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The N$^3$LO Twist-2 Matching of TMD Quark Transversity

Yu Jiao Zhu

TL;DR

The paper tackles the challenge of achieving high-precision predictions for quark transversity in TMD factorization by computing the first $N^{3}$LO twist-2 matching coefficients that relate TMD transversity to its collinear counterpart. Employing SCET-based operator definitions and an exponential regulator, the authors derive the full $N^{3}$LO coefficient functions and the complete $NNLO$ transversity DGLAP splitting functions in both space-like and time-like channels, with detailed RG and rapidity evolution frameworks. They provide analytic results and comprehensive numerical fits, including small-$x$ and threshold asymptotics, and perform cross-checks against existing literature, noting minor discrepancies in select channels. The results strengthen the perturbative foundation of spin-dependent TMD phenomenology and enable precise transversity extractions and improved SIDIS predictions for upcoming EIC data. Overall, the work places transversity on par with unpolarized and helicity sectors in precision QCD and paves the way for rigorous global analyses of spin-dependent observables.

Abstract

We present the first next-to-next-to-next-to-leading order (N$^3$LO) calculation of the twist-2 matching coefficients for transverse momentum dependent (TMD) quark transversity parton distribution and fragmentation functions in QCD. This matching relates the TMD quark transversity functions to their collinear counterparts in the large-transverse-momentum regime, and provides essential ingredients for precision TMD phenomenology involving transversely polarized beams. As part of our analysis, we also compute the complete set of next-to-next-to-leading order (NNLO) DGLAP splitting functions governing the QCD evolution of collinear transversity distributions. These results extend the perturbative toolkit for spin-dependent observables and establish the transversity sector on the same theoretical footing as unpolarized and helicity distributions. Our findings enable high-precision extractions of transversity PDFs and facilitate improved theoretical predictions for azimuthal asymmetries in semi-inclusive deep inelastic scattering (SIDIS), especially in light of forthcoming data from the Electron-Ion Collider (EIC).

The N$^3$LO Twist-2 Matching of TMD Quark Transversity

TL;DR

The paper tackles the challenge of achieving high-precision predictions for quark transversity in TMD factorization by computing the first LO twist-2 matching coefficients that relate TMD transversity to its collinear counterpart. Employing SCET-based operator definitions and an exponential regulator, the authors derive the full LO coefficient functions and the complete transversity DGLAP splitting functions in both space-like and time-like channels, with detailed RG and rapidity evolution frameworks. They provide analytic results and comprehensive numerical fits, including small- and threshold asymptotics, and perform cross-checks against existing literature, noting minor discrepancies in select channels. The results strengthen the perturbative foundation of spin-dependent TMD phenomenology and enable precise transversity extractions and improved SIDIS predictions for upcoming EIC data. Overall, the work places transversity on par with unpolarized and helicity sectors in precision QCD and paves the way for rigorous global analyses of spin-dependent observables.

Abstract

We present the first next-to-next-to-next-to-leading order (NLO) calculation of the twist-2 matching coefficients for transverse momentum dependent (TMD) quark transversity parton distribution and fragmentation functions in QCD. This matching relates the TMD quark transversity functions to their collinear counterparts in the large-transverse-momentum regime, and provides essential ingredients for precision TMD phenomenology involving transversely polarized beams. As part of our analysis, we also compute the complete set of next-to-next-to-leading order (NNLO) DGLAP splitting functions governing the QCD evolution of collinear transversity distributions. These results extend the perturbative toolkit for spin-dependent observables and establish the transversity sector on the same theoretical footing as unpolarized and helicity distributions. Our findings enable high-precision extractions of transversity PDFs and facilitate improved theoretical predictions for azimuthal asymmetries in semi-inclusive deep inelastic scattering (SIDIS), especially in light of forthcoming data from the Electron-Ion Collider (EIC).

Paper Structure

This paper contains 18 sections, 77 equations.

Table of Contents

  1. Introduction
  2. The transversely polarized quark TMD PDFs and FFs
  3. Operator definitions for TMD quark transeversity PDFs and FFs
  4. Collinear mass factorization and renormalization group equations
  5. The NNLO transversity splitting functions
  6. Space-like results in ${\overline{\text{MS}}}$
  7. Time-like results in ${\overline{\text{MS}}}$
  8. N$^3$LO coefficients for transversity TMDs in ${\overline{\text{MS}}}$
  9. Small-x and threshold limit
  10. Small-$x$ expansion of TMD transversity PDFs
  11. Small-$z$ expansion of TMD transversity FFs
  12. Numerical fits of the N3LO coefficients
  13. Numerical fit for TMD PDFs
  14. Numerical fit for TMD FFs
  15. Conclusion
  16. ...and 3 more sections