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Closed-Form Decomposition of One-Loop Massive Amplitudes

Ruth Britto, Bo Feng, Pierpaolo Mastrolia

Abstract

We present formulas for the coefficients of 2-, 3-, 4- and 5-point master integrals for one-loop massive amplitudes. The coefficients are derived from unitarity cuts in D dimensions. The input parameters can be read off from any unitarity-cut integrand, as assembled from tree-level expressions, after simple algebraic manipulations. The formulas presented here are suitable for analytical as well as numerical evaluation. Their validity is confirmed in two known cases of helicity amplitudes contributing to gg -> gg and gg -> gH, where the masses of the Higgs and the fermion circulating in the loop are kept as free parameters.

Closed-Form Decomposition of One-Loop Massive Amplitudes

Abstract

We present formulas for the coefficients of 2-, 3-, 4- and 5-point master integrals for one-loop massive amplitudes. The coefficients are derived from unitarity cuts in D dimensions. The input parameters can be read off from any unitarity-cut integrand, as assembled from tree-level expressions, after simple algebraic manipulations. The formulas presented here are suitable for analytical as well as numerical evaluation. Their validity is confirmed in two known cases of helicity amplitudes contributing to gg -> gg and gg -> gH, where the masses of the Higgs and the fermion circulating in the loop are kept as free parameters.

Paper Structure

This paper contains 29 sections, 170 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Decomposition in terms of master integrals
  3. The double cut phase space integration
  4. Formulas for the coefficients of master integrals
  5. Box coefficient
  6. Triangle coefficient
  7. Bubble coefficient
  8. Example I: $s_{12}$-channel cut of $A(1^+,2^+,3^+,H)$
  9. The reconstruction of the coefficients
  10. The box coefficient $c_4^{1m}$
  11. The triangle coefficient $c_{[1|2|3H]}$
  12. The coefficient $c_{[12|3|H]}$
  13. Example II: $s_{23}$-channel cut of $A(1^-,2^-,3^+,4^+)$
  14. The reconstruction of the coefficients
  15. The box coefficient $c_4^{0m}$
  16. ...and 14 more sections

Figures (5)

  • Figure 1: Double-cut of Box functions
  • Figure 2: Double-cut of a Triangle
  • Figure 3: Double-cut of a Bubble
  • Figure 4: Double-cut in the $s_{12}$-channel for $A(1^+,2^+,3^+,H)$.
  • Figure 5: Double-cut in the $s_{23}$-channel for $A(1^-,2^-,3^+,4^+)$.