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The Unresolved Behaviour of Polarized Scattering Matrix Elements at NNLO in QCD

Thomas Gehrmann, Markus Löchner

TL;DR

The paper addresses the challenge of NNLO QCD calculations for polarized processes and derives universal infrared-collinear behavior for longitudinally polarized real radiation. It develops a methodology based on Sudakov kinematics, angular averaging, and color ordering, and computes single-collinear limits up to one loop and triple-collinear limits at tree level using current-decay matrix elements in the Larin scheme. The authors provide explicit polarized splitting functions and cross-checks, including analytic continuation between time-like and space-like kinematics and consistency relations among polarized and unpolarized pieces. These results underpin process-independent subtraction schemes for NNLO polarized observables, enabling precision predictions for spin-dependent processes at future facilities like the Electron-Ion Collider.

Abstract

Spin asymmetries in collisions of spin-polarized hadrons probe polarized parton distributions, which encode the spin structure of the colliding hadrons. To perform precision physics studies with spin asymmetries, higher order QCD corrections to the underlying polarized cross sections are required. Their numerical implementation relies on the use of an infrared subtraction scheme, which extracts the infrared singular pieces from the real and virtual subprocesses. We derive the universal behaviour of longitudinally polarized real radiation matrix elements in their infrared-singular limits up to next-to-next-to-leading order (NNLO) in QCD, thereby enabling the construction of infrared subtraction schemes for generic polarized cross sections at NNLO.

The Unresolved Behaviour of Polarized Scattering Matrix Elements at NNLO in QCD

TL;DR

The paper addresses the challenge of NNLO QCD calculations for polarized processes and derives universal infrared-collinear behavior for longitudinally polarized real radiation. It develops a methodology based on Sudakov kinematics, angular averaging, and color ordering, and computes single-collinear limits up to one loop and triple-collinear limits at tree level using current-decay matrix elements in the Larin scheme. The authors provide explicit polarized splitting functions and cross-checks, including analytic continuation between time-like and space-like kinematics and consistency relations among polarized and unpolarized pieces. These results underpin process-independent subtraction schemes for NNLO polarized observables, enabling precision predictions for spin-dependent processes at future facilities like the Electron-Ion Collider.

Abstract

Spin asymmetries in collisions of spin-polarized hadrons probe polarized parton distributions, which encode the spin structure of the colliding hadrons. To perform precision physics studies with spin asymmetries, higher order QCD corrections to the underlying polarized cross sections are required. Their numerical implementation relies on the use of an infrared subtraction scheme, which extracts the infrared singular pieces from the real and virtual subprocesses. We derive the universal behaviour of longitudinally polarized real radiation matrix elements in their infrared-singular limits up to next-to-next-to-leading order (NNLO) in QCD, thereby enabling the construction of infrared subtraction schemes for generic polarized cross sections at NNLO.

Paper Structure

This paper contains 12 sections, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Kinematics of collinear splittings
  4. Angular averaging
  5. Colour ordering
  6. Derivation of the universal collinear behaviour
  7. Single-collinear behaviour up to one loop
  8. Tree-level spltting functions
  9. One-loop splitting functions
  10. Triple-collinear behaviour at tree level
  11. Conclusions and outlook
  12. Unpolarized triple-collinear splitting functions

Figures (3)

  • Figure 1: Factorization of a tree-level matrix element in the final-final state single-collinear limit $i\parallel j$.
  • Figure 2: Factorization of one-loop matrix element in single-collinear final-final limit $i\parallel j$.
  • Figure 3: Factorization of tree-level matrix element in final state triple-collinear limit $i\parallel j \parallel k$.