The soft-collinear bootstrap: \( \mathcal{N} = {4} \) Yang-Mills amplitudes at six- and seven-loops

  • J. L. Bourjaily
  • A. DiRe
  • A. Shaikh
  • M. Spradlin
  • A. Volovich
Article

Abstract

Infrared divergences in scattering amplitudes arise when a loop momentum becomes collinear with a massless external momentum p. In gauge theories, it is known that the L-loop logarithm of a planar amplitude has much softer infrared singularities than the L-loop amplitude itself. We argue that planar amplitudes in \( \mathcal{N} = {4} \) super-Yang-Mills theory enjoy softer than expected behavior as p already at the level of the integrand. Moreover, we conjecture that the four-point integrand can be uniquely determined, to any loop-order, by imposing the correct soft-behavior of the logarithm together with dual conformal invariance and dihedral symmetry. We use these simple criteria to determine explicit formulae for the four-point integrand through seven-loops, finding perfect agreement with previously known results through five-loops. As an input to this calculation, we enumerate all four-point dual conformally invariant (DCI) integrands through seven-loops, an analysis which is aided by several graph-theoretic theorems we prove about general DCI integrands at arbitrary loop-order. The six- and seven-loop amplitudes receive non-zero contributions from 229 and 1873 individual DCI diagrams respectively. PDF and Mathematica files with all of our results are provided at http://goo.gl/qIKe8.

Keywords

Supersymmetric gauge theory Scattering Amplitudes 

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Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • J. L. Bourjaily
    • 1
  • A. DiRe
    • 2
  • A. Shaikh
    • 3
  • M. Spradlin
    • 2
  • A. Volovich
    • 2
  1. 1.Department of PhysicsHarvard UniversityCambridgeU.S.A.
  2. 2.Department of PhysicsBrown UniversityProvidenceU.S.A.
  3. 3.School of EngineeringBrown UniversityProvidenceU.S.A.

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