Next-to-next-to-leading-order radiative corrections to e+e− → χcJ + γ at B factory

Abstract

Abstract Within the nonrelativistic QCD (NRQCD) factorization framework, we have computed the $$ \mathcal{O}\left({\alpha}_s^2\right) $$ O α s 2 corrections to the exclusive production of P-wave spin-triplet charmonia χcJ (J = 0, 1, 2) accompanied with a hard photon at B factory. For the first time, we have explicitly verified the validity of NRQCD factorization for exclusive P-wave quarkonium production to two-loop order. Unlike the χcJ electromagnetic decays, the $$ \mathcal{O}\left({\alpha}_s^2\right) $$ O α s 2 corrections are found to be smaller than the $$ \mathcal{O} $$ O (αs) corrections in all three channels e+e− → χc0,1,2 + γ. In particular, the $$ \mathcal{O}\left({\alpha}_s^2\right) $$ O α s 2 corrections appear moderate for χc1 + γ case, and marginal for χc0 + γ. Moreover, the predictions in next-to-next-to-leading order (NNLO) accuracy for the production rates of χc0,1 + γ are insensitive to the renormalization and factorization scales. All these features may indicate that perturbative expansion in these two channels exhibits a decent convergence behavior. By contrast, both the $$ \mathcal{O} $$ O (αs) and $$ \mathcal{O}\left({\alpha}_s^2\right) $$ O α s 2 corrections to the χc2 + γ production rate are sizable, which reduce the Born order cross section by one order of magnitude after including the NNLO perturbative corrections. Taking the values of the long-distance NRQCD matrix elements from nonrelativistic potential model, our prediction to χc1 + γ production rate is consistent with the recent Belle measurement. The NNLO predictions to the χc0,2 + γ production rates are much smaller than that for χc1 + γ, which seems to naturally explain why the e+e− → χc0,2 + γ channels have escaped experimental detection to date.