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The N$^3$LO Twist-2 Matching of Linearly Polarized Gluon TMDs

Yu Jiao Zhu

Abstract

We compute the twist-2 matching of the transverse-momentum-dependent (TMD) linearly polarized gluon parton distribution and fragmentation functions at next-to-next-to-next-to-leading order (N$^3$LO) in QCD, supplemented by next-to-next-to-leading logarithmic (NNLL) small-$x$ resummation for the gluon TMD fragmentation functions. These results provide high-precision fixed-order and resummed inputs to TMD phenomenology, and constitute essential theoretical ingredients for future studies of the spin structure and three-dimensional tomography of hadrons at the Electron-Ion Collider (EIC).

The N$^3$LO Twist-2 Matching of Linearly Polarized Gluon TMDs

Abstract

We compute the twist-2 matching of the transverse-momentum-dependent (TMD) linearly polarized gluon parton distribution and fragmentation functions at next-to-next-to-next-to-leading order (NLO) in QCD, supplemented by next-to-next-to-leading logarithmic (NNLL) small- resummation for the gluon TMD fragmentation functions. These results provide high-precision fixed-order and resummed inputs to TMD phenomenology, and constitute essential theoretical ingredients for future studies of the spin structure and three-dimensional tomography of hadrons at the Electron-Ion Collider (EIC).

Paper Structure

This paper contains 17 sections, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definition of linearly polarized gluon TMD PDFs and FFs
  3. N$^3$LO coefficients for linearly polarized gluon TMDs
  4. Renormalization group equations for renormalized coefficient functions
  5. Numerical fits of the N3LO coefficients
  6. Numerical fit for TMD PDFs
  7. Numerical fit for TMD FFs
  8. Perturbative convergence
  9. ${\cal N}=1$ supersymmetry sum rule for the linearly polarized gluon contribution
  10. Small $x$ expansion of the TMD coefficients and resummation for the TMD FFs
  11. Small-$x$ expansion of linearly polarized gluon TMD PDFs
  12. Small-$z$ expansion of linearly polarized gluon TMD FFs
  13. Resummation of small-$z$ logarithms for linearly polarized gluon TMD FFs
  14. Conclusion
  15. QCD Beta Function
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Integrated linearly polarized gluon TMD PDFs at various perturbative order.
  • Figure 2: Linearly polarized gluon contributions to the cumulative small $p_T$ cross section, plotted as functions of the artificial cut $p_T$.
  • Figure 3: Coefficient functions for the linearly polarized gluon TMD FFs. The plots show the fixed-order results at NLO, NNLO and N$^3$LO, together with the resummed predictions including higher-order terms truncated at order $\alpha_s^{15}$.