Pfaffian Diagrams for Gluon Tree Amplitudes

TL;DR

This paper introduces Pfaffian diagrams as a systematic and simplified way to compute gluon tree amplitudes from the CHY formula, reducing complexity and clarifying gauge invariance.

Contribution

It formulates Pfaffian diagrams for gluon amplitudes, providing a more efficient and gauge-invariant representation compared to traditional Feynman diagrams.

Findings

Fewer terms in amplitudes using Pfaffian diagrams (e.g., 19 vs. 35 for four-point)

Gauge invariance is more explicit in Pfaffian diagrams

Technical methods to handle poles and optimize calculations are developed.

Abstract

Pfaffian diagrams are formulated to represent gluon amplitudes computed from the Cachazo-He-Yuan (CHY) formula. They may be regarded as a systematic regrouping of Feynman diagrams after internal momenta are expanded and products of vertex factors are evaluated. This reprocessing enables gluon amplitudes expressed in Pfaffian diagrams to contain less terms. For example, there are 19 terms for the four-point amplitude in Pfaffian diagrams, and 35 terms in Feynman diagrams. Gauge invariance is simpler and more explicit in Pfaffian diagrams, in that subset of diagrams with the same root configuration are already gauge invariant in all lines but two. In getting to these results, several technical difficulties must be overcome. Double poles must be converted to simple poles, integrations must be carried out directly and formulated into simple rules, and the three \M constant lines must be suitably chosen to minimize the number of terms present.