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FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
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Complex algebraic calculations can be performed by reconstructing analytic results from numerical evaluations over finite fields. We describe FiniteFlow, a framework for defining and executing numerical algorithms over finite fields and reconstructing multivariate rational functions. The framework employs computational graphs, known as dataflow graphs, to combine basic building blocks into complex algorithms. This allows to easily implement a wide range of methods over finite fields in high-level languages and computer algebra systems, without being concerned with the low-level details of the numerical implementation. This approach sidesteps the appearance of large intermediate expressions and can be massively parallelized. We present applications to the calculation of multi-loop scattering amplitudes, including the reduction via integration-by-parts identities to master integrals or special functions, the computation of differential equations for Feynman integrals, multi-loop integrand reduction, the decomposition of amplitudes into form factors, and the derivation of integrable symbols from a known alphabet. We also release a proof-of-concept C++ implementation of this framework, with a high-level interface in Mathematica.
Forward citations
Cited by 28 Pith papers
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Machine learning discovers a tube-seeding strategy for IBP reduction of Feynman integrals that scales linearly with numerator power, demonstrated on rank-20 2-loop 5-point integrals.
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Learning to Unscramble Feynman Loop Integrals with SAILIR
A self-supervised transformer learns to unscramble Feynman integrals for online IBP reduction, delivering bounded memory use on complex two-loop topologies while matching Kira's speed on the hardest cases tested.
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Emergence of Calabi-Yau manifolds in high-precision black hole scattering
At 5PM-1SF order, Calabi-Yau three-fold periods emerge in radiation-reacted observables for classical black hole scattering computed with worldline QFT and advanced IBP/DE methods.
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Compact Syzygies for Feynman Integrals from Landau Singularities
Syzygy solutions for IBP reduction are systematically constructed as maximal minors of certificate matrices derived from the leading Landau singularities of Feynman diagrams.
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NNLO QCD predictions for $t\bar{t}W$ production at the LHC
NNLO QCD predictions for ttW production with two-loop amplitudes evaluated explicitly in the generalised leading-colour limit.
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Landau's Leviathans
New algorithm identifies complete Landau singularities of Feynman integrals via Euler characteristic drops over finite fields, applied to non-planar two-loop six-point and massive three-loop graphs.
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Gravitational Compton scattering at the fourth post-Minkowskian order
Derives gravitational Compton amplitude at O(G^4) and N-matrix element for scattering phase shift, verified by agreement with black-hole perturbation theory.
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Electroweak corrections to Higgs boson pair production: The quark channel
Mixed QCD-EW corrections to qqbar -> HH computed analytically via differential equations, matched to large-mass limit, implemented in POWHEG-BOX, showing up to +10% effect on invariant mass distribution near threshold.
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Taming Symbolic IBP Reduction with Intermediate Bases
An algorithm reconstructs symbolic IBP reduction coefficients via intermediate bases, demonstrated on massive box-triangle and pentagon-triangle integrals using 3289 and 13013 samplings versus over a million unknowns.
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NNLO QCD predictions for $t\bar t W$ production at hadron colliders
NNLO QCD predictions for ttW production at hadron colliders using direct two-loop amplitude computation in the generalised leading-colour limit.
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Magic Relations and Critical Varieties of Feynman Integrals
Magic relations in Feynman integral families coincide with higher-dimensional critical varieties, enabling a practical test to detect and handle them.
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Tensor decomposition of $e^+e^-\to\pi^+\pi^-\gamma$ to higher orders in the dimensional regulator
First beyond-NLO tensor decomposition and higher-order analytic one-loop amplitudes for e+e- to pi+pi-gamma, paired with a fast numerical five-point integral evaluator.
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Feynman integral reduction with intersection theory made simple
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Twisted Feynman Integrals: from generating functions to spin-resummed post-Minkowskian dynamics
Twisted Feynman integrals are introduced with graded Symanzik polynomials, classified as exponential periods, and shown to have geometry not inferable from generalized Baikov leading singularities.
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Two-loop all-plus helicity amplitudes for self-dual Higgs boson with gluons via unitarity cut constraints
Two-loop all-plus helicity amplitudes for self-dual Higgs plus gluons are obtained via four-dimensional unitarity cuts into one-loop and tree amplitudes plus finite-field tensor reduction.
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Double virtual QCD corrections to $t\bar{t}+$jet production at the LHC
Leading-colour two-loop virtual amplitudes for ttbar+jet are extracted analytically via finite-field evaluations and differential equations, then packaged in a C++ library with new numerical integration techniques.
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Integral Reduction with Kira 2.0 and Finite Field Methods
Kira 2.0 implements finite-field coefficient reconstruction for IBP reductions and improved user-equation handling, yielding lower memory use and faster performance on state-of-the-art problems.
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A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons
First numerical evaluation of planar two-loop helicity amplitudes for W-boson plus four partons using finite-field reduction and sector decomposition on a subset of master integrals.
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On the spanning cuts consistency problem in the IBP reductions of Feynman integrals
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HyperPrecision: A Mathematica package for High-Precision Numerical Evaluation of Multivariate Hypergeometric Functions
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All-order structure of static gravitational interactions and the seventh post-Newtonian potential
A closed formula computes static post-Newtonian corrections at arbitrary odd orders in gravity, yielding the explicit seventh post-Newtonian potential that matches an independent diagrammatic method.
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New algorithms for Feynman integral reduction and $\varepsilon$-factorised differential equations
A geometric order relation in IBP reduction yields a master-integral basis with Laurent-polynomial differential equations on the maximal cut that are then ε-factorized.
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Weak-field waveforms for generic relativistic orbits
Outlines a Schwinger-Keldysh path-integral framework that derives worldline equations of motion and computes weak-field gravitational waveforms independently for unspecified relativistic orbits.
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SubTropica
SubTropica is a software package that automates symbolic integration of linearly-reducible Euler integrals via tropical subtraction, supported by HyperIntica and an AI-driven Feynman integral database.
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The geometric bookkeeping guide for $\varepsilon$-factorised differential equations
Describes a geometric-ordering approach to the Laporta algorithm plus transformation matrices that produce ε-factorised differential equations for arbitrary Feynman integral families.
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One-loop amplitudes for $t\bar{t}j$ and $t\bar{t}\gamma$ productions at the LHC through $\mathcal{O}(\epsilon^2)$
Analytic expressions for one-loop helicity amplitudes in ttj and ttγ production are derived to O(ε²) as linear combinations of pentagon functions with rational coefficients in momentum-twistor variables, obtained via ...
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Linac: linear algebra with CUDA over finite fields
Linac provides a high-performance open-source CUDA implementation of Gaussian elimination over finite fields and floating-point arithmetic for analytic reconstruction of scattering amplitudes in quantum field theory.
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Les Houches 2023 -- Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
The report reviews progress since 2021 in fixed-order computations for LHC applications and identifies processes requiring missing higher-order corrections to match anticipated experimental precision.
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