  | $11806$ | $1$ | $0.1$ | $0.6$ | $28.0$ | $18.8$ | $44.4$ | $1$
| $2^{+}$ | $11765$ | $1$ | $2.6$ | $0.3$ | $1$ | $0.4$ | $0.3$ |
|  |  | $ccc{\otimes}cbs$ | $ccb{\otimes}ccs$
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Omega_{cb}^{*}$ | $\Omega_{ccc}^{*}\Omega_{cb}$ | $\Omega_{ccc}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{ccb}^{*}\Omega_{cc}^{*}$ | $\Omega_{ccb}^{*}\Omega_{cc}$ | $\Omega_{ccb}\Omega_{cc}^{*}$ | $\Omega_{ccb}\Omega_{cc}$
$ccccbs$ | $0^{+}$ | $11793$ | $\times$ |  |  | $\times$ |  |  | $1$
|  | $11941$ | $1$ |  |  | $19.5$ |  |  | $1$
| $1^{+}$ | $11802$ | $\times$ | $1$ | $\times$ | $\times$ | $3.5$ | $1.0$ | $1$
|  | $11871$ | $0.8$ | $1$ | $1.0$ | $6.2$ | $3.5$ | $2.5$ | $1$
|  | $11907$ | $28.3$ | $1$ | $26.6$ | $34.9$ | $6.6$ | $31.4$ | $1$
| $2^{+}$ | $11855$ | $1$ | $3.5$ | $0.3$ | $1$ | $0.5$ | $0.4$ |
|  |  | $bbb{\otimes}cbn$ | $bbc{\otimes}bbn$
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Xi_{cb}^{*}$ | $\Omega_{bbb}^{*}\Xi_{cb}$ | $\Omega_{bbb}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{bbc}^{*}\Xi_{bb}^{*}$ | $\Omega_{bbc}^{*}\Xi_{bb}$ | $\Omega_{bbc}\Xi_{bb}^{*}$ | $\Omega_{bbc}\Xi_{bb}$
$bbbbcn$ | $0^{+}$ | $21306$ | $1$ |  |  | $\times$ |  |  | $\times$
|  | $21473$ | $1$ |  |  | $2.6$ |  |  | $1$
| $1^{+}$ | $21251$ | $\times$ | $1$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $21303$ | $3.3$ | $\sim 0$ | $1$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $21448$ | $0.2$ | $1$ | $0.1$ | $11.2$ | $9.0$ | $8.5$ | $1$
| $2^{+}$ | $21286$ | $14.4$ | $1$ | $41.6$ | $\times$ | $\times$ | $\times$ |
|  |  | $bbb{\otimes}cbs$ | $bbc{\otimes}bbs$
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Omega_{cb}^{*}$ | $\Omega_{bbb}^{*}\Omega_{cb}$ | $\Omega_{bbb}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{bbc}^{*}\Omega_{bb}^{*}$ | $\Omega_{bbc}^{*}\Omega_{bb}$ | $\Omega_{bbc}\Omega_{bb}^{*}$ | $\Omega_{bbc}\Omega_{bb}$
$bbbbcs$ | $0^{+}$ | $21398$ | $1$ |  |  | $\times$ |  |  | $\times$
|  | $21550$ | $1$ |  |  | $2.5$ |  |  | $1$
| $1^{+}$ | $21341$ | $\times$ | $1$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $21395$ | $3.5$ | $\sim 0$ | $1$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $21524$ | $2.5$ | $2.1$ | $1$ | $11.0$ | $9.1$ | $7.9$ | $1$
| $2^{+}$ | $21383$ | $9.8$ | $1$ | $18.0$ | $\times$ | $\times$ | $\times$ |
---
(a) $cccbbn$ states
(b) $cccbbs$ states
Figure 4: Masses of the $cccbbq$ dibaryons. The dotted lines indicate various
baryon-baryon thresholds, where the baryon masses are calculated in the same
model [93]. Possible scattering states are marked with a dagger ($\dagger$),
along with the proportion of their dominant components. The masses are all in
units of MeV.
---
(a) $bbbccn$ states
(b) $bbbccs$ states
Figure 5: Masses of the $bbbccq$ dibaryons. The dotted lines indicate various baryon-baryon thresholds, where the baryon masses are calculated in the same model [93]. Possible scattering states are marked with a dagger ($\dagger$), along with the proportion of their dominant components. The masses are all in units of MeV. Table 13: The eigenvectors of the $cccbbq$ dibaryon states. The scattering states are marked with a dagger ($\dagger$). The masses are all in units of MeV. |  |  | $ccc{\otimes}bbn$ | $ccb{\otimes}cbn$ | $ccn{\otimes}bbc$
---|---|---|---|---|---
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Xi_{bb}^{*}$ | $\Omega_{ccc}^{*}\Xi_{bb}$ | $\Omega_{ccb}^{*}\Xi_{cb}^{*}$ | $\Omega_{ccb}^{*}\Xi_{cb}$ | $\Omega_{ccb}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{ccb}\Xi_{cb}^{*}$ | $\Omega_{ccb}\Xi_{cb}$ | $\Omega_{ccb}\Xi_{cb}^{\prime}$ | $\Xi_{cc}^{*}\Omega_{bbc}^{*}$ | $\Xi_{cc}^{*}\Omega_{bbc}$ | $\Xi_{cc}\Omega_{bbc}^{*}$ | $\Xi_{cc}\Omega_{bbc}$
$cccbbn$ | $0^{+}$ | $14887$ | $0.369$ |  | $0.064$ |  |  |  | $0.253$ | $-0.212$ | $0.541$ |  |  | $-0.322$
|  | $14981^{\dagger}$ | $0.899$ |  | $-0.017$ |  |  |  | $0.220$ | $0.284$ | $-0.308$ |  |  | $-0.209$
|  | $15093$ | $0.237$ |  | $-0.505$ |  |  |  | $-0.003$ | $-0.242$ | $-0.142$ |  |  | $-0.034$
| $1^{+}$ | $14833$ | $0.092$ | $0.073$ | $0.177$ | $0.029$ | $-0.209$ | $0.166$ | $0.029$ | $-0.090$ | $0.086$ | $0.049$ | $-0.487$ | $-0.437$
|  | $14884$ | $-0.266$ | $-0.241$ | $-0.000$ | $-0.120$ | $0.083$ | $0.190$ | $-0.234$ | $0.030$ | $-0.471$ | $-0.251$ | $-0.288$ | $0.174$
|  | $14953^{\dagger}$ | $0.357$ | $-0.890$ | $0.010$ | $-0.075$ | $-0.080$ | $0.035$ | $0.102$ | $0.320$ | $0.199$ | $-0.261$ | $-0.009$ | $-0.023$
|  | $14984$ | $-0.859$ | $-0.336$ | $-0.036$ | $-0.096$ | $-0.206$ | $-0.173$ | $-0.178$ | $-0.021$ | $0.295$ | $0.148$ | $-0.138$ | $0.159$
|  | $15039$ | $-0.236$ | $0.177$ | $0.374$ | $-0.265$ | $0.187$ | $-0.170$ | $0.159$ | $0.112$ | $-0.051$ | $0.071$ | $-0.078$ | $0.014$
| $2^{+}$ | $14872$ | $-0.097$ | $-0.303$ | $0.166$ | $-0.264$ | $-0.015$ | $0.117$ |  |  | $-0.365$ | $-0.433$ | $-0.300$ |
|  | $14962^{\dagger}$ | $0.234$ | $-0.933$ | $-0.139$ | $-0.047$ | $0.272$ | $0.125$ |  |  | $0.355$ | $-0.034$ | $-0.095$ |
|  | $14976^{\dagger}$ | $0.967$ | $0.195$ | $0.165$ | $0.197$ | $0.021$ | $0.211$ |  |  | $-0.008$ | $-0.265$ | $0.222$ |
| $3^{+}$ | $14974^{\dagger}$ | $1$ |  | $0.333$ |  |  |  |  |  | $0.333$ |  |  |
|  |  | $ccc{\otimes}bbs$ | $ccb{\otimes}cbs$ | $ccs{\otimes}bbc$
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Omega_{bb}^{*}$ | $\Omega_{ccc}^{*}\Omega_{bb}$ | $\Omega_{ccb}^{*}\Omega_{cb}^{*}$ | $\Omega_{ccb}^{*}\Omega_{cb}$ | $\Omega_{ccb}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{ccb}\Omega_{cb}^{*}$ | $\Omega_{ccb}\Omega_{cb}$ | $\Omega_{ccb}\Omega_{cb}^{\prime}$ | $\Omega_{cc}^{*}\Omega_{bbc}^{*}$ | $\Omega_{cc}^{*}\Omega_{bbc}$ | $\Omega_{cc}\Omega_{bbc}^{*}$ | $\Omega_{cc}\Omega_{bbc}$
$cccbbs$ | $0^{+}$ | $14986$ | $0.507$ |  | $0.058$ |  |  |  | $0.275$ | $-0.181$ | $0.485$ |  |  | $-0.351$
|  | $15069$ | $0.835$ |  | $-0.040$ |  |  |  | $0.194$ | $0.297$ | $-0.393$ |  |  | $-0.156$
|  | $15187$ | $0.214$ |  | $-0.504$ |  |  |  | $-0.022$ | $-0.248$ | $-0.133$ |  |  | $-0.028$
| $1^{+}$ | $14934$ | $-0.120$ | $-0.084$ | $-0.181$ | $-0.020$ | $0.206$ | $-0.164$ | $-0.036$ | $0.090$ | $-0.092$ | $-0.051$ | $0.477$ | $0.445$
|  | $14984$ | $-0.371$ | $-0.271$ | $-0.005$ | $-0.123$ | $0.055$ | $0.173$ | $-0.255$ | $0.033$ | $-0.428$ | $-0.227$ | $-0.316$ | $0.181$
|  | $15041$ | $0.465$ | $-0.844$ | $0.010$ | $-0.065$ | $-0.045$ | $0.065$ | $0.134$ | $0.313$ | $0.142$ | $-0.287$ | $0.007$ | $-0.035$
|  | $15072$ | $-0.773$ | $-0.411$ | $-0.004$ | $-0.120$ | $-0.208$ | $-0.197$ | $-0.117$ | $0.034$ | $0.373$ | $0.145$ | $-0.116$ | $0.123$
|  | $15127$ | $0.182$ | $-0.195$ | $-0.374$ | $0.248$ | $-0.220$ | $0.154$ | $-0.171$ | $-0.105$ | $0.083$ | $-0.055$ | $0.070$ | $-0.011$
| $2^{+}$ | $14976$ | $-0.154$ | $-0.382$ | $0.143$ | $-0.280$ | $0.019$ | $0.109$ |  |  | $-0.337$ | $-0.413$ | $-0.322$ |
|  | $15051$ | $0.764$ | $-0.631$ | $-0.015$ | $0.095$ | $0.226$ | $0.225$ |  |  | $0.285$ | $-0.180$ | $0.060$ |
|  | $15061$ | $-0.626$ | $-0.676$ | $-0.231$ | $-0.149$ | $0.154$ | $-0.107$ |  |  | $0.254$ | $0.237$ | $-0.203$ |
| $3^{+}$ | $15053^{\dagger}$ | $1$ |  | $0.333$ |  |  |  |  |  | $0.333$ |  |  |
Table 14: The values of $k\cdot|c_{i}|^{2}$ for the $cccbbq$ dibaryon states (in units of MeV). The scattering states are marked with a dagger ($\dagger$). |  |  | $ccc{\otimes}bbn$ | $ccb{\otimes}cbn$ | $ccn{\otimes}bbc$
---|---|---|---|---|---
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Xi_{bb}^{*}$ | $\Omega_{ccc}^{*}\Xi_{bb}$ | $\Omega_{ccb}^{*}\Xi_{cb}^{*}$ | $\Omega_{ccb}^{*}\Xi_{cb}$ | $\Omega_{ccb}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{ccb}\Xi_{cb}^{*}$ | $\Omega_{ccb}\Xi_{cb}$ | $\Omega_{ccb}\Xi_{cb}^{\prime}$ | $\Xi_{cc}^{*}\Omega_{bbc}^{*}$ | $\Xi_{cc}^{*}\Omega_{bbc}$ | $\Xi_{cc}\Omega_{bbc}^{*}$ | $\Xi_{cc}\Omega_{bbc}$
$cccbbn$ | $0^{+}$ | $14887$ | $\times$ |  | $\times$ |  |  |  | $\times$ | $\times$ | $\times$ |  |  | $72.3$
|  | $14981^{\dagger}$ | $167.0$ |  | $\times$ |  |  |  | $34.5$ | $45.5$ | $66.8$ |  |  | $44.0$
|  | $15093$ | $49.3$ |  | $218.2$ |  |  |  | $0.01$ | $62.8$ | $21.3$ |  |  | $1.4$
| $1^{+}$ | $14833$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $32.4$ | $83.5$
|  | $14884$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $22.5$ | $45.4$ | $20.8$
|  | $14953^{\dagger}$ | $\times$ | $\times$ | $\times$ | $1.4$ | $\times$ | $\times$ | $5.7$ | $33.9$ | $23.1$ | $48.8$ | $0.1$ | $0.5$
|  | $14984$ | $180.1$ | $49.1$ | $\times$ | $5.0$ | $13.7$ | $11.5$ | $23.0$ | $0.3$ | $62.0$ | $18.1$ | $17.6$ | $25.6$
|  | $15039$ | $36.0$ | $23.2$ | $79.9$ | $58.8$ | $25.2$ | $21.5$ | $24.4$ | $10.9$ | $2.3$ | $5.1$ | $6.6$ | $0.2$
| $2^{+}$ | $14872$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $47.2$ | $43.5$ |
|  | $14962^{\dagger}$ | $\times$ | $194.6$ | $\times$ | $0.8$ | $\times$ | $\times$ |  |  | $78.6$ | $0.9$ | $7.7$ |
|  | $14976^{\dagger}$ | $99.8$ | $14.3$ | $\times$ | $19.0$ | $0.1$ | $13.7$ |  |  | $0.04$ | $56.4$ | $44.6$ |
| $3^{+}$ | $14974^{\dagger}$ | $0.05$ |  | $\times$ |  |  |  |  |  | $75.2$ |  |  |
|  |  | $ccc{\otimes}bbs$ | $ccb{\otimes}cbs$ | $ccs{\otimes}bbc$
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Omega_{bb}^{*}$ | $\Omega_{ccc}^{*}\Omega_{bb}$ | $\Omega_{ccb}^{*}\Omega_{cb}^{*}$ | $\Omega_{ccb}^{*}\Omega_{cb}$ | $\Omega_{ccb}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{ccb}\Omega_{cb}^{*}$ | $\Omega_{ccb}\Omega_{cb}$ | $\Omega_{ccb}\Omega_{cb}^{\prime}$ | $\Omega_{cc}^{*}\Omega_{bbc}^{*}$ | $\Omega_{cc}^{*}\Omega_{bbc}$ | $\Omega_{cc}\Omega_{bbc}^{*}$ | $\Omega_{cc}\Omega_{bbc}$
$cccbbs$ | $0^{+}$ | $14986$ | $\times$ |  | $\times$ |  |  |  | $\times$ | $\times$ | $\times$ |  |  | $87.1$
|  | $15069$ | $225.8$ |  | $\times$ |  |  |  | $26.9$ | $43.2$ | $97.9$ |  |  | $24.1$
|  | $15187$ | $42.7$ |  | $219.8$ |  |  |  | $0.6$ | $65.3$ | $18.5$ |  |  | $1.0$
| $1^{+}$ | $14934$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $42.3$ | $90.8$
|  | $14984$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $15.8$ | $55.9$ | $22.9$
|  | $15041$ | $\times$ | $\times$ | $\times$ | $1.1$ | $\times$ | $\times$ | $9.9$ | $16.5$ | $9.9$ | $53.5$ | $0.03$ | $1.1$
|  | $15072$ | $210.9$ | $71.5$ | $\times$ | $7.9$ | $6.9$ | $13.5$ | $10.0$ | $0.6$ | $89.9$ | $16.3$ | $12.2$ | $15.2$
|  | $15127$ | $23.1$ | $28.1$ | $76.6$ | $52.0$ | $32.0$ | $17.4$ | $28.5$ | $9.0$ | $6.0$ | $2.9$ | $5.2$ | $0.1$
| $2^{+}$ | $14976$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $37.8$ | $53.6$ |
|  | $15051$ | $\times$ | $79.7$ | $\times$ | $3.4$ | $\times$ | $\times$ |  |  | $44.2$ | $22.3$ | $3.0$ |
|  | $15061$ | $90.3$ | $150.3$ | $\times$ | $10.4$ | $\times$ | $2.3$ |  |  | $38.4$ | $41.2$ | $35.6$ |
| $3^{+}$ | $15053^{\dagger}$ | $\times$ |  | $\times$ |  |  |  |  |  | $61.8$ |  |  |
Table 15: The partial width ratios for the $cccbbq$ dibaryon states. 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. For a dibaryon state decays into two channels through two different quark rearrangements, the $\gamma_{i}$’s, which depend on the wave functions of final states, may not equal (or approximately equal). Thus we compare only the decay width ratios between channels through same quark rearrangement type. The scattering states are marked with a dagger ($\dagger$). The masses are all in unit of MeV. |  |  | $ccc{\otimes}bbn$ | $ccb{\otimes}cbn$ | $ccn{\otimes}bbc$
---|---|---|---|---|---
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Xi_{bb}^{*}$ | $\Omega_{ccc}^{*}\Xi_{bb}$ | $\Omega_{ccb}^{*}\Xi_{cb}^{*}$ | $\Omega_{ccb}^{*}\Xi_{cb}$ | $\Omega_{ccb}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{ccb}\Xi_{cb}^{*}$ | $\Omega_{ccb}\Xi_{cb}$ | $\Omega_{ccb}\Xi_{cb}^{\prime}$ | $\Xi_{cc}^{*}\Omega_{bbc}^{*}$ | $\Xi_{cc}^{*}\Omega_{bbc}$ | $\Xi_{cc}\Omega_{bbc}^{*}$ | $\Xi_{cc}\Omega_{bbc}$
$cccbbn$ | $0^{+}$ | $14887$ | $\times$ |  | $\times$ |  |  |  | $\times$ | $\times$ | $\times$ |  |  | $1$
|  | $14981^{\dagger}$ | $1$ |  | $\times$ |  |  |  | $0.8$ | $1$ | $1.5$ |  |  | $1$
|  | $15093$ | $1$ |  | $3.5$ |  |  |  | $\sim 0$ | $1$ | $14.7$ |  |  | $1$
| $1^{+}$ | $14833$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $0.4$ | $1$
|  | $14884$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $1.1$ | $2.2$ | $1$
|  | $14953^{\dagger}$ | $\times$ | $\times$ | $\times$ | $1$ | $\times$ | $\times$ | $3.9$ | $23.6$ | $1$ | $2.1$ | $\sim 0$ | $\sim 0$
|  | $14984$ | $1$ | $0.3$ | $\times$ | $1$ | $2.7$ | $2.3$ | $4.6$ | $0.1$ | $2.4$ | $0.7$ | $0.7$ | $1$
|  | $15039$ | $1$ | $0.6$ | $7.4$ | $5.4$ | $2.3$ | $2.0$ | $2.2$ | $1$ | $1$ | $2.2$ | $2.8$ | $0.1$
| $2^{+}$ | $14872$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $1.1$ | $1$ |
|  | $14962^{\dagger}$ | $\times$ | $1$ | $\times$ | $1$ | $\times$ | $\times$ |  |  | $10.3$ | $0.1$ | $1$ |
|  | $14976^{\dagger}$ | $7.0$ | $1$ | $\times$ | $1$ | $\sim 0$ | $0.7$ |  |  | $\sim 0$ | $1.3$ | $1$ |
| $3^{+}$ | $14974^{\dagger}$ | $1$ |  | $\times$ |  |  |  |  |  | $1$ |  |  |
|  |  | $ccc{\otimes}bbs$ | $ccb{\otimes}cbs$ | $ccs{\otimes}bbc$
System | $J^{P}$ | Mass | $\Omega_{ccc}^{*}\Omega_{bb}^{*}$ | $\Omega_{ccc}^{*}\Omega_{bb}$ | $\Omega_{ccb}^{*}\Omega_{cb}^{*}$ | $\Omega_{ccb}^{*}\Omega_{cb}$ | $\Omega_{ccb}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{ccb}\Omega_{cb}^{*}$ | $\Omega_{ccb}\Omega_{cb}$ | $\Omega_{ccb}\Omega_{cb}^{\prime}$ | $\Omega_{cc}^{*}\Omega_{bbc}^{*}$ | $\Omega_{cc}^{*}\Omega_{bbc}$ | $\Omega_{cc}\Omega_{bbc}^{*}$ | $\Omega_{cc}\Omega_{bbc}$
$cccbbs$ | $0^{+}$ | $14986$ | $\times$ |  | $\times$ |  |  |  | $\times$ | $\times$ | $\times$ |  |  | $1$
|  | $15069$ | $1$ |  | $\times$ |  |  |  | $0.6$ | $1$ | $4.1$ |  |  | $1$
|  | $15187$ | $1$ |  | $3.4$ |  |  |  | $0.01$ | $1$ | $18.0$ |  |  | $1$
| $1^{+}$ | $14934$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $0.5$ | $1$
|  | $14984$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $0.7$ | $2.4$ | $1$
|  | $15041$ | $\times$ | $\times$ | $\times$ | $1$ | $\times$ | $\times$ | $9.3$ | $15.5$ | $8.7$ | $47.0$ | $0.03$ | $1$
|  | $15072$ | $2.9$ | $1$ | $\times$ | $1$ | $0.9$ | $1.7$ | $1.3$ | $0.1$ | $5.9$ | $1.1$ | $0.8$ | $1$
|  | $15127$ | $0.8$ | $1$ | $8.5$ | $5.7$ | $3.5$ | $1.9$ | $3.2$ | $1$ | $2.1$ | $1$ | $1.8$ | $0.05$
| $2^{+}$ | $14976$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $0.7$ | $1$ |
|  | $15051$ | $\times$ | $1$ | $\times$ | $1$ | $\times$ | $\times$ |  |  | $14.9$ | $7.5$ | $1$ |
|  | $15061$ | $0.6$ | $1$ | $\times$ | $1$ | $\times$ | $0.2$ |  |  | $1.1$ | $1.2$ | $1$ |
| $3^{+}$ | $15053^{\dagger}$ | $\times$ |  | $\times$ |  |  |  |  |  | $1$ |  |  |
Table 16: The eigenvectors of the $bbbccq$ dibaryon states. The scattering states are marked with a dagger ($\dagger$). The masses are all in units of MeV. |  |  | $bbb{\otimes}ccn$ | $bbc{\otimes}cbn$ | $bbn{\otimes}ccb$
---|---|---|---|---|---
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Xi_{cc}^{*}$ | $\Omega_{bbb}^{*}\Xi_{cc}$ | $\Omega_{bbc}^{*}\Xi_{cb}^{*}$ | $\Omega_{bbc}^{*}\Xi_{cb}$ | $\Omega_{bbc}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{bbc}\Xi_{cb}^{*}$ | $\Omega_{bbc}\Xi_{cb}$ | $\Omega_{bbc}\Xi_{cb}^{\prime}$ | $\Xi_{bb}^{*}\Omega_{ccb}^{*}$ | $\Xi_{bb}^{*}\Omega_{ccb}$ | $\Xi_{bb}\Omega_{ccb}^{*}$ | $\Xi_{bb}\Omega_{ccb}$
$bbbccn$ | $0^{+}$ | $18003^{\dagger}$ | $0.993$ |  | $0.051$ |  |  |  | $0.061$ | $0.328$ | $-0.081$ |  |  | $-0.322$
|  | $18167$ | $-0.077$ |  | $-0.361$ |  |  |  | $0.395$ | $0.049$ | $-0.269$ |  |  | $0.122$
|  | $18280$ | $0.090$ |  | $0.356$ |  |  |  | $0.120$ | $-0.111$ | $-0.573$ |  |  | $0.171$
| $1^{+}$ | $17941^{\dagger}$ | $-0.009$ | $0.996$ | $-0.063$ | $0.082$ | $0.127$ | $0.032$ | $-0.289$ | $-0.030$ | $-0.140$ | $0.271$ | $0.066$ | $-0.119$
|  | $18004^{\dagger}$ | $-0.994$ | $-0.013$ | $0.004$ | $-0.033$ | $-0.146$ | $0.149$ | $-0.035$ | $-0.259$ | $0.019$ | $0.152$ | $-0.152$ | $0.254$
|  | $18161$ | $0.071$ | $-0.073$ | $0.252$ | $-0.311$ | $0.153$ | $-0.218$ | $-0.124$ | $-0.224$ | $0.041$ | $-0.017$ | $-0.196$ | $-0.167$
|  | $18233$ | $0.074$ | $0.003$ | $0.320$ | $0.101$ | $0.142$ | $0.052$ | $0.016$ | $-0.091$ | $-0.118$ | $-0.063$ | $0.476$ | $0.365$
|  | $18250$ | $-0.024$ | $-0.040$ | $0.054$ | $-0.192$ | $0.040$ | $0.224$ | $-0.040$ | $0.135$ | $0.568$ | $0.244$ | $0.232$ | $-0.087$
| $2^{+}$ | $17942^{\dagger}$ | $0.003$ | $-0.999$ | $0.102$ | $0.278$ | $-0.101$ | $-0.116$ |  |  | $0.204$ | $-0.130$ | $-0.231$ |
|  | $18006^{\dagger}$ | $1.000$ | $0.003$ | $-0.115$ | $0.028$ | $0.217$ | $-0.224$ |  |  | $0.103$ | $-0.229$ | $0.219$ |
|  | $18230$ | $-0.017$ | $0.041$ | $-0.225$ | $-0.041$ | $-0.220$ | $-0.103$ |  |  | $-0.455$ | $-0.436$ | $-0.217$ |
| $3^{+}$ | $18006^{\dagger}$ | $1$ |  | $-0.333$ |  |  |  |  |  | $0.333$ |  |  |
|  |  | $bbb{\otimes}ccs$ | $bbc{\otimes}cbs$ | $bbs{\otimes}ccb$
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Omega_{cc}^{*}$ | $\Omega_{bbb}^{*}\Omega_{cc}$ | $\Omega_{bbc}^{*}\Omega_{cb}^{*}$ | $\Omega_{bbc}^{*}\Omega_{cb}$ | $\Omega_{bbc}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{bbc}\Omega_{cb}^{*}$ | $\Omega_{bbc}\Omega_{cb}$ | $\Omega_{bbc}\Omega_{cb}^{\prime}$ | $\Omega_{bb}^{*}\Omega_{ccb}^{*}$ | $\Omega_{bb}^{*}\Omega_{ccb}$ | $\Omega_{bb}\Omega_{ccb}^{*}$ | $\Omega_{bb}\Omega_{ccb}$
$bbbccs$ | $0^{+}$ | $18109^{\dagger}$ | $0.992$ |  | $0.047$ |  |  |  | $0.091$ | $0.323$ | $-0.095$ |  |  | $-0.317$
|  | $18251$ | $-0.098$ |  | $-0.343$ |  |  |  | $0.401$ | $0.012$ | $-0.296$ |  |  | $0.140$
|  | $18362$ | $0.083$ |  | $0.374$ |  |  |  | $0.091$ | $-0.122$ | $-0.557$ |  |  | $0.168$
| $1^{+}$ | $18040^{\dagger}$ | $-0.011$ | $0.997$ | $-0.063$ | $0.089$ | $0.122$ | $0.031$ | $-0.290$ | $-0.016$ | $-0.142$ | $0.270$ | $0.064$ | $-0.118$
|  | $18110^{\dagger}$ | $-0.994$ | $-0.016$ | $0.006$ | $-0.046$ | $-0.142$ | $0.152$ | $-0.050$ | $-0.255$ | $0.030$ | $0.156$ | $-0.149$ | $0.251$
|  | $18256$ | $0.069$ | $-0.075$ | $0.226$ | $-0.315$ | $0.160$ | $-0.216$ | $-0.137$ | $-0.204$ | $0.065$ | $-0.006$ | $-0.229$ | $-0.200$
|  | $18323$ | $0.078$ | $0.004$ | $0.322$ | $0.119$ | $0.136$ | $-0.015$ | $0.006$ | $-0.138$ | $-0.231$ | $-0.114$ | $0.397$ | $0.360$
|  | $18328$ | $0.010$ | $-0.033$ | $0.119$ | $-0.159$ | $0.080$ | $0.228$ | $-0.029$ | $0.120$ | $0.529$ | $0.224$ | $0.331$ | $-0.012$
| $2^{+}$ | $18041^{\dagger}$ | $0.003$ | $-0.9995$ | $0.104$ | $0.272$ | $-0.114$ | $-0.115$ |  |  | $0.209$ | $-0.125$ | $-0.228$ |
|  | $18112^{\dagger}$ | $0.999$ | $0.004$ | $-0.120$ | $0.038$ | $0.210$ | $-0.226$ |  |  | $0.093$ | $-0.239$ | $0.214$ |
|  | $18308$ | $-0.039$ | $0.030$ | $-0.221$ | $-0.051$ | $-0.223$ | $-0.099$ |  |  | $-0.455$ | $-0.432$ | $-0.224$ |
| $3^{+}$ | $18112^{\dagger}$ | $1$ |  | $-0.333$ |  |  |  |  |  | $0.333$ |  |  |
Table 17: The values of $k\cdot|c_{i}|^{2}$ for the $bbbccq$ dibaryon states (in units of MeV). The scattering states are marked with a dagger ($\dagger$). |  |  | $bbb{\otimes}ccn$ | $bbc{\otimes}cbn$ | $bbn{\otimes}ccb$
---|---|---|---|---|---
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Xi_{cc}^{*}$ | $\Omega_{bbb}^{*}\Xi_{cc}$ | $\Omega_{bbc}^{*}\Xi_{cb}^{*}$ | $\Omega_{bbc}^{*}\Xi_{cb}$ | $\Omega_{bbc}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{bbc}\Xi_{cb}^{*}$ | $\Omega_{bbc}\Xi_{cb}$ | $\Omega_{bbc}\Xi_{cb}^{\prime}$ | $\Xi_{bb}^{*}\Omega_{ccb}^{*}$ | $\Xi_{bb}^{*}\Omega_{ccb}$ | $\Xi_{bb}\Omega_{ccb}^{*}$ | $\Xi_{bb}\Omega_{ccb}$
$bbbccn$ | $0^{+}$ | $18003^{\dagger}$ | $\times$ |  | $\times$ |  |  |  | $\times$ | $\times$ | $\times$ |  |  | $\times$
|  | $18167$ | $5.9$ |  | $\times$ |  |  |  | $129.0$ | $1.6$ | $\times$ |  |  | $3.9$
|  | $18280$ | $10.4$ |  | $123.4$ |  |  |  | $18.5$ | $14.6$ | $259.7$ |  |  | $30.4$
| $1^{+}$ | $17941^{\dagger}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $18004^{\dagger}$ | $\times$ | $0.1$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $18161$ | $4.8$ | $6.0$ | $\times$ | $58.6$ | $9.0$ | $21.2$ | $12.3$ | $32.5$ | $\times$ | $\times$ | $\times$ | $3.9$
|  | $18233$ | $6.3$ | $0.01$ | $75.3$ | $10.0$ | $17.7$ | $2.4$ | $0.3$ | $8.4$ | $6.2$ | $2.7$ | $138.8$ | $107.8$
|  | $18250$ | $0.7$ | $2.1$ | $2.5$ | $39.3$ | $1.5$ | $49.2$ | $1.9$ | $19.7$ | $191.9$ | $47.6$ | $39.1$ | $6.9$
| $2^{+}$ | $17942^{\dagger}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$ |
|  | $18006^{\dagger}$ | $\times$ | $0.01$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$ |
|  | $18230$ | $0.3$ | $2.2$ | $36.5$ | $1.7$ | $41.6$ | $9.4$ |  |  | $87.4$ | $128.7$ | $28.0$ |
| $3^{+}$ | $18006^{\dagger}$ | $\sim 0$ |  | $\times$ |  |  |  |  |  | $\times$ |  |  |
|  |  | $bbb{\otimes}ccs$ | $bbc{\otimes}cbs$ | $bbs{\otimes}ccb$
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Omega_{cc}^{*}$ | $\Omega_{bbb}^{*}\Omega_{cc}$ | $\Omega_{bbc}^{*}\Omega_{cb}^{*}$ | $\Omega_{bbc}^{*}\Omega_{cb}$ | $\Omega_{bbc}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{bbc}\Omega_{cb}^{*}$ | $\Omega_{bbc}\Omega_{cb}$ | $\Omega_{bbc}\Omega_{cb}^{\prime}$ | $\Omega_{bb}^{*}\Omega_{ccb}^{*}$ | $\Omega_{bb}^{*}\Omega_{ccb}$ | $\Omega_{bb}\Omega_{ccb}^{*}$ | $\Omega_{bb}\Omega_{ccb}$
$bbbccs$ | $0^{+}$ | $18109^{\dagger}$ | $\times$ |  | $\times$ |  |  |  | $\times$ | $\times$ | $\times$ |  |  | $\times$
|  | $18251$ | $8.9$ |  | $\times$ |  |  |  | $129.7$ | $0.1$ | $\times$ |  |  | $2.6$
|  | $18362$ | $8.5$ |  | $130.1$ |  |  |  | $10.6$ | $17.0$ | $251.8$ |  |  | $28.4$
| $1^{+}$ | $18040^{\dagger}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $18110^{\dagger}$ | $\times$ | $0.2$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $18256$ | $4.4$ | $6.3$ | $\times$ | $64.4$ | $8.5$ | $21.9$ | $15.7$ | $25.7$ | $\times$ | $\times$ | $\times$ | $9.8$
|  | $18323$ | $7.0$ | $0.02$ | $75.2$ | $14.2$ | $15.5$ | $0.2$ | $0.04$ | $18.7$ | $29.3$ | $9.9$ | $96.8$ | $105.4$
|  | $18328$ | $0.1$ | $1.5$ | $10.7$ | $26.0$ | $5.5$ | $48.0$ | $0.9$ | $14.5$ | $165.0$ | $39.8$ | $71.5$ | $0.1$
| $2^{+}$ | $18041^{\dagger}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$ |
|  | $18112^{\dagger}$ | $\times$ | $0.01$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$ |
|  | $18308$ | $1.7$ | $1.2$ | $30.6$ | $2.4$ | $37.2$ | $8.0$ |  |  | $84.0$ | $124.7$ | $24.6$ |
| $3^{+}$ | $18112^{\dagger}$ | $\times$ |  | $\times$ |  |  |  |  |  | $\times$ |  |  |
Table 18: The partial width ratios for the $bbbccq$ dibaryon states. 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. For a dibaryon state decays into two channels through two different quark rearrangements, the $\gamma_{i}$’s, which depend on the wave functions of final states, may not equal (or approximately equal). Thus we compare only the decay width ratios between channels through same quark rearrangement type. The scattering states are marked with a dagger ($\dagger$). The masses are all in unit of MeV. |  |  | $bbb{\otimes}ccn$ | $bbc{\otimes}cbn$ | $bbn{\otimes}ccb$
---|---|---|---|---|---
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Xi_{cc}^{*}$ | $\Omega_{bbb}^{*}\Xi_{cc}$ | $\Omega_{bbc}^{*}\Xi_{cb}^{*}$ | $\Omega_{bbc}^{*}\Xi_{cb}$ | $\Omega_{bbc}^{*}\Xi_{cb}^{\prime}$ | $\Omega_{bbc}\Xi_{cb}^{*}$ | $\Omega_{bbc}\Xi_{cb}$ | $\Omega_{bbc}\Xi_{cb}^{\prime}$ | $\Xi_{bb}^{*}\Omega_{ccb}^{*}$ | $\Xi_{bb}^{*}\Omega_{ccb}$ | $\Xi_{bb}\Omega_{ccb}^{*}$ | $\Xi_{bb}\Omega_{ccb}$
$bbbccn$ | $0^{+}$ | $18003^{\dagger}$ | $\times$ |  | $\times$ |  |  |  | $\times$ | $\times$ | $\times$ |  |  | $\times$
|  | $18167$ | $1.0$ |  | $\times$ |  |  |  | $79.6$ | $1$ | $\times$ |  |  | $1$
|  | $18280$ | $1.0$ |  | $8.4$ |  |  |  | $1.3$ | $1$ | $8.5$ |  |  | $1$
| $1^{+}$ | $17941^{\dagger}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $18004^{\dagger}$ | $\times$ | $1$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $18161$ | $1$ | $1.3$ | $\times$ | $6.5$ | $1$ | $2.4$ | $1.4$ | $3.6$ | $\times$ | $\times$ | $\times$ | $1$
|  | $18233$ | $1$ | $\sim 0$ | $31.0$ | $4.1$ | $7.3$ | $1$ | $0.1$ | $3.5$ | $1$ | $0.4$ | $22.4$ | $17.4$
|  | $18250$ | $1$ | $3.0$ | $1.6$ | $26.2$ | $1$ | $32.8$ | $1.3$ | $13.1$ | $27.8$ | $6.9$ | $5.7$ | $1$
| $2^{+}$ | $17942^{\dagger}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$ |
|  | $18006^{\dagger}$ | $\times$ | $1$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$ |
|  | $18230$ | $0.2$ | $1$ | $22.1$ | $1$ | $25.2$ | $5.7$ |  |  | $3.1$ | $4.6$ | $1$ |
| $3^{+}$ | $18006^{\dagger}$ | $1$ |  | $\times$ |  |  |  |  |  | $\times$ |  |  |
|  |  | $bbb{\otimes}ccs$ | $bbc{\otimes}cbs$ | $bbs{\otimes}ccb$
System | $J^{P}$ | Mass | $\Omega_{bbb}^{*}\Omega_{cc}^{*}$ | $\Omega_{bbb}^{*}\Omega_{cc}$ | $\Omega_{bbc}^{*}\Omega_{cb}^{*}$ | $\Omega_{bbc}^{*}\Omega_{cb}$ | $\Omega_{bbc}^{*}\Omega_{cb}^{\prime}$ | $\Omega_{bbc}\Omega_{cb}^{*}$ | $\Omega_{bbc}\Omega_{cb}$ | $\Omega_{bbc}\Omega_{cb}^{\prime}$ | $\Omega_{bb}^{*}\Omega_{ccb}^{*}$ | $\Omega_{bb}^{*}\Omega_{ccb}$ | $\Omega_{bb}\Omega_{ccb}^{*}$ | $\Omega_{bb}\Omega_{ccb}$
$bbbccs$ | $0^{+}$ | $18109^{\dagger}$ | $\times$ |  | $\times$ |  |  |  | $\times$ | $\times$ | $\times$ |  |  | $\times$
|  | $18251$ | $1$ |  | $\times$ |  |  |  | $1$ | $\sim 0$ | $\times$ |  |  | $1$
|  | $18362$ | $1$ |  | $12.2$ |  |  |  | $1$ | $1.6$ | $8.9$ |  |  | $1$
| $1^{+}$ | $18040^{\dagger}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $18110^{\dagger}$ | $\times$ | $1$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
|  | $18256$ | $1$ | $1.4$ | $\times$ | $7.6$ | $1$ | $2.6$ | $1.9$ | $3.0$ | $\times$ | $\times$ | $\times$ | $1$
|  | $18323$ | $1$ | $\sim 0$ | $4.9$ | $0.9$ | $1$ | $0.01$ | $\sim 0$ | $1.2$ | $3.0$ | $1$ | $9.8$ | $10.7$
|  | $18328$ | $0.1$ | $1$ | $1.9$ | $4.7$ | $1$ | $8.7$ | $0.2$ | $2.6$ | $4.1$ | $1$ | $1.8$ | $\sim 0$
| $2^{+}$ | $18041^{\dagger}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$ |
|  | $18112^{\dagger}$ | $\times$ | $1$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$ |
|  | $18308$ | $1.4$ | $1$ | $12.6$ | $1$ | $15.2$ | $3.3$ |  |  | $3.4$ | $5.1$ | $1$ |
| $3^{+}$ | $18112^{\dagger}$ | $\times$ |  | $\times$ |  |  |  |  |  | $\times$ |  |  |

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