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

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-$\alpha_s^2$ 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 $\chi_{c2}\to\gamma\gamma$ 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 $\chi_{c0}$ and $\chi_{c2}$ toward the wrong direction. We also present the order-$\alpha_s^2$ predictions to $\chi_{b0,2}\to\gamma\gamma$, which await the future experimental test.
Comment: 6 pages, 2 figures, 1 table; v2, typos fixed, minor changes in text