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Tropical Feynman integration in the Minkowski regime

Michael Borinsky, Henrik J. Munch, Felix Tellander

TL;DR

The paper introduces feyntrop, a numerical framework that evaluates quasi-finite Feynman integrals in the Minkowski regime using tropical geometry. It develops a fully projective, iε-deformed parametric representation and leverages tropical sampling on base polytopes, notably generalized permutahedra, to achieve efficient Monte Carlo integration. Key contributions include a detailed regime classification, a contour-deformation scheme preserving projective symmetry, fast matrix-based evaluation of Symanzik polynomials and their derivatives, and extensive demonstrations on high-loop, multi-scale diagrams. The approach offers a practical tool for high-precision Feynman integral computations in physically relevant regimes, with clear pathways to handle subdivergences and exceptional kinematics in future work.

Abstract

We present a new computer program, $\texttt{feyntrop}$, which uses the tropical geometric approach to evaluate Feynman integrals numerically. In order to apply this approach in the physical regime, we introduce a new parametric representation of Feynman integrals that implements the causal $i\varepsilon$ prescription concretely while retaining projective invariance. $\texttt{feyntrop}$ can efficiently evaluate dimensionally regulated, quasi-finite Feynman integrals, with not too exceptional kinematics in the physical regime, with a relatively large number of propagators and with arbitrarily many kinematic scales. We give a systematic classification of all relevant kinematic regimes, review the necessary mathematical details of the tropical Monte Carlo approach, give fast algorithms to evaluate (deformed) Feynman integrands, describe the usage of $\texttt{feyntrop}$ and discuss many explicit examples of evaluated Feynman integrals.

Tropical Feynman integration in the Minkowski regime

TL;DR

The paper introduces feyntrop, a numerical framework that evaluates quasi-finite Feynman integrals in the Minkowski regime using tropical geometry. It develops a fully projective, iε-deformed parametric representation and leverages tropical sampling on base polytopes, notably generalized permutahedra, to achieve efficient Monte Carlo integration. Key contributions include a detailed regime classification, a contour-deformation scheme preserving projective symmetry, fast matrix-based evaluation of Symanzik polynomials and their derivatives, and extensive demonstrations on high-loop, multi-scale diagrams. The approach offers a practical tool for high-precision Feynman integral computations in physically relevant regimes, with clear pathways to handle subdivergences and exceptional kinematics in future work.

Abstract

We present a new computer program, , which uses the tropical geometric approach to evaluate Feynman integrals numerically. In order to apply this approach in the physical regime, we introduce a new parametric representation of Feynman integrals that implements the causal prescription concretely while retaining projective invariance. can efficiently evaluate dimensionally regulated, quasi-finite Feynman integrals, with not too exceptional kinematics in the physical regime, with a relatively large number of propagators and with arbitrarily many kinematic scales. We give a systematic classification of all relevant kinematic regimes, review the necessary mathematical details of the tropical Monte Carlo approach, give fast algorithms to evaluate (deformed) Feynman integrands, describe the usage of and discuss many explicit examples of evaluated Feynman integrals.
Paper Structure (42 sections, 6 theorems, 48 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 42 sections, 6 theorems, 48 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Feynman integrals
  3. Momentum and parametric representations
  4. Combinatorial Symanzik polynomials
  5. Kinematic regimes
  6. Euclidean regime
  7. Pseudo-Euclidean regime
  8. Minkowski regime
  9. Generic and exceptional kinematics
  10. Contour deformation
  11. Landau singularities
  12. Dimensional regularization and expansions
  13. Tropical geometry
  14. Tropical approximation
  15. Tropical sampling
  16. ...and 27 more sections

Key Result

Theorem 3.1

For a homogeneous $p\in\mathbb{C}[x_0,\ldots,x_{|E|-1}]$ that is completely non-vanishing on $\mathbb{P}_+^{E}$ there exist constants $C_1,C_2>0$ such that

Figures (3)

  • Figure 1: Partition of kinematics into different regimes.
  • Figure 2: Massless box with different external legs on- or off-shell. On-shell ($p^2=0$) legs are drawn as dashed lines and off-shell ($p^2\neq 0$) legs with solid lines. Internal propagators are massless.
  • Figure 3: Triangle Feynman graph relevant in QED. The two solid propagators have mass $m$ and the solid legs have incoming squared momentum $m^2$. The dashed propagator is massless and the doubled leg has incoming squared momentum $Q^2$.

Theorems & Definitions (10)

  • Theorem 3.1: Borinsky:2020rqs
  • Theorem 3.3: aguiar2017hopf and the references therein
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • proof
  • Conjecture 3.7
  • Theorem 3.9