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Non-relativistic effective theory for Quarkonium Production in Hadron Collisions

M. Beneke

TL;DR

Non-relativistic QCD (NRQCD) provides a factorization framework to describe inclusive quarkonium production in hadron collisions by separating short-distance heavy-quark production from long-distance hadronization. The approach introduces color-octet intermediate states and a velocity-scaling power counting, enabling predictions for cross sections and polarization across Tevatron and fixed-target experiments. The paper surveys how NRQCD matches to QCD calculations, discusses polarization signatures and practical data fits, and addresses universality and potential higher-twist corrections. Polarization measurements and χ-state production emerge as crucial tests of the NRQCD factorization picture.

Abstract

I review recent progress in understanding inclusive quarkonium production in hadron collisions. The first part focuses on non-relativistic QCD as an effective theory. I discuss its differences from and similarities with effective theories describing bound states of a single heavy quark, as far as matching calculations beyond tree-level and power counting are concerned. The second part summarizes predictions for charmonium and bottomonium production at collider and fixed target experiments and their comparison with data. The emphasis here is on novel signatures due to color octet production, polarization of quarkonia and the $χ_1/χ_2$ ratio in fixed target collisions.

Non-relativistic effective theory for Quarkonium Production in Hadron Collisions

TL;DR

Non-relativistic QCD (NRQCD) provides a factorization framework to describe inclusive quarkonium production in hadron collisions by separating short-distance heavy-quark production from long-distance hadronization. The approach introduces color-octet intermediate states and a velocity-scaling power counting, enabling predictions for cross sections and polarization across Tevatron and fixed-target experiments. The paper surveys how NRQCD matches to QCD calculations, discusses polarization signatures and practical data fits, and addresses universality and potential higher-twist corrections. Polarization measurements and χ-state production emerge as crucial tests of the NRQCD factorization picture.

Abstract

I review recent progress in understanding inclusive quarkonium production in hadron collisions. The first part focuses on non-relativistic QCD as an effective theory. I discuss its differences from and similarities with effective theories describing bound states of a single heavy quark, as far as matching calculations beyond tree-level and power counting are concerned. The second part summarizes predictions for charmonium and bottomonium production at collider and fixed target experiments and their comparison with data. The emphasis here is on novel signatures due to color octet production, polarization of quarkonia and the ratio in fixed target collisions.

Paper Structure

This paper contains 15 sections, 55 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Effective theory and factorization in quarkonium production
  3. Non-Relativistic QCD
  4. Factorization and matching
  5. Power counting (velocity scaling)
  6. Quarkonium Polarization
  7. Quarkonium production at the Tevatron
  8. Cross sections
  9. Polarization
  10. Quarkonium production at fixed target
  11. Cross sections
  12. The $\chi_1/\chi_2$ ratio
  13. Polarization
  14. Beyond NRQCD
  15. Conclusion

Figures (9)

  • Figure 1: Coulomb correction to point-like $\psi^\dagger\chi$ creation.
  • Figure 2: Trivial example of factorization.
  • Figure 3: Contribution to mixing of $\langle {\cal O}_8^\psi({}^3\!S_1)\rangle$ into $\langle {\cal O}_1^\psi({}^3\!P_0)\rangle$.
  • Figure 4: Diagrammatic representation of factorization in $\gamma^*+A\to \psi+X$. For the cross section the diagram has to be cut.
  • Figure 5: Fit of color octet contributions to direct $J/\psi$ production data from CDF ($\sqrt{s}=1.8\,$TeV, pseudo-rapidity cut $|\eta|<0.6$). Theory: CTEQ4L parton distribution functions, the corresponding $\Lambda_4=235\,$MeV, factorization scale $\mu=(p_t^2+4 m_c^2)^{1/2}$ and $m_c=1.5\,$GeV. Taken from Ref. bkr.
  • ...and 4 more figures