Models and potentials in hadron spectroscopy

Regge trajectories of hadrons were introduced by T Regge in 1959 [89, 90]. The mass and angular momentum of hadrons are related by the equation [40],

$$\begin{eqnarray}&&{m}_{J,n}^{2}={aJ}+{bn}+c,\end{eqnarray} \tag{ 12 }$$

where n is the principal quantum number and a, b, and c are the slope and intercept parameters. The hadrons lie on a line in the (J, M₂) plane called Regge trajectories. Usually, string picture of hadrons is the most popular model used to explain the Regge trajectories. Regge trajectory can be used to obtain the mass spectra of hadrons. As explained earlier in string models by Goebel, LaCourse, Olsson, and Soloviev formation of Regge trajectories was discussed [87, 88].

A study by Semay and Brac used the relativistic flux tube model and obtained a linear Regge trajectory for mesons in the ultra-relativistic limit [91]. The Coulomb-like potential, instanton effect, and constant potential lead to the flux tube picture. The constant potential was taken as a negative value to reproduce the data. The spin effect is added to the potential with the help of the instanton interaction. Here, they assumed the instanton effect acts only for L = 0 meson state. The study showed that the nature of constant potential is important in getting good results. Satisfactory results for many meson spectra were obtained. The constant potential coming from the extremities of the flux tube has given the most satisfying results with theories and experiments.

Sergeenko combined heavy and light quarkonia to develop an analytic expression [92]. Regge trajectories of heavy and light quarkonia in all regions were developed. They used Cornell potential with a constant term. For the square of mass, an analytical equation was developed. The formula incorporates spin–orbit and spin–spin interactions. The interpolating mass formula showed good accuracy in obtaining spectra of bound states of quarkonia.

Ebert, Faustov, and Galkin used the relativistic quark model to study the mass and Regge trajectories of light mesons using a quasipotential [93]. The quasipotential was derived by assuming the resultant interaction was the combination of one gluon exchange with long-range scalar and vector linear confining potential. They were able to calculate masses and the linear Regge trajectories of light mesons.

Nandan and Ranjan investigated the Regge trajectories of pentaquarks with different possible configurations using the flux tube model [94]. Regge trajectories for pentaquarks showed deviation from linear behavior. The result was compared with the available experimental results. At low rotational speed, the mass of pentaquark showed a linear increase, but at the high rotational speed of string, the Regge trajectories became highly nonlinear. They also observed that two different pentaquarks could show the same mass and angular momentum.

It can be clearly seen that more studies are needed in the case of Regge trajectories of exotic candidates.

Micu proposed the model that discussed the decays of meson resonance [104]. According to the model, each hadron undergoes a decay that produces a quark pair with the quantum number 0⁺⁺ from vacuum, which is then mixed with the quark pairs from the parent hadrons to create the daughter hadrons. The model was used to obtain the strong decays of 2⁺, 1⁺, and 0⁺ meson. Later in 1970s, this model was further modified by Orsay group [105, 106]. This model is now extensively applied on hadron strong decays. Following the production of the quark pair, the initial quarks are rearranged to produce an initial mock state. The decay amplitude was obtained by combining the overlap integrals in spin, color, flavor, and orbital spaces for the initial state and the final state.

In a separate work, Feng et al used this model to get the decay width of ${S}_{1}^{3}$ mesons [107]. Mass calculated from the Regge trajectories was compared with the experimental mass to identify the possible candidates. Then using ${P}_{0}^{3}$ model decay widths were calculated. One of the two model parameters represents effective radius of the particle. They compared the theoretical values with the experimental values. Mass and width of ρ(1900) and ω(1960) states matched with experimental values. For ϕ(2170), the calculated value and the experimental value for partial and total width showed a mismatch. K*(1410) state also contradicted with the experimental value in both mass and width.

The OGE model is one of the earliest approaches in hadron spectroscopy discussed by Rujula, Georgi, and Glashow [110], which is the basis of short-range quark interactions. Since all the hadrons are color singlets, the exchange of gluons can bind the quarks inside hadrons. The two-body OGE potential is given by,

$$\begin{eqnarray}&&{V}_{{ij}}=\displaystyle \frac{{k}_{s}{\alpha }_{s}}{r},\end{eqnarray} \tag{ 21 }$$

where α_s is the running coupling constant. The value of k_s will be −4/3 for $q\bar{q}$ and 2/3 for qq. Since the running coupling constant will decrease as the distance decreases, the potential V_ij will also approach the lowest order as r → 0. Therefore, at short interquark interactions, OGE can be applied.

The central part of the OGE which contains spin–spin interaction can also be written as [111],

$$\begin{eqnarray}&&{V}_{\mathrm{OGE}}^{C}(\vec{{r}_{{ij}}})=\frac{1}{4}{\alpha }_{s}\vec{{\lambda }_{i}^{c}}\cdot \vec{{\lambda }_{j}^{c}}\left\{\frac{1}{{r}_{{ij}}}-\frac{1}{6{m}_{i}{m}_{j}}\vec{{\sigma }_{i}}\cdot \vec{{\sigma }_{j}}{e}^{\tfrac{-{r}_{{ij}}/{r}_{0}(\mu )}{{r}_{{ij}}{r}_{0}^{2}(\mu )}}\right\}.\end{eqnarray} \tag{ 22 }$$

Rujula, Georgi, and Glashow [110] calculated hadron masses by one gluon exchange controlled by a small coupling which was Coulomb-like. The Hamiltonian was of the form,

$$\begin{eqnarray}&&H=L({r}_{1},{r}_{2}\cdots )+\displaystyle \sum _{i}\left({m}_{i}+\displaystyle \frac{{p}_{i}^{2}}{2{m}_{i}}+\cdots \right)+\displaystyle \sum _{i\gt j}\left(\alpha {Q}_{i}{Q}_{j}+{k}_{s}{\alpha }_{s}\right){S}_{{ij}},\end{eqnarray} \tag{ 23 }$$

where L represents the quark binding; p, Q, r, and m represent the momentum, charge, position, and mass of the quark respectively. In the case of two-body Coulomb interaction, S_ij includes spin-dependent and tensor terms. They obtained some of the mass relations successfully. Also, some new mass relations were derived in the light of renormalizable gauge field theory that imposed particular interaction and symmetry breaking mechanisms. Some features of the charmed hadron mass spectrum, including the origin of Σ−Λ mass splitting, were also discussed.

Isgur and Karl formulated a quark model framework that studied the spectrum of low-lying baryons with negative parity inspired by QCD [112]. The Hamiltonian was of the form,

$$\begin{eqnarray}&&H=\displaystyle \sum _{i}{m}_{i}+{H}_{0}+{H}_{{\rm{hyp}}},\end{eqnarray} \tag{ 24 }$$

and,

$$\begin{eqnarray}&&{H}_{0}=\displaystyle \sum _{i}\frac{{p}_{i}^{2}}{2{m}_{i}}+{V}_{\mathrm{conf}}.\end{eqnarray} \tag{ 25 }$$

The results agreed with the study by Rujula, Georgi, and Glashow [110]. Compared to the study by Rujula, Georgi, and Glashow [110] spin–orbit part of the ansatz was discarded. The results of Σ−Λ mass splitting supported the significance of confining potential in the Hamiltonian. Isgur and Karl also extended their study to low-lying baryons of positive parity [113]. Spin–orbit forces were neglected as earlier. Hamiltonian was again considered as the combination of the hyperfine and confining terms. The findings, which were mostly based on earlier research, agreed well with the attributes of these states.

Isgur and Karl also studied N, Σ, Σ*, Λ, Δ, Ω, Ξ, and Ξ* in their ground state in a similar approach [114] with color hyperfine and flavor independent confinement. Results were in good agreement with the observed masses.

Copley, Isgur, and Karl studied mass and decay rates of baryons with one charm quark [115]. The formulation was similar to previous studies [113, 114]. One of the most important results was the study of stability of ${{\rm{\Lambda }}}_{c}^{\tfrac{{1}^{-}}{2}}$ (P-wave) baryon against the strong decay.

It is one of the old and extensively used QCD motivated potential developed by the Cornell University group. The potential is mainly used for getting the mass spectrum of heavy quarkonia [123–125]. The potential is a combination of Coulomb and the confining term and has been extensively investigated. These terms may have different functional forms. One of the most extensively used forms of potential is,

$$\begin{eqnarray}&&V(r)=\displaystyle \frac{-a}{r}+{br},\end{eqnarray} \tag{ 26 }$$

where a and b are positive parameters and r is the interquark distance. The potential is quite helpful in addressing the nature of the QCD vacuum, which is paramagnetic and dielectric [126].

Kuchin and Maksimenko obtained an analytical solution for Cornell potential by applying the Nikiforov–Uvarov (NU) method and studied the mass spectrum of charmonium, bottomonium, and B_c mesons [124]. They modified the variable as $x=\tfrac{1}{r}$ and proposed some approximation scheme in the term $\tfrac{a}{x}$. They have assumed a characteristic radius r₀ for meson, and $\tfrac{a}{x}$ was expanded in a power series around r₀ till the second order. This helped to deform the centrifugal potential and this modification was solved by the NU method. The results showed a good agreement with the experimental values and other theoretical values.

Lahkar, Choudhury, and Hazarika used Cornell potential to analyze the mass and decay properties of heavy flavor mesons containing one heavy quark or antiquark with the help of the variational method using Coulomb, Gaussian, and Airy trial wave functions [127]. Properties like mass, decay constant, branching ratios of leptonic decays, and oscillation frequency of heavy mesons were determined. They compared the results with experimental, lattice, and QCD sum rule studies.

Vega and Flores used Cornell potential to describe properties of $c\bar{c}$, $b\bar{b}$, and $b\bar{c}$ states [128]. The Schroedinger equation was solved using an approximate method involving variational method and supersymmetric quantum mechanics (SUSY QM). The variational method is an effective tool to get the approximate ground state energies. Supersymmetric QM will help to get the solution of higher energy states. The energy and wave functions of s state of $c\bar{c}$, $b\bar{b}$, and $b\bar{c}$ were obtained. They calculated the energies for the first three states. Approximate wave functions for the ground, first, and second excited states were also provided. The calculated value showed good agreement with the computational value for the ground and first excited states. The decay properties and experimental values agreed equally well.

The charmonium mass spectra and decay properties were presented using Cornell potential by Chaturvedi and Rai [129]. Relative corrections were added to the Cornell and spin-dependent potential terms. They successfully determined the mass spectra and decay properties of charmonium. The Regge trajectories for charmonium states were also determined, which helped to associate some higher excited states with charmonium. Calculated charmonium masses fit the Regge planes perfectly. All the results were compared with experimental results. For 1S and 2S states, the splitting of the energy level was found to be higher than the experimental value. For 3S and 4S states, the results were in better agreement with the results from experiments and other theoretical studies. They have associated the X(3915) and X(3872) as 2⁰ P₃ and 2¹ P₃ respectively.

Karliner discussed the heavy baryons spectroscopy with bottom quark [130]. The hyperfine splitting ratio between meson and baryon was estimated with 5 potentials, including Coulomb, harmonic, linear, Cornell, and logarithmic. The coupling strength cancels the ratio between meson and baryon for all potentials with one coupling constant. Masses of Ξ_Q baryons with quark content Qsd or Qsu was found.

Patel, Shah, and Vinodkumar studied masses of hidden charm tetraquark state ${cq}\bar{c}\bar{q}$ (q ∈ u, d) using Cornell potential [131]. Spin-dependent interactions were added to the potential including spin–orbit, spin–spin, and tensor terms. The model parameters include constituent quark masses and string tension, which were chosen to fit with the ground state masses of experimentally observed X(3823), Z_c (3900), and Z_c (3885) states. The four-body system was considered as two two-body system, with one as a combination of diquark–antidiquark and the other as a cluster of quark–antiquark. Here they assume one quark moves in a static potential of other quarks like the hydrogen atom problem. The tetraquark states X(3823), X(3915), X(4160), Z_c (3900), Z_c (4025), and Ψ(4040) was interpreted as cluster of quark and antiquark. The tetraquark states X(3940), Y(4140), and Z_c (3885) were fitted with the diquark–antidiquark formalism. The X(3940) as diquark–antidiquark with 2⁺⁺ gives good agreement with experiment and study by Mainani et al [132], whereas Z₁(4050) agrees more with the formalism of cluster of quark–antiquark.

Martin potential has the form [133],

$$\begin{eqnarray}&&V(r)=a+{{br}}^{\alpha },\end{eqnarray} \tag{ 27 }$$

where α ∼ 0.1. Their study was motivated by the success of potential models found in the case of heavy quarkonia. The potential generates all the known levels of J/Ψ and ϒ systems studied by Martin previously [134]. One motivation for the study was realising the need for relativistic correction to the heavy quarkonium $c\bar{c}$ and $b\bar{b}$. Inspired by the accuracy of the fit, they considered an assumption that a Fermi-type term is controlling the hyperfine splitting. For $c\bar{c}$ system 1S, 2S, and 3S states lead to the value of α to be 0.1 with good accuracy. They calculated the masses and relative leptonic decay width of $c\bar{c}$, $b\bar{b}$ and $s\bar{s}$. They could only calculate the absolute decay width of ϕ from J/Ψ. The leptonic decay width of ϕ was obtained as 1.6 ± 0.23 KeV and the experimental result for the same was 1.43 ± 0.12 KeV. However, this is accepted because leptonic decay width is sensitive to the wave function and relativistic effects compared to the energy levels. Prediction of masses for these states also agreed with the experimental result.

Patel and Vinodkumar studied the $Q\bar{Q}$ system using the Coulomb plus power potential ${\left(\mathrm{CPP}\right)}_{\nu }$ using different values of ν [135],

$$\begin{eqnarray}&&V(r)=-\displaystyle \frac{{\alpha }_{c}}{r}+{{ar}}^{\nu }.\end{eqnarray} \tag{ 28 }$$

This potential is a part of the general form [136, 137],

$$\begin{eqnarray}&&V(r)=-{{cr}}^{\alpha }+{{dr}}^{\beta }+{V}_{0},\end{eqnarray} \tag{ 29 }$$

where α = − 1, β = ν and V₀ = 0. The study mainly used the power range 0.1 < ν < 2.0.

Different potential form will result from different ν values. The inappropriate choice of the radial wave function for heavy quarkonia can affect the decay width and spin splitting of J values because both are dependent on the radial wave function of $Q\bar{Q}$. Therefore, the value of A was limited as slightly varying according to the principal quantum number n. The Schroedinger equation was solved by the method given by Lucha and Schoeberl [138]. This potential gave the mass spectrum of $c\bar{c}$, $b\bar{c}$, and $b\bar{b}$ mesons up to a few excited states. The excited states with ν = 0.9 to 1.3 agreed with the results from experiments and other theoretical predictions. They also determined the decay constants of 1S to 6S states with and without QCD corrections. Without QCD corrections, the value of the decay constants for $c\bar{c}$ systems coincides very well with the experimental value, but not for $b\bar{b}$ systems. For the $b\bar{c}$ system, a comparison was not possible at that time due to the unavailability of experimental details. The di-gamma and leptonic decay widths were calculated. Di-gamma decay widths agree with experimental values for the range 1.1 to 1.3. However, in the same range leptonic decay widths were overestimated or underestimated with radiative corrections. They expect it may be because the decay occurs at some finite separation, not at zero separation.

Ciftci and Koru used power-law potential to study the leptonic decay widths and decay constants of mesons [139]. The potential considered was of the form,

$$\begin{eqnarray}&&V(r)=\displaystyle \frac{1}{2}(1+\beta )({{Ar}}^{\nu }+{V}_{0}),\end{eqnarray} \tag{ 30 }$$

where A and ν were greater than zero. Potential parameters were chosen as A = 0.68 GeV, V₀ = −0.3961 GeV, and ν = 0.2. When spin–spin and hyperfine interactions were added to the potential they could obtain good results for meson mass spectra.

Richard and Taxil used a set of power-law potential for baryons spectroscopy [140]. They estimated masses of baryons with a heavy quark (qsc and ssc). Three-body problem was solved with the help of hyperspherical harmonic expansion. Results were compared with the experimental findings.

They assumed that a quark and antiquark in a meson are confined in a potential of the form [141–143],

$$\begin{eqnarray}&&V(r)=\displaystyle \frac{1}{2}(1+{\gamma }_{0})({a}^{2}r+{V}_{0}),\end{eqnarray} \tag{ 31 }$$

where a > 0. The model parameters were a, V₀, and nonstrange quark mass. This model considers the spin-dependent forces as perturbation resulting from the one gluon exchange. The effect of the center of mass motion was also considered. They considered the quark-Lagrangian density for this model in zeroth order as,

$$\begin{eqnarray}&&{{ \mathcal L }}_{q}^{0}(x)={\bar{{\rm{\Psi }}}}_{q}(x)\left[\displaystyle \frac{i}{2}{\gamma }^{\mu }{\partial }_{\mu }-{m}_{q}-{V}_{q}(r)\right]{{\rm{\Psi }}}_{q}(x).\end{eqnarray} \tag{ 32 }$$

The mass and decay constant of the pion and the masses of the ρ and ω-mesons were calculated perturbatively. The value of the decay constant agreed with the experimental results.

Later Jena, Behera, and Tripathy extended the study to the radiative transition of light and heavy flavor mesons [143]. Model parameters were preserved as the same. They have included the momentum dependence due to the recoiling of the daughter meson. They found improvement in M1 transition for light mesons. The heavy meson decay width was also comparable with other results.

Authors proposed a potential for the heavy quarkonium of the form [144],

$$\begin{eqnarray}&&V(r)={{ar}}^{\tfrac{1}{2}}-{{br}}^{-\tfrac{1}{2}}.\end{eqnarray} \tag{ 33 }$$

The potential is a mixture of inverse square root and square root terms. The factors a and b were adjustable parameters. Relativistic effects were also included using spin–orbit and spin–spin terms. The Schroedinger equation was solved using the numerical method. Spin-dependent part of the potential was taken as perturbation. The numerical values for the calculated energy levels of $c\bar{c}$ and $b\bar{b}$ did not show much difference when compared with other potential model approaches because r does not vary much. However, it shows clear differences in the case of $t\bar{t}$ for very small distances (r < 0.1fm). The energy levels of $c\bar{c}$, $b\bar{b}$, and $t\bar{t}$ and the decay rates of $c\bar{c}$, $b\bar{b}$, $t\bar{t}$, and $s\bar{s}$ were calculated. Most of the results showed improvement compared with experiments and other potential model studies.

Lichtenberg et al proposed a new phenomenological potential which lies between Cornell and Song–Lin potential [145],

$$\begin{eqnarray}&&V(r)=-{{ar}}^{-\tfrac{3}{4}}+{{br}}^{\tfrac{3}{4}}+c.\end{eqnarray} \tag{ 34 }$$

The study by Lichtenberg restricted only to the bottomonium case because it is the least relativistic case among quarkonium. Because a study by Jacobs et al suggests relativistic corrections in bottomonium are minimal [146]. They have included higher energy levels assuming that the higher energy levels can give more knowledge about the long-range part of the potential. However, the decay rates were not considered for the study. They have done a comparative study using different potentials, including Indiana potential (which will be discussed later), Martin potential, Cornell potential, Song–Lin potential, and Turin potential (the name given in [136]). Turin potential was applied for the first time for the quarkonium system. They have assumed that the bottomonium interaction in these static potentials depends only on the distance between the particles (r). The energy levels for bottomonium were calculated using all these potentials. It fitted equally well for Cornell, Song–Lin, and Turin potential when b quark mass varied appropriately and with the vanishing constant term c. They also compared the energy level differences and spin-averaged energy levels for these potentials. They have obtained the χ² value by fitting the energy differences.

It is a very important potential in quantum mechanics because it is one of the few quantum mechanical systems for which there is a precise analytical solution. A harmonic oscillator potential takes the form,

$$\begin{eqnarray}&&V(r)={{kr}}^{2},\end{eqnarray} \tag{ 35 }$$

where k is force constant. Harmonic oscillator potential is an important term in many of the phenomenological potentials like Killingbeck potential.

Mansour and Gamal chose harmonic oscillator potential with linear and Yukawa potentials to study the mass spectra of quarkonium systems $c\bar{c}$, $b\bar{b}$ and $\bar{b}c$ using Nikiforov–Uvarov method [147]. They have also added the relativistic corrections with spin–spin, spin–orbit, and tensor interactions. The exponential term was expanded by Taylor series up to second order. Their results were in good agreement with experimental data.

This type potential has the form,

$$\begin{eqnarray}&&V(r)={{ar}}^{2}+{br}-\displaystyle \frac{c}{r},\end{eqnarray} \tag{ 36 }$$

and the potential is known as extended Cornell potential when an inverse quadratic term is added to it.

Abu-Shady et al studied the heavy meson system using the extended Cornell potential [148]. Here the N-dimensional Schroedinger equation was analytically solved by the Nikiforov–Uvarov method for N = 3. The obtained results were applied to $c\bar{c}$, $b\bar{b}$, $b\bar{c}$, and $c\bar{s}$ system to get the mass spectra. The parameters in the model were chosen to fit the experimental data. The energy eigenfunctions and eigenvalues were also obtained in higher dimensional space. For N = 3, heavy meson masses were obtained. The reported results showed well agreement with experimental result. In the case of $b\bar{c}$ mesons, enough experimental data was not available. In the case of $c\bar{s}$ mesons 1S, 2S, and 1D states were found close to the experimental value.

Omugbe et al have also studied the non-symmetric extended Cornell potential to get the mass spectra of $b\bar{b}$, $c\bar{c}$, $b\bar{c}$, and $c\bar{s}$ system [149]. The addition of the harmonic oscillator potential and inverse quadratic potential modifies the behavior when r → 0. The problem was solved using the WKB framework. The findings of this study were consistent with those of other analytical techniques and published experimental data.

Salehi investigated ground and excited states of some baryons including N, Δ, Σ, Ξ, Ω using the Killingbeck plus isotonic oscillator potential of the form [150],

$$\begin{eqnarray}&&V(r)={{ar}}^{2}+{br}+\displaystyle \frac{c}{r}+\displaystyle \frac{d}{{r}^{2}}+\displaystyle \frac{{hr}}{{r}^{2}+1}+\displaystyle \frac{{{kr}}^{2}}{{\left({r}^{2}+1\right)}^{2}}.\end{eqnarray} \tag{ 37 }$$

The Schroedinger equation was solved numerically to get energy eigenvalues. To obtain the baryon energies and determine the baryon masses, the values were fitted using the parameters of the generalised Gursey-Radicati mass formula. The potential model agreed well with the spectrum of octets and decuplets.

Mansour and Gamal used the polynomial potential to obtain the mass spectra $c\bar{c}$, $b\bar{b}$, and B_c mesons [151]. The mathematical form of the potential is,

$$\begin{eqnarray}&&V(r)=\displaystyle \sum _{m=0}^{m}{A}_{m-2}{r}^{m-2},m=0,1,2...\end{eqnarray} \tag{ 38 }$$

A special case of this potential used in the study was,

$$\begin{eqnarray}&&V(r)=\displaystyle \frac{b}{r}+{ar}+{{dr}}^{2}+{{pr}}^{4}.\end{eqnarray} \tag{ 39 }$$

The Schroedinger equation was solved using Nikiforov–Uvarov method to get the energy eigenstates. Here the first term corresponds to Coulomb potential. The term linear in r represents that V(r) is continuously growing as r → ∞  and leads to quark confinement. The third and fourth terms are harmonic and anharmonic terms responsible for quark confinement. The mass spectra for these three states were studied. The study also showed that the harmonic term gives more accuracy to results compared to other potentials.

Another potential form mainly used to study heavy quarkonia is the Kratzer potential. It is an extensively used potential to study molecular structure and interactions. The form of the potential is [152],

$$\begin{eqnarray}&&V(r)=\displaystyle \frac{a}{r}+\displaystyle \frac{b}{{r}^{2}}.\end{eqnarray} \tag{ 40 }$$

In molecular physics this potential is often written as [153],

$$\begin{eqnarray}&&V(r)=-2{D}_{e}\left(\displaystyle \frac{a}{r}-\displaystyle \frac{{a}^{2}}{2{r}^{2}}\right),\end{eqnarray} \tag{ 41 }$$

where a is the internuclear separation and D_e is the dissociation energy. The study by Bayrak et al presented a method for calculating the solution for non-zero angular momentum state by Kratzer potential using the asymptotic iteration method [154]. Kratzer potential can be combined with other potentials to solve the mass spectra of heavy quarkonia.

Inyang et al used Kratzer potential mixed with the screened Coulomb potential to solve the mass spectra of charmonium and bottomonium states [155]. The series expansion method was used to get the solution. The model parameters for charmonium were calculated by solving the two algebraic equations using the experimental result for the 2S and 2P states. Similarly, experimental value of 1S and 2S states was used for bottomonium. They have applied their results to calculate the masses for 1S, 2S, 1P, 2P, 3S, 4S, 1D, and 2D states. The findings were quite well in agreement with those of the experiments and other theoretical investigations.

Moazami, Hassanabadi, and Zarrinkamar gave a non-relativistic potential to get the mass spectrum of heavy–light mesons [156]. The proposed potential takes the form

$$\begin{eqnarray}&&V(r)=\displaystyle \frac{a}{r}+\displaystyle \frac{b}{{r}^{2}}+{k}_{0}{e}^{-\tfrac{{\alpha }^{2}{r}^{2}}{2}}+{cr},\end{eqnarray} \tag{ 42 }$$

where a, b, c, k₀, and α are constants. The model studied the S and P states of B, B_s , D, and D_s mesons. They have solved the Schroedinger equation by considering the inverse square term and Gaussian term as perturbation. The unperturbed part was solved using the Nikiforov–Uvarov method. They have obtained mass spectrum, decay constants, leptonic decay width, and semileptonic decay width for these mesons. The value of mass obtained was compared with other models and most of the values were in good agreement.

This is a QCD motivated potential. The potential shows perturbative QCD characteristics at a close range and linear confinement characteristics at a far range [157]. The potential has the form,

$$\begin{eqnarray}&&{V}_{W}(r)={V}_{I}(r)+{V}_{s}(r)+{V}_{L}(r),\end{eqnarray} \tag{ 43 }$$

where V_s (r) is a short-range potential which is regularized two-loop perturbative potential. V_I (r) is an intermediate potential, which has the form ${V}_{I}(r)=r({c}_{1}+{c}_{2}r){e}^{-\tfrac{r}{{r}_{0}}}$ vanishing for small and large quark separations. V_L (r) = ar is a long-range potential, which represents quark confinement.

Jacobs, Olsson, and Suchyta proposed a method to get the solution of the Schroedinger and spinless Salpeter equations with QCD inspired Cornell and Wisconsin potential [146]. The potential parameters and quark masses were varied to get good agreement with the experimental data for both Schroedinger and spinless Salpeter equations. They have obtained the charmonium and bottomonium energy levels. The ratios of charmonium and bottomonium energy levels, which were already satisfactory and found to be slightly improved by using relativistic kinetic energy and wave function corrections. The Wisconsin potential showed good results compared to the Cornell potential.

There are different forms of exponential type potential. Exponential potentials are important in nuclear physics, including the Woods–Saxon (WS) potential, the generalised WS potential [158], and the Yukawa potential [159]. Yukawa proposed this potential to study the interaction between nucleons. The form of Yukawa potential is,

$$\begin{eqnarray}&&V(r)=-{V}_{0}\displaystyle \frac{{e}^{-\alpha r}}{r},\end{eqnarray} \tag{ 44 }$$

where α is the screening parameter. This potential was mainly used to get the bound state normalization and energy levels of neutral atoms. Napsuciale and Rodriguez presented an analytical solution to the quantum Yukawa potential [160].

Yukawa potential was combined with linear or other potentials forms and solved for the mass spectra of quarkonia (already discussed in section 3.1.12) [155].

This type potential has long been used in molecular and nuclear physics to look at the anharmonicities of the vibrational spectra [161]. The Morse barrier potential takes the form,

$$\begin{eqnarray}&&V(r)={V}_{0}\left[2{e}^{\tfrac{r}{a}}-{e}^{\tfrac{2r}{a}}\right].\end{eqnarray} \tag{ 45 }$$

After a finite distance, the potential gives an asymptotically diverging attraction to the outgoing particle while offering a repulsion to an approaching particle at r < 0.

Jamel studied the heavy quarkonia properties using the trigonometric Rosen–Morse potential [162]. They have considered heavy quarkonia as a system confined in a hard wall potential formed by combining a cotangent and squared cosecant function. The potential of this combination is trigonometric Rosen–Morse potential. The potential was used to examine the state of conformal symmetry in the heavy flavor sector. They have obtained the energy eigenvalues and eigenfunctions with the help of Nikiforov–Uvarov method. These results have been applied to $c\bar{c}$ and $b\bar{b}$ quarkonia to obtain the mass spectra and root mean square radii. The results showed satisfactory agreement with the available experimental and theoretical results.

It is one of the short-range potentials in physics [163]. This potential is a modified form of the Eckart potential [164], which has been vastly applied in physics and whose bound state and scattering properties have been studied using different methods. The potential has the form,

$$\begin{eqnarray}&&V(r)=-{V}_{0}\displaystyle \frac{{e}^{-\tfrac{r}{a}}}{1-{e}^{-\tfrac{r}{a}}},\end{eqnarray} \tag{ 46 }$$

where V₀ = Ze² and a is a constant parameter. The Hulthen potential acts like a screened Coulomb potential in short ranges and declines exponentially at large ranges, therefore its bound state capacity is lower than the Coulomb potential.

Akpan et al presented the approximate solutions of the Schroedinger equation with Hulthen-Hellmann potentials for quarkonium systems [165]. The equation was solved by Nikiforov–Uvarov method. The wave functions were obtained in the form of Laguerre polynomials. The study was able to obtain the mass of charmonium and bottomonium states. Quarks were considered to be spinless. The result provides good agreement with the experimental studies and other theoretical studies. A plot of mass spectra with different potential parameters was also presented.

This potential was used for the calculation of $c\bar{c}$ and $b\bar{b}$ mesons spectra [166, 167]. The potential has the form,

$$\begin{eqnarray}&&\bar{V}(r)=\left(\bar{\sigma }r-\displaystyle \frac{4{\bar{\alpha }}_{s}}{3r}\right)\left(\displaystyle \frac{1-{e}^{-\mu r}}{\mu r}\right),\end{eqnarray} \tag{ 47 }$$

where μ is the screening parameter. The $\bar{\sigma }$ is provided to set it apart from non-screening case. The potential will behave like a Coulomb potential at r → 0 and at r → ∞ , it will be $\bar{\sigma }/\mu$. The form of the potential is suggested by quenched lattice QCD calculations. The confining part of the potential has the form,

$$\begin{eqnarray}&&\bar{V}{\left(r\right)}_{\mathrm{conf}}=\frac{\bar{\sigma }}{\mu }-{\bar{\sigma }}_{r}\frac{{e}^{-\mu r}}{r}.\end{eqnarray} \tag{ 48 }$$

They obtained the masses for low-lying hadrons. The model provided a rather accurate value for the $b\bar{b}$ leptonic decay width, masses, and radiative decays.

Quigg and Rosner discussed the potential of the form [168],

$$\begin{eqnarray}&&V(r)=C\mathrm{ln}\displaystyle \frac{r}{{r}_{0}},\end{eqnarray} \tag{ 49 }$$

with strength C ∼ 3/4 GeV. They showed that quarkonium level spacing becomes independent of quark mass in the non-relativistic limit. They presented features of this potential when it is applied to describe some properties of heavy quarkonium states like energy level spacing and leptonic decay width. They have found that the charmonium system has a denser spectrum than the modified Coulomb potential. They observed that a 4S charmonium level close to 4.25 GeV was essential for the quark–quark interaction for logarithmic potential.

Machacek and Tomozova discussed the energy spectra and leptonic decay width of the Ψ family with the help of either fractional power or logarithmic functions [169]. They used the following potentials,

$$\begin{eqnarray}&&V={{Ar}}^{0.1}+B,\end{eqnarray} \tag{ 50 }$$

$$\begin{eqnarray}&&V=A\mathrm{ln}r+B,\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&V=A\mathrm{ln}(1+r)+B,\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&V=A{\left[\mathrm{ln}(1+r)\right]}^{1/2}+B,\end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}&&V=A{\left[\mathrm{ln}(1+0.2{r}^{2})\right]}^{1/2}+B,\end{eqnarray} \tag{ 54 }$$

where A and B were adjustable parameters to produce a good experimental fit. The energy spectrum and leptonic decay widths were obtained. Results showed good agreement with the experimental results. However, the third potential (equation (40) (exhibited more agreement compared to other potentials. For second and third potential, introducing an additional Coulomb term caused the effect of reducing effective quark mass for good results.

Authors suggested a new potential for the bound states of heavy quarkonium [170]. The idea of new potential came from the thought that neither a solely Coulombic nor a simply linear potential would be sufficient. The suggested potential takes the form

$$\begin{eqnarray}V(r)=\left\{\begin{array}{ll}\displaystyle \frac{-4{\alpha }_{s}}{3r} & r\leqslant {r}_{1}\\ b\mathrm{log}\displaystyle \frac{r}{{r}_{0}} & {r}_{1}\leqslant r\leqslant {r}_{2}.\\ \displaystyle \frac{r}{{a}^{2}} & r\geqslant {r}_{2}\end{array}\right.\end{eqnarray} \tag{ 55 }$$

This potential has a logarithmic term that interpolates between a linear portion that confines at long distances and a Coulomb term that is asymptotically free at short distances. The study by Quigg and Rosner showed that a logarithmic potential for the $Q\bar{Q}$ could give mass splitting independent of quark mass m_Q [168]. Also, the asymptotic freedom of QCD satisfies the Coulombic and linear terms. The independence of 1S−2S splitting on m_Q for $Q\bar{Q}$ suggests a logarithmic part in between. When the three combinations of potentials were selected, the number of model parameters was also increased. They have found that the logarithmic part of this potential has a significant role in finding the properties of J/Ψ. The potential was applied to get the leptonic decay width for the J/Ψ and ϒ family. The results gave excellent agreement on the Ψ spectrum and leptonic decay width. The model could not test for the ϒ leptonic decay width due to the lack of experimental data during that time.

Lichtenberg and Wills studied mass spectra of Ψ, ϒ and ζ (bound state of $t\bar{t}$) family using a quark–antiquark potential named Indiana potential [171]. The potential has the form,

$$\begin{eqnarray}&&V=\displaystyle \frac{4{\alpha }_{0}{\left(1-\tfrac{r}{{r}_{0}}\right)}^{2}}{3r\mathrm{ln}\tfrac{r}{{r}_{0}}}+c,\end{eqnarray} \tag{ 56 }$$

where α₀ and r₀ are constants. This is a QCD motivated potential. The need for a logarithmic potential is already discussed by Quigg and Rosner [168] and Machacek and Tomozova [169]. Indiana potential behaves like a weakened logarithmic potential, $\tfrac{1}{r\mathrm{ln}\tfrac{r}{{r}_{0}}}$ at small distances. At large distances, the potential behaves like $\tfrac{r}{{r}_{0}^{2}\mathrm{ln}r}$ and the weakening of potential is comparable to linear potential. At r = r₀ the potential is vanishing. It has one more interesting property, $V\left(\tfrac{{r}_{0}^{2}}{r}\right)=-V(r)$, implying that the behavior at large and small distances cannot be adjusted separately. Spin dependence terms were also added to the potential. The wave function was also obtained to get the leptonic decay widths of vector mesons in this family using Van Royen and Weisskopf formula [172]. The results were compared with the experimental data for Ψ and ϒ. The leptonic decay width of Ψ showed a decrease with an increase in mass, which did not agree with the experimental results.

They developed a potential to get s-wave meson masses [173],

$$\begin{eqnarray}&&V(r)={V}_{{AF}}(r)+{V}_{\mathrm{INT}}(r)+{V}_{S}(r),\end{eqnarray} \tag{ 57 }$$

${V}_{{AF}}(r)=\left[\tfrac{-16\pi }{27}\tfrac{1}{\mathrm{ln}\left(\tfrac{1}{{r}^{2}{{\rm{\Lambda }}}^{2}{e}^{2\gamma }}\right)}+O\left(\tfrac{1}{{\mathrm{ln}}^{3}\left(\tfrac{1}{{r}^{2}{{\rm{\Lambda }}}^{2}{e}^{2\gamma }}\right)}\right)\right]\tfrac{1}{r}$, where γ is Euler-Mascheroni constant. When $r\ll \tfrac{1}{{\rm{\Lambda }}}$, they expect V_AF (r) to dominate. They have also taken into account that energy of the system raises linearly with the increase in separation, V_s (r) = kr. ${V}_{\mathrm{INT}}(r)={V}_{0}-{{kre}}^{-{ar}}$ was taken as a negligible function compared to the modified Coulombic interaction at short distances and negligible compared to the linear potential at large distances. The hyperfine splitting was taken into account. The parameters in the potential were a, V₀, and quark masses. A fit to the s-wave mass spectrum was done by choosing an appropriate quark mass. They demanded that their explanation applies to both light and heavy quark bound states in response to qualitative successes of QCD in describing the light hadron mass spectrum. When they used their model to predict quarks heavier than the charmed quark, the results were qualitatively similar but it was different in detail from those of Eichten and Gottfried's earlier work [174].

Authors developed a non-singular potential model to investigate the quarkonium spectra [175]. They have used the semi-relativistic Hamiltonian of the form,

$$\begin{eqnarray}&&H=2{\left({m}^{2}+{p}^{2}\right)}^{1/2}+{V}_{p}(r)+{V}_{c}(r),\end{eqnarray} \tag{ 58 }$$

where V_p and V_c are the perturbative and confining potentials. For confining potential they made use of a mixed scalar and vector exchange potential. The form of V_c is given by,

$$\begin{eqnarray}&&{V}_{c}={Ar}+\displaystyle \frac{{C}_{1}}{{m}^{2}r}(1-{e}^{-2{mr}}){S}_{1}\cdot {S}_{2}+\displaystyle \frac{{C}_{2}}{{m}^{2}r}\left(1-\displaystyle \frac{1}{2}{f}_{1}\right)L\cdot S+\displaystyle \frac{{C}_{3}}{{m}^{2}r}\left(1-\displaystyle \frac{3}{2}{f}_{2}\right){S}_{12},\end{eqnarray} \tag{ 59 }$$

here C₁, C₂ and C₃ are arbitrary constants. They have obtained the energy levels, leptonic decay widths, and E₁ transition width. The values agreed well with experimental data of $b\bar{b}$ and reasonably good for $c\bar{c}$.

Richardson proposed a potential incorporating the asymptotic freedom and linear quark confinement of QCD [176]. The potential generates the spectrum of the triplet $c\bar{c}$ and the triplet $b\bar{b}$. The form of the potential is,

$$\begin{eqnarray}&&\tilde{V}({q}^{2})=-\displaystyle \frac{4}{3}\displaystyle \frac{12\pi }{33-2{n}_{f}}\displaystyle \frac{1}{{q}^{2}}\displaystyle \frac{1}{\mathrm{ln}\left(1+\tfrac{{q}^{2}}{{{\rm{\Lambda }}}^{2}}\right)},\end{eqnarray} \tag{ 60 }$$

where the only parameters in the model are scale size Λ and the quark masses. The value of n_f was chosen as three, assuming the effect of heavy quark will be negligible at a distance they were studying. Here the Fourier transform for the coordinate space potential V(r) was taken by considering the one gluon exchange amplitude, which is proportional to $\tilde{V}({q}^{2})$. They have not considered spin-dependent effects on the potential. For ϒ and Ψ systems, experimental results showed a reasonably good agreement with the model.

Bagchi et al used Richardson potential to study energies and magnetic moments of Ω⁻ and Δ⁺⁺ [177]. They have modified the potential with a new set of scale parameter values for asymptotic freedom and confinement. Moreover, they expect this potential can be a good base for studying the baryon properties.

Grunfeld and Rocca presented a relativistic confining potential using the Klein–Gordon oscillator to get the mass spectra of $c\bar{c}$ and $b\bar{b}$ [178]. The KG oscillator was introduced by Bruce and Minning [179]. This oscillator will behave like a harmonic oscillator (HO) in the non-relativistic limit. The two-body problem was solved to obtain the mass spectra. A KG equation takes the form [179],

$$\begin{eqnarray}&&-\displaystyle \frac{{\partial }^{2}}{\partial {t}^{2}}{\rm{\Psi }}({\bf{q}},t)=\left({{\bf{p}}}^{2}+{m}^{2}{\bf{q}}\cdot {\hat{{\rm{\Omega }}}}^{2}\cdot {\bf{q}}+m\hat{\gamma }({tr}{\rm{\Omega }})+{m}^{2}\right){\rm{\Psi }}({\bf{q}},t),\end{eqnarray} \tag{ 61 }$$

where $\hat{{\rm{\Omega }}}$ is a 3 × 3 matrix and ω_i is the oscillator frequency.

$$\begin{eqnarray}&&{\hat{{\rm{\Omega }}}}_{{ij}}={\omega }_{i}{\delta }_{{ij}}.\end{eqnarray} \tag{ 62 }$$

The quark mass and ω are the two free parameters. The quarks were considered as spinless. The results were compared with the Klein–Gordon equation with linear and quadratic potentials [180, 181]. They have also compared their values with a four-dimensional harmonic oscillator model in a quantum relativistic frame [182]. The results have shown a good agreement with the theoretical and experimental data.

The hyperfine structure of hadron spectroscopy involves spin-related interaction between quarks or quarks and antiquarks, which have a color factor. The color-magnetic interaction that results in the mass splitting for ordinary hadrons is caused by the one gluon exchange potential. The Hamiltonian of the color-magnetic interaction, often known as the chromomagnetic interaction (CMI) model is an efficient way to describe hadron masses after the quark mass is included [116].

There are many types of CMI Hamiltonian. The general structure of the Hamiltonian for the CMI model is,

$$\begin{eqnarray}&&H=\displaystyle \sum _{i}{m}_{i}-\displaystyle \sum _{i\lt j}{v}_{{ij}}{\lambda }_{i}\cdot {\lambda }_{j}{\sigma }_{i}\cdot {\sigma }_{j},\end{eqnarray} \tag{ 63 }$$

where m_i is the effective mass, v_ij is the coupling parameter, and λ_i is the Gell-Mann matrices. CMI models contain coupling coefficients and effective masses as parameters. In a simple CMI model the coupling constant and effective quark mass can be extracted from known hadrons.

In the case of doubly heavy baryons the experimental studies are very few but its theoretical studies are done extensively. Weng, Chen, and Deng studied the masses of doubly heavy and triply heavy baryons using the chromomagnetic model with color interaction [117]. In 2017 LHCb reported the doubly charm ${{\rm{\Xi }}}_{{cc}}^{++}$ state [118]. The calculated value of Ξ_cc by this study was close to the LHCb result. For getting the model parameters (m_qq and v_qq ) of baryons, experimental data of the light and singly heavy baryons were used. They had extracted thirteen model parameters. The study did not discuss the decay properties.

Guo et al studied mass spectra of multi-heavy baryons and S-wave doubly heavy tetraquark ${QQ}\bar{q}\bar{q}$ (Q = c, b, q = u, d, s) with J^P = 0⁺, 1⁺, and 2⁺ in the improved CMI (ICMI) model, which includes chromomagnetic and chromoelectric interactions [119]. The parameters (m_ij and v_ij ) were extracted by fitting it with the conventional hadron spectra. Their study included doubly and triply heavy baryons. The study also proposed the mass of Ξ_cc baryon. The study gave similar result compared with the study by Weng [117]. However, they were able to predict the masses of tetraquark states also. The predicted mass of ${cc}\bar{u}\bar{d}$ tetraquark state agrees with the LHCb results. The mass of tetraquark states ${bb}\bar{n}\bar{s}$, ${bb}\bar{n}\bar{n}$, ${bc}\bar{n}\bar{n}$, ${bb}\bar{s}\bar{s}$, ${bc}\bar{s}\bar{s}$ , and ${bc}\bar{n}\bar{s}$ (n = u, d) were also predicted.

Chen et al used the simple chromomagnetic interaction model to study the triply heavy (${QQ}\bar{Q}\bar{q}$) tetraquark states [120]. They have used a diquark–antidiquark $[({QQ})(\bar{Q}\bar{q})]$ approach and a triquark–antiquark $[({QQ}\bar{Q})(\bar{q})]$ approach. Both methods gave same results. However, this approach could not predict the masses accurately. The reason comes from the effective coupling constant. Therefore, they suggested an improved model with color-Coulomb term, kinetic term, and confinement term instead of this simple model. They also predicted the decay properties of the states.

Heavy flavor pentaquark states are also studied using CMI models with chromomagnetic and chromoelectric contributions. An et al studied the mass spectra of heavy pentaquarks with four heavy quarks (${QQQQ}\bar{q}$) [121]. They calculated the relative partial decay width of ${cccc}\bar{q}$ and ${bbbb}\bar{q}$ pentaquark states. However, this type of pentaquark is not identified by any experiment till now. Therefore, more investigation on this type of pentaquarks states is necessary to identify its exotic nature and other properties. An et al extended their study to the fully heavy pentaquark states using the same formalism [122]. After the systematic calculation of the CMI Hamiltonian, mass spectra of ${QQQQ}\bar{Q}$ were calculated.

They proposed a non-relativistic potential for mesons [183]. The meson spectra were generated initially to get the ground state masses of baryons. It was assumed that the strength of qq interaction was half of $q\bar{q}$ interaction. The potential was of the form,

$$\begin{eqnarray}&&{V}_{{ij}}^{q\bar{q}}(r)=\displaystyle \frac{-\kappa }{r}+\lambda r-{\rm{\Lambda }}+\displaystyle \frac{\kappa }{{m}_{i}{m}_{j}}\displaystyle \frac{{\exp }^{\tfrac{-r}{{r}_{0}}}}{{{rr}}_{0}^{2}}{\sigma }_{i}\cdot {\sigma }_{j}.\end{eqnarray} \tag{ 64 }$$

Spin–orbit and tensor terms were neglected. They calculated the ground state baryon mass with S = 1/2 and S = 3/2. Later Brac used this potential for baryons with more than one heavy quark [184]. Faddeev equations were used for solving the three-body problem. Calculations were performed for static parameters, like wave functions at the origin and mass, charge, and magnetic radii. A harmonic oscillator basis with states up to 8 quanta is used to calculate the spectrum for each baryon.

The author presented a phenomenological non-relativistic quark–antiquark potential in the center of mass system. The potential has the form [185],

$$\begin{eqnarray}&&V={V}_{c}(r)+{V}_{d}(r)({\sigma }_{i}\cdot {\sigma }_{j})+{V}_{f}(r)(L\cdot S)+{V}_{t}(r)\left(\displaystyle \frac{3({\sigma }_{i}\cdot r)({\sigma }_{j}\cdot r)}{{r}^{2}}-{\sigma }_{i}\cdot {\sigma }_{j}\right),\end{eqnarray} \tag{ 65 }$$

where V_c (r) is a spherically symmetric infinite potential hole of radius a,

$$\begin{eqnarray}V(c)=\left\{\begin{array}{cc}\infty & r\gt a\\ 0 & r\lt a\end{array}\right.,\end{eqnarray} \tag{ 66 }$$

here the spin-dependent terms were considered as perturbation. The energy eigenvalues were given by the zero point Bessel function. They have obtained the energy eigenvalues for L = 0 and L = 1 states. The model was also able to predict the D wave boson masses. They have also observed that for a = 1/1.33m_π and M_q = 5 GeV/c² many other physical properties of mesons can be derived.

Weinstein and Isgur considered a non-relativistic potential model for tetraquarks [186]. This system was already studied with the help of the bag model and other relativistic potential models. Those studies concluded that dense discrete spectra exist for these states. Apart from previous potential model studies which were not considering the long-range color mixing effects, color confinement forces, and hyperfine interactions were considered here. They have solved the four-particle Schroedinger equation. The Hamiltonian was considered as,

$$\begin{eqnarray}&&H=\displaystyle \sum _{i=1}^{4}\left[{m}_{i}+\frac{{p}_{i}^{2}}{2{m}_{i}}\right]+\displaystyle \sum _{i\lt j}\left[{H}_{\mathrm{conf}}^{{ij}}+{H}_{\mathrm{hyp}}^{{ij}}\right],\end{eqnarray} \tag{ 67 }$$

where H_conf is harmonic confinement potential and H_hyp is color hyperfine interaction. The Hamiltonian did not consider the anharmonicities, the effect of possible $q\bar{q}$ annihilations via gluons, and some relatively small spin–orbit and tensor effects. The model did not give evidence for any denser discrete spectrum for the states. The model confirmed that light ${qq}\bar{q}\bar{q}$ states can exist with 0⁺⁺. Also, it allowed the existence of meson–meson-bound states like the nucleon–nucleon interaction of deuteron.