Optimized two-loop QCD correction to exclusive double-$J/{\psi}$ production at the B factories

We report the calculation of the process $e^+ e^- \to J/\psi J/\psi$ 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 $J/\psi$ decay constant, while the interference and non-fragmentation parts are computed at NNLO in $\alpha_s$ and lowest order in velocity. In this optimized scheme, both ${\cal O}(\alpha_s)$ and ${\cal O}(\alpha^2_s)$ 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 $J/\psi$ production are very promising.