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Prompt double-$J/\psi$ production at high-energy hadron colliders can beconsidered as a golden channel to probe double parton scatterings (DPS) --inparticular to study gluon-gluon correlations inside the proton-- and, at thesame time, to measure the distribution of linearly-polarised gluons inside theproton. Such studies however require a good control of both single and DPS inthe respective regions where they are carried out. In this context, we havecritically examined two mechanisms of single parton scatterings (SPS) that maybe kinematically enhanced where DPS are thought to be dominant, even thoughthey are either at higher orders in the strong-coupling or velocity expansion.First, we have considered a gauge-invariant and infrared-safe subset of theloop-induced contribution via Colour-Singlet (CS) transitions. We have found itto become the leading CS SPS contributions at large rapidity separation, yettoo small to account for the data without invoking the presence of DPS yields.Second, we have surveyed the possible Colour-Octet (CO) contributions usingboth old and up-to-date non-perturbative long distance matrix elements (LDMEs).We have found that the pure CO yields crucially depend on the LDMEs. Among allthe LDMEs we used, only two result into a visible modification of the NRQCD(CS+CO) yield, but only in two kinematical distributions measured by ATLAS,those of the rapidity separation and of the pair invariant mass. Thesemodifications however do not impact the control region used for their DPSstudy.

Prompt ${J/\psi}$-pair production at the LHC: impact of loop-induced contributions and of the colour-octet mechanism