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The failure of observing the $e^+ e^- \to J/\psi+J/\psi$ events at $B$factories to date is often attributed to the significant negativeorder-$\alpha_s$ correction. In this work we compute the ${\cal O}(\alpha^2_s)$correction to this process for the first time. The magnitude of thenext-to-next-to-leading order (NNLO) perturbative correction is substantiallynegative so that the standard NRQCD prediction would suffer from an unphysical,negative cross section. This dilemma may be traced in the fact that the bulkcontribution of the fixed-order radiative corrections stems from theperturbative corrections to the $J/\psi$ decay constant. We thus implement animproved NRQCD factorization framework, by decomposing the amplitude into thephoton-fragmentation piece and the non-fragmentation piece. With the measured$J/\psi$ decay constant as input, which amounts to resumming a specific classof radiative and relativistic corrections to all orders, thefragmentation-induced production rate can be predicted accurately and serves abenchmark prediction. The non-fragmentation type of the amplitude is thencomputed through NNLO in $\alpha_s$ and at lowest order in velocity. Both the${\cal O}(\alpha_s)$ and ${\cal O}(\alpha^2_s)$ corrections in the interferenceterm become positive and exhibit a decent convergence behavior. Our finestprediction is $\sigma(e^+ e^- \to J/\psi+J/\psi)= 2.13^{+0.30}_{-0.06}$ fb at$\sqrt{s}=10.58$ GeV. With the projected integrated luminosity of 50 ${\rmab}^{-1}$, the prospect to observe this exclusive process at \texttt{Belle} 2experiment appears to be bright.

Optimized ${\cal O}(\alpha_s^2)$ correction to exclusive double $J/\psi$ production at $B$ factories