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Within the framework of nonrelativistic QCD (NRQCD) factorization formalism,we compute the helicity amplitude as well as the decay width of $\eta_{Q2}$($Q=c,b$) electromagnetic decay into two photons up to next-to-next-to-leadingorder (NNLO) in $\alpha_s$ expansion. For the first time, we verify thevalidity of NRQCD factorization for the D-wave quarkonium decay at NNLO. Wefind that the $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(\alpha_s^2)$ correctionsto the helicity amplitude are negative and moderate, nevertheless bothcorrections combine to suppress the leading-order prediction for the decaywidth significantly. By approximating the total decay width of $\eta_{Q2}$ asthe sum of those for the hadronic decay and the electric $E1$ transition, weobtain the branching ratios ${\rm Br}(\eta_{c2}\to 2\gamma)\approx5\times10^{-6}$ and ${\rm Br}(\eta_{b2}\to 2\gamma)\approx 4\times10^{-7}$. Toexplore the potential measurement on $\eta_{Q2}$, we further evaluate theproduction cross section of $\eta_{Q2}$ at LHCb at the lowest order in$\alpha_s$ expansion. With the kinematic constraint on the longitudinalrapidity $4.5>y>2$ and transverse momentum $P_T>(2-4)m_Q$ for $\eta_{Q2}$, wefind the cross section can reach $2-50$ nb for $\eta_{c2}$, and $1-22$ pb for$\eta_{b2}$. Considering the integrated luminosity $\mathcal{L}=10\, {\rmfb}^{-1}$ at $\sqrt{s}=7$ TeV and $\sqrt{s}=13$ TeV, we estimate that there areseveral hundreds events of $pp\to \eta_{c2}\to 2\gamma$. Since the backgroundis relatively clean, it is promising to reconstruct $\eta_{c2}$ through itselectromagnetic decay. On the contrary, due to small branching ratio andproduction cross section, it is quite challenging to detect $\eta_{b2}\to2\gamma$ at LHCb.

Study on $\eta_{c2}(\eta_{b2})$ electromagnetic decay into double photons