1. Introduction
The existence of multiquark states with more than three quarks has been predicted by hadron scattering amplitudes [1, 2], where a tetraquark configuration has been suggested based on the MIT bag model with color and spin interaction. Different interpretations of the structure of the four-quark systems provide a unique environment for exploring exotic hadronic matter [3]. Exploring exotic hadrons will provide a new perspective on non-perturbative quantum chromodynamics (QCD).
Four-quark models were initially considered only with light flavors, but soon models with heavy quarks emerged [4, 5]. These exotic states may have two constitutional structures, tetraquark and meson ‘molecule' [6, 7]. The tetraquark includes a di-quark and a di-antiquark, while, the meson ‘molecule' is made of two meson clusters and it was expected that these structures give the same results in the same flavor, color, and spin representation [7].
In the past decade, many charmonium-like states such as X(3872) [8], Y(3940) [9], X(4350) [10], Y(4140) [11], Zc(4050) [12], Z+(4430) [13], and ${Z}_{c}^{+}(4200)$ [14] and bottomonium-like states Zb(10610), and Zb(10650) [15] were observed. The masses, decay modes, and quantum numbers of these particles prove they contain both heavy and light quarks. They are very good candidates for the tetraquarks configuration which have an important role to realize the physics of exotic hadrons. The hadron states with valence quarks with four different flavors would be a tetraquark if the interaction between quarks was not strong enough to form a molecule [16]. The ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$ four-quark configuration is allowed in QCD and, thus the QCD analogue of the hydrogen molecule would be applied for the states combined with heavy and light quarks and antiquarks [17].
Up to now, several methods have been employed to study the existence and stability of double-heavy tetraquarks such as the MIT bag model [18], QCD sum rule method [17], constituent quark model [19], the chromomagnetic interaction (CMI) model [20], quark-level models [21], chiral quark model [22], chiral perturbation theory [23], lattice QCD simulation [24], and holographic model [25].
In this paper, we study the tetraquark states with the configuration ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$ (Q = c, b, s and $\bar{Q}^{\prime} =\bar{c},\bar{b},\bar{s}$). Tetraquark masses are calculated by considering the Coulomb potential added to the hyperfine spin-spin term, and the confinement potential as the interaction between quarks and antiquarks. The confinement potential has been analyzed in two linear and quadratic forms This paper is organized as follows, section 2 gives a brief description of the Hamiltonian models, wave functions, and calculation method. The results and discussion are presented in section 3 . Finally, the paper ends with a short summary in section 4 .
2. Quark potential models
The non-relativistic Hamiltonian of quark-gluon model for the four interacting quark system is assumed to be
$\begin{eqnarray}H(4)=\displaystyle \sum _{i=1}^{4}\left({m}_{i}+\displaystyle \frac{{{p}_{i}}^{2}}{2{m}_{i}}\right)-{T}_{{\rm{cm}}}+\displaystyle \sum _{i\lt j=1}^{4}({V}_{{ij}}^{C}+{V}_{{ij}}^{G}).\end{eqnarray}$
Here Tcm is the kinetic energy of the center of mass and pi and mi are the momentum and the mass of each quark, respectively. As long as the interacting quarks are always in the same cluster, the long-range color-confinement interaction takes on a normal, unscreened form, quadratic or linear in distance between ith and jth quarks [26]. The linear and quadratic confinement models have been used many times to study dibaryon resonances, Nucleon-Nucleon scattering, hadron interaction, tetraquarks, pentaquark resonances and etc [17, 27-30], as well, the quadratic confinement model was sometimes more appropriate for a nonrelativistic model [27]. The color-confinement interaction in the linear [31] and quadratic [27] forms and the short-range potential, the Fermi-Breit approximation of the one-gluon exchange with the spin-spin interaction, are shown below, respectively $\begin{eqnarray}{V}_{{ij}}^{C}=\left\{\begin{array}{ll}-{\overrightarrow{\lambda }}_{i}\cdot {\overrightarrow{\lambda }}_{j}{{ar}}_{{ij}} & \mathrm{Linear}\\ -{\overrightarrow{\lambda }}_{i}\cdot {\overrightarrow{\lambda }}_{j}{{ar}}_{{ij}}^{2} & \mathrm{Quadratic}\end{array}\right.\end{eqnarray}$
$\begin{eqnarray}{V}_{{ij}}^{G}={\alpha }_{s}\displaystyle \frac{{\overrightarrow{\lambda }}_{i}\cdot {\overrightarrow{\lambda }}_{j}}{4}\left(\displaystyle \frac{1}{{r}_{{ij}}}-\displaystyle \frac{\pi \delta (\overrightarrow{{r}_{{ij}}})}{{m}_{i}{m}_{j}}\left(1+\displaystyle \frac{2}{3}{\overrightarrow{\sigma }}_{i}\cdot {\overrightarrow{\sigma }}_{j}\right)\right),\end{eqnarray}$
where $\overrightarrow{{r}_{{ij}}}=\overrightarrow{{r}_{i}}-\overrightarrow{{r}_{j}}$, λi is the color $SU_{3}^{c}$ generator (Gell-Mann matrix) and σ is the Pauli ${{SU}}_{2}^{\sigma }$ operator. a is the confinement potential strength determined by the zero nucleon binding, and αs is the quark-gluon coupling constant specified by the N − Δ mass difference [27, 31]. Since the constituent quark model explains hadron spectroscopy successfully, the parameters determined by nucleon mass, N − Δ mass splitting, and stability conditions for nucleon sizes are used directly for exotic hadrons.Based on the generator coordinate method, we define the wave function of the tetraquark system as follows
$\begin{eqnarray}{\rm{\Psi }}(1234)={ \mathcal A }{[{{\rm{\Psi }}}_{{B}_{1}}(12){{\rm{\Psi }}}_{{B}_{2}}(34)]}_{{CFIJ}},\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Psi }}}_{{B}_{1}}(12)={\chi }_{C}(12){\kappa }_{F}(12){\eta }_{{IJ}}(12){\psi }_{L}(1){\psi }_{L}(2),\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Psi }}}_{{B}_{2}}(34)={\chi }_{{CF}}(34){\kappa }_{F}(34){\eta }_{{IJ}}(34){\psi }_{R}(3){\psi }_{R}(4).\end{eqnarray}$
Here ${ \mathcal A }$ is the normalized antisymmetrization operator, χC and κF are respectively the color and flavor wave functions and ηIJ denote the spin-isospin wave function. For the usual quark cluster model, two single quark orbits are assumed due to the fact that the four-quark model extends to the delocalized single quark orbits [32]. The delocalized quark orbital wave functions have been defined using single quark orbits $\begin{eqnarray}{\psi }_{L}(\vec{r})=({\phi }_{L}(\vec{r})+\epsilon {\phi }_{R}(\vec{r}))/\sqrt{1+{\epsilon }^{2}+2\epsilon \langle {\phi }_{L}| {\phi }_{R}\rangle },\end{eqnarray}$
$\begin{eqnarray}{\psi }_{R}(\vec{r})=({\phi }_{R}(\vec{r})+\epsilon {\phi }_{L}(\vec{r}))/\sqrt{1+{\epsilon }^{2}+2\epsilon \langle {\phi }_{L}| {\phi }_{R}\rangle },\end{eqnarray}$
$\begin{eqnarray}{\phi }_{L}(\vec{r})={\left(\frac{1}{\pi {b}^{2}}\right)}^{\tfrac{3}{4}}{{\rm{e}}}^{-\tfrac{{\left(\vec{r}-\vec{{s}_{1}}\right)}^{2}}{2{b}^{2}}},\end{eqnarray}$
$\begin{eqnarray}{\phi }_{R}(\vec{r})={\left(\frac{1}{\pi {b}^{2}}\right)}^{\tfrac{3}{4}}{{\rm{e}}}^{-\tfrac{{\left(\vec{r}-\vec{{s}_{2}}\right)}^{2}}{2{b}^{2}}},\end{eqnarray}$
where ${\phi }_{L}(\vec{r})$ and ${\phi }_{R}(\vec{r})$ are left and right centered orbits, respectively, $\vec{s}=\vec{{s}_{1}}-\vec{{s}_{2}}$ describe the separation of two reference centers, and ε is the delocalization parameter which is determined by the four-quark system's dynamics for every $\vec{s}$. In this work, the tetraquark has been assumed as one bag of four quarks ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$. The wave function of a four-quark bag-model-like is described with ε = 1 which is spherical only for $\vec{s}=\vec{0}$ [33]. We used two sets of parameters in our calculation which are listed in table 1. In addition, they have been fixed by the baryon spectrum, NN scattering, and the stability condition for the tetraquark size [27, 31, 33]. The size parameter of four-quark b is determined by minimizing the tetraquark mass ∂M(b)/∂b = 0. The values shown here are similar to those found in constituents quark models. Generally, potential models use a Coulomb-like term for the short-range part potential by setting αs near αs = 1.71 and 1.54. Due to the fact that the interaction energy scale for heavy quarks in one gluon exchange potential is larger than quark masses, the αs parameters in table 1 are reasonable for this range of energy.Table 1. Fixed potential models parameters, a is strength of the confinement potential, unit in [MeV fm−1] for linear and [MeV fm−2] for quadratic , αs is quark-gluon coupling constant, mu,d,s,c,b are quark masses in [MeV] [27, 31], and b is tetraquark size parameter unit in [fm]. |
| Potential model | a | αs | mu | md | ms | mc | mb | b |
|---|---|---|---|---|---|---|---|---|
| Linear | 39.1 | 1.71 | 313 | 313 | 522 | 1758 | 4588 | 0.540 |
| Quadratic | 25.13 | 1.54 | 313 | 313 | 634 | 1870 | 4700 | 0.518 |
The wave functions of this configuration must be antisymmetric for the exchange of the two quarks and the two antiquarks. The allowed SU(3)c color representations for tetraquark ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$ states are given by the product of
$\begin{eqnarray}\begin{array}{l}[3]\otimes [\tilde{3}]\otimes [3]\otimes [\tilde{3}]=[1]\oplus [8]\oplus [8]\\ \quad \oplus [\bar{10}]\oplus [8]\oplus [8]\oplus [1]\oplus [10]\oplus [27].\end{array}\end{eqnarray}$
The color representation for the tetraquark is only one singlet. Therefore, in our work, we use only two below color representations of $\begin{eqnarray}\begin{array}{rcl}| {\chi }_{C}{\rangle }_{3\bar{3}} & = & \displaystyle \frac{1}{2\sqrt{3}}| {br}\bar{b}\bar{r}-{br}\bar{r}\bar{b}-{rb}\bar{b}\bar{r}+{rb}\bar{r}\bar{b}\\ & & +{gb}\bar{g}\bar{b}-{gb}\bar{b}\bar{g}-{bg}\bar{g}\bar{b}+{bg}\bar{b}\bar{g}\\ & & +{rg}\bar{r}\bar{g}-{rg}\bar{g}\bar{r}-{gr}\bar{r}\bar{g}+{gr}\bar{g}\bar{r}\rangle \end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}| {\chi }_{C}{\rangle }_{6\bar{6}} & = & \displaystyle \frac{1}{2\sqrt{6}}| 2{rr}\bar{r}\bar{r}+{rg}\bar{r}\bar{g}+{rg}\bar{g}\bar{r}\\ & & +{gr}\bar{r}\bar{g}+{gr}\bar{g}\bar{r}+2{gg}\bar{g}\bar{g}+{rg}\bar{r}\bar{g}\\ & & +{rb}\bar{b}\bar{r}+{br}\bar{r}\bar{b}+{br}\bar{b}\bar{r}+{gr}\bar{g}\bar{r}\\ & & +{gr}\bar{r}\bar{g}+{bg}\bar{g}\bar{b}+{bg}\bar{b}\bar{g}+2{bb}\bar{b}\bar{b}\rangle .\end{array}\end{eqnarray}$
The SU(2)s spin classifications of ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$ states are given by the product: $\begin{eqnarray}\begin{array}{l}[2]\otimes [2]\otimes [2]\otimes [2]\\ \quad =[1]\oplus [3]\oplus [1]\oplus [3]\oplus [3]\oplus [5].\end{array}\end{eqnarray}$
The total spin of the tetraquark can be Stot = 0, 1, 2. Among the all tetraquark spin states in the (${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$) configuration [34], the three below states have been used in this paper to solve the problem: $\begin{eqnarray}\begin{array}{rcl}| {\eta }_{0}\rangle & = & \displaystyle \frac{1}{2\sqrt{3}}| 2\uparrow \uparrow \downarrow \downarrow +2\downarrow \downarrow \uparrow \uparrow -\uparrow \downarrow \uparrow \downarrow -\downarrow \uparrow \downarrow \uparrow \\ & & -\uparrow \downarrow \downarrow \uparrow -\downarrow \uparrow \uparrow \downarrow \rangle \\ | {\eta }_{1}\rangle & = & \displaystyle \frac{1}{2}| \uparrow \uparrow \uparrow \downarrow +\uparrow \uparrow \downarrow \uparrow -\uparrow \downarrow \uparrow \uparrow -\downarrow \uparrow \uparrow \uparrow \rangle \\ | {\eta }_{2}\rangle & = & | \uparrow \uparrow \uparrow \uparrow \rangle ,\end{array}\end{eqnarray}$
where η indices represent the total spin quantum numbers of the tetraquark [34].The flavor representation for ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$ has different configurations depending on the number of flavor types that construct the tetraquark. We have studied the tetraquarks with different flavor structures. Therefore, each particle has different flavor states which are calculated with the SU(3)f and SU(4)f spectrum-generating algebra. The computation of the parity and the charge conjugation quantum numbers for the tetraquark (${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$) with the total spin S and the relative angular momentum L conclude to [35]
$\begin{eqnarray}\begin{array}{rcl}C & = & {(-1)}^{S-{L}_{{QQ}-\bar{Q^{\prime} }\bar{Q^{\prime} }}}\\ P & = & {(-1)}^{{L}_{{QQ}}+{L}_{\bar{Q^{\prime} }\bar{Q^{\prime} }}+{L}_{{QQ}-\bar{Q^{\prime} }\bar{Q^{\prime} }}}.\end{array}\end{eqnarray}$
The theoretical mass of the four-quark system has been given by
$\begin{eqnarray}M={ \mathcal M }+\langle T\rangle +\langle {V}_{C}\rangle +\langle {V}_{G}\rangle .\end{eqnarray}$
Where ${ \mathcal M }={\sum }_{i=1}^{4}({m}_{i})$ and $T={\sum }_{i=1}^{4}(\tfrac{{{\bf{p}}}_{i}^{2}}{2{m}_{i}})-{T}_{{\rm{cm}}}$. The confinement potential contribution to the tetraquark mass has been obtained with two models, linear model VLC and quadratic model VQC, and the contribution of short-range potential has been shown by ⟨VG⟩. The mass of ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$, generated by quarks c or b and s have been calculated in different quantum numbers Jp = 0+, 1+, 2+.3. Results and discussion
We have considered four-quark systems produced by two couples of quarks and anti-quarks, ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$, which are located in one bag. Based on the linear and quadratic confinement potentials, we have calculated the mass spectra of five four-quark systems using two sets of parameters in table 1. The results of ${cc}\bar{c}\bar{c}$, ${cc}\bar{s}\bar{s}$, and ${cc}\bar{b}\bar{b}$ with two allowed color configurations for three sets of quantum numbers Jp = 0+, 1+, 2+ have presented in table 2.
Table 2. The mass spectra of tetraquarks with different quantum numbers, generated from the linear and quadratic models. The values are given in MeV. |
| Color | ${cc}\bar{c}\bar{c}$ | ${cc}\bar{s}\bar{s}$ | |||
|---|---|---|---|---|---|
| Jp | ML | MQ | ML | MQ | |
| $| {\chi }_{C}{\rangle }_{3\bar{3}}$ | 0+ | 6924.4 | 7268.5 | 4858.3 | 5106.3 |
| 1+ | 6931.5 | 7275.0 | 4865.5 | 5112.7 | |
| 2+ | 6928.0 | 7271.7 | 4824.8 | 5084.6 | |
| | |||||
| $| {\chi }_{C}{\rangle }_{6\bar{6}}$ | 0+ | 7889.2 | 8174.8 | 5572.5 | 5844.7 |
| 1+ | 7885.6 | 8171.6 | 5569.0 | 5841.5 | |
| 2+ | 7887.4 | 8173.2 | 5589.3 | 5855.5 | |
| Mexp | 6905 ± 11 [40] | 4625.9 ± 0.4 [36] | |||
The mass of $({cc}\bar{c}\bar{c})$ in the state JP = 0+ for the color representations $| {\chi }_{C}{\rangle }_{3\bar{3}}$ measured with the linear model is 6924 MeV, while for the color representations $| {\chi }_{C}{\rangle }_{6\bar{6}}$ in JP = 0+ state with the same model it is about 7889 MeV. The mass splitting between these two 0+ states is about 965 MeV. The other two states JP = 1+ and JP = 2+ are also predicted the mass of $({cc}\bar{c}\bar{c})$ in a similar region, and the mass splitting between their result of two color configurations $| {\chi }_{C}{\rangle }_{6\bar{6}}$ and $| {\chi }_{C}{\rangle }_{3\bar{3}}$ are about 954 and 959, respectively. As one can see, the results of $({cc}\bar{c}\bar{c})$ with the linear model are closer to the experimental data of X(6900) than the quadratic one. As shown in table 2, the results of the linear and quadratic potential for $({cc}\bar{c}\bar{c})$ are in the mass range 6924-8175, and the predicted mass for X(6900) is in the range 5800-7400, thus both models computed acceptable results for this particle.
The first observation of a vector charmonium-like state Y(4626) decaying to a charmed-antistrange and anticharmed-strange meson ${D}_{s}^{+}{D}_{s1}{(2536)}^{-}+c.c.$ was reported by Belle Collaboration [36]. They measured the mass of Y(4626), with structure ${cs}\bar{c}\bar{s}$, ${M}_{\exp }={4625.9}_{-6.0}^{+6.2}\pm 0.4\,\mathrm{MeV}$. The masses of ${cc}\bar{s}\bar{s}$ for different quantum numbers are presented in table 2 using linear and quadratic models. As one can see, the differences between our results and the mass of Y(4626) are in the range of 100 to 1000 MeV. A possible explanation for this could be the differences in structure between the particles. In the quantum number 0+ for $[3]\otimes [\bar{3}]$ configuration the computed masses for this state are ML = 4858 MeV and MQ = 5572 MeV. One may find the estimated masses for charmanim-like tetraquark in figure 1.
Figure 1. The estimated masses of ${cc}\bar{c}\bar{c}$ (solid line), and ${cc}\bar{s}\bar{s}$ (dotted line) with linear ML and quadratic MQ models. The long dotted lines are experimental data Mexp. The values are given in MeV. |
The observations of two exotica states in ϒ(5S) decays and the hadron colliders in the ϒ(1S)ππ was the first evidence for the XYZ exotic state of the bottomonium sector [15, 37]. The Belle collaboration observed three candidates Yb(10890), Zb(10610), and Zb(10650) as the nonconventional hadronic states in the bottom sector [15]. The results of ${bb}\bar{b}\bar{b}$, ${bb}\bar{s}\bar{s}$, and ${cc}\bar{b}\bar{b}$ with two allowed color configurations for three sets of quantum numbers Jp = 0+, 1+, 2+ are presented in table 3.
Table 3. The mass spectra of tetraquarks with different quantum numbers, generated from the linear and quadratic models. The values are given in MeV. |
| Color classification | ${bb}\bar{b}\bar{b}$ | ${bb}\bar{s}\bar{s}$ | ${cc}\bar{b}\bar{b}$ | ||||
|---|---|---|---|---|---|---|---|
| Jp | ML | MQ | ML | MQ | ML | MQ | |
| $| {\chi }_{C}{\rangle }_{3\bar{3}}$ | 0+ | 18 121 | 18 467 | 10 492 | 10 738 | 12 535 | 12 879 |
| 1+ | 18 122 | 18 468 | 10 493 | 10 739 | 12 536 | 12 881 | |
| 2+ | 18 121 | 18 468 | 10 452 | 10 711 | 12 532 | 12 877 | |
| | |||||||
| $| {\chi }_{C}{\rangle }_{6\bar{6}}$ | 0+ | 19 113 | 19 398 | 11 213 | 11 483 | 13 507 | 13 792 |
| 1+ | 1.9112 | 19 397 | 11 213 | 11 482 | 13 506 | 13 791 | |
| 2+ | 19 113 | 19 398 | 11 233 | 11 496 | 13 508 | 13 793 | |
| | |||||||
| Mexp | 18 400 ± 100 [39] | — | — | ||||
The CMS collaboration reported an observation pair production of ϒ(1S) mesons in proton-proton collisions at $\sqrt{s}=8$ TeV with a mass 18.4 ± 0.1(stat) ± 0.2(syst) GeV [38, 39]. The invariant-mass distribution of ${\rm{\Upsilon }}{(1S)}_{{\mu }^{+}{\mu }^{-}}$ has been considered to seek a possible ${bb}\bar{b}\bar{b}$ exotic tetraquark [40]. In the ${bb}\bar{b}\bar{b}$ case for the 0+ state, the estimated mass is 18 121 MeV for the linear model and 18 467 MeV for the quadratic one in the $[3]\otimes [\bar{3}]$ configuration while the results for $[6]\otimes [\bar{6}]$ configuration for all quantum number are about 1000 MeV upper than them.
The estimated masses of ${bb}\bar{s}\bar{s}$ are listed in table 3. The obtained masses for 0+ in $[3]\otimes [\bar{3}]$ color configuration ML = 10 492 and MQ = 10 738 and these values are ML = 11 213 and MQ = 11 483 for $[6]\otimes [\bar{6}]$ . By comparing the ${bb}\bar{s}\bar{s}$ results with the meson-meson thresholds for Jp = 0+, 1+, 2+ in figure 2 one can find that the linear mass results for this tetraquark stats are below their relevant meson-meson thresholds. But the quadratic ones are above these thresholds.
Figure 2. Relative positions for ${bb}\bar{s}\bar{s}$ tetraquark states ML (solid lines) and MQ (dashed lines) and relevant meson-meson thresholds (dotted lines). The values are given in MeV. |
Although the existence of tetraquarks with two bottoms and two charms like ${bb}\bar{c}\bar{c}$ and $b\bar{c}b\bar{c}$ in nature has not been confirmed, the theoretical information [41-43] of the mass spectra of these states would contain a lot of useful knowledge of heavy tetraquarks explorations. The tetraquark ${cc}\bar{b}\bar{b}$ can appear in the resonances of the invariant mass of BcBc, ${B}_{c}{B}_{c}^{* }$, and ${B}_{c}^{* }{B}_{c}^{* }$ , with the Bc, the observed bound state of $c\bar{b}$ [44], and J/$\Psi$ϒ(1S) [45]. The theoretical thresholds for $2{B}_{c}^{-}$, ${B}_{c}^{-}{B}_{c}^{* -}$, and $2{B}_{c}^{* -}$ respectively are 12 683.6, 12 736.9, and 12 790.2 [41]. In table 3, we present results for quantum numbers 0+, 1+, and 2+ with linear confinement and quadratic potentials. According to figure 3, the linear results are lower than theoretical thresholds, but the quadratic results are higher than them.
Figure 3. Relative positions for ${cc}\bar{b}\bar{b}$ tetraquark states ML (solid lines) and MQ (dashed lines) and relevant meson-meson thresholds (dotted lines). The values are given in MeV. |
4. Conclusion
In this work, the masses of the ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$ (Q = c, b, s and $\bar{Q}^{\prime} =\bar{c},\bar{b},\bar{s}$) tetraquark states have been calculated with the non-relativistic model applying the linear and quadratic confinement models with the Fermi-Breit approximation of one-gluon potential. Two sets of allowed color, flavor, and spin states for the tetraquark have been chosen in our investigation. Furthermore, theoretical calculations have shown that the color configurations can significantly affect tetraquark mass. According to the linear confinement model, charmonium-like tetraquark states have masses closer to the experimental results, and bottomonium-like tetraquark states have masses below their meson-meson thresholds. These results indicate that the linear confinement model is suitable for studying full-heavy and heavy-light tetraquarks. The results of the quadratic model are also acceptable, but there is a noticeable difference between linear and quadratic models, and it seems that first-order confinement would be more appropriate for tetraquarks. We will examine the effects of linear and quadratic models using different approaches in future work in order to provide more convincing reasons for using linear models.
Due to significant uncertainties in parameters such as effective quark masses, many models cannot accurately estimate tetraquark masses. In our study, Hamiltonian model parameters have been applied to a wide variety of systems. There are no differences in parameter values between hadron systems. As a result, this model reduces the uncertainty associated with mass estimation.
All things considered, our results are generally in agreement with experimental observations, but to achieve more noteworthy results, we need more information and a more detailed understanding of the tetraquark. Apart from the spin-spin interaction, the short-range potential can also include spin-orbit, tensor, and other statements. These statements may reduce the discrepancy between results and experimental data.