Tropical Monte Carlo quadrature for Feynman integrals

Abstract

We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized integration algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.

arXiv comment6 figures, see http://github.com/michibo/tropical-feynman-quadrature for the referenced program code; v2: typos corrected, version accepted for publication in Annales de l'Institut Henri Poincaré D

References in corpus (37)

• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• Multiloop integrals in dimensional regularization made simple
• Cuba - a library for multidimensional numerical integration
• Differential Equations for Feynman Graph Amplitudes
• An automatized algorithm to compute infrared divergent multi-loop integrals
• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Lectures on differential equations for Feynman integrals
• pySecDec: a toolbox for the numerical evaluation of multi-scale integrals
• Minimally subtracted six loop renormalization of $O(n)$-symmetric ${\varphi }^4$ theory and critical exponents
• SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop
• Sector Decomposition
• Resolution of singularities for multi-loop integrals
• Four-Particle Phase Space Integrals in Massless QCD
• On Motives Associated to Graph Polynomials
• Feynman graph polynomials
• FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions
• Numbers and Functions in Quantum Field Theory
• On the periods of some Feynman integrals
• A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec
• Knots and Numbers in ${\varphi }^4$ Theory to 7 Loops and Beyond
• A K3 in phi4
• An Étude in non-linear Dyson--Schwinger Equations
• Loop Tree Duality for multi-loop numerical integration
• New differential equations for on-shell loop integrals
• A geometric method of sector decomposition
• GKZ-hypergeometric systems for Feynman integrals
• Recursion and growth estimates in renormalizable quantum field theory
• Spanning forest polynomials and the transcendental weight of Feynman graphs
• Feynman graph generation and calculations in the Hopf algebra of Feynman graphs
• Renormalized asymptotic enumeration of Feynman diagrams
• Perturbation expansions at large order: Results for scalar field theories revisited
• The Euler characteristic of $\mathrm{Out}(F_n)$
• Feynman amplitudes on moduli spaces of graphs
• Graph complexes and Feynman rules
• Multi-valued Feynman Graphs and Scattering Theory

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• Les Houches 2023 -- Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• Learning Feynman integrals from differential equations with neural networks
• Statistics of Feynman amplitudes in ${\varphi }^4$-theory
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