Color-kinematics Duality Research Articles

Self-dual cosmology

We construct cosmological spacetimes with a self-dual Weyl tensor whose dynamics are described by conformally coupled scalars with only cubic self-interactions. Similar to the previously discovered cases in flat and (Anti) de Sitter backgrounds, the interactions are characterized by a bracket that encodes a kinematic algebra. We discuss how the color-kinematics duality and double copy are realized in these cosmological backgrounds. If we further impose that the Ricci scalar is that of an FLRW spacetime, we find two new self-dual metrics corresponding to radiation-dominated and coasting (non-accelerating) FLRW backgrounds. Relaxing this requirement, we find an infinite family of solutions given by three different conformal classes of cosmological self-dual metrics. These solutions approximate those of FLRW as long as we impose a simple additional constraint on the scalar theory.

Higher-derivative relations between scalars and gluons

We extend the covariant color-kinematics duality introduced by Cheung and Mangan to effective field theories. We focus in particular on relations between the effective field theories of gluons only and of gluons coupled to bi-adjoint scalars. Maps are established between their respective equations of motion and between their tree-level scattering amplitudes. An additional rule for the replacement of flavor structures by kinematic factors realizes the map between higher-derivative amplitudes. As an example of new relations, the pure-gluon amplitudes of mass dimension up to eight, featuring insertions of the F3 and F4 operators which satisfy the traditional color-kinematics duality, can be generated at all multiplicities from just renormalizable amplitudes of gluons and bi-adjoint scalars. We also obtain closed-form expressions for the kinematic numerators of the dimension-six gluon effective field theory, which are valid in D space-time dimensions. Finally, we find strong evidence that this extended covariant color-kinematics duality relates the (DF)2+YM(+ϕ3) theories which, at low energies, generate infinite towers of operators satisfying the traditional color-kinematics duality, beyond aforementioned F3 and F4 ones.

Effective interactions of the open bosonic string via field theory

We describe a method to extract an effective Lagrangian description for open bosonic strings, at zero transcendentality. The method relies on a particular formulation of its scattering amplitudes derived from color-kinematics duality. More precisely, starting from a (DF)2 + YM quantum field theory, we integrate out all the massive degrees of freedom to generate an expansion in the inverse string tension α′. We explicitly compute the Lagrangian terms through O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(α′4), and target the sector of operators proportional to F4 to all orders in α′.

Manifest color-kinematics duality for point particles interacting with self-dual fields

We find that point particles interacting with a self-dual Yang-Mills field and self-dual gravity manifestly satisfy color-kinematics duality at the level of action. In a similar way color-kinematics duality also holds for a scalar field minimally coupled to a self-dual Yang-Mills field and self-dual gravity. By applying the appropriate limiting procedure to these scalar field theories we reproduce point particle theories we started from. This allows us to connect worldline color-kinematics duality to amplitude color-kinematics duality in field theory. Considering that point particles act as sources of classical solutions, our results may be regarded as a step towards establishing a precise relation between the amplitude and the classical double copies in the self-dual sector. Finally, we briefly mention that the extension of this discussion to the higher-spin case suggests that scalar point particles cannot interact with chiral higher-spin fields.

Massive covariant colour-kinematics in 3D

We explore topologically massive gauge theories using the covariant colour kinematics duality recently introduced in [1]. We show that the massive bi-adjoint scalar field is simply related to topologically massive gauge theory by the duality, and that enacting the same duality on the gauge theory produces topologically massive gravity coupled to a scalar or, equivalently, an antisymmetric field. We also show that different choices for the replacement of the colour structure constants with kinematic structure constants lead to different theories, including a topologically massive generalisation of Born-Infeld theory.

Gravitational wave double copy of radiation from gluon shockwave collisions

In [1], we showed that the gravitational wave spectrum from trans-Planckian shockwave scattering in Einstein gravity is determined by the gravitational Lipatov vertex expressed as the bilinear double copy Γμν=12CμCν−12NμNν where Cμ is the QCD Lipatov vertex and Nμ is the QED soft photon factor. We show here that this result can be directly obtained by careful application of the classical color-kinematic duality to the spectrum obtained in gluon shockwave collisions.

Color-kinematics duality and dual conformal symmetry for a four-loop form factor in $$\mathcal{N}$$ = 4 SYM

Color-kinematics duality and dual conformal symmetry for a four-loop form factor in $$\mathcal{N}$$ = 4 SYM

Recursive construction for expansions of tree Yang–Mills amplitudes from soft theorem

In this paper, we introduce a fundamentally different approach, based on a bottom-up methodology, for expanding tree-level Yang–Mills (YM) amplitudes into Yang–Mills-scalar (YMS) amplitudes and bi-adjoint-scalar (BAS) amplitudes. Our method relies solely on the intrinsic soft behavior of external gluons, eliminating the need for external aids such as Feynman rules or CHY rules. The recursive procedure consistently preserves explicit gauge invariance at every step, ultimately resulting in a manifest gauge-invariant outcome when the initial expression is already framed in a gauge-invariant manner. The resulting expansion can be directly analogized to the expansions of gravitational (GR) amplitudes using the double copy structure. When combined with the expansions of Einstein–Yang–Mills amplitudes obtained using the covariant color-kinematic duality method from existing literature, the expansions presented in this note yield gauge-invariant Bern–Carrasco–Johansson (BCJ) numerators.

Color-kinematics duality with minimal deformation: two-loop four-gluon amplitudes in pure Yang-Mills revisited

The conjectured duality between color and kinematics has significantly advanced our understanding of both gauge and gravitational theories. However, constructing numerators that manifest the color-kinematics (CK) duality, even for the two-loop four-gluon amplitude in pure Yang-Mills, has been challenging. In this paper, we revisit this amplitude and show that the difficulty of applying CK duality can be overcome by introducing a simple deformation. Our approach distinguishes itself from previous studies by maximizing the use of off-shell CK duality while maintaining a compact ansatz. In particular, the deformation we introduce satisfies a subset of off-shell dual Jacobi relations. The resulting numerators are presented in d-dimensionally Lorentz invariant local form and are applicable to all helicities of external gluons. The solution we provide can be directly employed to construct the corresponding gravitational amplitude through double copy. Our findings suggest a novel and efficient strategy for constructing high-loop gauge and gravitational amplitudes using CK duality.

Kinematic Hopf algebra for amplitudes from higher-derivative operators

Recently it has been shown that Bern-Carrasco-Johansson (BCJ) numerators of colour-kinematic duality for tree-level scattering amplitudes in Yang-Mills theory (coupled with scalars) can be determined using a quasi-shuffle Hopf algebra. In this paper we consider the same theory, but with higher-derivative corrections of the forms α′F3 and α′2F4, where F is the field strength. In the heavy mass limit of the scalars, we show that the BCJ numerators of these higher-derivative theories are governed by the same Hopf algebra. In particular, the kinematic algebraic structure is unaltered and the derivative corrections only arise when mapping the abstract algebraic generators to physical BCJ numerators. The underlying kinematic Hopf algebra enables us to obtain a compact expression for the BCJ numerators of any number of gluons and two heavy scalars for amplitudes with higher-derivative operators. The pure gluon BCJ numerators can also be obtained from our results by a simple factorisation limit where the massive particles decouple.

Expanding single-trace YMS amplitudes with gauge-invariant coefficients

In this note, we use the new bottom-up method based on soft theorems to recursively construct the expansion of single-trace Yang–Mills-scalar amplitudes. The resulting expansion manifests the gauge invariance for any polarization carried by external gluons, as well as the permutation symmetry among external gluons. Our result is equivalent to that found by Clifford Cheung and James Mangan via the so-called covariant color–kinematics duality approach.

Gravity Amplitudes from Double Bonus Relations.

In this Letter, we derive new expressions for tree-level graviton amplitudes in N=8 supergravity from Britto-Cachazo-Feng-Witten (BCFW) recursion relations combined with new types of bonus relations. These bonus relations go beyond the famous 1/z^{2} behavior under a large BCFW shift and use knowledge about certain zeros of graviton amplitudes in collinear kinematics. This extra knowledge can be used in the context of global residue theorems by writing the amplitude in a special form using canonical building blocks. In the next-to-maximally-helicity-violating case, these building blocks are dressed one-loop leading singularities, the same objects that appear in the expansion of Yang-Mills amplitudes, where each term corresponds to an R invariant. Unlike other approaches, our formula is not an expansion in terms of cyclic objects and does not manifest color-kinematics duality but rather preserves the permutational symmetry of its building blocks. We also comment on the possible connection to Grassmannian geometry and give some nontrivial evidence of such structure for graviton amplitudes.

Scattering on self-dual Taub-NUT

We derive exact solutions of massless free field equations and tree-level two-point amplitudes up to spin 2 on self-dual Taub-NUT (SDTN) spacetime, as well as on its single copy, the self-dual dyon (SDD). We use Killing spinors to build analogues of momentum eigenstates, finding that, in the spirit of color-kinematics duality, those for the SDD lift directly to provide states on the SDTN background if one replaces charge with energy. We discover that they are forced to have faster growth at infinity than in flat space due to the topological non-triviality of these backgrounds. The amplitudes for massless scalars and spinning particles in the and helicity configurations vanish for generic kinematics as a consequence of the integrability of the self-dual sector. The amplitudes are non-vanishing and we compute them exactly in the backgrounds, which are treated non-perturbatively. It is explained how spin is easily introduced via a Newman–Janis imaginary shift along the spin-vector leading directly to the well-known exponential factor in the dot product of the spin with the momenta. We also observe a double copy relation between the gluon amplitude on a SDD and graviton amplitude on a SDTN spacetime.

Tree-level color-kinematics duality from pure spinor actions

Tree-level color-kinematics duality from pure spinor actions

Double-copying self-dual Yang-Mills theory to self-dual gravity on twistor space

We construct a simple Lorentz-invariant action for maximally supersymmetric self-dual Yang-Mills theory that manifests colour-kinematics duality. We also show that this action double-copies to a known action for maximally supersymmetric self-dual gravity. Both actions live on twistor space and illustrate nicely the homotopy algebraic perspective on the double copy presented in [1]. This example is particularly interesting as the involved Hopf algebra controlling the momentum dependence is non-commutative and suggests a generalisation to gauged maximally supersymmetric self-dual gravity.

Color-kinematics dual representations of one-loop matrix elements in the open-superstring effective action

The α′-expansion of string theory provides a rich set of higher-dimension operators, indexed by ζ values, which can be used to study color-kinematics duality and the double copy. These two powerful properties, actually first noticed in tree-level string amplitudes, simplify the construction of both gauge and gravity amplitudes. However, their applicability and limitations are not fully understood. We attempt to construct a set of color-kinematics dual numerators at one loop and four points for insertions of operator combinations corresponding to the lowest four ζ2-free operator insertions from the open superstring: ζ3, ζ5, {zeta}_3^2 , and ζ7. We succeed in finding a representation for the first three in terms of box, triangle, and bubble numerators. In the case of ζ7 we find an obstruction to a fully color-dual representation related to the bubble-on-external-leg type diagrams. We discuss two paths around this obstruction, both of which signal an incompatability between color-kinematics duality and manifesting certain desired properties. Using the constructed color-dual numerators, we find two different Bern-Carrasco-Johansson double copies that produce candidate closed-string-insertion numerators. Both approaches to the double copy match the kinematics of the cuts, with relative normalization set by either summing over both double copies including degeneracy or by including an explicit prefactor on the double-copy numerator definitions.

Lagrangians manifesting color-kinematics duality in the NMHV sector of Yang-Mills

Scattering amplitudes in Yang-Mills theory are known to exhibit kinematic structures which hint to an underlying kinematic algebra that is dual to the gauge group color algebra. This color-kinematics duality is still poorly understood in terms of conventional Feynman rules, or from a Lagrangian formalism. In this work, we present explicit Lagrangians whose Feynman rules generate duality-satisfying tree-level BCJ numerators, to any multiplicity in the next-to-MHV sector of pure Yang-Mills theory. Our Lagrangians make use of at most three pairs of auxiliary fields (2, 1, 0-forms) — surprisingly few compared to previous attempts of Lagrangians at low multiplicities. To restrict the Lagrangian freedom it is necessary to make several non-trivial assumptions regarding field content, kinetic terms, and interactions, which we discuss in some detail. Future progress likely hinges on relaxing these assumptions.

Self-Dual Gravity and Color-Kinematics Duality in AdS_{4}.

We show that self-dual gravity in Euclidean four-dimensional anti-de Sitter space (AdS_{4}) can be described by a scalar field with a cubic interaction written in terms of a deformed Poisson bracket, providing a remarkably simple generalization of the Plebanski action for self-dual gravity in flat space. This implies a novel symmetry algebra in self-dual gravity, notably an AdS_{4} version of the so-called kinematic algebra. We also obtain the three-point interaction vertex of self-dual gravity in AdS_{4} from that of self-dual Yang-Mills by replacing the structure constants of the Lie group with the structure constants of the new kinematic algebra, implying that self-dual gravity in AdS_{4} can be derived from self-dual Yang-Mills in this background via a double copy. This provides a concrete starting point for defining the double copy for Einstein gravity in AdS_{4} by expanding around the self-dual sector. Moreover, we show that the new kinematic Lie algebra can be lifted to a deformed version of the w_{1+∞} algebra, which plays a prominent role in celestial holography.

Homotopy Double Copy of Noncommutative Gauge Theories

We discuss the double-copy formulation of Moyal–Weyl-type noncommutative gauge theories from the homotopy algebraic perspective of factorisations of L∞-algebras. We define new noncommutative scalar field theories with rigid colour symmetries taking the role of the zeroth copy, where the deformed colour algebra plays the role of a kinematic algebra; some of these theories have a trivial classical limit but exhibit colour–kinematics duality, from which we construct the double copy theory explicitly. We show that noncommutative gauge theories exhibit a twisted form of colour–kinematics duality, which we use to show that their double copies match with the commutative case. We illustrate this explicitly for Chern–Simons theory, and for Yang–Mills theory where we obtain a modified Kawai–Lewellen–Tye relationship whose momentum kernel is linked to a binoncommutative biadjoint scalar theory. We reinterpret rank-one noncommutative gauge theories as double copy theories and discuss how our findings tie in with recent discussions of Moyal–Weyl deformations of self–dual Yang–Mills theory and gravity.

Kinematic Lie Algebras from Twistor Spaces.

We analyze theories with color-kinematics duality from an algebraic perspective and find that any such theory has an underlying BV^{▪}-algebra, extending the ideas of Reiterer [A homotopy BV algebra for Yang-Mills and color-kinematics, arXiv:1912.03110.]. Conversely, we show that any theory with a BV^{▪}-algebra features a kinematic Lie algebra that controls interaction vertices, both on shell and off shell. We explain that the archetypal example of a theory with a BV^{▪}-algebra is Chern-Simons theory, for which the resulting kinematic Lie algebra is isomorphic to the Schouten-Nijenhuis algebra on multivector fields. The BV^{▪}-algebra implies the known color-kinematics duality of Chern-Simons theory. Similarly, we show that holomorphic and Cauchy-Riemann Chern-Simons theories come with BV^{▪}-algebras and that, on the appropriate twistor spaces, these theories organize and identify kinematic Lie algebras for self-dual and full Yang-Mills theories, as well as the currents of any field theory with a twistorial description. We show that this result extends to the loop level under certain assumptions.