Source-linked AI summary
Presenting LiteRed: a tool for the Loop InTEgrals REDuction
R. N. Lee
TL;DR
Multiloop IBP calculations require automation, while existing reductions can be slow or require substantial manual work to find reusable rules. LiteRed heuristically searches for symbolic reduction rules and provides tools for applying them and finding sector symmetries. When successful, the resulting rules provide an effective solution for the corresponding class of integrals, although the search is not guaranteed to terminate and depends on setup choices.
Problem
Multiloop IBP reduction is computationally demanding, while reusable symbolic rules are difficult to discover and existing approaches can generate redundant calculations and large databases.
Method
LiteRed uses the SolvejSector procedure for heuristic searches of symbolic IBP reduction rules, with additional tools for sector symmetries and related reduction tasks.
Results
When the heuristic search succeeds, LiteRed's rules provide an effective reduction solution for the given class of multiloop integrals.
Takeaways & Limitations
The saved symbolic rules can be applied in later reductions, while FindSymmetries can map equivalent sectors.
Takeaways & Limitations
The heuristic search is not proved to terminate and may fail or depend strongly on the ordering of denominators and choice of irreducible numerators.
Abstract
from arXiv · showhide
Mathematica package LiteRed is described. It performs the heuristic search of the symbolic IBP reduction rules for loop integrals. It implements also several convenient tools for the search of the symmetry relations, construction of the differential equations and dimensional recurrence relations.
1 Introduction
Multiloop IBP reduction is important but increasingly difficult to perform manually at scale. LiteRed addresses this bottleneck by heuristically searching for reusable symbolic reduction rules.
- Growing loop order makes multiloop calculations increasingly complicated and necessitates computer automation of stages such as IBP reduction.
- IBP reduction transforms multiloop-diagram calculations into the calculation of a finite set of master integrals and supports differential and difference equations.
- The Laporta algorithm is widely used because it is easy to implement, reliable given sufficient time, and open to programming improvements.
- Laporta reductions can be slow because they generate large rule databases and perform many unnecessary calculations caused by redundant IBP identities.
- Symbolic reduction rules avoid solving during reduction, making application fast and allowing compact rules to be saved for future calculations.
- LiteRed implements a heuristic approach that automatically searches for symbolic IBP reduction rules before applying them to reductions.
2 General setup
The paper formulates loop integrals as lattice-indexed functions and develops IBP, Lorentz-invariance, symmetry, sector, differential-equation, and dimensional-recurrence structures for their reduction.
- Integral representation: Loop integrals are represented by integer points whose scalar functions form a complete basis, including propagator denominators and irreducible numerators.The basis contains N=L(L+1)/2+LE scalar products for L loops and E external momenta.
- IBP identities: IBP identities follow from vanishing total derivatives in dimensional regularization and produce recurrence relations for the integral function.Differentiation and rewriting scalar products through Dα yield the recurrence relation.
- Lorentz-invariance identities: Lorentz-invariance identities arise because the integral is a Lorentz scalar and can be useful in practical reductions despite being expressible through IBP identities.Their differential operator is the generator of Lorentz transformations on scalar functions of external momenta.
- Sectors and ordering: Sectors partition the integer lattice by denominator presence, so integrals in one sector share the same denominator set.The corner point is the sector’s defining point, and integrals with fewer denominators are ordered as simpler.
- Scaleless integrals: A sector is zero when its corner-point IBP solution yields an identity setting the corner integral to zero, although this criterion can miss scaleless sectors.The massless one-loop on-shell propagator illustrates a scaleless integral that requires an auxiliary-vector construction to establish its vanishing.
- Symmetry relations: Symmetry transformations mapping denominator sets induce identities between integrals belonging to the same or different sectors.The transformations also map numerators into linear polynomials of the D_i.
- Further relations: Differential equations and dimensional recurrences provide additional relations for finding and calculating master integrals.Differential equations arise after differentiating with respect to masses or external invariants and applying IBP reduction; dimensional recurrences relate integrals at shifted dimension.
3 Using the LiteRed package
LiteRed usage separates searching for symbolic reduction rules from applying saved rules. The package also supports graph drawing, invariant differentiation, dimensional recurrences, and Feynman-parameter functions.
- Workflow: LiteRed workflows have separate search and application stages, with basis-related definitions and reduction rules saved to disk for later use.A sample program searches rules for a two-loop on-shell massive propagator.
- Basis construction: NewBasis defines a linearly independent, complete basis and generates objects used by subsequent package procedures.Generated objects include Ds, SPs, LMs, EMs, and Toj.
- Rule search: SolvejSector constructs symbolic rules for each unique sector and saves them for reduction.The resulting rules are associated with each js entry in the UniqueSectors list.
- Graph tools: LiteRed can attach graphs to highest sectors and determine graphs for subsectors, which can then be plotted with GraphPlot.Automatic graph determination from denominator sets was planned for future versions.
- Additional tools: Additional tools compute derivatives with respect to external invariants, raise or lower dimensional recurrences, and return Feynman-parametrization functions U and F.For Minkowskian metrics and odd loop number, the factor μ^L=−1 must be included manually in the dimensional-recurrence tools.
4 Implementation notes
LiteRed relies on auxiliary Mathematica packages and heuristic sector-analysis procedures. Its reduction-rule search is sensitive to basis choices and may require changing numerator or denominator ordering.
- Dependencies: LiteRed depends on Types, Numbers, Vectors, and LinearFunctions, which provide typed variables, transformation rules, and linear-function distribution.LinearFunctions distributes over sums and factors out numbers and Number-typed expressions.
- Sector and symmetry analysis: AnalyzeSectors uses a criterion from Ref. to identify zero sectors, while FindSymmetries combines Feynman parametrization with loop-momentum shifts.The first symmetry-search stage is reported to take a few minutes for four-loop on-shell mass-operator topologies.
- Heuristic limitations: The heuristic rule search can depend strongly on the ordering of Dα and the choice of irreducible numerators.If rule finding fails, the paper recommends trying different irreducible numerators and changing the Dα listing order.
5 Conclusion
LiteRed applies heuristic searches for symbolic IBP reduction rules and then uses the rules to reduce multiloop integrals. It also supports symmetry searches and succeeds across a broad class of physically interesting cases, although termination is not guaranteed.
- LiteRed performs IBP reduction through heuristic searches for symbolic rules, followed by application of those rules to the reduction problem.When the search succeeds, the resulting rules provide an effective solution for the corresponding class of integrals.
- The heuristic search is not proved to terminate and appears nonterminating in some complicated cases.
- Despite this limitation, LiteRed's implemented search appears to succeed for a larger class of physically interesting cases.
- FindSymmetries can identify mappings between equivalent sectors.