Next-to-Next-to-Leading-Order Radiative Corrections to $e^{+}{e}^{-}\to {\chi }_{cJ}+\gamma$ at $B$ factory

Abstract

Within the nonrelativistic QCD (NRQCD) factorization framework, we have computed the $\mathcal{O}({\alpha }_s^2)$ corrections to the exclusive production of $P$-wave spin-triplet charmonia ${\chi }_{cJ}(J=0,1,2)$ accompanied by a hard photon at $B$ factories. For the first time, we have explicitly verified the validity of NRQCD factorization for exclusive $P$-wave quarkonium production at two-loop order. Unlike the ${\chi }_{cJ}$ electromagnetic decay processes, the $\mathcal{O}({\alpha }_s^2)$ corrections are found to be smaller than the $\mathcal{O}({\alpha }_s)$ corrections in all three channels $e^{+}{e}^{-}\to {\chi }_{c0,1,2}+\gamma$. In particular, the $\mathcal{O}({\alpha }_s^2)$ corrections appear moderate for ${\chi }_{c1}$ case, and insignificant for ${\chi }_{c0}$. In addition, the next-to-next-to-leading order (NNLO) predictions for the production rates of ${\chi }_{c0,1}+\gamma$ are insensitive to the renormalization and factorization scales. All of these features may indicate that perturbative expansion in these two channels exhibits a decent convergence behavior. By contrast, both the $\mathcal{O}({\alpha }_s)$ and $\mathcal{O}({\alpha }_s^2)$ corrections to the ${\chi }_{c2}+\gamma$ production cross section are sizable, which even reduces the Born order cross section by one order of magnitude after including the NNLO perturbative correction. Taking the values of the long-distance NRQCD matrix elements from nonrelativistic potential model, our prediction to ${\chi }_{c1}+\gamma$ production rate is consistent with the recent {\tt Belle} measurement. The NNLO predictions for the ${\chi }_{c0,2}+\gamma$ production rates are much smaller than that for ${\chi }_{c1}+\gamma$, which seems to naturally explain why the $e^{+}{e}^{-}\to {\chi }_{c0,2}+\gamma$ channels have escaped experimental detection to date.

Comments: typos corrected, some references added