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Double collinear splitting amplitudes at next-to-leading order

German F. R. Sborlini, Daniel de Florian, German Rodrigo

TL;DR

This work tackles the IR structure of QCD at NLO by computing 1→2 splitting amplitudes across multiple DREG schemes and examining their consistency with Catani’s factorization. By introducing scalar-gluons, the authors relate results from different schemes and show how to incorporate these extra degrees of freedom to preserve the correct divergent structure. They provide explicit NLO splitting matrices for q→gq, g→qq̄, and g→gg, along with the corresponding Altarelli-Parisi kernels, and extend the analysis to QCD+QED splittings, including q→γq and γ→q q̄. The study demonstrates that scheme-dependent parts can be organized and related via universal IR pieces, enabling cross-checks and efficient computations across HV, FDH, and CDR schemes, with practical implications for precision collider predictions.

Abstract

We compute the next-to-leading order (NLO) QCD corrections to the $1 \to 2$ splitting amplitudes in different dimensional regularization (DREG) schemes. Besides recovering previously known results, we explore new DREG schemes and analyze their consistency by comparing the divergent structure with the expected behavior predicted by Catani's formula. Through the introduction of scalar-gluons, we show the relation among splittings matrices computed using different schemes. Also, we extended this analysis to cover the double collinear limit of scattering amplitudes in the context of QCD+QED.

Double collinear splitting amplitudes at next-to-leading order

TL;DR

This work tackles the IR structure of QCD at NLO by computing 1→2 splitting amplitudes across multiple DREG schemes and examining their consistency with Catani’s factorization. By introducing scalar-gluons, the authors relate results from different schemes and show how to incorporate these extra degrees of freedom to preserve the correct divergent structure. They provide explicit NLO splitting matrices for q→gq, g→qq̄, and g→gg, along with the corresponding Altarelli-Parisi kernels, and extend the analysis to QCD+QED splittings, including q→γq and γ→q q̄. The study demonstrates that scheme-dependent parts can be organized and related via universal IR pieces, enabling cross-checks and efficient computations across HV, FDH, and CDR schemes, with practical implications for precision collider predictions.

Abstract

We compute the next-to-leading order (NLO) QCD corrections to the splitting amplitudes in different dimensional regularization (DREG) schemes. Besides recovering previously known results, we explore new DREG schemes and analyze their consistency by comparing the divergent structure with the expected behavior predicted by Catani's formula. Through the introduction of scalar-gluons, we show the relation among splittings matrices computed using different schemes. Also, we extended this analysis to cover the double collinear limit of scattering amplitudes in the context of QCD+QED.

Paper Structure

This paper contains 21 sections, 185 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dimensional regularization and QCD
  3. DREG schemes definition
  4. Scalar-gluons: Lagrangian level decomposition
  5. Effective Feynman rules and other considerations for scalar-gluons
  6. Computational implementation
  7. Collinear limits of scattering amplitudes in QCD
  8. The $q\rightarrow q g$ splitting matrix
  9. Amplitude level results
  10. Scheme dependence and divergent structure
  11. NLO corrections to AP kernels
  12. The $g\rightarrow q \bar{q}$ splitting matrix
  13. Amplitude level results
  14. NLO corrections to AP kernels
  15. The $g \rightarrow gg$ splitting matrix
  16. ...and 6 more sections

Figures (14)

  • Figure 1: Available vertices involving scalar-gluons. Expanding QCD Lagrangian, we find six kind of vertices: A fermion-scalar-fermion, B gluon-scalar-gluon, C triple-scalar interaction, D scalar-gluon-scalar, E 2scalar-2gluon, and F 4scalar. Momenta associated with gluons and scalar-gluons are considered outgoing.
  • Figure 2: Usual $4$-dimensional QCD interaction vertices: A fermion-gluon-fermion, B triple-gluon and C quadruple-gluon vertex. Momenta associated with gluons are considered outgoing.
  • Figure 3: Typical contribution to the most divergent part of an $n$-particle scattering amplitude in the double collinear limit.
  • Figure 4: Feynman diagrams associated with $q(\tilde{P}) \rightarrow g(p_1) q(p_2)$ at NLO, including the self-energy correction to the parent parton. We show all the standard QCD contributions up to ${\cal O}(g_s^3)$.
  • Figure 5: Feynman diagrams associated with the scalar-gluon contributions to $q(\tilde{P}) \rightarrow g(p_1) q(p_2)$ at NLO. We show only SCA-nHV configurations.
  • ...and 9 more figures