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NGluon: A Package to Calculate One-loop Multi-gluon Amplitudes

Simon Badger, Benedikt Biedermann, Peter Uwer

TL;DR

NGluon provides a fully numerical implementation of one-loop, color-ordered $n$-gluon amplitudes in pure Yang-Mills theory using generalized unitarity. It decomposes amplitudes into cut-constructible and rational parts, extracting scalar-integral coefficients through on-shell cuts and Fourier projection, with double-precision accuracy up to $n=14$ gluons and selective use of extended precision. The library demonstrates high numerical stability and practical runtime performance suitable for LHC phenomenology, and includes cross-checks against established analytic results. It also offers a sample implementation to guide future developments toward full QCD applications, including fermion-loop extensions.

Abstract

We present a computer library for the numerical evaluation of colour-ordered n-gluon amplitudes at one-loop order in pure Yang-Mills theory. The library uses the recently developed technique of generalised unitarity. Running in double precision the library yields reliable results for up to 14 gluons with only a small fraction of events requiring a re-evaluation using extended floating point arithmetic. We believe that the library presented here provides an important contribution to future LHC phenomenology. The program may also prove useful in cross checking results obtained by other methods. In addition, the code provides a sample implementation which may serve as a starting point for further developments.

NGluon: A Package to Calculate One-loop Multi-gluon Amplitudes

TL;DR

NGluon provides a fully numerical implementation of one-loop, color-ordered -gluon amplitudes in pure Yang-Mills theory using generalized unitarity. It decomposes amplitudes into cut-constructible and rational parts, extracting scalar-integral coefficients through on-shell cuts and Fourier projection, with double-precision accuracy up to gluons and selective use of extended precision. The library demonstrates high numerical stability and practical runtime performance suitable for LHC phenomenology, and includes cross-checks against established analytic results. It also offers a sample implementation to guide future developments toward full QCD applications, including fermion-loop extensions.

Abstract

We present a computer library for the numerical evaluation of colour-ordered n-gluon amplitudes at one-loop order in pure Yang-Mills theory. The library uses the recently developed technique of generalised unitarity. Running in double precision the library yields reliable results for up to 14 gluons with only a small fraction of events requiring a re-evaluation using extended floating point arithmetic. We believe that the library presented here provides an important contribution to future LHC phenomenology. The program may also prove useful in cross checking results obtained by other methods. In addition, the code provides a sample implementation which may serve as a starting point for further developments.

Paper Structure

This paper contains 25 sections, 31 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Universal pole structure
  4. Tree-level amplitudes
  5. Cut integrals and the loop momentum parametrisation
  6. Cut-constructible contributions
  7. The box coefficients
  8. The triangle coefficients
  9. Bubble coefficients
  10. Rational Contributions
  11. The pentagon coefficients
  12. The box coefficients
  13. The triangle coefficients
  14. The bubble coefficients
  15. Installation
  16. ...and 10 more sections

Figures (7)

  • Figure 1: Figure 1(a) shows the labelling of the propagators in the integral representation of the amplitude. Figure 1(b) shows a generic quadruple cut where propagators $D_1,D_3,D_4$ and $D_6$ are replaced by on-shell delta functions and the amplitude factorises into a product of four tree amplitudes.
  • Figure 2: Test of the accuracy estimated from the scale test.
  • Figure 3: Accuracy for the $\frac{1}{ \epsilon^2}$-pole, $\frac{1}{\epsilon}$-pole and the finite part.
  • Figure 4: The average accuracy as estimated from the scale test as function of the number of gluons. The band gives a measure for the width of the distribution
  • Figure 5: Accuracy for a NMHV amplitude for 6 gluons.
  • ...and 2 more figures