Tropical Feynman integration in the Minkowski regime

Abstract

We present a new computer program, 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 iε prescription concretely while retaining projective invariance. 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 feyntrop and discuss many explicit examples of evaluated Feynman integrals. Program summaryProgram title:feyntrop.CPC Library link to program files:https://doi.org/10.17632/k6r62hdgvd.1Developer's repository link:https://github.com/michibo/feyntrop.Licensing provisions: MIT License.Programming language: The tropical Monte Carlo code is written in C++. The high-level interface is written in python.Supplementary material: The repository includes installation and usage instructions (README.md), a jupyter notebook tutorial (tutorial_2L_3pt.ipynb), the collection of examples presented in section 6 (see the folder /examples), and a test suite to ensure a successful installation (see the folder /tests).Nature of problem: Sufficiently fast numerical integration of (dimensionally regularized) Feynman integrals (also in the Minkowski regime of phase space).Solution method: Tropical Monte Carlo integration of a manifestly iε-free parametric representation of Feynman integrals.Additional comments: The program feyntrop is based on previous code available at https://github.com/michibo/tropical-feynman-quadrature, which was published as a proof-of-concept with, Michael Borinsky, ‘Tropical Monte Carlo quadrature for Feynman integrals’, Ann. Inst. Henri Poincaré Comb. Phys. Interact. (in press)[1]. This previous code did not have features which are required for phenomenological studies in high-energy physics. In particular, it only allowed for phase space points in the Euclidean regime, and only computed the leading term in the ϵ expansion.Restrictions: The Feynman integral must be quasi-finite and the momentum configuration must be sufficiently generic. Numerators of Feynman integrals are not implemented. ReferencesEigen3[2]. The xoshiro256+ pseudo-random-number generator [3]. python[4]. pybind11[5]. sympy[6].