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Unitarity Cuts of Integrals with Doubled Propagators

Mads Sogaard, Yang Zhang

TL;DR

The paper generalizes the unitarity-cut framework to loop integrals with repeated propagators by recasting degenerate multivariate residues as the computational core, allowing direct extraction of master-integral coefficients at one and two loops. It develops a systematic procedure using Gröbner-basis-based transformations and explicit master-integral projectors to reduce both planar and nonplanar double-box topologies to a minimal scalar/tensor basis, including cases with doubled and tripled propagators. The approach yields compact coefficient relations and is consistent with IBP identities in four dimensions, offering a path to automate reductions that were previously inaccessible with standard cuts. The method promises extensions to D-dimensional regularization and massive external legs, and can leverage Bezoutian techniques to speed residue computations.

Abstract

We extend the notion of generalized unitarity cuts to accommodate loop integrals with higher powers of propagators. Such integrals frequently arise in for example integration-by-parts identities, Schwinger parametrizations and Mellin-Barnes representations. The method is applied to reduction of integrals with doubled and tripled propagators and direct extract of integral coefficients at one and two loops. Our algorithm is based on degenerate multivariate residues and computational algebraic geometry.

Unitarity Cuts of Integrals with Doubled Propagators

TL;DR

The paper generalizes the unitarity-cut framework to loop integrals with repeated propagators by recasting degenerate multivariate residues as the computational core, allowing direct extraction of master-integral coefficients at one and two loops. It develops a systematic procedure using Gröbner-basis-based transformations and explicit master-integral projectors to reduce both planar and nonplanar double-box topologies to a minimal scalar/tensor basis, including cases with doubled and tripled propagators. The approach yields compact coefficient relations and is consistent with IBP identities in four dimensions, offering a path to automate reductions that were previously inaccessible with standard cuts. The method promises extensions to D-dimensional regularization and massive external legs, and can leverage Bezoutian techniques to speed residue computations.

Abstract

We extend the notion of generalized unitarity cuts to accommodate loop integrals with higher powers of propagators. Such integrals frequently arise in for example integration-by-parts identities, Schwinger parametrizations and Mellin-Barnes representations. The method is applied to reduction of integrals with doubled and tripled propagators and direct extract of integral coefficients at one and two loops. Our algorithm is based on degenerate multivariate residues and computational algebraic geometry.

Paper Structure

This paper contains 17 sections, 1 theorem, 84 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalized Feynman Integrals
  3. Direct Extraction of Integral Coefficients
  4. Multivariate Residues
  5. Example: One-Loop Box
  6. Example: Planar Double Box
  7. Parametrization of Hepta-Cut Solutions
  8. Residues of the Loop Integrand
  9. Master Integral Projectors
  10. Example: Nonplanar Double Box
  11. Parametrization of Hepta-Cut Solutions
  12. Residues of the Loop Integrand
  13. Master Integral Projectors
  14. A Scalar Integral Basis
  15. Discussion
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\{f_1,\dots,f_n\}$ and $\{u_1,\dots,u_n\}$ be two sets of holomorphic functions and $u_i = a_{ij} f_j$, where $a_{ij}$ are holomorphic functions. Assume that for each set, the common zeros are discrete points. Let $A$ be the matrix of the $a_{ij}$'s, then

Figures (4)

  • Figure 1: The massless four-point planar double box.
  • Figure 2: Global structure of the hepta-cut of the two-loop planar (left) and nonplanar (right) double box with purely massless kinematics and four external legs. The straight lines should be interpreted as genus-0 Riemann surfaces. Each branch may have an additional residue at $z = \infty$ which is eliminated here.
  • Figure 3: The nonplanar double box topology with four external particles.
  • Figure 4: We use an integral basis for the massless four-point nonplanar double box that contains no tensor numerators. Instead we have (left) a scalar integral with single propagators and (right) a scalar integral with a doubled propagator in the subbox.

Theorems & Definitions (1)

  • Theorem 1: Transformation law