Reference $pp$ cross-sections for $J/\psi$ studies in proton-lead collisions at $\sqrt{s_{NN}} = 5.02$~TeV and comparisons between ALICE and LHCb results

The ALICE and LHCb collaborations have studied $J/\psi$ production at forward rapidities in proton-lead collisions at $\sqrt{s}=5.02$~TeV using the dimuon channel. The evaluation of cold nuclear matter effects requires the knowledge of the corresponding $J/\psi$ cross-section in $pp$ collisions at the same centre-of-mass energy and in the same kinematic range. In this note the interpolation procedure used to determine this quantity and the related consistency checks are described. Finally, the results from the two experiments on nuclear modification factors are shortly summarized and found to be in good agreement.