Optimized QCD two-loop correction to exclusive double $J/ψ$ production at B factories

We report the calculation of the process $e^+ e^- \to J/ψJ/ψ$ up to next-to-next-to-leading order (NNLO) at a center-of-mass (CM) energy of $\sqrt{s}=10.58$ GeV. We employ an improved NRQCD factorization approach, decomposing the amplitude into photon-fragmentation and non-fragmentation components. The fragmentation contribution is determined using the measured $J/ψ$ decay constant, while the interference and non-fragmentation parts are computed at NNLO in $α_s$ and lowest order in velocity. In this optimized scheme, both ${\cal O}(α_s)$ and ${\cal O}(α^2_s)$ corrections in the interference part are positive and exhibit good convergence. The non-fragmentation part is numerically insignificant. Our results indicate that with the projected 50 ${\rm ab}^{-1}$ dataset at \texttt{Belle 2}, the prospects for observing exclusive double $J/ψ$ production are very promising.