FIESTA 2: parallelizeable multiloop numerical calculations

A. V. Smirnov; V. A. Smirnov; M. Tentyukov

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FIESTA 2: parallelizeable multiloop numerical calculations

A. V. Smirnov, V. A. Smirnov, M. Tentyukov

arXiv:0912.0158v1Dec 1, 2009hep-ph

TL;DR

FIESTA 2 addresses the need to evaluate and asymptotically expand difficult Feynman integrals across momentum and mass limits. It combines sector decompositions with Mellin–Barnes representations and adds parallel, high-precision, and broader integration capabilities. The authors demonstrate calculations previously unreachable with FIESTA 1 and numerical results reproducing analytical results to several digits.

Problem
Feynman-integral asymptotic expansions can be difficult because relevant regions are not always simple to identify, while numerical evaluation can suffer from severe instability.
Method
FIESTA 2 combines Mellin–Barnes representations with sector decompositions for automatic expansions and adds parallelization, high-precision arithmetic, new integrators, and Speer sectors.
Results
FIESTA 2 evaluates classes of integrals unreachable with the original version and reproduces analytical results with 4–5 digits using 1 500 000 sampling points.
Takeaways & Limitations
The updated program supports practical numerical calculations, asymptotic expansions, and cross-checks of analytical Feynman-integral results.
Takeaways & Limitations
Integration-by-parts methods produce more terms, worsen integrands, and fail for complicated cases; Taylor-based expressions can retain uncanceled large terms involving logarithms.

Abstract

from arXiv · show

The program FIESTA has been completely rewritten. Now it can be used not only as a tool to evaluate Feynman integrals numerically, but also to expand Feynman integrals automatically in limits of momenta and masses with the use of sector decompositions and Mellin-Barnes representations. Other important improvements to the code are complete parallelization (even to multiple computers), high-precision arithmetics (allowing to calculate integrals which were undoable before), new integrators and Speer sectors as a strategy, the possibility to evaluate more general parametric integrals.

1 Introduction

FIESTA 2 extends sector-decomposition software from numerical Feynman-integral evaluation to automatic asymptotic expansions, while adding parallelization, high-precision arithmetic, and further computational strategies.

Asymptotic expansion of Feynman integrals: FIESTA 2 automatically expands Feynman integrals in limits of momenta and masses by combining Mellin–Barnes representations with practical sector decompositions.Modern sector decompositions also apply when some kinematic invariants vanish or share a sign, beyond purely Euclidean configurations.
Parallelization: Parallel execution through Mathematica subkernels and TCP/IP across computers provides speed-ups that are about linear or better.The main machine also performs database-related tasks, contributing to better-than-linear scaling in examples.
New methods to deal with numerical instability: High-precision arithmetic addresses numerical instability caused by large intermediate values after Taylor treatment of sector singularities.The authors report that this enables calculations unreachable with the original FIESTA.
Speer sectors: Speer sectors provide an alternative strategy for complicated cases where Strategy S may practically fail because its running time grows exponentially.The paper supplies an example motivating their use.
Other features: Additional features include threshold integrals, analytic integration of highest poles, multiple integrators, and more general parametric integrals.The new code is illustrated with numerical results for a complicated massless four-loop propagator master integral.

2 Expanding Feynman integrals

FIESTA expands Feynman integrals in momentum and mass limits by combining sector decomposition with Mellin–Barnes representations. The algorithm extracts expansion terms from Mellin–Barnes residues and evaluates the resulting sector integrals numerically.

2 Expanding Feynman integrals: FIESTA introduces an automated expansion procedure for limits where one group of kinematic invariants and masses is much smaller than another.The small group is scaled by an expansion parameter λ, and a onefold Mellin–Barnes integral separates the two groups.
2 Expanding Feynman integrals: Mellin–Barnes analysis reduces the λ → 0 expansion to locating poles in z and taking residues after closing the contour to the right.Relevant poles arise both from Γ(−z) and from the parametric integration.
2 Expanding Feynman integrals: Sector decomposition factorizes U-hat, W1, and W2 into powers of sector variables times positive functions, with exponents depending on ε, z, and integer shifts.This produces parametric integrals over variables t_i from 0 to 1.
2 Expanding Feynman integrals: Taylor subtractions expose ε and z singularities analytically, while the subtraction remainder contributes to the expansion remainder and decreases with higher expansion order.The resulting residues are proper sector integrals treated numerically within FIESTA.
2 Expanding Feynman integrals: For a massless on-shell box and double box with t tending to zero, FIESTA computed expansions through order t^0 modulo logarithms; the box took almost no time and the double box under half an hour.The ε expansion order was set to zero.

3 Parallelization

FIESTA 2 parallelizes both Mathematica processing and numerical integration across multicore systems and multiple computers. Benchmarks show strong scalability for demanding jobs, while communication and disk access can limit gains.

3 Parallelization: FIESTA 2 parallelizes both the Mathematica part and numerical integration, including across multiple computers.The C integration communicates through Mathlink, which supports shared memory and TCP/IP data exchange.
3 Parallelization: Parallelization can slow tasks lasting only a few minutes because queueing and disk access become bottlenecks.For complicated examples, however, the paper reports that this parallelization is efficient.
3 Parallelization: FIESTA scales worse on Xeon than Opteron hardware despite using the same algorithm and software.The paper attributes the difference to Xeon’s shared Front-Side bus versus Opteron’s point-to-point HyperTransport connection.
3 Parallelization: 0.16% of the fitted Opteron runtime is non-parallelizable, while Xeon communication costs about 14% of total time with 8 cores.The fitted Xeon communication coefficient is 428.305 seconds per CPU core, with negligible b1.
3 Parallelization: More than twofold speedup is obtained by moving from one 8-kernel machine to two 8-kernel machines because disk work is shared.The second machine performs computation without also handling the main machine’s hard-drive activity.

4 Numerical instability

FIESTA addresses numerical instability caused by singular-looking integrands through Taylor or integration-by-parts resolutions, with high-precision arithmetic used when cancellation exceeds double precision.

4 Numerical instability: FIESTA 1's IfCut workaround used local Taylor replacements, but it slowed evaluation, produced uncontrollable error estimates, and could fail even with a cut of 0.5.The replacement can effectively integrate a function different from the original one.
4.1 Resolving numerical instability by integration by parts: At small x1, Taylor expressions subtract two divergent large terms whose smooth difference can lose all reliable digits in double precision.For x1 = 10^-20, terms of order 10^20 differ by about 10^-1, while double precision retains only 14-15 reliable decimal digits.
4.1 Resolving numerical instability by integration by parts: Taylor resolution produces a smooth integrand well suited for numerical integration, whereas IBP resolution yields a more complicated shape containing log(x1).The comparison is illustrated by the 3D plots for the Taylor and IBP1 resolutions.
4.1 Resolving numerical instability by integration by parts: IBP methods generate more terms and generally worse-behaved integrands than Taylor resolution, failing in complicated cases despite removing non-integrable terms.Both resolution methods address numerical instability, but their efficiency can become very poor.
4.2 High-precision arithmetics: High-precision arithmetic recovers reliable digits in cancellation-prone evaluations but is slower, creating decisions about which points to evaluate and how much precision to use.The authors identify point selection and precision setting as the two central practical problems.
4.2 High-precision arithmetics: FIESTA estimates a worst-case cancellation monomial to select high precision, using SmallX and PrecisionShift or allowing a fixed DefaultPrecision.The precision is chosen from estimated lost bits plus the desired reliable bits; the default reliable-bit shift is 38.

5 Speer sectors

FIESTA 2 adds Speer sectors as an alternative sector-decomposition strategy for complicated cases where the default strategy may become impractical. The strategy requires graph information and is especially useful when strategy S encounters exponential growth or memory problems.

5 Speer sectors: Speer sectors provide an alternative iterative strategy for sector decomposition when strategy S may practically fail on complicated examples.The paper notes that strategy S can suffer from exponentially growing time, motivating Speer sectors for such cases.
5 Speer sectors: For Euclidean external momenta, Speer sectors produce the same set of sectors as FIESTA’s strategy S.Speer sectors additionally use more information about the graph.
5 Speer sectors: The comparison of strategies S and SS accounts for diagram symmetries by retaining only independent primary-sector contributions through PrimarySectorCoefficients.For the hexagon, the option assigns nonzero coefficients only to symmetry-equivalent representative sectors.
5 Speer sectors: Strategy S could not produce a level-6 result because of memory overflow on a 2GB machine.The paper contrasts this failure with the motivation for using Speer sectors on complicated diagrams.
5 Speer sectors: Using strategy SS requires the diagram structure in addition to the propagators and functions U and F.Evaluation starts with SDEvaluateG, whose graph input lists connected vertex pairs and external vertices.

6 Additional features

FIESTA 2 broadens the classes of integrals it can handle, including threshold cases and more general parametric integrals, while adding integration, precision, and singularity-resolution features. These capabilities come with practical conditions and cautions for above-threshold poles and analytical integration timeouts.

6 Additional features: FIESTA can sometimes evaluate threshold integrals by finding squared differences inside F and decomposing integration regions before sector decomposition.This extends sector-decomposition use beyond cases where all terms of F have the same sign.
6 Additional features: Above-threshold first poles require UsingC=False and are provided without a correctness guarantee.The paper explicitly advises using such results with care.
6 Additional features: Analytical integration is controlled by ExactIntegrationOrder and ExactIntegrationTimeout, after which Mathematica falls back to numerical evaluation.The timeout default is 10, and the required timeout depends on problem complexity and CPU speed.
6 Additional features: FIESTA 2 integrates the Cuba library, offering Vegas, Suave, Divonne, and Cuhre, with Vegas selected by default.Vegas uses importance sampling and is usually preferred because sector decomposition produces relatively good integrands.
6 Additional features: The code can evaluate more general parametric integrals whose polynomial factors are non-negative and whose exponents depend linearly on ε.The interface is SDEvaluateDirect[x,P[1],P[2],...,P[n],r[1],r[2],...,r[n], order].
6 Additional features: The new parametric-integral feature was applied to Wilson-loop integrals to check analytic results numerically.This demonstrates use beyond the standard Feynman-integral evaluation examples.

7 Code installation

Installing FIESTA 2 requires its package, supporting binaries and libraries, and configured executable and data paths. The CIntegrateMP executable adds MPFR-based multiprecision support, while the Mathematica component requires Mathematica 6.0 or 7.0.

7 Code installation: FIESTA 2 installation involves unpacking the package and following the INSTALL instructions, then configuring the Mathematica file’s default paths.The relevant paths are QLinkPath, CIntegratePath, and DataPath.
7 Code installation: DataPath should be placed on a fast local disk because FIESTA creates substantial I/O traffic there.
7 Code installation: The code requires Wolfram Mathematica 6.0 or 7.0 and will not work correctly with lower versions.Version 7.0 is recommended.
7 Code installation: Nontrivial integrals require QLink and CIntegrate, with QLink available as a binary or compilable source package.The Mathematica file must point to the corresponding executable locations.
7 Code installation: CIntegrate depends on MPFR and Cuba, and compilation produces native and multiprecision executables.CIntegrate uses IEEE floating-point arithmetic, whereas CIntegrateMP uses MPFR when necessary and is slightly slower.
7 Code installation: Windows compilation should be performed under Cygwin, producing executable files with the .exe extension.

8 Algorithm usage

FIESTA 2 exposes commands and options for evaluating, expanding, and parallelizing sector-decomposed integrals. Its configuration covers integrators, decomposition strategies, singularity handling, remote jobs, precision, and compatibility settings.

8 Algorithm usage: SDEvaluate evaluates a Feynman integral specified through U, F, loop count, indices, and the requested ε-expansion order.UF can construct U and F automatically from the graph input.
8 Algorithm usage: The evaluation interface accepts numerical substitutions for external momenta, masses, and other values, with U and F required to depend only on integration variables.The paper illustrates this with a massless on-shell box at s = −3 and t = −1.
8 Algorithm usage: SDExpand expands an integral in a variable tending to zero and automatically accounts for possible logarithmic powers.The command specifies the expansion variable and its required order.
8 Algorithm usage: Parallel execution is configured through NumberOfLinks, NumberOfSubkernels, QLink, DataPath, and remote-link timeout options.For complicated problems, the authors recommend matching link and subkernel counts to the computer’s kernels and avoiding network drives for DataPath.
8 Algorithm usage: FIESTA supports multiple decomposition strategies, including guaranteed-termination strategies B, the default strategy S, Speer sectors SS, and heuristic strategy X.Strategy S is described as producing better results than the preceding strategies, while X is not guaranteed to terminate.
8 Algorithm usage: ResolutionMode selects IBP0, IBP1, or Taylor for handling singularities after sector decomposition, with Taylor as the default.The paper discusses Taylor and integration-by-parts methods as alternatives for numerical instability.
8 Algorithm usage: Several FIESTA 1 options were removed or retained only for backward compatibility, including IntegrationCut and IfCut.The paper recommends avoiding the backward-compatibility options.
8 Algorithm usage: Complicated integrals may require QLink to avoid memory overflow and more than the default 50000 evaluations to adapt to all integrand peaks.The default setting is described as fast but usually only a rough estimate.

9 Numerical results

FIESTA evaluates a difficult four-loop propagator master integral with Cuba VEGAS, matching known analytical results to several reliable digits while also providing an extra ε term.

The numerical integration uses the Cuba VEGAS integrator with different parameters.
Table 1 compares known analytical values with FIESTA results using Cuba VEGAS at 500 000 and 1 500 000 sampling points.
Table 2 reports numerical-integration times and total evaluation time, including the Mathematica part.
With 1 500 000 sampling points, FIESTA reproduces the analytical results with 4–5 digits.The calculation also evaluates one extra ε-expansion term unavailable analytically.
With 500 000 sampling points, the numerical result has 3–4 reliable digits in a reasonable time.

10 Conclusion

FIESTA 2 is an updated algorithm that evaluates Feynman integrals and expands them in momentum and mass limits. The authors demonstrate its new features and present it as a practical tool for calculations and analytical cross-checks.

10 Conclusion: FIESTA 2 extends sector-decomposition evaluation of Feynman integrals with expansions in various limits of momenta and masses.
10 Conclusion: Examples demonstrate the new features, and the authors describe FIESTA 2 as an important tool for practical calculations and analytical cross-checks.