We investigate the Magnus expansion of the $\textit{N}$-operator in relativistic quantum field theory, which is related to the $\textit{S}$-matrix via $\textit{S}$ = $\textit{e}$ . We develop direct methods to compute matrix elements of the $\textit{N}$-operator, which we refer to as Magnus amplitudes, bypassing scattering amplitudes entirely. At tree level, Magnus amplitudes are expressed in terms of retarded and advanced propagators, with each diagram weighted by factors that we identify as Murua coefficients. At loop level this structure is augmented by the Hadamard cut function, and we establish remarkable relations between loop- and tree-level Magnus amplitudes. Among these, we find that $\textit{n}$-point one-loop Magnus amplitudes are entirely determined by phase-space integrals of forward limits of ($\textit{n}$+2)-point tree-level amplitudes, and hence related to Murua coefficients, and we generalise this to a class of higher-loop contributions. Furthermore, in the case of heavy particles interacting via massless mediators, we conjecture that Magnus diagrams that contribute to the classical limit are always given by forward limits of trees, and we show this explicitly in a one-loop example. We derive these results studying theories of scalar fields with cubic interactions, but our methods are applicable to general theories as well as to integral functions appearing in gravitational-wave computations. Given that Magnus amplitudes are free of hyper-classical terms, and the known relations between Magnus amplitudes and the radial action, our results lay the groundwork for systematic and efficient calculations of classical observables from quantum field theory.

United Kingdom

Andreas Brandhuber

Graham Brown

Paolo Pichini

Gabriele Travaglini

Pablo Vives Matasan

The Magnus expansion in relativistic quantum field theory

We derive analytic all-order-in-spin expressions for the leading-order time-domain waveforms generated in the scattering of two Kerr black holes with arbitrary masses and spin vectors in the presence of all independent cubic deformations of Einstein-Hilbert gravity. These are the two parity-even interactions $\textit{I}$ and $\textit{G}$ , and the parity-odd ones $\textit{Ĩ}$ and $$ {\overset{\sim }{G}}_3 $$ . Our results are obtained using three independent methods: a particularly efficient direct integration and tensor reduction approach; integration by parts combined with the method of differential equations; and finally a residue computation. For the case of the $\textit{G}$ and $$ {\overset{\sim }{G}}_3 $$ deformations we can express the spinning waveform in terms of the scalar waveform with appropriately shifted impact parameters, which are reminiscent of Newman-Janis shifts. For $\textit{I}$ and $\textit{Ĩ}$ similar shifts occur, but are accompanied by additional contributions that cannot be captured by simply shifting the scalar $\textit{I}$ and $\textit{Ĩ}$ waveforms. We also show the absence of leading-order corrections to gravitational memory. Our analytic results are notably compact, and we compare the effectiveness of the three methods used to obtain them. We also briefly comment on the magnitude of the corrections to observables due to cubic deformations.

Denmark

Gang Chen

Spinning waveforms in cubic effective field theories of gravity

We study the binary dynamics of two Kerr black holes with arbitrary spin vectors in the presence of parity-even and parity-odd cubic deformations of gravity. We first derive the tree-level Compton amplitudes for a Kerr black hole in cubic gravity, which we then use to compute the two-to-two amplitudes of the massive bodies to leading order in the deformation and the post-Minkowskian expansion. The required one-loop computations are performed using the leading singularity approach as well as the heavy-mass effective field theory (HEFT) approach. These amplitudes are then used to compute the leading-order momentum and spin kick in cubic gravity in the KMOC formalism. Our results are valid for generic masses and spin vectors, and include all the independent parity-even and parity-odd cubic deformations of Einstein-Hilbert gravity. We also present spin-expanded expressions for the momentum and spin kicks, and the all-order in spin deflection angle in the case of aligned spins.

Spinning binary dynamics in cubic effective field theories of gravity

We compute the classical tree-level five-point amplitude for the two-to-two scattering of spinning celestial objects with the emission of a graviton. Using this five-point amplitude, we then turn to the computation of the leading-order time-domain gravitational waveform. The method we describe is suitable for arbitrary values of classical spin of Kerr black holes and does not require any expansion in powers of the spin. In this paper we illustrate it in the simpler case of the scattering of one Kerr and one Schwarzschild black hole. An important ingredient of our calculation is a novel form of the Compton amplitude with spinning particles including contact terms derived from matching to black-hole perturbation theory calculations. This ensures that our waveform is valid up to at least fourth order in the spin. Our method can be applied immediately to generate improved waveforms once higher-order contact terms in the Compton amplitude become available. Finally, we show the formula for the gravitational memory to all orders in the spin, which is in agreement with our results.

Joshua Gowdy

Resummed spinning waveforms from five-point amplitudes

Using a heavy-mass effective field theory (HEFT), we study gravitational-wave emission in the scattering of two spinless black holes or neutron stars of arbitrary masses at next-to-leading order in the Post-Minkowskian expansion. We compute the contributions to the one-loop scattering amplitude with four scalars and one graviton which are relevant to the calculation of the waveforms, also presenting expressions of classical tree-level amplitudes with four scalars and up to two radiated gravitons. The latter are obtained using a novel on-shell recursion relation for classical amplitudes with four scalars and an arbitrary number of gravitons. Our one-loop five-point amplitude is expressed in terms of a single family of master integrals with the principal value prescription for linearised massive propagators, which we evaluate using differential equations. In our HEFT approach, soft/heavy-mass expansions of complete integrands are avoided, and all hyper-classical iterations and quantum corrections are dropped at the diagrammatic level, thereby computing directly contributions to classical physics. Our result exhibits the expected factorisation of infrared divergences, the correct soft limits, and highly nontrivial cancellations of spurious poles. Finally, using our amplitude result we compute numerically the corresponding next-to-leading corrections to the spectral waveforms and the far-field time-domain waveforms using the Newman-Penrose scalar Ψ4.

United States

France

Stefano Angelis

One-loop gravitational bremsstrahlung and waveforms from a heavy-mass effective field theory

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 for gravitational HEFT amplitudes, we show that the coproducts of the kinematic algebra are in correspondence with factorisations of BCJ numerators on massive poles. We then study an extension of the standard quasi-shuffle Hopf algebra to a non-abelian version describing BCJ numerators with all possible gluon orderings. This is achieved by tensoring the original algebra with a particular Hopf algebra of orderings. In this extended version, a specific choice of the coproduct in the algebra of orderings leads to an antipode in the resulting Hopf algebra that has the interpretation of reversing the gluons’ order within each BCJ numerator.

Congkao Wen

Amplitudes, Hopf algebras and the colour-kinematics duality

We present a closed formula for all Bern-Carrasco-Johansson (BCJ) numerators describing <math><mi>D</mi></math>-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 quasishuffle Hopf algebra. The BCJ numerators thus obtained have a compact form and intriguing features: gauge invariance is manifest, locality is respected for massless exchange, and they contain poles corresponding to massive exchange. Counting the number of terms in a BCJ numerator for <math><mi>n</mi><mo>−</mo><mn>2</mn></math> gluons gives the Fubini numbers <math><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></math>, reflecting the underlying quasishuffle Hopf algebra structure. Finally, by considering an appropriate factorization limit, the massive particles decouple, and we thus obtain a kinematic algebra and all tree-level BCJ numerators for <math><mi>D</mi></math>-dimensional pure Yang-Mills theory.

Sweden

Henrik Johansson

Kinematic Hopf Algebra for Bern-Carrasco-Johansson Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory

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 the construction of the required D-dimensional loop integrands and allow for a direct extraction of contributions relevant for classical physics. In particular the soft/heavy-mass expansions of full integrands is circumvented, and all iterating contributions can be dropped from the get go. We use this method to compute the scattering angle up to third post-Minkowskian order in four dimensions, including radiation reaction contributions, also providing the expression of the corresponding integrand in D dimensions.

Classical gravitational scattering from a gauge-invariant double copy

We study celestial amplitudes in (super) Yang-Mills theory using a parametrization of the spinor helicity variables where their overall phase is not fixed by the little group action. In this approach the spin constraint <math><mi>h</mi><mo>−</mo><mover><mi>h</mi><mo>¯</mo></mover><mo>=</mo><mi>J</mi></math> for celestial conformal primaries emerges naturally from a new Mellin transform, and the action of conformal transformations on celestial amplitudes is derived. Applying this approach to <math><mi>N</mi><mo>=</mo><mn>4</mn></math> super Yang-Mills, we show how the appropriate definition of on-shell superspace coordinates leads naturally to a formulation of chiral celestial superamplitudes and a representation of the generators of the four-dimensional superconformal algebra on the celestial sphere, which by construction annihilate all tree-level celestial superamplitudes.

Graham R. Brown

Bill Spence

Celestial superamplitudes

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 the BCJ numerators in the heavy-mass effective theory with those in pure Yang-Mills amplitudes.