Hyperspherical-coordinate approach to the spectra and decay widths of hybrid quarkonia

In this paper, we consider the possibility that Y(4260), Y(4360), $ \psi(4415) $ or X(4660) is the ground state or the first excited state of a hybrid charmonium and examine whether $ \Upsilon ( 10860) $ is the ground state or the first excited state of a hybrid bottomonium. Under a constituent quark model, we carry out numerical calculations to obtain the spectra of both electric gluon and magnetic gluon hybrid quarkonia. We see that the ground state of a magnetic gluon hybrid charmonium and the first excited state of an electric gluon hybrid are comparable in energy to $ \psi(4415) $ and that the first excited states of an electric gluon hybrid bottomonium and the ground state of a magnetic gluon hybrid appear a few hundred MeV above the mass of $ \Upsilon (10860 ) $. Also, we evaluate the widths of the decays of a hybrid charmonium into $D^{(*)} \bar{D}^{(*) } $ and those of a hybrid bottomonium into $ B^{(*) } \bar{B}^{(* ) } $. We show that if the exotic meson candidates are hybrid charmonia, then they are magnetic gluon ones and argue that it seems unlikely that $ \Upsilon (10860) $ is the ground state of an electric gluon hybrid bottomonium.


I. INTRODUCTION
Over the past decades, research on exotic hadrons has been one of the most important areas of research in terms of Quantum Chromodynamics (QCD), given that the quantum field theory of the strong interaction does not deny the existence of colour-singlet particles of this sort.
Year 2003 marked the beginning of a new chapter in the history of QCD, because X(3872), the first exotic meson candidate, was observed at Belle [1]. The discovery of X(3872) accelerated the trend of research on exotic hadrons: since then, various exotic meson candidates have been found, and the possibility of existence of further exotic hadrons has been raised. This has provided a huge motivation to further search for exotic hadrons [2], and led to a better understanding of how the experimentallyverified exotic hadron candidates are described.
Among those exotic hadron candidates, Y(4260), which was first observed in 2005, is singled out for special treatment. Using initial state radiation (ISR), BaBar Collaboration revealed the existence of the particle [3], and it was subsequently confirmed by the CLEO-c [4] and Belle experiments [5]. Also, it was confirmed that the particle has J P C = 1 −− , as the measured dipion mass distribution agreed with the distribution that was calculated in an S-wave phase space model [6]. There are several reasons why Y(4260) has been regarded as a highly exotic meson: the decay width of the process Y (4260) → J/ψ π + π − is significantly large compared to that of other vector charmonia such as ψ(4040); Y(4260)'s decay into DD has not been observed; the decay width of the process Y (4260) → ψπ + π − was larger than 5.0 MeV, much larger than that of other 1 −− charmonium mesons such as ψ(3770) (80-90 keV) and ψ(4040) (≤ 100 ± 30 keV) [7]; also Y(4260) decays into an exotic meson Z(3900) ± [8].
From a theoretical point of view, there are the following several arguments to explain the anomalous behaviour of Y(4260). Firstly, it could be argued that the particle is a ccss tetraquark, although the decay channel Y (4260) → D sDs has not been observed yet. Secondly, the possibility that Y(4260) is aDD 1 (2420) molecule has been raised [9], while there is an argument that the binding energy extracted from the experimental data is stronger than the energy scale that is related to the Yukawa meson exchange interaction. Also, the particle could be interpreted as a hadrocharmonium, that is, a charmonium surrounded by the cloud of light hadrons [10].
Another interpretation has been put on the experimental results: the particle is a hybrid charmonium which consists of cc and a constituent gluon [11]. As a hybrid meson must conform to its selection rule [12], the ground state of a hybrid charmonium could have either an electric or a magnetic gluon. The decay pattern of Y(4260) suggests that the constituent gluon should be magnetic.
In relation to a charmonium hybrid, let us note that several hybrid mesons have been considered; for the details of light hybrids see the review paper which was written by Meyer and Swanson [13].
Under the assumption that a charmonium hybrid exists, it is possible that there could exist a bottomoniumequivalent of the hybrid meson; several different theoretical studies have explored the possibility of a hybrid bottomonium. For instance, a lattice QCD calculation which employed a non-relativistic approach showed that the gluonic excitation could exist 1.542 GeV above the "ground" state (≈ 9.45 GeV) [13,14]. Another lattice calculation, together with the leading Born Oppenheimer (LBO) approximation, suggested that the lowest hybrid state could exist 1.49 GeV above the 1S state [15]. Also, the QCD sum rule analyses predicted that there exist several bottomonium hybrids: J P C = 0 −+ at around 10.6 ± 0.19 GeV [16]; 1 −− in the mass range of 10.24 to 11.15 GeV [17]; and 1 ++ at around 11.32 ± 0.32 GeV [18]. Furthermore, it has been reported that the spin-dependent structures of hybrid quarkonia were unveiled within the Born-Oppenheimer Effective Field Theory (BOEFT) framework [19].
Experimentalists have not yet obtained conclusive evidence that the gluonic degrees of freedom could appear in a bottomonium. However, we can spotlight a few particles which sit above the BB threshold: Y b (10890) is said to be a candidate for a bbg [20]; and the anomalously large partial width concerning the decays of Υ(10860) (or Υ(5S)) into Υ(nS)π − π + should be looked at more carefully.
Hybrid quarkonia have opened up an opportunity to not only study the gluonic degrees of freedom but also assess how a quark and its antiquark interacts in a qqg. While the qq of a usual quarkonium are bound together by the static quark potential, which is not directly measurable in an experiment, a hybrid quarkonium emerge from the excitations of dynamical gluon. The excitations were studied within the lattice framework [21]; at the same time, it is possible to interpret the gluonic excitations as the string vibration modes under the flux tube regime [22]. The Lüscher term approach can also be used to assess the gluonic excitations [23], which is helpful in placing the constraints on the effective interactions relating to the mixing between hybrids and heavy quarkonium states [24].
In the present paper, we consider the possibility that Y(4260) or one of the other related exotic meson candidates such as Y(4360) is the ground state or the first excited state of a hybrid charmonium, and examine whether Υ(10860) is the ground state or the first excited state of a hybrid bottomonium. Under a semirelativistic framework, we calculate the energy spectra of a charmonium and bottomonium hybrids, and obtain the quark-antiquark effective potentials. Recognising QCD's non-perturbative nature, we adopt a constituent quark model in which the whole system consists of a quark, its antiquark and a constituent gluon. Also, we consider the decays of a charmonium hybrid and bottomonium hybrid into D ( * )D( * ) and B ( * )B( * ) , respectively, evaluating these decay widths.
Section 2 consists of several subsections: the auxili-ary field approach, hyperspherical formalism, the quarkantiquark effective potentials and the decays of a hybrid charmonium into D ( * )D( * ) and those of a hybrid bottomonium into B ( * )B( * ) . In Section 3, we present our numerical results. The summary is given in the final section.

A. The model for a hybrid meson
In this subsection, we begin by introducing the Hamiltonian of the whole system qqg, where q andq stand for a heavy quark and its antiquark, respectively. According to earlier studies conducted by Kalashnikova and (subsequently Mathieu), the total Hamiltonian of a hybrid meson is given as the sum of the free Hamiltonian and the potential [25,26]: H = H f + V . The free Hamiltonian and the potential term are expressed as follows: where m q and p q are the mass and the momentum of a quark, respectively. Likewise, subscriptsq and g denote an antiquark and a constituent gluon. In this model, the mass m g of the constituent gluon is strictly zero.
When it comes to the potentials in the Hamiltonian, σr qg and σrq g are the linear potentials for the quark confinement, and V C is the colour-Coulomb potential for this system. In relations to the potentials, the Casimir scaling is considered: using the SU(3) generators {λ a i } a=1,...,8 , we have the following relations for a hybrid meson: Accordingly, we have the expressions for V C : The Salpeter-type Hamiltonian in (1) contains square roots, and so does the corresponding action. In general, this sort of square-root has two difficulties: quantisation and dealing with a massless particle [27]. Specifically, in the present paper, we need to consider the constituent gluon which is massless. For these reasons, we adopt the auxiliary field method [28] which involves an auxiliary field µ (or an einbein field) to reparametrise this kind of Hamiltonian: The relation (5) suggests that the physical meaning of the auxiliary field µ is that it is the effective mass of a particle [29]. Using the auxiliary field method, and choosing the centre-of-mass of the three-body system as the coordinate origin, we reparametrise the free Hamiltonian (1) by the auxiliary fields µ q , µq and µ g of the quark, antiquark and constituent gluon, respectively: where µ x and µ y are defined by Also, k x and k y are the momenta which are defined in the three-body Jacobi coordinates; the details of the coordinates are written in Appendix A. Strictly speaking, auxiliary fields are dynamical variables, and we need the fields before quantisation. But it is difficult to do that; therefore we here treat the fields (µ q , µq and µ g ) as an approximate variational parameters [28] in order to determine the ground state of the whole system. We also note that an auxiliary field method leads to, at worst, seven per cent disagreement between a numerical value and its true value in a numerical calculation [29].

B. Hyperspherical coordinates
Now we point out that we can take advantage of the auxiliary fields for the linear potentials of (2) in order to treat the reparametrised Hamiltonian (7) as a harmonic oscillator system [32]. However, this sort of analytical treatment does not work properly under the presence of the colour-Coulomb force V C , as the whole interactions cannot be expressed as harmonic oscillators. Admittedly, instead of doing that kind of analytic calculations, we need to carry out numerical calculations in the present case, adopting a hyperspherical-coordinate approach.
This framework provides a useful tool to study fewbody systems such as hadrons and atomic nuclei (see [30], [31] for more details); in fact it was adopted in earlier research on a hybrid charmonium [25]. In this section, we therefore explain the hyperspherical formalism to solve the boundary value problems involving the Hamiltonian.
For convenience, we label each of the constituents of the three-body system by Particle 1, 2 and 3, where Particle 1 and 2 correspond to a quark and an antiquark, respectively. Particle 3 refers to a constituent gluon in this case. Here we use the Jacobi coordinates for the three-body system; the details of the coordinates are shown in Appendix A.
Suppose, in the Jacobi coordinates, X is the distance between Particle 1 and 2, and Y is the distance between Particle 3 and the centre-of-mass of Particle 1 and 2. Then, we introduce the hyperspherical coordinates (ρ, α) as follows: where ρ is hyperradius and α (∈ [0, π 2 ]) is hyperangle. For later convenience, we use a shorthand notation Ω 5 to express the set of five angular variables: where θ x and φ x are the polar and azimuthal angles of the X vector, respectively. Similarly, we have for the Y vector.
The hyperspherical harmonics(HH) is equipped with its relating quantum numbers: K is called the hyperangular momentum; L x is the orbital angular momentum between Particle 1 and 2; L y is that between Particle 3 and the centre-of-mass of Particle 1 and 2. Also, an integer n is defined by: With these quantum numbers we introduce the HH for the three-body system: where we define the hyperangular part A LxLy n (α), by using the Jacobi polynomials {P α,β n }, as follows: A LxLy n (α) = (cos α) Lx (sin α) Ly P Ly+1/2,Lx+1/2 n (cos 2α) .
Here N (K, L x , L y ) which appears in (11) is a normalisation constant given by: In the present method, the total wave function is expanded with regard to the HH: where {χ Kγ (ρ)} K,γ are the hyperradial functions of the whole wave function, and |S tot M Stot is the spin part, and γ is a set of quantum numbers: KLtot ⊗ |S tot ] JM J . In actual calculations, we need to numerically obtain the hyperradial functions. To do this, we substitute (14) into the Schrödinger equation, which yields the following a set of inhomogeneous differential equations (hyperradial equations): where the matrix element of a potential is defined by Thus we see that the three-body problem is reduced to dealing with a one-dimensional problem, but we are still confronted with the inhomogeneous equations. Here we address the problem by expanding each of the hyperradial functions with regard to a generalised Laguerrebased basis function in one-dimensional space: where B is the number of the basis functions. The generalised Laguerre-based basis functions {u α,j } are defined by By virtue of this, our problem is reduced to solving the boundary value problems which involve a (m) j . We note that here m is a set of quantum numbers: m = {K, γ}. In the present case, we set α = 5.
The whole wave function Ψ is normalised as: C. Effective potential of a colour-octet quark-antiquark pair As mentioned earlier, standard quarkonium states emerge from the static quark potential, while the excitations of dynamic gluons are responsible for hybrid meson states. From a constituent quark model perspective, the gluonic excitations are described by the effective potential with which the constituent gluon of a qqg involves, keeping the quark-antiquark subsystem colour-octet. In other words, the information about the gluonic degrees of freedom can be accessed through studying the quarkantiquark effective potential.
In this subsection, therefore we calculate the quarkantiquark effective potentials in a qqg. The effective potential is derived from integrating over the coordinates Y relating to the constituent gluon of a hybrid meson: where coordinates X represent the qq relative position. Our approach to derive the effective potential is to take advantage of the Lagrange mesh method (LMM). Details of the LMM is given in Appendix C. Suppose that the eigen-energy E and the whole wave function of a physical system are given. Then, the procedure for calculating the potentials consists of two parts: we extract a twobody radial wave function R(X) from the whole wave function; then we obtain the effective potential relating the subsystem by solving the inverse problem.
The two-body radial wave function R(X) of a colouroctet subsystem qq by solving the following equation: where Ω X is a shorthand notation for the angular variables of the X vector. Then, after normalising R(X), we obtain a dimensionless wave function by φ X (X) = XR(X). By using the dimensionless two-body radial wave function, we solve the inverse problems in terms of the Hamiltonian of the subsystem in order to obtain the effective potential: where {T ij } i,j are the matrix elements of the Hamiltonian. {x i } i and {λ i } i are defined in Appendix C.

D. Decays of hybrid mesons
Now we consider the decays of a hybrid charmonium into D ( * )D( * ) . Under the selection rules, a magneticgluon ccg does not get involved in these processes [11], and therefore we consider only electric-gluon hybrids.
In the present case, we consider the lowest-order decay, as is shown in Figure 1. Its formalism was already presented in earlier research [12,33], where the wave functions of the initial and final particles were described by Gaussian-type functions. The final states are treated in a non-relativistic way, given the fact that the mass of a D meson is at lightest about 1.86 GeV. Also, we note that this method could be more suitable for calculating the widths of the decays of a bbg into B ( * )B( * ) , because the final mesons are much heavier so that the non-relativistic approximation is more reasonable.
The initial and final wave functions of a ccg influence the decay amplitude f A→BC (whose details are shown in [33]). In the present work, obtaining the wave functions of an initial state, we use the momentum representation to calculate the decay widths (for the details of the representation, see Appendix D.) Also, we take account of the effects of correlation between the charm-anticharm pair and the constituent gluon of a ccg. Suppose a ccg is particle A, and D andD mesons are particle B and C, respectively. Here we denote their masses by m A , m B and m C . Then, the decay width Γ A→BC regarding the process is expressed as where E B and E C are the energies of particle B and C, respectively: E B = P 2 B + m 2 B and E C = P 2 B + m 2 C . We also note that here P B is given by

III. RESULTS
In this section, we present the results of our numerical calculations with respect to hybrid quarkonia. We begin by distinguishing the magnetic gluon case from the electric, using the parity P and C-parity C of a hybrid meson: they are given by P = (−1) Lqq+Lg C = (−1) Lqq+Sqq+1 ; also we have the relation of L g = J g for a magnetic gluon and L g = J g ± 1 for an electric gluon [12]. Taking these things into account, we show the low-lying states which are allowed to exist for a hybrid meson qqg in Table I.
For the magnetic-gluon qqg, we restrict ourselves to the case where S qq = 0 and J P C = 1 −− , mentioning that S qq = 1 leads to J P C = (0, 1, 2) −+ [26]. The specific values of the other quantum numbers such as L qq are shown in Table I. Strictly speaking, we need to consider higher values for the quantum numbers, but here we stick to the approximations. Similarly, we consider only S qq = 1 for the electric gluon, pointing out that, in this case, J P C = 1 −− does not involve the spin singlet state of the quark-antiquark pair.

A. Charmonium hybrid
We now focus on the spectrum of a charmonium hybrid ccg. As to the parameters relating to the potentials in the Hamiltonian, we set the string tension σ and colour-Coulomb parameter α s 0.16 GeV 2 and 0.55, respectively. Here we used 1.48 GeV for the mass m c of the charm quark [25]. This parameter setting reproduced the energy spectrum of the charmonium to the extent that the difference between our calculation and the experimental data was less than 1 per cent.
Then, taking advantage of the auxiliary field method, we obtain the effective masses (µ c = µc and µ g ) of a charm, anticharm and constituent gluon. The results are shown in Figure 2, where the calculated ground state masses are plotted as functions of µ c and µ g . As to the values of γ = {L tot , S tot , L cc , L g }, we used Table I to determine it according to the types of the constituent gluon (i.e. electric or magnetic). Here we retained the maximum of the hyperangular momentum K up to 9, which allowed us to see a reasonable convergence (See Figure 3).
Then, the resulting basic parameters regarding the spectrum of a ccg are shown in Table II. In earlier research a single Gaussian was used as their basis function [25], while we used multi-component generalised Laguerre-based functions to express the hyperradial functions. For this reason, the ground state energy that (≈ 4.448 GeV) we obtained was lower than that (4.573 GeV) they did [26]. In relation to this, we mention that if we use a single-component basis function, then the ground state energy is consistent with their result.
When it comes to the Gaussian correlated approach, the ground state energy for the magnetic case was 4.445 GeV [26]. Figure 3 shows that our results are consistent with it.  Next we present the mass spectra of a ccg in the right part of Figure 4. From the graph, it is apparent that for the electric gluon case we have lower masses as a whole, compared to the electric. We can see that the ground state of an electric-gluon ccg appears about 200 MeV below the mass of Y(4260), while the first excited state of an electric-gluon hybrid and the ground state of a magnetic-gluon hybrid are comparable in energy to ψ(4415).
Then we consider the relation between the mass and RMS radius in terms of a charmonium hybrid for both the magnetic gluon and electric. In the left part of Figure 4, the RMS radii X 2 and Y 2 for a ccg are plotted against the masses of the hybrid, where X is the distance between c andc, and Y is the distance between the centre-of-mass of cc and a constituent gluon. We can see on this graph that X 2 for the electric is as a whole larger than that for the magnetic case; the same thing holds true for Y 2 . This result is consistent with our findings about the spectra. In the left graph, the rms radii X 2 and Y 2 regarding a ccg are plotted against its eigen-energies. In the right figure, the relevant experimental data (Y(4260), Y(4360), ψ(4415), and X(4660)) and the calculated energy spectra are shown [34], where E stands for the electric case and M for the magnetic.
Also, the probability densities of a ccg for its ground states are shown in Figure 5, where density distributions are scaled so that their peaks and bottoms are set unity and zero, respectively. On these graphs, we see the broad configuration in the Y direction for the magnetic case, while the density distribution has a rather long tail in the X direction for the electric. These spatial configurations are consistent with the fact that a constituent gluon sits in the P state and the orbital angular momentum between c andc is zero for the magnetic case, and that a constituent gluon sits in the S state and the orbital angular momentum between c andc is one for the electric.

B. Bottomonium hybrid
We then move on to consider a heavier hybrid meson bbg. Before calculating the energy spectra of a hybrid bottomonium, we need to obtain relevant parameters: the mass m b of a bottom quark and the colour-Coulomb parameter α s . These parameters are fitted to the experimental data in terms of the energy spectrum of a bottomonium. The details of the fitting procedures are explained in Appendix B.
Using the parameters m b and α s that were obtained, we calculate the ground state energies of a bbg through the auxiliary field method.  The basic inputs and outputs are summarised in Table  III. From the table, it is apparent that the calculated total energies of a bottomonium hybrid are about 10.6 and 11.2 GeV for the electric gluon case and magnetic, respectively. These results are consistent with those of the QCD sum rule analyses [17].
Also, the effective mass of the constituent quark of a bbg is in the range of 4.9 to 5.1 GeV, much heavier than that (∼ 1.6 GeV) of a ccg. In contrast, the constituent gluon of a hybrid bottomonium is comparable to that of a hybrid charmonium in terms of effective mass: µ g is in the range of 0.9 to 1.3 GeV, and µ g in the range of 0.7 to 1.0 GeV for a ccg.
Using these expectation values of the auxiliary fields µ b and µ g , we calculated the energy spectra of a bottomonium hybrid. The calculated energy spectra of a bbg are shown in the right part of Figure 6, where we can see that our theoretical values are deviated from the mass of Υ(10860) for both the magnetic and electric cases. In Figure 6. In the left graph, the RMS radii of X and Y are plotted against the eigen-energies for both magnetic and electric gluon cases. In the right graph, the theoretical energy spectra of a bbg are shown, where M and E means a magnetic gluon and electric, respectively. The experimental data about the spectra (of Υ(10860) and Υ(11020)) are shown together [34]. the left part of Figure 6, we plot the RMS radii of X and Y against the eigen-energies. From the graph, we can see that basically these RMS radii for the electric case are larger than those for the magnetic, which is consistent with the fact that the calculated eigen-energies of a bottomonium hybrid for the magnetic case is larger than those for the electric. The probability densities of a bottomonium hybrid for its ground states are shown in Figure 7, where we can see the similar spatial configurations to those of a ccg. This reflects the fact that L g = 1 and the orbital angular momentum between b andb is zero for the magnetic case, and that L g = 0 and the orbital angular momentum between b andb is one for the electric.
Also, we show the quark-antiquark effective potentials for the first three states of a bottomonium hybrid in Figure 8, where both the magnetic and the electric cases are considered, and the lattice calculations (Σ +,ex , Π g ) are shown together [21]. (For the details of this Greek letter notation and the related sub-or super-scripts, see [21].) Here we show the effective potentials only for set A as they are not qualitatively different from those for set B. The calculated potentials are plotted as a function of a dimensionless quantity X/R 0 . Here R 0 is the hadronic scale parameter which is defined by r 2 dVqq(r) dr | r=R0 = 1.65, where V qq is the static quark potential [35]. The value of the parameter is about 0.5 fm [35]; in the present case, we set the parameter 0.4933 fm. We also note that the vertical axis in those graphs is made dimensionless: We estimate that, for the magnetic cases, the our calculations' numerically reliable regions are X/R 0 ∈ [0.4, 0.8] for the first two states, and [0.5, 0.8] for the second excites state. For the electric cases, we estimate, the reliable regions are X/R 0 ∈ [0.4, 1.4]. The calculated potentials are smaller than those of the lattice calculations, especially for the electric cases.

C. Partial decay widths
Now we calculate the decay widths with regard to the processes of a ccg (whose constituent gluon is electric) Figure 8. The calculated quark-antiquark effective potentials (set A) for the first three states of a bbg. The left graph is for the magnetic gluon case and the right is for the electric. The lattice calculations (Σ +,ex , Πg) are shown together [21].
into D ( * )D( * ) . Before doing this, we set relevant parameters which are given in Table IV, where ω is the averaged energy difference between the ground state and the first excited state. µ D and µ B are the reduced masses which are defined by Also, introducing a Gaussian-related parameter R, we have the following relation [33]: Then, we summarise these parameters in Table IV. In this case, we set m q = mq = 0.35 GeV, where q = u, d, and m s = 0.5 GeV [33,36]. We show the results in Table V, where we consider the decays into D 0D0 , D + D − , D + s D − s , D * 0D0 , and D * 0D * 0 . From the table, we see that the widths of the decays from the ground state to DD are in the range of 0.02 to 0.30 GeV, and that the total of those widths is about 0.74 GeV, much larger than the experimental data (∼ 0.05 GeV) [34].
This result suggests that if the exotic meson candidate is a charmonium hybrid, it is unlikely that the ccg's constituent gluon is electric. Therefore, Y(4260) contains a magnetic gluon if it is a hybrid charmonium, which is in line with earlier research [11].
As to the decays from the first excited state of a ccg to D ( * )D( * ) , the sum of the partial widths is larger than 0.19 GeV. Although this value is smaller than that for the decay from the ground state, it is still much larger than the observed total widths (∼ 0.1 GeV, ∼ 0.06, ∼ 0.07 GeV) of Y(4360), ψ(4415) and X(4660), respectively. This result supports the view that these exotic meson candidates are not electric-gluon hybrid mesons.  Table VI, where the experimental data are shown together [34].
As to the decay of a bottomonium hybrid which is in the ground state into B ( * )B( * ) , it is apparent from the table that, for set A, the sum of the partial widths is much larger than Υ(10860)'s total decay width (∼ 51 MeV). Although, for the set B's ground state, the sum of the partial widths is comparable to Υ(10860)'s total width, the set of parameters does not allow a bbg to decay into the states that involve B * or B s . For the decay into B 0B0 or B + B − , there are significant differences between the calculated values and the experimental data.
When it comes to the first excited state into B ( * )B( * ) , for both set A and B, the sums of the partial widths are comparable to Υ(10860)'s total width. However, our partial widths are in the order of 10 3 times greater than the experimental data in terms of the decays into B sBs ; the calculated widths are approximately 10 −2 of the measured width for the B * sB * s channel. On top of that, the calculated B 0B0 and B + B − partial widths are much greater than the calculated B * B * partial widths, while we see the opposite in the experimental data.
Our results suggest that it seems unlikely that Υ(10860) is the ground state of an electric-gluon hybrid bottomonium, and that the argument that Υ(10860) is the first excited state of an electric-gluon bbg is weak. Having said that, we do not rule out the possibility that the first excited state of a bottomonium hybrid will be discovered as a state which is different from Υ(10860) since the sum of the partial widths (for set A) is around 35 MeV.
The parameter setting with regard to α s is a matter of debate because the α s used in (23) is identical to that used in (4). If we take the difference in energy scale into consideration, the deviations from the experimental data become smaller.  [34]. Here the hybrid meson's constituent gluon is electric.

IV. SUMMARY
In this work, we considered the possibility that Y(4260), Y(4360), ψ(4415) or X(4660) is the ground state or the first excited state of a hybrid charmonium, and also examined whether Υ(10860) is the ground state or the first excited state of a hybrid bottomonium.
For an electric-gluon charmonium hybrid, the calculated energy spectrum showed that the ground state appeared a few hundred MeV below the mass of Y(4260), while the energy of first excited state was comparable to that of ψ(4415). Also, the calculated widths for the decays of the ground state or the first excited state of a ccg into D ( * )D( * ) were much larger than the total widths of these exotic meson candidates. For a magnetic-gluon ccg, the calculated ground state energy was comparable to the energy of the first excited state of an electric-gluon ccg, only confirming that the calculated ground state was consistent with the results of earlier studies. Our results supported the view that if one of these exotic meson candidates is a hybrid charmonium, its constituent gluon is magnetic.
In relation to a ccg, similar conclusions could be reached regarding a bottomonium hybrid. The ground states of an electric-gluon bbg appeared about two hundred MeV below the mass of Υ(10860), while the first excited state was a few hundred MeV above the mass.
In addition, we evaluated the widths of the decays of the ground states and the first excited states of a bbg into B ( * )B( * ) . We found that, for the ground states, some of the B ( * )B( * ) channels could not appear in our calculations, and that the widths were large to the extent that it seems unlikely that Υ(10860) is the ground state of an electric gluon bbg. Still, there could be a possibility that the first excited state is the hybrid meson candidate, given the fact that α s used for the calculations of the decay widths could be smaller under more realistic conditions.
When it comes to the magnetic gluon, our calculations of the ground states of a bbg are above the lattice calculations, but were in good agreement with the QCD sum rule analysis. This could be another indirect evidence to suggest that the gluonic excitations could appear in a bottomonium.
This section covers the hyperspherical and the Jacobi coordinates for a three-body system which consists of particle 1, 2 and 3; their masses are denoted by m 1 , m 2 and m 3 , respectively.
For the present case, there exist three different types of the Jacobi coordinates: X-coordinates, Y-coordinates and T-coordinates (See Figure 9). In the figure, X 1 is the distance between particle 1 and 3, and Y 1 is the distance between particle 2 and the centre-of-mass of particle 1 and 3. Similar things hold true for X 2 , Y 2 and X 3 , Y 3 . Picking up one of these coordinates, we can express the Jacobi coordinates by [39]: where M = m j +m k +m i , and (i, j, k) is a cyclic permutation of (1, 2, 3). Then, we note that the three different Jacobi coordinates (X, Y and T) are connected to each other via the kinematic rotation: where φ ki is the angle of the kinematic rotation. Specifically, the angle is φ ki = arctan (−1) P M mj mim k , where P is determined according to the permutation of (123). P is even/odd when (kij) is an even/odd permutation of (123).
When it comes to a hyperspherical coordinate system, we introduce a quantity which does not depend on the three coordinates (X, Y and T) by the following identities: where the hyperradius ρ has already been defined in (9). We note that from the definition of the kinematic rotation, it is apparent that {x 2 i + y 2 i } j=1,2,3 are invariant quantities under the rotation. Then, x i and y i are expressed as [38]: where {α i } i=1,2,3 are hyperangles. By using the ρ and α i , we can define the hyperspherical harmonics (11). Accordingly, it is possible that we express the hyperspherical harmonics in three different ways: {Y Lx i ,Ly i KLM L (Ω 5,i )} i=1,2,3 . These different HHs are linked to each other via the Raynal-Revai coefficients: L x k L y k |L xi L yi KL [38].

Appendix B: Fitting procedures
In this section, we show the basic fitting procedures with regard to the relevant parameters of a bottomonium. We determine the values of the string tension σ and the colour-Coulomb parameter α s by fitting them to the energy spectrum of a bb. Firstly we need to set the Hamiltonian H bb tot for the two-body system: where H bb 0 is the free Hamiltonian for the bottomonium; V bb Lin+Cou is the term of the linear and colour-Coulomb potentials; V bb str is a string correction term. In addition, we use the LS coupling term V bb LS , the spin-spin interaction V bb ss and the tensor term V bb ST . These interactions are explicitly shown as follows: where we explicitly write c factors as well. We replace the delta function which appears in the spin-spin interaction with the following exponential function [41]: We make this sort of modification in part because we cannot treat the delta function numerically; also we need to address numerical instability issues relating to the delta function. Also, we modify the tensor interaction as follows [41]: where S 12 = 12(s b · n)(sb · n) − 4(s b · sb).
Secondly, we changed the mass m b of a bottom quark, and calculated the energy spectrum, fixing σ and α s . We used the auxiliary field method to determine the ground state and the value of a relevant auxiliary field. After obtaining these values, we calculated the higher states of a charmonium.
The results are shown in the set A column of Table  VII, where we can see that the calculated energies agree with the experimental data for the bottom two states. When it comes to higher states, the difference between our values and the experimental data is more than 1.5 per cent.
Then, we changed both m b and α s and searched the parametrisation under which the calculated energy well reproduces the bb energy spectrum. Accordingly we get α s = 0.45 and the corresponding energy spectrum which is shown in the set B column of Table VII. This parametrisation produces the theoretical values that are consistent with the experimental results.
Finally, we present the summary of the parameter sets in Table VIII.  In this section, we explain the Lagrange mesh method. Suppose the Lagrange functions {f j } j=1,...,N are real functions, and {z j } j=1,...,N are their mesh points; the set of these points is the quadrature-point-equivalent in the