Radiative transitions of $\chi_{_{cJ}}\to\psi\gamma$ and $\chi_{_{bJ}}\to\Upsilon\gamma$

In the framework of instantaneous Bethe-Salpeter equation, according to the $J ^ {PC}$ of quarkonia, we find that their wave functions all contain multiple partial waves, rather than pure waves. In the radiative electromagnetic transitions $\chi_{_{cJ}}$$\rightarrow$$\gamma\psi$ and $\chi_{_{bJ}}$$\rightarrow$$\gamma\Upsilon$ ($J=0,1,2$), the main wave of quarkonium gives the non-relativistic contribution, while other waves provide the relativistic corrections. Our results indicate that the relativistic effect of charmonium, especially highly excited states, is significant. Such as the relativistic effects of $\chi_{_{cJ}}(2P)\to\gamma\psi(1S)$ ($J=0,1,2$) are $\{49.7\%,~30.9\%,~37.5\%\}$, much larger than the corresponding $\{17.8\%,~7.08\%,~12.9\%\}$ of $\chi_{_{bJ}}(2P)\rightarrow\gamma\Upsilon(1S)$. The decay of $\chi_{_{cJ}}(2P)\to\gamma\psi$ can be used to distinguish between $\chi_{_{c0}}(3860)$ and $\chi_{_{c0}}(3915)$, which particle is the charmonium $\chi_{_{c0}}(2P)$. Although our result of $\chi_{_{c1}}(3872)$$\rightarrow$$\gamma\psi(2S)$ is consistent with data, but the one of $\chi_{_{c1}}(3872)$$\rightarrow$$\gamma\psi(1S)$ is much larger than data, so whether $\chi_{_{c1}}(3872)$ is the conventional $\chi_{_{c1}}(2P)$ remains an open question. The undiscovered $\Upsilon(1D)$ and $\Upsilon(2D)$ have large production rates in decays of $\chi_{_{b0}}(2P)\rightarrow\gamma\Upsilon(1D)$ and $\chi_{_{bJ}}(3P)\rightarrow\gamma\Upsilon(2D)$ ($J=0,1$), respectively. To search for $\chi_{_{bJ}}(3P)$ $(J=0,1,2)$, the most competitive channels are the decays $\chi_{_{bJ}}(3P)\rightarrow\gamma\Upsilon(3S)$. And the best way to find $\chi_{_{b2}}(1F)$ is to search for the decay of $\chi_{_{b2}}(1F)\rightarrow\gamma\Upsilon(1D)$.