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The Unitarity Method using a Canonical Basis Approach

David C. Dunbar, Warren B. Perkins, Edmund Warrick

TL;DR

This work develops and applies a canonical-basis Unitarity framework to compute one-loop gauge-theory amplitudes. By projecting two- and triple-cut data onto a finite set of canonical forms, the authors extract box, triangle, and bubble coefficients as explicit rational expressions, enabling closed analytic results for the ${\cal N}=1$ seven-gluon NMHV amplitude. The approach systematically reduces the problem to algebraic reconstruction from cuts, producing compact expressions without reparametrising integration measures and generalising to higher-point NMHV configurations via well-defined canonical forms. The results demonstrate the practicality and scalability of the method for complex one-loop amplitudes and offer explicit compact analytic results suitable for further analysis and phenomenology.

Abstract

Various implementations of the Unitarity method have been developed to compute one-loop amplitudes in gauge theories. In this paper we present an implementation which uses canonical forms to generate the rational coefficients of the basis integral functions. As an example, we present the results for the N=1 contribution to seven gluon scattering in closed, rational, analytic form.

The Unitarity Method using a Canonical Basis Approach

TL;DR

This work develops and applies a canonical-basis Unitarity framework to compute one-loop gauge-theory amplitudes. By projecting two- and triple-cut data onto a finite set of canonical forms, the authors extract box, triangle, and bubble coefficients as explicit rational expressions, enabling closed analytic results for the seven-gluon NMHV amplitude. The approach systematically reduces the problem to algebraic reconstruction from cuts, producing compact expressions without reparametrising integration measures and generalising to higher-point NMHV configurations via well-defined canonical forms. The results demonstrate the practicality and scalability of the method for complex one-loop amplitudes and offer explicit compact analytic results suitable for further analysis and phenomenology.

Abstract

Various implementations of the Unitarity method have been developed to compute one-loop amplitudes in gauge theories. In this paper we present an implementation which uses canonical forms to generate the rational coefficients of the basis integral functions. As an example, we present the results for the N=1 contribution to seven gluon scattering in closed, rational, analytic form.

Paper Structure

This paper contains 21 sections, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Organisation of the Amplitudes
  3. Canonical Basis for Bubble Coefficients from Two-Particle Cuts
  4. Order $\ell^0$ Terms
  5. Terms of Order $\ell^1$ and $\ell^2$
  6. Example: Seven-point ${\cal N} = 1$ Contributions
  7. $A^{{\cal N} = 1}_7(1^-,2^-,3^+,4^-,5^+,6^+,7^+)$
  8. $A^{{\cal N} = 1}_7(1^-,2^-,3^+,4^+,5^-,6^+,7^+)$
  9. $A^{{\cal N} = 1}_7(1^-,2^+,3^-,4^+,5^-,6^+,7^+)$
  10. $C_X$ Functions
  11. $D_X$ Functions
  12. Canonical Basis for Triangle Coefficients from Triple Cuts
  13. Example: $n$-point Three-Mass Triangles for ${\cal N} = 1$ and Scalar Contributions to NMHV Amplitudes
  14. Summary
  15. Integral Functions
  16. ...and 6 more sections

Figures (3)

  • Figure 1: The triple cut of an amplitude
  • Figure 2: The three mass triangle integral function appearing in the NMHV amplitudes indicating the labeling of the legs.
  • Figure 3: The different types of scalar box.