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Cuts from residues: the one-loop case

Samuel Abreu, Ruth Britto, Claude Duhr, Einan Gardi

TL;DR

This work provides a rigorous, residue-based definition of one-loop cut integrals in dimensional regularization by employing Leray's multivariate residues and a compactified projective-space formulation. It connects cuts to Landau singularities of both types, derives explicit maximal and next-to-maximal cut results, and shows that cuts and uncut integrals share a unified parametric representation. Through a detailed study of homology, it establishes linear relations among different cuts and constructs two natural bases for one-loop cut integrals, all while preserving the same differential equations as uncut integrals. The framework clarifies the relationship between cuts, discontinuities, and leading singularities via Picard-Lefschetz theory and sets the stage for generalization to higher-loop amplitudes and master-contour construction.

Abstract

Using the multivariate residue calculus of Leray, we give a precise definition of the notion of a cut Feynman integral in dimensional regularization, as a residue evaluated on the variety where some of the propagators are put on shell. These are naturally associated to Landau singularities of the first type. Focusing on the one-loop case, we give an explicit parametrization to compute such cut integrals, with which we study some of their properties and list explicit results for maximal and next-to-maximal cuts. By analyzing homology groups, we show that cut integrals associated to Landau singularities of the second type are specific combinations of the usual cut integrals, and we obtain linear relations among different cuts of the same integral. We also show that all one-loop Feynman integrals and their cuts belong to the same class of functions, which can be written as parametric integrals.

Cuts from residues: the one-loop case

TL;DR

This work provides a rigorous, residue-based definition of one-loop cut integrals in dimensional regularization by employing Leray's multivariate residues and a compactified projective-space formulation. It connects cuts to Landau singularities of both types, derives explicit maximal and next-to-maximal cut results, and shows that cuts and uncut integrals share a unified parametric representation. Through a detailed study of homology, it establishes linear relations among different cuts and constructs two natural bases for one-loop cut integrals, all while preserving the same differential equations as uncut integrals. The framework clarifies the relationship between cuts, discontinuities, and leading singularities via Picard-Lefschetz theory and sets the stage for generalization to higher-loop amplitudes and master-contour construction.

Abstract

Using the multivariate residue calculus of Leray, we give a precise definition of the notion of a cut Feynman integral in dimensional regularization, as a residue evaluated on the variety where some of the propagators are put on shell. These are naturally associated to Landau singularities of the first type. Focusing on the one-loop case, we give an explicit parametrization to compute such cut integrals, with which we study some of their properties and list explicit results for maximal and next-to-maximal cuts. By analyzing homology groups, we show that cut integrals associated to Landau singularities of the second type are specific combinations of the usual cut integrals, and we obtain linear relations among different cuts of the same integral. We also show that all one-loop Feynman integrals and their cuts belong to the same class of functions, which can be written as parametric integrals.

Paper Structure

This paper contains 36 sections, 198 equations, 3 figures.

Table of Contents

  1. Introduction
  2. One-loop integrals
  3. Cuts and residues
  4. Multivariate residues
  5. Definition of cut integrals
  6. The integration contour
  7. Polytope geometry and the Landau conditions
  8. Explicit results for some cut integrals
  9. Evaluation of the residues
  10. Vanishing cuts
  11. Explicit results for maximal cuts
  12. Explicit results for next-to-maximal cuts
  13. Compactification and singularities of the second type
  14. Compactification of one-loop integrals
  15. Homology groups associated to one-loop integrals
  16. ...and 21 more sections

Figures (3)

  • Figure 1: The simplex $\mathcal{K}_{C}$ whose base is the simplex $\mathcal{Q}_{C}$, with the transverse component $k^E_\bot$ of the loop momentum.
  • Figure 2: The simplex $\mathcal{K}_{[k^E,0,1,2,\ldots,j]}$ and its base simplex $\mathcal{Q}_{[0,1,2,\ldots,j]}$, whose respective altitudes are $r\sin\phi_j$ and $q^E_{jj}$.
  • Figure 3: Diagram representative of a triple cut.