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

Jean-Philippe Lansberg, Melih Arslan Ozcelik

TL;DR

The paper identifies the unphysical energy dependence observed at NLO in quarkonium production as arising from over-subtraction of collinear divergences in MSbar PDFs. It proposes a process-dependent factorisation-scale hat{μ}_F that absorbs high-energy real-emission contributions into PDFs, stabilising predictions for η_c, η_b, and a fictitious scalar across wide energy ranges. The authors demonstrate that this scale-corrected approach yields positive, slowly varying K^NLO factors and enables meaningful use of low-scale gluon PDFs to constrain the gluon content of the proton, potentially guiding future PDF fits. They also provide analytical expressions for σ and dσ/dy, explore gluon-luminosity sensitivities, and discuss experimental prospects to measure η_c (and η_b) to refine low-scale gluon PDFs, with broader implications for NRQCD and collinear factorisation at low scales.

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 MSbar 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, alphaS 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^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.

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

TL;DR

The paper identifies the unphysical energy dependence observed at NLO in quarkonium production as arising from over-subtraction of collinear divergences in MSbar PDFs. It proposes a process-dependent factorisation-scale hat{μ}_F that absorbs high-energy real-emission contributions into PDFs, stabilising predictions for η_c, η_b, and a fictitious scalar across wide energy ranges. The authors demonstrate that this scale-corrected approach yields positive, slowly varying K^NLO factors and enables meaningful use of low-scale gluon PDFs to constrain the gluon content of the proton, potentially guiding future PDF fits. They also provide analytical expressions for σ and dσ/dy, explore gluon-luminosity sensitivities, and discuss experimental prospects to measure η_c (and η_b) to refine low-scale gluon PDFs, with broader implications for NRQCD and collinear factorisation at low scales.

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 MSbar 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, alphaS 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^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.

Paper Structure

This paper contains 19 sections, 27 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. $\eta_Q$ production up to NLO in the collinear and NRQCD factorisations
  3. CSM, NRQCD and collinear factorisation
  4. $\eta_Q$ hadroproduction at LO
  5. $\eta_Q$ hadroproduction at NLO
  6. On the origin of unphysical $\eta_Q$ cross section at high energies
  7. The NLO partonic cross section and its HE behaviour
  8. From negative partonic cross sections to negative (or positive) hadronic cross sections
  9. A scale choice as solution
  10. Results and discussion
  11. A word on our PDF choice
  12. Assessing the perturbative convergence with $\hat{\mu}_F$ using the $K^{\rm NLO}$ factors
  13. A word on gluon luminosities at NLO and low scales
  14. A digression on the $\eta_b$ detectability
  15. Cross section predictions
  16. ...and 4 more sections

Figures (11)

  • Figure 1: Representative diagrams contributing to $\eta_Q$ hadroproduction via CS channels at orders $\alpha_s^2$ (a), $\alpha_s^3$ (b,c,d,e,f). The quark and antiquark attached to the ellipsis are taken as on-shell and their relative velocity $v$ is set to zero.
  • Figure 2: Gluon PDFs as encoded in PDF4LHC15_nlo_30 Butterworth:2015oua, JR14NLO08VF Jimenez-Delgado:2014twa, NNPDF31sx_nlonllx_as_0118 Ball:2017otu, CT14nlo Dulat:2015mca, MMHT14nlo Harland-Lang:2014zoa, NNPDF31_nlo_as_0118 Ball:2017nwa for two scale values : (a) 1.55 GeV and (b) 3 GeV. In addition, we have added on (a) (solid black lines) the resulting constraints on NNPDF3.0 obtained by Flett et al. under some asumptions Flett:2020duk from $J/\psi$ exclusive photoproduction. [These plots have been adapted from plots generated by APFEL web Bertone:2013vaaCarrazza:2014gfa].
  • Figure 3: $K^{\rm NLO}|_{y=0}$ for $\eta_c$ (top) and $\eta_b$ (bottom) (for PDF4LHC15_nlo_30 (left), JR14NLO08VF (middle) and NNPDF31sx_nlonllx_as_0118 (right)) as a function of $\sqrt{s}$ for the usual 7-point scale choices and our $\hat{\mu}_F$ scale with $\mu_R=\mu_F$.
  • Figure 4: $K^{\rm NLO}$ for (top) fictitious $\tilde{H}^0$ and (bottom) $H^0$ with different (fictitious) heavy-quark masses for PDF4LHC15_nlo_30 as a function of $\sqrt{s}/M_H$ for the usual 7-point scale choices and our $\hat{\mu}_F$ scale with $\mu_R=\mu_F$. [Only the case (f) is realistic, all the other are academical examples.]
  • Figure 5: $\tau_0 \frac{d\mathcal{L}}{d\tau dy}$ as function of energy $\sqrt{s}$ and at $y=0$ (top) and $\frac{d\mathcal{L}}{d\tau dy}$ as function of $y$ at $\sqrt{s}=14$ TeV (bottom) for $M=3$ GeV (for PDF4LHC15_nlo_30 (left), JR14NLO08VF (middle) and NNPDF31sx_nlonllx_as_0118 (right)) for 3 $\mu_F$ values ($0.5 M$, $M$ and $2 M$).
  • ...and 6 more figures