Next-to-Next-to-Leading-Order Radiative Corrections to $e^+e^-\toχ_{cJ}+γ$ at $B$ factory

Abstract

Within the nonrelativistic QCD (NRQCD) factorization framework, we have computed the ${\mathcal O}(α_s^2)$ corrections to the exclusive production of $P$-wave spin-triplet charmonia $χ_{cJ}(J=0,1,2)$ 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 $χ_{cJ}$ electromagnetic decay processes, the $\mathcal{O}(α_s^2)$ corrections are found to be smaller than the $\mathcal{O}(α_s)$ corrections in all three channels $e^+e^-\to χ_{c0,1,2}+γ$. In particular, the ${\mathcal O}(α_s^2)$ corrections appear moderate for $χ_{c1}$ case, and insignificant for $χ_{c0}$. In addition, the next-to-next-to-leading order (NNLO) predictions for the production rates of $χ_{c0,1}+γ$ 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 ${\mathcal O}(α_s)$ and ${\mathcal O}(α_s^2)$ corrections to the $χ_{c2}+γ$ 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 $χ_{c1}+γ$ production rate is consistent with the recent {\tt Belle} measurement. The NNLO predictions for the $χ_{c0,2}+γ$ production rates are much smaller than that for $χ_{c1}+γ$, which seems to naturally explain why the $e^+e^-\to χ_{c0,2}+γ$ channels have escaped experimental detection to date.