  Prompt double-$J/\psi$ production at high-energy hadron colliders can be
considered as a golden channel to probe double parton scatterings (DPS) --in
particular to study gluon-gluon correlations inside the proton-- and, at the
same time, to measure the distribution of linearly-polarised gluons inside the
proton. Such studies however require a good control of both single and DPS in
the respective regions where they are carried out. In this context, we have
critically examined two mechanisms of single parton scatterings (SPS) that may
be kinematically enhanced where DPS are thought to be dominant, even though
they are either at higher orders in the strong-coupling or velocity expansion.
First, we have considered a gauge-invariant and infrared-safe subset of the
loop-induced contribution via Colour-Singlet (CS) transitions. We have found it
to become the leading CS SPS contributions at large rapidity separation, yet
too small to account for the data without invoking the presence of DPS yields.
Second, we have surveyed the possible Colour-Octet (CO) contributions using
both 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 all
the 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. These
modifications however do not impact the control region used for their DPS
study.
