Mao Zeng

Motivated by the recent introduction of the Dirac bracket framework to compute spinning observables for the scattering of Kerr black holes, we initiate the study of conserved quantities from an on shell amplitude perspective. We establish new results for the conservation of energy, angular momentum, the Rüdiger invariant, and the quadrupolar Carter constant using the spinning radial action extracted from the literature both in the probe limit and beyond, up to third post-Minkowskian order in the conservative sector. Furthermore, we offer a new perspective on the spin-shift symmetry of the radial action, clarifying its role in the dynamics. Finally, we define a new on shell notion of asymptotic integrability in the Liouville sense and present strong evidence that it is surprisingly satisfied by a spinning probe in Kerr up to quartic order in the probe spin, to all orders in the post-Minkowskian expansion. We further establish integrability beyond the probe limit at low post-Minkowskian orders. Our results suggest important new implications for the dynamics of Kerr black holes.

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.

A bstract: We present a new algorithm for integration-by-parts (IBP) reduction of Feynman integrals with high powers of numerators or propagators, a demanding computational step in evaluating multi-loop scattering amplitudes. The algorithm starts with solving syzygy equations in individual sectors to produce IBP operators that turn seed integrals into IBP equations without artificially raised propagator powers. The IBP operators are expressed in terms of index-shift operators and number operators. We perform row reduction to systematically reshuffle the IBP operators and expose reduction rules with symbolic dependence on the powers of propagators and numerators. When this is insufficient, we produce more symbolic reduction rules by directly solving the linear system of IBP equations in which some propagator/numerator powers are kept symbolic. This linear system is kept small, as the equations are generated from a small set of seed integrals in the neighborhood of the target integral. We stress-test our algorithm against two highly non-trivial examples, namely rank-20 integrals for the double box with an external mass and the massless pentabox. As an application, we revisit the IBP reduction in a calculation of scattering amplitudes for spinning black hole binary systems, which involves two-loop Feynman integrals with complexity greater than 20, and achieve much faster IBP reduction than that of the original calculation.

We propose new methods for optimizing the integration-by-parts (IBP) reduction of Feynman integrals, an important computational bottleneck in modern perturbative calculations in quantum field theory. Using the simple example of one-loop massive bubble integrals, we pose the problem of minimizing the number of arithmetic operations in reducing a target integral to master integrals via the Laporta algorithm. This is a nontrivial combinatorial optimization problem over the ordering of IBP equation generation (from pairs of seed integrals and IBP operators) and the ordering of integral elimination. Our first proposed method is reinforcement learning, which involves an agent interacting with an environment in a step-by-step manner and learning the best actions to take given an observation of the environment (in this case, the current state of the IBP reduction process). The second method is using metaheuristics, e.g., simulated annealing, to minimize the computational cost as a black-box function of numerical priority values that control the orderings. For large-scale problems, the number of free parameters can be compressed by using a small neural network to assign priority values. Remarkably, with almost no human guidance, both methods lead to IBP reduction schemes that are competitive with the most efficient human-designed algorithms. We also found interpretable features in the AI results that may be applicable to more complicated problems.

We compute the conservative and radiation-reaction contributions to classical observables in the gravitational scattering between a spinning and a spinless black hole to the fourth order in spin and third order in the gravitational constant. The conservative results are obtained from two-loop amplitudes for the scattering process of a massive scalar with a massive spin- s field ( s = 0 , 1, 2) minimally coupled to gravity, employing the recently introduced spin interpolation method to resolve all spin-Casimir terms. The two-loop amplitude exhibits a spin-shift symmetry in both probe limits, which we conjecture to be a sign of yet unknown integrability of Kerr orbits through the quartic order in spin and to all orders in the gravitational constant. We obtain the radial action from the finite part of the amplitude and use it to compute classical observables, including the impulse and spin kick. This is done using the recently introduced covariant Dirac brackets, which allow for the computation of classical scattering observables for general (nonaligned) spin configurations. Finally, employing the radiation-reaction amplitude proposed by Alessio and Di Vecchia, together with the Dirac brackets, we obtain radiation-reaction contributions to observables at all orders in spin and beyond the aligned-spin limit. We find agreement with known results up to the quadratic order in spin for both conservative and radiation-reaction contributions. Our results advance the state of the art in the understanding of spinning binary dynamics in general relativity and demonstrate the power and simplicity of the Dirac bracket formalism for relating scattering amplitudes to classical observables.

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.

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 ℏ → 0 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.

Integration-by-parts (IBP) reduction is one of the essential steps in evaluating Feynman integrals. A modern approach to IBP reduction uses modular arithmetic evaluations at the specific numerical values of parameters with subsequent reconstruction of the analytic rational coefficients. Due to the large number of sample points needed, problems at the frontier of science require an application of supercomputers. In this article, we present a rational function reconstruction method that fully takes advantage of sparsity, combining the balanced reconstruction method and the Zippel method. Additionally, to improve the efficiency of the finite-field IBP reduction runs, at each run several numerical probes are computed simultaneously, which allows to decrease the resource overhead. We describe what performance issues one encounters on the way to an efficient implementation on supercomputers, and how one should co-design the algorithm and the supercomputer infrastructure. We present characteristic examples of IBP-reduction in the case of massless two-loop fourand five-point Feynman diagrams using a development version of FIRE and give illustrative examples mimicking the reduction of coefficients appearing in scattering amplitudes for post-Minkowskian gravitational binary dynamics. Редукция с использованием интегрирования по частям (IBP) является одним из существенных этапов при вычислении интегралов Фейнмана. Современный подход к IBP-редукции использует модулярную арифметику при конкретных числовых значениях параметров в пробных точках с последующей реконструкцией аналитических рациональных коэффициентов. Задачи, возникающие на переднем крае науки, требуют применения суперкомпьютеров из-за большого количества необходимых проб. В этой статье мы представляем алгоритм рациональной реконструкции, который в полной мере использует преимущества разреженности, объединяя сбалансированный алгоритм реконструкции и алгоритм Зиппеля. Кроме того, для повышения эффективности редукции в модулярной арифметике при каждом запуске одновременно вычисляется несколько числовых проб, что позволяет сокращать потребляемые ресурсы. Мы описываем, какие проблемы появляются на пути к эффективной реализации на суперкомпьютерах и как следует совместно проектировать алгоритмы и соответствующую суперкомпьютерную инфраструктуру. Представлены характерные примеры IBP-редукции в случае безмассовых двухпетлевых четырехточечных и пятиточечных диаграмм Фейнмана с использованием частной версии FIRE, а также показательные примеры редукции, имитирующие редукцию коэффициентов, появляющихся в амплитуде рассеяния в рамках постминковской гравитационной бинарной динамики.