Abstract
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 colour-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 $\kappa^{3}$.
Highlights
Scattering amplitudes involving only positive helicity gluon and graviton states take a very special role in minimally coupled gauge and gravitational theories: At tree-level they always vanish due to an effective supersymmetric Ward identity [1]Mtnr;eme ð1þa1 ; ...; nþan ; ðn þ 1Þþþ; ...; ðn þ mÞþþÞ 1⁄4 0: ð1ÞThis holds true for an amplitude in an arbitrary matter coupled Einstein-Yang-Mills (EYM) with n gluons and m gravitons and persists to all-loops in supersymmetric theories
The resulting one-loop pure gluon or pure graviton positive helicity amplitudes turn out to be remarkably simple rational functions of the kinematic invariants and spinor products. This is a mandatory property in order to have vanishing unitarity cuts in four dimensions for these amplitudes. These pure gluon or graviton one-loop amplitudes are identical to the ones one finds in self-dual YM or self-dual gravity, respectively
The adjoint generators ifabc are antisymmetric in their indices which implies that the color factors are antisymmetric under permutation: ci 1⁄4 −cj. (ii) The set of all reduced Feynman propagators 1=ðp2 − m2Þ associated to the jQth graph are denoted by the inverse of the product αjDαj. (iii) The numerators nj account for the remaining kinematical dependence of the amplitude
Summary
Scattering amplitudes involving only positive helicity gluon and graviton states take a very special role in minimally coupled gauge and gravitational theories: At tree-level they always vanish due to an effective (or hidden) supersymmetric Ward identity [1] This holds true for an amplitude in an arbitrary matter coupled Einstein-Yang-Mills (EYM) with n gluons and m gravitons and persists to all-loops in supersymmetric theories. Restricting to one particle species circulating the loop allowed them to use supersymmetric Ward–Takahashi identities to replace the virtual particles in the loop by complex scalars, since the all-plus and all-plus-but-one-minus helicity amplitudes vanish in supersymmetric and supergravitational theories [1] These identities do not hold for a mixed propagation of gravitons and gluons in the loop, and this technique is not applicable in the most general setting. The Appendix C contains all the integrands of YM þ φ3 which shall be used to determine the integrands of EYM for one particular ordering of the external states
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