Feynman integrals are completely monotonic (and often Stieltjes) functions, enabling a CM bootstrap for bounds from differential equations and Padé approximants with provable convergence.
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Hidding,DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, Comput
15 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
DiffExp is a Mathematica package for integrating families of Feynman integrals order-by-order in the dimensional regulator from their systems of differential equations, in terms of one-dimensional series expansions along lines in phase-space, which are truncated at a given order in the line parameter. DiffExp is based on the series expansion strategies that were explored in recent literature for the computation of families of Feynman integrals relevant for Higgs plus jet production with full heavy quark mass dependence at next-to-leading order. The main contribution of this paper, and its associated package, is to provide a public implementation of these series expansion methods, which works for any family of integrals for which the user provides a set of differential equations and boundary conditions (and for which the program is not computationally constrained.) The main functions of the DiffExp package are discussed, and its use is illustrated by applying it to the three loop equal-mass and unequal-mass banana graph families.
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representative citing papers
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.
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.
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.
Chebyshev polynomial approximations with adaptive sampling solve canonical differential equations for Feynman integrals, demonstrated to be stable and competitive for two-loop five-point cases in double precision.
A strategy is introduced to solve canonical differential equations for Feynman master integrals on arbitrary geometries by reducing numerical evaluation to an enlarged system of rational differential equations.
HyperPrecision is a new Mathematica package that evaluates general Horn-type multivariate hypergeometric functions and their ε-expansions to high precision by reducing Pfaffian PDE systems to solvable ODEs.
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.
AMFlow 2.0 cuts symbolic and numerical cost of multi-loop Feynman integral evaluation via an FT recursion mode, a C++ DE solver, and modern IBP reducers, demonstrated on a three-loop five-point family.
IterInt package evaluates iterated integrals by transforming them into solvable differential equation systems with built-in regularization.
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.
Describes a geometric-ordering approach to the Laporta algorithm plus transformation matrices that produce ε-factorised differential equations for arbitrary Feynman integral families.
A general numerical framework is described for high-precision evaluation and analytic continuation of multivariate hypergeometric functions via Pfaffian systems and the Frobenius method.
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 differential equations solved numerically by generalized power series expansion.
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.
citing papers explorer
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Approximating Feynman integrals using complete monotonicity and Stieltjes properties
Feynman integrals are completely monotonic (and often Stieltjes) functions, enabling a CM bootstrap for bounds from differential equations and Padé approximants with provable convergence.
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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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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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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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Chebyshev Approximations of Feynman Integrals for Collider Physics
Chebyshev polynomial approximations with adaptive sampling solve canonical differential equations for Feynman integrals, demonstrated to be stable and competitive for two-loop five-point cases in double precision.
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Solution of Canonical Differential Equations for Integrals on Arbitrary Geometries
A strategy is introduced to solve canonical differential equations for Feynman master integrals on arbitrary geometries by reducing numerical evaluation to an enlarged system of rational differential equations.
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HyperPrecision: A Mathematica package for High-Precision Numerical Evaluation of Multivariate Hypergeometric Functions
HyperPrecision is a new Mathematica package that evaluates general Horn-type multivariate hypergeometric functions and their ε-expansions to high precision by reducing Pfaffian PDE systems to solvable ODEs.
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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.
-
AMFlow 2.0: significant algorithmic and software improvements for Feynman integral evaluation
AMFlow 2.0 cuts symbolic and numerical cost of multi-loop Feynman integral evaluation via an FT recursion mode, a C++ DE solver, and modern IBP reducers, demonstrated on a three-loop five-point family.
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IterInt: Evaluating iterated integrals via differential equations
IterInt package evaluates iterated integrals by transforming them into solvable differential equation systems with built-in regularization.
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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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Numerical analytical continuation of multivariate hypergeometric functions
A general numerical framework is described for high-precision evaluation and analytic continuation of multivariate hypergeometric functions via Pfaffian systems and the Frobenius method.
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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 differential equations solved numerically by generalized power series expansion.
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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.