Partial wave effects in the heavy quarkonium radiative electromagnetic decays

In a previous paper \cite{Bc}, it was pointed out that the wave functions of all particles are not pure waves, besides the main partial waves, they all contain {other partial waves}. It is very interesting to know what role these different partial waves play in particle transitions. Therefore, by using the Bethe-Salpeter equation method, we study the radiative electromagnetic decays $\psi\rightarrow\gamma\chi_{_{cJ}}$ and $\Upsilon\rightarrow\gamma\chi_{_{bJ}}$ ($J=0,1,2$). We find that for the $S$ and $P$ wave dominated states, like the $\psi(2S)$, $\Upsilon(2S)$, $\chi_{_{cJ}}(1P)$, and $\chi_{_{bJ}}(1P)$ etc., the dominant $S$ and $P$ waves provide main and nonrelativistic contrition to the decays; other partial waves mainly contribute to the relativistic correction. For the states like the $\psi(1D)$, $\Upsilon(2D)$, $\chi_{c2}(1F)$, and $\chi_{b2}(1F)$ etc., they are the $S-P-D$ mixing state dominated by $D$ wave or the $P-D-F$ mixing state dominated by $F$ wave. Large decay widths are found in the transitions $\psi(2D)\to \chi_{c2}(1F)$, $\Upsilon(1D)\to \chi_{bJ}(1P)$, and $\Upsilon(2D)\to \chi_{bJ}(2P)$ etc., which may be helpful to study the missing states $\chi_{c2}(1F)$, $\Upsilon(1D)$, and $\Upsilon(2D)$.