Advances in High Energy Physics
Volume 2018 (2018), Article ID 7269657, 7 pages
https://doi.org/10.1155/2018/7269657

Research Article

Bound State of Heavy Quarks Using a General Polynomial Potential

Hesham Mansour and Ahmed Gamal

Physics Department, Faculty of Science, Cairo University, Giza, Egypt

Correspondence should be addressed to Ahmed Gamal; alnabolci2010@gmail.com

Received 3 October 2018; Revised 15 November 2018; Accepted 27 November 2018; Published 17 December 2018

Academic Editor: Sally Seidel

Copyright © 2018 Hesham Mansour and Ahmed Gamal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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  taking

we get

where both and are polynomials of degree at most one.

Also we one can take

where

And

So 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 form

Substituting from equation (9) in equation (7) and solving the quadratic equation for , we obtain

where

The 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 is

We will use a generalized polynomial potential

By substituting in equation (15), we get

Let

and hence,

where

; hence

And

By 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 define

And, hence, equation (26) becomes

Comparing with equation (1), we get

And, by substituting in equation (13), we get

Now 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 term

Hence, by substituting in equation (4), we get the following.

Substituting in equation (12), we obtain

Using equation (14), we obtain

Equalizing equations (34) and (35), we get

By substituting the values of in equation (36), we get

Equation (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 hence

Now, 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.

Table 1: Mass spectra of charmonium in comparison with other works .

Type
present work
[16, 17]
[18, 19]
[20, 21]
[22, 23]
[24, 25]
EXP  [26, 27]

1s
3.0969
3.078
3.096
3.096
3.096
3.078
3.0969

1p
-
3.415
3.433
3.433
3.255
3.415
-

2s
-
4.187
3.686
3.686
3.686
3.581
-

1d
3.6861
3.752
3.676
3.770
3.504
3.749
3.6861

2p
3.7702
4.143
3.910
4.023
3.779
3.917
3.773

3s
-
5.297
3.984
4.040
4.040
4.085
-

4s
-
6.407
4.150
4.358
4.269
4.589
-

2d
-
-
-
4.096
-
3.078
-

4d
4.160
-
-
-
-
-
4.159

1g
4.039
-
-
-
-
-
4.039

6d
4.263
-
-
-
-
-
4.263

Table 2: Mass spectra of bottomonium in comparison with other works .

Type
present work
[16, 17]
[18, 19]
[28, 29]
[22, 23]
[24, 25]
EXP  [26, 27]

1s
9.4600
9.510
9.460
9.460
9.460
9.510
9.4601

1p
-
9.612
9.840
9.811
9.916
9.862
-

2s
-
10.627
10.023
10.023
10.023
10.038
-

1d
-
10.214
10.140
10.161
9.864
10.214
-

2p
-
10.944
10.160
10.374
10.114
10.390
-

3s
-
11.726
10.280
10.355
10.355
10.655
-

2d
10.02
-
-
-
-
-
10.023

4s
10.571
12.834
10.420
10.655
10.567
11.094
10.579

3d
10.358
-
-
-
-
-
10.355

6S
11.0198
-
-
-
-
-
11.019

5d
10.873
-
-
-
-
-
10.876

Table 3: Mass spectra of in comparison with other works ].

Type
present work
[30–32]
[33, 34]
[35–37]
[22, 23]
EXP  [26, 27]

1s
6.227
6.349
6.264
6.270
6.278
6.277

1p
6.287
6.715
6.700
6.699
6.486
-

2s
6.714
6.821
6.856
6.853
6.866
-

1d
6.398
-
-
-
6.772
-

2p
6.759
7.102
7.108
7.091
6.973
-

3s
7.09
7.175
7.244
7.193
7.181
-

4s
7.386
-
-
-
7.369
-

2d
6.85
-
-
-
7.128
-

4d
7.469
-
-
-
-
-

1g
6.728
-
-
-
-
-

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.

References

F. Halzem and A. Martin, Quarks and Leptons an Introductory Course in Modern Particle Physics, Wiley Publication, 1984.

D. Griffiths, Introduction to Elementary Particles, John Wiley & Sons, New York, NY, USA, 1987.

D. H. Perkins, Introduction to High Energy Physics, Cambridge University Press, Cambridge, UK, 2000.

B. R. Martin and G. G. Shaw, Particle Physics, The Manchester Physics Series, 2nd edition, 1997.

A. Bettini, Introduction to Elementary Particle Physics, Cambridge University Press, 2018.

J. Rathsman, Quark and Lepton Interactions, Acta Universitatis Upsalensis Uppsala, 1996.

N. Mistry, Brief Introduction of Particle Physics, Cornell University, 2011.

A. T. Paul, Ralph Llewellyn Modern Physics, W.H.Freeman, 2012.

B. H. Bransden and C. J. Joachain, Quantum Mechanics, Prentice Hall, 2000.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Butterworth Heinemann, 1981.

D. Griffiths, Introduction to Elementary Particles, John Wiley & Sons, New York, NY, USA, 2008.

N. Zettili, Quantum Mechanics Concepts and Applications, John Wiley, 2009.

C. Berkdemir, Theoretical Concept of Quantum Mechanics, In Tech, 2012.

A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhäauser, Basel, Switzerland, 1988.

H. Karayer, D. Demirhan, and F. Büyükkılıç, “Extension of Nikiforov-Uvarov method for the solution of Heun equation,” Journal of Mathematical Physics, vol. 56, no. 6, Article ID 063504, 14 pages, 2015.

R. Kumar and F. Chand, “Asymptotic study to the N-dimensional radial schrödinger equation for the quark-antiquark system,” Communications in Theoretical Physics, vol. 59, no. 5, pp. 528–532, 2013.

M. Abu-Shady, “Heavy Quarkonia and Mesons in the Cornell Potential with Harmonic Oscillator Potential in the N-dimensional Schrödinger Equation,” International Journal of Applied Mathematics and Theoretical Physics, vol. 2, no. 2, p. 16, 2016.

A. Al-Oun, A. Al-Jamel, and H. Widyan, “Various Properties of Heavy Quarkonia from Flavor-Independent Coulomb Plus Quadratic Potential,” Jordan Journal of Physics, vol. 8, no. 4, pp. 199–203, 2015.

H. Ke, G. Wang, X. Li, and C. Chang, “The magnetic dipole transitions in the () binding system,” Science China Physics, Mechanics and Astronomy, vol. 53, no. 11, pp. 2025–2030, 2010.

N. V. Masksimenko and S. M. Kuchin, “Determination of the mass spectrum of quarkonia by the Nikiforov–Uvarov method,” Russian Physics Journal, vol. 54, no. 57, 2011.

E. Eichten, K. Gottfried, T. Kinoshita, J. Kogut, K. D. Lane, and T. M. Yan, “Spectrum of charmed quark-antiquark bound states,” Physical Review Letters, vol. 34, no. 6, pp. 369–372, 1975.

S. M. Kuchin and N. V. Maksimenko, “Theoretical Estimations of the Spin – Averaged Mass Spectra of Heavy Quarkonia and Bc Mesons,” Universal Journal of Physics and Application, vol. 7, pp. 295–298, 2013.

R. Faccini, BABAR-CONF-07/035 SLAC-PUB-13080.

R. Kumar and F. Chand, “Energy Spectra of the Coulomb Perturbed Potential in N-Dimensional Hilbert Space,” Phys. Scr, vol. 85, Article ID 055008, 2012.

Y. Li, P. Maris, and J. P. Vary, “Quarkonium as a relativistic bound state on the light front,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 96, no. 1, 2017.

J. Beringer, J. F. Arguin, and R. M. Barnett, “Review of particle physics,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 86, Article ID 010001, 2012.

B. Hong, T. Barillari, S. Bethke, S. Kluth, and S. Menke, “Overview of quarkonium production in heavy-ion collisions at LHC,” EPJ Web of Conferences, vol. 120, p. 06002, 2016.

Z. Ghalenovi, A. A. Rajabi, and A. Tavakolinezhad, “Masses of Charm and Beauty Baryons in the Constituent Quark Model,” International Journal of Modern Physics E, vol. 21, no. 6, Article ID 1250057, 2012.

J. Niu, L. Guo, H. Ma, and S. Wang, “Heavy quarkonium production through the top quark rare decays via the channels involving flavor changing neutral currents,” The European Physical Journal C, vol. 78, no. 8, 2018.

A. K. Ray and P. C. Vinodkumar, “Properties of Bc meson,” Pramana, vol. 66, no. 5, pp. 953–958, 2006.

S. Laachir and A. Laaribi, “Exact Solutions of the Helmholtz equation via the Nikiforov-Uvarov Method,” International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, vol. 7, no. 1, 2013.

H. Ciftci, R. L. Hall, and N. Saad, “Iterative solutions to the Dirac equation,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 72, no. 2, part A, 022101, 7 pages, 2005.

E. J. Eichten and C. Quigg, “Mesons with beauty and charm: Spectroscopy,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 49, no. 11, pp. 5845–5856, 1994.

M. Alberg and L. Wilets, “Exact solutions to the Schrödinger equation for potentials with Coulomb and harmonic oscillator terms,” Physics Letters A, vol. 286, no. 1, pp. 7–14, 2001.

D. Ebert, R. N. Faustov, and V. O. Galkin, “Properties of heavy quarkonia and,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 67, no. 1, 2003.

B. Ita, P. Tchoua, E. Siryabe, and G. E. Ntamack, “Solutions of the Klein-Gordon Equation with the Hulthen Potential Using the Frobenius Method,” International Journal of Theoretical and Mathematical Physics, vol. 4, no. 5, pp. 173–177, 2014.

H. Goudarzi and V. Vahidi, “Supersymmetric approach for Eckart potential using the {NU} method,” Advanced Studies in Theoretical Physics, vol. 5, no. 9-12, pp. 469–476, 2011.