Expansion of Einstein-Yang-Mills amplitude

Abstract

Abstract:In this paper, we study from various perspectives the expansion of tree level single trace Einstein-Yang-Mills amplitudes into linear combination of color-ordered YangMills amplitudes. By applying the gauge invariance principle, a programable recursive construction is devised to expand EYM amplitude with arbitrary number of gravitons into EYM amplitudes with fewer gravitons. Based on this recursive technique we write down the complete expansion of any single trace EYM amplitude in the basis of color-order Yang-Mills amplitude. As a byproduct, an algorithm for constructing a polynomial form of the BCJ numerator for Yang-Mills amplitudes is also outlined in this paper. In addition, by applying BCFW recursion relation we show how to arrive at the same EYM amplitude expansion from the on-shell perspective. And we examine the EYM expansion using KLT relations and show how to evaluate the expansion coefficients efficiently.

BCJ numerators from reduced Pfaffian

“…Up to a total sign caused by different conventions, the above numerators agree with those given in [16]. The four-point gravity amplitude in this representation:…”

“…This expansion was first derived in [16] by using Lam and Yao's cycle expansion [14]. In appendix A, we provide another proof using the Laplace expansion.…”

“…The coefficients can be very nicely evaluated from a set of graphic rules based on spanning trees. Then in [16], a very simple linear relation between YM amplitudes and YM-scalar amplitudes is derived, which enables us to write down a similar set of graphic rules to directly evaluate the BCJ numerators for YM. This construction will give polynomial form BCJ numerators in the DDM form, which will be studied in detail in section 4.…”

“…The main subject of this note is to provide a set of very straightforward graphic rules to read out these numerators directly, based on two previous papers [15,16]. 4 Interestingly, the single trace EYM integrands appear as intermediate steps in this construction.…”

Chiral splitting and N = 4 $$ \mathcal{N}=4 $$ Einstein-Yang-Mills tree amplitudes in 4d

“…Following the early work [11][12][13] there has been renewed interest recently in the study of Einstein-Yang-Mills amplitudes and their relation to pure Yang-Mills from the perspective of the double copy construction [14,15], string theory [16,17] and the CHY formulae [18][19][20][21]. We believe that the new formulas for N = 4 EYM scattering amplitudes (eqs.…”