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

Within the framework of nonrelativistic QCD (NRQCD) factorization formalism, we compute the helicity amplitude as well as the decay width of $η_{Q2}$ ($Q=c,b$) electromagnetic decay into two photons up to next-to-next-to-leading order (NNLO) in $α_s$ expansion. For the first time, we verify the validity of NRQCD factorization for the D-wave quarkonium decay at NNLO. We find that the $\mathcal{O}(α_s)$ and $\mathcal{O}(α_s^2)$ 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 $η_{Q2}$ as the sum of those for the hadronic decay and the electric $E1$ transition, we obtain the branching ratios ${\rm Br}(η_{c2}\to 2γ)\approx 5\times10^{-6}$ and ${\rm Br}(η_{b2}\to 2γ)\approx 4\times10^{-7}$. To explore the potential measurement on $η_{Q2}$, we further evaluate the production cross section of $η_{Q2}$ at LHCb at the lowest order in $α_s$ expansion. With the kinematic constraint on the longitudinal rapidity $4.5>y>2$ and transverse momentum $P_T>(2-4)m_Q$ for $η_{Q2}$, we find the cross section can reach $2-50$ nb for $η_{c2}$, and $1-22$ pb for $η_{b2}$. Considering the integrated luminosity $\mathcal{L}=10\, {\rm fb}^{-1}$ at $\sqrt{s}=7$ TeV and $\sqrt{s}=13$ TeV, we estimate that there are several hundreds events of $pp\to η_{c2}\to 2γ$. Since the background is relatively clean, it is promising to reconstruct $η_{c2}$ through its electromagnetic decay. On the contrary, due to small branching ratio and production cross section, it is quite challenging to detect $η_{b2}\to 2γ$ at LHCb.

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