Non-relativistic Mass Spectra Splitting of Heavy Mesons Under the Cornell Potential Perturbed by Spin–Spin, Spin–Orbit and Tensor Components

Abstract

In this paper, we have obtained the analytical and numerical mass spectra of the charmonium and bottomonium mesons using the non-relativistic Schrödinger equation under a spin–spin, spin–orbit and tensor coupled Cornell potential energy. We adopted the Wentzel–Kramers–Brilluoin approximation method to obtain the energy bound equation in closed form. We obtained the potential free parameters by fitting the mass spectra equation to the experimental data of the Particle Data Group. The hyperfine mass splitting of the mesons are obtained for different singlet ($s=0)$ and triplet ($s=1)$ quantum states ($n^{2s+1}l_{j})$. Also, the hyperfine multiplet splitting for $l>0$ and total angular momentum quantum number $j=l, j=l\pm 1$ were obtained. The results revealed that the charmonium masses $\psi (n^{3}S_{1})$ and $\eta _{c}(n^{1}S_{0})$ ($n=2, 3,4,5,6)$ and bottomonium masses ($\eta _{b}(n^{1}S_{0}))$ and $\Upsilon (n^{3}S_{1})$ ($n=2, 3, 4, 6)$ for the s-wave quantum states are in good agreement with the results obtained by other methods in the existing literature and available experimental data. For $l>0$, the charmonia masses $\chi _{c_{j}}(n^{3}P_{j})$ $\psi _{1}\left( {1^{3}D}_{1} \right)$ and $\psi _{2}\left( {2^{3}D}_{1} \right)$ agreed with the works obtained using other potential models and observed data. In comparison to experimental data, the total absolute deviation error of 3.21% and 1.06% was obtained for the respective charmonium and bottomonium masses. The proposed potential model provides a satisfying account for the mass spectra of the heavy mesons and may be extended to study other spectroscopic parameters.