Fully heavy pentaquarks

At the birth of the quark model GellMann:1964nj ; Zweig:1981pd ; Zweig:1964jf , Gell-Mann and Zweig indicated that hadronic states with the $qq\bar{q}\bar{q}$ and $qqqq\bar{q}$ quark configurations should exist in nature. Such exotic states were further investigated with some phenomenological models soon afterwards. For example, Jaffe adopted the quark-bag model to study $q^{2}\bar{q}^{2}$ hadrons, where the mass spectrum and dominant decay behavior were predicted Jaffe:1976ig . In 1979, Strottman calculated the masses of $q^{4}\bar{q}$ and $q^{5}\bar{q}^{2}$ in the framework of the MIT bag model Strottman:1979qu . The name pentaquark was firstly proposed by Gignoux et al. in 1987 Gignoux:1987cn ; Lipkin:1987sk , and they found that the states $P^{0}=\bar{c}uuds$ and $P^{-}=\bar{c}ddus$ with spin 1/2 and their beauty analogs are very likely to be stable pentaquarks. Since 2003, with the accumulation of experimental data, more and more charmonium-like $XYZ$ states were reported in experiments. Especially, the observation by the LHCb Collaboration confirms the existence of pentaquark $P_{c}$ states Aaij:2015tga ; Aaij:2016phn ; Aaij:2019vzc . In the past twenty years, progress has been made on studying exotic multiquarks Chen:2016qju ; Liu:2019zoy ; Guo:2017jvc ; Brambilla:2019esw .

Recently, the LHCb Collaboration studied the invariant mass spectrum of $J/\psi$ pairs, and they reported a narrow structure around 6.9 GeV and a broad structure in the mass range 6.2-6.8 GeV. The global significance for these two structures are larger than 5$\sigma$. Such distinct structures are expected to be with the $cc\bar{c}\bar{c}$ configuration Aaij:2020fnh .

The $cc\bar{c}\bar{c}$ tetraquark had been discussed a lot in the literature before its discovery by the LHCb Collaboration. In Ref. Iwasaki:1975pv , a $cc\bar{c}\bar{c}$ tetraquark was predicted at about 6.2 GeV. The $cc\bar{c}\bar{c}$ tetraquark system was systematically investigated based on a quark-gluon model in Ref. Chao:1980dv . In Ref. Ader:1981db Ader et al. found the lowest $cc\bar{c}\bar{c}$ state is not bound, and there are other discussions with a similar conclusion Chiu:2005ey ; Hughes:2017xie ; SilvestreBrac:1992mv ; Vega-Morales:2017pmm ; Chen:2016jxd ; Richard:2017vry .

This important signal from the LHCb Collaboration provides us a new ground to understand the nonperturbative behavior of QCD Chao:2020dml ; Richard:2020hdw ; Maiani:2020pur . The mass of the observed signal around 6.9 GeV is consistent with a previous QCD sum rule predictions Chen:2016jxd . After the LHCb Collaboration reported their results, the strong decay properties of $S$- and $P$-wave tetraquark states were further studied in Ref. Chen:2020xwe , and the observed structure at around 6.9 GeV is suggested to be with $J^{PC}=0^{-+}$ or $1^{-+}$. Becchi et al. have studied that the tetraquarks $cc\bar{c}\bar{c}$ with $J^{PC}=0^{++}$, $2^{++}$ decay into four mouns and into hidden- and open-charmed mesons and provide the decay widths of fully charmed tetraquarks Becchi:2020uvq . The inner structures of the fully charmed tetraquark state were studied Guo:2020pvt . The mass spectrum of $cc\bar{c}\bar{c}$ tetraquarks were studied in an extended relativized quark model, QCD sum rule, chromomagnetic model and so on Lu:2020cns ; Albuquerque:2020hio ; Wang:2020dlo ; Zhang:2020xtb ; Giron:2020wpx ; Faustov:2020qfm ; Gordillo:2020sgc ; Weng:2020jao . The production mechanism of $cc\bar{c}\bar{c}$ was also studied in various schemes Wang:2020gmd ; Wang:2020wrp ; Dong:2020nwy ; Maciula:2020wri ; Karliner:2020dta ; Feng:2020riv ; Ma:2020kwb ; Zhu:2020xni ; Zhu:2020xni .

The discoveries of fully heavy tetraquark states and $P_{c}$ states make one speculate that the pentaquark state with fully heavy quarks $QQQQ\bar{Q}$ may also exist. If the mass is above the baryon-meson thresholds, the heavy pentaquark state may allow the strong decays into the corresponding two body. The study of the masses and decay properties would help to search for the heavy pentaquark states in experiments.

The strong interaction in a fully heavy multiquark state is not clear at present. The chromomagnetic interaction (CMI) model provides us with a simple picture to quantitatively understand the spectrum of multiquark states. In the framework of the CMI model DeRujula:1975qlm , the strong interaction between quarks via gluon exchange force is parametrized into effective quark masses and quark coupling parameters. Despite its simple Hamiltonian, this model can catch the basic features of hadron spectra, since the mass splittings between hadrons reflect the basic symmetries of their inner structures SilvestreBrac:1992mv . This model has been widely adopted to study the mass spectra of multiquark systems Wu:2016vtq ; Wu:2016gas ; Chen:2016ont ; Wu:2017weo ; Luo:2017eub ; Zhou:2018pcv ; Li:2018vhp ; Wu:2018xdi ; An:2019idk ; Weng:2019ynva ; Cheng:2019obk ; Liu:2016ogz ; Hogaasen:2013nca ; Cheng:2020irt ; Cheng:2020nho ; Cheng:2020wxa ; Weng:2018mmf ; Weng:2020jao ; Zhao:2014qva ; Wu:2017weo ; Zhou:2018pcv ; Li:2018vhp ; An:2019idk . In this work, we systematically study the S-wave $QQQQ\bar{Q}$ pentaquark system within the framework of the CMI model to calculate the mass spectra and the relative partial decay widths, and find a possible stable pentaquark state.

This paper is organized as follows. In Sec. II, we introduce the CMI model and determine the relevant parameters used in the CMI model. The $flavor\otimes color\otimes spin$ wave functions are constructed and the CMI Hamiltonian elements are calculated for the $QQQQ\bar{Q}$ pentaquark system in Sec. III. In Sec. IV, we present the mass spectra, the mass splittings, the possible strong decay channels, and the relative partial decay widths, and also discuss the stability for the pentaquark states. A short summary is followed in Sec. V. Finally, some useful expressions are presented in the Appendix.

The masses of the ground hadrons can be obtained by the effective Hamiltonian at quark level

	$\displaystyle H$	$\displaystyle=$	$\displaystyle\sum_{i}m_{i}+H_{\textrm{CMI}}$		(1)
		$\displaystyle=$	$\displaystyle\sum_{i}m_{i}-\sum_{i<j}C_{ij}\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j},$

where the $H_{\textrm{CMI}}$ denotes the Hamiltonian of the chromomagnetic interaction DeRujula:1975qlm . $\sigma_{i}$ and $\lambda_{i}$ are the Pauli matrices and the Gell-Mann matrices, respectively. For the antiquark, the $\lambda_{i}$ should be replaced with $-\lambda_{i}^{*}$. $m_{i}$ is the effective mass of the $i$-th constituent quark. In the above Hamiltonian, the chromoelectric interaction and color confinement effect are also incorporated in the effective quark mass $m_{i}$. $C_{ij}$ is the effective coupling constant between the $i$-th quark and $j$-th quark. The effective quark mass $m_{i}$ and the coupling constant $C_{ij}$ can be determined from the experimental hadron masses.

As indicated in Refs. Wu:2016vtq ; Wu:2016gas ; Chen:2016ont ; Wu:2017weo ; Luo:2017eub ; Zhou:2018pcv ; Li:2018vhp ; Wu:2018xdi ; An:2019idk ; Liu:2016ogz , the predicted hadron masses obtained from Eq. (1) are generally overestimated. The main reason is that the dynamical effects inside hadrons can not simply be absorbed into the effective quark masses. Thus, in order to take such effective interaction into account, we replace the sum of the $m_{i}$ term in Eq. (1) with $M_{ref}-\langle H_{\rm CMI}\rangle_{ref}$, where $M_{ref}$ is a reference mass scale and $\langle H_{\rm CMI}\rangle$ is the corresponding CMI matrix element. The mass of the ground hadron can thus be written as

$\displaystyle M=M_{ref}-\langle H_{\rm CMI}\rangle_{ref}+\langle H_{\rm CMI}\rangle.$

(2)

Here, we choose the baryon-meson thresholds as the mass scales, where the reference baryon-meson system should have the same constituent quarks with the studied pentaquark state. In this way, the dynamical effects that are not incorporated in the original approach are somehow phenomenologically compensated for this procedure Zhou:2018pcv . We label this method as the reference mass scheme.

In addition, we introduce another scheme to estimate the masses of pentaquark states. We separate the two-body chromoelectric effects out of the effective quark masses and generalize the chromomagnetic interaction model by writing the chromoelectric term explicitly Hogaasen:2013nca ; Weng:2019ynva ; Weng:2018mmf ; Weng:2020jao i.e.,

	$\displaystyle H$	$\displaystyle=$	$\displaystyle\left(\sum_{i}m_{i}^{0}+H_{\textrm{CEI}}^{0}\right)+H_{\textrm{CMI}}^{0}$		(3)
		$\displaystyle=$	$\displaystyle\sum_{i}m_{i}^{0}-\sum_{i<j}A_{ij}\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}-\sum_{i<j}v_{ij}\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j},$
		$\displaystyle=$	$\displaystyle-\frac{3}{4}\sum_{i<j}m_{ij}\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}-\sum_{i<j}v_{ij}\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}+...$

where the omitted operator nullifies the color-singlet physical states, and

$\displaystyle m_{ij}=\frac{1}{4}(m_{i}^{0}+m_{j}^{0})+\frac{4}{3}A_{ij}.$

(4)

This treatment has been successfully adopted in Refs. Karliner:2014gca ; Karliner:2016zzc ; Karliner:2017qjm ; Hogaasen:2013nca ; Weng:2019ynva ; Weng:2018mmf ; Weng:2020jao . The parameters $m_{ij}$ and $v_{ij}$ are also determined from the experimental hadron masses. In this work, we label this method as the modified CMI model scheme.

To estimate the masses of the $QQQQ\bar{Q}$ pentaquark states, we need some hadron masses as input to fit the effective coupling parameters $C_{ij}$, $m_{ij}$, and $v_{ij}$ Tanabashi:2018oca . These conventional hadrons are listed in Table 1. Because some of the heavy flavor baryons are not yet observed, we introduce the theoretical results in Refs. Weng:2018mmf ; Godfrey:1985xj as our input, and enclose the theoretical values of masses for these baryons with parentheses in Table 1.

In principle, the values of $A_{ij}$ and $v_{ij}$ in the modified CMI model should be different for various systems. However, it is difficult to exactly calculate these parameters for a given system without knowing the spatial wave function. Thus, they are extracted from the masses of conventional hadrons by assuming that quark-(anti)quark interactions are the same for all the hadron systems. Of course, this assumption certainly leads to uncertainties on mass estimations for multiquark states. Since the size of a multiquark state is expected to be larger than that of a conventional hadron and the distance between quark components may be larger, the attraction between quark components should be weaker. Thus, our framework may produce a little more binding.

Now we fit the effective coupling parameters $C_{ij}$, $m_{ij}$, and $v_{ij}$ in the reference mass and modified CMI model schemes by applying Eq. (2) and Eq. (3), respectively. We present the obtained effective coupling parameters of $QQQQ\bar{Q}$ pentaquark states in Table 2. One can refer to Refs. Weng:2019ynva ; Weng:2018mmf ; Liu:2019zoy for more details.

In order to systematically study the mass spectra of the $QQQQ\bar{Q}$ pentaquark system, we need construct the wave function of $QQQQ\bar{Q}$ pentaquark first. We exhaust all the possible color $\otimes$ spin wave functions of pentaquark states, and combine them with the corresponding flavor wave functions. The constructed pentaquark wave functions should be constrained appropriately by the Pauli principle. After that, we can use these pentaquark wave functions to calculate the mass spectra of the corresponding pentaquark states.

The total wave function of the S-wave $QQQQ\bar{Q}$ pentaquark can be described by the direct product of flavor, color, and spin wave functions,

$\displaystyle\psi_{\textrm{tot}}=\psi_{flavor}\otimes\psi_{color}\otimes\psi_{spin}.$

(5)

Due to the Pauli principle, this wave function should be fully antisymmetric when exchanging identical quarks.

In the flavor space, we divide the $QQQQ\bar{Q}$ pentaquark system into three groups of subsystems according to their symmetries: (1) the first four quarks are identical: the $cccc\bar{c}$, $cccc\bar{b}$, $bbbb\bar{c}$, and $bbbb\bar{b}$ pentaquark subsystems, (2) the first three quarks are identical: the $cccb\bar{c}$, $cccb\bar{b}$, $bbbc\bar{c}$, and $bbbc\bar{b}$ pentaquark subsystems, (3) there are two pairs of identical quarks: the $ccbb\bar{c}$ and $ccbb\bar{b}$ pentaquark subsystems.

In the color space, the color wave functions are singlets due to the color confinement. The color wave functions can be deduced from the following direct product

$\displaystyle\left[3_{c}\otimes 3_{c}\otimes 3_{c}\otimes 3_{c}\right]\otimes\bar{3}_{c}$

(6)

Based on Eq. (6), the first four quarks are in three color triplet, the corresponding Young tableau [2,1,1] can be written as

$\displaystyle\begin{tabular}[]{|c|c|}\hline\cr 1&2\\ \cline{1-2}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\ \cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\ \cline{1-1}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$C_{1}$\hfil\lx@intercol\end{tabular}},\quad\begin{tabular}[]{|c|c|}\hline\cr 1&3\\ \cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\ \cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\ \cline{1-1}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$C_{2}$\hfil\lx@intercol\end{tabular}},\quad\begin{tabular}[]{|c|c|}\hline\cr 1&4\\ \cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\ \cline{1-1}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\ \cline{1-1}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$C_{3}$\hfil\lx@intercol\end{tabular}}.$

(19)

Then, by combining the antitriplet from the antiquark with the deduced three color triplets in Eq. (19), we obtain three color singlets for the $QQQQ\bar{Q}$ pentquark system.

In the spin space, the spin state can be represented in terms of a five-dimensional Young tableau [3,2] as

$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\ \cline{1-3}\cr 4&5\\ \cline{1-2}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{1}$\hfil\lx@intercol\end{tabular}},\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&4\\ \cline{1-3}\cr 3&5\\ \cline{1-2}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{2}$\hfil\lx@intercol\end{tabular}},\begin{tabular}[]{|c|c|c|}\hline\cr 1&3&4\\ \cline{1-3}\cr 2&5\\ \cline{1-2}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{3}$\hfil\lx@intercol\end{tabular}},\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&5\\ \cline{1-3}\cr 3&4\\ \cline{1-2}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{4}$\hfil\lx@intercol\end{tabular}},\begin{tabular}[]{|c|c|c|}\hline\cr 1&3&5\\ \cline{1-3}\cr 2&4\\ \cline{1-2}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{5}$\hfil\lx@intercol\end{tabular}}.$

(35)

for the pentaquark states with total spin $J=1/2$. Since particle 5 is an antiquark, we can isolate this antiquark and discuss the symmetry property of the first four quarks: 1, 2, 3, and 4 in color $\otimes$ spin space.

When antiquark 5 is separated from the spin wave functions, the spin states represented in Young tableaus without antiquark 5 can be directly obtained from Eq. (35) as

$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\ \cline{1-3}\cr 4\\ \cline{1-1}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{1}$\hfil\lx@intercol\end{tabular}},\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&4\\ \cline{1-3}\cr 3\\ \cline{1-1}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{2}$\hfil\lx@intercol\end{tabular}},\begin{tabular}[]{|c|c|c|}\hline\cr 1&3&4\\ \cline{1-3}\cr 2\\ \cline{1-1}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{3}$\hfil\lx@intercol\end{tabular}},\begin{tabular}[]{|c|c|}\hline\cr 1&2\\ \cline{1-2}\cr 3&4\\ \cline{1-2}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{4}$\hfil\lx@intercol\end{tabular}},\begin{tabular}[]{|c|c|}\hline\cr 1&3\\ \cline{1-2}\cr 2&4\\ \cline{1-2}\cr\end{tabular}_{\begin{tabular}[]{|c|}\lx@intercol\hfil$S_{5}$\hfil\lx@intercol\end{tabular}}.$

(51)

We can identify the spin states in Eq. (51) with the Young-Yamanouchi bases for Young tableauss [3,1] and [2,2]. The pentaquark states with total spin $J=5/2,3/2$ have similar situations.

With the above preparation, we can start to construct the flavor $\otimes$ color $\otimes$ spin wave functions of $QQQQ\bar{Q}$ pentaquark states Stancu:1999qr ; Park:2017jbn ; Park:2015nha ; Park:2016mez ; Park:2016cmg ; Park:2018oib . Then, based on the possible $\psi_{flavor}\otimes\psi_{color}\otimes\psi_{spin}$ bases of the $QQQQ\bar{Q}$ pentaquark system, we calculate the CMI matrices for the corresponding pentaquark states. In Table 7 of the Appendix, we only present the expressions of CMI Hamiltonians for the $cccc\bar{c}$, $cccb\bar{c}$, and $ccbb\bar{c}$ pentaquark subsystems. Moreover, we also get more tractable CMI matrices under the $ccc\otimes b\bar{c}$ ($cc\otimes bb\otimes\bar{c}$) bases for the $cccb\bar{c}$ ($ccbb\bar{c}$) subsystem. One can refer to Refs. Wu:2017weo ; Zhou:2018pcv for more details. As for the expressions of CMI matrices for the $cccc\bar{b}$, $bbbb\bar{c}$, $bbbb\bar{b}$, $cccb\bar{b}$, $bbbc\bar{c}$, $bbbc\bar{b}$, and $ccbb\bar{b}$ pentaquark subsystems, we can obtain them from the expressions of the $cccc\bar{c}$, $cccb\bar{c}$, and $ccbb\bar{c}$ pentaquark subsystems according to their similar symmetry properties.

The interacting Hamiltonians can be diagonalized and one can thus obtain the eigenvalues as well as eigenvectors for the corresponding pentaquark systems. According to our results, we discuss the mass gaps, decay behaviors, and stabilities of all the $QQQQ\bar{Q}$ pentaquark states.

Based on the two schemes proposed in Sec. II, we present the mass spectra for all the $QQQQ\bar{Q}$ pentaquark subsystems in Table 3. Take the $cccb\bar{c}$ pentaquark subsystem as an example. In the reference mass scheme, we use two types of baryon-meson reference systems ($\Omega_{ccc}+B_{c}$ and $\Omega_{ccb}+\eta_{c}$) to estimate the masses of $cccb\bar{c}$ states. Some results calculated from the two reference systems differ by more than a hundred MeV for the $cccb\bar{c}$ pentaquark states. However, the gaps with different reference systems are still the same. Thus, if one pentaquark state were observed, its partner states may be searched for with the relative positions presented in Table 3. Such a study can be used to test our calculation. Here, we need to emphasis that as a rough estimation, the dynamics and contributions from other terms in the interacting potential are not elaborately considered in Eq. (3) Zhou:2018pcv .

The modified CMI model scheme takes the chromoelectric interaction explicitly compared to the reference mass scheme, and therefore we use the results in this scheme for the following analysis. According to the modified CMI model scheme, we present the masses of the $QQQQ\bar{Q}$ pentaquark states and the relevant baryon-meson thresholds in Fig. 1. In Fig. 1, we label the possible total angular momenta of the S-wave baryon-meson states. When the spin of an initial pentaquark state is equal to the total angular momentum of the channel below, it may decay into that baryon-meson channel through the S wave.

Here we define the relatively ”stable” pentaquarks as those which cannot decay into the S wave baryon-meson states. We label these stable pentaquark states with “$\star$” in the figure and tables.

In addition to the mass spectra, the eigenvectors of pentaquark states will also provide important information about the two-body strong decay of multiquark states Jaffe:1976ig ; Strottman:1979qu ; Weng:2019ynva ; Weng:2020jao ; Zhao:2014qva ; Wang:2015epa . Thus we calculate the overlaps of wave functions between a fully heavy pentaquark state and a particular baryon $\otimes$ meson state, and show them in Table 4.

We can further study the decay of the fully heavy pentaquark states into the baryon $\otimes$ meson channels. Here, we take the $cccc\bar{c}$ pentaquark states as an example to describe our calculation. We transform the eigenvectors of the pentaquark states into the $ccc$ $\otimes$ $c\bar{c}$ configuration. Normally, the $ccc$ and $c\bar{c}$ components inside the pentaquark can be either of color-singlet or of color-octet. The former can easily dissociate into an $S$-wave baryon and meson [the so-called Okubo-Zweig-Iizuka-superallowed (OZI-superallowed] decays Jaffe:1976ig ), while the latter cannot fall apart without the gluon exchange force. For simplicity, in this work, we only focus on the OZI-superallowed pentaquark decay process. The color singlet can be described by the direct product of a meson wave function and a baryon wave function. For each decay mode, the branching fraction is proportional to the square of the coefficient of the corresponding component in the eigenvectors, and the strong decay phase space.

For the two-body decay via the $L$-wave process, the expression describing partial decay width can be parametrized as Weng:2019ynva ; Weng:2020jao

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

(52)

where $\alpha$ is an effective coupling constant, $m$ is the mass of the initial state, and $k$ is the momentum of the final states in the rest frame. $c_{i}$ is the coefficient related to the corresponding baryon-meson component, which is the overlap of the wave functions shown in Table 4. For the decay processes that we are interested in, $(k/m)^{2}$ is of $\mathcal{O}(10^{-2})$ or even smaller. Thus we only consider the $S$-wave decays.

$\gamma_{i}$ represents other factors that contribute to the decay widths $\Gamma_{i}$. For each process, $\Gamma_{i}$ also depends on the spatial wave functions of the initial pentaquark state and the final meson and baryon. In the quark model in the heavy quark limit, the spatial wave functions of the ground $S$-wave pseudoscalar and vector meson are the same Weng:2019ynva . As a rough estimation, we introduce the following approximations to calculate the relative partial decay widths of the pentaquark states:

	$\displaystyle\gamma_{\Omega_{ccc}J/\psi}=\gamma_{\Omega_{ccc}\eta_{c}},\quad\gamma_{\Omega_{ccc}B_{c}^{*}}=\gamma_{\Omega_{ccc}B_{c}},$
	$\displaystyle\gamma_{\Omega_{bbb}B_{c}^{*}}=\gamma_{\Omega_{bbb}B_{c}},\quad\gamma_{\Omega_{bbb}\Upsilon}=\gamma_{\Omega_{bbb}\eta_{b}},$
	$\displaystyle\gamma_{\Omega_{ccc}B^{*}_{c}}=\gamma_{\Omega_{ccc}B_{c}},\quad\gamma_{\Omega_{ccc}\Upsilon}=\gamma_{\Omega_{ccc}\eta_{b}},$
	$\displaystyle\gamma_{\Omega_{bbb}J/\psi}=\gamma_{\Omega_{bbb}\eta_{c}},\quad\gamma_{\Omega_{bbb}B^{*}_{c}}=\gamma_{\Omega_{bbb}B_{c}},$
	$\displaystyle\gamma_{\Omega^{*}_{ccb}J/\psi}=\gamma_{\Omega^{*}_{ccb}\eta_{c}}=\gamma_{\Omega_{ccb}J/\psi}=\gamma_{\Omega_{ccb}\eta_{c}},$
	$\displaystyle\gamma_{\Omega^{*}_{ccb}B^{*}_{c}}=\gamma_{\Omega^{*}_{ccb}B_{c}}=\gamma_{\Omega_{ccb}B^{*}_{c}}=\gamma_{\Omega_{ccb}B_{c}},$
	$\displaystyle\gamma_{\Omega^{*}_{bbc}B^{*}_{c}}=\gamma_{\Omega^{*}_{bbc}B_{c}}=\gamma_{\Omega_{bbc}B^{*}_{c}}=\gamma_{\Omega_{bbc}B_{c}},$
	$\displaystyle\gamma_{\Omega^{*}_{bbc}\Upsilon}=\gamma_{\Omega^{*}_{bbc}\eta_{b}}=\gamma_{\Omega_{bbc}\Upsilon}=\gamma_{\Omega_{bbc}\eta_{b}},$
	$\displaystyle\gamma_{\Omega^{*}_{ccb}B^{*}_{c}}=\gamma_{\Omega^{*}_{ccb}B_{c}}=\gamma_{\Omega_{ccb}B^{*}_{c}}=\gamma_{\Omega_{ccb}B_{c}},$
	$\displaystyle\gamma_{\Omega^{*}_{bbc}J/\psi}=\gamma_{\Omega^{*}_{bbc}}\eta_{c}=\gamma_{\Omega_{bbc}J/\psi}=\gamma_{\Omega_{bbc}}\eta_{c},$
	$\displaystyle\gamma_{\Omega^{*}_{ccb}\Upsilon}=\gamma_{\Omega^{*}_{ccb}\eta_{b}}=\gamma_{\Omega_{ccb}\Upsilon}=\gamma_{\Omega_{ccb}\eta_{b}},$
	$\displaystyle\gamma_{\Omega^{*}_{bbc}B^{*}_{c}}=\gamma_{\Omega^{*}_{bbc}B_{c}}=\gamma_{\Omega_{bbc}B^{*}_{c}}=\gamma_{\Omega_{bbc}B_{c}}.$		(53)

We present $k\cdot|c_{i}|^{2}$ for each decay process in Table 5. From Table 5, one can roughly estimate the relative decay widths between different decay processes of different initial pentaquark states if neglecting the $\gamma_{i}$ differences. As a rough estimation about the ratios of the decay widths, we neglect the recoil momenta of quarks in the hadrons and thus use the approximation of Eq. (52). A complete five-body study will modify the phase space factor.

Here, we divide the $QQQQ\bar{Q}$ pentaquark system into the following three groups:

1. A.

   The $cccc\bar{Q}$ and $bbbb\bar{Q}$ pentaquark subsystems;

2. B.

   The $cccb\bar{Q}$ and $bbbc\bar{Q}$ pentaquark subsystems;

3. C.

   The $ccbb\bar{Q}$ pentaquark subsystem.

We discuss the mass spectra and strong decay properties of the $QQQQ\bar{Q}$ pentaquark system group by group. For simplicity, we use $\rm P_{content}$(Mass, $I$, $J^{P}$) to label a specific pentaquark state.