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 $\psi \to \gamma {\chi }_{{}_{cJ}}$ and $\mathrm{{\rm Y}}\to \gamma {\chi }_{{}_{bJ}}$ ($J=0,1,2$). We find that for the $S$ and $P$ wave dominated states, like the $\psi (2S)$, $\mathrm{{\rm Y}}(2S)$, ${\chi }_{{}_{cJ}}(1P)$, 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 $\psi (1D)$, $\mathrm{{\rm Y}}(2D)$, ${\chi }_{c2}(1F)$, 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 $\psi (2D)\to {\chi }_{c2}(1F)$, $\mathrm{{\rm Y}}(1D)\to {\chi }_{bJ}(1P)$, and $\mathrm{{\rm Y}}(2D)\to {\chi }_{bJ}(2P)$ etc., which may be helpful to study the missing states ${\chi }_{c2}(1F)$, $\mathrm{{\rm Y}}(1D)$, and $\mathrm{{\rm Y}}(2D)$.