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Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation

  • Arezu Jahanshir EMAIL logo , Ekwevugbe Omugbe , Joseph Ngene Aniezi , Ifeanyi Jude Njoku , Clement Atachegbe Onate , Edwin Samson Eyube , Samuel Olugbade Ogundeji , Chinonso Mbamara , Raphael Mmaduka Obodo and Michael Chukwudi Onyeaju
Published/Copyright: April 11, 2024
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Open Physics
From the journal Open Physics Volume 22 Issue 1

Abstract

The energy bound-state solutions of the spinless Salpeter equation (SSE) have been obtained under a spin-dependent Cornell potential function via the Wentzel–Kramers–Brillouin approximation. The energy levels were applied to predict the mass spectra for the charmonium, bottomonium, and bottom-charmed mesons. The relativistic corrections for the angular momentum quantum number l > 0 , total angular momentum quantum numbers j = l , j = l ± 1 , and the radial quantum numbers n = 1–4 improve the mass spectra. The results agree fairly with experimental data and theoretic results reported in existing works, where the authors utilized different forms of the inter-quark potentials and methods. The deviation of the obtained masses for the charmonium and bottomonium from the observed data yields a total percentage error of 3.32 and 1.11%, respectively. The results indicate that the accuracy of the masses is correlated with the magnitude of masses for the charm and bottom quarks. The SSE together with the phenomenological spin-dependent Cornell potential provides an adequate account of the mass spectroscopy for the heavy mesons and may be used to predict other spectroscopic parameters.

Keywords: meson spectroscopy; spin–orbit coupling; Cornell potential; Salpeter equation

1 Introduction

The discovery of the electron by Thomson [1] in 1897 and the nucleus by Rutherford [2] in 1911 gave a fairly complete picture of the atomic structure. However, decades after these discoveries, several other elementary particles have been discovered with the aid of modern equipment such as particle colliders, detectors, and accelerators. These elementary particles that are the composite of much smaller particles are referred to as quarks. Quarks form the building blocks of matter, and their dynamical existence has been confirmed via experimental works on deep inelastic scattering of electrons and neutrons [3]. The standard model of elementary particles forms the basis for the understanding of particle physics, where the quarks have six flavors, namely, the up ( u ), down ( d ), top ( t ), bottom ( b ), strange ( s ), and charm ( c ) quarks; six leptons such as the electron ( e ), muon ( μ ), tau ( τ ), and their respective neutrinos ( ν e , ν μ , ν τ ); the gauge; and the Higgs bosons. The quarks and leptons interact via the unified electroweak forces and the strong quantum chromo-dynamics (QCD) force. The strong QCD force is responsible for the binding of quarks into protons and neutrons, while the interaction between the leptons is mediated by the electromagnetic force. The weak nuclear force is associated with particle decay.

The bound states of three quarks give rise to the baryons, while the quark–antiquark pairs constitute the mesons. Both the baryons and mesons are grouped into hadrons. The hadrons with ½ integer spin are fermions and obey the Fermi-Dirac statistics, while bosons possess spin 0, 1 and obey the Bose–Einstein statistics. The properties of these elementary particles have garnered research interest among particle physicists in recent years. These properties have been observed using heavy machinery [4], and some were predicted theoretically before their experimental discovery. In this regard, the bound-state solutions of the wave equations under the inter-quark potentials have been utilized to predict the mass spectroscopy and decay properties of the elementary particles. The most utilized potential energy is the Cornell potential, which is the combination of the Coulomb’s energy and a linear function. The Coulomb’s energy is responsible for the short-range gluon exchange interaction between a quark and its antiquark, while the linear function is in charge of quark confinement. The addition of spin components to the Cornell potential allows for relativistic corrections and results in the hyperfine splitting between the s-wave singlet and triplet states. The multiple triplet splitting occurs for any angular momentum quantum number l > 0 .

Li et al. [5] predicted the charmonium ( c c ̅ ) mass spectra using the coupled-channel model and the screened potential model in the mass region below 4 GeV . Their results agreed with the masses of the c c ̅ meson obtained with a quenched potential model and literature data [6]. Mutuk [7] investigated the mass spectra and decay constants of vector and pseudoscalar heavy-light mesons within the framework of the QCD sum rule and the quark model. The numerical results were in good agreement compared to observed data and other theoretical works. The charmonia spectra have been investigated using the non-relativistic quark model and matrix-Numerov method [8,9]. Chaturvedi and Rai [10], in a recent work, investigated the electromagnetic transitions, mass spectroscopy, and decay rates of the bottom-charm ( b c ̅ ) meson within the context of the non-relativistic QCD. Several authors [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] have carried out extensive studies on the mesons bound-state solutions and applied them to obtain their spectroscopic parameters. The results in the references therein were compared to experimental data of the particle data group [29,30,31], and the results were predicted using different QCD-inspired potentials, phenomenological potentials, and theoretical methods.

In this work, we investigate the hyperfine mass spectra splitting of the heavy mesons within the framework of the semi-relativistic spinless Salpeter equation (SSE). Previously, the mass spectra of the heavy mesons have been obtained in previous studies [32,33] using the SSE without considering the spin components and relativistic corrections for the inter-quark potentials. The results in the references therein revealed that the semi-relativistic equation provides a satisfying account for the meson mass spectroscopy. Motivated by these facts, we report for the first time the approximate analytical and numerical mass spectra splitting of the heavy mesons under the SSE with a phenomenological spin-dependent Cornell potential via the Wentzel–Kramers–Brillouin (WKB) approximation.

In this study, we considered the interactions potential function given by

(1) V ( r ) = V c ( r ) + V SS ( r ) + V LS ( r ) + V T ( r ) ,

where V c ( r ) is the Cornell potential. The functions V SS ( r ) , V LS ( r ) , and V T ( r ) are the spin–spin, spin–orbit, and tensor channels, respectively. The respective potential functions are represented as [22,24]

(2) V c ( r ) = − 4 α s 3 r + br ,

(3) V SS ( r ) = 32 π α s 9 m q m q ̅ δ ̅ σ ( r ) 〈 S ̅ · S ̅ 〉 ,

(4) V LS ( r ) = 1 m q m q ̅ 2 α s r 3 − b 2 r 〈 L ̅ · S ̅ 〉 ,

(5) V T ( r ) = 4 α s m q m q ̅ r 3 〈 T ̅ 〉 ,

where α s , b , m q , and m q ̅ are the coupling constant, linear confinement parameter, the mass of quark, and its anti-quark, respectively. In (3), the Dirac delta function has been used as a Gaussian function ( δ ̅ σ ( r ) = ( σ / π ) 3 e − σ 2 r 2 ). The operators in Eqs. (3)–(5) are diagonal in | j , l , s 〉 with the respective spin–spin (( S ̅ · S ̅ )) spin–orbit ( L ̅ · S ̅ ) and tensor ( T ̅ ) matrix elements given by refs [22,24]

(6) 〈 S ̅ · S ̅ 〉 = s ( s + 1 ) 2 − 3 4 ,

(7) 〈 L ̅ · S ̅ 〉 = 1 2 ( j ( j + 1 ) − l ( l + 1 ) − s ( s + 1 ) ) ,

(8) 〈 n 3 l j | T ̅ | n 3 l j 〉 = − 6 ( 〈 L ̅ · S ̅ 〉 ) 2 + 3 〈 L ̅ · S ̅ 〉 − 2 s ( s + 1 ) l ( l + 1 ) 6 ( 2 l − 1 ) ( 2 l + 3 ) .

The notations s , l , and j = l + s denote the spin number, orbital quantum number, and the total angular momentum quantum number, respectively. The spin–spin coupling gives rise to the s-wave ( l = 0 ) hyperfine splitting between the triplet ( s = 1 ) and singlet ( s = 0 ) states. For l > 0 , and j = l ± 1 , j = l , we have the multiplets splitting for the p , d , f , g , and h triplets quantum states. The n represents the principal quantum number. The spin-dependent potentials in (4) and (5) give the mass shifts and are obtained from leading-order perturbation theory [14,24]. The operator 〈 T ̅ 〉 can be described by the non-vanishing diagonal matrix element for l > 0 and correspond to the spin triplet states [21].

2 Energy spectrum of the SSE with spin-dependent Cornell potential via the WKB approximation

To obtain an approximate analytical solution, we truncate the Gaussian function to a harmonic function for r ≪ 1 fm via a Taylor series expansion around r = 0 . The Gaussian function can be expressed as e − σ 2 r 2 ∼ 1 − σ 2 r 2 . In the femto-meter scale, this approximation is important for quark interactions [28].

Using this approximation, the potential in (1) can be simplified as

(9) V ( r ) = − Q r + br + N r 3 − P r 2 + Ω s ,

where

(10) Q = 4 α s 3 + b ( L ̅ · S ̅ ) 2 m q m q ̅ ,

(11) N = 2 α s ( L ̅ · S ̅ ) m q m q ̅ + 4 α s ( T ̅ ) m q m q ̅ ,

(12) P = σ 2 Ω s ,

(13) Ω s = 16 π α s 9 m q m q ̅ ( σ / π ) 3 ( s ( s + 1 ) − 1.5 ) .

The spinless SSE equation for describing a two-body system is given as [34,35]

(14) ∑ i = 1 , 2 − ∆ + m i 2 + V ( r ) − E nl ξ ( r , θ , φ ) = 0 , ∆ = ∇ 2 ,

where

(15) ξ ( r , θ , φ ) = ψ nl ( r ) Y lm ( θ , φ ) .

The notations ∇ 2 , V ( r ) , E nl , and ξ ( r , θ , φ ) represent the Laplacian, the potential function, total energy, and the total wave function, respectively.

For interaction between particles, the summation in Eq. (14) can further be expanded via Taylor series to order two:

(16) ∑ i = 1 , 2 − ∆ + m i 2 = m 1 + m 2 − ∆ 2 μ − ∆ 2 8 η 3 ,

where

μ = m 1 m 2 m 1 + m 2 , η = μ m 1 m 2 m 1 m 2 − 3 μ 2 1 / 3 .

Eq. (14) can further be reduced to a Schrödinger-like equation [35]

(17) − 1 2 μ d 2 d r 2 + V ( r ) − E nl − 1 2 m ∼ ( V ( r ) − E nl ) 2 + ( l + 1 / 2 ) 2 2 μ r 2 ψ nl ( r ) = 0 , ℏ = 1 , m ∼ = η 3 μ 2 .

Eq. (17) can be written as a momentum eigenvalue equation

(18) ( ( P ˆ ) 2 + p μ 2 ( r ) ) ψ nl ( r ) = 0 ,

where P ˆ is the radial momentum operator and p μ ( r ) is the meson momentum eigenvalue given as

(19) p μ ( r ) = 2 μ 1 2 m ∼ ( V ( r ) − E nl ) 2 − ( V ( r ) − E nl ) − ( l + 1 / 2 ) 2 2 μ r 2 .

To obtain the energy equation for the modeled potential, we employed the WKB energy quantization condition for two real turning points r 1 and r 2 via the integral equation:

(20) ∫ r 1 r 2 p μ ( r ) d r = π n + 1 2 , ℏ = 1 .

It is worth stating that we have added a Langer’s correction [36] to the centrifugal potential using the transformation l ( l + 1 ) → ( l + 1 / 2 ) 2 . This correction in the WKB approximation admits the exact energy eigenvalues for soluble potentials and ensures that the wave function is well behaved near the origin.

Inserting the momentum into the WKB quantization integral in (20) with the potential energy given by (1), we obtained

(21) 2 μ ∫ r 1 r 2 A r 6 − B r 4 + C r 3 + D r 2 − E r + Λ r 4 − F r 3 + G r 2 + Hr + K d r = π n + 1 2 ,

where

A = N 2 2 m ∼ , B = 2 QN 2 m ∼ , C = 2 N Ω s − 2 N E nl 2 m ∼ − N , D = Q 2 + 2 N b 2 m ∼ − L , E = 2 Q Ω s + 2 PN − 2 Q E nl 2 m ∼ − Q , F = 2 Pb 2 m ∼ , G = b 2 − 2 P Ω s + 2 P E nl 2 m ∼ + P , H = 2 b Ω s + 2 PQ − 2 b E nl 2 m ∼ − b , K = ( E nl − Ω s ) 2 − 2 Qb 2 m ∼ + E nl − Ω s , Λ = P 2 2 m ∼ , L = ( l + 1 / 2 ) 2 2 μ .

Using coordinate transformation q = 1 / r , Eq. (21) reduces to

(22) − 2 μ ∫ q 1 q 2 A q 2 − B + C q + D q 2 − E q 3 + Λ q 8 − F q 7 + G q 6 + H q 5 + K q 4 d q = π n + 1 2 .

To solve Eq. (22) analytically, the multiple turning points need to be reduced to two via a Pekeris-type approximation around q = 0 . Let q = y + δ with δ ( 1 / q ) assumed to be the characteristic distance between the quark and anti-quark pairs. The inverse power terms can be obtained using the Taylor series expansion to the second order:

(23) q − k = δ − k 1 + y δ − k ∼ δ − k − k δ − k − 1 y + k ( k + 1 ) 2 ! δ − k − 2 y 2 + O ( y 3 ) ,

where

q − 1 = 3 δ − 3 q δ 2 + 3 q 2 δ 3 , q − 2 = 6 δ 2 − 8 q δ 3 + 3 q 2 δ 4 , q − 3 = 10 δ 3 − 15 q δ 4 + 6 q 2 δ 5 , q − 4 = 15 δ 4 − 24 q δ 5 + 10 q 2 δ 6 , q − 5 = 21 δ 5 − 35 q δ 6 + 15 q 2 δ 7 , q − 6 = 28 δ 6 − 48 q δ 7 + 21 q 2 δ 8 , q − 7 = 36 δ 7 − 63 q δ 8 + 28 q 2 δ 9 , q − 8 = 45 δ 8 − 80 q δ 9 + 36 q 2 δ 10 .

Inserting the terms q − k (k = 1–8)) into (22) with algebraic simplifications gives

(24) − 2 μ W ∫ q 1 q 2 − q 2 + ℵ q − T d q = 2 μ W ∫ q 2 q 1 ( q 2 − q ) ( q − q 1 ) d q = π n + 1 2 ,

where

(25) ℵ = 1 W 63 F δ 8 − 80 Λ δ 9 − 48 G δ 7 − 35 H δ 6 − 24 K δ 5 + 15 E δ 4 − 8 D δ 3 − 3 C δ 2 ,

(26) T = 1 W 36 F δ 7 − 45 Λ δ 8 − 28 G δ 6 − 21 H δ 5 − 15 K δ 4 + 10 E δ 3 − 6 D δ 2 − 3 C δ + B ,

(27) W = 28 F δ 9 − 36 Λ δ 10 − 21 G δ 8 − 15 H δ 7 − 10 K δ 6 + 6 E δ 5 − 3 D δ 4 − 3 C δ 3 − A .

The turning points ( q 1 , q 2 ) in (24) are obtained by solving the − q 2 + ℵ q − T = 0 , which is a requirement in the WKB approximation for the momentum to vanish at the classical turning points:

(28) q 1 = ℵ 2 − 1 2 ℵ 2 − 4 T ,

(29) q 2 = ℵ 2 + 1 2 ℵ 2 − 4 T .

Solving (24), we obtained

(30) ∫ q 2 q 1 ( q 2 − q ) ( q − q 1 ) d q = π 8 ( q 1 − q 2 ) 2 .

Comparing Eqs. (24) and (30), the condition for the energy-level equation is obtained as

(31) ℵ 2 4 − T = 2 μ W n + 1 2 .

The mass spectra are obtained from the relation between the quark masses and the energy eigenvalue:

(32) M nl = m q + m q ̅ + E nl .

3 Numerical results and discussion

The energy bound-state solution of the SSE under a spin-dependent Cornell potential has been obtained via the semi-classical WKB approximation method. The potential parameters ( α s , b , δ , σ ) were obtained by fitting the obtained mass spectra with the corresponding experimental masses of the particle data group in Tables 2–5. We present the parameters in Table 1. The quantum states in the asterisk correspond to the points where the potential parameters are fitted using (32). To check for accuracy, the potential parameters were obtained for random quantum numbers for the s-wave singlet and triplet states and also the hyperfine triplet states ( l > 0 , j ≥ 0 ) where we choose the parameters that reproduce minimum total errors. Also, to obtain potential parameters for the bottom-charmed meson, we assumed a constant characteristic distance ( δ ) and spin-dependent constant ( σ ) due to the limited availability of experimental data. For the bottomonium and charmonium mesons, we solved four polynomial equations simultaneously with the help of Maple software, while two non-linear equations were solved for the bottom-charmed meson to obtain the parameters α s and b . The variation of the potential function with distance for the heavy mesons is plotted in Figure 1(a–c). The potential curves for the triplet and singlet states account for linear confinement and short-range gluon exchange between the quark–antiquark pairs. The Coulomb part of the potential in the absence of tensor and spin orbit components dominates at short distances, whereas the linear component is prominent at large distances.

Table 1

Calculated parameters of interaction potential function

Parameters Potential free parameters
Charmonium Bottomonium Bc Meson
Mass ( GeV ) m c = m c ̅ = 1.530 a m b = m b ̅ = 5.250 a m c = 1.530 , m b = 5.25 0
α s 30.1602 27.7862 41.1745
δ ( GeV ) 0.05846 0.0796 0.0796
b ( GeV 2 ) 0.1416 0.1702 0.2710
σ ( GeV ) ‒0.1294 0.1356 0.1356

Note: aWe choose the bottom and charm masses from the obtained range [29] 1.2 < m c < 1.8 GeV and 4.5 < m b < 5.4 GeV .

Figure 1 (a–c) Variation of potential function with distance: (a) charmonium, (b) bottomonium, and (c) bottom-charm mesons for spin numbers s = 0 s=0 and s = 1 s=1 .
Figure 1

(a–c) Variation of potential function with distance: (a) charmonium, (b) bottomonium, and (c) bottom-charm mesons for spin numbers s = 0 and s = 1 .

The total errors are obtained using the following formula:

(33) χ = 100 Z ∑ i = 1 Z M nl exp − M nl theo M nl exp ,

where Z , M nl exp , and M nl theo are the respective number of experimental data points, experimental masses, and theoretically obtained masses. In Table 2, the mass spectra of charmonium for the s-wave increase as the quantum number increases with the singlet states bounded below the triplet states. The J / ψ ( 1 3 S 1 ) and η c ( 1 1 S 0 ) are higher than the experimentally determined values. However, the other s-wave masses are in agreement with the experimental masses [30] and the masses obtained using other inter-quark potential functions and methods reported in the existing literature [11,16,18,22,24]. In comparison with experimental data, the masses χ c 2 ( 1 3 P 2 ) , χ c 0 ( 1 3 P 0 ) , ψ 1 ( 1 3 D 1 ) , and ψ 2 ( 2 3 D 2 ) with hyperfine splitting and relativistic corrections l > 0 , j = l , j = l ± 1 and s = 1 as shown in Tables 2 and 3 were found to be more accurate compared to the masses ( h c ( 1 1 P 1 ) , η c 2 ( 1 1 D 2 ) , and η c 2 ( 2 1 D 2 ) for the quantum states ( l > 0 , s = 0 , j = l ). The charmonium masses for the s, p , d , and f states increase with the increase in the radial quantum number and were found to deviate from the experimental masses by a total error of 3.32%.

Table 2

S and P-states mass spectrum of charmonium meson in GeV

State Present [24] [18] [16] [11] [22] Expt. [30]
n 2 s + 1 L j J PC
J / ψ ( 1 3 S 1 ) 1 ‒ ‒ 3.520 3.094 3.0413 3.096 3.126 3.0851 3.097 ± ( 6 × 10 ‒ 6 )
η c ( 1 1 S 0 ) 0 ‒ + 3.293 2.989 3.1404 2.981 3.033 2.9904 2.984 ± 0.0004
ψ ( 2 3 S 1 ) 1 ‒ ‒ 3.902 3.681 3.7017 3.685 3.701 3.6821 3.686 ± ( 6 × 10 ‒ 5 )
η c ( 2 1 S 0 ) * 0 ‒ + 3.638 3.602 3.6610 3.635 3.666 3.6465 3.638 ± ( 1.1 × 10 ‒ 3 )
ψ ( 3 3 S 1 ) 1 ‒ ‒ 4.194 4.129 4.0502 4.039 4.055 4.1002 4.039 ± 10 ‒ 3
η c ( 3 1 S 0 ) 0 ‒ + 3.895 4.058 4.1347 3.989 4.158 4.0719
ψ ( 4 3 S 1 ) * 1 ‒ ‒ 4.421 4.514 4.4185 4.427 4.415 4.4394 4.421 ± ( 4 × 10 ‒ 3 )
η c ( 4 1 S 0 ) 0 ‒ + 4.091 4.448 4.4136 4.401 4.415 4.4209
ψ ( 5 3 S 1 ) 1 ‒ ‒ 4.600 4.863 4.6591 4.837 4.585 4.63 ± ( 6 × 10 ‒ 3 )
η c ( 5 1 S 0 ) 0 ‒ + 4.242 4.799 4.6618 4.811 4.607
ψ ( 6 3 S 1 ) 1 ‒ ‒ 5.804 5.185 4.8801 5.167 4.733
η c ( 6 1 S 0 ) 0 ‒ + 5.175 5.124 4.8825 5.155 4.754
χ c 1 ( 1 3 P 1 ) * 1 + + 3.511 3.468 3.5036 3.511 3.487 3.5004 3.511 ± ( 5 × 10 ‒ 5 )
χ c 2 ( 1 3 P 2 ) 2 + + 3.545 3.480 3.4888 3.555 3.522 3.5514 3.556 ± 7 × 10 ‒ 5
χ c 0 ( 1 3 P 0 ) 0 + + 3.466 3.428 3.4137 3.413 3.407 3.3519 3.415 ± 3 × 10 ‒ 4
h c ( 1 1 P 1 ) 1 + ‒ 3.298 3.470 3.5180 3.525 3.502 3.5146 3.525 ± ( 1.1 × 10 ‒ 4 )
χ c 1 ( 2 3 P 1 ) 1 + + 3.895 3.938 3.8072 3.906 3.786 3.9335 3.872 ± ( 6 × 10 ‒ 5 )
χ c 2 ( 2 3 P 2 ) * 2 + + 3.923 3.955 3.9151 3.949 3.905 3.9798 3.923 ± 1 × 10 ‒ 3
χ c 0 ( 2 3 P 0 ) 0 + + 3.856 3.897 3.7646 3.870 3.899 3.8357 3.922 ± 1.8 × 10 ‒ 3
h c ( 2 1 P 1 ) 1 + ‒ 3.642 3.943 3.8239 3.926 3.8210 3.9446
χ c 1 ( 3 3 P 1 ) 1 + + 4.188 4.338 4.1210 4.319 4.1230 4.3179 4.147 ± ( 3 × 10 ‒ 3 )
χ c 2 ( 3 3 P 2 ) 2 + + 4.212 4.358 4.1514 4.354 4.144 4.3834
χ c 0 ( 3 3 P 0 ) 0 + + 4.155 4.296 4.0804 4.301 4.120 4.2167
h c ( 3 1 P 1 ) 1 + ‒ 3.899 4.344 4.1368 4.337 4.1640 4.3339
χ c 1 ( 4 3 P 1 ) 1 + + 4.416 4.696 4.4005 4.728 4.3730 4.6203
χ c 2 ( 4 3 P 2 ) 2 + + 4.436 4.718 4.4298 4.763 4.411 4.7367
χ c 0 ( 4 3 P 0 ) 0 + + 4.388 4.653 4.3621 4.698 4.362 4.5518
h c ( 4 1 P 1 ) 1 + ‒ 4.094 4.704 4.1455 4.744 4.4200 4.6395
Table 3

D, F, and G-states mass spectrum of charmonium meson in GeV

State Present [24] [18] [16] [11] [22] Expt. [30]
n 2 s + 1 L j J PC
ψ 2 ( 1 3 D 2 ) 2 ‒ ‒ 3.520 3.772 3.46047 3.795 3.3480 3.8077
ψ 3 ( 1 3 D 3 ) 3 ‒ ‒ 3.575 3.755 3.51402 3.813 3.307 3.8146
ψ 1 ( 1 3 D 1 ) 1 ‒ ‒ 3.461 3.775 3.40228 3.783 3.374 3.7853 3.774 ± 4 × 10 ‒ 4
η c 2 ( 1 1 D 2 ) 2 ‒ + 3.306 3.765 3.47795 3.807 3.3760 3.8073
ψ 2 ( 2 3 D 2 ) 2 ‒ ‒ 3.902 4.188 3.81161 4.190 3.8010 4.1737 3.824 ± ( 5 × 10 ‒ 4 )
ψ 3 ( 2 3 D 3 ) 3 ‒ ‒ 3.949 4.176 3.86300 4.220 3.797 4.1829
ψ 1 ( 2 3 D 1 ) 1 ‒ ‒ 3.852 4.188 3.75606 4.150 3.800 4.1504
η c 2 ( 2 1 D 2 ) 2 ‒ + 3.649 4.182 3.82825 4.196 3.8360 4.1737
ψ 2 ( 3 3 D 2 ) 2 ‒ ‒ 4.195 4.557 4.12502 4.544 4.1350 4.5588
ψ 3 ( 3 3 D 3 ) 3 ‒ ‒ 4.234 4.549 4.17431 4.574 4.163 4.5725
ψ 1 ( 3 3 D 1 ) 1 ‒ ‒ 4.152 4.555 4.07208 4.507 4.113 4.5258
η c 2 ( 3 1 D 2 ) 2 ‒ + 3.905 4.553 4.14085 4.549 4.176 4.5597
χ c 3 ( 1 3 F 3 ) 3 + + 3.532 4.012 3.46746 4.068 3.375 4.0440
χ c 4 ( 1 3 F 4 ) 4 + + 3.609 4.036 3.54185 4.093 3.315 4.0374
χ c 2 ( 1 3 F 2 ) 2 + + 3.454 3.990 3.38961 4.041 3.403 4.0424
η c 3 ( 1 1 F 3 ) 3 + ‒ 3.319 4.017 3.48494 4.071 3.403 4.0411
χ c 3 ( 2 3 F 3 ) 3 + + 3.913 4.396 3.81815 4.400 3.823 4.3744
χ c 4 ( 2 3 F 4 ) 4 + + 3.978 4.415 3.88949 4.434 3.814 4.3711
χ c 2 ( 2 3 F 2 ) 2 + + 3.846 4.378 3.74376 4.361 3.812 4.3699
η c 3 ( 2 1 F 3 ) 3 + ‒ 3.660 4.400 3.83479 4.406 3.858 4.3723
ψ c 4 ( 1 3 G 4 ) 4 ‒ ‒ 3.549 4.343 4.2506
ψ c 5 ( 1 3 G 5 ) 5 ‒ ‒ 3.647 4.357 4.2369
ψ c 3 ( 1 3 G 3 ) 3 ‒ ‒ 3.450 4.321 4.2582
η c 4 ( 1 1 G 4 ) 4 ‒ + 3.336 4.345 4.2471
Total error ( χ ) 3.32% 1.98% 1.64% 1.30% 1.33% 1.53%

In Table 4, the bottomonium masses for the s and p -quantum states are presented. The low-lying quantum states masses are higher than the observed values. As the quantum number increases, the masses Y ( 5 3 S 1 ) and η b ( 6 1 S 0 ) were found to be in good agreement compared to other works in the existing literature [11,14,16,17,18] and the masses of the particle data group [30]. The p -states masses were found to agree with the ones obtained earlier. However, the masses for d quantum states presented in Table 5 fairly compare with the observed masses [30]. Generally, the masses increase with the increase in the radial quantum number. Using Eq. (33), the bottomonium mass spectra deviation from experimental data yields a total percentage error of approximately 1.11%, which indicates an improvement over the results in the study by Mansour and Gamal [18] and comparable to the total percentage error of 0.96% in the study by Mansour and Gamal [11] and 0.66% in the study by Soni et al. [24]. It can be seen that the percentage error is higher for the charmonium meson due to its light reduced mass. In Tables 6 and 7, the masses obtained for the bottom-charmed mesons increases with the increase in the radial quantum number and were found to be in good agreement with the results in previous studies [11,16,24]. It is important to state that the accuracy of the mass spectra is dependent on obtaining a good fit for the phenomenological potential function. This allows for the adjustment of the potential parameters such that the predictions correspond as good as possible to other theoretic works and observed data.

Table 4

S and P-states mass spectrum of bottomonium meson in GeV

State Present [17] [16] [18] [11] [14] [24] Expt. [30]
n 2 s + 1 L j J PC
Y ( 1 3 S 1 ) 1 ‒ ‒ 9.906 9.465 9.460 9.49081 9.525 9.4600 9.4600 9.460 ± ( 2.6 × 10 ‒ 4 )
η b ( 1 1 S 0 ) 0 ‒ + 9.916 9.402 9.398 9.43601 9.472 9.3900 9.4280 9.399 ± ( 2 × 10 ‒ 3 )
Y ( 2 3 S 1 ) 1 ‒ ‒ 10.240 10.003 10.023 10.01257 10.049 10.0150 9.9790 10.023 ± ( 3.1 × 10 ‒ 4 )
η b ( 2 1 S 0 ) 0 ‒ + 10.251 9.976 9.990 9.99146 10.028 9.9900 9.9550
Y ( 3 3 S 1 ) 1 ‒ ‒ 10.504 10.354 10.355 10.32775 10.371 10.3430 10.3590 10.355 ± ( 5 × 10 ‒ 4 )
η b ( 3 1 S 0 ) 0 ‒ + 10.517 10.336 10.329 10.1386 10.360 10.3260 10.3380
Y ( 4 3 S 1 ) 1 ‒ ‒ 10.715 10.635 10.586 10.5461 10.598 10.5970 10.6830 10.579 ± ( 1.2 × 10 ‒ 3 )
η b ( 4 1 S 0 ) 0 ‒ + 10.729 10.623 10.573 10.3236 10.592 10.5840 10.6630
Y ( 5 3 S 1 ) 1 ‒ ‒ 10.883 10.878 10.851 10.82628 10.870 10.8110 10.9750 10.885 ± (2.6 × 10‒3) ± (1.6 × 10‒3)
η b ( 5 1 S 0 ) 0 ‒ + 10.899 10.869 10.869 10.4977 10.790 10.8000 10.9560
Y ( 6 3 S 1 ) * 1 ‒ ‒ 11.020 11.102 11.061 10.97061 11.022 10.9970 11.2430 11.020 ± ( 4 × 10 ‒ 3 )
η b ( 6 1 S 0 ) 0 ‒ + 11.036 11.097 11.088 10.6615 10.961 10.9880 11.2260 11.014 [17]
χ b 1 ( 1 3 P 1 ) 1 + + 9.906 9.876 9.892 9.87371 9.875 9.9030 9.8190 9.893 ± ( 2.6 × 10 ‒ 4 ) ± ( 3.1 × 10 ‒ 4 )
χ b 2 ( 1 3 P 2 ) * 2 + + 9.912 9.897 9.912 9.89083 9.903 9.921 9.825 9.912 ± ( 2.6 × 10 ‒ 4 ) ± ( 3.1 × 10 ‒ 4 )
χ b 0 ( 1 3 P 0 ) 0 + + 9.898 9.847 9.859 9.8432 9.840 9.864 9.806 9.859 ± ( 4.2 × 10 ‒ 4 ) ± ( 3.1 × 10 ‒ 4 )
h b ( 1 1 P 1 ) 1 + ‒ 9.918 9.882 9.900 9.87919 9.884 9.9090 9.8210 9.899 ± ( 8 × 10 ‒ 4 )
χ b 1 ( 2 3 P 1 ) 1 + + 10.240 10.246 10.255 10.21695 10.229 10.249 10.2170 10.255 ± ( 2.2 × 10 ‒ 4 ) ± ( 5 × 10 ‒ 4 )
χ b 2 ( 2 3 P 2 ) 2 + + 10.245 10.261 10.268 10.22961 10.254 10.246 10.224 10.269 ± ( 2.2 × 10 ‒ 4 ) ± ( 5 × 10 ‒ 4 )
χ b 0 ( 2 3 P 0 ) * 0 + + 10.233 10.226 10.233 10.19625 10.202 10.220 10.205 10.233 ± ( 4 × 10 ‒ 4 ) ± ( 5 × 10 ‒ 4 )
h b ( 2 1 P 1 ) 1 + ‒ 10.254 10.250 10.260 10.22153 10.237 10.254 10.220 10.260 ± ( 1.2 × 10 ‒ 3 )
χ b 1 ( 3 3 P 1 ) 1 + + 10.504 10.538 10.541 10.1378 10.339 10.515 10.553 10.514 ± ( 7 × 10 ‒ 4 )
χ b 2 ( 3 3 P 2 ) 2 + + 10.509 10.550 10.550 10.1405 10.406 10.528 10.560 10.524 ± ( 8 × 10 ‒ 4 )
χ b 0 ( 3 3 P 0 ) 0 + + 10.498 10.522 10.521 10.1342 10.299 10.490 10.540 10.500 [17]
h b ( 3 1 P 1 ) * 1 + ‒ 10.519 10.541 10.544 4.14695 10.362 10.5190 10.556 10.519 [17]
χ b 1 ( 4 3 P 1 ) 1 + + 10.715 10.788 10.802 10.3229 10.571 10.853
χ b 2 ( 4 3 P 2 ) 2 + + 10.719 10.798 10.812 10.3255 10.637 10.860
χ b 0 ( 4 3 P 0 ) 0 + + 10.710 10.775 10.781 10.3193 10.532 10.840
h b ( 4 1 P 1 ) 1 + ‒ 10.731 10.790 10.804 10.3242 10.594 10.855
Table 5

D, F, and G-states mass spectrum of bottomonium meson in GeV

n 2 s + 1 L j J PC Present [17] [16] [18] [11] [14] [24] Expt. [30]
Y 2 ( 1 3 D 2 ) 2 ‒ ‒ 9.911 10.147 10.161 10.1126 10.096 10.153 10.075 10.164 ± ( 1.4 × 10 ‒ 3 )
Y 3 ( 1 3 D 3 ) 3 ‒ ‒ 9.921 10.155 10.166 9.73855 9.849 10.157 10.073 10.172 [17]
Y 1 ( 1 3 D 1 ) 1 ‒ ‒ 9.900 10.138 10.154 9.72905 9.666 10.146 10.074 10.155 [17]
η b 2 ( 1 1 D 2 ) 2 ‒ + 9.923 10.148 10.163 9.7355 9.767 10.153 10.074 10.165 [17]
Y 2 ( 2 3 D 2 ) 2 ‒ ‒ 10.244 10.449 10.443 9.94259 10.071 10.432 10.424
Y 3 ( 2 3 D 3 ) 3 ‒ ‒ 10.253 10.455 10.449 9.94704 10.175 10.436 10.423
Y 1 ( 2 3 D 1 ) 1 ‒ ‒ 10.235 10.441 10.435 9.93775 9.996 10.425 10.423
η b 2 ( 2 1 D 2 ) 2 ‒ + 10.258 10.450 10.445 9.94405 10.093 10.432 10.424
Y 2 ( 3 3 D 2 ) 2 ‒ ‒ 10.508 10.705 10.711 10.1391 10.345 10.733
Y 3 ( 3 3 D 3 ) 3 ‒ ‒ 10.516 10.711 10.717 10.1435 10.446 10.733
Y 1 ( 3 3 D 1 ) 1 ‒ ‒ 10.501 10.698 10.704 10.1344 10.272 10.731
η b 2 ( 3 1 D 2 ) 2 ‒ + 10.523 10.706 10.713 10.1405 10.368 10.733
χ b 3 ( 1 3 F 3 ) 3 + + 9.918 10.355 10.346 9.7361 9.754 10.340 10.287
χ b 4 ( 1 3 F 4 ) 4 + + 9.932 10.358 10.349 9.74242 9.896 10.340 10.291
χ b 2 ( 1 3 F 2 ) 2 + + 9.904 10.350 10.343 9.72948 9.642 10.338 10.283
η b 3 ( 1 1 F 3 ) 3 + ‒ 9.931 10.355 10.347 9.73759 9.778 10.339 10.288
χ b 3 ( 2 3 F 3 ) 3 + + 10.251 10.619 10.614 9.94462 10.081 10.607
χ b 4 ( 2 3 F 4 ) 4 + + 10.263 10.622 10.617 9.9508 10.219 10.609
χ b 2 ( 2 3 F 2 ) 2 + + 10.239 10.615 10.610 9.93815 9.971 10.604
η b 3 ( 2 1 F 3 ) 3 + ‒ 10.265 10.619 10.615 9.94608 10.104 10.607
Y 4 ( 1 3 G 4 ) 4 ‒ ‒ 9.929 10.531 10.512
Y 5 ( 1 3 G 5 ) 5 ‒ ‒ 9.946 10.532 10.514
Y 3 ( 1 3 G 3 ) 3 ‒ ‒ 9.911 10.529 10.511
η b 4 ( 1 1 G 4 ) 4 ‒ + 9.941 10.530 10.513
Total error ( χ ) 1.11% 0.20% 0.11% 3.85% 0.96% 0.13% 0.66%
Table 6

S and P states mass spectrum of bottom-charmed meson in GeV

State Present [16] [11] [24] Expt. [31]
n 2 s + 1 L j J PC
1 3 S 1 1 ‒ ‒ 6.225 6.333 6.313 6.321
( 1 1 S 0 ) * 0 ‒ + 6.275 6.272 6.276 6.272 6.275
2 3 S 1 1 ‒ ‒ 6.783 6.882 6.867 6.900
( 2 1 S 0 ) * 0 ‒ + 6.842 6.842 6.841 6.864 6.842
3 3 S 1 1 ‒ ‒ 7.216 7.258 7.308 7.338
3 1 S 0 0 ‒ + 7.283 7.226 7.281 7.306
4 3 S 1 1 ‒ ‒ 7.556 7.609 7.660 7.714
4 1 S 0 0 ‒ + 7.631 7.585 7.634 7.684
5 3 S 1 1 ‒ ‒ 7.826 7.947 7.941 8.054
5 1 S 0 0 ‒ + 7.907 7.928 7.917 8.025
6 3 S 1 1 ‒ ‒ 8.042 8.168 8.368
6 1 S 0 0 ‒ + 8.130 8.144 8.340
1 3 P 1 1 + + 6.218 6.743 6.281 6.705
1 3 P 2 2 + + 6.248 6.761 6.366 6.712
1 3 P 0 0 + + 6.178 6.699 6.223 6.686
1 1 P 1 1 + ‒ 6.280 6.750 6.290 6.706
2 3 P 1 1 + + 6.777 7.134 6.836 7.165
2 3 P 2 2 + + 6.803 7.157 6.917 7.173
2 3 P 0 0 + + 6.743 7.146 6.782 7.146
2 1 P 1 1 + ‒ 6.847 7.147 6.846 7.168
3 3 P 1 1 + + 7.211 7.500 7.278 7.555
3 3 P 2 2 + + 7.233 7.524 7.355 7.565
3 3 P 0 0 + + 7.182 7.474 7.227 7.536
3 1 P 1 1 + ‒ 7.287 7.510 7.287 7.559
4 3 P 1 1 + + 7.552 7.844 7.631 7.905
4 3 P 2 2 + + 7.571 7.867 7.704 7.915
4 3 P 0 0 + + 7.527 7.817 7.583 7.885
4 1 P 1 1 + ‒ 7.634 7.853 7.640 7.908
Table 7

D, F, and G-states mass spectrum of bottom-charmed meson in GeV

n 2 s + 1 L j J PC Present [16] [11] [24]
1 3 D 2 2 ‒ ‒ 6.228 7.025 6.299 6.997
1 3 D 3 3 ‒ ‒ 6.278 7.029 6.429 6.990
1 3 D 1 1 ‒ ‒ 6.177 7.021 6.200 6.998
1 1 D 2 2 ‒ + 6.291 7.026 6.308 6.994
2 3 D 2 2 ‒ ‒ 6.787 7.399 6.852 7.403
2 3 D 3 3 ‒ ‒ 6.829 7.405 6.975 7.399
2 3 D 1 1 ‒ ‒ 6.742 7.392 6.759 7.403
2 1 D 2 2 ‒ + 6.856 7.400 6.861 7.401
3 3 D 2 2 ‒ ‒ 7.220 7.741 7.29 7.764
3 3 D 3 3 ‒ ‒ 7.256 7.750 7.409 7.764
3 3 D 1 1 ‒ ‒ 7.181 7.732 7.205 7.762
3 1 D 2 2 ‒ + 7.296 7.743 7.302 7.762
1 3 F 3 3 + + 6.244 7.269 6.326 7.242
1 3 F 4 4 + + 6.312 7.277 6.501 7.244
1 3 F 2 2 + + 6.175 7.273 6.182 7.234
1 1 F 3 3 + ‒ 6.307 7.268 6.335 7.241
2 3 F 3 3 + + 6.801 7.616 6.876 7.615
2 3 F 4 4 + + 6.859 7.617 7.041 7.617
2 3 F 2 2 + + 6.740 7.618 6.741 7.607
2 1 F 3 3 + ‒ 6.870 7.615 6.885 7.614
1 3 G 4 4 ‒ ‒ 6.265 7.489
1 3 G 5 5 ‒ ‒ 6.353 7.482
1 3 G 3 3 ‒ ‒ 6.178 7.497
1 1 G 4 4 ‒ + 6.328 7.487

4 Conclusions

The energy bound-state solution of the SSE has been obtained under a spin-dependent Cornell potential energy function using the WKB approximation approach. The energy equation was then used to obtain the mass spectra for the heavy mesons. The mass spectra were found to be in good agreement with the ones obtained by other methods. The charmonium masses deviated from the experimental values with a total percentage error of 3.32%. Also, the bottomonium masses deviated by 1.11%, which indicates an improvement compared to the results in the study by Mansour Gamal [18] and comparable to the works reported in the existing literature [11,24]. The errors were found to reduce significantly for the bottomonium meson due to its heavier mass. The points at which the fitting was carried out may be responsible for the low accuracy of the masses for the low-lying quantum states. However, the SSE together with the phenomenological spin-dependent Cornell potential provides an adequate account of the mass spectroscopy for the heavy mesons and may be used to predict other spectroscopic parameters if the wave function can be obtained.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: The authors confirm contribution to this article as follows: study conception and design: Arezu Jahanshir, Ekwevugbe Omugbe; data collection: Ekwevugbe Omugbe, Joseph Ngene Aniezi, Ifeanyi Jude Njoku, Clement Atachegbe Onate; analysis and interpretation of results: Ekwevugbe Omugbe, Edwin Samson Eyube, Samuel Olugbade Ogundeji, Chinonso Mbamara; draft manuscript preparation: Arezu Jahanshir, Ekwevugbe Omugbe, Raphael Mmaduka Obodo, Michael Chukwudi Onyeaju. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2024-01-10
Revised: 2024-03-06
Accepted: 2024-03-07
Published Online: 2024-04-11

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2024-0004
Keywords for this article
meson spectroscopy; spin–orbit coupling; Cornell potential; Salpeter equation
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