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The N$^3$LO Twist-2 Matching of Helicity TMDs and SIDIS $q_\ast$ Spectrum

Yu Jiao Zhu

TL;DR

The paper delivers the first complete $N^3LO$ twist-2 matching for helicity TMD PDFs and FFs, together with the full set of NNLO polarized DGLAP splitting functions and $N^3LL$-level predictions for the SIDIS $q_*$ spectrum within TMD factorization. It systematically develops operator definitions, collinear factorization, and the treatment of $oldsymbol{iggamma}_5$ in different schemes, yielding NNLO space-like and new time-like helicity splitting functions. The work provides the boundary conditions and perturbative inputs needed for precision global analyses of spin-dependent TMDs and offers a robust framework for SIDIS observables at the EIC, including potential small-$x$ spin dynamics. Collectively, these results advance our understanding of the proton’s spin structure and the interplay between perturbative QCD evolution and nonperturbative TMD physics, with direct phenomenological relevance for future polarized collider experiments.

Abstract

We compute the twist-2 matching of transverse momentum dependent (TMD) helicity parton distribution and fragmentation functions at next-to-next-to-next-to-leading order (N$^3$LO) in QCD. This calculation entails the complete set of next-to-next-to-leading order (NNLO) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting functions govering the evolution of helicity-dependent parton distribution functions (PDFs) and fragmentation functions (FFs). Within TMD factorization framework, we quantify the impact of radiative corrections by completing the next-to-next-to-next-to-leading logarithmic (N$^3$LL) prediction for lepton-hadron transverse momentum imbalance in semi-inclusive deep inelastic scattering (SIDIS). Our results provide the most precise theoretical input for probing the helicity structure and confined motion of quarks and gluons at future electron-ion collider (EIC).

The N$^3$LO Twist-2 Matching of Helicity TMDs and SIDIS $q_\ast$ Spectrum

TL;DR

The paper delivers the first complete twist-2 matching for helicity TMD PDFs and FFs, together with the full set of NNLO polarized DGLAP splitting functions and -level predictions for the SIDIS spectrum within TMD factorization. It systematically develops operator definitions, collinear factorization, and the treatment of in different schemes, yielding NNLO space-like and new time-like helicity splitting functions. The work provides the boundary conditions and perturbative inputs needed for precision global analyses of spin-dependent TMDs and offers a robust framework for SIDIS observables at the EIC, including potential small- spin dynamics. Collectively, these results advance our understanding of the proton’s spin structure and the interplay between perturbative QCD evolution and nonperturbative TMD physics, with direct phenomenological relevance for future polarized collider experiments.

Abstract

We compute the twist-2 matching of transverse momentum dependent (TMD) helicity parton distribution and fragmentation functions at next-to-next-to-next-to-leading order (NLO) in QCD. This calculation entails the complete set of next-to-next-to-leading order (NNLO) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting functions govering the evolution of helicity-dependent parton distribution functions (PDFs) and fragmentation functions (FFs). Within TMD factorization framework, we quantify the impact of radiative corrections by completing the next-to-next-to-next-to-leading logarithmic (NLL) prediction for lepton-hadron transverse momentum imbalance in semi-inclusive deep inelastic scattering (SIDIS). Our results provide the most precise theoretical input for probing the helicity structure and confined motion of quarks and gluons at future electron-ion collider (EIC).

Paper Structure

This paper contains 17 sections, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The TMD Factorization of $q_\ast$ Spectrum
  3. Renormalization Group Evolution and $q_\ast$ Spectrum with RG Improved Perturbation Theory
  4. Twist-2 Matching of helicity TMDs
  5. Operator definitions for the TMD helicity PDFs and FFs
  6. Collinear mass factorization and renormalization group equations
  7. $\gamma_5$ in dimensional regularization and scheme transformations
  8. The NNLO helicity-dependent splitting functions
  9. Space-like results in ${\overline{\text{MS}}}$
  10. Time-like results in ${\overline{\text{MS}}}$
  11. N$^3$LO coefficient functions and small-x expansion
  12. Conclusion
  13. Flavor decomposition
  14. Renormalization Group Consistency
  15. QCD Beta Function
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Resummed $q_\ast$ distributions at EIC energies.
  • Figure 2: Resummed $q_\ast$ distributions at COMPASS energies.
  • Figure 3: Impact of non-perturbative model function of BLNY18 Sun:2014dqm for pion $q_\ast$-distribution. Here, all the results are implemented with $b_\ast$ prescription, for otherwise the integration diverges.