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Loop amplitudes in gauge theories: modern analytic approaches

Ruth Britto

TL;DR

This review surveys on-shell, unitarity-based methods for analytic loop amplitudes in gauge theories, emphasizing the master-integral basis and the extraction of coefficients via unitarity cuts. It details how four-dimensional cuts, spinor-helicity techniques, and generalized unitarity (quadruple, triple, single cuts) yield box, triangle, and bubble coefficients, with $D$-dimensional unitarity extending to rational terms. The article also covers massive particle treatments, MHV-diagram approaches, recursion for coefficients and rational parts, and a comprehensive survey of recent one-loop results, while outlining challenges and prospects for extending these techniques beyond one loop. Overall, the framework provides a powerful, analytic route to one-loop amplitudes that complements traditional Feynman-diagram calculations and informs multiloop progress, including applications to QCD and electroweak processes.

Abstract

This article reviews on-shell methods for analytic computation of loop amplitudes, emphasizing techniques based on unitarity cuts. Unitarity techniques are formulated generally but have been especially useful for calculating one-loop amplitudes in massless theories such as Yang-Mills theory, QCD, and QED.

Loop amplitudes in gauge theories: modern analytic approaches

TL;DR

This review surveys on-shell, unitarity-based methods for analytic loop amplitudes in gauge theories, emphasizing the master-integral basis and the extraction of coefficients via unitarity cuts. It details how four-dimensional cuts, spinor-helicity techniques, and generalized unitarity (quadruple, triple, single cuts) yield box, triangle, and bubble coefficients, with -dimensional unitarity extending to rational terms. The article also covers massive particle treatments, MHV-diagram approaches, recursion for coefficients and rational parts, and a comprehensive survey of recent one-loop results, while outlining challenges and prospects for extending these techniques beyond one loop. Overall, the framework provides a powerful, analytic route to one-loop amplitudes that complements traditional Feynman-diagram calculations and informs multiloop progress, including applications to QCD and electroweak processes.

Abstract

This article reviews on-shell methods for analytic computation of loop amplitudes, emphasizing techniques based on unitarity cuts. Unitarity techniques are formulated generally but have been especially useful for calculating one-loop amplitudes in massless theories such as Yang-Mills theory, QCD, and QED.

Paper Structure

This paper contains 26 sections, 98 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. One-Loop Master Integrals
  3. Simple theories
  4. Unitarity methods
  5. Cuts of master integrals
  6. Cut bubble
  7. Cut triangle
  8. Cut box
  9. Unitarity cut of the amplitude
  10. Stokes' Theorem on the complex plane
  11. Partial fraction identities for splitting denominator factors
  12. Solutions for coefficients
  13. Box coefficients
  14. Triangle coefficients
  15. Bubble coefficient
  16. ...and 11 more sections

Figures (3)

  • Figure 1: One-loop master integrals: box, triangle, bubble and tadpole. The Lorentz vectors $K_i$ are sums of external momenta, all directed outward.
  • Figure 2: Unitarity cut of a one-loop amplitude in the $K$ momentum channel. The two propagators are constrained to their respective mass shells. The disks represent the sum of all Feynman diagrams linking the fixed external lines and the two cut propagators.
  • Figure 3: A quadruple cut puts four propagators on shell. It is a trivial integral isolating a single box coefficient.