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

Abstract

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

14 pages, 1 figures, 2 tables

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