Masses and magnetic moments of exotic fully heavy pentaquarks

Abstract

Inspired by the observation of a resonant state X(6600) of fully charm tetraquark by the CMS experiment of LHCb Collaboration in double $J/\psi$ decay channel, we perform a systematical study of all configurations of fully heavy pentaquarks $P_{Q_{1}Q_{2}Q_{3}Q_{{4}}\bar{Q_{5}}}$ ($Q_{i}=c,b$, $i=1,2,3,4,5$) in their ground states in unified framework of MIT bag model. The color-spin wavefunctions of pentaquarks, classified via Young tableau and presented in terms of the Young–Yamanouchi bases, are used to compute masses and magnetic moments of fully heavy pentaquarks via numerical variational method, predicting a set of masses ranging from 8.229 GeV for the $P_{cccc\bar{c}}$ to 24.770 GeV for the $P_{bbbb\bar{b}}$. Combining with computed masses of fully heavy mesons and baryons, we find that masses of fully heavy hadrons(mesons, baryons, tetraquarks and pentaquarks) with identical flavor rise almost linearly with the number of valence quarks in hadrons, being consistent with the heavy quark symmetry in the heavy-quark limit.

1 Introduction

Possible existence of multiquark hadrons like tetraquarks ($q^{2}\bar{q}^{2}$) and pentaquarks ($q^{4}\bar{q}$) was suggested earlier at the birth of quark model [1, 2]. Later in the 1970s, multiquark states are studied by Jaffe via the MIT bag model [3, 4]. Since observation of the X(3872) [5] in 2003 by the Belle, many candidates of tetraquarks have been observed, some of which, such as the $Z_{c}(3900)$ [6] and the $T_{cc}(3875)$, are undoubtedly exotic. In 2020, a first candidate of fully charm tetraquark, the X(6900)/the X(6600), has been observed by LHCb in the di-$J/\psi$ invariant mass spectrum and later  confirmed by CMS and ATLAS of the LHC at CERN [7,8,9]. For experimental searches of light-flavor pentaquark hadrons, it has been a long and nontrivial history, with no undisputed candidates found in over 50 years. Convincing evidence for pentaquark-like structures $P_{c}(4450)^{+}$ and $P_{c}(4380)^{+}$ with a minimal quark content of $uudc\bar{c}$ was reported by LHCb in a study of $\Lambda _{b}^{0}\rightarrow J/\psi pK^{-}$($J/\psi \rightarrow \mu ^{+}\mu ^{-}$) decays [10] in 2015, for which the former peak(state) $P_{c}(4450)^{+}$ was resolved further into two states $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ (over $5.4\sigma$) in 2019. More recently, LHCb collaboration reported evidences of two new charmonium pentaquarks with strangeness, $P_{cs}(4459)$ [11], in the $J/\psi \Lambda$ distribution(in $\Xi _{b}^{-}\rightarrow J/\psi \Lambda K^{-}$ decays) in 2021 and $P_{\psi s}^{\Lambda }(4338)$ [12], in flavour-untagged $B^{-}\rightarrow J/\psi \Lambda \bar{p}$ decays in 2022. After observation and measurement of two $P_{c}$ states, many theoretical groups explained the $P_{c}(4450)^{+}$ and $P_{c}(4380)^{+}$ states as compact pentaquarks with diquarks and triquarks as building blocks [13,14,15,16,17,18,19], except for some few attempts via full five-body dynamics [20] which lead to states below the lowest threshold for spontaneous dissociation. On the other hand, there are molecular picture, proposed [21,22,23,24] before the first LHCb results [10], which interprets these narrow $P_{c}$ states in terms of deuteron-like loosely bound states of the baryon and meson, such as the $\Sigma _{c}\bar{D}^{(*)}$ states. This picture favors the narrow $P_{c}$ states, which have measured widths about 6–10 MeV for $P_{c}(4457)^{+}$ as well as $P_{c}(4312)^{+}$ and about 21 MeV for $P_{c}(4440)^{+}$, while the compact picture attributes narrow width to spin-orbit interaction (via spatial separation of c and $\bar{c}$ quarks). As the observed state $P_{c}(4380)^{+}$ is wider (with width about 205 MeV), the issue of width suppression remains to be explored. Till now, there are different pictures and approaches employed to analyze the hidden-charm pentaquarks ($P_{c}$ and $P_{cs}$) [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42], the hidden-strange and hidden-bottom pentaquarks [43,44,45,46]. One can infer Refs. [47, 48] for the pre-2019 reviews and Refs. [49, 50] for recent reviews. The purpose of this work is to systematically study all possible ground-states of fully heavy pentaquarks, $P_{Q_{1}Q_{2}Q_{3}Q_{{4}}\bar{ Q_{5}}}$ ($Q_{i}=c,b$, $i=1,2,3,4,5$) in unified framework of MIT bag model. For this, we construct all color-spin wavefunctions of pentaquarks based on $SU(2)_{s}\times SU(3)_{c}$ group, classified via Young tableau and expressed in terms of the Young–Yamanouchi bases, to compute masses and magnetic moments of fully heavy pentaquarks in ground states and thereby predict their masses. With the help of numerical variation upon the bag radius, the chromomagnetic mixing among the color-spin configurations are explored in details. Computing further spin-independent masses of the mesons, baryons and tetraquarks all made of identical heavy flavors, we find a linear dependence of the hadron mass upon the heavy quark number N in hadrons. This linear rise of hadron masses is notable for the bottom sector and is demonstrated analytically via a variational method of bag model, being consistent with heavy quark effective theory (HQET) in heavy quark limit. After introduction, the wavefunctions of the fully heavy pentaquarks with various configurations are classified in Sect. 2 by the Young tableau and expressed in terms of the Young–Yamanouchi bases. In Sect. 3, we briefly review basic relations of MIT bag model including magnetic moments for each spin state, and describe how to apply the variational method to solve the bag model for the heavy hadrons. In Sect. 4, we perform numerical analysis of mass and magnetic moment for fully heavy pentaquarks. Similar computation are given to fully heavy mesons and baryons to show a linear relation for these fully heavy hadrons in Sect. 5. The paper ends with summary and conclusions in Sect.  6.

2 Wavefunctions of fully heavy pentaquarks

Fully heavy pentaquarks $P_{Q_{1}Q_{2}Q_{3}Q_{{4}}\bar{Q_{5}}}$ may contain two or more identical particles in the component $Q_{1}Q_{2}Q_{3}Q_{{4}}$, for which the corresponding parts of the wavefunctions have to be antisymmetric due to Pauli principle and colorlessness of $P_{Q_{1}Q_{2}Q_{3}Q_{{4}}\bar{Q_{5}}}$. Consider the pentaquark $P_{cccb\bar{ b}}$, for instance. Since the component ccc in $P_{cccb\bar{b}}$ are identical in flavor, the antisymmetry requirement for a fermionic hadron (color singlet) implies that the wavefunction of $P_{cccb\bar{b}}$ has to be antisymmetric in orbital, color and spin under exchange of each cc pair among the component ccc. To respect the overall symmetry of the hadronic wavefunctions mentioned above, we utilize the Young tableau to represent the irreducible representations (bases) of the permutation group and thereby classify the pentaquark configurations with certain symmetry. For a given classification, we use the Young–Yamanouchi basis, which corresponds to the Young tableau, to construct explicitly the configuration describing the pentaquark we consider.

First of all, we consider color wavefunctions of the pentaquark $P_{Q_{1}Q_{2}Q_{3}Q_{{4}}\bar{Q_{5}}}$, which are singlets in color space. Utilizing direct product of the fundamental (color) representations $[3]_{c}$ and $[\bar{3}]_{c}$, one can classify, in the language of group theory, the color wavefunctions of a pentaquark as below:

$$\begin{aligned}{} & {} [[3]_{c}\otimes [3]_{c}\otimes [3]_{c}\otimes [3]_{c}]\otimes [\bar{3}]_{c} \nonumber \\{} & {} \quad =[(1_{c}\oplus 8_{c}\oplus 8_{c}\oplus 10_{c})\otimes 3_{c}]\otimes \bar{3} _{c} \nonumber \\{} & {} \quad \rightarrow (1_{c}\otimes 3_{c}\otimes \bar{3}_{c})\oplus (8_{c}\otimes 3_{c}\otimes \bar{3}_{c})\oplus (8_{c}\otimes 3_{c}\otimes \bar{3}_{c}) \nonumber \\{} & {} \quad \rightarrow (3_{c}\otimes \bar{3}_{c})\oplus (3_{c}\otimes \bar{3} _{c})\oplus (3_{c}\otimes \bar{3}_{c}), \end{aligned}$$

(1)

in which the allowed color singlet is $\{3_{c}\otimes \bar{3}_{c}\}$ only.

One is then forced to consider the color triplet in the direct product, $[[3]_{c}\otimes [3]_{c}\otimes [3]_{c}\otimes [3]_{c}]$, of the component $Q_{1}Q_{2}Q_{3}Q_{{4}}$, which corresponds to Young tableau [2,1,1] expressed by

(2)

Here, the subscript labels the irreducible representation of the color group $SU(3)_{c}$. This yields three color-singlet configurations of $P_{Q_{1}Q_{2}Q_{3}Q_{{4}}\bar{Q_{5}}}$ if one combines each of color-triplets ($3_{c}$) listed in Eq. (2) with the antitriplet ($\bar{3}_{c}$) of the remaining antiquark $\bar{Q}_{5}$. We express the Young tableau representing the obtained colorless configurations as

(3)

and correspond them to the following color wavefunctions explicitly:

$$\begin{aligned} \phi _{1}^{P}= & {} \frac{1}{4\sqrt{3}}\Big [(2bbgr-2bbrg+gbrb-gbbr+bgrb-bgbr \nonumber \\{} & {} -rbgb+rbbg-brgb+brbg)\bar{b}+(2rrbg-2rrgb \nonumber \\{} & {} +rgrb-rgbr+grrb-grbr+rbgr-rbrg+brgr \nonumber \\{} & {} -brrg)\bar{r}+(2ggrb-2ggbr-rggb+rgbg-grgb \nonumber \\{} & {} +grbg+gbgr-gbrg+bggr-bgrg)\bar{g}\Big ], \end{aligned}$$

(4)

$$\begin{aligned} \phi _{2}^{P}= & {} \frac{1}{12}\Big [(3bgbr-3gbbr-3brbg+3rbbg-rbgb-2rgbb \nonumber \\{} & {} +2grbb+brgb+gbrb-bgrb)\bar{b}+(3grrb-3rgrb \nonumber \\{} & {} -3brrg+3rbrg-rbgr-2gbrr+2bgrr-grbr \nonumber \\{} & {} +rgbr+brgr)\bar{r}+(3grgb-3rggb+3bggr-3gbgr \nonumber \\{} & {} -grbg+rgbg+2rbgg-2brgg+gbrg-bgrg)\bar{g}\Big ], \end{aligned}$$

(5)

$$\begin{aligned} \phi _{3}^{P}= & {} \frac{1}{3\sqrt{2}}\Big [(grbb-rgbb+rbgb-brgb+bgrb-gbrb)\bar{ b} \nonumber \\{} & {} +(grbr-rgbr+rbgr-brgr+bgrr-gbrr)\bar{r} \nonumber \\{} & {} +(grbg-rgbg+rbgg-brgg+bgrg-gbrg)\bar{g}\Big ]. \end{aligned}$$

(6)

In the spin space, the direct product of five fermions represented in terms of Young tableau can be written as

(7)

For the spin $J=5/2,3/2$ and 1/2 of the pentaquarks, the configurations can be similarly represented in terms of Young tableau [5],[4,1], and [3,2] of one, four, and five dimensions. For the pentaquark with $J=5/2$, we write the spin wavefunction as,

(8)

In the case of the $J=3/2$, the spin wavefunctions become

(9)

while for $J=1/2$, they become,

(10)

We see from the above analysis that to describe fully heavy pentaquarks $P_{Q_{1}Q_{2}Q_{3}Q_{{4}}\bar{Q_{5}}}$ with one antiquark $\bar{Q}_{5}$ having a given flavor and color, the antiquark can be temporarily omitted from the spin degrees of freedom. Thus, we shall identify the spin states shown in Eqs. (8) through (10) via the Young–Yamanouchi bases associated with the Young tableau [4], [3,1], and [2,2]. Given that the color and spin states represented in terms of the Young tableau respect certain symmetry, one can construct the combined color-spin state of a pentaquark which is fully antisymmetric under the exchange of any pair among the heavy quarks $1=Q_{1}$, $2=Q_{2}$, $3=Q_{3}$, and $4=Q_{4}$(for short).

We start from the color singlets in Eq. (3), and combine them with spin states by the outer product of the permutation group, $S_{4}$, resulting in the color $\otimes$ spin states for the quarks 1, 2, 3 and 4. After isolating spin of the antiquark, one can deduce the outer product between Young tableaus [2,1,1] of the color singlets and that of the spin states[4], [3,1], [2,2]. For the state expression in terms of the first four quarks (1234) mentioned above, one can obtain the Young tableau reps. of color $\otimes$ spin states and write them as that in Ref. [51] (see Eq. (17))

According to the method described in Ref. [52], all the possible Young–Yamanouchi bases (i.e. $\psi =\phi _{2}^{P}\chi _{1}^{P}$) for the pentaquarks we shall address in this work can be obtained from the couplings ($\otimes$) of the color and spin degrees of freedom. Collectively, we express these color-spin coupled bases, again, in terms of the Young tableau:

(11)

in which (for $J=5/2$)

$$\begin{aligned} \psi _{1}^{*}=\phi _{2}^{P}\chi _{1}^{P},\quad \psi _{2}^{*}=\phi _{3}^{P}\chi _{1}^{P}\text {, } \end{aligned}$$

(12)

and (for $J=3/2$)

$$\begin{aligned}&\psi _{1}^{\prime }=\frac{1}{\sqrt{3}}\phi _{1}^{P}\chi _{5}^{P}-\frac{1}{ \sqrt{3}}\phi _{2}^{P}\chi _{4}^{P}+\frac{1}{\sqrt{3}}\phi _{3}^{P}\chi _{3}^{P}, \nonumber \\&\psi _{2}^{\prime }=-\frac{1}{\sqrt{6}}\phi _{1}^{P}\chi _{5}^{P}+\frac{1}{ \sqrt{6}}\phi _{2}^{P}\chi _{4}^{P}+\sqrt{\frac{2}{3}}\phi _{3}^{P}\chi _{3}^{P}, \nonumber \\&\psi _{3}^{\prime }=\frac{1}{\sqrt{3}}\phi _{1}^{P}\chi _{5}^{P}-\frac{1}{ \sqrt{6}}\phi _{2}^{P}\chi _{3}^{P}+\frac{1}{\sqrt{3}}\phi _{2}^{P}\chi _{4}^{P}+\frac{1}{\sqrt{6}}\phi _{3}^{P}\chi _{4}^{P}, \nonumber \\&\psi _{4}^{\prime }=-\frac{1}{\sqrt{6}}\phi _{1}^{P}\chi _{5}^{P}-\frac{1}{ \sqrt{3}}\phi _{2}^{P}\chi _{3}^{P}-\frac{1}{\sqrt{6}}\phi _{2}^{P}\chi _{4}^{P}+\frac{1}{\sqrt{3}}\phi _{3}^{P}\chi _{4}^{P}, \nonumber \\&\psi _{5}^{\prime }=\phi _{3}^{P}\chi _{2}^{P}, \nonumber \\&\psi _{6}^{\prime }=\phi _{2}^{P}\chi _{2}^{P}. \end{aligned}$$

(13)

In the case of $J=1/2$, the color $\otimes$ spin bases in Eq. (11) are explicitly,

$$\begin{aligned}&\psi _{1}=\frac{1}{\sqrt{3}}\phi _{1}^{P}\chi _{8}^{P}-\frac{1}{\sqrt{3}} \phi _{2}^{P}\chi _{7}^{P}+\frac{1}{\sqrt{3}}\phi _{3}^{P}\chi _{6}^{P}, \nonumber \\&\psi _{2}=-\frac{1}{\sqrt{6}}\phi _{1}^{P}\chi _{8}^{P}+\frac{1}{\sqrt{6}} \phi _{2}^{P}\chi _{7}^{P}+\sqrt{\frac{2}{3}}\phi _{3}^{P}\chi _{6}^{P}, \nonumber \\&\psi _{3}=\frac{1}{\sqrt{3}}\phi _{1}^{P}\chi _{8}^{P}-\frac{1}{\sqrt{6}} \phi _{2}^{P}\chi _{6}^{P}+\frac{1}{\sqrt{3}}\phi _{2}^{P}\chi _{7}^{P}+ \frac{1}{\sqrt{6}}\phi _{3}^{P}\chi _{7}^{P}, \nonumber \\&\psi _{4}=-\frac{1}{\sqrt{6}}\phi _{1}^{P}\chi _{8}^{P}-\frac{1}{\sqrt{3}} \phi _{2}^{P}\chi _{6}^{P}-\frac{1}{\sqrt{6}}\phi _{2}^{P}\chi _{7}^{P}+ \frac{1}{\sqrt{3}}\phi _{3}^{P}\chi _{7}^{P}, \nonumber \\&\psi _{5}=\frac{1}{\sqrt{2}}\phi _{1}^{P}\chi _{10}^{P}-\frac{1}{\sqrt{2}} \phi _{2}^{P}\chi _{9}^{P}, \nonumber \\&\psi _{6}=\frac{1}{2}\phi _{1}^{P}\chi _{10}^{P}+\frac{1}{2}\phi _{2}^{P}\chi _{9}^{P}-\frac{1}{\sqrt{2}}\phi _{3}^{P}\chi _{9}^{P}. \end{aligned}$$

(14)

Based on the Young tableau in Eq. (11) and Pauli principle for each pair of two fermions, we can construct all allowed color-spin wavefunctions and list them in Table 1 for pentaquarks consisting of four heavy quarks and a heavy antiquark. Based on similar principle, one can construct the states involved in chromomagnetic interactions (CMI) (18) with the help of color and spin factors (19,20) via the color bases (3) and spin bases (8–10), respectively. Following Ref.  [53], we shall use the scaling ratios of color factors to evaluate the matrices of the binding energy, as discussed in Sect. 3.

Table 1 Color-spin bases of pentaquarks $q_{1}q_{2}q_{3}q_{4}\bar{q}_{5}$ with $J^{P}$ quantum number

3 MIT bag model and CMI

MIT bag model describes hadron as a sphere bag containing confined valence quarks. The model also includes perturbative inter-quark interactions via considering the lowest-order gluon exchange among quarks, which is referred as chromomagnetic interaction [54, 55]. Mass of hadrons for a bag of radius R is [54, 55],

$$\begin{aligned} M\left( R\right)= & {} \sum _{i}\omega _{i}+\frac{4}{3}\pi R^{3}B-\frac{Z_{0}}{R} +M_{BD}+M_{CMI}, \end{aligned}$$

(15)

$$\begin{aligned} \omega _{i}= & {} \left( m_{i}^{2}+\frac{x_{i}^{2}}{R^{2}}\right) ^{1/2}, \end{aligned}$$

(16)

where the first term is total sum of the kinematic energy of relativistic quark i with mass $m_{i}$, the second is bag volume energy with B the bag constant, the third is zero point energy with coefficient $Z_{0}$, the forth is binding energy ($M_{BD}$) between heavy quarks or between heavy and strange quarks [56, 57], and the fifth is chromomagnetic interaction. In Eq. (15), the bag radius R is bag radius to be determined variationally, and the quark momentum $x_{i}$ in unit of $R^{-1}$ satisfies a boundary condition on bag surface,

$$\begin{aligned} \tan x_{i}=\frac{x_{i}}{1-m_{i}R-\left( m_{i}^{2}R^{2}+x_{i}^{2}\right) ^{1/2}}\text {,} \end{aligned}$$

(17)

which is to be solved iteratively in this work.

The interaction in this work enters among quarks via two parts, $\mathrm {\Delta } M=M_{BD}+M_{CMI}$. The first part, the binding energy $M_{BD}= \sum _{M<N}B_{MN}$ ($M,N=s$, c and b), proposed in Refs. [56, 57], rises mainly from short-range chromoelectric interactions between heavy quarks or within heavy-strange pairs as they are (relatively) massive and move nonrelativistically. This part was supported by our previous work [53], which reconciles bag dynamics of the light and heavy hadrons. Five binding energies $B_{cs}$, $B_{cc}$, $B_{bs}$, $B_{bc}$ and $B_{bb}$ in the color $\bar{3}_{c}$ rep., extracted from the bag mass corrections to the baryons when heavy pairs (cs, cc, bs, bc and bb) involved, can be scaled to the binding energies $B_{MN}$ in $6_{c}$ reps. via scaling factor ratios between two reps. [53, 56]. As there are two reps. ($\bar{3} _{c}$ and $6_{c}$) for color wavefunctions of quark pairs, as discussed in Sect. 2, the binding energies could be matrices in color-spin space given in Table 1, where spin bases are orthogonal.

The second part of interaction in Eq. (15) is chromomagnetic interaction $M_{CMI}$, which can be due to the short-range gluon exchange [58]. Similar to the interaction of magnetic moments of the quark spins, this part has the form of

$$\begin{aligned} M_{CMI}=-\sum _{i<j}\left( \mathbf {\lambda }_{i}\cdot \mathbf {\lambda } _{j}\right) \left( \mathbf {\sigma }_{i}\cdot \mathbf {\sigma }_{j}\right) C_{ij}\text {,} \end{aligned}$$

(18)

where i and j denote the quark (anti-quark) indices, $\lambda$ the Gell-Mann matrices, $\sigma$ the Pauli matrices, and $C_{ij}$ the coupling parameters of the CMI.

To compute the color and spin factor in Eq. (18), we employ the following formula of the matrix element:

$$\begin{aligned} {\left\langle \mathbf {\lambda }_{i}\cdot \mathbf {\lambda }_{j}\right\rangle } _{nm}=\sum _{\alpha =1}^{8}\textrm{Tr}\left( c_{in}^{\dagger }\lambda ^{\alpha }c_{im}\right) \textrm{Tr}\left( c_{jn}^{\dagger }\lambda ^{\alpha }c_{jm}\right) \text {,} \end{aligned}$$

(19)

$$\begin{aligned} {\left\langle \mathbf {\sigma }_{i}\cdot \mathbf {\sigma }_{j}\right\rangle } _{xy}=\sum _{\alpha =1}^{3}\textrm{Tr}\left( \chi _{ix}^{\dagger }\sigma ^{\alpha }\chi _{iy}\right) \textrm{Tr}\left( \chi _{jx}^{\dagger }\sigma ^{\alpha }\chi _{jy}\right) \text {,} \end{aligned}$$

(20)

where n, m and x, y indicate components of the basis vectors of color and spin wavefunctions of a hadron, respectively. The symbols c and $\chi$ denote bases of color and spin vectors associated with a quark, respectively. The matrices of two factors can be calculated with the help of Eqs. (19)–(20) once color-spin wavefunctions $\psi =\sum \phi ^{P}\chi ^{P}$ were determined.

In the MIT bag model, the CMI parameter $C_{ij}$ is known analytically, given by [54]

$$\begin{aligned} C_{ij}=3\frac{\alpha _{s}\left( R\right) }{R^{3}}\bar{\mu }_{i}\bar{\mu } _{j}I_{ij}\text {,} \end{aligned}$$

(21)

where the reduced magnetic moment $\bar{\mu }_{i}$ and the (running) strong coupling $\alpha _{s}$ are given by

$$\begin{aligned} \bar{\mu }_{i}=\frac{R}{6}\frac{4\alpha _{i}+2\lambda _{i}-3}{2\alpha _{i}\left( \alpha _{i}-1\right) +\lambda _{i}}\text {,} \end{aligned}$$

(22)

$$\begin{aligned} \alpha _{s}(R)=\frac{0.296}{ln\left[ 1+{\left( 0.281R\right) }^{-1}\right] } \text {,} \end{aligned}$$

(23)

and

$$\begin{aligned} I_{ij}=1+2\int _{0}^{R}\frac{dr}{r^{4}}\bar{\mu }_{i}\bar{\mu } _{j}=1+F(x_{i},x_{j})\text {.} \end{aligned}$$

(24)

Here, $\alpha _{i}=\omega _{i}R$ and $\lambda _{i}=m_{i}R$ [54]. The function of $\alpha _{s}(R)$ in Eq. (23) is discussed in Ref. [53]. The function $F(x_{i},x_{j})$ in Eq. (24) is rational in terms of parameters $x_{i}$ and $x_{j}$,

(25)

where $y_{i}=x_{i}-\textrm{sin}(x_{i})\textrm{cos}(x_{i})$ and

$$\begin{aligned} \textrm{Si}(x)=\int _{0}^{x}\ \frac{\textrm{sin}(t)}{t}\textrm{d}t\text {.} \end{aligned}$$

(26)

From Eq. (22), one can write the magnetic moment $\mu _{i}$ of the quark (anti-quark) i as, in bag model,

$$\begin{aligned} \mu _{i}=Q_{i}\bar{\mu }_{i}=Q_{i}\frac{R}{6}\frac{4\alpha _{i}+2\lambda _{i}-3}{2\alpha _{i}\left( \alpha _{i}-1\right) +\lambda _{i}}\text {,} \end{aligned}$$

(27)

with electric charge $Q_{i}$. As a result, the magnetic moment of a hadron with color-spin wavefunction $\psi$ is,

$$\begin{aligned} \mu =\left\langle \psi \left| \sum \nolimits _{i}g_{i}\mu _{i}S_{iz}\right| \psi \right\rangle \text {,} \end{aligned}$$

(28)

where $g_{i}=2$, and $S_{iz}$ is the third component of spin of the individual quark i [59].

We shall calculate the magnetic moments of the pentaquarks in unit of the magnetic moment of the proton $\mu _{p}$ and further transform the obtained results into that in unit of $\mu _{N}$ with the help of measured proton magnetic moment $\mu _{p}=2.79285\mu _{N}$ [60, 61]. Note that Eq. (28) also holds true in the case of chromomagnetic mixing, for which one can expand $\psi$ in terms of the color-spin bases to find magnetic moment by Eq. (28). For the spin wavefunctions in Sect. 2, one can derive each diagonal element of magnetic moment. The results are listed in Table 2 in detail.

Table 2 Sum rule of magnetic moments for pentaquarks $q_{1}q_{2}q_{3}q_{4}{ \bar{q}}_{5}$

For the model parameters, we use the values in our previous work [53] which reconcile dynamics of the light and heavy hadrons. These values are $Z_{0}$, the constant B and masses of the quarks (the nonstrange quarks $n=u,d$, the strange quark s, the charm quark c and the bottom quark b) in this work,

$$\begin{aligned} \begin{Bmatrix} Z_{0}=1.83,&B^{1/4}=0.145\,\text {GeV,} \\ m_{n}=0\,\text {GeV,}&m_{s}=0.279\,\text {GeV,} \\ m_{c}=1.641\,\text {GeV,}&m_{b}=5.093\,\text {GeV.} \end{Bmatrix} \end{aligned}$$

(29)

Meanwhile, using the binding energies $B_{QQ^{\prime }}$ for the quark pair $QQ^{\prime }$ in color antitriplet rep. [53],

$$\begin{aligned} \bar{3}_{c}: \begin{Bmatrix} B_{cs}=-0.025\,\text {GeV,}&B_{cc}=-0.077\,\text {GeV,} \\ B_{bs}=-0.032\,\text {GeV,}&B_{bb}=-0.128\,\text {GeV,} \\ B_{bc}=-0.101\,\text {GeV,}&\end{Bmatrix} \end{aligned}$$

(30)

one can obtain the corresponding values of the binding energies for color-$\varvec{6}_{c}$ rep., via multiplying the color-factor ratios (scaling factors). The results of the binding energies between quark pairs (denoted in subscripts) are:

$$\begin{aligned} 6_{c}: \begin{Bmatrix} B_{cs}=0.013\,\text {GeV,}&B_{cc}=0.039\,\text {GeV,} \\ B_{bs}=0.016\,\text {GeV,}&B_{bb}=0.064\,\text {GeV,} \\ B_{bc}=0.051\,\text {GeV. }&\end{Bmatrix} \end{aligned}$$

(31)

Given the parameter inputs in Eqs. (29), (30) and (31), one can apply variational method to Eq. (15) to determine the bag radius R and the respective momentum $x_{i}$ via Eq. (17) for a given hadron with color-spin wavefunction $\psi$, given in Table 1. It is then straightforward to calculate masses and magnetic moments of fully heavy pentaquarks we address in this work, as detailed in Sect. 4.

4 Masses and magnetic moments of fully heavy hadrons

Table 3 Computed masses (MeV) of triply heavy baryons, compared to other works cited. Bag radius $R_{0}$ is in GeV$^{-1}$

Before discussing fully heavy pentaquarks, we first consider triply (fully) heavy baryons, the fully heavy binding systems of three quarks, which include the five baryons $\Omega _{ccc}$, $\Omega _{ccb}$, $\Omega _{ccb}^{*}$, $\Omega _{cbb}$, $\Omega _{cbb}^{*}$ and $\Omega _{bbb}$, as listed in Table 3. For these baryons, we apply the same framework of MIT bag model (15) with the inputs in Eqs. (29), (30) and (31) to compute the mass M and the bag radii $R_{0}$ of the fully heavy baryons, with other works cited for comparison.

Given the color-spin bases [53] (see Eq. (15)), defined as $\phi ^{B}\chi _{1}^{B}$ and $\phi ^{B}\chi _{2}^{B}$ for $J^{P}=3/2^{+}$ and $1/2^{+}$, respectively, the CMI matrices and binding energies are well determined. When pentaquarks included, the computed results could help us further understand the general features of fully heavy systems, as we shall discuss in Sect. 5.

In Table 3, the second and third rows present bag radii $R_{0}$ and masses of triplet heavy baryons, and the data followed rows present the masses (in unit of MeV) cited from various works. One sees that the predicted masses of the $\Omega _{ccc}$, for instance, ranges from $4520\,$MeV to $5130\,$MeV for which various methods including QCD sum rules [62, 63, 71, 78] are employed. Up to $610\,$MeV, our prediction is close to the prediction $4810\pm 100\,$MeV in Ref. [71]. The lattice QCD was also applied to study the $\Omega _{ccc}$ in Ref. [67], with the prediction of $4796\pm 8\pm 18\,$MeV. Comparing with the bag model computation of the doubly heavy baryons and tetraquarks [53] and the fully charm tetraquarks $cc\bar{c}\bar{c}$ in Ref. [80], our results for fully heavy baryons and pentaquarks are reasonable, considering that the inputs of the parameters are fitted only from light baryons and heavy mesons.

Table 4 Predicted spectra of pentaquarks $P_{bbbb\bar{b}}$ and $P_{cccc\bar{ c}}$ with comparisons from various works. Bag radius $R_{0}$ is in GeV$^{-1}$. Masses are in GeV. Magnetic moments $\mu$ are in unit of $\mu _N$

Using the method described in Sects. 2 and 3, one can construct a total mass formula (15) for the fully heavy hadrons containing the CMI and binding energies and calculate masses, magnetic moments and eigenvectors for all fully heavy systems containing the fully heavy mesons, baryons and pentaquarks, especially of the ten pentaquarks $P_{bbbb\bar{b}}$, $P_{bbbb\bar{c}}$, $P_{cccc\bar{b}}$, $P_{cccc\bar{c}}$, $P_{bbbc\bar{b}}$, $P_{bbbc\bar{c}}$, $P_{cccb\bar{b}}$, $P_{cccb\bar{c}}$, $P_{ccbb\bar{b}}$ and $P_{ccbb\bar{c}}$, whose color-spin bases have been listed in Table 1. Among these heavy hadrons, more attention is given to fully bottom and charm systems, which attracts extensive attention [81,82,83,84,85,86].

In Table 4, we present our computed results for the masses and magnetic moments of the fully bottom system $P_{bbbb\bar{b}}$ and the fully charm system $P_{cccc\bar{c}}$, compared to other works via the CMI model [81] and constituent quark model [82], which adopt the same color-spin wavefunctions, and via the lattice-QCD inspired model [83] and the chiral quark model [84]. The comparison is also given with the predictions via the QCD sum rules [85, 86]. Agreement of our predictions with that by the constituent quark model [82] is achieved. Given the masses of triply heavy baryons computed in Table 3 and that of the measured heavy mesons [60], one finds that our predicted masses of fully heavy pentaquarks are all above the thresholds of the heavy baryons and mesons, as shown in Fig. 1.

Fig. 1

Computed masses (in GeV, short solid lines) of pentaquarks $P_{bbbb \bar{b}}$ and $P_{cccc\bar{c}}$ for $J^{P}=5/2^{-}, 3/2^{-}, 1/2^{-}$, compared to the respective thresholds plotted as long dotted lines

Further more, we examine the respective mass splittings $\mathrm {\Delta } M$ between spin multiplets due to the lowest two quantum numbers J for the fully heavy mesons, tetraquarks and pentaquarks and show them collectively in Table 5. Here, $J=\{0,1\}$ or $J=\{1/2,3/2\}$. For the fully heavy systems with identical quarks, it is found that $\mathrm {\Delta } M$ are suppressed significantly with the heavy quark number N increasing and hadron becoming heavier. We see that the mass splittings are narrower for the bottom sector compared to the charm sector.

For the fully heavy pentaquarks with both bottom and charm flavors, we list the corresponding results computed via the MIT bag model in Tables 6, 7, 8, and 9, in comparison with other similar works cited [81, 82]. The obtained masses are also compared to the respective thresholds and plotted in Fig. 2. All hadrons computed are above the thresholds associated with the baryons and mesons, meaning that these fully heavy pentaquarks are unstable against strong decays to two-hadron final states.

The instability of pentaquarks can be investigated by analyzing the eigenvectors obtained in this study. In order to examine the color-spin wavefunction $\psi$ of pentaquark state, we utilize the corresponding eigenvectors and bases defined in Table 1, and express it in the form as shown below:

$$\begin{aligned} \psi = c_{1} \left( q_{1}q_{2}q_{3}\right) _{\varvec{1}_{c}} \otimes \left( q_{4}\bar{q_{3}}\right) _{\varvec{1}_{c}} + \dots . \end{aligned}$$

(32)

Here, $c_{1}$ represents the overlap between the wavefunctions of the pentaquark and the specific baryon $\otimes$ meson component, which corresponds to a scattering state. If the pentaquark state couples strongly to a scattering state, indicating that the probability $|c_1|^2$ of the collapse of $\psi$ into a color-singlet (6) approaches 1, we can infer that it is unstable against strong decay with a very broad width.

For the pentaquark states presented in Tables 4 and 6, the values of $|c_1|^2$ are determined to be 1/3. Similarly, in Table 9, all the calculated values are approximately 1/3. Notably, the pentaquark states denoted by asterisks in Tables 7 and 8 have dominant components of scattering states with $|c_1|^2$ values greater than 0.83, while the remaining states range from 0.34 to 0.80. Based on the above discussions, we conclude that the pentaquark states $P_{bbbb\bar{Q}}$, $P_{cccc\bar{Q}}$, and $P_{ccbb\bar{Q}}$ (Q = c, b) are likely to possess a compact structure with relatively narrow widths. On the other hand, the bag radius $R_{0}$ ranging from $3.48\,$GeV$^{-1}=0.7$ fm to $5.08\,$GeV$^{-1}=1.0$ fm, which are in the order of the typical hadrons in size (about $1\,$fm), implies that these compact hadrons of pentaquarks may exist during hadronization in experiments of LHCb.

Table 5 Calculated masses M (in GeV) of fully heavy mesons [53], tetraquarks [80] and pentaquarks, with their mass splittings $\mathrm {\Delta } M$ (in GeV) due to the lowest two $J^P$ quantum numbers

Table 6 Predicted spectra of pentaquarks $P_{bbbb\bar{c}}$ and $P_{cccc\bar{ b}}$ with comparisons from various works. Bag radius $R_{0}$ is in GeV$^{-1}$. Masses are in GeV. Magnetic moments $\mu$ are in unit of $\mu _N$

Table 7 Predicted spectra of pentaquarks $P_{bbbc\bar{b}}$ and $P_{bbbc\bar{ c}}$ with comparisons from various works. Bag radius $R_{0}$ is in GeV$^{-1}$. Masses are in GeV. Magnetic moments $\mu$ are in unit of $\mu _N$. The states denoted by asterisks couple strongly to scattering states

Table 8 Predicted spectra of pentaquarks $P_{cccb\bar{b}}$ and $P_{cccb\bar{ c}}$ with comparisons from various works. Bag radius $R_{0}$ is in GeV$^{-1}$. Masses are in GeV. Magnetic moments $\mu$ are in unit of $\mu _N$. The states denoted by asterisks couple strongly to scattering states

Table 9 Predicted spectra of pentaquarks $P_{ccbb\bar{b}}$ and $P_{ccbb\bar{ c}}$ with comparisons from various works. Bag radius $R_{0}$ is in GeV$^{-1}$. Masses are in GeV. Magnetic moments $\mu$ are in unit of $\mu _N$

Fig. 2

Computed mass spectra (in GeV) of pentaquarks $P_{bbbb\bar{c}}$, $P_{cccc\bar{b}}$, $P_{bbbc\bar{b}}$, $P_{bbbc\bar{c}}$, $P_{cccb\bar{b}}$, $P_{cccb\bar{c}}$, $P_{ccbb\bar{b}}$, and $P_{ccbb\bar{c}}$ for $J^{P}=5/2^{-}, 3/2^{-}, 1/2^{-}$, plotted by short solid lines, with respective threshold energy as blue (labeled right) and red (labeled left) dotted lines

5 Mass pattern and linearity upon quark number

To explain the mass pattern obtained in this work, we consider question as to if the bag radius R leads to the larger volume energy $M_{V}=4\pi R^{3}B/3$ for the larger heavy quark number N, so that the hadron masses tend to be of suprathreshold. For this, we ignore the mass splittings due to the CMI and rewrite Eq. (15) in the form that hadrons consist of the identical flavor Q($=c,b$). The result is,

$$\begin{aligned} \bar{M}\left( R,N\right){} & {} \!=\!N\left( m_{Q}^{2} \!+\!\frac{x_{Q}^{2}}{R^{2}}\right) ^{1/2}\!+\!\frac{4}{3}\pi R^{3}B\!-\!\frac{Z_{0}}{R}\!+\!NB_{QQ}, \nonumber \\ \end{aligned}$$

(33)

where $\bar{M}$ stands for the spin-independent mass, N the total number of valence heavy quarks, and $B_{QQ}$ the binding energy given in Eq. (30).

Owing to the scaling of the color factor, as mentioned in Sect. 3, the fully heavy hadrons with N identical quarks happen to have total binding energy of $NB_{QQ}$. Given Eq. (33), one can use it to examine each part of the energy for the fully heavy systems with identical quark number $N=2\sim 5$. The numerical results are listed in Table 10. One sees that the volume energy $M_{V}$ grows quickly while the suppression of zero point energy $M_{Z}=-Z_{0}/R$ becomes weaker significantly with more valence quarks involved. This implies that fully heavy hadrons tend to be heavier when N becomes larger and they are all above thresholds of the decaying final states.

We further examine this mass pattern for fully heavy systems of the mesons, baryons, tetraquarks and pentaquarks within the same framework of MIT bag model. Beside the masses of baryons and pentaquarks which are computed via the MIT bag model, we cite the results for the mesons and tetraquarks in Refs. [53] and [80], respectively. The obtained results are plotted in Fig. 3. For the fully bottom and charm hadrons one sees that hadron masses rise almost linearly with N ranging from 2 to 5. It is notable that our mass predictions of the mesons and tetraquark $cc\bar{c}\bar{c}$ are in good agreements with experimental data, especially, the computed mass $6572\,$MeV of the fully charm tetraquark $cc\bar{c}\bar{c}$ agrees well with the measured mass $6552\pm 10\pm 12\,$MeV of the newly-discovered resonance X(6600) [8].

Inspired by Eq. (33) and the numerical results above, we promote the heavy quark number N to be as large as 20 mathematically, despite that a hadron with 20 valence quarks, if exist, may result in a very large and thereby unphysical bag radius R. We apply this procedure to $N=2,3,4$ and 5 to test if the linear dependence of the hadron mass holds true as far as this work is involved. Similar to Fig. 3, we plot the numerical results in Fig. 4a for the spin-independent masses v.s. quark number N for fully bottom and charm systems respectively, with the data from $N=2\sim 20$. From these results, we find the following numerical fits for the fully bottom (denoted by subscript b) and charm (denoted by subscript c) systems (in GeV, $2\le N\le 5$):

$$\begin{aligned} \begin{aligned} \bar{M}_{b}\left( N\right)&=5.1004 N-0.7229, \\ \bar{M}_{c}\left( N\right)&=1.7079 N-0.3241. \end{aligned} \end{aligned}$$

(34)

In Fig. 4a, one observes the linear lines numerically, corresponding to Eq. (34), but the marching of it with the data of the spin-independent masses in Table 10 is approximated. To see this, we apply Eq. (33) to the fully light hadron systems with binding energy ignored, $B_{QQ}=0$, and obtain the results plotted in Fig. 4b. The linear dependence breaks slightly when N is small, as it should be for the light hadrons. Notice that the (solid) line for the bottom sector is nearly linear in contrast with that for the charm sector, this implies that linearity of mass dependence holds true at heavy quark limit.

In the following, we show such a mass linearity upon N in heavy quark limit analytically. We expand the mass formula (33) in term of the inverse heavy-quark mass $m_{Q}^{-1}$,

$$\begin{aligned}{} & {} \bar{M}\left( R, N\right) \approx Nm_{Q}\left( 1+\frac{x_{Q}^{2}}{2m^{2}_{Q}R^{2}}\right) \nonumber \\{} & {} \quad +\frac{4}{3}\pi R^{3}B-\frac{Z_{0}}{R}+NB_{QQ}, \end{aligned}$$

(35)

and minimize the mean mass Eq. (35) to find the bag radius,

$$\begin{aligned} \frac{\partial \bar{M}\left( R,N\right) }{\partial R}=0\Rightarrow R=R_{0}. \end{aligned}$$

(36)

The solution of Eq. (36) in the large-N limit is $R_{0}\sim N^{1/5}$. This gives rise to the following mass formula for fully heavy hadrons

$$\begin{aligned} \bar{M}\left( N\right) =Nm_{Q}+\bar{\Lambda }_{N}+\frac{x_{Q}^{2}N^{\frac{3}{5 }}}{2m_{Q}}, \end{aligned}$$

(37)

with $\bar{\Lambda }_{N}=NB_{QQ}+\left( 4\pi B/3\right) N^{3/5}-Z_{0}N^{-1/5}$, by analogy with the mass relation $m_{H_{Q}}=m_{Q}+\bar{\Lambda }+\frac{\mathrm {\Delta } m^{2}}{2m_{Q}}$ in heavy quark effective theory (HQET). This explains the linear mass dependence of fully heavy hadrons with heavy quark flavor N, as demonstrated in Fig. 3.

6 Conclusions and remarks

Table 10 Numerical results for the mass components (all in GeV), the kinematic energy $\omega _{Q}$ (in GeV), the volume energy $M_{V}$, zero point energy $M_{Z}$ and spin-independent mass $\bar{M}$ of fully heavy systems (mesons, baryons and pentaquarks) in detail. The parameters of the bag radius $R_{0}$ (GeV$^{-1}$) and $x_{Q}$ are also shown for the heavy quark Q.

Fig. 3

Masses of fully heavy systems ranging from mesons ($N=2$) to pentaquarks ($N=5$) with N the number of quarks and antiquarks, calculated in the unified framework of MIT bag model. The results of mesons are quoted from Ref. [53], and that of tetraquarks are from Ref.  [80]

Fig. 4

The calculated spin-independent masses of fully heavy and light systems by Eq. (33). Quark number N is from 2 up to 20 for mathematical extrapolation

In this work, we employ a unified framework of MIT bag model to perform a systematical study of all ground-state configurations of fully heavy pentaquarks and other fully heavy hadrons. With the help of the color-spin wavefunctions of pentaquarks, which are classified via Young tableaux which respect symmetry of the $SU(2)_{s}\otimes SU(3)_{c}$ group and present in terms of the Young–Yamanouchi bases, we have computed the masses and magnetic moments of the fully charm, bottom and bottom-charm pentaquarks in ground states via numerical variational method that applies to the MIT bag model. Our computation predicts a set of masses of fully heavy pentaquarks, ranging from 8.229 GeV for the $P_{cccc\bar{c}}$ to 24.770 GeV for the $P_{bbbb\bar{b}}$. Combining with the similar bag-model computation of masses of fully heavy systems, we find that the masses of fully heavy hadrons (mesons, baryons, tetraquarks and pentaquarks) with identical flavors depend almost linearly upon the number N of the valence quarks in hadrons, and we demonstrate this linear behavior of the hadron masses using the MIT bag model, being consistent with the heavy quark symmetry at the heavy-quark limit. Our results also indicate that the heavier the fully heavy hadrons, the narrower the mass splittings of their lowest two multiplets (in J).

We remark that with the increase of the heavy quark number N, the bag radius $R_{0}$ of the fully heavy hadrons can rise up as high as $1\,$fm, and it results in the larger volume energy $M_{V}$ and the less suppression of zero point energy $M_{Z}$, which makes them suprathreshold and unstable against strong decay. On the other hand, the shorter bag radius of the compact pentaquarks implies some larger possibilities for their formation via hadronization in experiments, compared to the formation of the hadrons containing light degrees of freedom. We hope that our predictions can help the future LHCb experiments to search the fully heavy pentaquarks or baryons addressed in this work.