Higher mass spectra of the fully-charmed and fully-bottom tetraquarks

In this work, we calculate the higher mass spectra for the $2S$- and $1D$-wave fully-charmed and fully-bottom tetraquark states in a nonrelativistic potential quark model. The $2S$-wave fully-charmed/bottom tetraquark states lie in the mass range of $\sim (6.9,7.1)$/$(19.7,19.9)$ GeV, apart for the highest $0^{++}$ state $T_{(cc\bar{c}\bar{c})0^{++}}(7185)$/ $T_{(bb\bar{b}\bar{b})0^{++}}(19976)$. Most of the $2S$-wave states highly overlap with the high-lying $1P$-wave states. The masses for the $1D$-wave fully-charmed/bottom tetraquarks are predicted to be in the range of $\sim (6.7,7.2)/(19.5,20.0)$ GeV. The mass range for the $D$-wave tetraquark states cover most of the mass range of the $P$-wave states and the whole mass range of the $2S$-wave states. The narrow structure $X(6900)$ recently observed at LHCb in the di-$J/\psi$ invariant mass spectrum may be caused by the $1P$-, or $2S$-, or $1D$-wave $T_{cc\bar{c}\bar{c}}$ states. The vague structure $X(7200)$ may be caused by the highest $2S$-wave state $T_{(cc\bar{c}\bar{c})0^{++}}(7185)$, two low-lying $3S$-wave states $T_{(cc\bar{c}\bar{c})0^{++}}(7240)$ and $T_{(cc\bar{c}\bar{c})2^{++}}(7248)$, and/or the high-lying $1D$-wave states with masses around 7.2 GeV and $J^{PC}=0^{++},1^{++},2^{++},3^{++}$, or $4^{++}$. While it is apparent that the potential quark model calculations predict more states than the structures observed in the di-$J/\psi$ invariant mass spectrum, our calculations will help further understanding of the properties of these fully-heavy tetraquark states in their strong and magnetic interactions with open channels based on explicit quark model wave functions.

Attention is also paid to the vague structure X(7200) in the di-J/ψ invariant mass spectrum [1]. In the spectrum studies, it is assigned as higher excitation states, such as the second radial (3S ) excitations [36,40], some 2P and/or 2D excitations [52].
Apart from the interpretations which treat these signals as genuine tetraquark states, there are also other explanations proposed in the literature. For instance, Refs. [67][68][69] describe the structures appearing in the di-J/ψ invariant mass spectrum by coupled-channel effects of double-charmonium rescatterings via an effective potential. In Ref. [70], it is proposed that the Pomeron exchange between two vector charmonium states can provide strong near-threshold couplings and then dynamically generate pole structures above the two vector charmonium thresholds. Other solutions include treat-ing the narrow X(6900) as a light Higgs-like boson [73], and X(6900) and X(7200) as gluonic tetracharm states [72], or Ξ cc Ξ cc molecules [74].
In our previous works [28,32], we have systematically studied the mass spectra of the 1S and 1P-wave T (cccc) states within a nonrelativistic potential quark model (NRPQM). Their masses are predicted to be within the ranges of ∼ (6445, 6550) MeV and ∼ (6600, 7000) MeV, respectively. Our results show that the broad structure X(6200 − 6800) is consistent with the 1S -wave states around 6.5 GeV, which is similar to the conclusions from Refs. . It should be mentioned that some low-lying 1P-wave states with a mass around 6.7 GeV may contribute to X(6200 − 6800) as well. Furthermore, it is found that the narrow structure X(6900) may correspond to some 1P-wave states around 6.9 GeV with J PC = 0 −+ , 1 −+ , 2 −+ . Such a possibility was also discussed by the recent potential quark model calculations [29,51] and the previous QCD sum rule predictions [26]. Actually, since quite a lot of states are predicted by the tetraquark picture, we will see later that some other possibilities, such as 2S -, and 1D-wave states, cannot be eliminated. The vague structure X(7200) lies about 200 MeV above the high-lying 1Pwave states according to our potential quark model predictions [32]. Given that higher excited states should exist in the quark model, the structure of X(7200) may be an indications of some higher excitations. This is also one of our motivations to investigate the higher T (cccc) mass spectrum in order to understand the nature of X(7200).
In this work, we continue to study the mass spectra for the higher 2S -, 3S -, and 1D-wave fully-heavy tetraquark states, i.e. T (QQQQ) (Q = c, b). The main purposes are: (i) to investigate the possibility of assigning X(6900) as the 2S -and/or 1Dwave states in the mass spectrum within a consistent framework; (ii) to know whether or not the X(7200) structure can be associated with the high 2S -, 3S -, and/or 1D-wave T (cccc) excitations; (iii) to give relatively complete T (QQQQ) spectra up to the second orbital excitation. It should be mentioned that many studies of the spectrum of the radial (S -wave) excitations have been carried out [36,39,40,46,51,52]. However, because of the complexity of dealing with the four-body system, the study of higher excited states progresses rather slowly. So far, the theoretical predictions on some special 1Dwave states are based on either the diquark-antidiquark picture, or spin-flavor symmetries as a scheme for parametrization [15,29,36,52]. For the tetraquark orbital excitations, the spin-orbital and tensor couplings will contribute to the Hamiltonian which will lead to configuration mixings in the wave functions. Thus, the calculation of an orbital excitation tetraquark system turns out to be more challenging than that for a radial excitation.
The NRPQM adopted in the present work is based on the Hamiltonian from the Cornell model [75], which contains a linear confinement potential, a Coulomb-like potential, spin-spin interactions, spin-orbital interactions, and tensor potentials. With the NRPQM, we have studied both the 1S -and 1P-wave fully-heavy and fully-strange tetraquark states [28,32,76] with a Gaussian expansion method [77,78]. In our calculations, we deal with the fully-heavy tetraquark systems without the diquark-antidiquark approximation. With the same parameter set as that for our calculations of the 1Sand 1P-wave T (QQQQ) (Q = c, b) states in Refs. [28,32], we obtain self-consistent predictions of the T (QQQQ) mass spectrum for 12 2S /3S -wave radial excitations, and 80 1D-wave orbital excitations.
The paper is organized as follows: a brief introduction to the methodology for the tetraquark spectrum is given in Sec. II. In Sec. III, the numerical results and discussions are presented. A short summary is given in Sec. IV.

A. Hamiltonian
We adopt a NRPQM to calculate the masses of the tetraquark states. In this model, the Hamiltonian is given by where m i and T i stand for the constituent quark mass and kinetic energy of the ith quark, respectively; T G stands for the center-of-mass (c.m.) kinetic energy of the tetraquark system; r i j ≡ |r i −r j | is the distance between the ith and jth quarks; and V i j (r i j ) stands for the effective potential between them. In this work the V i j (r i j ) adopts a widely used form [75,76,[79][80][81][82][83][84][85][86][87][88][89][90]: where the confinement potential adopts the standard form of the Cornell potential [75], which includes the spinindependent linear confinement potential V Lin i j (r i j ) ∝ r i j and Coulomb-like potential V Coul i j (r i j ) ∝ 1/r i j : The constant C 0 stands for the zero point energy. While the spin-dependent potential V sd i j (r i j ) is the sum of the spin-spin contact hyperfine potential V S S i j , the spin-orbit potential V S S i j , and the tensor term V T i j : with In the above equations, S i stands for the spin of the ith quark, and L i j stands for the relative orbital angular momentum between the ith and jth quarks. If the interaction occurs between two quarks or antiquarks, the λ i · λ j operator is defined as λ i · λ j ≡ 8 a=1 λ a i λ a j , while if the interaction occurs between a quark and an antiquark, the λ i · λ j operator is defined as λ i · λ j ≡ 8 a=1 −λ a i λ a * j , where λ a * is the complex conjugate of the Gell-Mann matrix λ a . The parameters b i j and α i j denote the strength of the confinement and strong coupling of the one-gluon-exchange potential, respectively. The quark model parameter sets {m c , α cc , σ cc , b cc } and {m b , α bb , σ bb , b bb } are taken the same as those in Refs. [28,32,86], they are determined by fitting the charmonium and bottomonium spectra. The quark model parameters adopted in this work are collected in the Table I.  The wave function for a qqqq system can be constructed as a product of the flavor, color, spin, and spatial configurations.
In the color space, there are two color-singlet bases |66 c and |33 c , their wave functions are given by |66 c = 1 2 √ 6 (rb + br)(br +rb) + (gr + rg)(ḡr +rḡ) In the spin space, there are six spin bases, which are denoted by χ S 12 S 34 S S z . Where S 12 stands for the spin quantum number for the diquark (q 1 q 2 ) (or antidiquark (q 1q2 )), while S 34 stands for the spin quantum number for the antidiquark (q 3q4 ) (or diquark (q 3 q 4 ) ). S is the total spin quantum number of the tetraquark qqqq system, while S z stands for the third component of the total spin S. The spin wave functions χ S 12 S 34 S S z with a determined S z can be explicitly expressed as follows: In the spatial space, we define the relative Jacobi coordinates with the single-partial coordinates r i (i = 1, 2, 3, 4): Note that ξ 1 and ξ 2 stand for the relative Jacobi coordinates between two quarks q 1 and q 2 (or antiquarksq 1 andq 2 ), and two antiquarksq 3 andq 4 (or quarks q 3 and q 4 ), respectively. While ξ 3 stands for the relative Jacobi coordinate between diquark qq and antidiquarkqq.
Taking into account the Pauli principle and color confinement for the four-quark system qqqq, one has 12 configurations for the 2S -wave radial excitations, and 80 configurations for the 1D-wave orbital excitations. The higher radially excited configurations corresponding to the same excited modes of these 2S -wave configurations are also easily obtained. The spin-parity quantum numbers, notations, and total wave functions for these 2S /3S -and 1D-wave configurations are presented in Tables II, III and IV. C. Numerical method To work out the matrix elements in the coordinate space, we follow the same method adopted in our previous works [28,32,76]. As we know, the relative-motion wave functions ψ α i (ξ i ) can be expressed as where Y l ξ i m ξ i (ξ i ) are the standard spherical harmonic functions. The unknown radial parts, R n ξ i l ξ i (ξ i ), are expanded with a series of Gaussian basis functions [32,88]: with It should be pointed out that if there are no radial excitations, the expansion method with Gaussian basis functions are just the same as the expansion method with harmonic oscillator wave functions. The parameter d ξ i ℓ in Eq. (31) can be related to the harmonic oscillator frequency ω ξ i ℓ with 1/d 2 ξ i ℓ = µ ξ i ω ξ i ℓ . For a tetraquark state T (qqqq) containing fully-equal mass quarks, if we ensure that the spatial wave function with Jacobi coordinates can transform into the single particle coordinates, the harmonic oscillator frequencies ω ξ i ℓ (i = 1, 2, 3) can be related to the harmonic oscillator stiffness factor K ℓ with ω ξ 1 ℓ = 2K ℓ /µ ξ 1 , ω ξ 2 ℓ = 2K ℓ /µ ξ 2 , and ω ξ 3 ℓ = 4K ℓ /µ ξ 3 . Taking the reduced masses µ ξ 1 = µ ξ 2 = m q /2, µ ξ 3 = m q for T (qqqq) , one has ω ξ 1 ℓ = ω ξ 2 ℓ = ω ξ 3 ℓ = ω ℓ and d ξ i ℓ = (m q /µ ξ i ) 1/2 d ℓ with d ℓ = (4m q K ℓ ) −1/4 . Then, the expansion of 3 i=1 R n ξ i l ξ i (ξ i ) can be simplified as Following the method of Refs. [77,78], we let the d ℓ parameters form a geometric progression where n is the number of Gaussian basis functions, and a is the ratio coefficient. There are three parameters {d 1 , d n , n} to be determined through the variation method. It is found that with the parameter sets, {0.068 fm, 2.711 fm, 15} and {0.050 fm, 2.016 fm, 15}, for the cccc and bbbb systems, respectively, we can obtain stable solutions. The numerical results should be independent of the parameter d 1 . To confirm this point, as done in the literature [91][92][93] we scale the parameter d 1 of the basis functions as d 1 → αd 1 . The mass of a T (QQQQ) (Q = c, b) state should be stable at a resonance energy insensitive to the scaling parameter α. As an example, we plot the masses of 80 1D-wave T (cccc) configurations as a function of the scaling factor α in Fig. 1. It is found that the numerical results are nearly independent of the scaling factor α. The stabilization of other states predicted in this work has also been examined by the same method. With the mass matrix elements ready for every configuration, the mass of the tetraquark configuration and its spacial wave function can be determined by solving a generalized eigenvalue problem. The details can be found in our previous works [28,88]. The physical states can be obtained by diagonalizing the mass matrix of different configurations with the same J PC numbers. Finally, we give some discussions of the numerical method adopted in present work. Usually the trial spatial wave functions are expanded by a series of Gaussian functions. To keep the completeness of the Gaussian basis set, and to precisely treat an N-body system, one can involve several different sets of Jacobi coordinates as those done in Refs. [77,78,[91][92][93][94], or adopt a single set of Jacobi coordinates X = (ξ 1 , ξ 2 , ..., ξ N−1 ) with non diagonal Gaussians e −XAX T as those done in Refs. [95][96][97][98], where A is a symmetric matrix. In our work, we deal with the cccc and bbbb systems which are composed equal mass constituent quarks and antiquarks. With equal quark masses, a non diagonal term with exp{... + βξ i · ξ j ...} (i j) implies another one with exp{... − βξ i · ξ j ...}, so that the contribution of first order in β is eliminated. Thus, the equal mass symmetry can let us select a single set of Jacobi coordinates with a diagonal Gaussian basis set as an approximation in our study. In this work, the advantage of adopting a single Jacobi coordinates is that: (i) all the configurations are orthogonal compact multiquark configurations; (ii) its basis functions of are significantly less than those of several different sets of Jacobi coordinates. However, with several different sets of Jacobi coordinates, even the continuum states corresponding to the hadron-hadron scattering solutions come out as discrete states, therefore, one needs adopt a method, such as the real-scaling method, to distinguish the genuine resonances from the discretized scattering states [92]. It should be mentioned that if one deals with a multiquark system composed of mass-different quarks, a single set of Jacobi coordinates with a diagonal Gaussian basis set may be not a good approximation.

III. RESULTS AND DISCUSSIONS
Our predicted mass spectra for the 2S /3S -wave T (cccc) and T (bbbb) states are presented in Tables V and VI, respectively. The predictions for the 1D-wave T (cccc) states are given in Tables VII and VIII, and the predictions for the 1D-wave T (bbbb) states are given in Tables IX and X. From these tables the components of different configurations for a physical state can be seen. To see the contributions from each part of the Hamiltonian to the mass of different configurations, we also present our results in Tables XI-XVI.
We find that both the kinetic energy term T and the lin-ear confinement potential term V Lin contribute large positive values to the masses, while the Coulomb type potential V Coul has a large cancellation with these two terms. The spin-spin interaction term V S S , the tensor potential term V T , and/or the spin-orbit interaction term V LS have also sizeable contributions to some configurations. It suggests that a reliable calculation should include both the spin-independent and spindependent potentials in the calculations for the tetraquarks. For illustration, our predicted T (cccc) and T (bbbb) spectra are plotted in Figs. 2 and 3, respectively.
A. 2S and 3S states In the radially excited (2S -, 3S -wave, etc.) states, apart from the 1S ground states with the same quantum numbers, i.e., J PC = 0 ++ , 1 +− , 2 ++ , some states with additional quantum numbers, i.e., J PC = 0 +− , 1 ++ , 2 +− , can also be accessed. It should be mentioned that by fully expanding 3 i=1 R n ξ i l ξ i (ξ i ) with the GEM, one cannot distinguish the ξ 1 and ξ 2 excitation modes which are defined for the 2S /3S configurations listed in Table II. As a consequence, it is not possible to numerically work out the masses for the configurations with J PC = 0 +− , 1 ++ , 2 +− . To overcome this problem, following the method adopted in Ref. [76] we only expand the spatial wave functions containing the radial excitations with series of Gaussian basis functions, while for those spatial wave functions containing no excitations we adopt the single Gaussian function as an approximation.
There often exist configuration mixings among the states with the same J PC quantum numbers, which can be seen from the results listed in Tables V and VI. For example, there is an obvious mixing between the two 0 +− states 2 1 S 0 +− (66) c (ξ 1 ,ξ 2 ) and 2 1 S 0 +− (33) c (ξ 1 ,ξ 2 ) in the T (cccc) family due to the sizeable offdiagonal elements contributed by the spin-spin interaction together with the nearly equal masses for these two 0 +− configurations. In contrast, the mixing between the two 0 +− T (bbbb) states due to the spin-spin interaction is strongly suppressed by the heavy bottom quark mass. The sizeable mixing between the two 2 ++ states 2 5 S 2 ++ (33) c (ξ 1 ,ξ 2 ) and 2 5 S 2 ++ (33) c (ξ 3 ) is mainly caused by the Coulomb type potential. The Coulomb type potential together with the linear confinement potential will cause a sizeable mixing between the two 1 +− configura- In the configurations with the same J PC , the Coulomb type potentials V Coul for the |33 c structure are always more attractive than that for the |66 c structure. However, in some cases a |33 c configuration may have a larger mass than the |66 c configuration due to the larger contributions from the linear confinement potential term V Lin and kinetic energy term T . For example, for the T (cccc) sector the |66 c configuration 2 1 S 0 ++ (66) c (ξ 1 ,ξ 2 ) has a mass of 6954 MeV, which is about 46 MeV smaller than that for the |33 c configuration 2 1 S 0 ++ (33) c (ξ 1 ,ξ 2 ) of which the mass is 7000 MeV. It is seen that the contributions T = 725 MeV and V Lin = 883 MeV for 2 1 S 0 ++ (66) c (ξ 1 ,ξ 2 ) are smaller than T = 774 MeV and V Lin = 919 MeV for 2 1 S 0 ++ (33) c (ξ 1 ,ξ 2 ) , while V Coul = −622 MeV for 2 1 S 0 ++ (33) c (ξ 1 ,ξ 2 ) is more attractive than V Coul = −598 MeV for 2 1 S 0 ++ (66) c (ξ 1 ,ξ 2 ) . It should be mentioned that in the configurations with the same J PC , the lowest mass state is the radial excitation between the diqaurk (qq) and antidiquark (qq) (i.e. the ξ 3 -mode excitation) with the |33 c structure due to the most attractive Coulomb type potential term V Coul together with the smallest linear confinement potential term V Lin .
Including configuration mixing effects, the physical masses for the 2S T (cccc) are predicted to be in the range of ∼ (6900, 7000) MeV, except for one 0 ++ state T (cccc)0 ++ (7185) which has a mass of M = 7185 MeV. The masses for the 3S T (cccc) are predicted to be in the range of ∼ (7200, 7400) MeV, except for the highest 0 ++ state T (cccc)0 ++ (7720). The masses for most of the 2S T (bbbb) are predicted to be in the range of ∼ (19720, 19840) MeV, except for the highest 0 ++ state T (bbbb)0 ++ (19976). Similarly, the masses for most of the 3S T (bbbb) are predicted to be in the range of ∼ (19980, 20130) MeV, except for the highest 0 ++ state T (bbbb)0 ++ (20405). The gap between the lowest 2S T (cccc) state and the highest 1S T (cccc) state is about 358 MeV, which is very close to the value ∼ 368 MeV for the T (bbbb) sector. Our predictions for the 2S T (cccc) states are close to those predicted in Refs. [30,40,46,51]. Also, our predictions for the 3S T (cccc) states are close to those predicted in Ref. [40]. The predicted masses for the T (bbbb) states in our work are systematically (∼ 100 MeV) higher than the results from Refs. [40,46].
Since the predicted masses for the radially excited T (cccc) /T (bbbb) states are far above the low-lying twocharmonium/two-bottomonium mass thresholds, most of these 2S and 3S T (cccc) and T (bbbb) states could be rather broad since they may easily fall apart and decay into two heavy quarkonium channels. Quantitative study of their decays is crucial for better understanding the properties of these radially excited T (cccc) and T (bbbb) states.
From Tables XIII-XVI, it shows that in the two L = 0 (or L = 1) configurations containing the same J PC quantum numbers and excitation mode (ξ 1 ξ 3 , ξ 2 ξ 3 ), the low mass configuration has a |66 c structure. In contrast, in the two L = 2 configurations containing the same J PC quantum numbers and excitation mode (ξ 1 ξ 3 , ξ 2 ξ 3 ), the low mass configuration has a |33 c structure. For example, in the two 0 +− T (cccc) configurations , the former has a low mass 6868 MeV due to the strong attraction from the Coulomb type potential V Coul together with the relatively smaller contributions from the linear confinement potential V Lin . For the two 1 ++ T cccc configurations with L = 2, the mass of It should be mentioned that in the L = 0 (or L = 1) configurations, a |66 c structure has larger attractions from the Coulomb type potential than |33 c although all of the color factors λ i ·λ j are negative for |33 c . The reason is that for a |66 c structure the color factors λ 1 ·λ 3 = λ 2 ·λ 4 = λ 1 ·λ 4 = λ 2 ·λ 3 = −10/3 are a factor of 2.5 larger than Including configuration mixing effects, the physical masses for the 1D-wave T (cccc) states are predicted to be in the range of ∼ (6700, 7200) MeV, and the masses for the 1Dwave T (bbbb) states are predicted to be in the range of ∼ (19500, 20000) MeV. The mass range for the 1D-wave states covers most of the mass range of the 1P-wave states and the whole mass range of the 2S -wave states. Figure 2 shows that in the mass range ∼ (6700, 7000) MeV, many low-lying 1Dwave T (cccc) states highly overlap with the 1P-wave states, while in the range of ∼ (6900, 7050) MeV, many 1D-wave T (cccc) states highly overlap with the 2S -wave states. Such a phenomenon will complicate the experimental analysis if one wants to disentangle their quantum numbers.

IV. SUMMARY
In this work, we further calculate the higher mass spectra for 2S -, 3S -, and 1D-wave fully charmed and bottom tetraquark states in a nonrelativistic potential quark model, which is a continuation of our previous study of the 1S -and 1P-wave states [28,32].
Our calculation suggests that within the range of ∼ (6.9, 7.2) GeV, it scatters the 2S -wave fully-charmed T (cccc) states, while the 3S -wave fully-charmed T (cccc) states lie in the mass range of ∼ (7.2, 7.4) GeV with one 0 ++ state T (cccc)0 ++ (7720) locating at a mass of M = 7720 MeV. We also find that the masses for the 1D-wave T (cccc) states are located in the range of ∼ (6.7, 7.2) GeV. Notice that this is also the mass region that 1P-wave states sit [32]. We actually obtain a busy spectrum for the fully-charmed tetraquark states with the low-excitation quantum numbers. Similar phenomenon occurs for the fully bottomed tetraquark states which can be seen in Fig. 3.
Our study shows that both the kinetic energy T and the linear confinement potential V Lin contribute a large positive value to the tetraquark masses, while the Coulomb type potential V Coul has a large cancellation with the these two terms. Although some 2S /3S and 1D configurations have a similar mass, the contributions of T , V Lin and V Coul are usually very different from each other. As a consequence of these in-teractions, most of the physical states are mixing states with different configurations.
To summarize, we have carried out quantitative calculations of the fully-heavy charmed and bottom tetraquark states in order to gain insights into the four-body system with equal-mass heavy quarks and antiquarks. We disentangle the important role played by the confinement potential which implies that the fully-heavy tetraquark states should exist above two heavy quarkonium thresholds. Meanwhile, we find that the potential quark model will predict an extremely rich spectrum with significant configuration mixings. This raises questions on the understanding of the experimental observations since apparently only a few states have been seen in experiment. It should be noted that the rich spectrum predicted in the quark model may be significantly affected by the strong S -wave couplings to the nearby open channels. A combined analysis of the role played by the nearby S -wave continuum channels should be the direction for a better understanding of the multiquark dynamics in the future. Moreover, to uncover the nature of the structures observed in the di-J/ψ invariant mass spectrum, further studies of the decays of these candidates of T (cccc) states are desired. II: Configurations for the NS -wave tetraquark qqqq system, where N = n + 1 (n = 1, 2, ...). ξ 1 , ξ 2 , ξ 3 are the Jacobi coordinates. (ξ 1 , ξ 2 ) stands for a configuration containing both ξ 1 -and ξ 2 -mode excitations.

J PC Configuration
Wave function III: Configurations for the 1D-wave tetraquark qqqq system. ξ 1 , ξ 2 , ξ 3 are the Jacobi coordinates. (ξ 1 , ξ 2 ) stands for a configuration containing both ξ 1 -and ξ 2 -mode orbital excitations. (ξ i ξ j ) stand for a coupling mode between two different excited modes ξ i and ξ j .            (7053) 7053 1          87 87 7062 19917 68 68 19869  The average contributions of each part of the Hamiltonian to the 2S -and 3S -wave T (cccc) configurations. T stands for the contribution of the kinetic energy term. V Lin and V Coul stand for the contributions from the linear confinement potential and Coulomb type potential, respectively. V S S , V T , and V LS stand for the contributions from the spin-spin interaction term, the tensor potential term, and the spin-orbit interaction term, respectively. (66)   The average contributions of each part of the Hamiltonian to the 1D-wave T (cccc) configurations. T stands for the contribution of the kinetic energy term. V Lin and V Coul stand for the contributions from the linear confinement potential and Coulomb type potential, respectively. V S S , V T , and V LS stand for the contributions from the spin-spin interaction term, the tensor potential term, and the spin-orbit interaction term, respectively.