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

TL;DR

The work tackles spurious higher-order poles that arise in CHY amplitudes built from Pfaffians. It introduces a diagrammatic, cycle-based expansion of the reduced Pfaffian and uses cross-ratio identities to prove systematic cancellation of double poles in Yang–Mills and gravity, ensuring only single poles contribute to physical amplitudes. It then demonstrates that dimensional reduction from gravity to daughter theories such as Einstein–Yang–Mills and squared-Pfaffian constructions produces CHY integrands with the correct pole structure, extending the method to nonlinear sigma model and DBI-type theories. The results provide a constructive, gauge-invariant framework for controlling pole structures in CHY formulas and offer a unified pathway to derive CHY representations across a family of theories.

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.

Figures (6)

• Figure 1: Three types of lines.
• Figure 2: If a node i) belongs to a $W$-cycle but is not an end of $W$-cycle or ii) belongs to an $U$-cycle with more than one element, it gets contribution from the corresponding $W$- or $U$-cycle as shown by Figures (a) and (b). The two structures are related by flipping a sign because they are corresponding to the $p^{\mu}\epsilon^{\nu}$ and $-p^{\nu}\epsilon^{\mu}$ of $F^{\mu\nu}$.
• Figure 3: All possible structures in the four-element example.
• Figure 4: Diagrams contributing to double pole in $s_{12}$ channel.
• Figure 5: Diagrams contributing to double pole, which only contain type-2 lines.