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

TL;DR

This work develops explicit all-multiplicity single-trace Yang-Mills-Einstein (YME) tree amplitudes with up to five external gravitons by leveraging the BCJ double-copy: amplitudes are linear combinations of YM tree amplitudes with coefficients given by color/kinematics-satisfying numerators from a YM+φ^3 theory. The authors formulate a Del Duca–Dixon–Maltoni (DDM) based decomposition, derive a constructive procedure to fix YM+φ^3 numerators using gauge invariance and BCJ relations, and provide explicit results for k ≤ 5 gravitons, including compact recursive structures for the numerators. They extend the framework to loop level via relations among YM integrands and all-loop amplitude relations, and show how the approach generalizes to other theories such as DBI, NLSM, and string theory through various double-copy constructions. The results reveal a recursive, gauge-invariant structure underlying single-trace YME amplitudes and establish a practical bridge from gauge-theory data to gravity amplitudes, with potential extensions to Coulomb branches and gauged R-symmetry scenarios.

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+φ^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.

Figures (3)

• Figure 1: A multiperipheral (or half-ladder) graph for YM theory. The particles are labeled by the word $w$, where the first and last element are kept fixed.
• Figure 2: A typical multiperipheral (or half-ladder) graph for the YM+$\phi^3$ theory. The gluons are labeled as $1,2,\ldots, k$ and the remaining $m$ particles are canonically ordered scalars. Reading from left to right, these form a word $w$ as explained in fig. \ref{['figmulti']}. The $z_i$ is the internal scalar momenta to the right of each gluon $i$.
• Figure 3: A schematic representation of the $(m+1)$-point $L$-loop YM amplitude where a cubic-graph vertex has been isolated. The shaded blob stands for all the graphs contributing to this amplitude. In all such graphs, we uniformly name the internal momenta flowing in and out of the isolated vertex such that $p_1 = q_1+q_2$.