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

We resolve subtleties in calculating the post-Minksowskian dynamics of binary systems, as a spin expansion, from massive scattering amplitudes of fixed finite spin. In particular, the apparently ambiguous spin Casimir terms can be fully determined from the gradient of the spin-diagonal part of the amplitudes with respect to = − ( +1) , using an interpolation between massive amplitudes with different spin representations. From two-loop amplitudes of spin-0 and spin-1 particles minimally coupled to gravity, we extract the spin Casimir terms in the conservative scattering angle between a spinless and a spinning black hole at $$ \mathcal{O} $$ ( ), finding agreement with known results in the literature. This completes an earlier study [Phys. Rev. Lett. 130 (2023), 021601] that calculated the non-Casimir terms from amplitudes. We also illustrate our methods using a model of spinning bodies in electrodynamics, finding agreement between scattering amplitude predictions and classical predictions in a electromagnetic background up to $$ \mathcal{O} $$ ( ). For both gravity and electrodynamics, the finite part of the amplitude coincides with the two-body radial action in the aligned spin limit, generalizing the amplitude-action relation beyond the spinless case. Surprisingly, the two-loop amplitude displays a hidden spin-shift symmetry in the probe limit, which was previously observed at one loop. We conjecture that the symmetry holds to all orders in the coupling constant and is a consequence of integrability of Kerr orbits in the probe limit at the first few orders in spin.

Dogan Akpinar

Fernando Febres Cordero

Mexico

Manfred Kraus

Spinning black hole scattering at

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

We describe an approach to incorporating the physical effects of the absorption of energy by the event horizon of black holes in the scattering amplitudes based post-Minkowskian, point-particle effective description. Absorptive dynamics are incorporated in a model-independent way by coupling the usual point-particle description to an invisible sector of gapless internal degrees-of-freedom. The leading order dynamics of this sector are encoded in the low-energy expansion of a spectral density function obtained by matching an absorption cross section in the ultraviolet description. This information is then recycled using the scattering amplitudes based Kosower-Maybee-O’Connell in-in formalism to calculate the leading absorptive contribution to the impulse and change in rest mass of a Schwarzschild black hole scattering with a second compact body sourcing a massless scalar, electromagnetic or gravitational field. The results obtained are in complete agreement with previous worldline Schwinger-Keldysh calculations and provide an alternative on-shell scattering amplitudes approach to incorporating horizon absorption effects in the gravitational two-body problem.

Callum Jones

Absorptive effects and classical black hole scattering

We compute classical gravitational observables for the scattering of two spinless black holes in general relativity and   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$ =8 supergravity in the formalism of Kosower, Maybee, and O’Connell (KMOC). We focus on the gravitational impulse with radiation reaction and the radiated momentum in black hole scattering at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (G 3) to all orders in the velocity. These classical observables require the construction and evaluation of certain loop-level quantities which are greatly simplified by harnessing recent advances from scattering amplitudes and collider physics. In particular, we make use of generalized unitarity to construct the relevant loop integrands, employ reverse unitarity, the method of regions, integration-by-parts (IBP), and (canonical) differential equations to simplify and evaluate all loop and phase-space integrals to obtain the classical gravitational observables of interest to two-loop order. The KMOC formalism naturally incorporates radiation effects which enables us to explore these classical quantities beyond the conservative two-body dynamics. From the impulse and the radiated momentum, we extract the scattering angle and the radiated energy. Finally, we discuss universality of the impulse in the high-energy limit and the relation to the eikonal phase.

Julio Parra-Martinez

Germany

Radiative classical gravitational observables at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (G 3) from scattering amplitudes

We use   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 8 supergravity as a toy model for understanding the dynamics of black hole binary systems via the scattering amplitudes approach. We compute the conservative part of the classical scattering angle of two extremal (half-BPS) black holes with minimal charge misalignment at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (G 3) using the eikonal approximation and effective field theory, finding agreement between both methods. We construct the massive loop integrands by Kaluza-Klein reduction of the known D-dimensional massless integrands. To carry out integration we formulate a novel method for calculating the post-Minkowskian expansion with exact velocity dependence, by solving velocity differential equations for the Feynman integrals subject to modified boundary conditions that isolate conservative contributions from the potential region. Motivated by a recent result for universality in massless scattering, we compare the scattering angle to the result found by Bern et. al. in Einstein gravity and find that they coincide in the high-energy limit, suggesting graviton dominance at this order.

Switzerland

Extremal black hole scattering at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (G 3): graviton dominance, eikonal exponentiation, and differential equations