The dipole picture and the non-relativistic expansion

We study exclusive quarkonium production in the dipole picture at next-to-leading order (NLO) accuracy, using the non-relativistic expansion for the quarkonium wavefunction. The quarkonium light cone wave functions needed in the dipole picture have typically been available only at tree level, either in phenomenological models or in the nonrelativistic limit. Here, we discuss the compatibility of the dipole approach and the non-relativistic expansion and compute NLO relativistic corrections to the quarkonium light-cone wave function in light-cone gauge.

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 $$\alpha _s$$ α s which we attribute to an over-subtraction in the factorisation of the collinear singularities inside the PDFs in the $$\overline{\text {MS}}$$ 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 $$P_T$$ P T -integrated charmonium hadroproduction which renders a wide class of NLO results essentially unusable. Indeed, in such processes, $$\alpha _s$$ α 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 $$\eta _{c,b}$$ η c , b but also for a fictitious light elementary scalar boson. Having provided stable NLO predictions for $$\eta _{c,b}$$ η c , b $$P_T$$ P T -integrated cross sections, $$\sigma ^{\mathrm{NLO}}_{\eta _Q}$$ σ η Q NLO , and discussed the options to study $$\eta _{b}$$ η 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 $$x=0.001$$ x = 0.001 at scales below 2 GeV is physical or not.