Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

BCJ Identities and $d$-Dimensional Generalized Unitarity

Amedeo Primo, William J. Torres Bobadilla

Abstract

We present a set of relations between one-loop integral coefficients for dimensionally regulated QCD amplitudes. Within dimensional regularization, the combined use of color-kinematics duality and integrand reduction yields the existence of relations between the integrand residues of partial amplitudes with different orderings of the external particles. These relations can be established for the cut-constructible contributions as well for the ones responsible for rational terms. Starting from the general parametrization of one-loop residues and applying Laurent expansion in order to extract the coefficients of the amplitude decomposition in terms of master integrals, we show that the full set of relations can be obtained by considering BCJ identities between d-dimensional tree-levels. We provide explicit examples for multi-gluon scattering amplitudes at one-loop.

BCJ Identities and $d$-Dimensional Generalized Unitarity

Abstract

We present a set of relations between one-loop integral coefficients for dimensionally regulated QCD amplitudes. Within dimensional regularization, the combined use of color-kinematics duality and integrand reduction yields the existence of relations between the integrand residues of partial amplitudes with different orderings of the external particles. These relations can be established for the cut-constructible contributions as well for the ones responsible for rational terms. Starting from the general parametrization of one-loop residues and applying Laurent expansion in order to extract the coefficients of the amplitude decomposition in terms of master integrals, we show that the full set of relations can be obtained by considering BCJ identities between d-dimensional tree-levels. We provide explicit examples for multi-gluon scattering amplitudes at one-loop.

Paper Structure

This paper contains 18 sections, 116 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Color-kinematics duality in $d$-dimensions
  3. Coefficient relations for one-loop amplitudes in $d$-dimensions
  4. Relations for pentagon coefficients
  5. Relations for box coefficients
  6. Relations for triangle coefficients
  7. Relations for bubble coefficients
  8. Examples
  9. Pentagons
  10. Boxes
  11. Triangles
  12. Bubbles
  13. Conclusions
  14. Coefficient relations from 5-point BCJ identities
  15. Relations for pentagon coefficients
  16. ...and 3 more sections

Figures (13)

  • Figure 1: Feynman diagrams for $g^{\bullet}g^{\bullet} \to gg$.
  • Figure 2: Pentagon topologies for the cuts $C_{12|3\ldots k|\left(k+1\right)\ldots l|\left(l+1\right)\ldots m|\left(m+1\right)\ldots n}$ and $C_{21|3\ldots k|\left(k+1\right)\ldots l|\left(l+1\right)\ldots m|\left(m+1\right)\ldots n}$.
  • Figure 3: Box topologies for the cuts $C_{12|3\ldots k|k+1\ldots l|l+1\ldots n}$ and $C_{21|3\ldots k|k+1\ldots l|l+1\ldots n}$.
  • Figure 4: Triangle topologies for the cuts $C_{12|3\ldots k|\left(k+1\right)\ldots n}$ and $C_{21|3\ldots k|\left(k+1\right)\ldots n}$.
  • Figure 5: Bubble topologies for the cuts $C_{12|3\ldots n}$ and $C_{21|3\ldots n}$.
  • ...and 8 more figures