Study on $η_{c2}(η_{b2})$ 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 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 O(α s ) and O(α 2 s ) 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 Br(η c2 → 2γ) ≈ 5 × 10 −6 and Br(η b2 → 2γ) ≈ 4 × 10 −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 L = 10 fb −1 at √ s = 7 TeV and √ s = 13 TeV, we estimate that there are several hundreds events of pp → η c2 → 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 → 2γ at LHCb.