# Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy

## TL;DR

This paper derives explicit formulas for Yang-Mills-Einstein amplitudes using the double-copy method, providing compact, dimension-independent expressions that extend to supergravity and reveal recursive structures.

## Contribution

It introduces a new compact, explicit formula for Yang-Mills-Einstein amplitudes with up to five gravitons, extending the double-copy construction and revealing recursive amplitude structures.

## Key findings

- Explicit formulas for up to five gravitons
- Simpler presentations for low-graviton amplitudes
- New loop-level amplitude relations for Yang-Mills

## Abstract

Using the double-copy construction of Yang-Mills-Einstein theories formulated in our earlier work, we obtain compact presentations for single-trace Yang-Mills-Einstein tree amplitudes with up to five external gravitons and an arbitrary number of gluons. These are written as linear combinations of color-ordered Yang-Mills trees, where the coefficients are given by color/kinematics-satisfying numerators in a Yang-Mills+\phi^3 theory. The construction outlined in this paper holds in general dimension and extends straightforwardly to supergravity theories. For one, two, and three external gravitons, our expressions give identical or simpler presentations of amplitudes already constructed through string-theory considerations or the scattering equations formalism. Our results are based on color/kinematics duality and gauge invariance, and strongly hint at a recursive structure underlying the single-trace amplitudes with an arbitrary number of gravitons. For the single-graviton case, we give amplitudes to any loop order and obtain, through gauge invariance, new loop-level amplitude relations for Yang-Mills theory.