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.
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"value": "We address the unphysical energy dependence of quarkonium-hadroproduction cross sections at Next-to-Leading Order (NLO) in $$\\alpha _s$$ <math> <msub> <mi>\u03b1</mi> <mi>s</mi> </msub> </math> which we attribute to an over-subtraction in the factorisation of the collinear singularities inside the PDFs in the $$\\overline{\\text {MS}}$$ <math> <mover> <mtext>MS</mtext> <mo>\u00af</mo> </mover> </math> 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 $$P_T$$ <math> <msub> <mi>P</mi> <mi>T</mi> </msub> </math> -integrated charmonium hadroproduction which renders a wide class of NLO results essentially unusable. Indeed, in such processes, $$\\alpha _s$$ <math> <msub> <mi>\u03b1</mi> <mi>s</mi> </msub> </math> 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 $$\\eta _{c,b}$$ <math> <msub> <mi>\u03b7</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </math> but also for a fictitious light elementary scalar boson. Having provided stable NLO predictions for $$\\eta _{c,b}$$ <math> <msub> <mi>\u03b7</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </math> $$P_T$$ <math> <msub> <mi>P</mi> <mi>T</mi> </msub> </math> -integrated cross sections, $$\\sigma ^{\\mathrm{NLO}}_{\\eta _Q}$$ <math> <msubsup> <mi>\u03c3</mi> <msub> <mi>\u03b7</mi> <mi>Q</mi> </msub> <mi>NLO</mi> </msubsup> </math> , and discussed the options to study $$\\eta _{b}$$ <math> <msub> <mi>\u03b7</mi> <mi>b</mi> </msub> </math> 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 $$x=0.001$$ <math> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </math> at scales below 2 GeV is physical or not."
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