Spin Splitting Spectroscopy of Heavy Quark and Antiquarks Systems

Phenomenological potentials describe the quarkonium systems like c c, bb, and bc where they give a good accuracy for the mass spectra. In the present work, we extend one of our previous works in the central case by adding spin-dependent terms to allow for relativistic corrections. By using such terms, we get better accuracy than previous theoretical calculations. In the present work, the mass spectra of the bound states of heavy quarks c c, b b, and Bc mesons are studied within the framework of the nonrelativistic Schrödinger equation. First, we solve Schrödinger’s equation by Nikiforov-Uvarov (NU) method. The energy eigenvalues are presented using our new potential. The results obtained are in good agreement with the experimental data and are better than the previous theoretical estimates.


Methodology
In the quarkonium system which deals with quark and antiquark interaction in the center of mass frame, the masses of the quark and antiquark are bigger than chromodynamics scaling, i.e., M q, q ≫ Λ QCD . So, this allows for nonrelativistic treatment and is considered as heavy bound systems. Using Schrödinger's equation of the two-body system in a spherical symmetric potential one obtains the following: where μ is reduced mass, E is energy eigenvalue, l is orbital quantum number, V tot ðrÞ is total potential of the system, QðrÞ = rRðrÞ, and RðrÞ is a radial wave function solution of Schrödinger's equation.Our radial potential is taken as follows: Also, we use spin-dependent splitting terms: spin-spin interaction, spin-orbital interaction, and tensor interaction, respectively [35][36][37][38][39][40][41][42][43]. where V V is a vector potential term, and V s is a scalar potential term.
So, the total potential becomes where (1), we get

By substituting in equation
where in natural units, Let x = 1/r, and by substituting in equation (6), we obtain In equation (11), one can use the Nikiforov-Uvarov method (NU) to get eigenvalue and eigenfunction equations.
Due to the singularity point in equation (11), put y + δ = x, and using the Taylor series to expand to second order terms, one obtains where − 6ε We get By substituting equations (13)- (15) in equation (16) and arrange it, we get We substitute equations (8)-(10) into equation (17) to obtain the energy eigenvalue equation.
where δ = 1/r 0 , Advances in High Energy Physics Start Process (1): after we deduced the energy eigenvalues and substituted the mass parameters in the different states using the quantum numbers n, L. We get a set of equations to be compared with the experimental data and save them in MATLAB file like equations.m. In this file one has five undefined parameters; we should have five equations synchronized with them.
Process (2): we will initialize the undefined parameters and use a built in function "fsolve" to get the suggested values of the coefficients of the potential (a, b, d, and p) and the characteristic radius r 0 with the condition that they must be real.
Process (3): substitute the suggested values of the coefficients of the potential in the energy eigenvalue equation then get the theoretical data and calculate the error between the experimental data and the theoretical one.
Process (4): put processes (2) Process (2): take the best suggested values of the coefficients of the potential form MATLAB program to put them in Origin lab and make small changes in the parameters columns and calculate the output (the theoretical data) Process (3): calculate the chi-square test for the output (the theoretical data), also one can calculate chi-square for other literature theoretical works Process (4): if the chi-square test for output (the theoretical data) is bigger than chi-square for other literature theoretical works, return to process (2), else, save the values of chi-squared and parameters columns.

Start
Process (5): make a small change in values of the columns manually, individually and watch chi-square columns, and observe that chi-square values of the output are smaller with each change. Until reaching the smallest chisquare value. Figure 1: This is a flow chart about how one can calculate the theoretical data. It consists of three stages to do that. We found that after some iteration processes in the loops, the previous suggested values of the coefficients are the same in all last processes, as there is no change in the data. So, we used another program which gives a small change in the parameter values which was Origin lab program. In the third stage, after we got the values of the coefficients of the potential with the smallest chi-square values from the Origin program, we will substitute them in the energy eigenvalue equation and calculate the theoretical data (present work) for the different n and L states by using "fsolve" built in function in MATLAB like first flow chart.