| Summary: | Abstract Within the nonrelativistic QCD (NRQCD) factorization framework, we have computed the O α s 2 $$ \mathcal{O}\left({\alpha}_s^2\right) $$ corrections to the exclusive production of P-wave spin-triplet charmonia χ cJ (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 χ cJ electromagnetic decays, the O α s 2 $$ \mathcal{O}\left({\alpha}_s^2\right) $$ corrections are found to be smaller than the O $$ \mathcal{O} $$ (α s ) corrections in all three channels e + e − → χ c0,1,2 + γ. In particular, the O α s 2 $$ \mathcal{O}\left({\alpha}_s^2\right) $$ corrections appear moderate for χ c1 + γ case, and marginal for χ c0 + γ. Moreover, the predictions in next-to-next-to-leading order (NNLO) accuracy for the production rates of χ c0,1 + 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 O $$ \mathcal{O} $$ (α s ) and O α s 2 $$ \mathcal{O}\left({\alpha}_s^2\right) $$ corrections to the χ c2 + 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 χ c1 + γ production rate is consistent with the recent Belle measurement. The NNLO predictions to the χ c0,2 + γ production rates are much smaller than that for χ c1 + γ, which seems to naturally explain why the e + e − → χ c0,2 + γ channels have escaped experimental detection to date.
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