Curing the unphysical behaviour of NLO quarkonium production at the LHC and its relevance to constrain the gluon PDF at low scales

We address the unphysical energy dependence of quarkonium-hadroproduction cross sections at Next-to-Leading Order (NLO) in alpha_S which we attribute to an over-subtraction in the factorisation of the collinear singularities inside the PDFs in the MSbar scheme. Such over- or under-subtractions have a limited phenomenological relevance in most of the scattering processes in particle physics. On the contrary, it is particularly harmful for P_T-integrated charmonium hadroproduction which renders a wide class of NLO results essentially unusable. Indeed, in such processes, alphaS is not so small, the PDFs are not evolved much and can be rather flat for the corresponding momentum fractions and, finally, some process-dependent NLO pieces are either too small or too large. We propose a scale-fixing criterion which avoids such an over-subtraction. We demonstrate its efficiency for eta(c,b) but also for a fictitious light elementary scalar boson. Having provided stable NLO predictions for eta(c,b) P_T-integrated cross sections, sigma^NLO(eta(Q)), and discussed the options to study eta(b) hadroproduction, we argue that their measurement at the LHC can help better determine the gluon PDF at low scales and tell whether the local minimum in conventional NLO gluon PDFs around x=0.001 at scales below 2 GeV is physical or not.