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 present a systematic diagrammatic investigation of the classical limit of observables computed from scattering amplitudes in quantum field theory through the Kosower-Maybee-O’Connell formalism, motivated by the study of gravitational waves from black hole binaries. We achieve the manifest cancellation of divergences in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℏ</mi><mo stretchy="false">→</mo><mn>0</mn></math> limit at the integrand level beyond one loop by employing the Schwinger parametrization to rewrite both cut and uncut propagators in a worldlinelike representation before they are combined. The resulting finite classical integrand takes the same form as the counterpart in the worldline formalisms such as post-Minkowskian effective field theory and worldline quantum field theory, and in fact exactly coincides with the latter in various examples, showing explicitly the equivalence between scattering amplitude and worldline formalisms. The classical causality flow, as expressed by the retarded propagator prescription, appears as an emergent feature. Examples are presented for impulse observables in electrodynamics and a scalar model at two loops, as well as certain subclasses of diagrams to higher orders and all orders.</p>

Switzerland

Zeno Capatti

Classical worldlines from scattering amplitudes

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 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

We explore inequality constraints as a new tool for numerically evaluating Feynman integrals. A convergent Feynman integral is non-negative if the integrand is non-negative in either loop momentum space or Feynman parameter space. Applying various identities, all such integrals can be reduced to linear sums of a small set of master integrals, leading to infinitely many linear constraints on the values of the master integrals. The constraints can be solved as a semidefinite programming problem in mathematical optimization, producing rigorous two-sided bounds for the integrals which are observed to converge rapidly as more constraints are included, enabling high-precision determination of the integrals. Positivity constraints can also be formulated for the ϵ expansion terms in dimensional regularization and reveal hidden consistency relations between terms at different orders in ϵ. We introduce the main methods using one-loop bubble integrals, then present a nontrivial example of three-loop banana integrals with unequal masses, where 11 top-level master integrals are evaluated to high precision.

Feynman integrals from positivity constraints

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

Fei Teng

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

We compute the conservative two-body Hamiltonian of a compact binary system with a spinning black hole through <math><mi>O</mi><mo>(</mo><msup><mi>G</mi><mn>3</mn></msup><mo>)</mo></math> to all orders in velocity, including linear and quadratic spin terms. To obtain our results we calculate the classical limit of the two-loop amplitude for the scattering of a massive scalar particle with a massive spin-1 particle minimally coupled to gravity. We employ modern scattering amplitude and loop integration techniques, in particular numerical unitarity, integration-by-parts identities, and the method of regions. The conservative potential in terms of rest-frame spin vectors is extracted by matching to a nonrelativistic effective field theory. We also apply the Kosower-Maybee-O’Connell (KMOC) formalism to calculate the impulse in the covariant spin formalism directly from the amplitude. We work systematically in conventional dimensional regularization and explicitly evaluate all divergent integrals that appear in full- and effective-theory amplitudes, as well as in the phase-space integrals that arise in the KMOC formalism.

China

Guanda Lin

Conservative Binary Dynamics with a Spinning Black Hole at <math><mi>O</mi><mo>(</mo><msup><mi>G</mi><mn>3</mn></msup><mo>)</mo></math> from Scattering Amplitudes

Quantum Electrodynamics (QED) serves as a useful toy model for classical observables in gravitational two-body systems with reduced complexity due to the linearity of QED. We investigate scattering observables in scalar QED at the sixth order in the charges (two-loop order) in a classical regime analogous to the post-Minkowskian expansion in General Relativity. We employ modern scattering amplitude tools and extract classical observables by both eikonal methods and the formalism of Kosower, Maybee, and O’Connell (KMOC). In addition, we provide a simplified approach to extracting the radial action beyond the conservative sector.

Juan Gatica

Denmark

Andres Luna

Scalar QED as a toy model for higher-order effects in classical gravitational scattering

We complete the calculation of conservative two-body scattering dynamics at fourth post-Minkowskian order, i.e., <math><mi>O</mi><mo>(</mo><msup><mi>G</mi><mn>4</mn></msup><mo>)</mo></math> and all orders in velocity, including radiative contributions corresponding to the tail effect in general relativity. As in previous calculations, we harness powerful tools from the modern scattering amplitudes program including generalized unitarity, the double copy, and advanced multiloop integration methods, in combination with effective field theory. The classical amplitude involves complete elliptic integrals, and polylogarithms with up to transcendental weight 2. Using the amplitude-action relation, we obtain the radial action directly from the amplitude, and match the known overlapping terms in the post-Newtonian expansion.

Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at <math><mi>O</mi><mo>(</mo><msup><mi>G</mi><mn>4</mn></msup><mo>)</mo></math>