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An automatized algorithm to compute infrared divergent multi-loop integrals

T. Binoth, G. Heinrich

TL;DR

This work introduces an automated, constructive subtraction scheme based on sector decomposition in Feynman parameter space to isolate infrared divergences in arbitrary multi-loop integrals. It yields a Laurent expansion in $\epsilon$ with pole coefficients given by finite, numerically integrable parameter functions, and is validated against known analytic results while extending to complex 2-loop and 3-loop diagrams. Implemented in symbolic software and demonstrated on off-shell two-loop 4-point and on-shell/off-shell three-loop 3-point cases, the method shows reliable numerical performance and potential for NNLO QCD applications such as $e^+e^- \to 3$ jets. Overall, the approach provides a practical framework to tackle otherwise analytically intractable multi-loop integrals by combining local subtractions, sector decomposition, and numerical integration.

Abstract

We describe a constructive procedure to separate overlapping infrared divergences in multi-loop integrals. Working with a parametric representation in D=4-2*epsilon dimensions, adequate subtractions lead to a Laurent series in epsilon, where the coefficients of the pole- and finite terms are sums of regular parameter integrals which can be evaluated numerically. We fully automatized this algorithm by implementing it into algebraic manipulation programs and applied it to calculate numerically some nontrivial 2-loop 4-point and 3-loop 3-point Feynman diagrams. Finally, we discuss the applicability of our method to phenomenologically relevant multi--loop calculations such as the NNLO QCD corrections for e^+e^- --> 3 jets.

An automatized algorithm to compute infrared divergent multi-loop integrals

TL;DR

This work introduces an automated, constructive subtraction scheme based on sector decomposition in Feynman parameter space to isolate infrared divergences in arbitrary multi-loop integrals. It yields a Laurent expansion in with pole coefficients given by finite, numerically integrable parameter functions, and is validated against known analytic results while extending to complex 2-loop and 3-loop diagrams. Implemented in symbolic software and demonstrated on off-shell two-loop 4-point and on-shell/off-shell three-loop 3-point cases, the method shows reliable numerical performance and potential for NNLO QCD applications such as jets. Overall, the approach provides a practical framework to tackle otherwise analytically intractable multi-loop integrals by combining local subtractions, sector decomposition, and numerical integration.

Abstract

We describe a constructive procedure to separate overlapping infrared divergences in multi-loop integrals. Working with a parametric representation in D=4-2*epsilon dimensions, adequate subtractions lead to a Laurent series in epsilon, where the coefficients of the pole- and finite terms are sums of regular parameter integrals which can be evaluated numerically. We fully automatized this algorithm by implementing it into algebraic manipulation programs and applied it to calculate numerically some nontrivial 2-loop 4-point and 3-loop 3-point Feynman diagrams. Finally, we discuss the applicability of our method to phenomenologically relevant multi--loop calculations such as the NNLO QCD corrections for e^+e^- --> 3 jets.

Paper Structure

This paper contains 7 sections, 31 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The algorithm
  3. Applications
  4. Comparison with analytical results
  5. Two--loop massless 4--point functions, one leg off-shell
  6. Three--loop massless 3--point functions, one leg off-shell
  7. Discussion

Figures (5)

  • Figure 1: The planar double box with leg 4 off-shell.
  • Figure 2: The non-planar double box with leg 1 off-shell.
  • Figure 3: The non-planar double box with leg 3 off-shell.
  • Figure 4: A 3--loop Mercedes-Star (MS) topology.
  • Figure 5: The 3--loop box-box-triangle (BBT) graph