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Feynman Integral Reduction and Landau Singularities

Federico Coro, Pavel P. Novichkov, Ben Page, Qian Song

TL;DR

The paper tackles the computational challenge of Feynman integral reduction by linking syzygy relations to Landau singularities through the Baikov representation. It develops a Landau-decomposition framework that expresses the critical syzygy module as a sum over Landau-geometry components, implemented via a determinantal, saturation-based construction of Fitting syzygies. The approach is demonstrated on two-loop, five-point examples, including the double-box and a pentabox family relevant to $pp \rightarrow t\\overline{t}H$, showing compact analytic generators organized by infrared loci and enabling cross-topology reuse. This provides a physically transparent reduction mechanism with potential for stable, efficient computations in high-precision collider amplitudes.

Abstract

We propose that Feynman integral reduction is controlled by solutions of the Landau equations. We study integral relations with prescribed propagator powers using syzygy methods and discuss how syzygies can be expressed as a sum over components of the Landau singularity locus. This leads to a determinantal approach to solving the syzygy problem, giving rise to highly compact and physically transparent solutions. We demonstrate the method in applications to planar two-loop five-point integrals relevant for the $pp \rightarrow t\overline{t}H$ process. Our results suggest an efficient method of Feynman integral reduction and provide a novel physical perspective on the problem.

Feynman Integral Reduction and Landau Singularities

TL;DR

The paper tackles the computational challenge of Feynman integral reduction by linking syzygy relations to Landau singularities through the Baikov representation. It develops a Landau-decomposition framework that expresses the critical syzygy module as a sum over Landau-geometry components, implemented via a determinantal, saturation-based construction of Fitting syzygies. The approach is demonstrated on two-loop, five-point examples, including the double-box and a pentabox family relevant to , showing compact analytic generators organized by infrared loci and enabling cross-topology reuse. This provides a physically transparent reduction mechanism with potential for stable, efficient computations in high-precision collider amplitudes.

Abstract

We propose that Feynman integral reduction is controlled by solutions of the Landau equations. We study integral relations with prescribed propagator powers using syzygy methods and discuss how syzygies can be expressed as a sum over components of the Landau singularity locus. This leads to a determinantal approach to solving the syzygy problem, giving rise to highly compact and physically transparent solutions. We demonstrate the method in applications to planar two-loop five-point integrals relevant for the process. Our results suggest an efficient method of Feynman integral reduction and provide a novel physical perspective on the problem.

Paper Structure

This paper contains 14 sections, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Singularities and Integral Relations
  3. Notation and Variable Choice
  4. A Five-Point Double Box Example
  5. $pp \rightarrow t\overline{t}H$ Pentabox Study
  6. Discussion and Outlook
  7. Proof of Fitting Decomposition of $A_0^\Gamma$
  8. Remaining $pp \rightarrow t\overline{t}H$ Five-Point Examples
  9. Pentabox
  10. Pentatriangle
  11. First Box-Triangle
  12. Second Box-Triangle
  13. Pentabubble
  14. Relation Matrix Ancillary Files

Figures (3)

  • Figure 1: Double box topology from $pp \rightarrow t\overline{t}H$ and its leading Landau singularities. Black lines are massless particles; red, the top quark; blue, the Higgs boson; dashed, soft particles; dotted, collinear particles.
  • Figure 2: Diagrams of Landau singularities arising in the computation of $\sqrt{J_{\text{syz}}^\Gamma}$. Notation matches \ref{['fig:FivePointExamples']}.
  • Figure 3: Remaining five-point topologies contributing to the leading-color, light-quark-loop contributions to $pp \rightarrow t\overline{t}H$ pentabox topology.