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The Infrared Behavior of One-Loop QCD Amplitudes at Next-to-Next-to-Leading Order

Z. Bern, V. Del Duca, W. B. Kilgore, C. R. Schmidt

TL;DR

This work derives universal, all-orders-in-ε factorization formulas for one-loop QCD amplitudes in soft and collinear limits, enabling NNLO jet calculations by supplying the necessary infrared building blocks. It introduces a primitive amplitude decomposition to cleanly separate color from kinematics and provides explicit factorizing and non-factorizing contributions for gluon and quark splitting, along with soft amplitudes. The results are validated through supersymmetry and Higgs-amplitude checks, and renormalization is addressed within the MS-bar scheme. The framework offers essential tools for constructing NNLO multi-jet predictions and outlines pathways to extend the analysis to two-loop infrared structure, advancing perturbative QCD precision.

Abstract

We present universal factorization formulas describing the behavior of one-loop QCD amplitudes as external momenta become either soft or collinear. Our results are valid to all orders in the dimensional regularization parameter, $\eps$. Terms through $\Ord(\eps^2)$ can contribute in infrared divergent phase space integrals associated with next-to-next-to-leading order jet cross-sections.

The Infrared Behavior of One-Loop QCD Amplitudes at Next-to-Next-to-Leading Order

TL;DR

This work derives universal, all-orders-in-ε factorization formulas for one-loop QCD amplitudes in soft and collinear limits, enabling NNLO jet calculations by supplying the necessary infrared building blocks. It introduces a primitive amplitude decomposition to cleanly separate color from kinematics and provides explicit factorizing and non-factorizing contributions for gluon and quark splitting, along with soft amplitudes. The results are validated through supersymmetry and Higgs-amplitude checks, and renormalization is addressed within the MS-bar scheme. The framework offers essential tools for constructing NNLO multi-jet predictions and outlines pathways to extend the analysis to two-loop infrared structure, advancing perturbative QCD precision.

Abstract

We present universal factorization formulas describing the behavior of one-loop QCD amplitudes as external momenta become either soft or collinear. Our results are valid to all orders in the dimensional regularization parameter, . Terms through can contribute in infrared divergent phase space integrals associated with next-to-next-to-leading order jet cross-sections.

Paper Structure

This paper contains 32 sections, 114 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Review of color decompositions and primitive amplitudes
  3. Review of collinear and soft properties of QCD amplitudes
  4. The collinear behavior
  5. Soft behavior
  6. One-loop soft and collinear splitting amplitudes
  7. Factorizing contributions to the one-loop splitting amplitudes
  8. Factorizing contributions to one-loop $g\to gg$ splitting amplitudes
  9. Factorizing contributions to one-loop $g\to \overline{q}q$ splitting amplitudes
  10. Factorizing contributions to one-loop $q\to qg$ splitting amplitudes
  11. Non-factorizing contributions
  12. Non-factorizing contributions to one loop $g\to gg$ splitting amplitudes
  13. Non-factorizing contributions to one-loop $g\to \overline{q}q$ splitting amplitudes
  14. Non-factorizing contributions to one loop $q\to qg$ splitting amplitudes
  15. Full one-loop splitting amplitudes for leading color partial amplitudes to all orders in $\epsilon$.
  16. ...and 17 more sections

Figures (5)

  • Figure 1: The generic behavior of one-loop amplitudes in the limit as two external momenta become collinear. The shaded disc represents the sum over tree diagrams and the annulus the sum over one-loop diagrams.
  • Figure 2: The three vertices describing (a) ${\cal D}^{\mu,\,{\rm tree}}_{g \to g_1g_2}$, (b) ${\cal D}^{\mu,\,{\rm tree}}_{g \to {\bar{q}}_1 q_2}$ and (c) ${\cal D}^{j,\,{\rm tree}}_{q \to q_1g_2}$.
  • Figure 3: The diagrams in a massless theory (ignoring tadpoles) that need to be calculated to obtain the factorizing contribution to the loop splitting function. The dotted line represents the off-shell leg on which the collinear factorization is performed.
  • Figure 4: The scalar integrals that can appear in the collinear splitting amplitudes. Legs 1 and 2 are collinear.
  • Figure 5: The particular box integral under consideration in this appendix. Leg 3 is composed of a sum over massless momenta so that $K_3^2 \not = 0$.