From the Holomorphic Wilson Loop to `d log' Loop-Integrands of Super-Yang-Mills Amplitudes

Abstract

The S-matrix for planar N = 4 super Yang-Mills theory can be computed as the correlation function for a holomorphic polygonal Wilson loop in twistor space. In an axial gauge, this leads to the construction of the all-loop integrand via MHV diagrams in twistor space. We show that at MHV, this formulation leads directly to expressions for loop integrands in d log form; i.e., the integrand is a product of exterior derivatives of logarithms of rational functions. For higher MHV degree, it is in d log form multiplied by delta functions. The parameters appearing in the d log form arise geometrically as the coordinates of insertion points of propagators on the holomorphic Wilson loop or on MHV vertices. We discuss a number of examples at one and two loops and give a preliminary discussion of the evaluation of the 1-loop MHV amplitude.

v3. updated discussion of Kermit

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• Leading singularities and off-shell conformal integrals
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• The four-loop cusp anomalous dimension in N=4 super Yang-Mills and analytic integration techniques for Wilson line integrals
• ABJM amplitudes and the positive orthogonal grassmannian
• The Double Pentaladder Integral to All Orders
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• The Correlahedron
• From dlogs to dilogs; the super Yang-Mills MHV amplitude revisited
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• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Twistor actions for gauge theory and gravity
• Worldsheet factorization for twistor-strings
• A planar four-loop form factor and cusp anomalous dimension in QCD
• The four-loop six-gluon NMHV ratio function
• $\mathcal{N}=7$ On-Shell Diagrams and Supergravity Amplitudes in Momentum Twistor Space
• Lectures on twistor theory
• The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills Theory