# Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism

## TL;DR

This paper demonstrates that higher-order poles in the Pfaffian within CHY formulations cancel out, ensuring only single poles remain in physical amplitudes for Yang-Mills and gravity theories, through a diagrammatic and algebraic approach.

## Contribution

It introduces a diagrammatic expansion and cross-ratio identities to prove the cancellation of higher-order poles in the Pfaffian of CHY formulas.

## Key findings

- Higher-order poles cancel out in Yang-Mills and gravity CHY amplitudes.
- A diagrammatic rule for expanding the reduced Pfaffian is developed.
- Cancellation of higher-order poles is shown in other field theories with suitable truncations.

## Abstract

For a physical field theory, the tree-level amplitudes should possess only single poles. However, when computing amplitudes with Cachazo-He-Yuan (CHY) formulation, individual terms in the intermediate steps will contribute higher-order poles. In this paper, we investigate the cancelation of higher-order poles in CHY formula with Pfaffian as the building block. We develop a diagrammatic rule for expanding the reduced Pfaffian. Then by organizing diagrams in appropriate groups and applying the cross-ratio identities, we show that all potential contributions to higher-order poles in the reduced Pfaffian are canceled out, i.e., only single poles survive in Yang-Mills theory and gravity. Furthermore, we show the cancelations of higher-order poles in other field theories by introducing appropriate truncations, based on the single pole structure of Pfaffian.