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Kira - A Feynman Integral Reduction Program

Philipp Maierhoefer, Johann Usovitsch, Peter Uwer

TL;DR

Kira presents a new C++ implementation of the Laporta algorithm for reducing scalar multi-loop Feynman integrals to a compact master set. It introduces a modular-arithmetic based method to identify and remove linearly dependent equations, coupled with an optimized Gauss-type forward elimination and a parallelized back-substitution routine, all while storing intermediates in SQLite3. The implementation leverages GiNaC, Fermat, and a YAML-based workflow to handle multi-topology reductions and is benchmarked against Reduze 2 and FIRE 5, showing competitive or superior performance on complex, multi-scale problems. The work demonstrates substantial speedups and memory efficiency, enabling NNLO and higher-loop reductions and suggesting applicability beyond two loops.

Abstract

In this article, we present a new implementation of the Laporta algorithm to reduce scalar multi-loop integrals---appearing in quantum field theoretic calculations---to a set of master integrals. We extend existing approaches by using an additional algorithm based on modular arithmetic to remove linearly dependent equations from the system of equations arising from integration-by-parts and Lorentz identities. Furthermore, the algebraic manipulations required in the back substitution are optimized. We describe in detail the implementation as well as the usage of the program. In addition, we show benchmarks for concrete examples and compare the performance to Reduze 2 and FIRE 5. In our benchmarks we find that Kira is highly competitive with these existing tools.

Kira - A Feynman Integral Reduction Program

TL;DR

Kira presents a new C++ implementation of the Laporta algorithm for reducing scalar multi-loop Feynman integrals to a compact master set. It introduces a modular-arithmetic based method to identify and remove linearly dependent equations, coupled with an optimized Gauss-type forward elimination and a parallelized back-substitution routine, all while storing intermediates in SQLite3. The implementation leverages GiNaC, Fermat, and a YAML-based workflow to handle multi-topology reductions and is benchmarked against Reduze 2 and FIRE 5, showing competitive or superior performance on complex, multi-scale problems. The work demonstrates substantial speedups and memory efficiency, enabling NNLO and higher-loop reductions and suggesting applicability beyond two loops.

Abstract

In this article, we present a new implementation of the Laporta algorithm to reduce scalar multi-loop integrals---appearing in quantum field theoretic calculations---to a set of master integrals. We extend existing approaches by using an additional algorithm based on modular arithmetic to remove linearly dependent equations from the system of equations arising from integration-by-parts and Lorentz identities. Furthermore, the algebraic manipulations required in the back substitution are optimized. We describe in detail the implementation as well as the usage of the program. In addition, we show benchmarks for concrete examples and compare the performance to Reduze 2 and FIRE 5. In our benchmarks we find that Kira is highly competitive with these existing tools.

Paper Structure

This paper contains 23 sections, 30 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integration-by-parts and Lorentz-invariance identities:
  4. Sectors and sub-sectors:
  5. Identification of trivial sectors:
  6. Symmetry relations between integrals:
  7. Laporta Algorithm -- Implementation
  8. Ordering of integrals and equations
  9. Ordering of the Integrals:
  10. Ordering of the Equations:
  11. Reduction procedure
  12. Selection of linearly independent equations
  13. Gauss type forward elimination
  14. Back substitution
  15. Simplifying multivariate rational functions with Fermat
  16. ...and 8 more sections

Figures (3)

  • Figure 1: Planar double box with one massive propagator and one massive external momentum (double lines). All external momenta are counted ingoing. Momentum conservation reads $q_{1}+q_{2}+p_{1}+p_{2}=0$.
  • Figure 2: topo4 is a non planar double box with two massive propagators and one massive external momentum. Momentum conservation reads $q_{1}=p_{1}+p_{2}-q_{2}$.
  • Figure 3: topo5 is a non planar double box with three massive propagators and one massive external momentum. The momentum conservation reads $q_{1}=p_{1}+p_{2}-q_{2}$.