# Nonrelativistic Arbitraryl-States of Quarkonium through Asymptotic Iteration Method

Authors: Hakan Ciftci; Hasan Fatih Kisoglu
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

The energy eigenvalues with anyl≠0 states and mass of heavy quark-antiquark system (quarkonium) are obtained by using Asymptotic Iteration Method in the view of nonrelativistic quantum chromodynamics, in which the quarks are considered as spinless for easiness and are bounded by Cornell potential. A semianalytical formula for energy eigenvalues and mass is achieved via the method in scope of the perturbation theory. The accuracy of this formula is checked by comparing the eigenvalues with the ones numerically obtained in this study and with exact ones in literature. Furthermore, semianalytical formula is applied to cc-, bb-, and cb- meson systems for comparing the masses with the experimental data.


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## Body

## 1. Introduction

Investigation of an atomic or subatomic system is done by achieving an energy spectrum of the system. This is carried out, generally, for the events in which the system is bounded by a potential function. Besides, the scattering states or resonance states can also be observed in the investigation of the system. The eigenvalues (or eigenenergies) of Hamiltonian of this system is obtained for a given potential function. In order to do this, various mathematical methods are used in quantum mechanics. One of these, named Asymptotic Iteration Method (AIM), has been commonly used since 2003 [1]. AIM can be used for analytically as well as numerically (or approximately) solvable problems [2–4]. Moreover, it can be used for obtaining the perturbative energy eigenvalues of the system without any need of the unperturbative eigenstate [5, 6].As a subatomic system, a quarkonium that is composed of a heavy quark-antiquark (qq-) pair has attracted attention of particle physicists since the first half of 1970, and [7–11] are just a few studies of them. In most of these studies, for easiness, the system is examined via Schrödinger equation in nonrelativistic quantum chromodynamics (NRQCD), assuming that the quarks are spinless [12–15]. Cornell potential is one of the potential functions that represent interactions between the quarks in such a qq- system. It is used for obtaining the mass and energy spectrum of the quarkonium and obtaining the hadron decay widths [7–9, 16]. Cornell potential is given as(1)Vr=-Ar+B2rwhere A and B are positive constants (B is in energy dimension). As is seen in (1), Cornell potential has two parts: one is the Coulombic term and the other is the linear part. For obtaining the energy levels and mass of the quarkonium, A and B may be fitted to the first-few states. Therefore, the full spectrum of the quarkonium can be constructed through these potential parameters.In literature, it is possible to find many studies in which the solutions of Schrödinger equation for Cornell potential have been obtained. For example, in [17], Hall has found an approximate energy formula to construct an energy spectrum of Schrödinger equation for Cornell potential, under some conditions. Jacobs et al. [13] have compared the eigenvalues of Schrödinger and spinless Salpeter equations in the cases of Cornell potential and Wisconsin potential [18]. Vega and friends have obtained, forl=0 states, the energy spectrum, mass, and wavefunctions at the origin for cc-, bb-, and bc- mesons by using the usual variation method in the scope of supersymmetric quantum mechanics (SUSYQM) [19, 20], in [12]. They have also compared their results with the exact ones in literature and with the experimental data.In this study, we attempted to get the energy eigenvalues (for anyl≠0 states) and masses of heavy mesons by using Asymptotic Iteration Method in the view of NRQCD, in which the quarks are considered as spinless for easiness and are bounded by Cornell potential. We achieved a semianalytical formula for constructing the energy spectrum and obtaining the masses of the mesons, using the method in scope of the perturbation theory. The accuracy of this formula was cross-checked by comparing the eigenvalues with the ones numerically obtained in this study and with the exact ones in literature. Furthermore, semianalytical formula was applied to cc-, bb-, and cb- heavy mesons for comparing the masses with the experimental data.AIM has been firstly applied to Schrödinger equation for Cornell potential by Hall and Saad in [21]. They have used Airy function as an asymptotic form of the wavefunction and have got highly-accurate numerical results in their study. Alternatively, we obtained a semianalytical mass-energy formula for quarkonium by having differential equation which gives polynomial solutions for asymptotic forms of the wavefunction of the system.This paper is organized as follows: we give a short summary of AIM in Section2, while Section 3 includes the main problem. In Section 4, we give numerical results for the eigenenergies and obtain semianalytical energy formula by applying perturbation theory to our problem in the view of AIM. Furthermore, in Section 4, we compare our energy spectrum and masses with the exact ones in literature and with the experimental data. Finally, Section 5 includes some comments about our results.


## 2. The Asymptotic Iteration Method (AIM)

According to the organization of the paper, we summed up AIM in this section, while it is comprehensively introduced in [1]. The AIM is used to solve second-order homogeneous linear differential equations in the following form:(2)y′′x=λ0xy′x+s0xyxwhere λ0(x) and s0(x) have continuous derivatives in the defined interval of the x independent variable. If there is an asymptotic condition such as(3)snsn-1=λnλn-1≡αfor n∈Z+, where n is large enough, the general solution of (2) is obtained as(4)yx=exp⁡-∫xαtdtC2+C1∫xexp⁡∫tλ0τ+2ατdτdtwith the functions(5)λn=λn-1′+sn-1+λ0λn-1sn=sn-1′+s0λn-1As a field of application, AIM can be used to deal with Schrödinger equation (or energy eigenvalue problem) in mathematical physics. The eigenvalues can be obtained through the following quantization condition:(6)δnx,E=snx,Eλn-1x,E-λnx,Esn-1x,E=0If the energy eigenvalues (E) can be obtained from (6), independently from the x variable, the problem is exactly solvable. In this case, the eigenvalue and eigenfunction of nth energy level can be derived in explicit algebraic form via n iterations. However, there are limited numbers of suitable potentials for this case.As for the approximately (or numerically) solvable problems,δn depends on both x and E. In this case, an appropriate value, x≡x0, should be determined to solve δn(x,E)=0 with respect to E [2, 6]. The energy eigenvalue of an nth level is obtained through q iterations where q≥n.


## 3. Formulation of the Problem

Consider the following Cornell potential:(7)Vr=-Ar+B2rwhere A, B are real and positive constants (B is in energy dimension) and r∈(0,∞). If we substitute V(r) into Schrödinger equation in three dimensions, we have(8)d2dr2+ε--αr+ρr+ll+1r2Ψr=0where ε=2μEn, α=2μA, and ρ=2μB2. En and μ=m1m2/m1+m2 are energy eigenvalue of nth level and reduced mass of the qq- system, respectively (m1 and m2 are quark masses). Besides, we study in natural units (i.e., ħ, c=1) for the system. After changing the variable, in (8), as r=u2, then substituting Ψ(u)=u1/2g(u), we get(9)g′′u+4εu2+4α-4ρu4-4ll+1+3/4u2gu=0If one putsg(z)=zγ+1e-z3/3f(z) into (9), in accordance with the domain of the problem, we have(10)f′′z=2z2-γ+1zf′z+2γ+2z-σz2-ωfzwhere ω=4α/(4ρ)1/3, σ=4ε/(4ρ)2/3, γ=2l+1/2, and z=(4ρ)1/6u. The final equation is suitable for applying AIM. After this point, we can apply AIM to the problem in two different ways: one is direct application (i.e., approximate solution) to get the numerical results and the other is usage of the method in scope of perturbation theory to obtain perturbative energies through a perturbation expansion as follows: (11)σ=σ0+ωσ1+ω2σ2+…where σ0, σ1, σ2,… are perturbation expansion coefficients. These can be obtained independently from the potential parameters. Thus, we can get a semianalytical formula for the energy eigenvalues. One can also achieve the mass-energy of the system by using this formula, as given in Section 4.


### 3.1. Numerical Results

In this section, we directly apply AIM to (10) to get the energy eigenvalues for different potential parameters, and we compare our results with the perturbative energies, for which (28) in the next section has been used.(12)f′′z=2z2-γ+1zf′z+2γ+2z-σz2-ωfzFrom (12), it is easily seen that λ0(z)=2z2-γ+1/z and s0(z)=2(γ+2)z-σz2-ω according to (2). We tabulate the results of direct application of AIM in Tables 1, 2, and 3. For simplicity, in the calculations, the reduced mass has been considered μ=1/2 GeV. In Table 1 the potential parameters have been chosen as A=1 and B=1 GeV while A=1, B=0.1 GeV in Table 2, and A=1, B=10 GeV in Table 3. Epert, seen in the tables, is for the comparison and has been obtained by using (28).Table 1
Comparisons of the perturbative energy eigenvalues (Epert) (in GeV) with those obtained by direct application of AIM (Enl) for the potential parameters A=1 and B=1 GeV, and for the reduced mass μ=1/2 GeV.
n E n0 E pert l E 0l E pert 0 1.39788 1.41015 0 1.39788 1.41015  1 3.47509 3.47509 1 2.82565 2.82690  2 5.03291 5.03224 2 3.85058 3.85089  3 6.37015 6.36948 3 4.72675 4.72687  4 7.57493 - 4 5.51698 -  5 8.68791 - 5 6.24840 -Table 2
Comparisons of the perturbative energy eigenvalues (Epert) (in GeV) with those obtained by direct application of AIM (Enl) for the potential parameters, A=1 and B=0.1 GeV, and for the reduced mass μ=1/2 GeV.
n E n0 E pert l E 0l E pert 0 -0.221031 -0.164433 0 -0.221031 -0.164433  1 0.0347222 0.033627 1 0.0174006 0.023501  2 0.141913 0.138477 2 0.102472 0.104008  3 0.220287 0.217229 3 0.159831 0.160406  4 0.286111 - 4 0.206238 -  5 0.344602 - 5 0.246681 -Table 3
Comparisons of the perturbative energy eigenvalues (Epert) (in GeV) with those obtained by direct application of AIM (Enl) for the potential parameters, A=1 and B=10 GeV, and for the reduced mass μ=1/2 GeV.
n E n0 E pert l E 0l E pert 0 46.4022 46.4047 0 46.4022 46.4047  1 85.3393 85.3394 1 70.0161 70.0165  2 116.729 116.729 2 89.7154 89.7154  3 144.315 144.315 3 107.334 107.334  4 169.461 - 4 123.562 -  5 192.851 - 5 138.761 -As can be seen from Tables1–3, the perturbative energy eigenvalues are in very good agreement with the numerically obtained ones, even for small values of the parameter B. Furthermore, they are in accordance with each other for B≥1 GeV, while A=1 (see in Tables 1 and 3). Additionally, this agreement is much better for higher quantum states. The perturbative eigenvalues are a little bit different from that obtained as numerically, for B<1 GeV, A=1, and the lower quantum states (see in Table 2). However, they are in agreement for the higher levels.


## 3.1. Numerical Results

In this section, we directly apply AIM to (10) to get the energy eigenvalues for different potential parameters, and we compare our results with the perturbative energies, for which (28) in the next section has been used.(12)f′′z=2z2-γ+1zf′z+2γ+2z-σz2-ωfzFrom (12), it is easily seen that λ0(z)=2z2-γ+1/z and s0(z)=2(γ+2)z-σz2-ω according to (2). We tabulate the results of direct application of AIM in Tables 1, 2, and 3. For simplicity, in the calculations, the reduced mass has been considered μ=1/2 GeV. In Table 1 the potential parameters have been chosen as A=1 and B=1 GeV while A=1, B=0.1 GeV in Table 2, and A=1, B=10 GeV in Table 3. Epert, seen in the tables, is for the comparison and has been obtained by using (28).Table 1
Comparisons of the perturbative energy eigenvalues (Epert) (in GeV) with those obtained by direct application of AIM (Enl) for the potential parameters A=1 and B=1 GeV, and for the reduced mass μ=1/2 GeV.
n E n0 E pert l E 0l E pert 0 1.39788 1.41015 0 1.39788 1.41015  1 3.47509 3.47509 1 2.82565 2.82690  2 5.03291 5.03224 2 3.85058 3.85089  3 6.37015 6.36948 3 4.72675 4.72687  4 7.57493 - 4 5.51698 -  5 8.68791 - 5 6.24840 -Table 2
Comparisons of the perturbative energy eigenvalues (Epert) (in GeV) with those obtained by direct application of AIM (Enl) for the potential parameters, A=1 and B=0.1 GeV, and for the reduced mass μ=1/2 GeV.
n E n0 E pert l E 0l E pert 0 -0.221031 -0.164433 0 -0.221031 -0.164433  1 0.0347222 0.033627 1 0.0174006 0.023501  2 0.141913 0.138477 2 0.102472 0.104008  3 0.220287 0.217229 3 0.159831 0.160406  4 0.286111 - 4 0.206238 -  5 0.344602 - 5 0.246681 -Table 3
Comparisons of the perturbative energy eigenvalues (Epert) (in GeV) with those obtained by direct application of AIM (Enl) for the potential parameters, A=1 and B=10 GeV, and for the reduced mass μ=1/2 GeV.
n E n0 E pert l E 0l E pert 0 46.4022 46.4047 0 46.4022 46.4047  1 85.3393 85.3394 1 70.0161 70.0165  2 116.729 116.729 2 89.7154 89.7154  3 144.315 144.315 3 107.334 107.334  4 169.461 - 4 123.562 -  5 192.851 - 5 138.761 -As can be seen from Tables1–3, the perturbative energy eigenvalues are in very good agreement with the numerically obtained ones, even for small values of the parameter B. Furthermore, they are in accordance with each other for B≥1 GeV, while A=1 (see in Tables 1 and 3). Additionally, this agreement is much better for higher quantum states. The perturbative eigenvalues are a little bit different from that obtained as numerically, for B<1 GeV, A=1, and the lower quantum states (see in Table 2). However, they are in agreement for the higher levels.


## 4. Perturbation Theory

Although the usage of perturbation method in the frame of AIM is comprehensively introduced in [5], we give a summary about the methodology in this section, assuming that the potential of a system is written as(13)Vx=V0x+ηVpxwhere V0(x) is solvable (unperturbed Hamiltonian) potential. Vp(x) and η are potential of the perturbed Hamiltonian and perturbation expansion parameter, respectively. The Schrödinger equation then reads(14)-d2dx2+V0x+ηVpxΨx=EΨxwhere En eigenvalues are written as a series expansion of jth-order correction En(j) as follows:(15)En=En0+ηEn1+η2En2+…=∑j=0∞ηjEnjAfter substitutingψ(x)=ψ0(x)f(x) in (14), one can obtain the following equation for f(x):(16)f′′x=λ0x,η,Ef′x+s0x,η,Efxand the termination condition in this case can be written as(17)δnx,η,E=snx,η,Eλn-1x,η,E-λnx,η,Esn-1x,η,E=0Onceδn(x,η,E) is expanded about η=0, we obtain(18)δnx,η,E=δnx,0,E+η1!∂δnx,η,E∂ηη=0+η22!∂2δnx,η,E∂η2η=0+…=∑k=0∞ηkδnkx,E=0 where δn(k)(x,E)=1/k!∂kδn(x,η,E)/∂ηkη=0.According to perturbation method in the framework of AIM, solving the equationδn(x,0,E)=0 with respect to (unknown) E gives En(0) (eigenvalues of unperturbed Hamiltonian), equation δn(1)(x,E)=0 gives En(1) (first-order correction to En), δn(2)(x,E) gives En(2) (second-order correction to En), and so on. Besides, the perturbative eigenfunctions can be achieved in the same vein with the eigenvalues. This is an alluring feature of the AIM usage in the perturbation theory for obtaining the eigenfunctions fn(x) given as follows:(19)fnx=exp⁡-∫xαnt,ηdtwhere αn(t,η)≡sn(t,η)/λn(t,η). αn(t,η) is expanded about η=0 in a similar manner, done for obtaining the eigenvalues. So,(20)αnt,η=∑k=0∞ηkαnktwhere αn(x)(k)=1/k!∂kαn(x,η)/∂ηkη=0. Thus, perturbation expansion of fn(x) is written as follows:(21)fnx=exp⁡∑k=0∞ηk-∫xαnktdt=∏k=0∞fnkxwhere kth-order correction fn(k)(x) to fn(x) is(22)fnkx=ηk-∫xαnktdt


### 4.1. Perturbation Theory for the Cornell Potential

For our problem, we may apply the perturbation expansion which has been elucidated in previous section to the following differential equation:(23)f′′z=2z2-γ+1zf′z+2γ+2z-σz2-ωfzSuppose thatσ is written as follows:(24)σn,l=σ0n,l+σ1n,lω+σ2n,lω2+…where ω is the perturbation expansion parameter. So, the energy eigenvalue is yielded as(25)Epert=4ρ2/38μ2/3σn,land more clearly(26)Epert=4ρ2/38μσ0n,l+4ρ1/32μασ1n,l+2α2μσ2n,l+…In the above expansion, the general form of the zeroth-order correctionσ0 is obtained via(27)δ0z,0,σ0=0The first-order correction,σ1, is obtained by using the equation δ(1)(z,0,σ1)=0 in the same manner with σ0, while δ(2)(z,0,σ2)=0 is used for σ2. Numerical results of σ0, σ1, and σ2 coefficients, obtained by AIM, are reported in Table 4 for some energy levels. Besides, for μ=1/2 GeV, comparisons of the perturbative energy eigenvalues with the ones obtained by direct application of AIM have been given in Tables 1, 2, and 3, in previous section. We emphasize, in Table 4, that corrections to the perturbation expansion do not depend on the potential parameters.Table 4
Perturbation coefficients of the expansion given as (24) and (26). Notice that corrections to the perturbation expansion do not depend on the potential parameters.
l n σ 0 (n,l) σ 1 (n,l) σ 2 (n,l) 0 0 3.71151 -0.525933 -0.0232729 1 6.48922 -0.366743 -0.00767365 2 8.76334 -0.297538 -0.00400191 3 10.7732 -0.256486 -0.00251618  1 0 5.33566 -0.322683 -0.00554189 1 7.75358 -0.258925 -0.00282569 2 9.85399 -0.222298 -0.00176295 3 11.7558 -0.197751 -0.00122526  2 0 6.74357 -0.244191 -0.00241586 1 8.93661 -0.208300 -0.00148846 2 10.9037 -0.184664 -0.00102765 3 12.7146 -0.167585 -0.000761053  3 0 8.01784 -0.200753 -0.00134507 1 10.0516 -0.177251 -0.000921458 2 11.9129 -0.160449 -0.000679139 3 13.6471 -0.147666 -0.000525832As a practice, we have applied our perturbation expansion formula (up to second-order correction) to get the ground-state energies of quarkonium in Table5, for various values of the parameter A, while B=1 GeV and μ=1/2 GeV. In Table 5, we also report comparisons of the perturbative energy eigenvalues with the ones of s-wave heavy quarkonium from [10, 21].Table 5
Comparisons of energy eigenvalues (in GeV) obtained by using the perturbation expansion formula in (26) (Epert) with the ones of s-wave heavy quarkonium from [10, 21]. The potential parameter B is taken as B=1 GeV, while the reduced mass is μ=1/2 GeV in this case. The eigenvalues of [10, 21] are exact results.
A E 00 (Ref. [10]) E 00 (Ref. [21]) E pert A E 00 (Ref. [10]) E 00 (Ref. [21]) E pert 0.2 2.16732 2.16732 2.16741 0.1 2.25368 2.25368 2.25369  0.4 1.98850 1.98850 1.98923 0.3 2.07895 2.07895 2.07927  0.6 1.80107 1.80107 1.80367 0.5 1.89590 1.89590 1.89740  0.8 1.60441 1.60441 1.61063 0.7 1.70394 1.70393 1.70808  1 1.39788 1.39788 1.41015 0.9 1.50242 1.50242 1.51132  1.2 1.18084 1.18083 1.20221 1.1 1.29071 1.29071 1.30711  1.4 0.95264 0.95264 0.98683 1.3 1.06817 1.06817 1.09545  1.6 0.71266 0.71266 0.76400 1.5 0.83416 0.83416 0.87635  1.8 0.46027 0.46026 0.53373 1.7 0.58805 0.58805 0.64980As is seen from Table5, the results for which our perturbation expansion (up to second-order correction) has been used are in very good agreement with [10, 21] for small values of A. However, our analytical results are a little bit different from the exact ones as A gets larger values. It seems that the perturbation expansion, which includes third-order correction, may give more accurate results. The more correction term we add to the perturbative expansion, the more compatible results we get. Nevertheless, we can say that (26) can be used as an eigenvalue formula of the Schrödinger equation in case of Cornell potential, for practical purposes. So, one can use the following formula:(28)Epert=4ρ2/38μσ0n,l+4ρ1/32μασ1n,l+2α2μσ2n,lfor obtaining the eigenvalues and mass of the quarkonium for Cornell potential. Besides, it can be fit to mass formula of experimental values for determining the potential parameters A and B. The advantage of (28) is that the coefficients σ0, σ1, and σ2 are independent of the potential parameters.


### 4.2. Energy Eigenvalues and Mass Spectrum for Heavy Quarkonium

In this section, we tested our formula through cross-checking with the exact results in literature and with the experimental data. For comparing our energy eigenvalues with the exact ones, the parameters of Cornell potential have been consideredA=0.52 and B=0.43 GeV. Besides, we have chosen the quark masses as mc=1.84 GeV and mb=5.18 GeV, in this case [12].Also, we tested our formula by comparing our results, for the masses of heavy mesons, with the experimental data. For doing this, we have taken the quark masses asmc=1.44 GeV and mb=4.87 GeV and the potential parameters as A=0.64 and B=0.39 GeV. All these values have been obtained by fitting our formula to the experimental data in [22].In Table6, we compared our energy eigenvalues calculated by using (28) with the ones of [12]. Furthermore, in Table 7, we gave our results for the masses of the mesons obtained by the same equation. Table 7 also includes the experimental data got from [22].Table 6
Comparisons of the energy eigenvalues (in GeV) of the mesons cc-, bb-, and bc- calculated by using (28) with the exact ones of [12]. The parameters of Cornell potential are A=0.52 and B=0.43 GeV, while the quark masses are mc  =1.84 GeV and mb  =5.18 GeV.
cc- bb- bc- E n Exact [12] Ref. [12] AIM Exact [12] Ref. [12] AIM Exact [12] Ref. [12] AIM 1s 0.2575 0.2578 0.2660 -0.1704 -0.1702 -0.1216 0.1110 0.1113 0.1269  2s 0.8482 0.8096 0.8481 0.4214 0.3579 0.4203 0.6813 0.6324 0.6803  3s 1.2720 1.1427 1.2715 0.7665 0.5612 0.7635 1.0686 0.9065 1.0668Table 7
Comparisons of the masses (in GeV), obtained via AIM, of the heavy mesons cc-, bb-, and cb- with the ones of [12], and with the experimental data from [22]. In this case, we have taken the quark masses as mc  =1.44 GeV and mb  =4.87 GeV, and the potential parameters as A=0.64 and B=0.39 GeV, for our calculations. All these parameters have been obtained by fitting our formula, given in (28), to the experimental data.
cc- bb- cb- M n Exp. Ref. [12] AIM Exp. Ref. [12] AIM Exp. Ref. [12] AIM 1s 3.097 3.097 3.096 9.460 9.350 9.462 6.275 6.291 6.362  2s 3.686 3.649 3.672 10.023 9.878 10.027 6.842 6.812 6.911  3s 4.039 3.963 4.085 10.355 10.081 10.361 - 7.087 7.284  4s - - 4.433 10.579 - 10.624 - - 7.593  1p 3.511 - 3.521 9.899 - 9.963 - - 6.792  2p 3.927 - 3.951 10.260 - 10.299 - - 7.178  3p - - 4.310 10.512 - 10.564 - - 7.494  1d - - 3.800 10.164 - 10.209 - - 7.051It can be seen from Table6 that the energy eigenvalues of the mesons cc-, bb-, and bc-, obtained by (28), are more compatible with the exact ones, than those of [12]. The difference between AIM and [12] becomes clearer as the energy level increases. Similar things can be said for the masses in Table 7: the results obtained via AIM are closer to the experimental data than those of [12].


## 4.1. Perturbation Theory for the Cornell Potential

For our problem, we may apply the perturbation expansion which has been elucidated in previous section to the following differential equation:(23)f′′z=2z2-γ+1zf′z+2γ+2z-σz2-ωfzSuppose thatσ is written as follows:(24)σn,l=σ0n,l+σ1n,lω+σ2n,lω2+…where ω is the perturbation expansion parameter. So, the energy eigenvalue is yielded as(25)Epert=4ρ2/38μ2/3σn,land more clearly(26)Epert=4ρ2/38μσ0n,l+4ρ1/32μασ1n,l+2α2μσ2n,l+…In the above expansion, the general form of the zeroth-order correctionσ0 is obtained via(27)δ0z,0,σ0=0The first-order correction,σ1, is obtained by using the equation δ(1)(z,0,σ1)=0 in the same manner with σ0, while δ(2)(z,0,σ2)=0 is used for σ2. Numerical results of σ0, σ1, and σ2 coefficients, obtained by AIM, are reported in Table 4 for some energy levels. Besides, for μ=1/2 GeV, comparisons of the perturbative energy eigenvalues with the ones obtained by direct application of AIM have been given in Tables 1, 2, and 3, in previous section. We emphasize, in Table 4, that corrections to the perturbation expansion do not depend on the potential parameters.Table 4
Perturbation coefficients of the expansion given as (24) and (26). Notice that corrections to the perturbation expansion do not depend on the potential parameters.
l n σ 0 (n,l) σ 1 (n,l) σ 2 (n,l) 0 0 3.71151 -0.525933 -0.0232729 1 6.48922 -0.366743 -0.00767365 2 8.76334 -0.297538 -0.00400191 3 10.7732 -0.256486 -0.00251618  1 0 5.33566 -0.322683 -0.00554189 1 7.75358 -0.258925 -0.00282569 2 9.85399 -0.222298 -0.00176295 3 11.7558 -0.197751 -0.00122526  2 0 6.74357 -0.244191 -0.00241586 1 8.93661 -0.208300 -0.00148846 2 10.9037 -0.184664 -0.00102765 3 12.7146 -0.167585 -0.000761053  3 0 8.01784 -0.200753 -0.00134507 1 10.0516 -0.177251 -0.000921458 2 11.9129 -0.160449 -0.000679139 3 13.6471 -0.147666 -0.000525832As a practice, we have applied our perturbation expansion formula (up to second-order correction) to get the ground-state energies of quarkonium in Table5, for various values of the parameter A, while B=1 GeV and μ=1/2 GeV. In Table 5, we also report comparisons of the perturbative energy eigenvalues with the ones of s-wave heavy quarkonium from [10, 21].Table 5
Comparisons of energy eigenvalues (in GeV) obtained by using the perturbation expansion formula in (26) (Epert) with the ones of s-wave heavy quarkonium from [10, 21]. The potential parameter B is taken as B=1 GeV, while the reduced mass is μ=1/2 GeV in this case. The eigenvalues of [10, 21] are exact results.
A E 00 (Ref. [10]) E 00 (Ref. [21]) E pert A E 00 (Ref. [10]) E 00 (Ref. [21]) E pert 0.2 2.16732 2.16732 2.16741 0.1 2.25368 2.25368 2.25369  0.4 1.98850 1.98850 1.98923 0.3 2.07895 2.07895 2.07927  0.6 1.80107 1.80107 1.80367 0.5 1.89590 1.89590 1.89740  0.8 1.60441 1.60441 1.61063 0.7 1.70394 1.70393 1.70808  1 1.39788 1.39788 1.41015 0.9 1.50242 1.50242 1.51132  1.2 1.18084 1.18083 1.20221 1.1 1.29071 1.29071 1.30711  1.4 0.95264 0.95264 0.98683 1.3 1.06817 1.06817 1.09545  1.6 0.71266 0.71266 0.76400 1.5 0.83416 0.83416 0.87635  1.8 0.46027 0.46026 0.53373 1.7 0.58805 0.58805 0.64980As is seen from Table5, the results for which our perturbation expansion (up to second-order correction) has been used are in very good agreement with [10, 21] for small values of A. However, our analytical results are a little bit different from the exact ones as A gets larger values. It seems that the perturbation expansion, which includes third-order correction, may give more accurate results. The more correction term we add to the perturbative expansion, the more compatible results we get. Nevertheless, we can say that (26) can be used as an eigenvalue formula of the Schrödinger equation in case of Cornell potential, for practical purposes. So, one can use the following formula:(28)Epert=4ρ2/38μσ0n,l+4ρ1/32μασ1n,l+2α2μσ2n,lfor obtaining the eigenvalues and mass of the quarkonium for Cornell potential. Besides, it can be fit to mass formula of experimental values for determining the potential parameters A and B. The advantage of (28) is that the coefficients σ0, σ1, and σ2 are independent of the potential parameters.


## 4.2. Energy Eigenvalues and Mass Spectrum for Heavy Quarkonium

In this section, we tested our formula through cross-checking with the exact results in literature and with the experimental data. For comparing our energy eigenvalues with the exact ones, the parameters of Cornell potential have been consideredA=0.52 and B=0.43 GeV. Besides, we have chosen the quark masses as mc=1.84 GeV and mb=5.18 GeV, in this case [12].Also, we tested our formula by comparing our results, for the masses of heavy mesons, with the experimental data. For doing this, we have taken the quark masses asmc=1.44 GeV and mb=4.87 GeV and the potential parameters as A=0.64 and B=0.39 GeV. All these values have been obtained by fitting our formula to the experimental data in [22].In Table6, we compared our energy eigenvalues calculated by using (28) with the ones of [12]. Furthermore, in Table 7, we gave our results for the masses of the mesons obtained by the same equation. Table 7 also includes the experimental data got from [22].Table 6
Comparisons of the energy eigenvalues (in GeV) of the mesons cc-, bb-, and bc- calculated by using (28) with the exact ones of [12]. The parameters of Cornell potential are A=0.52 and B=0.43 GeV, while the quark masses are mc  =1.84 GeV and mb  =5.18 GeV.
cc- bb- bc- E n Exact [12] Ref. [12] AIM Exact [12] Ref. [12] AIM Exact [12] Ref. [12] AIM 1s 0.2575 0.2578 0.2660 -0.1704 -0.1702 -0.1216 0.1110 0.1113 0.1269  2s 0.8482 0.8096 0.8481 0.4214 0.3579 0.4203 0.6813 0.6324 0.6803  3s 1.2720 1.1427 1.2715 0.7665 0.5612 0.7635 1.0686 0.9065 1.0668Table 7
Comparisons of the masses (in GeV), obtained via AIM, of the heavy mesons cc-, bb-, and cb- with the ones of [12], and with the experimental data from [22]. In this case, we have taken the quark masses as mc  =1.44 GeV and mb  =4.87 GeV, and the potential parameters as A=0.64 and B=0.39 GeV, for our calculations. All these parameters have been obtained by fitting our formula, given in (28), to the experimental data.
cc- bb- cb- M n Exp. Ref. [12] AIM Exp. Ref. [12] AIM Exp. Ref. [12] AIM 1s 3.097 3.097 3.096 9.460 9.350 9.462 6.275 6.291 6.362  2s 3.686 3.649 3.672 10.023 9.878 10.027 6.842 6.812 6.911  3s 4.039 3.963 4.085 10.355 10.081 10.361 - 7.087 7.284  4s - - 4.433 10.579 - 10.624 - - 7.593  1p 3.511 - 3.521 9.899 - 9.963 - - 6.792  2p 3.927 - 3.951 10.260 - 10.299 - - 7.178  3p - - 4.310 10.512 - 10.564 - - 7.494  1d - - 3.800 10.164 - 10.209 - - 7.051It can be seen from Table6 that the energy eigenvalues of the mesons cc-, bb-, and bc-, obtained by (28), are more compatible with the exact ones, than those of [12]. The difference between AIM and [12] becomes clearer as the energy level increases. Similar things can be said for the masses in Table 7: the results obtained via AIM are closer to the experimental data than those of [12].


## 5. Conclusion

We have used AIM to obtain both, the eigenvalues of Schrödinger equation and mass of qq- system for Cornell potential, in three dimensions. AIM has some advantages such as being used for either exactly or numerically (or approximately) solvable problems. Furthermore, one can use AIM in the frame of perturbation theory. Once it is performed to obtain perturbative solutions, the wavefunction of unperturbed Hamiltonian is not needed to get the corrections to the perturbation expansion.In the present study, the energy eigenvalues in the case of Cornell potential have been achieved by direct application of the method. Besides, we have performed perturbation theory in the view of AIM for the problem and found a semianalytical formula for energy eigenvalues. Numerical results obtained by using this formula, for the reduced massμ=1/2 GeV, conform with the exact results of [10, 21], in a wide spectrum of the potential parameters A and B (especially for B>A). Furthermore, the results are compatible with the ones obtained directly, in Section 3. It is also possible to see from the results that the perturbative eigenvalues fit in with the exact ones for higher quantum states, even for the large values of A. For any values of A and B, the higher quantum states are more consonant with the exact ones than the lower states. The perturbation expansion, which includes third-order correction, may give more accurate results. The more correction terms we add to the perturbative expansion, the more compatible results we may get.We have also tested our semianalytical formula, by cross-checking it with the exact results in literature and with the experimental data. It can be seen, from Table6, that our energy eigenvalues calculated by using (28) are more compatible with the exact ones than those of [12]. Furthermore, the difference between our results and [12] becomes clearer as the energy level increases. By using AIM, we have also obtained mass results which are closer to the experimental data than [12].As a consequence, semianalytical formula achieved for energy eigenvalues and mass of quarkonium can be used for practical purposes in the case of Cornell potential. If our formula is fitted to the experimental data, the potential parameters (and masses of the quarks, if it is needed) can also be obtained.



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