Evaluating EYM amplitudes in four dimensions by refined graphic expansion

Abstract

The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes induces a refined graphic expansion, by which any tree-level EYM amplitude can be expressed as a summation over all possible refined graphs. Each graph contributes a unique coefficient as well as a proper combination of color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to evaluate EYM amplitudes through YM amplitudes, the latter have much simpler structures in four dimensions than the former. In this paper, we classify the refined graphs for the expansion of EYM amplitudes into N k MHV sectors. Amplitudes in four dimensions, which involve k + 2 negative-helicity particles, at most get non-vanishing contribution from graphs in N k′ (k′ ≤ k) MHV sectors. By the help of this classification, we evaluate the non-vanishing amplitudes with two negative-helicity particles in four dimensions. We establish a correspondence between the refined graphs for single-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − or $$ \left({h}_i^{-},{g}_j^{-}\right) $$ h i − g j − configuration and the spanning forests of the known Hodges determinant form. Inspired by this correspondence, we further propose a symmetric formula of double-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − configuration. By analyzing the cancellation between refined graphs in four dimensions, we prove that any other tree amplitude with two negative-helicity particles has to vanish.

A note on multi-trace EYM amplitudes in four dimensions

“…Among the above progresses on the expansion of EYM amplitudes, a symmetric formula of maximally-helicity-violating (MHV) double-trace EYM amplitudes in four dimensions was proposed [15]. In this formula, the two gluon traces are given by two Parke-Taylor [19] factors, while gravitons are included by spanning forests that are rooted at gluons.…”

“…Along this line, the tree-level MHV amplitudes of YM, GR and the single-trace MHV EYM amplitudes have been already evaluated [23,29]. Nevertheless, there are still gaps between the symmetric formula proposed in [15] and the direct evaluation of the CHY formula for double-trace MHV amplitudes. First, the two traces in the CHY formula do not stand in an equal status, in the sense that the Pfaffian part in CHY formula only involves gluons from one trace.…”

“…General patterns of the expansions based on a recursive expression were established for not only single-trace amplitudes [4][5][6] but also multi-trace ones [7]. These expansion formulas provided a new angle for understanding several related problems, including: constructing local Bern-Carrasco-Johansson (BCJ) [8] numerators [4,9,10], inducing new amplitude relations by gauge symmetries [4,[9][10][11][12], providing an off-shell approach to BCJ duality (and amplitude relations) [13,14] and evaluating EYM amplitudes in four dimensions [15]. Several discussions at loop amplitudes can also be found (see [16][17][18] for example).…”

Off-shell extended graphic rule and the expansion of Berends-Giele currents in Yang-Mills theory

“…In particular, the relationship between symmetry induced identities and BCJ relations in YM was founded through the graphs [26,27]. In addition, sectors of EYM (YM) amplitudes were naturally constructed through graphic rule, which further inspired a symmetric formula [28] of double trace maximally-helicity-violeting (MHV) amplitude in EYM theory.…”

Note on graph-based BCJ relation for Berends-Giele currents

“…As shown in [4,7,8] the expansion coefficients correspond to local Bern-Carrasco-Johansson (BCJ) numerators [9,10] in Del Duca-Dixon-Maltoni (DDM) basis [11]. Systematic graphic rule has been proposed [7,8,12,13] to construct these coefficients and has been applied in many situations such as studying the relationship between the symmetry induced identities [12,13] and BCJ relations [9], evaluating amplitudes in four-dimensions [14].…”