In the maximally-helicity-violating (MHV) configuration, tree-level single-trace Einstein-Yang-Mills (EYM) amplitudes with one or two gravitons have been shown to satisfy a formula in which each graviton splits into a pair of collinear gluons. In this study, we extend this formula to more general cases. We present a general formula that expresses tree-level single-trace MHV amplitudes in terms of pure gluon amplitudes. In this formula, each graviton turns into a pair of collinear gluons.

China

Zhirun Li

Yi-Jian Du

Note on single-trace EYM amplitudes with MHV configuration

One-loop integrands in Cachazo-He-Yuan (CHY) formula, which is based on the forward limit of tree-level amplitudes, involves linear propagators that are different from quadratic ones in traditional Feynman diagrams. In this paper, we provide a general approach to converting linear propagators in one-loop CHY formula into quadratic propagators, by refined graphic rule stemming from the recursive expansion of tree-level Einstein-Yang-Mills amplitudes. Particularly, we establish the correspondence between refined graphs and bi-adjoint scalar (BS) Feynman diagrams with linear propagators. Using this correspondence and graph-based relations of Berends-Giele currents in BS theory, the nonlocal terms accompanied by refined graphs can either be canceled out or be collected into local ones. Once the locality has been achieved, the integrand with linear propagators can be directly arranged into that with quadratic propagators. Following this approach, we first convert the linear propagators in single-trace Yang-Mills-scalar (YMS) integrands (with a pure-scalar loop) into quadratic ones. This result is then demonstrated to match the traditional one-loop Feynman diagrams. The discussions on single-trace YMS integrands are generalized to multi-trace YMS and Yang-Mills integrands.

Chongsi Xie

Extracting quadratic propagators by refined graphic rule

Tree level multi-trace Yang-Mills-scalar (YMS) amplitudes have been shown to satisfy a recursive expansion formula, which expresses any YMS amplitude by those with fewer gluons and/or scalar traces. In an earlier work, the single-trace expansion formula has been shown to be determined by the universality of soft behavior. This approach is nevertheless not extended to multi-trace case in a straightforward way. In this paper, we derive the expansion formula of tree-level multi-trace YMS amplitudes in a bottom-up way: we first determine the simplest amplitude, the double-trace pure scalar amplitude which involves two scalars in each trace. Then insert more scalars to one of the traces. Based on this amplitude, we further obtain the double-soft behavior when the trace contains only two scalars is soft. The multi-trace amplitudes with more scalars and more gluons finally follow from the double-soft behavior as well as the single-soft behaviors which has been derived before.

Kang Zhou

Multi-trace YMS amplitudes from soft behavior

It is well known that gravity amplitudes in four dimensions can be reconstructed by the inverse soft limit (ISL) method. According to ISL, a tree level n-graviton maximally-helicity-violating (MHV) amplitude is expressed in terms of deformed (n − 1)-graviton amplitudes accompanied by soft graviton factors. On another hand, single- and double-trace tree-level Einstein-Yang-Mills (EYM) MHV amplitudes have been proven to satisfy spanning forest formulas, where each edge in a forest has the same form with a term in soft graviton factor. It is not transparent that the formulas satisfied by EYM amplitudes can be constructed with ISL. In this paper, we construct the single- and double-trace MHV amplitudes in EYM, by the ISL and show that the known formulas can be precisely reproduced. Interesting identities which are based on Schouten identity and characterized by graphs are also introduced.

Shiquan Ma

United States

Rongyu Dong

Constructing EYM amplitudes by inverse soft limit

In four dimensions, a tree-level double-trace Einstein-Yang-Mills (EYM) amplitude with two negative-helicity gluons (the (g  −  , g  − )-configuration) satisfies a symmetric spanning forest formula, which was derived from the graphic expansion rule. On another hand, in the framework of Cachazo-He-Yuan (CHY) formula, the maximally-helicity-violating (MHV) amplitudes are supported by the MHV solution of scattering equations. The relationship between the symmetric formula for double-trace amplitudes, and the MHV sector of Cachazo-He-Yuan (CHY) formula in four dimensions is still not clear. In this note, we promote a series of transformations of the spanning forests in four dimensions and then show a systematic way for decomposing the MHV sector of the CHY formula of double-trace EYM amplitudes. Along this line, the symmetric formula of double-trace MHV amplitudes is directly obtained by the MHV sector of CHY formula. We then prove that EYM amplitude with an arbitrary total number of negative-helicity particles (gravitons and gluons) has to vanish when the number of negative- (or positive-) helicity gluons is less than the number of traces.

A note on multi-trace EYM amplitudes in four dimensions

Graph-based Bern-Carasso-Johansson (BCJ) relation for Berends-Giele currents in bi-adjoint scalar (BS) theory, which is characterized by connected tree graphs, was proposed in an earlier work. In this note, we provide a systematic study of this relation. We first prove the relations based on two special types of graphs: simple chains and star graphs. The general graph-based BCJ relation established by an arbitrary tree graph is further proved, through Berends-Giele recursion. When combined with proper off-shell extended numerators, this relation induces the graph-based BCJ relation for Berends-Giele currents in Yang-Mills theory. The corresponding relations for amplitudes are obtained via on-shell limits.

Konglong Wu

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

Tree-level color-ordered Yang-Mills (YM) amplitudes can be decomposed in terms of (n − 2)! bi-scalar (BS) amplitudes, whose expansion coefficients form a basis of Bern-Carrasco-Johansson (BCJ) numerators. By the help of the recursive expansion of Einstein-Yang-Mills (EYM) amplitudes, the BCJ numerators are given by polynomial functions of Lorentz contractions which are conveniently described by graphic rule. In this work, we extend the expansion of YM amplitudes to off-shell level. We define different types of off-shell extended numerators that can be generated by graphs. By the use of these extended numerators, we propose a general decomposition formula of off-shell Berends-Giele currents in YM. This formula consists of three terms: (i). an effective current which is expanded as a combination of the Berends-Giele currents in BS theory (The expansion coefficients are one type of off-shell extended numerators) (ii). a term proportional to the total momentum of on-shell lines and (iii). a term expressed by the sum of lower point Berends-Giele currents in which some polarizations and momenta are replaced by vectors proportional to off-shell momenta appropriately. In the on-shell limit, the last two terms vanish while the decomposition of effective current precisely reproduces the decomposition of on-shell YM amplitudes with the expected coefficients (BCJ numerators in DDM basis). We further symmetrize these coefficients such that the Lie symmetries are satisfied. These symmetric BCJ numerators simultaneously satisfy the relabeling property of external lines and the algebraic properties (antisymmetry and Jacobi identity).

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

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   <math> <mfenced> <msubsup> <mi>g</mi> <mi>i</mi> <mo>−</mo> </msubsup> <msubsup> <mi>g</mi> <mi>j</mi> <mo>−</mo> </msubsup> </mfenced> </math>  $$ \left({g}_i^{-},{g}_j^{-}\right) $$  or   <math> <mfenced> <msubsup> <mi>h</mi> <mi>i</mi> <mo>−</mo> </msubsup> <msubsup> <mi>g</mi> <mi>j</mi> <mo>−</mo> </msubsup> </mfenced> </math>  $$ \left({h}_i^{-},{g}_j^{-}\right) $$  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   <math> <mfenced> <msubsup> <mi>g</mi> <mi>i</mi> <mo>−</mo> </msubsup> <msubsup> <mi>g</mi> <mi>j</mi> <mo>−</mo> </msubsup> </mfenced> </math>  $$ \left({g}_i^{-},{g}_j^{-}\right) $$  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.

Hongxiang Tian

Sweden

Enze Gong

Evaluating EYM amplitudes in four dimensions by refined graphic expansion

Symmetries of Einstein-Yang-Mills (EYM) amplitudes, together with the recursive expansions, induce nontrivial identities for pure Yang-Mills amplitudes. In the previous work [1], we have already proven that the identities induced from tree level single-trace EYM amplitudes can be precisely expanded in terms of BCJ relations. In this paper, we extend the discussions to those identities induced from all tree level multi-trace EYM amplitudes. Particularly, we establish a refined graphic rule for multi-trace EYM amplitudes and then show that the induced identities can be fully decomposed in terms of BCJ relations.

Linghui Hou

A graphic approach to identities induced from multi-trace Einstein-Yang-Mills amplitudes

All tree-level amplitudes in Einstein-Yang-Mills (EYM) theory and gravity (GR) can be expanded in terms of color ordered Yang-Mills (YM) ones whose coefficients are polynomial functions of Lorentz inner products and are constructed by a graphic rule. Once the gauge invariance condition of any graviton is imposed, the expansion of a tree level EYM or gravity amplitude induces a nontrivial identity between color ordered YM amplitudes. Being different from traditional Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations, the gauge invariance induced identity involves polarizations in the coefficients. In this paper, we investigate the relationship between the gauge invariance induced identity and traditional BCJ relations. By proposing a refined graphic rule, we prove that all the gauge invariance induced identities for single trace tree-level EYM amplitudes can be precisely expanded in terms of traditional BCJ relations, without referring any property of polarizations. When further considering the transversality of polarizations and momentum conservation, we prove that the gauge invariance induced identity for tree-level GR (or pure YM) amplitudes can also be expanded in terms of traditional BCJ relations for YM (or bi-scalar) amplitudes. As a byproduct, a graph-based BCJ relation is proposed and proved.