Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame

Using the Cachazo-He-Yuan (CHY) formalism, we prove a recursive expansion of tree level single trace Einstein-Yang-Mills (EYM) amplitudes with arbitrary number of gluons and gravitons, which is valid for general spacetime dimensions and any helicity configurations. The recursion is written in terms of fewer-graviton EYM amplitudes and pure Yang-Mills (YM) amplitudes, which can be further carried out until we reach an expansion in terms of pure YM amplitudes in Kleiss-Kuijf (KK) basis. Our expansion then generates naturally a spanning tree structure rooted on gluons whose vertices are gravitons. We further propose a set of graph theoretical rules based on spanning trees that evaluate directly the pure YM expansion coefficients.Comment: 36 pages, 3 captioned figures; v2: more details added, revised and published versio

Einstein-Yang-Mills from pure Yang-Mills amplitudes

We present new relations for scattering amplitudes of color ordered gluons and gravitons in Einstein-Yang-Mills theory. Tree-level amplitudes of arbitrary multiplicities and polarizations involving up to three gravitons and up to two color traces are reduced to partial amplitudes of pure Yang-Mills theory. In fact, the double-trace identities apply to Einstein-Yang-Mills extended by a dilaton and a B-field. Our results generalize recent work of Stieberger and Taylor for the single graviton case with a single color trace. As the derivation is made in the dimension-agnostic Cachazo-He-Yuan formalism, our results are valid for external bosons in any number of spacetime dimensions. Moreover, they generalize to the superamplitudes in theories with 16 supercharges.Comment: 28 pages, v2: references and appendix added, published versio

Note on differential operators, CHY integrands, and unifying relations for amplitudes

An elegant unified web for amplitudes of various theories was given by Cachazo, He and Yuan in the CHY framework a few years ago. Recently, similar web has also been constructed by Cheung, Shen and Wen, which relies on a set of differential operators. In this note, by acting these differential operators on CHY-integrands systematically, we have established the relation between these two approaches. Thus, amplitudes for all theories which have CHY representations, include gravity theory, Einstein-Yang-Mills theory, Einstein-Maxwell theory, pure Yang-Mills theory, Yang-Mills-scalar theory, Born-Infeld theory, Dirac-Born-Infeld theory and its extension, bi-adjoint scalar theory, $\phi^4$ theory, non-linear sigma model, as well as special Galileon theory, have been included in the unified web rooted from gravity theory.Comment: 20 page

Chiral Splitting and N=4\mathcal N = 4 Einstein--Yang--Mills Tree Amplitudes in 4d

We present a world-sheet formula for all tree level scattering amplitudes, in all trace sectors, of four dimensional $\mathcal{N} \leq 4$ supersymmetric Einstein-Yang-Mills theory, based on the refined scattering equations. This generalizes previously known formulas for all-trace purely bosonic, or supersymmetric single-trace amplitudes. We find this formula by applying a new chiral splitting formula for all CHY Pfaffians in 4d, into two determinants, of positive and negative helicity respectively. The splitting of CHY Pfaffians is shown to be a special case of the splitting of $T\mathbb{M}$ valued fermion correlators on the sphere, which does not require the scattering equations to hold, and is a consequence of the isomorphism $T\mathbb{M} \simeq \mathbb{S}^+ \otimes \mathbb{S}^-$ between the tangent bundle of Minkowski space and the left- and right-handed spin bundles. We present and prove this general splitting formula.Comment: 21 page

Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes

In this paper, we derive generalized Bern-Carrasco-Johansson relations for color-ordered Yang-Mills amplitudes by imposing gauge invariance conditions and dimensional reduction appropriately on the new discovered graphic expansion of Einstein-Yang-Mills amplitudes. These relations are also satisfied by color-ordered amplitudes in other theories such as color-scalar theory, bi-scalar theory and nonlinear sigma model (NLSM). As an application of the gauge invariance induced relations, we further prove that the three types of BCJ numerators in NLSM , which are derived from Feynman rules, Abelian Z-theory and Cachazo-He- Yuan formula respectively, produce the same total amplitudes. In other words, the three distinct approaches to NLSM amplitudes are equivalent to each other.Comment: 40pages, 2 figure

Properties of scattering forms and their relation to associahedra

We show that the half-integrands in the CHY representation of tree amplitudes give rise to the definition of differential forms -- the scattering forms -- on the moduli space of a Riemann sphere with $n$ marked points. These differential forms have some remarkable properties. We show that all singularities are on the divisor $\overline{\mathcal M}_{0,n} \backslash {\mathcal M}_{0,n}$. Each singularity is logarithmic and the residue factorises into two differential forms of lower points. In order for this to work, we provide a threefold generalisation of the CHY polarisation factor (also known as reduced Pfaffian) towards off-shell momenta, unphysical polarisations and away from the solutions of the scattering equations. We discuss explicitly the cases of bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes.Comment: 40 pages, version to be publishe

One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations

In this paper we reconsider the Cachazo-He-Yuan construction (CHY) of the so called scattering amplitudes at one-loop, in order to obtain quadratic propagators. In theories with colour ordering the key ingredient is the redefinition of the Parke-Taylor factors. After classifying all the possible one-loop CHY-integrands we conjecture a new one-loop amplitude for the massless Bi-adjoint $\Phi^3$ theory. The prescription directly reproduces the quadratic propagators from of the traditional Feynman approach.Comment: 43 pages, new appendix added, few typos corrected. Accepted for publication in JHE