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 √s=10.6 GeV, mc=1.5 GeV and μR=2mc, we obtain an approximate NNLO cross section of 0.324+0.041-0.066 pb, with scale uncertainties from mc≤μR≤4mc. 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 mc=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.