We compute the potential-graviton contributions to the conservative scattering angle of two nonspinning bodies in maximal supergravity at fifth order in Newton’s constant, including second-order self-force effects. Our goal is to tackle the challenging integrals arising at this order in Einstein gravity, but within the technically simpler framework of supergravity. The calculation employs the scattering-amplitude framework, effective field theory, and multiloop integration techniques based on integration by parts and differential equations. The final result is expressed as a series expansion around the static limit, thereby avoiding the explicit evaluation of intricate special functions. This series solution for the master integrals applies, as well, to the corresponding computation in general relativity. Remarkably, we observe nontrivial cancellations among contributions associated with Calabi-Yau integrals, alongside a distinct contribution governed by a Heun differential equation.

United States

Zvi Bern

Enrico Herrmann

Switzerland

Radu Roiban

Michael S. Ruf

Russia

Alexander V. Smirnov

Vladimir A. Smirnov

United Kingdom

Mao Zeng

Second-Order Self-Force Potential-Region Binary Dynamics at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>5</mn></mrow></msup><mo stretchy="false">)</mo></mrow></math> in Supergravity

We study the algebraic structure of one-loop Bern-Carrasco-Johansson numerators in Yang-Mills and related theories. Starting from the propagator matrix that connects color-ordered integrands to numerators, we identify the consistency conditions that ensure the existence of Jacobi-satisfying numerator solutions and determine the unique construction. The relation between one-loop numerators and forward-limit tree numerators is clarified, together with the additional physical conditions required for a consistent double-copy interpretation. We propose a two-step expansion strategy for obtaining explicit one-loop numerators. The Yang-Mills integrand is first decomposed into scalar-loop Yang-Mills-scalar building blocks, which are then expanded into biadjoint scalar integrands. We derive explicit results for up to three external gluons, showing how the kinematic consistency conditions uniquely determine the coefficients in each case. Similar results for Einstein-Yang-Mills and gravity amplitudes are also presented.

China

Yi-Jian Du

Chih-Hao Fu

Yihong Wang

Chongsi Xie

Algebraic consistency and explicit construction of one-loop Bern-Carrasco-Johansson numerators of Yang-Mills and related theories

In a previous Letter, we showed that physical scattering observables for compact spinning objects in general relativity can depend on additional degrees of freedom beyond those described by the spin vector, which can be packaged into an unconstrained spin tensor. In this paper, we provide further details on the physics of these additional degrees of freedom, whose commutation relations and Poisson brackets are inherited from the underlying Lorentz symmetry, and on their consequence on observables. In particular, we give the waveform at leading order in Newton’s constant and up to second order in the components of the spin tensor, and the conservative impulse, boost and spin kick, exhibiting spin magnitude change, through next-to-leading-order in Newton’s constant and third order in the components of the spin tensor. We provide explicit examples — a Newtonian two-particle bound system and a certain black-hole solution in an exotic matter-coupled gravitational theory — that exhibit these degrees of freedom and are described by our four-dimensional and worldline field theories. We also discuss connections between these degrees of freedom and dynamical worldline multipole moments. We construct effective two-body Hamiltonians, we demonstrate explicitly that the extra degrees of freedom beyond the spin vector are necessary to describe the complete dynamics, and we explicitly remove certain unphysical singularities. Moreover, we show that the previously proposed eikonal (or radial action) formula correctly captures observables derived from the classical Hamiltonian. Finally, we comment on possible descriptions of the additional degrees of freedom from the perspective of Goldstone’s theorem.

Mark Alaverdian

Dimitrios Kosmopoulos

Denmark

Andres Luna

Germany

Trevor Scheopner

Fei Teng

Observables and unconstrained spin tensor dynamics in general relativity from scattering amplitudes

We introduce a constructive method for defining a global loop-integrand basis for scattering amplitudes, encompassing both planar and nonplanar contributions. Our approach utilizes a graph-based framework to establish a well-defined, non-redundant basis of integrands. This basis, constructed from a chosen set of non-redundant graphs together with a selection of irreducible scalar products, provides clear insights into various physical properties of scattering amplitudes and proves useful in multiple contexts, such as on-shell Ward identities and manifesting gauge-choice independence. A key advantage of our integrand basis is its ability to streamline the generalized unitarity method. Specifically, we can directly read off the coefficients of basis elements without resorting to ansätze or solving linear equations. This novel approach allows us to lift generalized unitarity cuts — expressed as products of tree amplitudes — to loop-level integrands, facilitating the use of the tree-level double copy to generate complete gravitational integrands at any loop order. This method circumvents the difficulties in identifying complete higher-loop-order gauge-theory integrands that adhere to the color-kinematics duality. Additionally, our cut-based organization is well-suited for expansion in hard or soft limits, aiding in the exploration of ultraviolet or classical limits of scattering amplitudes.

Michael Ruf

Global bases for nonplanar loop integrands, generalized unitarity, and the double copy to all loop orders

We show that physical scattering observables for compact spinning objects in general relativity can depend on additional degrees of freedom in the spin tensor beyond those described by the spin vector alone. The impulse, spin kick, and leading-order waveforms exhibit such a nontrivial dependence. A signal of this additional structure is the change in the magnitude of the spin vector under conservative Hamiltonian evolution, similar to our previous studies in electrodynamics. These additional degrees of freedom describe dynamical mass multipoles of compact objects and decouple for black holes. We also show that the conservative impulse, spin kick, and change of the additional degrees of freedom are encoded in the eikonal phase.

Conservative Spin-Magnitude Change in Orbital Evolution in General Relativity

We compute the potential-graviton contribution to the scattering amplitude, the radial action, and the scattering angle of two extremal black holes in <math> <mi>N</mi> </math> $$ \mathcal{N} $$ = 8 supergravity at the fifth post-Minkowskian order and to next-to-leading order in a large mass expansion (first self-force order). Properties of classical unitarity cuts allow us to focus on the integration-by-parts reduction of planar integrals, while nonplanar integrals at this order are obtained from the planar ones by straightforward manipulations. We present the solution to the differential equations for all master integrals necessary to evaluate the classical scattering amplitudes of massive scalar particles at this order in all gravitational theories, in particular in <math> <mi>N</mi> </math> $$ \mathcal{N} $$ = 8 supergravity, and in general relativity. Despite the appearance of higher-weight generalized polylogarithms and elliptic functions in the solution to the differential equation for master integrals, the final supergravity answer is remarkably simple and contains only (harmonic) polylogarithmic functions up to weight 2. The systematic analysis of elliptic integrals discussed here, as well as the particular organization of boundary integrals in <math> <mi>N</mi> </math> $$ \mathcal{N} $$ = 8 observables are independent of supersymmetry and may have wider applications, including to aspects of collider physics.

Alexander Smirnov

Vladimir Smirnov

Amplitudes, supersymmetric black hole scattering at <math> <mi>O</mi> <mfenced> <msup> <mi>G</mi> <mn>5</mn> </msup> </mfenced> </math> $$ \mathcal{O}\left({G}^5\right) $$ , and loop integration

We significantly extend recent work on the kinematic Hopf algebra, a structure that was shown to underlie the color-kinematics duality in Yang-Mills (YM) theory coupled to two heavy scalars, known as the heavy-mass effective theory (HEFT) limit. First, staying in the HEFT limit, we show the same kinematic Hopf algebra can be used to obtain Bern-Carrasco-Johansson (BCJ) numerators in <math><msup><mi>DF</mi><mn>2</mn></msup><mo>+</mo><mrow><mi>YM</mi></mrow></math> theory, a theory containing an infinite number of higher derivative corrections to YM, used to obtain string amplitudes via the double copy. Second, we exploit the intricate structure induced by the massive poles to obtain an efficient and direct expression for local BCJ numerators in pure YM based on the same kinematic Hopf algebra. This demonstrates that the kinematic Hopf algebra works even beyond the HEFT limit, strongly suggesting this structure universally underlies the color-kinematics duality.

Gang Chen

Laurentiu Rodina

Congkao Wen

Kinematic Hopf algebra and Bern-Carrasco-Johansson numerators at finite <math><msup><mi>α</mi><mo>′</mo></msup></math>

We compute the potential-photon contributions to the classical relativistic scattering angle of two charged nonspinning bodies in electrodynamics through fifth order in the coupling. We use the scattering amplitudes framework, effective field theory, and multiloop integration techniques based on integration by parts and differential equations. At fifth order, the result is expressed in terms of cyclotomic polylogarithms. Our calculation demonstrates the feasibility of the corresponding calculations in general relativity, including the evaluation of the encountered four-loop integrals.

Conservative Binary Dynamics at Order <math><msup><mi>α</mi><mn>5</mn></msup></math> in Electrodynamics

We introduce a novel approach for deriving one-loop Bern-Carrasco-Johansson (BCJ) numerators and reveal the world-sheet origin of the one-loop double copy. Our work shows that expanding Cachazo-He-Yuan half-integrands into generalized Parke-Taylor factors intrinsically generates BCJ numerators on quadratic propagators satisfying Jacobi identities. We validate our methodology by successfully reproducing one-loop BCJ numerators for the nonlinear sigma model as well as those of pure Yang-Mills theory in four dimensions with all-plus or single-minus helicities.

Jin Dong

Yao-Qi Zhang

Canada

Yong Zhang

One-loop Bern-Carrasco-Johansson numerators on quadratic propagators from the worldsheet

A previous paper [Z. Bern et al., Binary dynamics through the fifth power of spin at <math><mi>O</mi><mo>(</mo><msup><mi>G</mi><mn>2</mn></msup><mo>)</mo></math>, Phys. Rev. Lett. 130, 201402 (2023)] identified a puzzle stemming from the amplitudes-based approach to spinning bodies in general relativity: additional Wilson coefficients appear compared to current worldline approaches to conservative dynamics of generic astrophysical objects, including neutron stars. In this paper we clarify the nature of analogous Wilson coefficients in the simpler theory of electrodynamics. We analyze the original field-theory construction, identifying definite-spin states some of which have negative norms, and relating the additional Wilson coefficients in the classical theory to transitions between different quantum spin states. We produce a new version of the theory which also has additional Wilson coefficients, but no negative-norm states. We match, through <math><mrow><mi>O</mi><mo>(</mo><msup><mi>α</mi><mn>2</mn></msup><mo>)</mo></mrow></math> and <math><mrow><mi>O</mi><mo>(</mo><msup><mi>S</mi><mn>2</mn></msup><mo>)</mo></mrow></math>, the Compton amplitudes of these field theories with those of a modified worldline theory with extra degrees of freedom introduced by releasing the spin supplementary condition. We build an effective two-body Hamiltonian that matches the impulse and spin kick of the modified field theory and of the worldline theory, displaying additional Wilson coefficients compared to standard worldline approaches. The results are then compactly expressed in terms of an eikonal formula. Our key conclusion is that, contrary to standard approaches, while the magnitude of the spin tensor is still conserved, the magnitude of the spin vector can change under conserved Hamiltonian dynamics and this change is governed by the additional Wilson coefficients. For specific values of Wilson coefficients the results are equivalent to those from a definite spin obeying the spin supplementary condition, but for generic values they are physically inequivalent. These results warrant detailed studies of the corresponding issues in general relativity.

Justin Vines

Quantum field theory, worldline theory, and spin magnitude change in orbital evolution