Abstract
In the framework of instantaneous Bethe-Salpeter equation, according to the JPC of quarkonia, we find that their wave functions all contain multiple partial waves, rather than pure waves. In the radiative electromagnetic transitions χcJ→γψ and χbJ→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 χcJ(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 χbJ(2P) → γΥ(1S). The decay of χcJ(2P) → γψ can be used to distinguish between χc0(3860) and χc0(3915), which particle is the charmonium χc0(2P). Although our result of χc1(3872)→γψ(2S) is consistent with data, but the one of χc1(3872)→γψ(1S) is much larger than data, so whether χc1(3872) is the conventional χc1(2P) remains an open question. The undiscovered Υ(1D) and Υ(2D) have large production rates in decays of χb0(2P) → γΥ(1D) and χbJ(3P) → γΥ(2D) (J = 0, 1), respectively. To search for χbJ(3P) (J = 0, 1, 2), the most competitive channels are the decays χbJ(3P) → γΥ(3S). And the best way to find χb2(1F) is to search for the decay of χb2(1F) → γΥ(1D).
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Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (NSFC) under the Grants Nos. 12075073 and 12375085, the Natural Science Foundation of Hebei province under the Grant No. A2021201009, Post-graduate’s Innovation Fund Project of Hebei University under the Grant No. HBU2022BS002.
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Pei, SY., Li, W., Wang, T. et al. Radiative transitions of χcJ → ψγ and χbj → Υγ. J. High Energ. Phys. 2024, 191 (2024). https://doi.org/10.1007/JHEP08(2024)191
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DOI: https://doi.org/10.1007/JHEP08(2024)191