Study On The Radiative Correction Of η_Q2→γγ

The heavy quarkonium systems contain multiple energy scales,including both perturbation scale and non-perturbation scale,so it is an ideal platform for the study of quantum chromodynamics（QCD）.The quark model predicts heavy quarkonium of various quantum numbers,and most ground-state quarkonium have been discovered experimentally.However,the D-wave heavy quark pair η_c2/η_b2(with ^2s+1L_J quantum number of ¹D₂)has not been found yet.It is helpful for the experiment to study the production and decay modes of this kind of particles and find out the mode which is favorable for detection.Although η_c2 has a mass larger than twice that of D meson,the total decay width of η_c2 is relatively small due to the limitation of quantum number.In this paper,we study the electromagnetic decay of η_c2/η_b2 to two photon processes by using the nonrelativistic quantum chromodynamics factorization theorem.The decay amplitudes of the process are decomposed into the product of several formal factors and Lorentz tensors according to the Lorentz covariance,parity conservation,the symmetry of identical particles and the transverse conditions of photons,then the helicity amplitude of the process is obtained by helicity projection.According to the factorization theorem,these helicity amplitudes can be expanded as the product of the perturbation coefficient and the non-perturbation matrix elements.The short-distance coefficient of the perturbation can be expanded according to the strong coupling constant and calculated step by step.In this paper,the short-distance coefficients of the leading order and the next-leading order are calculated analytically,and the short-distance coefficients of the next-next-leading order are obtained numerically.Our calculations show for the first time that the factorization theorem is still valid for the D-wave even-prime electromagnetic decay at the order of the two-loop diagram.In addition,it is found that the next-leading-order and the next-next-leading order radiative corrections are not very large and both are negative,but the decay width will be greatly reduced when the next-leading-order and the next-next-leading order corrections are combined.By properly selecting the long-distance matrix elements and the mass of the heavy quark,the theoretical prediction of the decay width is obtained,and the corresponding theoretical error is given.In the end we found that r(η_c2→γγ)=2.4eV,Γ（η_b→γγ）=0.012eV.The total decay width of η_c2/η_b2 can be approximated by the calculation and analysis in the literature.The total fraction ratio of the decay to two-photon process is also predicted,and the results are Br(η_c2→γγ)=5×10^-6,Br(η_b2→γγ)=4×10^-7.In order to investigate the detection potential of η_c2/η_b2 at the collider,the cross sections of η_c2/η_b2 at the collider and at the LHC are calculated.The scattering cross section of η_c2 with photon is calculated to be 1.5fb.Given the brightness of the currentη_c2 plant,the number of cases of η_c2 would be very small,and given the fractional ratio of η_c2 to two photons,it would be difficult to detect the particle at the B plant.Even with the prospect of a Super B factory in Japan,η_c2 is hard to detect through a two photon process.In addition,we also consider the production cross section of η_c2/η_b2 on LHC.Under suitable kinematic truncation conditions,we find that the production cross section of η_c2 on LHCb is 2-50nb（depending on the transverse momentum truncation condition）,and the production cross section of η_b2 is 1-22pb.A conservative consideration of LHCb with an integral brightness of 10 fb-1 would result in 107-108 η_c2 particles and 104-105 η_b2 particles,so the LHC is an excellent platform for detecting η_c2/η_b2 particles.There will be thousands of η_c2 cases if we construct it from the two photon end state,so it is very promising to detect the decay channel.For η_b2,on the other hand,it is very difficult to reconstruct it from the two photon end state,it needs to be detected by other decay states.