Abstract
All tree-level amplitudes in Einstein-Yang-Mills (EYM) theory and gravity (GR) can be expanded in terms of color ordered Yang-Mills (YM) ones whose coefficients are polynomial functions of Lorentz inner products and are constructed by a graphic rule. Once the gauge invariance condition of any graviton is imposed, the expansion of a tree level EYM or gravity amplitude induces a nontrivial identity between color ordered YM amplitudes. Being different from traditional Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations, the gauge invariance induced identity involves polarizations in the coefficients. In this paper, we investigate the relationship between the gauge invariance induced identity and traditional BCJ relations. By proposing a refined graphic rule, we prove that all the gauge invariance induced identities for single trace tree-level EYM amplitudes can be precisely expanded in terms of traditional BCJ relations, without referring any property of polarizations. When further considering the transversality of polarizations and momentum conservation, we prove that the gauge invariance induced identity for tree-level GR (or pure YM) amplitudes can also be expanded in terms of traditional BCJ relations for YM (or bi-scalar) amplitudes. As a byproduct, a graph-based BCJ relation is proposed and proved.
Highlights
Gauge invariance has been shown to provide a strong constraint on scattering amplitudes in recent years
One interesting application is that a recursive expansion of single-trace Einstein-Yang-Mills (EYM) amplitudes can be determined by gauge invariance conditions of external gravitons in addition with a proper ansatz [5, 6] (the approach based on Cachazo-He-Yuan (CHY) formula [7,8,9,10] is provided in [11])
By expressing the coefficients in the gauge invariance induced identity according to the refined graphic rule and collecting those terms with the same structure of the lines corresponding to coefficients · and · k in particular examples, we find that the gauge invariance induced identity can always be expressed by a combination of BCJ relations
Summary
Gauge invariance has been shown to provide a strong constraint on scattering amplitudes in recent years. Applying the recursive expansion repeatedly, one expresses an arbitrary tree level single-trace EYM amplitude in terms of pure Yang-Mills (YM) ones whose coefficients are conveniently constructed by a graphic rule [12]. This expansion can be regarded as a generalization of the earlier studies on EYM amplitudes with few gravitons [13,14,15,16]. We investigate the relationship between an arbitrary gauge invariance induced identity, which is derived from the expansion of tree level single-trace EYM amplitudes or pure GR amplitudes, and general BCJ relations. We review the recursive expansion of single-trace EYM amplitudes, the graphic rule, the gauge invariance induced identity from single-trace EYM amplitudes as well as the BCJ relations for Yang-Mills amplitudes
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