Abstract

In the present work, the mass spectra of the bound states of heavy quarks , and meson are studied within the framework of the nonrelativistic Schrödinger’s equation. First, we solve Schrödinger’s equation with a general polynomial potential by Nikiforov-Uvarov (NU) method. The energy eigenvalues for any L- value is presented for a special case of the potential. The results obtained are in good agreement with the experimental data and are better than previous theoretical studies.

1. Introduction

The study of quarkonium systems provides a good understating of the quantitative description of quantum chromodynamics (QCD) theory, the standard model and particle physics [1–7]. The quarkonia with a heavy quark and antiquark and their interaction are well described by Schrödinger’s equation. The solution of this equation with a spherically symmetric potentials is one of the most important problems in quarkonia systems [8–11]. These potentials should take into account the two important features of the strong interaction, namely, asymptotic freedom and quark confinement [2–6].

In the present work, an interaction potential in the quark-antiquark bound system is taken as a general polynomial to get the general eigenvalue solution. In the next step, we chose a specific potential according to the physical properties of the system. Several methods are used to solve Schrödinger’s equation. One of them is the Nikiforov-Uvarov (NU) method [12–14], which gives asymptotic expressions for the eigenfunctions and eigenvalues of the Schrödinger’s equation. Hence one can calculate the energy eigenstates for the spectrum of the quarkonia systems [12–15].

The paper is organized as follows: In Section 2, the Nikiforov-Uvarov (NU) method is briefly explained. In Section 3, the Schrödinger equation with a general polynomial potential is solved by the Nikiforov-Uvarov (NU) method. In Section 4, results and discussion are presented. In Section 5, the conclusion is given.

2. The Nikiforov-Uvarov (NU) Method [12–15]

The Nikiforov-Uvarov (NU) method is based on solving the hypergeometric-type second-order differential equation.Here and are second-degree polynomials, is a first-degree polynomial, and ψ(s) is a function of the hypergeometric-type.

By taking and substituting in equation (1), we get the following equation By  takingwe getwhere both and are polynomials of degree at most one.

Also we one can take whereAndSo equation (2) becomes An algebraic transformation from equation (1) to equation (8) is systematic. Hence one can divide by to obtain a constant ; i.e.,Equation (8) can be reduced to a hypergeometric equation in the formSubstituting from equation (9) in equation (7) and solving the quadratic equation for , we obtainwhereThe possible solutions for depend on the parameter according to the plus and minus signs of [13]. Since π(s) is a polynomial of degree at most one, the expression under the square root has to be the square of a polynomial. In this case, an equation of the quadratic form is available for the constant . To determine the parameter , one must set the discriminant of this quadratic expression to be equal to zero. After determining the values of one can find the values of .

Applying the same systematic way for equation (10), we get where is the principle quantum number.

By comparing equations (12) and (14), we get an equation for the energy eigenvalues.

3. The Schrödinger Equation with a General Polynomial Potential

The radial Schrödinger equation of a quark and antiquark system isWe will use a generalized polynomial potential By substituting in equation (15), we getLet and hence,where; henceAndBy substituting in equation (19), we get We propose the following approximation scheme on the term . Let us assume that there is a characteristic radius (residual radius) of the quark and antiquark system (which is the smallest distance between the two quarks where they cannot collide with each other). This scheme is based on the expansion of in a power series around , i.e., around in the x-space, up to the second order, so that the , dependent term, preserves the original form of equation (23). This is similar to Pekeris approximation [14, 15], which helps to deform the centrifugal potential such that the modified potential can be solved by the Nikiforov-Uvarov (NU) method. Setting around (the singularity), one can expand into a power series as follows By substituting from equation (25) in equation (23), dividing by where , and rearranging this equation, we get We defineAnd, hence, equation (26) becomes Comparing with equation (1), we get And, by substituting in equation (13), we getNow one can obtain the value of the parameter , by knowing that is a polynomial of degree at most one and by putting the discriminant of this expression under the square root equal to zero.By substituting in equation (30) and taking the negative value of , for bound state solutions, one finds that the solution is in agreement with the free hydrogen atom spectrum, because of the Coulomb termHence, by substituting in equation (4), we get the following.Substituting in equation (12), we obtainUsing equation (14), we obtain Equalizing equations (34) and (35), we getBy substituting the values of in equation (36), we getEquation (37) is the desired equation of the energy eigenvalues in spherical symmetric coordinates with a general polynomial radial potential using the Nikiforov-Uvarov (NU) method.

A special case of the above potential was chosen to describe the interaction, namely,where the first term is the Coulomb potential because the two quarks are charged and the second term is the linear term in which means that continues growing as . It is this linear term that leads to quark confinement. One of the striking properties of QCD asymptotic freedom is that the interaction strength between quarks becomes smaller as the distance between them gets shorter.

The third term is a harmonic term and the fourth is an anharmonic term and they are responsible also for quark confinement.

For the above chosen potential, we put and , and henceNow, we can rewrite equation (39) in a different form which depends on the parameters of the potential as follows

4. Results and Discussion

In this section, we will calculate the spectra for the bound states of heavy quarks such as charmonium, bottomonium, and . To determine the mass spectra in three dimensions, we use the following relation. By substituting in equation (40), we get It is clear that equation (42) depends on the potential parameters which will be obtained from the experimental data.

In the case of charmonium , the rest mass equation is as follows. In the case of bottomonium , the rest mass equation is as follows. And in the case of the meson , the rest mass equation is as follows. Comparing our theoretical work with the experimental data, we found that the maximum errors are 0.229% for the charmonium, 0.0742% for the bottomonium, and 0.00123% for . These may be due to errors in the measurements of the device. The spin can also be taken into account if one uses relativistic corrections and the appropriate relativistic Schrödinger’s equation. Our results are shown in Tables 1, 2, and 3, with a comparison between our results and those obtained in previous calculations in the literature. In the charmonium system, the maximum distance where a quark and antiquark can approach each other is . Similarly the maximum distances in the cases of the bottomonium system and system are and = , respectively. The positive and negative signs of the coefficient of the harmonic potential refer to the direction of motion of the oscillation. The negative sign of the coefficient of the Coulomb potential refers to the charges of the two quarks, but the positive sign refers to the existence of another negative contribution. The positive and negative signs of the coefficient of the anharmonic potential give a correction to the linear potential.

5. Conclusion

The mass spectra of the quarkonia (charmonium, bottomonium, and meson) using our potential were studied within the framework of the nonrelativistic Schrödinger’s equation by using the Nikiforov-Uvarov (NU) method. In [16, 17], the authors used an iteration method and the same potential when . It is found that adding the anharmonic potential gives a good accuracy and our work is comparable with them. In [18, 19], the authors used also the same potential when . We found that our work gives better results in comparison with experimental data. In [20, 21, 28, 29], the authors used the same method and the same potential when . We noticed that our work is in better agreement with the experimental data. In [22, 23], the authors used the Cornell potential only and used the same method. It is found that their results are comparable with our work. In [24, 25], the authors used the same potential when and the same method used in the present work. We found that our work is comparable with them. In [22, 23, 30–37] for the mass spectra of , there is no enough experimental data to compare with. In conclusion, comparing with the experimental data, we found that our results are better than those given by previous theoretical estimates.

Data Availability

The information given in our tables is available for readers in the original references listed in our work.

Disclosure

Hesham Mansour is a Fellow of the Institute of Physics (FInstP).

Conflicts of Interest

The authors declare that they have no conflicts of interest.