Modified Pad\'e Approach to the S-Wave Charmonium Spectroscopy in QCD

We calculate the $S$-wave charmonium spectroscopy using the Hamiltonian with the non-relativistic QCD (NRQCD) potential. The logarithmic factor $\ln \mu r$, appearing in the next-to-leading order QCD loop corrections to the potential, is expanded about $r=1/\mu$, where $\mu$ corresponds to the typical charmonium scale. The resulting potential characterized by the Coulombic and linear components is consistent with the form of the Cornell potential. We obtain $\chi^2$ fitting results for the masses of the $S$-wave charmonium states, $\eta_c(1^1S_0)$, $J/\psi(1^3S_1)$, $\eta_c(2^1S_0)$, and $\psi(2^3S_1)$ in remarkable accordance with data. Our results successfully account for the hyperfine splitting for the 1S state as well as for the 2S state. We further use the three best fit parameters: the charm quark mass $m_c$, coupling constant $\alpha_s$ and the corresponding scale $\mu$ to predict the $S$-wave mass spectrum with $n\leq 6$. The hints for results are discussed.