Abstract
The energy eigenvalues with any 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 c, b, and c meson systems for comparing the masses with the experimental data.
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 (q) 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 q 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 aswhere and are positive constants ( 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, and 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, for l=0 states, the energy spectrum, mass, and wavefunctions at the origin for c, b, and b 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 any 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 c, b, and c 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 Section 2, 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:where and have continuous derivatives in the defined interval of the independent variable. If there is an asymptotic condition such asfor , where is large enough, the general solution of (2) is obtained aswith the functions
As 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:
If the energy eigenvalues () can be obtained from (6), independently from the variable, the problem is exactly solvable. In this case, the eigenvalue and eigenfunction of th energy level can be derived in explicit algebraic form via iterations. However, there are limited numbers of suitable potentials for this case.
As for the approximately (or numerically) solvable problems, depends on both and . In this case, an appropriate value, , should be determined to solve with respect to [2, 6]. The energy eigenvalue of an th level is obtained through iterations where .
3. Formulation of the Problem
Consider the following Cornell potential:where , are real and positive constants ( is in energy dimension) and . If we substitute into Schrödinger equation in three dimensions, we havewhere , , and . and are energy eigenvalue of th level and reduced mass of the q system, respectively ( and are quark masses). Besides, we study in natural units (i.e., , ) for the system. After changing the variable, in (8), as , then substituting , we get
If one puts into (9), in accordance with the domain of the problem, we havewhere , , , and . 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: where , , ,… 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.
From (12), it is easily seen that and 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 GeV. In Table 1 the potential parameters have been chosen as and GeV while , GeV in Table 2, and , GeV in Table 3. , seen in the tables, is for the comparison and has been obtained by using (28).
As can be seen from Tables 1–3, the perturbative energy eigenvalues are in very good agreement with the numerically obtained ones, even for small values of the parameter . Furthermore, they are in accordance with each other for GeV, while (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 GeV, , 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 aswhere is solvable (unperturbed Hamiltonian) potential. and are potential of the perturbed Hamiltonian and perturbation expansion parameter, respectively. The Schrödinger equation then readswhere eigenvalues are written as a series expansion of th-order correction as follows:
After substituting in (14), one can obtain the following equation for :and the termination condition in this case can be written as
Once is expanded about , we obtain where .
According to perturbation method in the framework of AIM, solving the equation with respect to (unknown) gives (eigenvalues of unperturbed Hamiltonian), equation gives (first-order correction to ), gives (second-order correction to ), 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 given as follows:where . is expanded about in a similar manner, done for obtaining the eigenvalues. So,where . Thus, perturbation expansion of is written as follows:where th-order correction to is
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:
Suppose that is written as follows:where is the perturbation expansion parameter. So, the energy eigenvalue is yielded asand more clearly
In the above expansion, the general form of the zeroth-order correction is obtained via
The first-order correction, , is obtained by using the equation in the same manner with , while is used for . Numerical results of , , and coefficients, obtained by AIM, are reported in Table 4 for some energy levels. Besides, for 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.
As a practice, we have applied our perturbation expansion formula (up to second-order correction) to get the ground-state energies of quarkonium in Table 5, for various values of the parameter , while GeV and GeV. In Table 5, we also report comparisons of the perturbative energy eigenvalues with the ones of s-wave heavy quarkonium from [10, 21].
As is seen from Table 5, 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 . However, our analytical results are a little bit different from the exact ones as 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:for 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 and . The advantage of (28) is that the coefficients , , and 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 considered and GeV. Besides, we have chosen the quark masses as GeV and 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 as GeV and GeV and the potential parameters as and GeV. All these values have been obtained by fitting our formula to the experimental data in [22].
In Table 6, 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].
It can be seen from Table 6 that the energy eigenvalues of the mesons c, b, and b, 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 q 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 mass GeV, conform with the exact results of [10, 21], in a wide spectrum of the potential parameters and (especially for ). 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 . For any values of and , 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 Table 6, 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This academic work was supported by the Mersin Technology Transfer Office Academic Writing Center of Mersin University.