Source-linked AI summary
Kira 3: integral reduction with efficient seeding and optimized equation selection
Fabian Lange, Johann Usovitsch, Zihao Wu
TL;DR
High-precision particle and gravitational-wave calculations require reducing large systems of Feynman integrals to master integrals, but conservative seeding becomes impractical for complex problems. Kira 3 revises seeding and equation selection while adding reduction features, and benchmarks report major improvements over Kira 2.3. Its analytic-coefficient performance remains limited when interpolation and reconstruction become bottlenecks.
Problem
Reducing Feynman integrals to master integrals is central to high-precision particle and gravitational-wave calculations, while conservative seeding can become infeasible for many-loop or high-rank problems.
Method
Kira 3 uses refined seeding and equation selection, with additional relations, finite-field sampling, symbolic reductions, and master-integral basis checks.
Results
Kira 3 produces substantially fewer equations and improves performance across two- and three-loop topologies compared with Kira 2.3.
Takeaways & Limitations
Kira 3 provides a broadly improved reduction program with features supporting numerical, symbolic, and user-extended integral-reduction workflows.
Takeaways & Limitations
For complicated rational functions, interpolation and reconstruction can become bottlenecks that diminish the achieved performance improvements.
Abstract
from arXiv · showhide
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
Kira 3 supersedes the previous version with improved algorithms, especially seeding and equation selection, plus expanded reduction features. Its reduction workflow combines equation generation from physical and symmetry relations with finite-field identification and Laporta-style solving.
- Kira 3 supersedes the previous version and delivers improved algorithms with significant performance gains across problems.
- Improved seeding and equation selection are the primary new algorithmic features.
- Additional capabilities include numerical integration-by-parts applications, symbolic integration-by-parts reductions, and user-defined additional relations.
- Feynman-integral reduction supports high-precision observables in theoretical particle and gravitational-wave physics and helps calculate master integrals.
- Kira generates relations from integration-by-parts, Lorentz-invariance, and symmetries, then identifies master integrals over finite fields and solves with Laporta’s algorithm.
1 Introduction
High-precision particle-physics and gravitational-wave calculations increasingly involve complex multi-loop and multi-leg integral reductions. Kira 3 addresses this challenge with improved seeding and selection strategies alongside numerical, symbolic, and basis-checking features.
- Feynman integrals underpin high-precision predictions for particle-physics experiments and gravitational-wave studies.
- Increasing precision demands more complex loop orders and external legs, motivating continued development of integration-by-parts reduction strategies.
- Kira 3 reduces reduction complexity through improved seeding and equation-selection strategies, especially for high tensor ranks.
- The release also supports additional relations, user-provided finite-field sampling, symbolic reductions, and master-integral basis checks.
2 Preliminaries
Kira reduces Feynman integrals by generating and solving linear relations among integrals, using sector organization, bounded seeding, and finite-field or symbolic algebraic strategies. Its earlier workflow generated independent equations broadly, then discarded equations irrelevant to target reductions.
- Integral setup: Feynman integrals depend on propagator momenta, external scales, masses, and dimensional regularization, with integer propagator exponents.
- Relations: Integration-by-parts and Lorentz-invariance identities, together with symmetry relations, produce linear equations that express integrals through master integrals.
- Reduction strategy: Laporta reduction generates equations for different propagator-power choices called seeds and solves the resulting system with elimination algorithms.
- Integral organization: Kira organizes integrals into topologies and sectors, using sector structure to order integrals and constrain which seeds generate equations.
- Workflow: The workflow separates equation generation from solving, offering analytic Fermat arithmetic or modular finite-field solving with coefficient reconstruction.
- Previous workflow: The previous algorithm generated equations within user bounds, retained numerically independent equations, and later removed equations that did not contribute to target solutions.
3 New features
Kira 3 targets seed combinatorics by selecting relevant equations more efficiently and revising how sectors, symmetries, subsectors, and truncation bounds are handled. These changes can substantially reduce generated work, while some options require tuning to avoid unreduced integrals.
- 3.1 Improved seeding: For many-loop or high-rank reductions, conservative seeding can make equation generation infeasible because seed combinatorics exceed runtime and memory limits.
- 3.1 Improved seeding: Kira 3 identifies a relevant subset of equations and revises seeding for symmetry-related sectors, preferred-master sectors, and subsectors.
- 3.1.1 Changed sector behavior: Subsectors inherit rmax, smax, and dmax from higher sectors independently of symmetries and preferred master integrals.
- 3.1.1 Changed sector behavior: Kira 3 generates IBP equations for sectors mapped away by symmetries because symmetry equations can become more expensive with complicated numerator coefficients.
- 3.1.2 Truncating the seeds: Restricting s in lower sectors addresses factorial growth in irreducible-scalar-product assignments, and suitable parameters can produce speedups of orders of magnitude.
- 3.1.2 Truncating the seeds: Using select_mandatory_recursively with truncate_sp can leave many integrals unreduced because required seeds are removed.
3.2 Improved selection algorithm
Kira 3 replaces earlier equation-selection behavior with a forward-elimination-based algorithm that removes hidden-zero and irrelevant equations more effectively. In the topo7 comparison, it selected fewer equations despite generating more.
- Problem: Previous selection could retain equations that cancel during intermediate steps, creating hidden zeros and unnecessary work.For x[4], equations (i) and (ii) were unnecessary after inserting equation (iv) into equation (iii).
- Algorithm: Kira 3 selects relevant equations after forward elimination and repeats numerical checks until all requested integrals reduce to the minimal master-integral basis.The procedure also applies to user-defined systems.
- Limitation: The algorithm does not guarantee the minimal number of equations because stealthy zeros can remain undetected during forward elimination.Random reordering was considered, but dense systems made it computationally expensive and batch reordering reduced equations by at most 20%.
- Results: 22% more equations are generated at smax = 5, but Kira 3 selects 50% fewer equations than Kira 2.3.The comparison uses the topo7 system and the same requested integrals.
- Results: Increasing smax to 6 no longer changes the selected equations with the new selection algorithm.Earlier sector-by-sector selection increased selected equations, terms, and solution time when smax increased.
3.3 Extra relations
Kira 3 allows additional relations to be added during the normal reduction setup, addressing relations that Kira may not discover automatically.
- Extra relations: Users can add relations arising from non-trivial symmetries or four-dimensional kinematic constraints during normal reduction setup.Earlier Kira versions allowed additional relations only in user-defined system mode.
3.4 Numerical sampling
Kira 3 supports numerical sampling of reductions at user-chosen phase-space points, including user-specified finite fields and input values. The output is organized into Mathematica-readable result files.
- Purpose: Numerical sampling can be used before obtaining full analytic expressions or instead of analytic reduction expressions.The feature targets complicated reductions and selected phase-space points.
- Input: Users provide the finite-field prime and sample points through the numerical_points option.The input includes symbols such as kinematic invariants, the dimension d, and any additional user-defined symbols.
- Equation generation: Kira accounts for phase-space slices where symbols are not independent when generating equations for numerical reductions.Default pseudo-random sampling can miss cancellations that occur on a restricted phase-space slice.
- Output: Results are written in Mathematica-readable files containing the prime, numerical-point identifiers, and reduction results.Files are split after 1000 entries to avoid overly large outputs.
3.5 Symbolic integration-by-parts reductions
Kira 3 can treat selected propagator powers symbolically and use the Laporta strategy to derive symbolic recursion relations. The feature is practical for one or a few powers, while fully algebraic relations for all indices remain expensive.
- Feature: Kira 3 treats selected propagator powers as symbols and derives symbolic recursion relations with the Laporta strategy.The feature is enabled through symbolic_ibp in integralfamilies.yaml.
- Scope: Every symbolic power becomes an additional symbol, making fully algebraic recursion relations in all indices computationally expensive.Constructing relations for one or very few powers works reasonably well in many problems.
- Setup: Users specify symbolic propagators by listing their positions in the integral-family definition.The allowed indices range from 1 to the total number of propagators.
- Example: The supplied example uses symbolic IBP for the first propagator and generates lowering relations for its symbolic power.The example selects topo7 sector b011111100 and uses the lowering job configuration.
- Relation direction: Selected integrals and preferred master integrals determine whether Kira constructs lowering or raising relations.Sectors may be merged with a subsector when symbolic indices are zero or negative, generating fewer equations.
- Output: The resulting expressions use b0 through b(N-1) for symbolic propagator powers and encode recurrence relations between integrals.The example includes a relation for topo7 with coefficients depending on d and b0.
3.6 Check master integral basis
Kira lets users specify a master-integral basis, including linear combinations, before reduction. This basis choice can be combined with new seeding features to generate an optimal equation system.
- Users can provide their preferred master-integral basis through the job-file option preferred_masters.Since version 2.0, the basis may also contain linear combinations of master integrals.
- The basis-selection feature is especially useful when users have already determined a basis and want to exploit Kira 3’s new seeding features.These features are used to generate an optimal system of equations.
4 Further changes
Kira 3 adds internal propagator reordering, wider integral weights, and user-controlled numerical sampling during equation generation. These changes improve equation selection while retaining a stated 31-line topology boundary from 32-bit sector representation.
- 4.1 Internal reordering of propagators: 29% faster finite-field solution time accompanies 3% fewer equations and 6.5% fewer terms when a favorable propagator order is used.The comparison concerns two permutation choices for a four-loop propagator topology.
- 4.1 Internal reordering of propagators: Kira 3 enables a propagator-ordering scheme by default, while users can override it with permutation_option or a manual permutation.The ordering scheme is intended to improve equation selection before reduction.
- 4.2 Support for 128-bit weights: 128-bit integral weights are available through the kira128 executable, but sectors remain 32-bit integers.The stated consequence is that topologies with more than 31 lines are not yet supported.
- 4.3 Numerical sampling: User-defined numerical points are now respected during equation generation and selection.This behavior is combined with symbols2num: false when constructing the independent system.
5 Benchmarks
Benchmarks cover two-loop double boxes, a three-loop TennisCourt, double-pentagon families, and a pentagon-hexagon family. Across these examples, Kira 3 substantially reduces generated equations, memory use, generation time, and solution time versus Kira 2.3, though some comparisons remain future work.
- Benchmark setup: The benchmark suite includes double-box, TennisCourt, double-pentagon, and pentagon-hexagon integral families, with reductions specified by rmax.smax.dmax.Comparisons are performed against Kira 2.3 using top-level-sector integrals.
- 5.1 Double-box topologies: 27 times fewer equations and 4.0 times faster finite-field solution characterize Kira 3 for topo7.Generation becomes 11 times faster and memory use improves by a factor of 12.
- 5.1 Double-box topologies: 15 times fewer equations and 3.9 times faster pyRed solution characterize Kira 3 for topo5.Generation is 16 times faster and requires 9.5 less memory.
- Solver comparison: Kira 3’s equations can contain more equations overall yet remain smaller on disk because they use fewer terms and simpler coefficients.Solving runtimes are comparable across the compared systems, while only Ratracer and pyRed were tested as solvers.
- Scope of the benchmarks: Testing whether Kira 3 matches NeatIBP-level efficiency for another three-loop 2 →3 topology is left for future work.The paper describes the observed efficiency for the studied two-loop 2 →3 topology as an open question for other topologies.
6 Conclusions
Kira 3 improves reduction performance and memory use through refined seeding and equation selection, while adding numerical, symbolic, and user-defined-relation capabilities. Benchmarks cover two- and three-loop topologies, and the authors identify remaining opportunities in selection, interpolation, solving, and symmetry detection.
- Kira 3 brings substantial performance improvements and reduces main-memory requirements compared with Kira 2.3.The gains are mainly driven by refined seeding and improved equation selection.
- The refined seeding strategy generates significantly fewer equations while still fully reducing target integrals.
- For a two-loop 2 →3 topology, Kira 3 produces a system of equations on par with those generated by dedicated strategies.
- New features include additional relations, finite-field sampling at chosen phase-space points, symbolic IBP reductions, and master-integral-basis checks.
- Further gains may come from filtering irrelevant equations, machine-learning-based seeding and selection, new symmetry detection, improved interpolation and reconstruction, and better linear-system solvers.The authors also mention possible GPU acceleration and further computer-algebra-system improvements.