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Reduze 2 - Distributed Feynman Integral Reduction
A. von Manteuffel, C. Studerus
TL;DR
Reduze 2 addresses the computational task of reducing Feynman integrals to master integrals. It combines a distributed Laporta-based reduction with load balancing and graph- and matroid-based topology relations. The program supports reductions across sectors and within single sectors, and has been used for two-loop heavy-quark pair-production corrections.
Problem
Reducing dimensionally regularized Feynman integrals to a small set of master integrals requires solving the associated identity system efficiently.
Method
Reduze 2 uses MPI-distributed Laporta reduction, dynamic load balancing, and graph- and matroid-based identification of equivalent sectors and integrals.
Results
Reduze 2 supports distributed reduction of sectors and individual sectors’ integrals and was used for two-loop leading-color heavy-quark pair-production corrections in gluon fusion.
Takeaways & Limitations
Reduze 2 provides an open-source, non-proprietary framework for large-scale Feynman-integral reduction across multiple processor cores.
Abstract
from arXiv · showhide
Reduze is a computer program for reducing Feynman integrals to master integrals employing a variant of Laporta's reduction algorithm. This article describes version 2 of the program. New features include the distributed reduction of single topologies on multiple processor cores. The parallel reduction of different topologies is supported via a modular, load balancing job system. Fast graph and matroid based algorithms allow for the identification of equivalent topologies and integrals.
1 Introduction
Reduze 2 is a public C++ program for reducing dimensionally regularized Feynman integrals to master integrals. It extends Reduze with distributed reduction, topology matching, and tools for broader amplitude and integral workflows.
- Reduze 2 rewrites and extends its predecessor as a public C++ reduction program.
- The program uses a fully distributed variant of Laporta’s algorithm to reduce sectors and integrals within a single sector via MPI.
- Graph- and matroid-theory algorithms identify relations between sectors across integral families, including possible crossings of external momenta.
- Reduze supports differential-equation generation, amplitude-interference computation up to master integrals, Berkeley DB storage, and exports to common computer-algebra formats.
- Reduze 2 was used for two-loop leading-color corrections to heavy-quark pair production in the gluon-fusion channel and is released under GPL v3 without proprietary dependencies.
2 Basic concepts and notations
Reduze organizes loop integrals through integral families and sectors, then derives IBP and LI identities to reduce them and detects zero, redundant, and symmetry-related sectors.
- Integral families and sectors: An integral family is a minimal, complete ordered set of propagators that represents all loop-momentum scalar products using inverse propagators and kinematic invariants.
- Integral families and sectors: Selecting t propagators defines a sector; sectors are related by subset inclusion, with each sector assigned an identification number.
- Integral representation: Reduze represents integrals by a family, sector identifier, propagator-exponent data, and numerator and denominator exponent sums.
- IBP identities: IBP identities arise because dimensionally regularized integrals of total derivatives vanish, producing l(l+m) equations from one seed integral.
- LI identities: LI identities are not linearly independent of IBPs but can accelerate reduction because same-seed IBPs generally cannot reproduce them alone.
- Sector relations and symmetries: Graph methods detect trivially zero sectors, while sector relations and symmetries eliminate redundant sectors or express integrals through related sectors and subsectors.
3 Graph and matroid based algorithms for sectors
Reduze identifies physical sectors by constructing graph representations, then uses graph isomorphisms, twists, and matroid isomorphisms to detect sector relations and symmetries efficiently.
- Physical sectors: A physical sector has propagators represented as graph edges with momentum conservation, enabling graph-based analysis.Reduze constructs the graph by starting from independent loop-momentum propagators and successively cleaving vertices to match remaining propagator momenta.
- Physical sectors: Reduze automatically constructs graphs for representable sectors, identifies physical sectors, and exposes fast algorithms for graph isomorphism, sector relations, and symmetries.These graph representations support both topology equivalence detection and symmetry analysis.
- Sector relations: Graph-isomorphic physical sectors yield loop-momentum shifts, including cases with crossed external legs when |det M| = 1.Canonical labels of graph adjacencies are used to identify the relevant graph correspondences.
- Sector relations: For non-isomorphic graphs described by the same propagators, matroid isomorphism captures equivalence, with twists relating the corresponding biconnected vacuum graphs.Whitney’s theorem provides the graph-theoretic basis for this relation.
- Sector relations: Reduze detects vacuum-sector shifts by decomposing graphs into biconnected and triconnected components and minimizing canonical labels after possible twists.This extends graph-isomorphism-based shift detection for cases represented by matroid equivalence.
- Sector relations: A combinatorial linear-algebra procedure finds all shifts between arbitrary sectors, but is usually less efficient than graph-based methods.The alternative also covers sectors without graph representations.
4 Distributed reduction algorithm
Reduze 2 distributes Laporta reduction at both the sector and within-sector levels, using bottom-up equation solving and dynamic worker assignment. Performance depends strongly on the problem and algebra system, with substantial scaling in favorable cases.
- Distributed reduction algorithm: Reduze 2 implements two MPI-based parallelization levels: independent sector jobs and distributed reduction of integrals within a single sector.Sector reductions proceed bottom-up because subsector results are required first.
- Distributed reduction algorithm: IBP and LI identities form a sparse homogeneous linear system whose equation blocks are reduced bottom-up through back substitution and forward elimination.Integrals are replaced by combinations of lower integrals according to results from lower blocks and subsectors.
- Load balancing: Worker processes perform block reductions while a manager tracks the complete equation system and balances work among workers.The manager assigns work when workers request it, enabling distributed execution of a single sector.
- Load balancing: Block-reduction times can differ by more than 6 orders of magnitude, motivating dynamic load balancing and bottom-up evaluation order.The authors state that both factors affect full-reduction execution time.
- Performance results: Scaling is problem specific: an example scales well to 22 workers, while larger worker counts eventually reduce gains and can increase runtime.The observed limits are attributed to serial code, communication overhead, and potentially less ideal evaluation order.
- Performance results: Using Fermat instead of GiNaC for GCD computations can produce a speed-up by an order of magnitude for sector B.The comparison used Fermat with equations in RAM and GiNaC with a database for the displayed benchmarks.
5 Job system
Reduze’s modular job system schedules dependent computations across MPI processes and dynamically reallocates workers between jobs. In a heavy-quark production example, reductions benefited considerably from up to 96 cores, although scaling remained application dependent.
- Job system: A Reduze job is a computation sequence enabled when its input-file dependencies are fulfilled, allowing different jobs to execute in parallel.The job queue resolves dependencies and selects the next executable job.
- Job system: A job center dynamically assigns jobs to MPI clients, which act as managers or workers depending on the assigned task.Idle clients request work, and workers assist the responsible job manager.
- Dynamic balancing: The job center reallocates workers using performance measurements, releasing them from inefficient employers and assigning them to efficient employers.This mechanism is intended to avoid low efficiency from overloaded managers.
- Performance results: Up to 96 cores provided considerable benefit in a realistic heavy-quark pair-production reduction example.For smaller core counts, runtime decrease was close to optimal 1/n_cores behavior.
- Performance results: The authors stress that scaling may be worse for other application types.The benchmark compared Fermat with equations in RAM against GiNaC with a database on a different cluster.
6 Other features
Reduze provides additional workflows beyond integral reduction, including differential-equation generation and automated computation of scalar amplitude interferences from QGRAF diagrams.
- Differential equations: Reduze can derive differential equations for Feynman integrals, substitute reduction results, and support user-selected master-integral choices.A change of master-integral basis may be required for sectors with several independent integrals.
- Interference terms: Starting from QGRAF diagrams, Reduze computes scalar QED or QCD amplitude interferences in dimensional regularization and can reduce the occurring integrals to master integrals.The workflow includes user-defined rules, Lorentz-index contraction, Dirac traces, and color evaluation; individual diagram interferences are MPI-parallelizable jobs.
7 Usage
Reduze is distributed as a downloadable package containing installation and usage documentation together with example files.
- Usage: The Reduze distribution includes a tutorial with detailed installation and usage instructions and several example files.The package is available from the project web page.