Radiative transitions of χcJ → ψγ and χbj → Υγ

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