(γ, χcJ(J = 0,1)) and (γ, ηc) production in electron-positron annihilation at s=10.6 GeV and 4.6 GeV in the framework of Bethe-Salpeter equation

In present work we study the production of ground and excited charmonium states in $e^{-}{e}^{+}\to \gamma {\chi }_{cJ}\left(nP\right)(J=0,1)$, and $e^{-}{e}^{+}\to \gamma {\eta }_c\left(nS\right)$, through leading order (LO) diagrams, which proceed through exchange of a virtual photon that couples to γ and ${\eta }_c/{\chi }_{cJ}$ through the triangular quark loop diagram, in the framework of $4\times 4$ Bethe-Salpeter equation (BSE), at center of mass energies $\sqrt{s}=10.6$ GeV (Belle energy), and 4.6 GeV (BESIII energy). The amplitude simplifies to a general form required by Lorentz-covariance, in terms of its form factors. The cross sections for these processes with leading order diagrams alone at $\sqrt{s}=10.6$ GeV provide a sizable contribution, which might be mainly due to the BSE being a fully relativistic approach that incorporates the relativistic effect of quark spins and can also describe internal motion of constituent quarks within the hadron in a relativistically covariant manner. Our results are compared with recent Belle data at 10.6 GeV, and BESIII data at 4.6 GeV, as well as other models. Plots of cross sections versus the center-of mass energy, $\sqrt{s}$ reveal mild fluctuations in their behaviour for all the three processes in the low energy ($\sqrt{s}=4\text{–}6$ GeV) region, and are analyzed in terms of the form factors.