Permutation in the CHY-Formulation

TL;DR

This paper introduces a cycle representation of permutations to better understand the CHY formulation of bi-adjoint scalar amplitudes, linking permutation cycles to Feynman diagram features and PT-factors.

Contribution

It develops a novel cycle representation for permutations in the CHY framework, connecting permutation cycles with Feynman diagram structures and PT-factors.

Findings

Cycle representation characterizes Feynman diagram poles and vertices.

Permutation cycles can be reconstructed from Feynman diagrams.

Relations among PT-factors are elucidated through cycle analysis.

Abstract

The CHY-integrand of bi-adjoint cubic scalar theory is a product of two PT-factors. This pair of PT-factors can be interpreted as defining a permutation. We introduce the cycle representation of permutation in this paper for the understanding of cubic scalar amplitude. We show that, given a permutation related to the pair of PT-factors, the pole and vertex information of Feynman diagrams of corresponding CHY-integrand is completely characterized by the cycle representation of permutation. Inversely, we also show that, given a set of Feynman diagrams, the cycle representation of corresponding PT-factor can be recursively constructed. In this sense, there exists a deep connection between cycles of a permutation and amplitude. Based on these results, we have investigated the relations among different independent pairs of PT-factors in the context of cycle representation as well as the multiplication of cross-ratio factors.