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We address the unphysical energy dependence of quarkonium-hadroproductioncross sections at Next-to-Leading Order (NLO) in alpha_S which we attribute toan over-subtraction in the factorisation of the collinear singularities insidethe PDFs in the MSbar scheme. Such over- or under-subtractions have a limitedphenomenological relevance in most of the scattering processes in particlephysics. On the contrary, it is particularly harmful for P_T-integratedcharmonium hadroproduction which renders a wide class of NLO resultsessentially unusable. Indeed, in such processes, alphaS is not so small, thePDFs are not evolved much and can be rather flat for the corresponding momentumfractions and, finally, some process-dependent NLO pieces are either too smallor too large. We propose a scale-fixing criterion which avoids such anover-subtraction. We demonstrate its efficiency for eta(c,b) but also for afictitious light elementary scalar boson. Having provided stable NLOpredictions for eta(c,b) P_T-integrated cross sections, sigma^NLO(eta(Q)), anddiscussed the options to study eta(b) hadroproduction, we argue that theirmeasurement at the LHC can help better determine the gluon PDF at low scalesand tell whether the local minimum in conventional NLO gluon PDFs aroundx=0.001 at scales below 2 GeV is physical or not.

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