Next-to-next-to-leading-order QCD corrections to $χ_{c0,2}\rightarrow γγ$

We calculate the next-to-next-to-leading-order (NNLO) perturbative corrections to $P$-wave quarkonia annihilation decay to two photons, in the framework of nonrelativistic QCD (NRQCD) factorization. The order-$α_s^2$ short-distance coefficients associated with each helicity amplitude are presented in a semi-analytic form, including the "light-by-light" contributions. With substantial NNLO corrections, we find disquieting discrepancy when confronting our state-of-the-art predictions with the latest \textsf{BESIII} measurements, especially fail to account for the measured $χ_{c2}\toγγ$ width. Incorporating the effects of spin-dependent forces would even exacerbate the situation, since it lifts the degeneracy between the nonperturbative NRQCD matrix elements of $χ_{c0}$ and $χ_{c2}$ toward the wrong direction. We also present the order-$α_s^2$ predictions to $χ_{b0,2}\toγγ$, which await the future experimental test.

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