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

We calculate the next-to-next-to-leading-order perturbative corrections to $P$-wave quarkonia annihilation decay to two photons, in the framework of nonrelativistic QCD factorization. The order-${\alpha }_s^2$ short-distance coefficients associated with each helicity amplitude are presented in a semianalytic form, including the “light-by-light” contributions. With sizable next-to-next-to-leading-order corrections, we find a disquieting discrepancy when confronting our state-of-the-art predictions with the latest BESIII measurements, which 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 nonrelativistic QCD 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.

Figure 1

Representative Feynman diagrams for $c\overline{c}({{}^{3}P}_J^{(1)})\to \gamma \gamma$ through order ${\alpha }_s^2$.

Figure 2

The dependence of $|\overline{R_{{\chi }_{cJ}}^{\prime }}({\mu }_{\mathrm{\Lambda }}){|}^2$, which is fitted from the BESIII data using the NNLO formula, as a function of ${\mu }_R$. The blue and green bands are obtained by varying ${\mu }_{\mathrm{\Lambda }}$ from 1 GeV to $m$, and the two horizontal lines correspond to the respective values given by B-T and Cornell potential models [45].