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… Click to show full abstract
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.
               
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