We study double prompt $J/psi$ hadroproduction within the nonrelativistic-QCD factorization formalism adopting the parton Reggeization approach to treat initial-state radiation in a gauge invariant and infrared-safe way. We present first predictions for the cross section distributions in the transverse momenta of the subleading $J/psi$ meson and the $J/psi$ pair. Already at leading order in $alpha_s$, these predictions as well as those for the total cross section and its distributions in the invariant mass $m_{psipsi}$ and the rapidity separation $|Y|$ of the $J/psi$ pair nicely agree with recent ATLAS and CMS measurements, except for the large-$m_{psipsi}$ and large-$|Y|$ regions, where the predictions substantially undershoot the data. In the latter regions, BFKL resummation is shown to enhance the cross sections by up to a factor of two and so to improve the description of the data.
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