# Off-Shell Yang-Mills Amplitude in the CHY Formalism

## TL;DR

This paper explores the extension of the CHY formalism to off-shell Yang-Mills amplitudes, revealing partial gauge invariance and a split into two components, which simplifies calculations but does not fully replicate gauge invariance.

## Contribution

It demonstrates that a M"obius invariant off-shell amplitude can be constructed in the CHY formalism, but requires an additional amplitude to restore gauge invariance, highlighting partial invariance properties.

## Key findings

- Modified CHY amplitude $M_P$ is M"obius invariant but not gauge invariant.
- Adding $M_Q$ restores gauge invariance but depends on the Lagrangian.
- Splitting amplitudes into $M_P$ and $M_Q$ simplifies calculations.

## Abstract

M\"obius invariance is used to construct gluon tree amplitudes in the Cachazo, He, and Yuan (CHY) formalism. If it is equally effective in steering the construction of off-shell tree amplitudes, then the S-matrix CHY theory can be used to replace the Lagrangian Yang-Mills theory. In the process of investigating this possibility, we find that the CHY formula can indeed be modified to obtain a M\"obius invariant off-shell amplitude, but unfortunately this modified amplitude $M_P$ is not the Yang-Mills amplitude because it lacks gauge invariance. A complementary amplitude $M_Q$ must be added to restore gauge invariance, but its construction relies on the Lagrangian and not M\"obius invariance. Although neither $M_P$ nor $M_Q$ is fully gauge invariant, both are partially gauge invariant in a sense to be explained. This partial gauge invariance turns out to be very useful for checking calculations. A Feynman amplitude so split into the sum of $M_P$ and $M_Q$ also contains fewer terms.