Radiative transitions of _₂₉ and _₁₉

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 _₂₉ and _₁₉ (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 _₂₉ (2P) (1S) (J=0, 1, 2) are \49. 7\%, ~30. 9\%, ~37. 5\%\, much larger than the corresponding \17. 8\%, ~7. 08\%, ~12. 9\%\ of _₁₉ (2P) (1S). The decay of _₂₉ (2P) can be used to distinguish between _₂₀ (3860) and _₂₀ (3915), which particle is the charmonium _₂₀ (2P). Although our result of _₂₁ (3872) (2S) is consistent with data, but the one of _₂₁ (3872) (1S) is much larger than data, so whether _₂₁ (3872) is the conventional _₂₁ (2P) remains an open question. The undiscovered (1D) and (2D) have large production rates in decays of _₁₀ (2P) (1D) and _₁₉ (3P) (2D) (J=0, 1), respectively. To search for _₁₉ (3P) (J=0, 1, 2), the most competitive channels are the decays _₁₉ (3P) (3S). And the best way to find _₁₂ (1F) is to search for the decay of _₁₂ (1F) (1D).