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Integration-by-parts reductions from unitarity cuts and algebraic geometry

Kasper J. Larsen, Yang Zhang

TL;DR

The paper addresses the computational bottleneck of deriving IBP reductions at two loops for multi-scale processes. It introduces a method that leverages unitarity cuts of subgraphs and syzygy equations in a polynomial z-parameterization to generate IBP identities, avoiding doubled propagators and enabling efficient merging of cut results. Demonstrated on the massless double-box topology, the approach yields a complete reduction to eight master integrals and delivers fast analytic reductions, highlighting practical potential for complex NNLO calculations. The work suggests broad extensions to higher multiplicity, masses, non-planar diagrams, and higher-loop cases, offering a scalable alternative to traditional IBP techniques.

Abstract

We show that the integration-by-parts reductions of various two-loop integral topologies can be efficiently obtained by applying unitarity cuts to a specific set of subgraphs and solving associated polynomial (syzygy) equations.

Integration-by-parts reductions from unitarity cuts and algebraic geometry

TL;DR

The paper addresses the computational bottleneck of deriving IBP reductions at two loops for multi-scale processes. It introduces a method that leverages unitarity cuts of subgraphs and syzygy equations in a polynomial z-parameterization to generate IBP identities, avoiding doubled propagators and enabling efficient merging of cut results. Demonstrated on the massless double-box topology, the approach yields a complete reduction to eight master integrals and delivers fast analytic reductions, highlighting practical potential for complex NNLO calculations. The work suggests broad extensions to higher multiplicity, masses, non-planar diagrams, and higher-loop cases, offering a scalable alternative to traditional IBP techniques.

Abstract

We show that the integration-by-parts reductions of various two-loop integral topologies can be efficiently obtained by applying unitarity cuts to a specific set of subgraphs and solving associated polynomial (syzygy) equations.

Paper Structure

This paper contains 3 sections, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Method
  3. Example

Figures (1)

  • Figure 1: The massless double-box diagram, along with our labelling conventions for its internal lines, is shown on top. The lower part shows the subset of the master integrals in Eq. (\ref{['eq:master_integrals']}) with the property that their graphs cannot be obtained by adding internal lines to the graph of some other master integral. The corresponding cuts $\{2,5,7\}$, $\{1,4,7\}$, $\{2,4,6,7\}$ and $\{1,3,4,6\}$ are the cuts required for deriving complete IBP relations for the double-box diagram.