Expansion of Einstein-Yang-Mills theory by differential operators

The factorization form of the integrands in the Cachazo-He-Yuan formalism makes the generalized Kawai-Lewellen-Tye relations manifest; thus, amplitudes of one theory can be expanded in terms of the amplitudes of another theory. Although this claim seems a rather natural consequence of the above structure, finding the exact expansion coefficients to express an amplitude in terms of another amplitudes is, nonetheless, a nontrivial task despite many efforts devoted to it in the literature. In this paper, we propose a new strategy based on using the differential operators introduced by Cheung, Shen, and Wen, and taking advantage of the fact that these operators already relate the amplitudes of different theories. Using this new method, expansion coefficients can be found effectively. Although the method should be general, to demonstrate the idea, we focus on the expansion of single trace Einstein-Yang-Mills (sEYM) amplitudes in the Kleiss-Kuijf basis and Bern-Carrasco-Johansson (BCJ) basis of Yang-Mills theory. Using the new method, the general recursive expansion to the Kleiss-Kuijf basis has been reproduced. The expansion to the BCJ basis is a more difficult problem. Using the new method, we have worked out the details for sEYM with one, two, and three gravitons. As a by-product, profound relations among two kinds of expansion coefficients, i.e., the expansion of sEYM amplitudes to the BCJ basis of Yang-Mills theory and the expansion of any color ordered Yang-Mills amplitudes to its BCJ basis, have been observed.