Positive helicity Einstein-Yang-Mills amplitudes from the double copy method

All positive helicity four-point gluon-graviton amplitudes in Einstein-Yang-Mills theory coupled to a dilaton and axion field are computed at the leading one-loop order using color-kinematics duality. In particular, all relevant contributions in the gravitational and gauge coupling are established. This extends a previous generalized unitarity based computation beyond the leading terms in the gravitational coupling $\kappa$. The resulting purely rational expressions take very compact forms. The previously seen vanishing of the single-graviton-three-gluon amplitude at leading order in $\kappa$ is seen to be lifted at order ${\kappa }^3$.

Figure 1

Graphs of these topologies are the only ones that have to be considered in $\mathrm{YM}+{\varphi }^3$ at order ${\lambda }^2{g}^4$. The other graphs do not contain the color structures (14). Curly lines represent propagating gluons and dashed lines represent scalar fields. The internal momenta $Q_i$ are $d$ dimensional.

Figure 2

At order $g^4$ the box, triangle and bubble topologies are nonvanishing after double copying.

Figure 3

To obtain ${⟨1^{+}{2}^{+}{3}^{+}{4}^{++}⟩|}_{{\kappa }^3{g}_{\mathrm{YM}}}$ we have to analyze this type of graphs in $\mathrm{YM}+{\varphi }^3$.

Figure 4

This pair of graph topologies represent graphs which appear at leading order in ${\lambda }^2{g}^2$ for $⟨1^{+}{2}^{+}{3}^{++}{4}^{++}⟩$.

Figure 5

At $g^4$ these graph topologies shall contribute to the gravity amplitude ${⟨1^{+}{2}^{+}{3}^{++}{4}^{++}⟩|}_{{\kappa }^4}$.

Figure 6

These topologies in $\mathrm{YM}+{\varphi }^3$ have to be evaluated to obtain ${⟨1^{++}{2}^{++}{3}^{++}{4}^{++}⟩|}_{{\kappa }^4}$.