arXiv subjects

Mao Zeng

Publications and source records attributed to Mao Zeng.

At least 19 recordsLinked to original sources

arXiv2026

Five-point spinor-helicity Compton amplitudes in gauge theory and gravity

We calculate the five-point tree-level Compton amplitudes for minimally coupled spinning matter in gauge theory and gravity in the massive spinor-helicity formalism, using BCFW recursion and KLT relations. The results are written in terms of symmetrized spinor products and are simultaneously valid for a range of matter spins, which directly generalizes the amplitudes of Arkani-Hamed, Huang, and Huang to the five-point case. Unlike the three- and four-point cases, each amplitude can be a sum of two or more distinct spin structures. Spurious poles are manifestly canceled through a mechanism that holds for all massive spins $s\leq 1$ in gauge theory and $s\leq 2$ in gravity. While we keep the amplitudes exact, their classical limits are expected to be relevant for binary systems of Kerr black holes in the post-Minkowskian expansion.

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arXiv2026

Efficient AI-Inspired Reduction of Feynman Integrals via Tube Seeding

In this paper, we use machine learning to discover a new seeding strategy for integration-by-parts reduction of Feynman integrals, which is a frequent bottleneck in state-of-the-art calculations in theoretical particle and gravitational-wave physics. Our strategy allows us to reduce multi-loop integrals with large numerator powers via essentially the standard Laporta algorithm but with a sparse selection of seed integrals that grows only linearly with the numerator power, whereas existing strategies lead to growth with a polynomial power that increases with the complexity of the integral being reduced. The seeds are restricted to a thin tube-like region that connects the target integral to the master integrals along a zigzag path. We demonstrate the power of our approach by reducing non-planar 2-loop 5-point integrals of rank 20 with numerical kinematics over a finite field, which is prohibitively difficult for the Laporta algorithm with conventional seeding. Going beyond individual integrals, we further demonstrate the reduction of a complete set of top-level rank-10 integrals by dividing the target integrals into several chunks, each of which can be solved by our sparse seeding strategy with considerably less time and a significantly lower memory footprint than other state-of-the-art strategies, making the approach well-suited for phenomenological applications. We provide a proof-of-principle implementation on GitHub at https://github.com/andreslunagodoy/tube_seeding.

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arXiv2025

Scattering Amplitudes and Conservative Binary Dynamics at $O(G^5)$ without Self-Force Truncation

We compute the complete potential-graviton contributions to the conservative radial action and scattering angle for two non-spinning bodies in general relativity, accurate through fifth order in Newton's constant and including second-order self-force (2SF) effects. The calculation is carried out in the scattering-amplitude framework, combining the double copy, effective field theory, and multi-loop integration techniques based on integration by parts and differential equations. To address a major computational bottleneck, we develop improved integration-by-parts algorithms that render calculations at this order tractable. The post-Minkowskian amplitude is presented as a series expansion, following the strategy used earlier in maximal supergravity. For the first self-force sector, which involves only polylogarithmic functions, we also provide a closed-form analytic expression. For the second self-force sector, as in earlier supergravity work, we find nontrivial cancellations among contributions related to integrals supported on Calabi-Yau geometry.

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arXiv2025

FIRE 7: Automatic Reduction with Modular Approach

FIRE7 is a major update to the FIRE program for integration-by-parts (IBP) reduction of Feynman integrals. A large part of improvements is related to the automatic reduction and reconstruction with the modular arithmetic approach, while the performance of the classical rational polynomial approach is also significantly increased. An improved presolve algorithm performs Gaussian elimination to simplify IBP identities before substituting numerical indices as in the Laporta algorithm. Various new command line tools are included to facilitate tasks such as applying an IBP reduction table to reduce a loop integrand as a linear combination of individual integrals.

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arXiv2025

Second-order self-force potential-region binary dynamics at $O(G^5)$ in supergravity

We compute the potential-graviton contributions to the conservative scattering angle of two non-spinning 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 multi-loop 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.

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arXiv2025

Unexpected Symmetries of Kerr Black Hole Scattering

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 R\"udiger 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 PM orders. Our results suggest important new implications for the dynamics of Kerr black holes.

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arXiv2025

Feynman Integral Reduction using Syzygy-Constrained Symbolic Reduction Rules

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.

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arXiv2025

Reinforcement Learning and Metaheuristics for Feynman Integral Reduction

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.

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arXiv2025

First Look at Quartic-in-Spin Binary Dynamics at Third Post-Minkowskian Order

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 (non-aligned) 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.

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arXiv2024

Classical worldlines from 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 (KMOC) formalism, motivated by the study of gravitational waves from black hole binaries. We achieve the manifest cancellation of divergences in the $\hbar \to 0$ limit at the integrand level beyond one loop by employing the Schwinger parametrisation to rewrite both cut and uncut propagators in a worldline-like 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 (PMEFT) and worldline quantum field theory (WQFT), 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.

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arXiv2024

Feynman integral reduction: balanced reconstruction of sparse rational functions and implementation on supercomputers in a co-design approach

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 with parameters set to numerical values at sample points, followed by 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. Benchmarks are presented for IBP reductions for massless two-loop four- and five-point integrals using a development version of FIRE, as well as synthetic examples mimicking the coefficients involved in scattering amplitudes for post-Minkowskian gravitational binary dynamics.

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arXiv2024

Global Bases for Nonplanar Loop Integrands, Generalized Unitarity, and the Double Copy to All Loop Orders

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 ansatze 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.

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arXiv2024

Spinning Black Hole Scattering at $\mathcal{O}(G^3 S^2)$: Casimir Terms, Radial Action and Hidden Symmetry

We resolve subtleties in calculating the post-Minkowskian 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 $S^2 = -s(s+1)\hbar^2$, 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 $\mathcal{O}(G^3 S^2)$, 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 root-Kerr electromagnetic background up to $\mathcal{O}(\alpha^3 S^2)$. 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.

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arXiv2024

Amplitudes, Supersymmetric Black Hole Scattering at $\mathcal{O}(G^5)$, and Loop Integration

We compute the potential-graviton contribution to the scattering amplitude, the radial action, and the scattering angle of two extremal black holes in N = 8 supergravity at the fifth post-Minkowskian order 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 all master integrals and solve their associated differential equations necessary to evaluate the classical scattering amplitudes of massive scalar particles at this order in all gravitational theories, in particular in 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 N = 8 observables are independent of supersymmetry and may have wider applications, including to aspects of collider physics.

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arXiv2023

FIRE 6.5: Feynman Integral Reduction with New Simplification Library

FIRE is a program which performs integration-by-parts (IBP) reduction of Feynman integrals. Originally, the C++ version of FIRE relies on the computer algebra system Fermat by Robert Lewis to simplify rational functions. We present an upgrade of FIRE which incorporates a new library FUEL initially described in a separate publication, which enables a flexible choice of third-party computer algebra systems as simplifiers, as well as efficient communications with some of the simplifiers as C++ libraries rather than through Unix pipes. We achieve significant speedups for IBP reduction of Feynman integrals involving many kinematic variables, when using an open source backend based on FLINT newly added in this work, or the Symbolica backend developed by Ben Ruijl as a potential successor of FORM.

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arXiv2023

Rational Function Simplification for Integration-by-Parts Reduction and Beyond

We present FUEL (Fractional Universal Evaluation Library), a C++ library for performing rational function arithmetic with a flexible choice of third-party computer algebra systems as simplifiers. FUEL is an outgrowth of a C++ interface to Fermat which was originally part of the FIRE code for integration-by-parts (IBP) reduction for Feynman integrals, now promoted to be a standalone library and with access to simplifiers other than Fermat. We compare the performance of various simplifiers for standalone benchmark problems as well as IBP reduction runs with FIRE. A speedup of more than 10 times is achieved for an example IBP problem related to off-shell three-particle form factors in $\mathcal N=4$ super-Yang-Mills theory.

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arXiv2023

Feynman Integrals from Positivity Constraints

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.

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arXiv2023

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

We compare numerical self-force results and analytical fourth-order post-Minkowskian (PM) calculations for hyperbolic-type scattering of a point-like particle carrying a scalar charge $Q$ off a Schwarzschild black hole, showing a remarkably good agreement. Specifically, we numerically compute the scattering angle including the full $O(Q^2)$ scalar-field self-force term (but ignoring the gravitational self-force), and 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 $O(Q^2 G^3)$. 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.

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