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.

United States

Zvi Bern

Enrico Herrmann

Radu Roiban

Michael Ruf

United Kingdom

Mao Zeng

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

<p>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.</p>

Mark Alaverdian

Switzerland

Dimitrios Kosmopoulos

Denmark

Andres Luna

Trevor Scheopner

Fei Teng

Conservative Spin-Magnitude Change in Orbital Evolution in General Relativity

We show that tree-level and one-loop Mellin space correlators in anti-de Sitter space obey certain difference equations, which are the direct analog to the differential equations for Feynman loop integrals in the flat space. Finite-difference relations, which we refer to as “summation-by-parts relations”, in parallel with the integration-by-parts relations for Feynman loop integrals, are derived to reduce the integrals to a basis. We illustrate the general methodology by explicitly deriving the difference equations and summation-by-parts relations for various tree-level and one-loop Witten diagrams up to the four-point bubble level.

Aidan Herderschee

Difference equations and integral families for Witten diagrams

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.

Russia

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

Scattering amplitudes in Yang-Mills-Einstein theories have been investigated mostly for compact gauge groups. While non-compact gauge groups are not physically viable in Yang-Mills theory, non-compact gaugings feature prominently in the supergravity literature, where any choice of perturbative vacuum spontaneously breaks the gauge group to a compact subgroup. In this paper, we formulate double-copy constructions for several five-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supergravities with non-compact gauge groups. On one side of the double copy, we employ amplitudes from a super-Yang-Mills theory with a massive hypermultiplet. On the other, we use amplitudes from particular non-supersymmetric Yang-Mills-scalar theories with massive fermions, chosen to obey constraints coming from color/kinematics duality. Supergravities with massive self-dual tensors in five dimensions are also considered, showing that tensors are straightforwardly realized as double copies of gauge-theory fermions with suitable choices of signs in the corresponding solutions of the Dirac equation. We present several examples of these constructions, noting in particular the appearance of Heisenberg groups in the supergravity gauge symmetry and, in some cases, the possibility of exotic tensor-vector matter couplings.

Sweden

Marco Chiodaroli

Murat Günaydin

Henrik Johansson

Non-compact gauge groups, tensor fields and Yang-Mills-Einstein amplitudes

We revisit the quantum-amplitude-based derivation of the gravitational waveform emitted by the scattering of two spinless massive bodies at the third order in Newton’s constant, <math><mi>h</mi><mo>∼</mo><mi>G</mi><mo>+</mo><msup><mi>G</mi><mn>2</mn></msup><mo>+</mo><msup><mi>G</mi><mn>3</mn></msup></math> (one-loop level), and correspondingly update its comparison with its classically derived multipolar-post-Minkowskian counterpart. A spurious-pole-free reorganization of the one-loop five-point amplitude substantially simplifies the post-Newtonian expansion. We find complete agreement between the two results up to the fifth order in the small velocity expansion after taking into account three subtle aspects of the amplitude derivation: (1) in agreement with [A. Georgoudis et al., J. High Energy Phys. 03 (2024) 089], the term quadratic in the amplitude in the observable-based formalism [D. A. Kosower et al., J. High Energy Phys. 02 (2019) 137] generates a frame rotation by half the classical scattering angle; (2) the dimensional regularization of the infrared divergences of the amplitude introduces an additional <math><mo>(</mo><mi>d</mi><mo>−</mo><mn>4</mn><mo>)</mo><mo>/</mo><mo>(</mo><mi>d</mi><mo>−</mo><mn>4</mn><mo>)</mo></math> finite term; and (3) zero-frequency gravitons are found to contribute additional terms both at order <math><mi>h</mi><mo>∼</mo><msup><mi>G</mi><mn>1</mn></msup></math> and at order <math><mi>h</mi><mo>∼</mo><msup><mi>G</mi><mn>3</mn></msup></math> when including disconnected diagrams in the observable-based formalism.

Italy

Donato Bini

France

Thibault Damour

Stefano De Angelis

Andrea Geralico

Gravitational waveforms: A tale of two formalisms

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.

Michael S. Ruf

Alexander V. Smirnov

Vladimir A. Smirnov

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

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

We compare numerical self-force results and analytical fourth-order post-Minkowskian (PM) calculations for hyperbolic-type scattering of a pointlike particle carrying a scalar charge <math><mi>Q</mi></math> off a Schwarzschild black hole, showing a remarkably good agreement. Specifically, we numerically compute the scattering angle including the full <math><mi>O</mi><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> scalar-field self-force term (but ignoring the gravitational self-force), and we compare with analytical expressions obtained in a PM framework using scattering-amplitude methods. This example provides a nontrivial, high-precision test of both calculation methods and illustrates the complementarity of the two approaches in the context of the program to provide high-precision models of gravitational two-body dynamics. Our PM calculation is carried out through 4PM order, i.e., including all terms through <math><mi>O</mi><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><msup><mi>G</mi><mn>3</mn></msup><mo>)</mo></math>. At the fourth post-Minkowskian order the point-particle description involves two a priori undetermined coefficients, due to contributions from tidal effects in the model under consideration. These coefficients are chosen to align the post-Minkowskian results with the self-force ones.

Leor Barack

Germany

Oliver Long

Julio Parra-Martinez

Chia-Hsien Shen

Mikhail P. Solon

Comparison of post-Minkowskian and self-force expansions: Scattering in a scalar charge toy model

We compute the next-to-leading order term in the scattering waveform of uncharged black holes in classical general relativity and of half-BPS black holes in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 8 supergravity. We propose criteria, generalizing explicit calculations at next-to-leading order, for determining the terms in amplitudes that contribute to local observables. For general relativity, we construct the relevant classical integrand through generalized unitarity in two distinct ways, (1) in a heavy-particle effective theory and (2) in general relativity minimally-coupled to scalar fields. With a suitable prescription for the matter propagator in the former, we find agreement between the two methods, thus demonstrating the absence of interference of quantum and classically-singular contributions. The classical   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 8 integrand for massive scalar fields is constructed through dimensional reduction of the known five-point one-loop integrand. Our calculation exhibits novel features compared to conservative calculations and inclusive observables, such as the appearance of master integrals with intersecting matter lines and the appearance of a classical infrared divergence whose absence from classical observables requires a suitable definition of the retarded time.

The sub-leading scattering waveform from amplitudes