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Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA)

A. V. Smirnov, M. N. Tentyukov

arXiv:0807.4129v1hep-ph

TL;DR

Feynman-integral sector decomposition requires resolving singularities before epsilon expansion and numerical integration, while existing strategies trade termination guarantees against sector counts. FIESTA implements existing strategies and introduces a new terminating strategy, successfully evaluating a triple-box benchmark within the stated error estimate, though numerical instability can produce indeterminate results.

  • Problem

    Sector decomposition must resolve singularities in general Feynman-integral functions before epsilon expansion and numerical integration, while available strategies differ in termination guarantees and sector counts.

  • Method

    FIESTA performs sector decomposition, singularity resolution, epsilon expansion, and numerical integration, implementing Strategies A, B and X and introducing terminating Strategy S.

  • Results

    62.4 hours: FIESTA evaluated the triple-box diagram through the finite part, with all answers within the 1 percent error estimate of an existing analytical result.

  • Takeaways & Limitations

    FIESTA provides a practical implementation of sector-decomposition strategies for numerical Feynman-integral evaluation, including a terminating alternative to heuristic Strategy X.

  • Takeaways & Limitations

    Numerical instability at small integration-variable values can cause FIESTA to return Indeterminate.

Abstract

from arXiv · show

Up to the moment there are two known algorithms of sector decomposition: an original private algorithm of Binoth and Heinrich and an algorithm made public lastyear by Bogner and Weinzierl. We present a new program performing the sector decomposition and integrating the expression afterwards. The program takes a set of propagators and a set of indices as input and returns the epsilon-expansion of the corresponding integral.

PROGRAM SUMMARY

FIESTA is a software program for sector decomposition and numerical evaluation of Feynman integrals, with Mathematica and C components and external numerical routines.

  • Program profile: FIESTA is implemented in Wolfram Mathematica 6.0 and C under GPLv2 licensing.The program is designed for Unix and Windows systems, from desktop computers to supercomputers.
  • Program scope: The sector-decomposition workflow includes sector minimization, pole resolution and epsilon expansion, followed by numerical integration.The program addresses these stages as parts of evaluating Feynman integrals.
  • External software: QLink and Vegas are listed as external routines or libraries used by the program.Vegas provides the numerical-integration algorithm, while QLink is an open-source associated routine.

1 Introduction

The paper positions sector decomposition as a practical numerical method for Feynman integrals and introduces FIESTA with a new terminating strategy alongside existing strategies.

  • Background: Sector decomposition transforms Feynman-parametric integration domains into sectors where the integrand factorizes into a monomial times a nonsingular function.The method originated in convergence and renormalization analyses and later became practical for numerical evaluation.
  • Prior algorithms: Bogner and Weinzierl’s public algorithm offers four strategies, with Strategies A, B and C guaranteed to terminate while Strategy X generally produces fewer sectors without that guarantee.Strategy A is conceptually simplest but can generate too many sectors; Strategy C improves on Strategy B.
  • Contribution: FIESTA reproduces Strategies A, B and X and introduces Strategy S, which is guaranteed to terminate but produces more sectors than Strategy X.The sector-count difference between Strategies S and X is reported as less significant than that between Strategies B and X.
  • Benchmark: 62.4 hours: FIESTA evaluated the triple-box diagram through its finite part, with all answers within the 1 percent error estimate of an existing analytical result.The calculation began from ε−6 poles and used Strategy X while accounting for diagram symmetries.
  • Platform scope: FIESTA was tested on Windows and Linux, while compilation on other Unix platforms was considered possible.Precompiled binaries were provided only for Windows x86 and Linux x86-64.

2 Theoretical background

The theoretical workflow converts dimensionally regularized Feynman integrals into sector integrals whose singularities can be isolated before epsilon expansion and numerical integration.

  • Integral representation: FIESTA evaluates scalar dimensionally regularized Feynman integrals using the alpha representation after Dirac and Lorentz algebra.The integrals are expressed through loop momenta, propagators, indices, and the space-time dimension d = 4 − 2ε.
  • Sector construction: Sector decomposition partitions the integration region into primary sectors and applies variable replacements to obtain a linear combination of sector integrals.The strategy is an essential algorithmic component because it should constructively minimize the number of resulting sectors.
  • Singularity resolution: General U and F functions can retain singularities, so singularity resolution is required before epsilon expansion and numerical integration.Sector decomposition aims to produce equivalent expressions whose U and F functions are nonsingular, leaving singular behavior in the monomial factors.
  • Evaluation sequence: After decomposition, Taylor expansion resolves singularities by integrating selected terms analytically; the nonsingular remainder is then epsilon-expanded and numerically integrated.This sequence returns the evaluated integral from the resulting sector expressions.
  • Scope condition: The approach is numerical only, requiring all invariant values to be substituted early after generating the U and F functions.This condition limits the workflow’s use with symbolic invariant values.

3 Overview of the software structure

FIESTA separates symbolic preprocessing in Mathematica from numerical integration in C, with the C component using the Vegas algorithm for more powerful integration of complicated expressions.

  • Numerical backend: The C component performs numerical integration with the Vegas algorithm implemented as a FORTRAN program.It is launched from Mathematica through the Mathlink protocol rather than used directly by the user.
  • Component roles: The Mathematica component can independently perform the workflow, but C integration is recommended for complicated integrals because it is more powerful than Mathematica’s built-in integrator.The software therefore combines symbolic preparation with an optional high-performance numerical backend.

4 Description of the individual software components

FIESTA combines Mathematica-based symbolic processing with C-based numerical integration, optionally using QLink for disk-backed intermediate data and parallel execution.

  • Mathematica component: The Mathematica component accepts loop momenta, propagators, indices, and epsilon-expansion order, then performs symbolic preparation through numerical-integration setup.It handles differentiation, sector decomposition, singularity resolution, epsilon expansion, and expression preparation.
  • Numerical integration: Numerical integration can run either in Mathematica or through the C component, with Mathematica supporting complex numbers.The C integrator is invoked through Mathlink.
  • Parallel execution: C integration can run in parallel by launching multiple CIntegrate copies, while Mathematica distributes tasks and collects results.The number of launched copies is specified by the user.
  • Numerical integration: The C integration uses Vegas and internally compiles Mathematica-generated expression strings for faster evaluation without external compilers or exchange files.Vegas performs the numerical integration after internal compilation of the input expression.
  • Data management: QLink stores intermediate data on disk instead of RAM for complicated cases.The QLink executable and database paths must be specified beforehand.

5 Installation instructions

Installing FIESTA requires the Mathematica package plus, for nontrivial integrals, the QLink and CIntegrate components with configured executable and data paths.

  • Basic installation: The installation package is downloaded, unpacked, and installed according to the included INSTALL instructions.The paper gives a project download URL for the package.
  • Mathematica component: The Mathematica component requires copying FIESTA1.0.0.m and editing QLinkPath, CIntegratePath, and DataPath.These paths identify the QLink executable, CIntegrate executable, and database directory.
  • Data management: DataPath should be placed on a fast local disk because FIESTA creates substantial I/O traffic there.QLink uses this directory for stored data.
  • Requirements: FIESTA requires Wolfram Mathematica 6.0 and will not work correctly with lower versions.The requirement applies to the code itself.
  • Requirements: Nontrivial integrals require installation of QLink and the CIntegrate program.Users may use precompiled CIntegrate binaries or compile the executable with the Mathematica Developer Kit.
  • Platform-specific setup: Unix compilation uses Makefile and make after ensuring WIN is undefined, whereas Windows compilation uses CIntegrate.mak and nmake with Microsoft Visual C++.The platform-specific instructions configure the WIN macro differently.

6 Test run description

The test-run section demonstrates command-based evaluation of a massless on-shell box and documents configuration options, strategy choices, and common failure conditions.

  • Evaluation workflow: FIESTA evaluates an integral through SDEvaluate[{U,F,l},indices,order], where U and F are parametric functions and order specifies epsilon-expansion order.The UF function can construct U and F automatically from loop momenta, propagators, and substitutions.
  • Test example: The example evaluates the massless on-shell box diagram with S = 3 and T = 1.A complete log is provided in Appendix B.
  • Configuration: Options control C integration, the number of CIntegrate processes, QLink disk storage, data paths, and Vegas sampling settings.PrimarySectorCoefficients can also exploit diagram symmetries.
  • Strategy selection: Strategies 0, B, S, and X differ in whether decomposition is skipped, guaranteed to terminate, or heuristic and potentially nonterminating.Strategy S is the default option described as producing better results than preceding strategies.
  • Failure conditions: FIESTA may return Indeterminate because of negative F, complex-valued integrands, special singularities, or numerical instability.The documented remedies include changing propagator signs, disabling C integration, varying ResolveNegativeTerms, or using integration-cut options.

A Sector decomposition strategies

The paper introduces a sector strategy based on weight-polynomial geometry, extends the allowed sector replacements, and compares termination and sector-count behavior across strategies.

  • Goal and setup: Sector decomposition partitions a unit hypercube so variable changes factor a polynomial into a monomial times a polynomial containing 1 as a summand.This form often prevents polynomial zeros beyond those caused by the monomial.
  • Goal and setup: The procedure is recursive, with each single decomposition step defined by a sector decomposition strategy.All known strategies partition the integration region using a selected index subset and variable replacements.
  • Strategy S: The new strategy extends possible regions and replacements by using any positive-quadrant vector with at least two non-zero coordinates.The non-zero coordinates determine the index set used to partition the hypercube.
  • Strategy S: The strategy chooses a facet of the convex hull of monomial weights and uses its normal vector to reduce the number of lowest weights.The chosen vector makes relevant weight vectors orthogonal to the facet, potentially leaving one lowest-weight vertex after a step.
  • Termination: The new step is guaranteed not to increase the Zeillinger-strategy norm, and fallback steps guarantee termination; fallback occurs in no more than five percent of cases in practice.The fallback is used when no facet-based step decreases the norm.
  • Comparative behavior: Strategy comparisons count sectors produced for examples including D420, where sector decomposition can fail or exhaust memory under some strategies.The paper suspects D420 is a counterexample to the nonterminating heuristic strategy X.

B An example

The massless on-shell box example demonstrates FIESTA's sector decomposition, pole resolution, epsilon expansion, and numerical integration workflow, producing an epsilon-expanded result with a finite numerical term.

  • B An example: FIESTA evaluates a massless on-shell box diagram with S = 3 and T = 1 through its Mathematica session.The example uses four unit indices and reports the program's integration workflow.
  • B An example: 12 sectors are generated during sector decomposition before variable substitution and decomposition of the epsilon-independent term.The reported sector decomposition takes 0.0138 seconds.
  • B An example: 1.333333 is obtained for the order -2 term, giving 1.333333/ep^2.This result is reported before the order -1 and order 0 numerical integrations.
  • B An example: -2.065743 +- 5.*^-6 is obtained for the order -1 term, contributing (-0.73241 +- 5.*^-6)/ep alongside the double-pole term.The order -1 calculation uses 28 two-fold integrals and takes 0.862786 seconds for numerical integration.
  • B An example: -3.417375 +- 0.000012 is obtained for the order 0 integration, with finite contribution (-4.386496 +- 0.000013).The total time used for the calculation is 1.52991 seconds.

C Dealing with negative terms in F

FIESTA addresses extra integration singularities caused by negative monomials in the Symanzik function F by automatically partitioning variable regions and changing variables before sector decomposition.

  • C Dealing with negative terms in F: Three-loop static-quark-potential master integrals can produce extra integration singularities when F contains terms such as (x_i − x_j)^2.These terms make not all monomials of F have positive signs.
  • C Dealing with negative terms in F: FIESTA attempts to remove negative summands in F before sector decomposition because the basic approach may not properly treat the resulting additional singularities.The feature is designed for cases where negative terms create singular behavior beyond the standard decomposition treatment.
  • C Dealing with negative terms in F: The code searches for two variables, x_i and x_j, splits the unit cube into x_i ≤ x_j and x_i ≥ x_j regions, and applies a variable replacement.For x_i ≤ x_j, the replacement begins with x′_i = x_j − x_i.
  • C Dealing with negative terms in F: The automatic treatment is presented as original because users need not specify these singularities for the code to resolve them.Related special operations before sector decomposition had appeared in earlier work by Binoth and Heinrich.

D Avoiding memory limits in Mathematica

FIESTA avoids Mathematica memory limits by storing sector expressions on disk and processing them incrementally, while shifting numerical integration to C-based routines. Its data-driven interpreter reduces translation and evaluation overhead for large, numerous sector expressions.

  • Disk-backed processing: FIESTA stores sector expressions in a QLink database and processes them one sector at a time instead of loading the full set into RAM.This supports singularity resolution, epsilon expansion, symbolic expansion, and conditional operations without retaining all expressions in memory.
  • Epsilon expansion: Explicit differentiation replaces Mathematica’s Series function for epsilon expansion, because it is faster for the considered expressions and permits delayed expansion.The Expand operation can be postponed until numerical integration, reducing Mathematica’s memory burden.
  • Numerical instability: Numerical instability arises when a large term and a small remainder must be subtracted at finite precision, especially as integration variables approach zero.Expanding expressions first reduces the instability, but highly degenerate examples can still prevent evaluation at small variables.
  • Numerical instability: FIESTA offers IntegrationCut and a slower variable-expansion method for unstable regions, with the latter integrating expanded and original functions on different ranges.The IfCut and VarExpansionDegree options control the variable-expansion approach.
  • Numerical integration: Vegas uses importance sampling to concentrate integration points where the absolute integrand contributes most strongly.The C implementation is linked to Mathematica and evaluates the resulting expressions numerically.
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