Curing the unphysical behaviour of NLO quarkonium production at the LHC and its relevance to constrain the gluon PDF at low scales
Résumé
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 $\overline{\text {MS}}$ 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, $\alpha _s$ 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, $\sigma ^{\mathrm{NLO}}_{\eta _Q}$, 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.