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Hidden Simplicity of Gauge Theory Amplitudes

J. M. Drummond

TL;DR

The notes argue that planar $ ext{N}=4$ super Yang–Mills theory exhibits hidden, highly constraining symmetries that organize its scattering amplitudes. Tree-level amplitudes can be constructed recursively via BCFW shifts, revealing nonlocal dual superconformal invariants that together generate a Yangian symmetry. At loop level, dual conformal symmetry constrains loop integrals and underpins a deep amplitude–Wilson loop duality, with a finite remainder beyond the BDS ansatz emerging at higher points. These structures point to an underlying integrability of the planar S-matrix and motivate further exploration of symmetry-based methods beyond MHV and planar limits.

Abstract

These notes were given as lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2010. We describe the structure of scattering amplitudes in gauge theories, focussing on the maximally supersymmetric theory to highlight the hidden symmetries which appear. Using the BCFW recursion relations we solve for the tree-level S-matrix in N=4 super Yang-Mills theory, and describe how it produces a sum of invariants of a large symmetry algebra. We review amplitudes in the planar theory beyond tree-level, describing the connection between amplitudes and Wilson loops, and discuss the implications of the hidden symmetries.

Hidden Simplicity of Gauge Theory Amplitudes

TL;DR

The notes argue that planar super Yang–Mills theory exhibits hidden, highly constraining symmetries that organize its scattering amplitudes. Tree-level amplitudes can be constructed recursively via BCFW shifts, revealing nonlocal dual superconformal invariants that together generate a Yangian symmetry. At loop level, dual conformal symmetry constrains loop integrals and underpins a deep amplitude–Wilson loop duality, with a finite remainder beyond the BDS ansatz emerging at higher points. These structures point to an underlying integrability of the planar S-matrix and motivate further exploration of symmetry-based methods beyond MHV and planar limits.

Abstract

These notes were given as lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2010. We describe the structure of scattering amplitudes in gauge theories, focussing on the maximally supersymmetric theory to highlight the hidden symmetries which appear. Using the BCFW recursion relations we solve for the tree-level S-matrix in N=4 super Yang-Mills theory, and describe how it produces a sum of invariants of a large symmetry algebra. We review amplitudes in the planar theory beyond tree-level, describing the connection between amplitudes and Wilson loops, and discuss the implications of the hidden symmetries.

Paper Structure

This paper contains 7 sections, 160 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Tree-level gauge theory scattering amplitudes
  3. BCFW recursion relations
  4. Dual superconformal symmetry
  5. Loop corrections
  6. Summary
  7. Conventions

Figures (15)

  • Figure 1: Schematic form for the propagators and vertices for Yang-Mills theory. The detailed index structure is not important for us. We just note that the colour structure enters only via the structure constants, while the momentum dependence appears in the propagators as $1/p^2$ and in the three-point vertices as a positive power of $p$.
  • Figure 2: Colour rules.
  • Figure 3: Different amplitudes classified according to their helicity structures. Parity acts as reflection about the vertical axis of the diagram, swapping MHV and $\overline{\rm MHV}$ amplitudes for example.
  • Figure 4: An example of a Feynman diagram showing the propagators affected by the BCFW shift.
  • Figure 5: The sum over states giving a particular residue.
  • ...and 10 more figures