Abstract
The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes induces a refined graphic expansion, by which any tree-level EYM amplitude can be expressed as a summation over all possible refined graphs. Each graph contributes a unique coefficient as well as a proper combination of color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to evaluate EYM amplitudes through YM amplitudes, the latter have much simpler structures in four dimensions than the former. In this paper, we classify the refined graphs for the expansion of EYM amplitudes into N k MHV sectors. Amplitudes in four dimensions, which involve k + 2 negative-helicity particles, at most get non-vanishing contribution from graphs in N k′ (k′ ≤ k) MHV sectors. By the help of this classification, we evaluate the non-vanishing amplitudes with two negative-helicity particles in four dimensions. We establish a correspondence between the refined graphs for single-trace amplitudes with left({g}_i^{-},{g}_j^{-}right) or left({h}_i^{-},{g}_j^{-}right) configuration and the spanning forests of the known Hodges determinant form. Inspired by this correspondence, we further propose a symmetric formula of double-trace amplitudes with left({g}_i^{-},{g}_j^{-}right) configuration. By analyzing the cancellation between refined graphs in four dimensions, we prove that any other tree amplitude with two negative-helicity particles has to vanish.
Highlights
The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes induces a refined graphic expansion, by which any tree-level EYM amplitude can be expressed as a summation over all possible refined graphs
We further propose a symmetric formula of double-trace amplitudes with configuration
We summarize some critical features of the above evaluations, which will inspire a symmetric formula of double-trace amplitude with the configuration in the coming section:
Summary
We illustrate that an N kMHV amplitude in four dimensions can at most get nonzero contribution from graphs in the N k (k ≤k)MHV sector In four dimensions, both the momentum of a graviton/gluon and the (half) polarization of a graviton can be expressed according to the spinor-helicity formalism [32] (see appendix B). When the reference momentum of all positive-helicity gravitons are chosen as the momentum of one of the negative-helicity gravitons, say hi, the maximal number of type-1 lines in a graph is further reduced by one because. The nonvanishing EYM amplitudes with k + 2 negative-helicity particles can at most get nonzero contributions from the graphs in the N k (k ≤k)MHV sector. According to BCFW recursion, an EYM amplitude A with k + 2 negative-helicity particles (gravitons and gluons) can be expressed as A ∼. The general construction rule of the N kMHV sector can be found in appendix A
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