$O(\alpha_s v^2)$ correction to $e^+e^-\to J/\psi+\eta_c$ at $B$ factories

We investigate the ${\mathcal O}(\alpha_s v^2)$ correction to the $e^+e^-\to J/\psi+\eta_c$ process in the nonrelativistic QCD (NRQCD) factorization approach. Within some reasonable choices of the relative order-$v^2$ NRQCD matrix elements, we find that including this new ingredient of correction only mildly enhances the existing NRQCD predictions. We have also deduced the asymptotic expressions for the ${\mathcal O}(\alpha_s v^2)$ short-distance coefficients, and reconfirm the early speculation that at next-to-leading order in $\alpha_s$, the double logarithm of type $\ln^2 (s/m_c^2)$ appearing in various NRQCD short-distance coefficients is always associated with the helicity-suppressed channels.