Abstract
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.
Highlights
The production of quarkonia (Q) in inclusive proton–proton and electron–proton collisions when the protons break apart is one of the most often studied processes at high-energy colliders
What we have learnt in the recent years with the advent of Next-toLeading Order (NLO) computations of PT -differential cross sections, dσ NLO/d PT, of J/ψ and Υ [22,23,24,25,26,27,28,29] is that the inclusion of NLO corrections in any data-theory comparison is absolutely mandatory to extract qualitatively reliable statements
When one integrates over PT, these NLO channels, which are PT -enhanced in the Colour-Singlet Model (CSM) and Non-Relativistic QCD (NRQCD) and which are precisely responsible for the large impact of the NLO corrections at mid and large PT, are just suppressed by one power of αs without any PT -enhancement factor
Summary
The production of quarkonia (Q) in inclusive proton–proton and electron–proton collisions when the protons break apart is one of the most often studied processes at high-energy colliders. We propose a solution to this issue which we attribute to an over-subtraction in the factorisation of the collinear singularities inside the PDF in the MS scheme As such, it may appear in any NLO computations once a couple of unfavourable factors combine. Having proposed a way to get sound NLO perturbative results, we discuss the interplay between the behaviour of the gluon PDFs at low scales and, in particular, the ηc production cross sections. For dσ /dy, the phase-space integration is slightly less straightforward to be performed analytically than for σwhere one can just integrate over the full phase-space without separating out the rapidity y and the transverse momentum PT For both σand dσ /dy, after combining the virtual with the real corrections, the soft singularities vanish and we are left, as usual, with the initial-state collinear divergences which originate from diagrams such as, e. The codes which we have derived from these expressions and which we have used to generate the results presented later have been successfully cross-checked versus the semi-automatic code FDC [50]
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