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Scattering Amplitudes, Wilson Loops and the String/Gauge Theory Correspondence

Luis F. Alday, Radu Roiban

TL;DR

The paper surveys both weak- and strong-coupling analyses of planar N=4 SYM scattering amplitudes, emphasizing a deep connection to light-like Wilson loops through dual conformal symmetry and AdS/CFT. It covers on-shell methods, unitarity, and IR factorization at weak coupling, and then maps partial amplitudes to minimal-area strings in AdS at strong coupling, including regularization schemes. A central theme is the amplitude/Wilson loop duality, supported by Ward identities and explicit multi-point tests, while also highlighting the BDS ansatz and its observed limitations via remainder functions starting at six points. The work outlines open problems, such as constructing higher-point strong-coupling minimal surfaces, understanding integrability implications, and extending the framework to less supersymmetric theories, with potential impact on broader gauge theories.

Abstract

We review, in a self-contained and pedagogical manner, recent developments and techniques for the evaluation of the scattering amplitudes of planar N=4 SYM theory at both weak and strong coupling. Special emphasis is placed on the newly discovered connection between a special class of amplitudes and the expectation values of particular cusped light-like Wilson loops.

Scattering Amplitudes, Wilson Loops and the String/Gauge Theory Correspondence

TL;DR

The paper surveys both weak- and strong-coupling analyses of planar N=4 SYM scattering amplitudes, emphasizing a deep connection to light-like Wilson loops through dual conformal symmetry and AdS/CFT. It covers on-shell methods, unitarity, and IR factorization at weak coupling, and then maps partial amplitudes to minimal-area strings in AdS at strong coupling, including regularization schemes. A central theme is the amplitude/Wilson loop duality, supported by Ward identities and explicit multi-point tests, while also highlighting the BDS ansatz and its observed limitations via remainder functions starting at six points. The work outlines open problems, such as constructing higher-point strong-coupling minimal surfaces, understanding integrability implications, and extending the framework to less supersymmetric theories, with potential impact on broader gauge theories.

Abstract

We review, in a self-contained and pedagogical manner, recent developments and techniques for the evaluation of the scattering amplitudes of planar N=4 SYM theory at both weak and strong coupling. Special emphasis is placed on the newly discovered connection between a special class of amplitudes and the expectation values of particular cusped light-like Wilson loops.

Paper Structure

This paper contains 43 sections, 201 equations, 30 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Scattering amplitudes at weak coupling
  3. Organization, presentation and general properties
  4. Spinor helicity and color ordering
  5. General properties of color ordered amplitudes
  6. Some simple examples
  7. Soft/Collinear factorization
  8. Loop amplitudes; generalized unitarity-based method
  9. Calculations at higher loops
  10. Some explicit higher loop results at low multiplicity
  11. A possible integral basis at higher loops; Conformal integrals
  12. The BDS conjecture and potential departures form it
  13. The six-point MHV amplitude at two-loops and the BDS ansatz
  14. Scattering amplitudes at strong coupling through the AdS/CFT duality
  15. The general construction
  16. ...and 28 more sections

Figures (30)

  • Figure 1: The planar three-loop open string diagram contributing to the five-gluon scattering. The single-trace structure is manifest.
  • Figure 2: Soft/Collinear factorization and its planar limit.
  • Figure 3: Singlet and nonsinglet cuts of a one-loop four-gluon amplitude.
  • Figure 4: Box, triangle and bubble scalar integrals. The clusters at each corner are constructed from color-adjacent external legs. If more than one external leg is present at a corner, then that corner is "massive" as the total momentum is no longer light-like.
  • Figure 5: The one-mass (a) and easy two-mass (b) integrals.
  • ...and 25 more figures