Radiative transitions of χ_cJ→ψγ and χ_bJ→Υγ

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→γΥ ($J=0,1,2$), 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(2P)→γψ(1S) ($J=0,1,2$) are \49.7%,~30.9%,~37.5%\, much larger than the corresponding \17.8%,~7.08%,~12.9%\ of χ_bJ(2P)→γΥ(1S). The decay of χ_cJ(2P)→γψ can be used to distinguish between χ_c0(3860) and χ_c0(3915), which particle is the charmonium χ_c0(2P). Although our result of χ_c1(3872)→γψ(2S) is consistent with data, but the one of χ_c1(3872)→γψ(1S) is much larger than data, so whether χ_c1(3872) is the conventional χ_c1(2P) remains an open question. The undiscovered Υ(1D) and Υ(2D) have large production rates in decays of χ_b0(2P)→γΥ(1D) and χ_bJ(3P)→γΥ(2D) ($J=0,1$), respectively. To search for χ_bJ(3P) $(J=0,1,2)$, the most competitive channels are the decays χ_bJ(3P)→γΥ(3S). And the best way to find χ_b2(1F) is to search for the decay of χ_b2(1F)→γΥ(1D).