Calculating Scattering Amplitudes Efficiently

TL;DR

This paper surveys a coherent toolkit for calculating scattering amplitudes in gauge theories with improved efficiency, focusing on tree and one-loop multi-parton QCD amplitudes. It argues that organizing calculations by total quantum numbers (color and helicity), employing color-ordered (partial/primitive) amplitudes, and using spinor-helicity formalism dramatically reduce intermediate complexity. The author details recursive constructions for tree amplitudes, SUSY Ward identities, and factorization properties, and then extends these ideas to loops via supersymmetric rearrangements, background-field gauge, loop-integral reduction, and unitarity cuts, with concrete examples such as five-gluon amplitudes. Collectively, these methods enable compact, analytically tractable expressions and scalable approaches to NLO computations, while acknowledging ongoing challenges for more complex processes.

Abstract

We review techniques for more efficient computation of perturbative scattering amplitudes in gauge theory, in particular tree and one-loop multi-parton amplitudes in QCD. We emphasize the advantages of (1) using color and helicity information to decompose amplitudes into smaller gauge-invariant pieces, and (2) exploiting the analytic properties of these pieces, namely their cuts and poles. Other useful tools include recursion relations, special gauges and supersymmetric rearrangements.

Figures (11)

• Figure 1: Diagrammatic equations for simplifying $SU(N_c)$ color algebra. Curly lines ("gluon propagators") represent adjoint indices, oriented solid lines ("quark propagators") represent fundamental indices, and "quark-gluon vertices" represent the generator matrices $(T^a)_i^{~{\bar{\jmath}}}$.
• Figure 2: A sample diagram for tree-level five-gluon scattering, reduced to a single trace.
• Figure 3: A diagram for one-loop four-gluon scattering, reduced to single and double traces.
• Figure 4: Diagrammatic evaluation of color sums for the tree-level four-gluon cross-section.
• Figure 5: Color-ordered Feynman rules, in Lorentz-Feynman gauge, omitting ghosts. Straight lines represent fermions, wavy lines gluons. All momenta are taken outgoing.