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In this paper, we provide a thorough study on the expansion of single traceEinstein-Yang-Mills amplitudes into linear combination of color-orderedYang-Mills amplitudes, from various different perspectives. Using the gaugeinvariance principle, we propose a recursive construction, where EYM amplitudewith any number of gravitons could be expanded into EYM amplitudes with lessnumber of gravitons. Through this construction, we can write down the completeexpansion of EYM amplitude in the basis of color-ordered Yang-Mills amplitudes.As a byproduct, we are able to write down the polynomial form of BCJ numerator,i.e., numerators satisfying the color-kinematic duality, for Yang-Millsamplitude. After the discussion of gauge invariance, we move to the BCFWon-shell recursion relation and discuss how the expansion can be understoodfrom the on-shell picture. Finally, we show how to interpret the expansion fromthe aspect of KLT relation and the way of evaluating the expansion coefficientsefficiently.

Expansion of Einstein-Yang-Mills Amplitude