Analytical two-loop amplitudes of $e^{+}{e}^{-}⟶\mathit{J}\mathrm{/}\mathit{\psi }+{\mathit{\eta }}_c$ at $B$ factories

In double charmonium production, a long-standing challenge is that the theoretical predictions are not consistent with the measurements at B factories. Within the NRQCD framework, the next-to-leading order (NLO) calculation has proved its power to cut down the discrepancy between theory and experiments. To further clarify this puzzle, we have performed the next-to-next-to-leading order (NNLO) calculation. The amplitude is obtained as an analytical asymptotic expansion in the ratio of the squared charm-quark mass over the squared center-of-mass energy, $m_c^2\mathrm{/}s$. We investigate the origin of the leading logarithms by performing a region analysis, revealing the intricate factorization structure in this process. We provide numerical predictions on the total cross sections of $J\mathrm{/}\psi +{\eta }_c$ production, which agree with the experimental results. Extension of our computation to $\mathrm{{\rm Y}}+{\eta }_b$ production is also discussed.