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Automated computation of spin- and colour-correlated Born matrix elements

Stefan Weinzierl

TL;DR

This paper introduces an automated algorithm and a C++ implementation for computing spin- and colour-correlated Born matrix elements in QCD, which are essential for NLO calculations using the dipole subtraction method. It combines dipole subtraction, colour-flow (double-line) notation, colour decomposition, spinor helicity techniques, and Berends-Giele-type recurrence relations to efficiently produce helicity amplitudes and their colour-structured squared norms, even with multiple quark–antiquark pairs and massive quarks. The method precomputes colour structures symbolically at initialization and evaluates momentum-dependent parts numerically, enabling flexible handling of massless/massive quarks and extensions to QCD amplitudes with one electroweak boson. Validation against MadGraph and cross-checks with established NLO codes demonstrate accurate, scalable performance up to seven external particles, paving the way for automated NLO predictions in LHC processes; the remaining challenge lies in automating the one-loop interference term.

Abstract

I report on an implementation of an algorithm for the automated numerical calculation of spin- and colour-correlated Born matrix elements in QCD. These spin- and colour-correlated matrix elements are needed for NLO calculations in combination with the subtraction method. Both massless and massive quarks are considered. There are no restrictions on the number of external particles. As a trivial sub-case, the algorithm also applies to Born matrix elements without any correlations. These are sufficient for leading order calculations.

Automated computation of spin- and colour-correlated Born matrix elements

TL;DR

This paper introduces an automated algorithm and a C++ implementation for computing spin- and colour-correlated Born matrix elements in QCD, which are essential for NLO calculations using the dipole subtraction method. It combines dipole subtraction, colour-flow (double-line) notation, colour decomposition, spinor helicity techniques, and Berends-Giele-type recurrence relations to efficiently produce helicity amplitudes and their colour-structured squared norms, even with multiple quark–antiquark pairs and massive quarks. The method precomputes colour structures symbolically at initialization and evaluates momentum-dependent parts numerically, enabling flexible handling of massless/massive quarks and extensions to QCD amplitudes with one electroweak boson. Validation against MadGraph and cross-checks with established NLO codes demonstrate accurate, scalable performance up to seven external particles, paving the way for automated NLO predictions in LHC processes; the remaining challenge lies in automating the one-loop interference term.

Abstract

I report on an implementation of an algorithm for the automated numerical calculation of spin- and colour-correlated Born matrix elements in QCD. These spin- and colour-correlated matrix elements are needed for NLO calculations in combination with the subtraction method. Both massless and massive quarks are considered. There are no restrictions on the number of external particles. As a trivial sub-case, the algorithm also applies to Born matrix elements without any correlations. These are sufficient for leading order calculations.

Paper Structure

This paper contains 23 sections, 85 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. General setup and basic tools
  3. The dipole formalism
  4. Double line notation
  5. Colour decomposition
  6. Spinor techniques
  7. Recurrence relations
  8. The method
  9. Helicity amplitudes and spin correlations
  10. Amplitudes with more than one quark-antiquark pair
  11. The colour structure
  12. The partial amplitudes
  13. The pure gluon amplitude
  14. QCD amplitudes with one electro-weak boson
  15. Numerical implementation
  16. ...and 8 more sections

Figures (3)

  • Figure 1: The recurrence relation for the gluon current. An off-shell current with $n$ legs can be computed recursively from off-shell currents with fewer legs.
  • Figure 2: The cyclic order of a partial amplitude. Without loss of generality we can assume that quarks follow immediately antiquarks in the clockwise order.
  • Figure 3: An example for the decomposition into colour clusters.