Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Triple-Cut of Scattering Amplitudes

Pierpaolo Mastrolia

TL;DR

This work introduces a triple-cut technique for dimensional-reg regularised one-loop amplitudes, defining the triple-cut as a difference of two double-cuts with opposite i0 on a shared propagator. The authors show how the four-dimensional spinor integration, combined with a Feynman parameter, reduces to residues at branch points of a quadratic function, with the remaining (-2ε)-dimensional integration maps to master-cuts in shifted dimensions. They provide explicit results for several master integrals (1m-triangle, 0m-box, linear triangle, and linear box), extracting triangle- and higher-point coefficients and verifying consistency with known decompositions. The method offers a path to reconstruct amplitudes from generalised cuts in any dimension and has potential applications to no-triangle analyses in gravity and deeper insights into master-integral structures.

Abstract

It is analysed the triple-cut of one-loop amplitudes in dimensional regularisation within spinor-helicity representation. The triple-cut is defined as a difference of two double-cuts with the same particle content, and a same propagator carrying, respectively, causal and anti-causal prescription in each of the two cuts. That turns out into an effective tool for extracting the coefficients of the three-point functions (and higher-point ones) from one-loop-amplitudes. The phase-space integration is oversimplified by using residues theorem to perform the integration over the spinor variables, via the holomorphic anomaly, and a trivial integration on the Feynman parameter. The results are valid for arbitrary values of dimensions.

On Triple-Cut of Scattering Amplitudes

TL;DR

This work introduces a triple-cut technique for dimensional-reg regularised one-loop amplitudes, defining the triple-cut as a difference of two double-cuts with opposite i0 on a shared propagator. The authors show how the four-dimensional spinor integration, combined with a Feynman parameter, reduces to residues at branch points of a quadratic function, with the remaining (-2ε)-dimensional integration maps to master-cuts in shifted dimensions. They provide explicit results for several master integrals (1m-triangle, 0m-box, linear triangle, and linear box), extracting triangle- and higher-point coefficients and verifying consistency with known decompositions. The method offers a path to reconstruct amplitudes from generalised cuts in any dimension and has potential applications to no-triangle analyses in gravity and deeper insights into master-integral structures.

Abstract

It is analysed the triple-cut of one-loop amplitudes in dimensional regularisation within spinor-helicity representation. The triple-cut is defined as a difference of two double-cuts with the same particle content, and a same propagator carrying, respectively, causal and anti-causal prescription in each of the two cuts. That turns out into an effective tool for extracting the coefficients of the three-point functions (and higher-point ones) from one-loop-amplitudes. The phase-space integration is oversimplified by using residues theorem to perform the integration over the spinor variables, via the holomorphic anomaly, and a trivial integration on the Feynman parameter. The results are valid for arbitrary values of dimensions.

Paper Structure

This paper contains 13 sections, 153 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Double-Cut
  3. Logarithmic Terms of 4D-Double-Cut
  4. Triple-Cut
  5. Examples
  6. Scalar $1m$-Triangle
  7. Scalar $0m$-Box
  8. Linear Triangle
  9. Linear Box
  10. Triple-cut ${\cal N}_{1|2|34}$
  11. Triple-cut ${\cal N}_{12|3|4}$
  12. Triple-Cut in Four Dimensions
  13. Conclusions

Figures (8)

  • Figure 1: Double-Cut.
  • Figure 2: Triple-cut in terms of two double-cuts, respectively with a causal propagator and an anti-causal propagator: ${\cal A}_L, {\cal A}_M$, and ${\cal A}_R$ are tree level amplitudes, respectively depending on the external momenta $K, K_2, K_3$.
  • Figure 3: Triple-cut of a $1m$-Triangle.
  • Figure 4: Triple-cut of a $0m$-Box.
  • Figure 5: Triple-cut of a linear triangle in terms of the master triple-cut.
  • ...and 3 more figures