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SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop

S. Borowka, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk, T. Zirke

arXiv:1502.06595v2hep-ph

TL;DR

Precision collider calculations require automated treatment of higher-order multi-scale loop integrals and dimensional-regularization poles. SecDec 3.0 extends sector-decomposition-based numerical evaluation with terminating geometric strategies, broader propagator support, and an improved interface. The authors report lower sector counts for G2 in typical cases and broader applicability beyond NLO, while light-cone-gauge integrals remain restricted to the Euclidean region.

  • Problem

    Higher-order precision calculations need multi-loop integrals with several mass scales, while dimensional regularization requires factorizing regulator poles from multidimensional parameter integrations.

  • Method

    SecDec 3.0 automates sector decomposition to factorize poles and numerically integrates Laurent-series coefficients, adding two terminating geometric strategies and support for inverse and linear propagators.

  • Results

    G2 usually generates the lowest number of sectors, while strategy X can reduce spurious poles despite producing more sectors; SecDec 3.0 extends numerical evaluation beyond NLO and multi-scale analytic limits.

  • Takeaways & Limitations

    SecDec 3.0 is intended for calculations beyond NLO involving several mass scales and cases where analytical loop-integral calculations reach their limits.

  • Takeaways & Limitations

    Because the principal-value prescriptions used for light-cone-gauge integrals are incompatible with SecDec’s automated contour deformation, those integrals can only be evaluated in the Euclidean region.

Abstract

from arXiv · show

SecDec is a program which can be used for the factorization of dimensionally regulated poles from parametric integrals, in particular multi-loop integrals, and the subsequent numerical evaluation of the finite coefficients. Here we present version 3.0 of the program, which has major improvements compared to version 2: it is faster, contains new decomposition strategies, an improved user interface and various other new features which extend the range of applicability.

PROGRAM SUMMARY

SecDec 3.0 is a numerical high-energy-physics computing program for perturbative calculations involving Feynman diagrams and multi-loop integrals.

  • SecDec 3.0 supports perturbation theory, Feynman diagrams, multi-loop calculations, and numerical integration.It uses Wolfram Mathematica, Perl, Fortran, and C++ and can run from a single PC to a cluster.

1 Introduction

Precision collider physics requires higher-order, multi-scale loop integrals, while dimensional regularization makes pole factorization a central computational challenge. SecDec addresses this through automated sector decomposition and numerical integration, and version 3.0 adds terminating decomposition strategies and a restructured interface.

  • Higher-order Standard Model and beyond-Standard-Model calculations require two- and higher-loop integrals with several mass scales.This is especially important for precision studies of Higgs Yukawa couplings while retaining heavy-quark masses without low-energy approximations.
  • Dimensional regularization creates the challenge of factorizing poles in the regulator ϵ from multidimensional parameter integrations.The integrations may involve Feynman parameters for loop integrals or subtraction-term parameters for infrared-divergent radiation.
  • SecDec automates pole factorization and numerically integrates the finite coefficients of the resulting Laurent series in ϵ using sector decomposition.The program applies the sector decomposition algorithm to multi-loop and more general parametric integrals.
  • SecDec-2.0 extended multi-scale numerical integration beyond Euclidean kinematics by combining sector decomposition with complex contour deformation.Earlier versions had restricted multi-scale integrals to Euclidean kinematics.
  • Version 3.0 implements two decomposition strategies based on a method guaranteed to avoid infinite recursion and restructures the user interface.The paper describes these strategies, new program features, usage, and examples.

2 Theoretical framework

The paper represents multi-loop Feynman integrals parametrically through graph polynomials and regulator-dependent exponents, while extending SecDec to inverse and linear propagators. The resulting numerical output follows the defined integral and prefactor conventions, with singularities depending on graph functions and kinematics.

  • The theoretical framework expresses scalar multi-loop integrals with arbitrary propagator powers in terms of Feynman parameters.The momentum-space integral is defined for L loops, N propagators, and propagator powers ν_j that need not be integer.
  • SecDec 3.0 handles linear propagators whose denominators depend linearly rather than quadratically on loop momenta.Analytic continuation requires care for these propagators and is discussed separately.
  • The graph polynomials U and F can be constructed from the topology of the corresponding Feynman graph and are used by SecDec.For massless propagators, the relevant form is F=F0.
  • U vanishing is associated with ultraviolet subdivergences, whereas F vanishing is associated with infrared divergences.Whether F produces a divergence depends on the kinematics, not only on the topology.
  • SecDec reports the integral with its default prefactor and μ=1 unless the user specifies a different prefactor.The user may choose which prefactor is factored out of the numerical result.

3 Structure and new features of SecDec version 3.0

SecDec 3.0 reorganizes its program structure and interface while adding broader integral support and geometric decomposition strategies. The new strategies terminate by construction, and strategy G2 usually minimizes sector counts, whereas strategy X can reduce spurious poles.

  • 3.1 Program structure: SecDec has separate loop-integral and general-parametric-integral parts sharing pole factorization and subtraction procedures.Automated contour deformation is supported for loop integrals, where analytic continuation follows Feynman’s iδ prescription.
  • 3.1 Program structure: The program’s operational sequence produces a numerical result as a Laurent series in ϵ, with its internal directory structure documented separately.Figures 1 and 2 present the sequence and directory organization.
  • 3.2.1 New user interface: Version 3.0 launches with secdec rather than launch and restructures the loop interface around simpler, more flexible input.The redesign targets large kinematic scans, cluster use, reduction-code interfaces, user-defined invariant names, and minimal mandatory input.
  • 3.2.1 New user interface: Kinematic values are moved into kinem.input, where each line defines a new kinematic point for evaluation.This decouples numerical kinematics from paramloop.input and facilitates scans over multiple points.
  • 3.2.1 New user interface: The interface accepts zero or negative propagator indices, user-defined invariant symbols, and explicit scalar-product rules.Negative indices correspond to inverse propagators and provide a straightforward interface to reduction programs supplying master integrals as index lists.
  • 3.2.1 New user interface: The algebraic output can be archived and transferred to another machine or cluster for parallel numerical evaluation across kinematic points.Selected pole coefficients may also be evaluated individually.
  • 3.2.2 Decomposition strategies: The original iterative decomposition can enter infinite recursion when variables are rescaled inconveniently.A simple variable choice can reproduce the original functional form, while another choice avoids the recursion.
  • 3.2.2 Decomposition strategies: Two additional geometric decomposition strategies in SecDec-3.0 are guaranteed to stop and use convex-polytope calculations and triangulations.Normaliz supplies the required polyhedral computations; G1 implements the Kaneko–Ueda algorithm, while G2 uses the Cheng–Wu theorem instead of primary sector decomposition.

Compilation times

SecDec 3.0 improves reusable numerical compilation and broadens supported integral structures, including inverse and linear propagators and epsilon-dependent dummy functions. Its contour-deformation assumptions restrict light-cone-gauge integrals to the Euclidean region.

  • Compilation times: SecDec 3.0 compiles resulting functions once and passes kinematic values as numerical arguments, avoiding separate integration files for each point.This removes repeated numerical-file generation and avoids redoing algebraic steps unless special kinematics alter the singularity structure.
  • Compilation times: 34 mass configurations of several two-loop two-point integral topologies were computed automatically in an application using a preliminary version of SecDec 3.0.The authors report that the reusable compilation approach had a substantial performance effect in this calculation.
  • Inverse propagators: Negative integer propagator powers are supported through a generalized Feynman parametrization that uses derivatives for corresponding parameters.The functions F and U include propagators with negative powers, while the resulting numerator polynomial is built from their partial derivatives.
  • Linear propagators: SecDec 3.0 supports propagators with linear rather than quadratic loop-momentum dependence, as encountered in heavy quark effective theory and non-covariant gauges.The implementation assumes a +iδ term for each propagator when constructing the automated contour deformation.
  • Linear propagators: SecDec’s automated contour deformation is incompatible with the principal-value and Mandelstam-Leibbrandt prescriptions, so light-cone-gauge integrals can only be evaluated in the Euclidean region.The principal-value prescription is not compatible with Wick rotation, while the Mandelstam-Leibbrandt prescription introduces an additional dual vector and more invariants.
  • Epsilon-dependent dummy functions: Epsilon-dependent dummy functions can mask large finite expressions while their epsilon orders are automatically combined with sector-decomposition pole coefficients.Dummy functions are inserted during numerical integration after automatic rescaling of integration variables.

4 Installation and usage

SecDec 3.0 is installed by compiling its numerical libraries and, when needed, the Normaliz decomposition tool. Users configure input templates, run staged calculations, and can select general or user-defined integral formats.

  • Installation: Installation compiles Cuba, Bases, Cquad, and Sobol after checking for Mathematica and perl.The package includes the necessary GSL files for Cquad; Normaliz 2.10.1 is additionally required for geometric strategies G1 and G2.
  • Installation: Mathematica 10.0.2 may hang when parallel kernels are launched non-interactively, so SecDec defaults nbmathsubkrnls to zero.A different Mathematica version and a non-zero nbmathsubkrnls can improve algebraic speed for complicated integrands.
  • Installation: Users can verify installation with make check, which compares test outputs against pre-calculated results.
  • Input setup: SecDec generates param.input, math.m, and kinem.input templates with secdec -prep, which users edit to define the integral and numerical parameters.The only mandatory param.input fields are graphname and epsord; unspecified options use defaults.
  • Execution: The output directory is created automatically as a graph-named subdirectory of the input directory unless configured otherwise.
  • Input setup: The -u option skips primary decomposition for user-defined functions without a standard Feynman-parameter delta constraint.This mode requires a specified input format for the functions to be decomposed.

math.m

The math.m file defines the graph to be calculated in Mathematica syntax, using a generated one-loop box template as the default starting point.

  • math.m: math.m contains the graph definition in Mathematica syntax, with secdec -prep generating a one-loop box template.

kinem.input

SecDec evaluates integrals through staged algebraic, numerical, and collection workflows, while version 3 allows users to target selected pole structures and epsilon orders.

  • kinem.input: kinem.input supplies numerical values for symbolic kinematic invariants and masses used during numerical evaluation.Each line represents one kinematic point, and values follow the order defined in KinematicInvariants and Masses.
  • Execution stages: The algebraic, numerical, and collection stages can be run separately with secdec -algebraic, secdec -numerics, and secdec -collectresults.In single-machine mode, setting exeflag to 3 runs all stages automatically.
  • Execution stages: Cluster mode requires separate algebraic, numerical, and collection commands because result completion is not automatically detectable.The algebraic step is performed on a Mathematica-equipped machine before numerical jobs are transferred to a cluster.
  • Execution stages: Detailed and basic commands must follow the execution order, while a specific command such as secdec -subexpand runs only that task.
  • Selected coefficients: Version 3 lets users calculate selected pole structures and epsilon orders individually.Pole structures encode logarithmic, linear, and higher-than-linear poles; for example, 2l1h0 means two logarithmic, one linear, and zero higher-than-linear poles.
  • Selected coefficients: Users can select pole structures, epsilon orders, or both through the -polestructs and -epsords options.If neither is specified, SecDec loops over all contributing pole structures and epsilon orders; together combines structures into one folder while preserving order selection.

5 Examples

The examples demonstrate SecDec-3.0 across multi-loop topologies, masses, numerators, many external legs, epsilon-dependent powers, linear propagators, alternative integrators, and user-defined integrands. Reported runtimes range from 13 seconds to 14 minutes, with input-specific constraints and strategy choices.

  • 5.1 One-loop box: 41 seconds produced results for two kinematic points in the one-loop box example.The graph has one off-shell leg and one internal mass.
  • 5.2 Two-loop triangle: 144 seconds was required for the two-loop three-point function containing a massive triangle loop.The example is identified as P126 in SecDec version 2.
  • 5.3 Two-loop non-planar box with internal masses: 186 seconds was required for the massive seven-propagator non-planar two-loop box evaluated at m = 50, M = 90, and s23 = −104.The mass assignments and momentum labeling are specified for the non-planar graph.
  • 5.4 Tensor integrals: 14 minutes, below 3 minutes, and 19 seconds were required for the inverse-propagator, specified-numerator, and pinched-graph treatments, respectively.The two numerator methods cannot be combined when a negative propagator index is specified.

6 Conclusions

SecDec-3.0 extends automated multi-loop and general parametric-integral calculations with terminating decomposition strategies, broader integral support, improved timing and cluster use, additional integration options, and a redesigned interface. The authors expect it to be useful beyond NLO, especially for multi-scale cases where analytical loop integration reaches its limits.

  • New capabilities: SecDec-3.0 adds two geometric decomposition strategies guaranteed to stop, alongside support for tensor integrals, negative or zero indices, and linear propagators.The geometric strategies are introduced as alternatives to the heuristic algorithm.
  • Usability and performance: The release improves timings, simplifies numerical integration over many kinematic points on a cluster, and adds Mathematica integration alongside Cuba.The interface and code structure were also redesigned to be more user-friendly.
  • General setup: The general setup can preserve dummy functions that are not decomposed and may depend on ϵ.This feature applies to pole factorization for parameter integrals not related to loop integrals.
  • Conclusion: The authors expect SecDec-3.0 to support calculations beyond NLO involving several mass scales when analytical loop calculations reach their limits.This is presented as the program’s intended practical usefulness rather than a measured benchmark result.

A User Manual

SecDec-3.0 provides separate loop and general setups for standard loop integrals and more general parametric functions. The general setup lacks contour deformation, whereas user-defined non-standard integrals remain available in the broader framework.

  • Setups: The loop setup handles standard loop integrals, while the general setup handles general parametric functions.The two setups correspond to the loop and general directories.
  • Contour deformation: Contour deformation is unavailable for parametric functions in the general folder but remains available for integrals beyond standard loop integrals.Non-standard integrals can be defined by the user.
  • Invocation: The program is invoked through the secdec script, which can be added to the user’s search path.This allows calling it without specifying the script path each time.

A.1.1 Basic usage

Basic SecDec usage centers on preparing parameter, mathematics, and kinematics inputs, then running algebraic, numerical, and result-collection stages separately or together. Defaults reduce required configuration, while options control kinematic continuation, decomposition strategy, expansion order, and selected pole structures.

  • Basic workflow: The preparation command generates param.input, kinem.input, and math.m templates.These files hold parameters, kinematic points, and the integral definition.
  • Basic workflow: Users specify the graph and epsilon expansion order, define the integral, provide kinematic points, and run SecDec.Only graph and epsord are mandatory; unspecified parameters use defaults.
  • Execution stages: The algebraic, numerical, and result-collection stages can be run separately with secdec -algebraic, secdec -numerics, and secdec -collectresults.The algebraic stage can expose pole structure and F and U functions before numerical integration.
  • Multiple kinematic points: Additional kinematic points can be added to kinem.input and numerical runs can be submitted in cluster mode.Existing calculated points should be removed and new points given new labels.
  • Fine-grained control: Users can subdivide the workflow further, select pole structures and epsilon orders, and execute only a specified task.Detailed commands include constructing graph polynomials and restricting numerical calculations to selected poles and orders.
  • Input files: Numerators may be supplied as contracted momentum lists or represented by an additional propagator with a negative index.Propagator powers may take zero or negative integer values.
  • Key options: Contour deformation should be enabled for multi-scale or non-Euclidean problems, while strategies G1 or G2 avoid infinite recursion when strategy X does not stop.The program optimizes a user-supplied initial λ value, with values between 1 and 3 usually recommended.
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