1 Introduction

Along with baryons and mesons, multi-quark states were also proposed by Gell-Mann and Zweig in the early stage of quark models in 1964 [1, 2]. In recent years, great progress has been made on studying multi-quark states composed of heavy quarks, since such states with very large energy can be accessed experimentally and easily distinguished from other states, among which the fully heavy tetraquark states \((QQ\bar{Q}\bar{Q}, Q=c,b)\) attracted extensive attention.

In terms of experiment, the CMS collaboration observed pair production of \(\Upsilon (1S)\) mesons in proton-proton collisions at \(\sqrt{s}=\)8 TeV [3] and the existence of a fully bottom tetraquark \(bb\bar{b}\bar{b}\) in the \(\Upsilon (1\,S) \ell ^+ \ell ^-\) final states with a mass around 18.4 GeV and a global significance of 3.6 \(\sigma \) was claimed [4, 5]. In addition, an evidence of a significance peak around 18.12 GeV was reported by the ANDY Collaboration in Cu + Au collisions at the Relativistic Heavy Ion Collider [6]. However, some other experiments searched for the \(bb\bar{b}\bar{b}\) tetraquark but no observation was made [7, 8]. In 2020, the LHCb collaboration reported their result on the observation of the fully charmed state \(cc\bar{c}\bar{c}\). A narrow structure X(6900), matching the lineshape of a resonance and a broad structure next to the \(J/\psi J/\psi \) mass threshold was obtained [7]. At the recent ICHEP 2022 conference, the ATLAS collaboration reported on the di-charmonium resonance search in the four-muon final state with ATLAS 13 TeV data [10, 11]. Significant excess observed in the di-\(J/\psi \) invariant mass spectrum, and a broad structure at lower mass along with a resonance around 6.9 GeV was observed. They also observed excess at 6.9 GeV and 7.2 GeV in the \(J/\psi + \psi (2S)\) invariant mass spectrum. The existence of X(6900) was also confirmed by the CMS collaboration at the ICHEP 2022 conference [12]. Moreover, the signals of two new peaking structures was observed, which are X(6600) with \(6552 \pm 10 \pm 12\) MeV and X(7200) with \(7287 \pm 19 \pm 5\) MeV. These experimental breakthroughs have attracted a lot of attention and inspired a lot of enthusiasm for the theoretical researches on fully heavy states.

Theoretical work in regard to multi-quark states composed of heavy quarks began in 1976. In Ref. [13], Iwasaki predicted a fully charmed resonance state at about 6.2 GeV based on a model calculation. Since the LHCb and CMS collaborations reported the relevant experiments, theoretical studies of the fully heavy states have continued to be carried out and have become prosperous. The fully heavy tetraquark states have been investigated in the framework of lattice quantum chromodynamics(QCD) method [14], the QCD sum rules [15,16,17,18,19,20,21,22,23,24,25], relativistic quark model [26,27,28,29], various non-relativistic quark model [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55], QCD string model  [56], flux-tube model [57], holography inspired stringy hadron model [58], dynamical diquark model [59, 60], string junction picture [61, 62] and Bethe-Salpeter wave function [63,64,65]. There are also studies investigated the production mechanisms [66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82] and the decay properties  [83,84,85,86,87] of the fully heavy tetraquark states.

On the basis of heavy-antiquark-diquark symmetry (HADS) [88], which states that two heavy quarks within a doubly heavy baryon behave approximately as a heavy antiquark. If one replaces the anti-charm quark with the charmed di-quark, fully charmed pentaquark and dibaryon configurations can be obtained naturally. Therefore, the fully heavy pentaquark and dibaryon are quite valuable to study.

The fully heavy dibaryons were also studied in the framework of the lattice QCD method [89,90,91], the QCD sum rules [92], one-boson exchange model [93], non-relativistic constituent quark models [94,95,96,97], potential model [98], extended chromomagnetic model [99]. According to the results in Refs. [89,90,91,92,93,94,95], \(\Omega _{ccc}\Omega _{ccc}\) and \(\Omega _{bbb}\Omega _{bbb}\) are possible to form stable bound states. However, different opinions can be found in Refs. [96,97,98,99]. In their calculations, the energies of \(\Omega _{ccc}\Omega _{ccc}\) and \(\Omega _{bbb}\Omega _{bbb}\) are above the corresponding thresholds, respectively.

Accordingly, it is timely to study the existence of the fully heavy pentaquarks. The energies of \(cccc\bar{c}\) and \(bbbb\bar{b}\) bound states with \(J^P=\frac{1}{2}^-\) and \(J^P=\frac{3}{2}^-\) have been presented in Refs. [100, 101] via QCD sum rules approach. Besides, systematic studies on the mass spectra of the \(S-\)wave fully heavy pentaquarks \(QQQQ\bar{Q}\) have also been performed in the framework of the chromomagnetic interaction (CMI) model [102], constituent quark model with variational method [103], a lattice-QCD inspired quark model with complex scaling method [104], quark models with resonating group method [105] and MIT bag model [106]. Several conclusions concerning resonance states can be found in the Refs. [102, 104]. In Refs. [102, 103, 106], the author systematically investigated not only fully charmed (bottom) pentaquarks, but also all possible charmed-bottom pentaquarks with color singlet-singlet structure (\(\textbf{1}\) \( _{[QQQ]} \mathbf {\otimes } \textbf{1}\) \(_{[Q \bar{Q}]}\)). In Ref. [104], the authors take into consideration different types of color (\(\textbf{1}\) \( _{[QQQ]} \mathbf {\otimes } \textbf{1}\) \(_{[Q \bar{Q}]}\) and \(\textbf{8}\) \( _{[QQQ]} \mathbf {\otimes } \textbf{8}\) \(_{[Q \bar{Q}]}\)) and spatial structures to investigate \(cccc\bar{c}\) and \(bbbb\bar{b}\) states.

The quantum chromodynamics is the underlying theory of the strong interaction. However, it is difficult to study the structure of the hadrons and the hadron-hadron interaction directly because of the nonperturbative properties of QCD in the low energy region. Although lattice QCD has made impressive progresses on nucleon-nucleon interactions and multiquark systems [107,108,109], the QCD-inspired quark model is still the main approach to study the hadron-hadron interactions and multiquark states. A common approach is the chiral quark model (ChQM) [110], in which the constituent quarks interact with each other through colorless Goldstone bosons exchange in addition to the colorful one-gluon-exchange and confinement, and the chiral partner \(\sigma \) meson-exchange is introduced to obtain the immediate-range attraction of hadron-hadron interaction. As for fully heavy pentaquark systems, since there lacks a light-meson exchange mechanism, we mainly consider the one-gluon-exchange and confinement terms.

Our purpose is to find out if there exists any fully heavy pentaquark states by investigating the possible charmed-bottom pentaquark systems. This paper is organized as follows. After introduction, we briefly introduce the quark model used in Sect. 2. The way we construct the wave functions is also presented. Then, the numerical results and discussions are presented in Sect. 3. Finally, the paper ends with summary in Sect. 4.

2 Theoretical framework

Herein, ChQM is employed to investigate the properties of pentaquark systems, and the channel coupling effect is considered. Both color singlet-singlet structure (color singlet channel) and color octet-octet structure (hidden-color channel) are taken into consideration. In this sector, we will introduce this model and the way of constructing wave functions.

2.1 Chiral quark model (ChQM)

The Salamanca version was chosen as the representative of the chiral quark models, because the Salamanca group’s work covers the hadron spectra, nucleon-nucleon interaction, and multiquark states. The model details can be found in Ref. [110]. Here, we mainly present the salient features of the model. The general form of the five-body complex Hamiltonian is given by

$$\begin{aligned} H=&\sum _{i=1}^5\left( m_i+\frac{\varvec{p}_{i}^{2}}{2m_i}\right) -T_{CM} +\sum _{j>i=1}^5 V(\varvec{r}_{ij}) \end{aligned}$$
(1)

where \(m_i\) is the quark mass, \(\varvec{p}_{i}\) is the momentum of the quark, and \(T_{CM}\) is the center-of-mass kinetic energy. The dynamics of the pentaquark system is driven by a two-body potential

$$\begin{aligned} V(\varvec{r}_{ij})=&V_{CON}(\varvec{r}_{ij})+V_{OGE}(\varvec{r}_{ij}) \end{aligned}$$
(2)

The most relevant features of QCD at its low energy regime: color confinement (\(V_{CON}\)) and perturbative one-gluon exchange interaction (\(V_{OGE}\)) have been taken into consideration. For the fully-heavy systems, there is no \(\sigma -\)exchange or the Goldstone boson exchange.

Here, a phenomenological quadratic form confinement potential(\(V_{CON}\)) is used as

$$\begin{aligned} V_{CON}(\varvec{r}_{ij}) =&-a_{c}\varvec{\lambda }_{i}^{c} \cdot \varvec{\lambda }_{j}^{c}\left[ f(\varvec{r}_{ij})+V_{0}\right] \end{aligned}$$
(3)

In the present work, we mainly focus on the low-lying negative parity \(QQQQ\bar{Q}\) pentaquark systems of S-wave, so the spin-orbit and tensor interactions are not included. The one-gluon exchange potential (\(V_{OGE}\)), which includes coulomb and color-magnetic interactions, is written as

$$\begin{aligned} V_{OGE}(\varvec{r}_{ij})&= \frac{1}{4}\alpha _{s} \varvec{\lambda }_{i}^{c} \cdot \varvec{\lambda }_{j}^{c} \nonumber \\&\quad \cdot \left[ \frac{1}{r_{i j}}-\frac{\pi }{2} \delta \left( \textbf{r}_{i j}\right) \left( \frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}+\frac{4 \varvec{\sigma }_{i} \cdot \varvec{\sigma }_{j}}{3 m_{i} m_{j}}\right) \right] \end{aligned}$$
(4)

where \(\varvec{\sigma }\) is the Pauli matrices and \(\alpha _{s}\) is the quark-gluon coupling constant. In order to cover the wide energy range from light to strange to heavy quarks, an effective scale-dependent quark-gluon coupling \(\alpha _{s}(\mu )\) between quarks was introduced  [111]:

$$\begin{aligned} \alpha _{s}(\mu ) =&\frac{\alpha _{0}}{\ln \left( \frac{\mu ^{2}+\mu _{0}^{2}}{\Lambda _{0}^{2}}\right) } \end{aligned}$$
(5)

The other symbols in the above expressions have their usual meanings. All parameters, which are fixed by fitting to the masses of baryons with light flavors and heavy flavors, are taken from our previous works. To examine the dependence of our numerical results on parameters, we calculate the fully heavy pentaquark systems with two sets of parameters. The first set of parameters are derived from the research on \(N \Omega _{ccc}\) and \(N \Omega _{bbb}\) systems [112], and the second one is from the research on the fully heavy dibaryons [94].

2.2 Wave functions

The resonating group method (RGM) [113, 114] and generating coordinates method [115, 116] are used to carry out a dynamical calculation. The main feature of the RGM for two-cluster systems is that it assumes that two clusters are frozen inside, and only considers the relative motion between the two clusters. So the conventional ansatz for the two-cluster wavefunctions is

$$\begin{aligned} \psi _{5q} = \mathcal{A}\left[ [\phi _{B_{1}}\phi _{B_{2}}]^{[\sigma ]IS}\otimes \chi _{L}(\varvec{R})\right] ^{J} \end{aligned}$$
(6)

where the symbol \(\mathcal{A}\) is the anti-symmetrization operator. \([\sigma ]=[222]\) gives the total color symmetry and all other symbols have their usual meanings. \(\phi _{B_{1}}\) and \(\phi _{B_{2}}\) are the \(q^{3}\) and \(\bar{q} q\) cluster wavefunctions, respectively. From the variational principle, after variation with respect to the relative motion wave function \(\chi (\varvec{\textbf{R}})=\sum _{L}\chi _{L}(\varvec{\textbf{R}})\), one obtains the RGM equation:

$$\begin{aligned} \int H(\varvec{\textbf{R}},\varvec{\mathbf {R'}})\chi (\varvec{\mathbf {R'}})d\varvec{\mathbf {R'}}=E\int N(\varvec{\textbf{R}},\varvec{\mathbf {R'}})\chi (\varvec{\mathbf {R'}})d\varvec{\mathbf {R'}} \end{aligned}$$
(7)

where \(H(\varvec{\textbf{R}},\varvec{\mathbf {R'}})\) and \(N(\varvec{\textbf{R}},\varvec{\mathbf {R'}})\) are Hamiltonian and norm kernels. By solving the RGM equation, we can get the energies E and the wave functions. In fact, it is not convenient to work with the RGM expressions. Then, we expand the relative motion wave function \(\chi (\varvec{\textbf{R}})\) by using a set of gaussians with different centers

$$\begin{aligned} \chi _{L}(\varvec{R})&= \frac{1}{\sqrt{4 \pi }}\left( \frac{6}{5 \pi b^{2}}\right) ^{3 / 4} \sum _{i = 1}^{n} C_{i} \nonumber \\&\quad \cdot \int \exp \left[ -\frac{3}{5 b^{2}}\left( \varvec{R}-\varvec{S}_{i}\right) ^{2}\right] Y^{L}\left( \hat{\varvec{S}}_{i}\right) d \hat{\varvec{S}}_{i} \end{aligned}$$
(8)

where L is the orbital angular momentum between two clusters, and \(\varvec{S_{i}}\), \(i=1,2,\ldots ,n\) are the generator coordinates, which are introduced to expand the relative motion wavefunction. By including the center of mass motion:

$$\begin{aligned} \phi _{C} (\varvec{R}_{C}) = \left( \frac{5}{\pi b^{2}}\right) ^{3/4}e^{-\frac{5\varvec{R}^{2}_{C}}{2b^{2}}} \end{aligned}$$
(9)

the ansatz Eq. (6) can be rewritten as

$$\begin{aligned} \psi _{5 q}&= \mathcal {A} \sum _{i=1}^{n} C_{i} \int \frac{d \hat{\varvec{S}}_{i}}{\sqrt{4 \pi }} \prod _{\alpha =1}^{3} \phi _{\alpha }\left( \varvec{S}_{i}\right) \prod _{\beta =4}^{5} \phi _{\beta }\left( -\varvec{S}_{i}\right) \nonumber \\&\quad \cdot \left[ \left[ \chi _{I_{1} S_{1}}\left( B_{1}\right) \chi _{I_{2} S_{2}} \left( B_{2}\right) \right] ^{I S} Y_{L M}\left( \hat{\varvec{S}}_{i}\right) \right] ^{J} \nonumber \\&\quad \cdot \left[ \chi _{c}\left( B_{1}\right) \chi _{c}\left( B_{2}\right) \right] ^{[c]} \end{aligned}$$
(10)

where \(\chi _{I_{1}S_{1}}\) and \(\chi _{I_{2}S_{2}}\) are the product of the flavor and spin wave functions, and \(\chi _{c}\) is the color wave function. These will be shown in detail later. \(\phi _{\alpha }(\varvec{S}_{i})\) and \(\phi _{\beta }(-\varvec{S}_{i})\) are the single-particle orbital wavefunctions with different reference centers:

$$\begin{aligned} \phi _{\alpha }\left( \varvec{S}_{i}\right)&= \left( \frac{1}{\pi b^{2}}\right) ^{3 / 4} e^{-\frac{1}{2 b^{2}}\left( r_{\alpha }-\frac{2}{5} \varvec{S}_{i}\right) ^{2}} \nonumber \\ \phi _{\beta }\left( \varvec{-S}_{i}\right)&= \left( \frac{1}{\pi b^{2}}\right) ^{3 / 4} e^{-\frac{1}{2 b^{2}}\left( r_{\beta }+\frac{3}{5} \varvec{S}_{i}\right) ^{2}} \end{aligned}$$
(11)

With the reformulated ansatz Eq. (10), the RGM Eq. (7) becomes an algebraic eigenvalue equation:

$$\begin{aligned} \sum _{j} C_{j}H_{i,j}= E \sum _{j} C_{j}N_{i,j} \end{aligned}$$
(12)

where \(H_{i,j}\) and \(N_{i,j}\) are the Hamiltonian matrix elements and overlaps, respectively. By solving the generalized eigen problem, we can obtain the energy and the corresponding wavefunctions of the pentaquark systems.

For the spin wave function, we first construct the spin wave functions of the \(q^{3}\) and \(\bar{q}q\) clusters with SU(2) algebra, and then the total spin wave function of the pentaquark system is obtained by coupling the spin wave functions of two clusters together. The spin wave functions of the \(q^{3}\) and \(\bar{q}q\) clusters are

(13)
(14)

For pentaquark system, the total spin quantum number can be 1/2, 3/2 or 5/2, so the wave function of each spin quantum number can be written as follows, according to Clebsch-Gordan (CG) coefficient

$$\begin{aligned} \chi _{\frac{5}{2}, \frac{5}{2}}^{\sigma }(5)&= \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma }(3) \chi _{1,1}^{\sigma }(2) \nonumber \\ \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma 1}(5)&= \sqrt{\frac{3}{5}} \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma }(3) \chi _{1,0}^{\sigma }(2)-\sqrt{\frac{2}{5}} \chi _{\frac{3}{2}, \frac{1}{2}}^{\sigma }(3) \chi _{1,1}^{\sigma }(2)\nonumber \\ \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma 2}(5)&= \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma }(3) \chi _{0,0}^{\sigma }(2) \nonumber \\ \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma 1}(5)&= \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma }(3) \chi _{0,0}^{\sigma }(2) \nonumber \\ \chi _{\frac{1}{2}, \frac{1}{2}}^{\sigma 2}(5)&= \frac{1}{\sqrt{6}} \chi _{\frac{3}{2},-\frac{1}{2}}^{\sigma }(3) \chi _{1,1}^{\sigma }(2)-\frac{1}{\sqrt{3}} \chi _{\frac{3}{2}, \frac{1}{2}}^{\sigma }(3) \chi _{1,0}^{\sigma }(2)\nonumber \\&\quad +\frac{1}{\sqrt{2}} \chi _{\frac{3}{2}, \frac{3}{2}}^{\sigma }(3) \chi _{1,-1}^{\sigma }(2) \end{aligned}$$
(15)

Similar to constructing spin wave functions, we first write down the flavor wave functions of the \(q^{3}\) and \(\bar{q} q\) clusters, which are

$$\begin{aligned} \chi ^{f 1}(3)&= ccc \nonumber \\ \chi ^{f 2}(3)&= \frac{1}{\sqrt{3}}(ccb+cbc+bcc) \nonumber \\ \chi ^{f 3}(3)&= \frac{1}{\sqrt{3}}(bbc+bcb+cbb) \nonumber \\ \chi ^{f 4}(3)&= bbb \nonumber \\ \chi ^{f 5}(3)&= \frac{1}{\sqrt{6}}(2ccb-cbc-bcc) \nonumber \\ \chi ^{f 6}(3)&= \frac{1}{\sqrt{2}}(cbc-bcc) \nonumber \\ \chi ^{f 7}(3)&= \frac{1}{\sqrt{6}}(cbb+bcb-2bbc) \nonumber \\ \chi ^{f 8}(3)&= \frac{1}{\sqrt{2}}(cbb-bcb) \nonumber \\ \chi ^{f 1}(2)&= \bar{c} c \nonumber \\ \chi ^{f 2}(2)&= \bar{c} b \nonumber \\ \chi ^{f 3}(2)&= \bar{b} c \nonumber \\ \chi ^{f 4}(2)&= \bar{b} b \end{aligned}$$
(16)

As for the flavor degree-of-freedom, since the quark content of the investigated pentaquark system is \(QQQQ\bar{Q}\) \((Q = c, b)\), the isospin I of pentaquark states can only be 0. So the CG coefficient of this part is

$$\begin{aligned} \chi _{0,0}^{f}(5)=\chi _{0,0}^{f}(3) \chi _{0,0}^{f}(2) \end{aligned}$$
(17)

For the color singlet channel, which means two clusters are color singlet-singlet structure, the color wave function can be obtained by \(\textbf{1}\) \( _{[QQQ]} \mathbf {\otimes } \textbf{1}\) \(_{[Q \bar{Q}]}\)

$$\begin{aligned} \chi ^{c 1}&= \frac{1}{\sqrt{6}}(r g b-r b g+g b r-g r b+b r g-b g r) \nonumber \\&\quad \cdot \frac{1}{\sqrt{3}}(r \bar{r}+g \bar{g}+b \bar{b}) \end{aligned}$$
(18)

In addition, it is possible for the pentaquark systems to have hidden-color channel, that is, the baryon and the meson are color octet-octet structure and then become colorless through the coupling of the two clusters (\(\textbf{8}\) \( _{[QQQ]} \mathbf {\otimes } \textbf{8}\) \(_{[Q \bar{Q}]}\)). The corresponding color wave function of the pentaquark systems can be obtained through the following coupling

(19)

Then, the colorless pentaquark wave functions can be written as

$$\begin{aligned} \chi ^{c 2}&= \frac{1}{\sqrt{48}}(2 r r g-r g r-g r r) \bar{r} b\nonumber \\&\quad +\frac{1}{\sqrt{48}}(r g g+g r g-2 g g r) \bar{g} b \nonumber \\&\quad -\frac{1}{\sqrt{48}}(2 r r b-r b r-b r r) \bar{r} g\nonumber \\&\quad -{\frac{1}{\sqrt{4}8}}(r b b+b r b-2 b b r) \bar{b} g\nonumber \\&\quad + \frac{1}{\sqrt{48}}(2 g g b-g b g-b g g) \bar{g} r\nonumber \\&\quad +{\frac{1}{\sqrt{4}8}}(g b b+b g b-2 b b g) \bar{b} r \nonumber \\&\quad + \frac{1}{\sqrt{192}}(r b g-g b r+b r g-b g r)(2 \bar{b} b-\bar{r} r-\bar{g} g) \nonumber \\&\quad + \frac{1}{\sqrt{192}}(2 r g b-r b g+2 g r b\nonumber \\&\quad -g b r-b r g-b g r)(\bar{r} r-\bar{g} g) \nonumber \\ \chi ^{c 3}&= \frac{1}{\sqrt{16}}(r g r-g r r) \bar{r} b+\frac{1}{\sqrt{16}}(r g g-g r g) \bar{g} b \nonumber \\&\quad -\frac{1}{\sqrt{16}}(r b r-b r r) \bar{r} g-\frac{1}{\sqrt{16}}(r b b-b r b) \bar{b} g \nonumber \\&\quad +\frac{1}{\sqrt{16}}(g b g-b g g) \bar{g} r+\frac{1}{\sqrt{16}}(g b b-b g b) \bar{b} r \nonumber \\&\quad +\frac{1}{\sqrt{64}}(r b g+g b r-b r g-b g r)(\bar{r} r-\bar{g} g) \nonumber \\&\quad +\frac{1}{\sqrt{576}}(2 r g b+r b g-2 g r b-g b r-b r g+b g r) \nonumber \\&\quad \cdot (2 \bar{b} b-\bar{g} g-\bar{r} r) \end{aligned}$$
(20)

Finally, we can acquire the total wave function by combining the wave functions of the orbital, spin, flavor and color parts together according to the quantum number of the pentaquark systems. More details on the construction of the wave function can be found in Ref. [117].

3 The results and discussions

In this work, we perform a systematical investigation of the \(S-\)wave fully heavy pentaquark systems \(QQQQ\bar{Q}\) (\(Q=c,~b\)) in the framework of chiral quark model. Two sets of parameters are used to calculate the systems. The numerical results of two sets of parameters are consistent with each other. To save place, the discussions below are mainly based on the results of the first set of parameters. At the end of this section, we present the binding energies and the masses of all bound states with both sets of parameters, which may give an approximate energy range for all predicted fully heavy pentaquarks.

The quantum numbers of these systems are \(J^P = 1/2^-\), \(3/2^-\), and \(5/2^-\), respectively. The effective potentials are calculated to study the interaction between two hadron clusters. The dynamic bound state calculations are carried out to find if there is any bound state. Both color singlet (with color coupling \(\textbf{1}\) \( _{[QQQ]} \mathbf {\otimes } \textbf{1}\) \(_{[Q \bar{Q}]}\)) and hidden color channels (with color coupling \(\textbf{8}\) \( _{[QQQ]} \mathbf {\otimes } \textbf{8}\) \(_{[Q \bar{Q}]}\)), as well as the channel-coupling of all channels are included in the calculation. Moreover, the calculation of root mean square (RMS) is helped to explore the structure of the bound states

All the calculating energies for fully heavy pentaquark systems are listed in Tables 1, 4 and 6, respectively. The second column, headed with channel, denotes the physical contents of each channel; \(E_{th}\) and \(E_{sc}\) represent the threshold and the energy of each single channel. Regarding the results of the channel coupling calculation, \(E_{cc1}\) stands for the coupling of only color singlet channels, and \(E_{cc2}\) stands for the coupling of both the color singlet channels and the hidden-color channels. The subscript ‘8’ is marked to represent the hidden-color channel. The last column, headed with \(E_B\), stands for the binding energy of the pentaquark state, which is defined as \(E_B = E_{cc2} - M_{1} - M_{2}\). \(M_{1}\) and \(M_{2}\) are the theoretical masses of the baryon and meson, respectively.

3.1 \(cccc\bar{b}\) and \(bbbb\bar{c}\) systems

Fig. 1
figure 1

The effective potentials of \(cccc\bar{b}\) systems with the first set of parameters

Full size image

Since the attractive potential is necessary for forming bound states, we first calculate the effective potential between the baryon and meson, which is defined as \(V(S_{i}) = E(S_{i}) - E(\infty )\), where \(E(S_{i})\) is the diagonal matrix element of the Hamiltonian of the system in the generator coordinate. We present the effective potentials of color singlet channels and channel coupling to investigate the interactions between baryons and mesons.

The effective potentials of \(cccc\bar{b}\) systems with different quantum numbers are shown in Fig. 1, where ‘coupling’ stands for the effective potential with coupling of both color singlet and hidden-color channels. The behavior of effective potentials of \(bbbb\bar{c}\) systems is similar to \(cccc\bar{b}\) systems because of the heavy flavor symmetry. To save place, we only present the effective potentials of \(cccc\bar{b}\) systems. For the \(J^P=1/2^-\) system, one sees that the potential for the channel \(\Omega _{ccc} B_{c}^*\) is attractive. And it is obvious that the attraction becomes deeper after channel-coupling calculation. For the \(J^P=3/2^-\) system, we can see that the effective potentials for the \(\Omega _{ccc} B_{c}\) and \(\Omega _{ccc} B_{c}^*\) are repulsive. Therefore, it is impossible for these two single channels to form any bound state. However, the effective potential appears attractive after the channel coupling. This shows that the influence of hidden-color channel and channel coupling is significant in this system. As for the \(J^P=5/2^-\) system, the effective potential is much more repulsive, comparing to the previous two systems. Moreover, the repulsion increases greatly when the two hadrons \(\Omega _{ccc}\) and \(B_{c}^*\) get very close. This is due to the Pauli exclusion principle. Considering that the four charm quarks in the pentaquark system with \(J^P=5/2^-\) are identical particles, they have the same state in the orbital, flavor, and spin wave functions. But there are only three color states, which are red, green and blue. Then two quarks must be in the same state, which is prohibited by the Pauli exclusion principle and these four quarks are difficult to stay together. That’s why strong repulsion appears between \(\Omega _{ccc}\) and \(B_{c}^*\) and the \(cccc\bar{b}\) pentaquark state with \(J^P=5/2^-\) is difficult to form a bound state.

Table 1 The energies of the \(cccc\bar{b}\) and \(bbbb\bar{c}\) pentaquark systems with the first set of parameters (unit: MeV)
Full size table

In order to investigate whether there is any bound state, a dynamic calculation based on the RGM has been performed. The energies of \(cccc\bar{b}\) and \(bbbb\bar{c}\) systems are listed in Table 1. For \(cccc\bar{b}\) systems, the energy of each single channel is above the corresponding theoretical threshold, which means that each color singlet channel is unbound. The energy of each hidden-color channel is much higher than that of the color singlet channel. After the channel coupling including hidden-color channels, the energy of the \(J^P=1/2^-\) state is lower than the threshold of \(\Omega _{ccc} B^*_c\), the binding energy of which is \(-0.9\) MeV. For \(J^P=3/2^-\) state, we first consider only the channel coupling of color singlet channels. In this case, the calculation results show that the energy is still higher than the theoretical threshold. However, after taking into account the channel coupling that includes the hidden-color channel, a bound state with binding energy of \(-0.5\) MeV is obtained. Besides, there is only one channel \(\Omega _{ccc} B^*_c\) for the \(J^P=5/2^-\) state, and this state is unbound. These numerical results are consistent with the behavior of two clusters in the effective potential. Both \(J^P=1/2^-\) and \(J^P=3/2^-\) systems exist attraction between baryon and meson, which leads to the formation of bound states. As for \(J^P=5/2^-\) systems, no bound state is formed, which is consistent with the repulsion in effective potential and Pauli exclusion principle.

One might notice that the binding energy is very small, compared to the mass of the pentaquark system. In fact, the large mass of the pentaquark system is mainly due to the constituent quark masses (\(m_c\) = 1744 MeV, \(m_b\) = 5100 MeV). In the process of calculating the binding energy, we subtract the corresponding threshold energy, which also subtracts the influence of the quark mass term. Therefore, compared with other residual terms, this binding energy is not so small.

Intuitively, as the mass of quarks increases, various effects may decrease to some extent. However, the channel-coupling effect of the hidden-color channel is not directly affected by the quark mass. It is influenced by the distance between quarks in quark model calculation. Actually, at the close distance, the system allows color structures other than color singlet-singlet structure to exist, as long as the system as a whole remains colorless. In this case, the channel-coupling effect of the hidden-color channel can interact with the color-singlet channels, and further may play a role in obtaining the system energy. While at the large distance, for instance, at 2 fm, the color octet-octet structure will result in a colored cluster being observed experimentally, which is forbidden. Therefore, when dealing with fully heavy quark systems, as long as there is a probability that the clusters in the system are at close distance, the channel-coupling effect of the hidden-color channel is still existent or even non-negligible.

Besides, in the effective potential figures, the channel-coupling effect of the hidden-color channel can be more directly observed. Here, we take the \(cccc\bar{b}\) systems as an example. In Fig. 1a, the attraction is deepened after coupling the hidden-color channels. Meanwhile, in Fig. 1b, the hidden-color channel causes an otherwise purely repulsive system to become attractive. In Fig. 1c, there exists no hidden-color channel. Moreover, one can find that, the attractions deepened or caused by the coupling of the hidden-color channels are located at 1 fm. This is consistent with the mechanism by which the hidden-color channel works as explained in the previous paragraph. In a word, the formation of bound states cannot be separated from the attraction at close distance caused by the channel-coupling effect of the hidden-color channels.

From Table 1, one may note that the mass difference between \(B_c\) and \(B^*_c\) is too small. This is mainly due to the parameters we used in this work. Since some of the heavy flavor hadrons have not been observed yet, such as \(B^*_c\) and \(\Omega _{ccb}\), we choose to use existing parameters instead of fitting new ones. As we mentioned in Sect. 2, the parameters we used here are taken from our previous work of Refs. [112] directly to avoid adjusting the parameters. The work of Ref. [112] is about the \(N\Omega _{ccc}\) and \(N\Omega _{bbb}\) systems, in which the parameters were determined by fitting the masses of both the light baryons and the heavy baryons. For instance, the masses of the \(\Omega _{ccc}\) and \(\Omega _{bbb}\) are respectively 5069.1 MeV and 15, 111.6 MeV in this work. When extending to the mesons, none of the parameters were readjusted. Therefore, the masses of \(B_c\) and \(B^*_c\) are not well reproduced here, which are 6711.6 MeV and 6712.3 MeV, respectively.

In order to minimize the theoretical errors and to compare calculated results to the experimental data in future, we shift the mass of the pentaquark states here. Generally, the mass of a pentaquark can be written as \(M^{theo.}=M^{theo.}_{1}+M^{theo.}_{2}+E_{B}\), where \(M^{theo.}_{1}\) and \(M^{theo.}_{2}\) stand for the theoretical masses of a baryon and a meson, and \(E_{B}\) is the binding energy of this state. To minimize the theoretical errors, we can shift the mass of pentaquark to \(M=M^{exp.}_{1}+M^{exp.}_{2}+E_{B}\), where the experimental values of a baryon and a meson are used. Since there lacks experimental values for some of the fully heavy hadrons, we use the values predicted by the lattice QCD calculations [118, 119], which can be seen in Table 2. In Ref. [119], the masses of baryons containing one, two, or three heavy quarks were calculated systematically. In Ref. [118], the \(B_c^*\) mass was predicted in full lattice QCD.

Table 2 The masses of hadrons taken from lattice QCD, that have not yet been experimentally observed (unit: MeV)
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Taking the pentaquark state \(cccc\bar{b}\) with \(J^P=1/2^-\) as an example, the calculated mass of this state is 11,776.4 MeV, then the binding energy \(E_B\) is obtained by subtracting the theoretical masses of \(\Omega _{ccc}\) and \(B_c^*\), \(11{,}776.4-5077.3-6700.0=-0.9\) (MeV). Using the Lattice QCD masses of \(\Omega _{ccc}\) and \(B_c^*\), the corrected mass of this pentaquark is \(M=4796+6330+(-0.9)=11{,}125.1\) (MeV). In this way, we finally obtain a fully heavy pentaquark state \(cccc\bar{b}\) with \(J^P=1/2^-\), the mass of which is 11,125.1 MeV. Utilizing the same operation, we can obtain the other three bound states, that are \(cccc\bar{b}\) with \(J^P=3/2^-\), \(bbbb\bar{c}\) with \(J^P=1/2^-\) and \(bbbb\bar{c}\) with \(J^P=3/2^-\), and the masses of these states are 11,070.0 MeV, 20,688.6 MeV and 20,633.5 MeV, respectively. The results of \(cccc\bar{b}\) and \(bbbb\bar{c}\) systems are similar, which is mainly due to the heavy flavor symmetry.

Table 3 The corrected masses and RMS of \(cccc\bar{b}\) and \(bbbb\bar{c}\) bound states with the first set of parameters
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In addition, to further confirm the existence of any bound state and investigate the structure of the state, we can calculate the RMS of the states discussed above. It is worth noting that, the scattering states have no real RMS since the relative motion wave functions of the scattered states are not integrable in the infinite space. If we calculate the RMS of scattering states in a limited space, we can only obtain a value that increases with the expansion of computing space. So we can calculate the RMS of various states to identify the nature of these states by keep expanding the computing space. For the \(\Omega _{ccc} B^*_c\) with \(J^P=5/2^-\), the RMS in a limited space is larger than 3 fm, and it continues to increase as the computational space increases. This indicates that the \(J^P=5/2^-\) \(\Omega _{ccc} B^*_c\) is not a bound state. For the \(cccc\bar{b}\) and \(bbbb\bar{c}\) systems with \(J^P = 1/2^-\) or \(3/2^-\), the value of RMS is around 1.8 fm, and it is stable as the computational space changes. This indicates that these systems are all loosely bound states. The results of RMS of the bound states are listed in the Table 3. Besides, one might feel that the explanation that the hidden-color channel works at close distance contradicts the RMS about 1.8 fm. In fact, they are not in conflict. Quantum mechanics tells us that the distance between two clusters is probabilistic from close distance to large distance. The function of the hidden-color channel is mainly at close distance, which does not affect the probability that the distance between the two clusters is large. On the other hand, given that the states we obtained are loosely bound states, the RMS result is reasonably large.

3.2 \(cccb\bar{c}\), \(cccb\bar{b}\), \(bbbc\bar{c}\) and \(bbbc\bar{b}\) systems

Fig. 2
figure 2

The effective potentials of \(cccb\bar{c}\) systems with the first set of parameters

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The behavior of effective potential of \(cccb\bar{c}\), \(cccb\bar{b}\), \(bbbc\bar{c}\) and \(bbbc\bar{b}\) pentaquark systems is similar. To save space, we only show the effective potential of \(cccb\bar{c}\) system in Fig. 2. One can see that, the potential of each channel is repulsive except for the one of \(\Omega _{ccb} J/\psi \) and \(\Omega _{ccb}^* J/\psi \) channels. However, the attraction of \(\Omega _{ccb} J/\psi \) and \(\Omega _{ccb}^* J/\psi \) channels is quite feeble. After channel coupling calculation, the attraction of \(cccb\bar{c}\) system with \(J^P=1/2^-\) is not reinforced, and there is still no obvious attraction in \(cccb\bar{c}\) system \(J^P=3/2^-\). As for \(cccb\bar{c}\) system of \(J^P=5/2^-\), although the single channels are repulsive, attraction appears after coupling the hidden-color channel. So, it is possible for this system to form a bound state.

Table 4 The energies of the \(cccb\bar{c}\), \(bbbc\bar{b}\), \(cccb\bar{b}\) and \(bbbc\bar{c}\) pentaquark systems with the first set of parameters (unit: MeV)
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The energies of the \(cccb\bar{c}\), \(bbbc\bar{b}\), \(cccb\bar{b}\) and \(bbbc\bar{c}\) pentaquark systems are presented in Table 4. On the basis of the numerical results, all the single channels and channel coupling of color singlet channels are unbound. After coupling hidden-color channels, energies of systems with \(J^P=1/2^-\) and \(J^P=3/2^-\) remain above the corresponding thresholds. In regard to systems with \(J^P=5/2^-\), unlike \(cccc\bar{b}\) and \(bbbb\bar{c}\) systems, the Pauli principle doesn’t rule out the possibility of bound states here. Four bound states are formed after channel coupling, which are \(cccb\bar{c}\) system with binding energy of \(-1.0\) MeV, \(bbbc\bar{b}\) system with binding energy of \(-7.3\) MeV, \(cccb\bar{b}\) system with binding energy of \(-1.9\) MeV and \(bbbc\bar{c}\) system with binding energy of \(-3.7\) MeV.

Same as previous sector, to minimize the theoretical errors and make sense of the results, we perform the mass correction here. Meanwhile, the RMS of the bound states are also calculated, which are listed in the Table 5. According to the RMS results, the spatial structure of these four bound states are not compact. In other words, high-spin \(cccb\bar{c}\), \(bbbc\bar{b}\), \(cccb\bar{b}\) and \(bbbc\bar{c}\) pentaquark states are loosely bound states in our calculations.

3.3 \(ccbb\bar{c}\) and \(ccbb\bar{b}\) systems

In this sector, we investigate the \(ccbb\bar{c}\) and \(ccbb\bar{b}\) systems with quantum numbers of \(J^P=1/2^-\), \(J^P=3/2^-\) and \(J^P=5/2^-\). The effective potential of \(ccbb\bar{c}\) is shown in the Fig. 3. As one can see that the effective potentials of both \(\Omega _{ccb}^* B_c^*\) and \(\Omega _{bbc}^* J/\psi \) channels are slightly attractive, while those of other single channels are all repulsive. After the channel coupling calculation, the attraction generated in the three \(J^P=1/2^-\), \(J^P=3/2^-\) and \(J^P=5/2^-\) systems is still very weak. Based on the bound-state calculation, we find that all the color-singlet channels are unbound, and the energies of all hidden-color channels are much higher than the theoretical thresholds. In addition, no bound state is formed in either color singlet channel coupling or channel coupling including hidden-color channels. The energies of both \(ccbb\bar{c}\) and \(ccbb\bar{b}\) systems with different quantum numbers are listed in the Table 6.

Table 5 The corrected masses and RMS of \(cccb\bar{c}\), \(bbbc\bar{b}\), \(cccb\bar{b}\) and \(bbbc\bar{c}\) bound states with the first set of parameters
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Fig. 3
figure 3

The effective potentials of \(ccbb\bar{c}\) systems with the first set of parameters

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Table 6 The energies of the \(ccbb\bar{c}\) and \(ccbb\bar{c}\) pentaquark systems with the first set of parameters (unit: MeV)
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As mentioned in the beginning of Sect. 3, we use two sets of parameters to calculate the systems and find that the results are consistent with each other. Here, we present the binding energies and the masses of all bound states with both sets of parameters in Table 7, which may give an approximate energy range for all predicted fully heavy pentaquarks. The theoretical thresholds ‘\(E_{th}\)’, channel-coupling energies (including hidden-color channels) ‘\(E_{cc}\)’, the binding energies ‘\(E_B\)’ and the corrected masses ‘Mass’ of bound states are concluded in the table. Because two sets of parameters are used, the thresholds and energies of the same system are different. However, in bound state calculations, these differences are subtracted through mass correction. As one can see, the binding energies and the corrected masses demonstrate that the results of the two sets of parameters are consistent. This indicates that although the binding energy is relatively small, it is relatively stable within a reasonable parameter range.

Table 7 The results of two sets of parameters (unit: MeV)
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Encouraged by the discovery of the fully charmed tetraquark states in experiments, some theoretical work has begun to search for the existence of fully heavy pentaquark states. By using the QCD sum rule approach, both the fully charm pentaquark and fully bottom pentaquark candidates are obtained [100, 101]. The calculated mass is \(7.41^{+0.27}_{-0.31}\) GeV for \(cccc\bar{c}\) and \(21.60^{+0.73}_{-0.22}\) GeV for \(bbbb\bar{b}\) in Ref. [100], while it is \(7.93\pm 0.15\) GeV for \(cccc\bar{c}\) and \(23.91\pm 0.15\) GeV for \(bbbb\bar{b}\) in Ref. [101]. In the chiral quark model and quark delocalization color screening model, the \(cccc\bar{c}\) state is obtained with \(J^P = 1/2^-\) and the mass of \(7891.9 \sim 7892.7\) MeV, the \(bbbb\bar{b}\) state is obtained with \(J^P=1/2^-\) and the mass of \(23{,}810.1 \sim 23{,}813.8\) MeV, and with \(J^P = 3/2^-\) and the mass of \(23{,}748.2 \sim 23{,}752.3\) MeV [105]. In a lattice-QCD inspired quark model, several resonances were obtained in each spin-parity channel for the fully charm and bottom systems, with mass above 8 GeV and 24 GeV, respectively [104]. In the framework of CMI model, two stable candidates were obtained, which are \(J^P = 3/2^-\) \(ccbb\bar{b}\) with mass of 17,416 MeV and \(J^P = 5/2^-\) \(ccbb\bar{b}\) with mass of 17,477 MeV [102]. However, in the work of Ref. [103], no any stable fully heavy pentaquark state was found within the constituent quark model.

For fully heavy pentaquark systems, due to the absence of light quark exchange, the forms of potentials used in different models have become simpler and less different. However, there are still some differences between various approaches that may lead to the different conclusions. (1) In Ref. [102, 103], only the color singlet-singlet structure was considered when constructing wave functions. Whereas in present work and Ref. [104], both the color singlet-singlet structure and the color octet-octet structure were considered. Besides, a more specific spatial configuration, diquark-diquark-antiquark configuration is also considered in the work of Ref. [104]. (2) In the chromomagnetic interaction (CMI) model [102], the effective quark mass was used, instead of the constituent quark mass. The quark mass term, confinement potential term, and kinetic term were absorbed by the effective quark mass term in the CMI model, yet these different terms have their own concrete forms in other models. In addition, even with the same terms, there may be subtle differences. For example, the confinement potential term could be used as a linear form of \(r_{ij}\) [104], a squared form of \(r_{ij}\) (this work), or some other reasonable forms [103]. Therefore, these differences may lead to different conclusions. All in all, since the fully heavy pentaquark still needs experimental results to provide more guidance for theoretical works, a variety of different theoretical works are needed and worthy of investigation at this stage to better understand the exotic states. We hope to have more theoretical and experimental work to search for possible fully heavy pentaquark states, and further test and improve theoretical methods.

4 Summary

In this work, we systematically investigate the low-lying fully heavy pentaquark systems in the chiral quark model. First, an adiabatic calculation of the effective potential is performed to explore the interaction between the baryon and meson clusters. Both the single channel and the channel coupling calculation are performed to explore the effect of the multi-channel coupling. Two types of color structure \(\textbf{1}\) \( _{[QQQ]} \mathbf {\otimes } \textbf{1}\) \(_{[Q \bar{Q}]}\) and \(\textbf{8}\) \( _{[QQQ]} \mathbf {\otimes } \textbf{8}\) \(_{[Q \bar{Q}]}\) are taken into consideration. The dynamic bound-state calculation is carried out to search for any bound state in the fully heavy pentaquark systems. Meanwhile, the RMS of the fully heavy pentaquark states is calculated to confirm the existence of any bound state and explore the internal spatial structure of the bound states.

The numerical results show that the effect of the channel coupling with hidden-color channels is important for forming bound states of the fully heavy pentaquark systems. With the help of the channel-coupling, we obtain eight fully heavy pentaquark states, which are \(cccc\bar{b}\) with \(J^P = 1/2^-\) and mass of 11,125.1\(\sim \)11,125.2 MeV, \(cccc\bar{b}\) with \(J^P = 3/2^-\) and mass of 11,070.0\(\sim \)11,070.1 MeV, \(bbbb\bar{c}\) with \(J^P = 1/2^-\) and mass of 20,688.6\(\sim \)20,690.0 MeV, \(bbbb\bar{c}\) with \(J^P = 3/2^-\) and mass of 20,633.5\(\sim \)20,634.7 MeV, \(cccb\bar{c}\) with \(J^P = 5/2^-\) and mass of 11,125.0\(\sim \)11,125.1 MeV, \(bbbc\bar{b}\) with \(J^P = 5/2^-\) and mass of 20681.8\(\sim \) 20,683.3 MeV, \(cccb\bar{b}\) with \(J^P = 5/2^-\) and mass of 14,254.4\(\sim \)14,254.6 MeV and \(bbbc\bar{c}\) with \(J^P = 5/2^-\) and mass of 17,459.2\(\sim \)17,460.2 MeV. Since the binding energies of these states are all below 10 MeV and the RMS are around 1.8 fm, we tend to interpret these states as molecular states, which are worth looking for in future experiments.