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Internal Reduction method for computing Feynman Integrals

Costas G. Papadopoulos, Christopher Wever

TL;DR

The paper introduces Internal Reduction, a method to compute complex multi-loop Feynman Integrals by rewriting a target five-point two-loop MI as a one-dimensional integral over simpler MIs. It applies the approach to planar and nonplanar pentaboxes with one off-shell leg, deriving one-fold integral representations for ten previously unknown nonplanar MIs by relating them to four-point double-box integrals; these are themselves reduced to known analytic MIs and expressed via Goncharov polylogs. Numerical checks against SecDec in the Euclidean region show excellent agreement, and the authors provide strategies to accelerate evaluation and to handle physical-region configurations. The work broadens the toolkit for NNLO QCD calculations, offering a scalable path to completing five-point two-loop MI bases and enabling improved predictions for processes like $H+2$ jets and $V+jets$ at the LHC.

Abstract

A new approach to compute Feynman Integrals is presented. It relies on an integral representation of a given Feynman Integral in terms of simpler ones. Using this approach, we present, for the first time, results for a certain family of non-planar five-point two-loop Master Integrals with one external off-shell particle, relevant for instance for $H+2$ jets production at the LHC, in both Euclidean and physical kinematical regions.

Internal Reduction method for computing Feynman Integrals

TL;DR

The paper introduces Internal Reduction, a method to compute complex multi-loop Feynman Integrals by rewriting a target five-point two-loop MI as a one-dimensional integral over simpler MIs. It applies the approach to planar and nonplanar pentaboxes with one off-shell leg, deriving one-fold integral representations for ten previously unknown nonplanar MIs by relating them to four-point double-box integrals; these are themselves reduced to known analytic MIs and expressed via Goncharov polylogs. Numerical checks against SecDec in the Euclidean region show excellent agreement, and the authors provide strategies to accelerate evaluation and to handle physical-region configurations. The work broadens the toolkit for NNLO QCD calculations, offering a scalable path to completing five-point two-loop MI bases and enabling improved predictions for processes like jets and at the LHC.

Abstract

A new approach to compute Feynman Integrals is presented. It relies on an integral representation of a given Feynman Integral in terms of simpler ones. Using this approach, we present, for the first time, results for a certain family of non-planar five-point two-loop Master Integrals with one external off-shell particle, relevant for instance for jets production at the LHC, in both Euclidean and physical kinematical regions.

Paper Structure

This paper contains 6 sections, 31 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Internal Reduction
  3. Application to Pentabox Integrals
  4. Results and checks
  5. Conclusions
  6. Relating the five-point MI to four-point MI

Figures (4)

  • Figure 1: The three planar pentabox families: $P_1$ (left), $P_2$ (middle) and $P_3$ (right) with one external massive leg.
  • Figure 2: The five non-planar families with one external massive leg: $N_1$ (top left), $N_2$ (top middle), $N_2$ (top right), $N_4$ (bottom left), $N_5$ (bottom right)
  • Figure 3: The two-loop planar pentabox family $P_1$. All external momenta are incoming.
  • Figure 4: The new two-loop nonplanar five-point diagrams in the family $N_1$. All external momenta are incoming.