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

Abstract

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.

Comments: 6 pages, 2 figures, 1 table; v2, typos fixed, minor changes in text