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SecDec: A general program for sector decomposition

Jonathon Carter, Gudrun Heinrich

TL;DR

SecDec provides an automated framework to numerically evaluate multi-dimensional parametric integrals arising in perturbative quantum field theory by iteratively sector-decomposing endpoint singularities and extracting a Laurent series in $\epsilon$. The approach handles both loop integrals (including tensor numerators and non-standard propagator powers) and general polynomial phase-space or parameter integrals, with coefficients computed via Monte Carlo integration using BASES or Cuba. The paper demonstrates the method on a suite of loop and phase-space examples, illustrating robustness against potential recursion issues, and extending applicability to hypergeometric representations and phase-space integrals. The tool is publicly available and designed to be extensible, with future plans including non-linear transformations and contour methods for physical-region thresholds, making it a practical asset for NNLO/NLO verifications and beyond.

Abstract

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.

SecDec: A general program for sector decomposition

TL;DR

SecDec provides an automated framework to numerically evaluate multi-dimensional parametric integrals arising in perturbative quantum field theory by iteratively sector-decomposing endpoint singularities and extracting a Laurent series in . The approach handles both loop integrals (including tensor numerators and non-standard propagator powers) and general polynomial phase-space or parameter integrals, with coefficients computed via Monte Carlo integration using BASES or Cuba. The paper demonstrates the method on a suite of loop and phase-space examples, illustrating robustness against potential recursion issues, and extending applicability to hypergeometric representations and phase-space integrals. The tool is publicly available and designed to be extensible, with future plans including non-linear transformations and contour methods for physical-region thresholds, making it a practical asset for NNLO/NLO verifications and beyond.

Abstract

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.

Paper Structure

This paper contains 25 sections, 52 equations, 11 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Feynman integrals
  4. General parameter integrals
  5. General parameter integrals
  6. Iterated sector decomposition
  7. Structure of the program
  8. Installation and usage
  9. Description of Examples
  10. Loop integrals
  11. Example 1: Non-planar massless two-loop box
  12. Example 2: Planar two-loop ladder diagram with massive on-shell legs
  13. Example 3: Non-planar two-loop diagrams with two massive on-shell legs
  14. Example 4: A rank one tensor two-loop box
  15. Example 5: A three-loop vertex diagram with $\epsilon$-dependent propagator powers
  16. ...and 10 more sections

Figures (11)

  • Figure 1: Directory structure of the SecDec program.
  • Figure 2: Flowchart showing the main steps the program performs to produce the result files. In each of the subdirectories loop or general, the file Template.m can be used to define the integrand. The produced files are written to a subdirectory created according to the settings given in param.input. By default, a subdirectory with the name of the graph or integrand is created to store the produced functions. This directory will contain subdirectories according to the pole structure of the integrand. The perl scripts (extension .pl) are steering the various steps to be performed by the program.
  • Figure 3: Example for a directory structure created by running the loop demo programs NPbox, QED, ggtt1, A61. A four-loop example defined by the user to be written to the scratch disk is also shown.
  • Figure 4: Example for a directory tree corresponding to the pole structure of the graph QED contained in the demo programs.
  • Figure 5: The non-planar two-loop box, called NPbox in example 1.
  • ...and 6 more figures