ftint: Calculating gradient-flow integrals with pySecDec

Computer Physics Communications, vol. 306, pp. 109384

Abstract

The program ftint is introduced which numerically evaluates dimensionally regularized integrals as they occur in the perturbative approach to the gradient-flow formalism in quantum field theory . It relies on sector decomposition in order to determine the coefficients of the individual orders in ϵ = ( 4 − D ) / 2 , where D is the space-time dimension. For that purpose, it implements an interface to the public library pySecDec . The current version works for massive and massless integrals up to three-loop level with vanishing external momenta, but the underlying method is extendable to more general cases. PROGRAM SUMMARY Program title: ftint CPC Library link to program files: https://doi.org/10.17632/bt6bkffxh9.1 Developer's repository link: https://gitlab.com/ftint/ftint Licensing provisions: MIT license Programming language: Python Supplementary material: README.md Nature of problem: The perturbative approach to the gradient-flow formalism in quantum field theory leads to integrals which closely resemble regular Feynman integrals. However, they involve exponential factors which depend on the loop and external momenta, as well as on so-called flow-time variables. In general, the latter are also integrated over a finite interval. These integrals cannot be solved immediately with standard tools. Solution method: The flow-time integrals are transformed to integrals over a hypercube by introducing Schwinger parameters. The latter are numerically evaluated using the public program pySecDec , which performs a sector decomposition and calculates the coefficients of the poles in the parameter ϵ = ( 4 − D ) / 2 by numerical integration, where D is the space-time dimension which occurs in dimensional regularization . Additional comments including restrictions and unusual features: In its current form, ftint is restricted to one-, two-, and three-loop integrals with vanishing external momenta. It does allow for massive propagators though, raised to virtually arbitrary integer powers.

Authors 4

1. Robert V. Harlander corresponding Aachen

   RWTH Aachen University

   Affiliation as printed

   TTK, RWTH Aachen University, 52056 Aachen, Germany

2. Theodoros Nellopoulos Aachen

   RWTH Aachen University

   Affiliation as printed

   TTK, RWTH Aachen University, 52056 Aachen, Germany

3. Anton Olsson

   Karlsruhe Institute of Technology

   Affiliation as printed

   Institute for Theoretical Physics, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany

4. Marius Wesle

   University of Tübingen

   Affiliation as printed

   Department of Mathematics, University of Tübingen, 72076 Tübingen, Germany