The factorization form of the integrands in the Cachazo-He-Yuan (CHY) formalism makes the generalized Kawai-Lewellen-Tye (KLT) 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 in using the differential operators introduced by Cheung, Shen and Wen, and taking advantage of the fact 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 (KK)-basis and Bern-Carrasco-Johansson (BCJ)-basis of Yang-Mills theory. Using the new method, the general recursive expansion to the KK-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 YM theory and the expansion of any color ordered Yang-Mills amplitudes to its BCJ-basis, have been observed. . The corresponding author is Kang Zhou. 6.2.1 Another derivation 42 6.3 The case with three gravitons 44 7. Conclusion and discussion 49 A. Terms with index circle structure 51 B. General discussions of manifestly gauge invariant functions 53 B.1 Having only one polarization vector 54 B.2 Having two polarization vectors 55 B.3 Having three polarization vectors 56 -1 -C. Some calculation details using differential operators 59 1 See section 5 of [18] for a comprehensive up-to-date list of the theories connected by double-copy relations.-2 -copy form (1.3). However, such a triviality is just an illusion. When trying to get the exact expansion coefficients C, one will find it is very difficult task. If we use the expression (1.4) directly, one needs to do the sum over (n − 3)! terms. Since analytical expressions for both parts A L (n − 1, n, σ, 1) and S[σ| σ] are very complicated, it is no wonder why in the practice no one uses this method except for some special cases 2 . Many efforts have been devoted to avoid the mentioned technical difficulties. In [21], using the string theory, the single trace Einstein-Yang-Mills (sEYM) amplitudes with just one graviton have been expanded in terms of the color ordered Yang-Mills (YM) amplitudes. In [22], using the heterotic string theory, evidence of the expansion of general EYM amplitudes in terms of YM amplitudes has also been presented, but explicit expansions are done only for some special cases, such as up to three gravitons, or up to two color traces. In [23], using the fa...

We provide a leading singularity analysis protocol in Baikov representation, for the searching of Feynman integrals with uniform transcendental (UT) weight. This approach is powered by the recent developments in rationalizing square roots and syzygy computations, and is particularly suitable for finding UT integrals with multiple mass scales. We demonstrate the power of our approach by determining the UT basis for a two-loop diagram with three external mass scales.

Motivated by the problem of expanding the single-trace tree-level amplitude of Einstein-Yang-Mills theory to the BCJ basis of Yang-Mills amplitudes, we present an alternative expansion formula in gauge invariant vector space. Starting from a generic vector space consisting of polynomials of momenta and polarization vectors, we define a new sub-space as a gauge invariant vector space by imposing constraints on the gauge invariant conditions. To characterize this sub-space, we compute its dimension and construct an explicit gauge invariant basis from it. We propose an expansion formula in this gauge invariant basis with expansion coefficients being linear combinations of the Yang-Mills amplitude, manifesting the gauge invariance of both the expansion basis and coefficients. With the help of quivers, we compute the expansion coefficients via differential operators and demonstrate the general expansion algorithm using several examples.