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 $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − or $$ \left({h}_i^{-},{g}_j^{-}\right) $$ 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 $$ \left({g}_i^{-},{g}_j^{-}\right) $$ 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.

Foreign trade in subdeveloped regions may lead to serious environmental problems. The direct and possible indirect ways in which foreign trade correlates with green economic efficiency in subdeveloped regions are qualitatively analyzed. Empirical tests are conducted based on the data of 80 prefecture-level cities in Central China. Results show that: (1) Foreign trade in Central China has a significant parabolic type of relationship with green economic efficiency. (2) Foreign trade in Central China will indirectly correlate with green economic efficiency through stimulating industries agglomerated in big cities, changing the level of industrial specialization and adjusting policy and technical environments. Whether the indirect relation between foreign trade and green growth is positive is determined by the scale of foreign trade itself. (3) The relation between foreign trade and green economic efficiency in different cities of Central China is heterogeneous. From the perspective of spatial heterogeneity, the negative relation of cities along major rivers and cities not belonging to capital city groups is more significant. From the perspective of city characteristic heterogeneity, a parabolic type of relationship of foreign trade with green economic efficiency is significant in large cities or cities with a pleasant policy environment, whereas the relationship is negative in small and medium sized cities or cities with a high proportion of manufacturing. Policy implications are proposed.

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Foreign trade in sub-developed regions may lead to serious environmental problems. Impact of foreign trade in sub-developed regions on green economic efficiency and its possible impact channels are qualitatively analyzed, empirical tests are conducted based on the data of 80 Prefecture-level cities in Central China. Results show that: 1. Foreign trade of Central China will have a significant direct negative impact on green economic efficiency, which is not conducive to green growth. 2. Expanding the scale of foreign trade in Central China will stimulate industries agglomerate to big cities and promote the level of industrial specialization within the region, which will have indirect negative impacts on green economic efficiency. 3. Foreign trade in Central China will improve the policy environment for industrial development, then have an indirect positive impact on green economic efficiency, whereas its impact on technological innovation is not significant. 4. Impact of foreign trade in different cities in Central China on green economic efficiency are heterogeneous from the perspective of spatial heterogeneity and from the perspective of city characteristic heterogeneity, and when environmental regulation exceeds a certain threshold, the negative impact of foreign trade on green economic efficiency will turn to positive. Policy implications are proposed.