Radiative decay widths of ground and excited states of vector charmonium and bottomonium

In this work we study the radiative decay widths of vector quarkonia for the process of $J/\psi(nS)\rightarrow\eta_{c}(nS)\gamma$ and $\Upsilon(nS)\rightarrow\eta_{b}(nS)\gamma$ (for principal quantum numbers $n=1, 2, 3$) in the framework of Bethe-Salpeter equation under the covariant instantaneous ansatz using a $4\times 4$ form of BSE. The parameters of the framework were determined by a fit to the mass spectrum of ground states of pseudoscalar and vector quarkonia, such as; $\eta_{c}$, $\eta_{b}$, $J/\psi$ and $\Upsilon$. These input parameters so fixed were found to give good agreements with data on mass spectra of ground and excited states of pseudoscalar and vector quarkonia, leptonic decay constants of pseudoscalar and vector quarkonia, two photon decays and two gluon decays of pseudoscalar quarkonia in our recent paper. With these input parameters so fixed, the radiative decay widths of ground (1S) and excited (2S, 3S) states of heavy vector quarkonia ($J/\Psi$ and $\Upsilon$) are calculated and found to be in reasonable agreement with data.