Journal of Nuclear and Particle Physics

p-ISSN: 2167-6895    e-ISSN: 2167-6909

2020;  10(1): 13-22

doi:10.5923/j.jnpp.20201001.03

 

Spectroscopy of the Quarkonium Systems for Heavy Quarks

Hesham Mansour, Ahmed Gamal

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

Correspondence to: Hesham Mansour, Physics Department, Faculty of Science, Cairo University, Giza, Egypt.

Email:

Copyright © 2020 The Author(s). Published by Scientific & Academic Publishing.

This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

Abstract

The study of the spectroscopy of bound states of quarkonium systems like and meson in the quark model framework with phenomenological potentials is motivated by quantum chromodynamics (QCD). It is found that analysis of the mass spectra of these systems is effectively given by the nonrelativistic Schrödinger's equation. There are several methods which are used to solve Schrödinger's equation with a general polynomial potential one of them is the Nikiforov-Uvarov (NU) method. It’s one of the effective methods which gives the energy eigenvalues and eigenstates for our potential. The results obtained are in good agreement with the experimental data and are better than previous theoretical studies.

Keywords: Quarkonium spectroscopy, Meson Mass Spectrum

Cite this paper: Hesham Mansour, Ahmed Gamal, Spectroscopy of the Quarkonium Systems for Heavy Quarks, Journal of Nuclear and Particle Physics, Vol. 10 No. 1, 2020, pp. 13-22. doi: 10.5923/j.jnpp.20201001.03.

Article Outline

1. Introduction
2. The Schrödinger Equation with Polynomial Potential
3. Results and Discussion

1. Introduction

In quark and anti-quark system, the quantitively description is given by (quantum chromodynamics (QCD) spectroscopy and the standard model theory) [1-4] is important to specify the mechanism of that system and its nature of being bound systems.
The Schrödinger's equation describes quarkonium systems [5-9] with a heavy quark and anti-quark interaction (two-body problem). We solve the Schrödinger equation in a spherical-symmetric coordinate and using radial potentials which can be described by the asymptotic limits of QCD which has been qualitatively verified by Lattice QCD calculations [2-4].
The main purpose of this paper is that an interaction potential in the quark-antiquark bound system is taken as a general polynomial to get the general eigenvalue and eigenfunction solution then choosing a specific potential according to the description of the physical mechanism of the system.
The Nikiforov-Uvarov (NU) method [10-15], gives asymptotic expressions for the eigenfunctions and eigenvalues of the Schrödinger's equation. Hence one can calculate the energy eigenvalues and eigenstates for the spectrum of the quarkonium systems [17-26].

2. The Schrödinger Equation with Polynomial Potential

The Schrödinger equation reads
(1)
We use the generalized potential
(2)
(3)
By substituting in Schrödinger equation (1), we get
(4)
(5)
Where
(6)
and
(7)
Let and substituting in equation (5), we get
(8)
Because of singularity, we expand the polynomial function by Taylor’s series around x=0
Let
(9)
(10)
Neglecting higher order terms, we get
(11)
By substituing , we get
(12)
By substituting in equation (9), we obtain
(13)
(14)
By substituting in equation (8), we get
(15)
We rearrange equation (15), and divide by where to obtain,
(16)
(17)
Equation (16) becomes
(18)
We use the Nikiforov-Uvarov (NU) method [12,14] as mentioned before,
are symbols used in the Nikiforov-Uvarov (NU) method.
(19)
(20)
(21)
We chose the value of which make the square root in equation (21) as a quadratic term
(22)
(23)
Substituting in equation (21), we get
(24)
(25)
We take the negative value
(26)
(27)
(28)
(29)
(30)
(31)
By equating equations (29) and (31), we get the eigen value equation
(32)
(33)
(34)
By substituting equation (17) in equation (34), we get
(35)
By substituting equation (7) in equation (35), we get
(36)
Putting and substituting in equation (36), we obtain
(37)
So, the total energy eigen value is
(38)
To find the eigenfunctions for the general potential form
(39)
First, we calculate
(40)
(41)
(42)
Second, we calculate
(43)
(44)
(45)
(46)
(47)
By substituting in equation (39), we obtain
(48)
Where is a normalization constant.
So, the radial wavefunction of Schrödinger equation (1) for a polynomial potential is
(49)
To find the normalization constant
(50)
The angular part of the spherical symmetric potential is
(51)
So, the total wavefunction in spherical symmetric potential is
(52)
In our interquark potential, and using the natural units
(53)
Putting
(54)
The energy eigen values equation according to such potentials become
(55)

3. Results and Discussion

In this section, we will calculate the spectra for the bound states of heavy quarks such as charmonium, bottomonium and meson. The mass spectra equation is
(56)
By substituting equation (55) in equation (56), we get
(57)
Equation (57) depends on the potential parameters which will be obtained from the experimental data.
In the case of charmonium the rest mass equation is
(58)
And we get the following masses in GeV
The charmonium system
     
     
In the following we draw the radial wave functions as a function of the radius,
Figure 1. Graphs represent the relation between the radial wave function and the radius in different n, L states according to experimental energy states in the charmonium system
In the case of bottomonium , the rest mass equation is
(59)
And we get the following masses in GeV
The bottomonium system
     
     
In the following we draw the radial wave functions as a function of the radius,
Figure 2. Graphs represent the relation between the radial wave function and the radius in different n, L according to experimental energy states in the bottomonium system
Figure 3. Graphs represent the relation between the radial wave function and the radius in different n, L according to experimental energy states in the bottomonium system
In the case of the meson the rest mass equation is
(60)
And we get the following masses in GeV
The meson
     
     
In the following we draw the radial wave functions as a function of the radius,
Figure 4. Graphs represent the relation between the radial wave function and the radius in different n, L according to experimental energy states in the meson system
In conclusion, form the tables, we found that our theoretical work is comparable with the experimental data and explains the behavior of the quarkonium systems. The difference between the experimental data and theoretical work may be because we neglect the spin terms, so, the spin can also be considered if one uses relativistic corrections and the appropriate relativistic Schrödinger's equation. From the figures which represent the quarkonium radial state wave functions, one can calculate physical parameters like the decay parameter.

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