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FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions
Alexander V. Smirnov
TL;DR
FIESTA 3 addresses limitations of earlier sector-decomposition software, especially calculations in physical regions and difficult multiloop asymptotic expansions. It introduces new algorithms and a cluster-parallel workflow, extending automatic numerical and expansion capabilities while requiring extra regularization in some regions.
Problem
Earlier FIESTA versions could not perform calculations in physical regions, while some complicated integrals required asymptotic expansion because direct evaluation or IBP reduction could fail.
Method
FIESTA 3 combines new sector-decomposition and asymptotic-expansion algorithms with cluster parallelization that separates Mathematica algebra from MPI-parallelized integration.
Results
The new release provides calculations in physical regions, automatic asymptotic expansion, and cluster-parallel execution for numerical Feynman-integral evaluation.
Takeaways & Limitations
FIESTA 3 extends automatic numerical evaluation of Feynman integrals to physical-region calculations and distributed cluster workflows.
Takeaways & Limitations
The regions-based asymptotic-expansion approach may require an extra regularization parameter because its integrals are not always well defined.
Abstract
from arXiv · showhide
The goal of this paper is to present a new major release of the program FIESTA (Feynman Integral Evaluation by a Sector decomposiTion Approach). This version presents features like cluster-parallelization, new asymptotic expansion algorithms, calculations in physical regions, new sector-decomposition strategies, as well as multiple speed, memory, and stability improvements.
PROGRAM SUMMARY
FIESTA 3 is a cross-platform, parallelized program for automatic numerical evaluation of multiloop Feynman integrals using sector decomposition. Its practical limits are primarily determined by the CPU time needed for integration and sufficient precision.
- FIESTA 3 evaluates multiloop Feynman integrals numerically through sector decomposition.
- The program combines Wolfram Mathematica with C++ and supports systems ranging from desktop computers to supercomputers.
- The code is parallelized, with tests performed using up to 1024 cores.
- The main restriction is CPU time required to perform integration and obtain proper precision.
1. Introduction
The introduction motivates FIESTA 3 as an extension of sector-decomposition methods to physical regions, automatic asymptotic expansions, and cluster-parallelized calculations. It addresses limitations in earlier FIESTA versions and in existing expansion approaches while retaining fully automatic numerical treatment.
- Sector decomposition automatically evaluates numerical Feynman integrals after masses and kinematic invariants are specified.
- Earlier FIESTA versions could not calculate in physical regions, limiting use of their compiler and multiprecision capabilities there.
- FIESTA 3 adds cluster parallelization by separating Mathematica-based algebra from MPI-parallelized integration on a cluster.
- Asymptotic expansion is motivated by integrals too complicated for direct evaluation or cases where IBP reduction fails.
- The paper incorporates two numerical asymptotic-expansion algorithms based on sector decomposition for complete automation.
- The earlier Mellin-Barnes algorithm is limited to two groups of large and small invariants and can make some three-loop problems too complicated.
- The proposed regions-based algorithm uses asy to identify regions, expands their contributions, and sector-decomposes the resulting coefficients, sometimes requiring extra regularization.
2. Integrating at and above the threshold
FIESTA 3 handles threshold and above-threshold integrals by combining sector decomposition with contour deformation and variable bisection. A K4-graph example demonstrates the numerical behavior and error scaling of the approach.
- Sector decomposition is guaranteed to terminate only when F has no negative-coefficient monomials; negative terms can cause failure.
- Threshold cases: At threshold, FIESTA 3 uses improved pre-resolution transformations to remove singularities of the form (x_i − x_j)^2.
- Above-threshold cases: Above threshold, contour deformation shifts integration away from internal zeros of F, where F changes sign and the integral becomes complex.
- Contour deformation: The deformation parameter λ is selected through random-point estimates and bounds on the complex-part behavior of F, with optional fixed-shift operation.
- Endpoint singularities: FIESTA 3 handles endpoint singularities at x → 1 by bisecting the integration interval and mapping the resulting singularities to x → 0.
- Example: The K4 example required less than 10 minutes for database preparation, while integration time grows linearly at large sampling scales and errors decrease as the square root of sample count.
3. Cluster-parallelization and internal structure of FIESTA 3
FIESTA 3 separates symbolic preparation from numerical integration, enabling threaded or MPI cluster execution through database-backed stages. Its parallel efficiency improves with larger sampling workloads and with SeparateTerms, while precision grows approximately with the square root of sampling points.
- Internal structure: FIESTA 3 processes propagator, momentum, index, and epsilon-order inputs through three major stages, beginning with preparation and sector decomposition.The first stage includes negative-index elimination, pre-resolution, sector decomposition, and Mathematica parallelization across primary sectors.
- Internal structure: The preparation pipeline performs variable substitution, contour transformation in complex mode, pole resolution, expression preparation, epsilon expansion, and string preparation.The resulting database stores integration strings that can be integrated without Mathematica when OnlyPrepare is enabled.
- Parallel integration: CIntegratePool uses threads, whereas CIntegratePoolMPI launches integrator processes for MPI-based cluster execution.The numerical integrators include basic, MPFR, and complex MPFR variants, while FIESTA analyzes expressions into an optimized internal form for evaluation.
- Performance evaluation: The timing tables use cores as columns, sampling points as rows, and seconds as the measured result.The test used a massive on-shell four-loop propagator database, with Table 2 reporting its MPI-parallelization timings.
- Performance evaluation: Parallelization is less effective at low sampling counts because database and MPI overhead dominates, but efficiency improves as sampling points increase.The fixed evaluation component becomes more favorable to parallelization for larger workloads.
- Performance evaluation: With SeparateTerms and enough sampling points, integration scales well for large clusters, while precision grows approximately as the square root of sampling points.The SeparateTerms configuration produced more MPI jobs and a better dependence of integration time on core count; the reported 50-million-point run with 1024 cores took 100 hours.
4. Numerical algorithms for asymptotic expansion of Feynman integrals
FIESTA 3 implements two sector-decomposition-based numerical algorithms for asymptotic expansion: a Mellin–Barnes approach and a regions approach. The latter introduces an auxiliary regularization parameter to handle region integrals that may not be well-defined, then evaluates the resulting epsilon and regularization expansions numerically.
- Mellin-Barnes method: The Mellin–Barnes algorithm separates kinematic contributions by introducing a small parameter and an additional contour integration over z.The resulting sector integrals retain z-dependent exponents, and the contour is closed to capture the relevant poles.
- Mellin-Barnes method: The Mellin–Barnes procedure reveals epsilon singularities from the z integration before returning to epsilon expansion and numerical Vegas integration.The analysis focuses on sector integrals whose z dependence is relevant to the asymptotic limit.
- Regions method: The regions algorithm identifies momentum-scaling regions, expands the integrand in each region, and evaluates the resulting integrals with sector decomposition.The regions formulation is expressed in alpha representation and drops scaling restrictions after expansion.
- Regions method: Because region integrals may be ill-defined, the algorithm introduces a regularization parameter λ whose poles cancel after summing contributions from all regions.The indices depend linearly on λ, without changing the determination of regions.
- Regions method: Sector decomposition then handles exponents depending on ε and λ, expands in λ through order zero, and numerically evaluates the coefficients in both expansions.After regional results are combined, λ poles should cancel, although numerical uncertainties can leave residual poles.
- Regions method: Some expanded region integrals can be simplified analytically when U or F becomes independent of an integration variable.The corresponding integration over that variable is taken out analytically before numerical integration.
5. Installation and usage
FIESTA 3 requires several numerical libraries and optional components, with Mathematica needed for database preparation but not cluster-side integration when databases are prepared elsewhere. Usage covers installation, testing, standard evaluation, expansions, and alternative sector-decomposition strategies.
- Installation requirements: FIESTA 3 depends on KyotoCabinet, Cuba, MPFR, and GMP; qhull is additionally required for selected strategies and SDExpandAsy.An MPI environment is recommended when building the cluster-oriented MPI version of CIntegratePool.
- Installation requirements: Mathematica 7.0 or higher is required for database preparation, but not for integration when databases are prepared elsewhere.The latter workflow is described as typical for clusters.
- Building and testing: The package is built with make, while make mpi builds the MPI version of CIntegratePool and make all builds it with the other packages.There is no make install target; make test or make testall checks whether the binaries are functional.
- Basic usage: FIESTA should be loaded through SetDirectory plus Get, or through FIESTAPath plus Get, rather than by specifying only a full path.The documented direct-full-path loading form will not work properly.
- Program syntax: SDEvaluate[{U,F,l},indices,order] evaluates an integral to the requested epsilon-expansion order, while UF can generate U and F from loop data, propagators, and substitutions.The numerical functions U and F must not retain external dependencies after substitutions.
- Expansion and sector strategies: SDExpand expands a Feynman integral in a small parameter, and SDExpandAsy supports asymptotic expansion when a regularization variable and shifted indices are supplied.Speer sectors use SDEvaluateG or SDExpandG with graph information whose line order matches the propagator order; SDEvaluateDirect handles non-Feynman integrals.
6. Conclusion
FIESTA 3 extends automatic numerical evaluation and analytic expansion of Feynman integrals with new decomposition and expansion algorithms, physical-region integration, and cluster parallelization. The authors present these capabilities as an essential improvement and identify GPU acceleration as future work.
- Conclusion: FIESTA 3 combines new sector-decomposition and expansion algorithms with physical-region integration and cluster parallelization.The program performs automatic numerical evaluation and analytic expansion of Feynman integrals.
- Future work: GPU use is identified as a future plan for speeding up integration.This is presented as ongoing development beyond the current release.