Source-linked AI summary
SecDec: A general program for sector decomposition
Jonathon Carter, Gudrun Heinrich
TL;DR
Higher-order perturbative calculations require numerical treatment of multi-dimensional parameter integrals with ultraviolet or infrared singularities. SecDec applies iterated sector decomposition to extract these singularities and numerically evaluate Laurent-series coefficients, while extending the approach to phase-space and more general parameter integrals. Its scope is bounded by practical resource demands, Euclidean restrictions for multi-scale integrals, endpoint-singularity assumptions, and a heuristic termination procedure that is not guaranteed in all cases.
Problem
SecDec addresses the need to extract ultraviolet and infrared singularities from parametric integrals arising in multi-loop Feynman and phase-space calculations.
Method
The program uses iterated sector decomposition to factorize singularities, subtract poles in dimensional regularisation, and numerically integrate the resulting finite coefficient functions.
Results
SecDec supports multi-loop Euclidean integrals and more general parameter integrals, including phase-space integrals, symbolic parameters, half-integer powers, and direct treatment of contracted tensor integrals.
Takeaways & Limitations
The program provides an automated numerical framework for isolating divergences and evaluating finite coefficients in a broad class of dimensionally regularised parameter integrals.
Takeaways & Limitations
The method is limited by memory and CPU time, multi-scale integrals are evaluable only at Euclidean points, general parameter integrals require endpoint singularities, and the heuristic termination procedure is not guaranteed.
Abstract
from arXiv · showhide
We present a program for the numerical evaluation of multi-dimensional polynomial parameter integrals. Singularities regulated by dimensional regularisation are extracted using iterated sector decomposition. The program evaluates the coefficients of a Laurent series in the regularisation parameter. It can be applied to multi-loop integrals in Euclidean space as well as other parametric integrals, e.g. phase space integrals.
PROGRAM SUMMARY
SecDec is a program for extracting ultraviolet and infrared singularities from multi-dimensional parameter integrals and numerically evaluating their finite coefficients. It extends sector decomposition to general parameter integrals, phase-space applications, symbolic parameters, half-integer powers, and contracted tensor integrals.
- Scope: SecDec extracts ultraviolet and infrared singularities from multi-loop Feynman, Wilson-loop, and phase-space parameter integrals using sector decomposition.The method is designed for numerical evaluation in dimensional regularisation.
- Numerical evaluation: The program evaluates finite parameter integrals numerically after singularities are isolated in dimensional regularisation.Its output is organized through the sector-decomposition treatment of divergent integrals.
- Applications: The program applies to Euclidean multi-scale integrals and to phase-space integrals involving soft and collinear real radiation.Phase-space applications are motivated particularly by double-real radiation in NNLO calculations with massive particles.
- Program features: Compared with existing packages, SecDec treats potential numerical instabilities and supports more general functions, symbolic parameters, and half-integer polynomial powers.These extensions include parameter integrals from real-radiation phase space.
- Feynman integrals: Contracted tensor integrals are handled directly by including additional Feynman-parameter numerator polynomials in sector decomposition.This avoids reducing them to scalar integrals first.
I. Generation of primary sectors (loop integrals only)
For loop integrals, SecDec partitions the integration domain into primary sectors and remaps each to the unit hypercube. Iterated subsector decomposition then factorizes endpoint-singular behavior until the rescaled polynomials acquire nonzero constant terms.
- Primary-sector construction: The integration domain is split into N primary sectors, each associated with the region where one Feynman parameter is largest.A delta constraint is then used to integrate out the largest parameter.
- Primary-sector construction: After remapping each primary sector, homogeneity of U and F allows the selected parameter to factorize completely.The remaining integrations run over unit-interval variables.
- Endpoint treatment: Endpoint singularities are mapped to variables approaching zero by remapping endpoint-one singularities and automatically splitting variables singular at both endpoints.The user specifies which variables should be split.
- Iterated decomposition: Iterated decomposition selects a minimal parameter set whose simultaneous vanishing makes at least one polynomial vanish, then decomposes its integration cube into subsectors.Variables are remapped to the unit hypercube in every new subsector.
- Stopping criterion: Iteration continues tree-wise until every rescaled polynomial contains a nonzero constant term, making the singular behavior readable from the exponents a_j and b_j.The resulting subsector integrals have a standardized factorized form.
III. Subtraction of the poles
SecDec extracts poles by subtracting Taylor expansions in variables with sufficiently singular exponents, producing a Laurent series in ǫ with finite coefficient integrals. The resulting coefficients can then be integrated numerically when the integrand remains nonvanishing in the domain.
- Subtraction procedure: If a_j > −1, no subtraction is needed; if a_j ≤ −1, a Taylor expansion around x_j = 0 isolates the singular contribution.The subtraction order is chosen according to the magnitude of the exponent, including half-integer cases through floor(|a_j|).
- Subtraction procedure: The subtraction separates pole terms proportional to 1/ǫ from a remainder with improved endpoint behavior.For a_j = −1, the expansion is equivalent to using plus distributions, with delta-function terms already integrated.
- Laurent expansion: After subtraction in every variable, all poles are extracted and the integral is expanded as a Laurent series in ǫ.The coefficients are finite parameter integrals whose dimensions depend on the Laurent index.
- Coefficient integrals: For n < 0, coefficient integrals have dimension N − 1 − |n|, while coefficients for n ≥ 0 have dimension N − 1.The leading pole can be at most 2L for an L-loop integral.
- Numerical evaluation: Monte Carlo integration can evaluate the finite coefficients when the chosen kinematics keep the integrand from vanishing inside the integration domain.This condition is stated for Mandelstam invariants or numerical constants in the general integrand.
Improving the numerical stability
SecDec improves numerical stability by using integration by parts to raise endpoint powers and reduce problematic cancellations before Monte Carlo integration. Its error estimates remain limited for complicated, non-square-integrable, or systematically biased integrations.
- Stabilizing singularities: Integration by parts raises powers of endpoint variables, trading higher-than-logarithmic singularities for additional logarithms that Monte Carlo integration handles more easily.The procedure iterates until the endpoint power reaches a >= -1 and remains linear in the nonsingular remainder.
- Stabilizing singularities: The IBP method reduces numerical instabilities caused by subtractive forms such as [f(x)−f(0)]/x and [f(x)−f(0)−x f ′(0)]/x2.These forms produce differences of large numbers as x approaches zero.
- Stabilizing singularities: Because the procedure is linear in R(x, ǫ), SecDec can split the remainder into smaller functions for easier numerical integration.This supports handling complicated integrands through separately treated components.
- Error treatment: Monte Carlo error estimates can underestimate true errors when decomposed and subtracted functions have complicated shapes or are not square integrable.The nominal estimate scales statistically with the number of sampling points but requires important features to be sampled sufficiently precisely.
- Error treatment: Adding many integration errors in quadrature is appropriate only when systematic numerical-integration errors are absent, and prefactor conversions can create further cancellations.Such cancellations may make the reported error estimate too optimistic.
- Program structure: The program separates algebraic sector decomposition and epsilon expansion from Fortran-based Monte Carlo integration of Laurent-series coefficients.Mathematica performs the algebraic preparation, while BASES or CUBA performs the numerical integrations.
4 Installation and usage
SecDec is installed from an archive and configured through parameter and template files before launch. Its workflow supports staged execution, single-machine or cluster calculations, result collection, cleanup, and rerunning numerical integration without repeating symbolic preparation.
- Installation: Installation requires Mathematica, perl, and a Fortran compiler on Unix or Linux systems, with gfortran used by default when no compiler is specified.The installation script checks Mathematica and perl but leaves compiler specification to param.input.
- Configuration: Users configure paths, integrand type, epsilon order, output, and numerical options in param.input and define the integrand or propagators in Template.m.Copies of these files can be renamed and supplied through launch options.
- Execution workflow: The launch command runs scripts that construct loop functions when needed, perform iterated sector decomposition, subtract singularities, expand in epsilon, and generate Fortran functions.The selected exe flag determines how far this workflow proceeds.
- Results and cleanup: Results are collected automatically on a single machine or with results.pl after cluster jobs finish, and launchclean can remove obsolete files while preserving selected outputs.The cleanup behavior differs between the default mode and the all option.
- Execution workflow: The exe flag provides staged execution from sector decomposition through executable runs, allowing completed intermediate stages to be reused.finishnumerics.pl continues incomplete calculations, while justnumerics.pl repeats only numerical integration for changed points or sampling settings.
- Execution workflow: SecDec starts with the most complicated pole structure because those jobs are expected to take the longest, especially when calculations are distributed across a cluster.This ordering is intended to improve cluster job scheduling.
5 Description of Examples
The examples demonstrate SecDec for multi-loop integrals, general polynomial functions, and phase-space-related calculations, including nontrivial singularity structures and numerical Laurent coefficients.
- Example 1: Non-planar massless two-loop box: The non-planar massless two-loop box exhibits non-logarithmic poles under standard sector decomposition, motivating alternative parametrisations and integration orderings.Such poles require complicated subtraction terms and can slow the calculation; integrating one loop first can yield a faster representation.
- Example 1: Non-planar massless two-loop box: 384 subsectors are produced for the non-planar massless two-loop box, whose numerical results are reported for two Euclidean kinematic points.The points are (s, t, u) = (−1, −1, −1) and (−1, −2, −3).
- Loop-integral examples: The examples cover massive on-shell ladder integrals, non-planar graphs for gg →t¯t at NNLO, tensor numerators, and non-standard propagator powers.The ladder example uses massive lines, while the gg →t¯t examples provide numerical results at Euclidean points.
- General polynomial functions: The general-integrand examples include recursively represented hypergeometric functions, with 5F4 evaluated at β = 0.5 and compared with an analytic result.The parameter choices are a1 = ǫ, a2 = −ǫ, a3 = −3ǫ, a4 = −5ǫ, a5 = −7ǫ and b1 = 2ǫ, b2 = 4ǫ, b3 = 6ǫ, b4 = 8ǫ.
- Phase-space examples: For NNLO real-radiation phase-space integrals, the authors recommend combining sector decomposition with convenient parametrisations or transformations.The examples deliberately include complicated denominators that cannot always be removed by transformations, while hybrid treatments can use SecDec for the remaining integrals.
Three massless particles in the final state
The massless-particle phase-space example combines a toy matrix element with the phase-space parametrisation, producing endpoint singularities that sector decomposition handles numerically.
- Three massless particles in the final state: For a massless p5, the limit β →1 combined with 1/(s35s23) produces singularities at x1 = 0, x2 = 0, and x3 = 0.The denominator structure can arise from interference of diagrams.
- Three massless particles in the final state: Splitting the x3 integration at 1/2 transforms a factor that vanishes at x3 →1, x2 →0 into a form preferred for numerical stability.The program performs this automatically when configured in the template file.
- Three massless particles in the final state: Sector decomposition of the example yields the numerical result reported in Table 8.The calculation is run with params23s35.input and templates23s35.m.
Two massless and one massive particles in the final state
The two-massless-particle, one-massive-particle examples illustrate selective decomposition and conclude with the program’s broader support for dimensional-regularisation integrals.
- Selective decomposition: The program can exclude selected integrand terms from decomposition when their vanishing does not create a singularity.Terms with powers ≥0 are excluded by default, and the option is demonstrated for a line-singularity-remapped integral.
- Selective decomposition: The integral in the second example is evaluated numerically for β = 0.75, with the factor 2 Cǫ omitted.The result is given in Table 9.
- Conclusions and outlook: SecDec evaluates multi-loop integrals in Euclidean space and more general dimensionally regulated parameter integrals by automatically extracting poles in 1/ǫ.Iterated sector decomposition produces finite functions that become Laurent-series coefficients evaluated by Monte Carlo integration.
- Conclusions and outlook: The program supports arbitrary tensor integrals, ǫ-dependent propagator powers, half-integer powers, and later numerical assignment of symbolic constants.This permits one sector decomposition to be reused while changing numerical constant values.
- Conclusions and outlook: The code is publicly available with examples, documentation, and support for the Bases and Cuba numerical-integration packages.The authors also plan alternative tools for generating optimised functions and automated non-linear transformations.
8 Appendix
The appendix documents timing, symmetry, alternative parametrisation, phase-space construction, and troubleshooting details for SecDec calculations.
- Timing and parallelisation: The four-loop example parallelises integrations of pole-order functions, with Table 10 comparing parallel and serial timings.The longest job determines the total time for a pole order when contributing functions run in parallel, using 500000 sampling points per pole order.
- Timing and parallelisation: Symmetry reduces the four-loop calculation to four primary sectors, whose multiplicities are included automatically when specified in param.input.The corresponding numerical and analytical results are reported in Table 11.
- Alternative box parametrisation: A specially parametrised non-planar two-loop box factorises one integration parameter, allowing analytic integration and considerably faster automated evaluation.This procedure requires an analytical step to introduce a convenient parametrisation.
- Alternative box parametrisation: The alternative box construction integrates one loop first, introduces Feynman parameters for the remaining propagators, and feeds the resulting function to the general-integrand routines.The parametrisation is defined through transformations of the x_i variables and subsequent combinations of propagator factors.
- Phase-space parametrisation: The phase-space singular limits include x3 →0 and x3 →1 soft limits and x1 or x2 endpoint limits corresponding to collinear configurations.The listed limits identify which variables approach boundaries for the relevant soft or collinear behaviour.
- Troubleshooting: Troubleshooting covers invalid on-shell conditions, zero functions, undefined symbols, and long numerical integrations, with checks based on F, U, and numerator expressions.Numerical values for external invariants can be assigned in the template or deferred to the numerical-integration stage.