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In present work we study the production of ground and excited charmoniumpairs in $e^- e^+ \rightarrow \Psi(nS)+ \chi_{cJ}(nP)$ for $J=0,1$ and $n=1,2$,through leading order (LO) tree-level diagrams $\sim O(\alpha_{em} \alpha_s)$,which proceed through exchange of a virtual photon and an internal gluon lineconnecting two quark lines (in the triangle quark loop part of the diagram), inthe framework of $4\times 4$ Bethe-Salpeter equation, at center of mass energy,$\sqrt{s}=10.6 GeV.$ We have tried to show that the cross sections for thisprocess calculated with use of the four leading order $\sim O(\alpha_{em}\alpha_s)$ diagrams using BSE approach come close to experimental data, whichmight be mainly due to the consistent treatment of motion of quarks inside thehadrons in the BSE framework.

$J/\Psi$ and $\chi_{cJ}(J=0,1)$ production in electron-positron annihilation at $\sqrt{s}=10.6$ GeV in the framework of Bethe-Salpeter equation