LAUSR

Within the framework of nonrelativistic QCD factorization formalism, we compute the helicity amplitude as well as the decay width of ${\ensuremath{\eta}}_{Q2}$ ($Q=c$, $b$) electromagnetic decay into two photons up to next-to-next-to-leading order in ${\ensuremath{\alpha}}_{s}$ expansion. For the first time, we verify the validity of nonrelativistic QCD factorization for the $D$-wave quarkonium decay at next-to-next-to-leading order. We find that the $\mathcal{O}({\ensuremath{\alpha}}_{s})$ and $\mathcal{O}({\ensuremath{\alpha}}_{s}^{2})$ corrections to the helicity amplitude are negative and moderate, nevertheless both corrections combine to suppress the leading-order prediction for the decay width significantly. By approximating the total decay width of ${\ensuremath{\eta}}_{Q2}$ as the sum of those for the hadronic decay and the electric $E1$ transition, we obtain the branching ratios $\text{Br}({\ensuremath{\eta}}_{c2}\ensuremath{\rightarrow}2\ensuremath{\gamma})\ensuremath{\approx}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and $\text{Br}({\ensuremath{\eta}}_{b2}\ensuremath{\rightarrow}2\ensuremath{\gamma})\ensuremath{\approx}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$. To explore the potential measurement on ${\ensuremath{\eta}}_{Q2}$, we further evaluate the production cross section of ${\ensuremath{\eta}}_{Q2}$ at LHCb at the lowest order in ${\ensuremath{\alpha}}_{s}$ expansion. With the kinematic constraint on the longitudinal rapidity $4.5gyg2$ and transverse momentum ${P}_{T}g(2--4){m}_{Q}$ for ${\ensuremath{\eta}}_{Q2}$, we find the cross section can reach 2--50 nb for ${\ensuremath{\eta}}_{c2}$, and 1--22 pb for ${\ensuremath{\eta}}_{b2}$. Considering the integrated luminosity $\mathcal{L}=10\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at $\sqrt{s}=7$, 13 TeV, we estimate that there are several hundreds events of $pp\ensuremath{\rightarrow}{\ensuremath{\eta}}_{c2}\ensuremath{\rightarrow}2\ensuremath{\gamma}$. Since the background is relatively clean, it is promising to reconstruct ${\ensuremath{\eta}}_{c2}$ through its electromagnetic decay. On the other hand, due to the small branching ratio and production cross section, it is quite challenging to detect ${\ensuremath{\eta}}_{b2}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ at LHCb.