We generalize the scattering equations to include both massless and massive particles. We construct an expression for the tree-level n-point amplitude with n-2 gluons or gravitons and a pair of massive scalars in arbitrary spacetime dimension as a sum over the (n-3)! solutions of the scattering equations, a la Cachazo, He, and Yuan. We derive the BCJ relations obeyed by these massive amplitudes.
We write down closed formulas for all necessary steps to obtain multiparticle super Yang--Mills superfields in the so-called BCJ gauge. The superfields in this gauge have obvious applications in the quest for finding BCJ-satisfying representations of amplitudes. As a benefit of having these closed formulas, we identify the explicit finite gauge transformation responsible for attaining the BCJ gauge. To do this, several combinatorial maps on words are introduced and associated identities rigorously proven.
Tree-level $n$-point gauge-theory amplitudes with $n-2k$ gluons and $k$ pairs of (massless or massive) particles in the fundamental (or other) representation of the gauge group are invariant under a set of symmetries that act as momentum-dependent shifts on the color factors in the cubic decomposition of the amplitude. These symmetries lead to gauge-invariant constraints on the kinematic numerators. They also directly imply the BCJ relations among the Melia-basis primitive amplitudes previously obtained by Johansson and Ochirov.
We introduce a new set of symmetries obeyed by tree-level gauge-theory amplitudes involving at least one gluon. The symmetry acts as a momentum-dependent shift on the color factors of the amplitude. Using the radiation vertex expansion, we prove the invariance under this color-factor shift of the $n$-gluon amplitude, as well as amplitudes involving massless or massive particles in an arbitrary representation of the gauge group with spin zero, one-half, or one. The Bern-Carrasco-Johansson relations are a direct consequence of this symmetry. We also introduce the cubic vertex expansion of an amp…
We explain a procedure to manifest the Bern-Carrasco-Johansson duality between color and kinematics in $n$-point one-loop amplitudes of a variety of supersymmetric gauge theories. Explicit amplitude representations are constructed through a systematic reorganization of the integrands in the Cachazo-He-Yuan formalism. Our construction holds for any nonzero number of supersymmetries and does not depend on the number of spacetime dimensions. The cancellations from supersymmetry multiplets in the loop as well as the resulting power counting of loop momenta is manifested along the lines of the corr…
We find simple expressions for the kinematic numerators of one-loop MHV amplitudes in maximally supersymmetric Yang-Mills theory and supergravity, for any multiplicity. The gauge theory numerators satisfy the Bern-Carrasco-Johansson (BCJ) duality between color and kinematics, so that the gravity numerators are simply the square of the gauge theory ones. The duality holds because the numerators can be written in terms of structure constants of a kinematic algebra, which is familiar from the BCJ organization of self-dual gauge theory and gravity. The close connection that we find between one-loo…
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 includes polarizations in the…
Recent progress on scattering amplitudes in super Yang--Mills and superstring theory benefitted from the use of multiparticle superfields. They universally capture tree-level subdiagrams, and their generating series solve the non-linear equations of ten-dimensional super Yang--Mills. We provide simplified recursions for multiparticle superfields and relate them to earlier representations through non-linear gauge transformations of their generating series. In this work we discuss the gauge transformations which enforce their Lie symmetries as suggested by the Bern-Carrasco-Johansson duality bet…
We present a closed formula for all Bern-Carrasco-Johansson (BCJ) numerators describing $D$-dimensional tree-level scattering amplitudes in a heavy-mass effective field theory with two massive particles and an arbitrary number of gluons. The corresponding gravitational amplitudes obtained via the double copy directly enter the computation of black-hole scattering and gravitational-wave emission. Our construction is based on finding a kinematic algebra for the numerators, which we relate to a quasi-shuffle Hopf algebra. The BCJ numerators thus obtained have a compact form and intriguing feature…
The duality between color and kinematics present in scattering amplitudes of Yang-Mills theory strongly suggest the existence of a hidden kinematic Lie algebra that controls the gauge theory. While associated BCJ numerators are known on closed forms to any multiplicity at tree level, the kinematic algebra has only been partially explored for the simplest of four-dimensional amplitudes: up to the MHV sector. In this paper we introduce a framework that allows us to characterize the algebra beyond the MHV sector. This allows us to both constrain some of the ambiguities of the kinematic algebra, a…
We propose a new form of the colour-kinematics/double-copy duality for heavy-mass effective field theories, which we apply to construct compact expressions for tree amplitudes with heavy matter particles in Yang-Mills and in gravity to leading order in the mass. In this set-up, the new BCJ numerators are fixed uniquely and directly written in terms of field strengths, making gauge invariance manifest. Furthermore, they are local and automatically satisfy the Jacobi relations and crossing symmetry. We construct these BCJ numerators explicitly up to six particles. We also discuss relations of th…
In this letter, we extend the tree-level Kawai--Lewellen--Tye (KLT) and Bern--Carrasco--Johansson (BCJ) amplitude relations to loop integrands of gauge theory and gravity. By rearranging the propagators of gauge and gravity loop integrands, we propose the first manifestly gauge- and diffeomorphism invariant formulation of their double-copy relations. The one-loop KLT formula expresses gravity integrands in terms of more basic gauge invariant building blocks for gauge-theory amplitudes, dubbed partial integrands. The latter obey a one-loop analogue of the BCJ relations, and both KLT and BCJ rel…
We present new formulas for one-loop ambitwistor-string correlators for gauge theories in any even dimension with arbitrary combinations of gauge bosons, fermions and scalars running in the loop. Our results are driven by new all-multiplicity expressions for tree-level two-fermion correlators in the RNS formalism that closely resemble the purely bosonic ones. After taking forward limits of tree-level correlators with an additional pair of fermions/bosons, one-loop correlators become combinations of Lorentz traces in vector and spinor representations. Identities between these two types of trace…
We extend the double copy picture of scattering amplitudes to a class of matrix elements (so-called form factors) that involve local gauge invariant operators. Both the Bern, Carrasco and Johansson (BCJ) and the Kawai, Lewellen and Tye (KLT) formalisms are considered and novel properties are observed. One remarkable feature is that through the double-copy construction, certain spurious poles hidden in the gauge form factors become physical propagators in gravity. This mechanism also reveals new hidden relations for form factors which can be understood as a generalization of the BCJ relations.
In this paper, we propose an improved method for directly calculating double-copy-compatible tree numerators in (super-)Yang-Mills and Yang-Mills-scalar theories. Our new scheme gets rid of any explicit dependence on reference orderings, restoring a form of crossing symmetry to the numerators. This in turn improves the computational efficiency of the algorithm, allowing us to go well beyond the number of external particles accessible with the reference order based methods. Motivated by a parallel study of one-loop BCJ numerators from forward limits, we explore the generalization to include a p…
We take a major step towards computing $D$-dimensional one-loop amplitudes in general gauge theories, compatible with the principles of unitarity and the color-kinematics duality. For $n$-point amplitudes with either supersymmetry multiplets or generic non-supersymmetric matter in the loop, simple all-multiplicity expressions are obtained for the maximal cuts of kinematic numerators of $n$-gon diagrams. At $n=6,7$ points with maximal supersymmetry, we extend the cubic-diagram numerators to encode all contact terms, and thus solve the long-standing problem of \emph{simultaneously} realizing the…
Kinematic numerators of Yang-Mills scattering amplitudes possess a rich Lie algebraic structure that suggest the existence of a hidden infinite-dimensional kinematic algebra. Explicitly realizing such a kinematic algebra is a longstanding open problem that only has had partial success for simple helicity sectors. In past work, we introduced a framework using tensor currents and fusion rules to generate BCJ numerators of a special subsector of NMHV amplitudes in Yang-Mills theory. Here we enlarge the scope and explicitly realize a kinematic algebra for all NMHV amplitudes. Master numerators are…
We propose and study a BCJ double-copy of massive particles, showing that it is equivalent to a KLT formula with a kernel given by the inverse of a matrix of massive bi-adjoint scalar amplitudes. For models with a uniform non-zero mass spectrum we demonstrate that the resulting double-copy factors on physical poles and that up to at least 5-particle scattering, color-kinematics satisfying numerators always exist. For the scattering of 5 or more particles, the procedure generically introduces spurious singularities that must be cancelled by imposing additional constraints. When massive particle…
In this talk we review relations and representations of primitive QCD tree amplitudes. Topics covered include the BCJ relations, the CHY representation, and the KLT relations. We will put a special emphasis on how these relations and representations generalise from pure Yang-Mills theory to QCD. The generalisation of the KLT relations from pure Yang-Mills to QCD includes the case of massive quarks. On the gravity side we then obtain hypothetical particles interacting with gravitational strength, which can be massive and non-relativistic.
It was recently proposed that the kinematic algebra featuring in the colour-kinematics duality for scattering amplitudes in heavy-mass effective field theory (HEFT) and Yang-Mills theory is a quasi-shuffle Hopf algebra. The associated fusion product determines the structure of the Bern-Carrasco-Johansson (BCJ) numerators, which are manifestly gauge invariant and with poles corresponding to heavy-particle exchange. In this work we explore the deep connections between the quasi-shuffle algebra and general physical properties of the scattering amplitudes. First, after proving the double-copy form…
Recent explorations on how to construct a double copy of massive gauge fields have shown that, while any amplitude can be written in a form consistent with colour-kinematics duality, the double copy is generically unphysical. In this paper, we explore a new direction in which we can obtain a sensible double copy of massive gauge fields due to the special kinematics in three-dimensional spacetimes. To avoid the appearance of spurious poles at 5-points, we only require that the scattering amplitudes satisfy one BCJ relation. We show that the amplitudes of Topologically Massive Yang-Mills satisfy…
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 …
Based on the covariant color-kinematics duality, we investigate combinatorial and algebraic structures underlying their Bern-Carrasco-Johansson (BCJ) numerators of tree-level amplitudes in Yang-Mills-scalar (YMS) theory. The closed-formulae for BCJ numerators of YMS amplitudes and the pure-Yang-Mills (YM) ones exhibit nice quasi-shuffle Hopf algebra structures, and interestingly they can be viewed as summing over boundaries of all dimensions of a combinatorial permutohedron. In particular, the numerator with two scalars and $n{-}2$ gluons contains Fubini number ( $F_{n{-}2}$ ) of terms in one-…
We explore extensions of the double copy to massive theories and find a new cubic theory with a local double copy. We consider the nonlinear sigma model and the special galileon theory, massless versions of which are known to be related through the double copy. We show that by performing a Kaluza-Klein reduction of these theories from five dimensions down to four, a double copy relation exists between the resulting massive four-dimensional scalar field theories. This requires the vanishing contribution of new galileon terms arising in high dimensions. We further explore if other interactions t…
Motivated by the problem of expanding 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 the 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 gauge invariant vector space by imposing constraints of 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 the gauge in…
Whenever the integrand of a gauge-theory loop amplitude can be arranged into a form where the BCJ duality between color and kinematics is manifest, a corresponding gravity integrand can be obtained simply via the double-copy procedure. However, finding such gauge-theory representations can be challenging, especially at high loop orders. Here we show that we can instead start from generic gauge-theory integrands, where the duality is not manifest, and apply a modified double-copy procedure to obtain gravity integrands that include contact terms generated by violations of dual Jacobi identities.…
In flat space, the color/kinematics duality states that perturbative Yang-Mills amplitudes can be written in such a way that kinematic numerators obey the same Jacobi relations as their color factors. This remarkable duality implies BCJ relations for Yang-Mills amplitudes and underlies the double copy to gravitational amplitudes. In this paper, we find analogous relations for Yang-Mills amplitudes in AdS$_4$. In particular we show that the kinematic numerators of 4-point Yang-Mills amplitudes computed via Witten diagrams in momentum space enjoy a generalised gauge symmetry which can be used to…
Constraints and Generalized Gauge Transformations on Tree-Level Gluon and Graviton Amplitudes
将全色修饰的振幅和引力子振幅分别写为${\bf A}=<C|A> =<C|M|N> $和${\bf A}_{gr}= <\tilde N|M|N> $，其中$|A> $是Kleiss-Kuijf色序基的集合，$|N>, $| \tilde N> $ and $| C> $ are the similarly ordered numerators and color coefficients, we show that the propagator matrix $ M $ has $（n-3）（n-3）！$ independent eigenvectors $| λ^0_j> $ with zero eigenvalue, for $ n $-particle processes. The resulting equations $ <λ^0_j| A> = 0 $ are relations among the color ordered amplitudes. The freedom to shift $| N> \to| N> + \sum_j f_j| λ^0_j> $ and similarly for $| PDMATHBMQ N> $, where $ f_j PDMATHBNC（n-3）（n-3）！$ arbitrary functions, encodes gen…
We study recently proposed chiral higher spin theories - cubic theories of interacting massless higher spin fields in four-dimensional flat space. We show that they are naturally associated with gauge algebras, which manifest themselves in several related ways. Firstly, the chiral higher spin equations of motion can be reformulated as the self-dual Yang-Mills equations with the associated gauge algebras instead of the usual colour gauge algebra. We also demonstrate that the chiral higher spin field equations, similarly to the self-dual Yang-Mills equations, feature an infinite algebra of hidde…
We find double copy relations between classical radiating solutions in Yang-Mills theory coupled to dynamical color charges and their counterparts in a cubic bi-adjoint scalar field theory which interacts linearly with particles carrying bi-adjoint charge. The particular color-to-kinematics replacements we employ are motivated by the BCJ double copy correspondence for on-shell amplitudes in gauge and gravity theories. They are identical to those recently used to establish relations between classical radiating solutions in gauge theory and in dilaton gravity. Our explicit bi-adjoint solutions a…
We propose a method to compute the scattering angle for classical black hole scattering directly from two massive particle irreducible diagrams in a heavy-mass effective field theory approach to general relativity, without the need of subtracting iteration terms. The amplitudes in this effective theory are constructed using a recently proposed novel colour-kinematic/double copy for tree-level two-scalar, multi-graviton amplitudes, where the BCJ numerators are gauge invariant and local with respect to the massless gravitons. These tree amplitudes, together with graviton tree amplitudes, enter t…
In this paper, we provide a thorough study on the expansion of single trace Einstein-Yang-Mills amplitudes into linear combination of color-ordered Yang-Mills amplitudes, from various different perspectives. Using the gauge invariance principle, we propose a recursive construction, where EYM amplitude with any number of gravitons could be expanded into EYM amplitudes with less number of gravitons. Through this construction, we can write down the complete expansion of EYM amplitude in the basis of color-ordered Yang-Mills amplitudes. As a byproduct, we are able to write down the polynomial form…
