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Six-Photon Amplitudes in Scalar QED

C. Bernicot, J. -Ph. Guillet

TL;DR

This work delivers complete analytic one-loop six-photon helicity amplitudes across scalar QED, QED, and N=1 supersymmetric QED by leveraging unitarity-cut methods and a supersymmetric decomposition that connects the three theories. The amplitudes are reduced to a small basis of master integrals with a minimal set of nonzero coefficients, yielding compact spinor-helicity expressions and revealing structure such as the absence of rational terms and, in many cases, triangles. The authors provide explicit formulas for MHV and NMHV sectors, demonstrate consistency with known results, and present numerical studies that illustrate infrared/collinear safety, parity relations, and Landau-singularity effects including double-parton scattering. Their results offer a robust analytic and numerical framework for one-loop multi-photon processes and establish techniques transferable to higher-multiplicity QED and related theories.

Abstract

The analytical result for the six-photon helicity amplitudes in scalar QED is presented. To compute the loop, a recently developed method based on multiple cuts is used. The amplitudes for QED and $QED^{\caln=1}$ are also derived using the supersymmetric decomposition linking the three theories.

Six-Photon Amplitudes in Scalar QED

TL;DR

This work delivers complete analytic one-loop six-photon helicity amplitudes across scalar QED, QED, and N=1 supersymmetric QED by leveraging unitarity-cut methods and a supersymmetric decomposition that connects the three theories. The amplitudes are reduced to a small basis of master integrals with a minimal set of nonzero coefficients, yielding compact spinor-helicity expressions and revealing structure such as the absence of rational terms and, in many cases, triangles. The authors provide explicit formulas for MHV and NMHV sectors, demonstrate consistency with known results, and present numerical studies that illustrate infrared/collinear safety, parity relations, and Landau-singularity effects including double-parton scattering. Their results offer a robust analytic and numerical framework for one-loop multi-photon processes and establish techniques transferable to higher-multiplicity QED and related theories.

Abstract

The analytical result for the six-photon helicity amplitudes in scalar QED is presented. To compute the loop, a recently developed method based on multiple cuts is used. The amplitudes for QED and are also derived using the supersymmetric decomposition linking the three theories.

Paper Structure

This paper contains 25 sections, 59 equations, 7 figures.

Table of Contents

  1. Introduction and Notation
  2. Structure of the amplitudes
  3. Decomposition of the amplitudes $A_{6}^{fermion}$, $A_{6}^{scalar}$ and $A_{6}^{{\cal N}=1}$
  4. Tree amplitudes
  5. Additional reductions
  6. $A_{6}^{scalar}$ amplitudes
  7. $A_{N}^{scalar}(1^{-},2^{-},3^{+}...,N^{+}), \ N> 4$ helicity amplitudes
  8. $A_{6}^{scalar}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+})$ helicity amplitude
  9. $A_{6}^{fermion}$ and $A_{6}^{{\cal N}=1}$ amplitudes
  10. $A_{N}^{fermion}(1^{-},2^{-},3^{+}...,N^{+})$ and $A_{N}^{{\cal N}=1}(1^{-},2^{-},3^{+}...,N^{+})$ helicity amplitudes
  11. $A_{6}^{fermion}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+})$ and $A_{6}^{{\cal N}=1}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+})$ helicity amplitudes
  12. Absence of triangles in $A_6^{{\cal N}=1}$
  13. Numerical results
  14. The MHV amplitudes.
  15. The NMHV amplitudes
  16. ...and 10 more sections

Figures (7)

  • Figure 1: Tree amplitudes needed for the six-photon amplitudes. The particles associated with a plain line are scalars or fermions.
  • Figure 2: Spinor QED loop.
  • Figure 3: Spinor QED loop.
  • Figure 4: The MHV six-photon amplitudes with the Nagy and Soper nagy configuration for the three theories. Note that the curve for $A^{scalar}$ amplitude has a minimum which is not zero.
  • Figure 5: NMHV six-photon amplitudes with the Nagy and Soper nagy configuration for the three QED.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Remark 2.1
  • Remark 2.2