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Multi-gluon one-loop amplitudes using tensor integrals

A. van Hameren

TL;DR

The paper develops a fully numerical tensor-integral framework for computing one-loop QCD amplitudes with many external gluons, combining tensor reduction, tensor symmetrization, and ordered-gluon recursion. It extends tree-level Berends–Giele relations to include loop contributions via auxiliary configurations and introduces ghost/quark loops and the R2-term to ensure gauge invariance, yielding a complete workflow from tensor integrals to the one-loop amplitude $A_n^{(1)}$. The method is shown to be competitive with unitarity-based approaches up to $n=10$, with careful attention to numerical stability and precision, and is implemented in Fortran77 with validation against established results. The approach is fully numerical, automatable, and extendable to other theories, highlighting its potential as a practical alternative for high-multiplicity loop computations.

Abstract

An efficient numerical algorithm to evaluate one-loop amplitudes using tensor integrals is presented. In particular, it is shown by explicit calculations that for ordered QCD amplitudes with a number of external legs up to 10, its performance is competitive with other methods.

Multi-gluon one-loop amplitudes using tensor integrals

TL;DR

The paper develops a fully numerical tensor-integral framework for computing one-loop QCD amplitudes with many external gluons, combining tensor reduction, tensor symmetrization, and ordered-gluon recursion. It extends tree-level Berends–Giele relations to include loop contributions via auxiliary configurations and introduces ghost/quark loops and the R2-term to ensure gauge invariance, yielding a complete workflow from tensor integrals to the one-loop amplitude . The method is shown to be competitive with unitarity-based approaches up to , with careful attention to numerical stability and precision, and is implemented in Fortran77 with validation against established results. The approach is fully numerical, automatable, and extendable to other theories, highlighting its potential as a practical alternative for high-multiplicity loop computations.

Abstract

An efficient numerical algorithm to evaluate one-loop amplitudes using tensor integrals is presented. In particular, it is shown by explicit calculations that for ordered QCD amplitudes with a number of external legs up to 10, its performance is competitive with other methods.

Paper Structure

This paper contains 22 sections, 59 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tensor reduction
  3. Tensor symmetrization
  4. Ordered gluon one-loop amplitudes
  5. Recursive relations for ordered gluon tree-level amplitudes
  6. Recursive relations for ordered gluon one-loop amplitudes
  7. Including ghost and quark loops
  8. The $\EuScript{R}_{2}$-term
  9. Results
  10. Conclusion
  11. Reproduction of existing results
  12. $n=6$
  13. $n=7$
  14. $n=8$
  15. $n=9$
  16. ...and 7 more sections

Figures (2)

  • Figure 1: The distribution of the quantity in Eq.(\ref{['Result47']}) (left) and Eq.(\ref{['Result83']}) (right) for calculations at the double precision level. The lower graphs represent the same distributions as the upper ones, but with a logarithmic scale for the y-axis.
  • Figure 2: The right tail of the distributions for $n=10$ of Fig.\ref{['Fig1']} (evaluated at double precision level), and the distributions for the same phase space points when the tensor integrals are evaluated at quadruple precision, and when everything is evaluated at quadruple precision.