Tropical Feynman integration in the Minkowski regime

TL;DR

This paper introduces 'feyntrop', a new software that uses tropical geometry to numerically evaluate Feynman integrals in the physical regime, accommodating multiple scales and complex kinematics.

Contribution

The paper presents a novel parametric representation and a computational tool that efficiently evaluates Feynman integrals using tropical geometry, including the causal $i extbackslashvarepsilon$ prescription.

Findings

Efficient evaluation of multi-scale Feynman integrals.

Implementation of causal $i extbackslashvarepsilon$ prescription in tropical approach.

Systematic classification of kinematic regimes.

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.