Kira 3: Integral reduction with efficient seeding and optimized equation selection

Abstract

We present version 3 of Kira , a Feynman integral reduction program for high-precision calculations in quantum field theory and gravitational-wave physics. Building on previous versions, Kira 3 introduces optimized seeding and equation selection algorithms, significantly improving performance for multi-loop and multi-scale problems. New features include convenient numerical sampling, symbolic integration-by-parts reductions, and support for user-defined additional relations. We demonstrate its capabilities through benchmarks on two- and three-loop topologies, showcasing up to two orders of magnitude improvement over Kira 2.3 . Kira 3 is publicly available and poised to support ambitious projects in particle physics and beyond. New version program summary Program title: Kira CPC Library link to program files: https://doi.org/10.17632/v3cmsnfrnn.3 Developer’s repository link: https://gitlab.com/kira-pyred/kira Licensing provisions: GNU General Public License 3 Programming language: C++ Journal Reference of previous version: P. Maierhöfer, J. Usovitsch and P. Uwer, Kira—A Feynman integral reduction program , https://doi.org/10.1016/j.cpc.2018.04.012 Comput. Phys. Commun. 230 (2018) 99 [ https://arxiv.org/abs/1705.05610 1705.05610 ]. J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods , https://doi.org/10.1016/j.cpc.2021.108024 Comput. Phys. Commun. 266 (2021) 108024 [ https://arxiv.org/abs/2008.06494 2008.06494 ]. Does the new version supersede the previous version?: Yes. Reasons for the new version: Improved algorithms with significant performance gains for all problems and new features. Summary of revisions: The primary new feature is an improved seeding and selection of equations. Further improvements include the expanded support for numerical integration-by-parts applications, symbolic integration-by-parts reductions, and support for user-defined additional relations. Nature of problem: The reduction of Feynman integrals to a smaller set of master integrals is a central building block for high-precision calculations of observables in theoretical particle and gravitational-wave physics. Furthermore, the reduction is a key ingredient in many methods to calculate the master integrals themselves. Solution method: Kira generates a system of equations using integration-by-parts [1,2], Lorentz-invariance [3], and symmetry relations. It eliminates linearly dependent equations and identifies master integrals by solving the system over a finite field [4] and then solves the system of equations with Laporta’s algorithm [5]. Two solution methods are available: since version 1.0 , Kira algebraically solves the system using Fermat [6]; since version 2.0 , it numerically solves the system multiple times over finite fields, reconstructing master integral coefficients with FireFly [7,8]. Both approaches extend to any homogeneous linear system. New in this version, an optimized algorithm for seeding and selecting integration-by-parts identities demonstrates that a small subset of prior equations suffices for a full reduction. References: [1] F. V. Tkachov, A theorem on analytical calculability of 4-loop renormalization group functions, Phys. Lett. B 100 (1981) 65. https://doi.org/10.1016/0370-2693(81)90288-4 [2] K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate β -functions in 4 loops, Nucl. Phys. B192 (1981) 159. https://doi.org/10.1016/0550-3213(81)90199-1 [3] T. Gehrmann and E. Remiddi, Differential equations for two-loop four-point functions, Nucl. Phys. B580 (2000) 485. https://doi.org/10.1016/S0550-3213(00)00223-6 . [ https://arxiv.org/abs/hep-ph/9912329 hep-ph/9912329 ]. [4] P. Kant, Finding linear dependencies in integration-by-parts equations: A Monte Carlo approach, Comput. Phys. Commun. 185 (2014) 1473. https://doi.org/10.1016/j.cpc.2014.01.017 . [ https://arxiv.org/abs/1309.7287 1309.7287 ]. [5] S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087. https://doi.org/10.1142/S0217751X00002159 . [ https://arxiv.org/abs/hep-ph/0102033 hep-ph/0102033 ]. [6] R. H. Lewis, Computer Algebra System Fermat, https://home.bway.net/lewis . [7] J. Klappert and F. Lange, Reconstructing rational functions with FireFly, https://doi.org/10.1016/j.cpc.2019.106951 Comput. Phys. Commun. 247 (2020) 106951 [ https://arxiv.org/abs/1904.00009 1904.00009 ]. [8] J. Klappert, S. Y. Klein and F. Lange, Interpolation of dense and sparse rational functions and other improvements in FireFly, Comput. Phys. Commun. 264 (2021) 107968. https://doi.org/10.1016/j.cpc.2021.107968 . https://arxiv.org/abs/2004.01463 2004.01463 .

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