Evaluating EYM amplitudes in four dimensions by refined graphic expansion

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 $\text{N}^{\,k}$MHV sectors. Amplitudes in four dimensions, which involve $k+2$ negative-helicity particles, at most get non-vanishing contribution from graphs in $\text{N}^{\,k'(k'\leq k)}$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 $(g^-_i,g^-_j)$ or $(h^-_i,g^-_j)$ 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 $(g^-_i,g^-_j)$ 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.