NNLO QCD corrections to direct $J/ψ$ production with open charm at B factories

We calculate the next-to-next-to-leading-order (NNLO) QCD corrections to direct colour-singlet $J/ψ$ production with open charm at B factories, using nested soft-collinear subtraction for massive partons. At $\sqrt{s}=10.6$ GeV, $m_c=1.5$ GeV and $μ_R=2m_c$, we obtain an approximate NNLO cross section of $0.324^{+0.041}_{-0.066}$ pb, with scale uncertainties from $m_c\leqμ_R\leq4m_c$. The total cross section is $18.4\%$ above NLO and exhibits good perturbative convergence through NNLO, with reduced scale dependence. An NLO-based mass extrapolation to $m_c=1.4$ GeV, combined with estimated $ψ(2S)$ feed-down, raises the rate to about $0.59$ pb. The NNLO correction brings the absolute momentum spectrum near its peak closer to Belle's prompt open-charm data, but shifts the angular shape further away from Belle's prompt charm-associated data. The difference in angular shape motivates more precise measurements at Belle II.