CHARMONIUM PROPERTIES

We calculated the mass spectra of charmonium meson by using matrix method to make the predictions of ground and radially excited states of charmonium mesons via non-relativistic potential model. We compared our results with other theoretical approaches and recently published experimental data. The predictions are found to be in a good accordance with the latest experimental results of Particle data group and with the results of other theoretical approaches. Besides, we calculated the momentum width coefficients β of charmonium meson. Since, there are no experimental data for the momentum width coefficients β of charmonium meson yet. Consequently, our calculated coefficients β are compared with other theoretical studies and it is found to be in a good agreement with our results. The obtained results of coefficients β have implications for decay constants, decay widths and differential cross sections for charmonium system and generally for heavy mesons system. Our study is considered as theoretical calculation of some properties of charmonium meson.


Characteristics of charmonium mesons 1.1 The constituent potential model
One of the most successful ways of describing the mesonic system is to solve the non-relativistic Schrödinger equation for these quark-anti quark states with an appropriate potential model. In a non-relativistic constituent quark model, the non-relativistic description with the Schrödinger equation gives acceptable results. We used the standard color Coulomb plus linear scalar form, and also include contact hyperfine interaction then We treated the remaining spin-dependent terms as mass shifts using leading-order perturbation theory. Thus, the potential model used here can be written as [1]:

EEJP. 3 (2020)
Tarek A. Nahool, A.M. Yasser, et al where ⃗ √ , (2) and is the reduced mass of the quark and anti-quark . ( Here is the quark-gluon coupling, ( ) is the appropriate color factor and (b) is the parameter of the string tension, and ( ) is the distance between the quarks; is the mass of the charm quark, and the last term is for the spin-orbit potential with where is the total spin quantum number of the meson [24] and The last term is for the spin-orbit potential with The spin-orbit operator is diagonal in a |J, L, S basis with the matrix elements. The tensor operator [25] has non vanishing diagonal matrix elements only between 0 spin-triplet states, which are For the charmonium mesons, the parameters , b, σ, and are taken from [26] to be 0.4942, 0.1446 GeV 2 , 1.1412 GeV and 1.4619 GeV, respectively.

The Radial Wave Functions of Charmonium Mesons
Charmonium mesons can be described by the wave function of the bound state of quark-antiquark. This wave function can be found by solving the time independent Schrödinger equation for a particle of reduced mass moving in a spherically symmetric potential with a position vector r ⃗ and it is can be expressed as: In spherical coordinates the Schrodinger equation takes the form where is orbital angular momentum and ( ) is the distance between the quarks.
Dividing the last equation into two parts, the first part is the radial kinetic energy and the second term is the potential energy. The radial part can be written as Then The matrix method [27] is used to solve Schrödinger to get the spectrum of charmonium, the details of this method could be found in the following section. L  33 Charmonium Properties EEJP. 3 (2020)

The momentum width coefficient β
The meson wave function is characterized by a momentum width parameter β that is depending upon the angular momentum quantum numbers and related to the root mean square radius quark-anti quark separation of the meson by [28].
Since, the root mean square radius depends on the wave functions , which results from solving Schrödinger equation by using our method, it is expected that, the β values will be very accurate.

Matrix method for solving Schrödinger equation
Matrix method is a numerical approach for solving the Schrodinger Equation where the eigenvalues of a matrix gives the total energies of the particle (spectra) and the Eigen functions are the corresponding wave functions. Let us re write the Schrodinger equation in Eq. (9) using natural units. Then it will be expressed as: ( 1 2 ) Next, the second derivate of function takes the form here, where ℎ is our step between two points, and , are minimum and maximum values for the variable ) respectively with a given number of steps (N) all over the range. We can rewrite the Schrödinger equation in Eq. (12) for as follow: , where, ℎ, 1, 2, … … , 1 Let us use a compact way of writing the previous equation as: (16) where, , ℎ , ℎ The last equation could be written as: ( 1 7 ) Where , ( 1 8 ) Where and represent the diagonal and non-diagonal elements, respectively. Eq. (17) could be transformed into a matrix form. Thus, we can rewrite it as a group of linear equations as follows:

EEJP. 3 (2020)
Tarek A. Nahool, A.M. Yasser, et al Then the matrix takes the diagonal form as: This is a tridiagonal matrix of dimension 1 1 , then this matrix can be solved as an eigenvalue problem and yields 1 eigenvalues. We implemented our method to solve eigenvalue problems.

Results and discussion
In this article we have calculated the spectra of S and P wave states of charmonium meson by solving the Schrödinger equation numerically using matrix method. Coulomb plus linear plus hyperfine plus spin-dependent terms potential model are considered. By taking N = 200, = 20 fm , 0.4942, 0.1497 GeV , 1.1412 GeV and the charm quark mass 1.4619 GeV for S-states and for P-states, the spectra were summarized in Tables (1,2). We predict the masses of the twenty states of ̅ meson where we compared our theoretical predictions with those from [15,16,17]. By comparing present work with the recently published experimental data, we found that the maximum errors are 0.03 GeV for S-states and the maximum errors for P-states are 0.044 GeV. Similarly, we found that the maximum errors between our work and [15] are 0.03 GeV for S-states and P-states. In the same way, we calculate the maximum errors between our work and [16] and it was 0.02 GeV for S-states and 0.03 GeV for P-states. Hence, it can be seen that our calculated spectra are in close accordance with the PDG results and with the results of other theoretical studies, thus it can be concluded that this method gives satisfying results for predicting charmonium spectra. The normalized radial wave functions for charmonium mesons are graphically represented in Figures (1) and (2) respectively.   In Figure (3), the present potential has been plotted for S and P states of charmonium. The predictions about the values of momentum width (Coefficient β) for charmonium S and P States in comparison between our results and those obtained in previous calculations [20,21] are reported in Tables (3, 4 Table 2. Mass spectrum of radially excited states (P-States) of cc meson (in GeV) using matrix method.

State
Matrix method [15] [16] PDG [17] n L S J J PC Meson  Table 3. Momentum width (Coefficient β) of ground states of charmonium mesons using matrix method.

State
Our work [20]

Conclusion
The non-relativistic quark model is one of the most powerful frameworks to investigate the heavy meson spectra. The mass spectra of ̅ mesons were calculated in that framework based on the matrix method. Predictions from our method are found to be in good agreement with the PDG results and available theoretical results. Our results in comparison with the experimental data and other theoretical approaches results are reported in Tables (1, 2). It can be noticed that our calculated mass spectra are in close resemblance with the latest PDG results and also with the results of other theoretical studies, thus it can be concluded that this method gives satisfying results for S and P wave spectroscopy. On the other hand, we used the matrix method to obtain the radial wave functions of charmonium meson to calculate the coefficient β. Then, we compare our results with available published results which obtained from other studies and it was nearly commensurate. Our calculated coefficient β in comparison with other theoretical studies are reported in Tables (3,4). As a remarkable result, we can point out that it is recommended to use the obtained values of coefficient β to calculate the decay widths and differential cross sections for charmonium system. Overall, the obtained results from the present study are reasonable when compared with the latest experimental results and available theoretical results which obtained from other approaches for the mass spectra of ̅ meson. Finally, we may notice that the calculated values of coefficient β are the newer outputs where we did not find experimental data for comparison. So, we are looking forward to taking these data in consideration by other experimental and theoretical researchers.