Semi-analytical two-loop QCD corrections to $e^+e^-\to J/\psi+\chi_{cJ}$ at B factories

Abstract

In this work, we compute the next-to-next-to-leading-order (NNLO) QCD corrections to the process $e^+e^-\to J/\psi+\chi_{cJ}$ at B factories within the NRQCD factorization framework. The helicity amplitudes are obtained via asymptotic expansions around $r=0$ and $r=1$, with $r=16m_c^2/s$. Our asymptotic expressions reproduce the exact numerical results with high accuracy across the entire range $0\le r \le 1$, achieving a relative error below $10^{-5}$, which is sufficient for phenomenological applications. Notably, the large logarithmic terms are obtained analytically. We compute the unpolarized cross sections. The $\mathcal{O}(\alpha_s)$ correction is found to be large, while the $\mathcal{O}(\alpha_s^2)$ correction for $\chi_{c0}$ production amounts to $33\%$ of the leading-order (LO) cross section, significantly reducing the scale uncertainties. For $\chi_{c1}$, the $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(\alpha_s^2)$ corrections correspond to $35\%$ and $-15\%$, respectively. For $\chi_{c2}$, the corresponding corrections are $25\%$ and $-38\%$. The large cancellation between the corrections for $\chi_{c2}$ brings the NNLO cross section close to the LO prediction. Our prediction for $\chi_{c0}$ is consistent with the {\tt Belle} measurement and agrees with the {\tt BaBar} data within $2\sigma$. We also predict the angular distribution parameters $\alpha^J_\theta$, which are independent of nonperturbative inputs. A sharp discrepancy between the theory and the {\tt Belle} measurement is observed for $\alpha^0_\theta$, calling for further experimental and theoretical investigations. Moreover, future measurements of the angular distribution parameters for $\chi_{c1}$ and $\chi_{c2}$ will provide important tests of the theoretical framework.