The strong vertices of charmed mesons $D$, $D^{*}$ and charmonia $J/\psi$, $\eta_{c}$

In this work, the strong form factors and coupling constants of the vertices $DDJ/\psi$, $DD^{*}J/\psi$, $D^{*}D^{*}J/\psi$, $DD^{*}\eta_{c}$, $D^{*}D^{*}\eta_{c}$ are calculated within the framework of the QCD sum rule. For each vertex, we analyze the form factor considering all possible off-shell cases and the contributions of the vacuum condensate terms $\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma Gq\rangle$, $\langle g_{s}^{2}G^{2}\rangle$, $\langle f^{3}G^{3}\rangle$ and $\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$. Then, the form factors are fitted into analytical functions $g(Q^2)$ and are extrapolated into time-like regions to get the strong coupling constants. Finally, the strong coupling constants are obtained by using on-shell cases of the intermediate mesons($Q^2=-m^2$). The results are as follows, $g_{DDJ/\psi}=5.01^{+0.58}_{-0.16}$, $g_{DD^{*}J/\psi}=3.55^{+0.20}_{-0.21}$GeV$^{-1}$, $g_{D^{*}D^{*}J/\psi}=5.10^{+0.59}_{-0.43}$, $g_{DD^{*}\eta_{c}}=3.68^{+0.39}_{-0.11}$ and $g_{D^{*}D^{*}\eta_{c}}=4.87^{+0.42}_{-0.40}$GeV$^{-1}$.