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Cuts and coproducts of massive triangle diagrams

Samuel Abreu, Ruth Britto, Hanna Grönqvist

TL;DR

The paper extends the unitarity-cut and coproduct framework to Feynman diagrams with internal masses, revealing new branch cuts and mapping their discontinuities to coproduct entries via a generalized first-entry condition. It introduces two cut types—external-channel cuts and single-propagator mass cuts—across one-loop triangle diagrams, and establishes precise relations between cuts, discontinuities, and the coproduct, supported by extensive one-loop triangle examples. A reconstruction program is developed: starting from a channel cut, the symbol and then the full function can be determined by an ansatz constrained by first-entry, integrability, and symmetry, with practical algorithms for both symbol construction and function recovery. The results highlight the Hopf-algebraic structure as a powerful organizing principle for analytic properties of Feynman integrals with masses, and anticipate generalizations to higher-loop and multi-leg diagrams.

Abstract

Relations between multiple unitarity cuts and coproducts of Feynman integrals are extended to allow for internal masses. These masses introduce new branch cuts, whose discontinuities can be derived by placing single propagators on shell and identified as particular entries of the coproduct. First entries of the coproduct are then seen to include mass invariants alone, as well as threshold corrections for external momentum channels. As in the massless case, the original integral can possibly be recovered from its cuts by starting with the known part of the coproduct and imposing integrability contraints. We formulate precise rules for cuts of diagrams, and we gather evidence for the relations to coproducts through a detailed study of one-loop triangle integrals with various combinations of external and internal masses.

Cuts and coproducts of massive triangle diagrams

TL;DR

The paper extends the unitarity-cut and coproduct framework to Feynman diagrams with internal masses, revealing new branch cuts and mapping their discontinuities to coproduct entries via a generalized first-entry condition. It introduces two cut types—external-channel cuts and single-propagator mass cuts—across one-loop triangle diagrams, and establishes precise relations between cuts, discontinuities, and the coproduct, supported by extensive one-loop triangle examples. A reconstruction program is developed: starting from a channel cut, the symbol and then the full function can be determined by an ansatz constrained by first-entry, integrability, and symmetry, with practical algorithms for both symbol construction and function recovery. The results highlight the Hopf-algebraic structure as a powerful organizing principle for analytic properties of Feynman integrals with masses, and anticipate generalizations to higher-loop and multi-leg diagrams.

Abstract

Relations between multiple unitarity cuts and coproducts of Feynman integrals are extended to allow for internal masses. These masses introduce new branch cuts, whose discontinuities can be derived by placing single propagators on shell and identified as particular entries of the coproduct. First entries of the coproduct are then seen to include mass invariants alone, as well as threshold corrections for external momentum channels. As in the massless case, the original integral can possibly be recovered from its cuts by starting with the known part of the coproduct and imposing integrability contraints. We formulate precise rules for cuts of diagrams, and we gather evidence for the relations to coproducts through a detailed study of one-loop triangle integrals with various combinations of external and internal masses.

Paper Structure

This paper contains 75 sections, 175 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Hopf algebra for Feynman integrals
  3. Multiple polylogarithms for Feynman integrals.
  4. Hopf algebra, coproduct, and symbol.
  5. The symbol alphabet.
  6. One-loop triangles
  7. The first-entry condition
  8. Example 1:
  9. Example 2:
  10. Two types of cuts
  11. Cut in a kinematic channel
  12. First cut.
  13. Cut in an internal mass
  14. Calculation of cut diagrams
  15. Cuts in external channels.
  16. ...and 60 more sections

Figures (5)

  • Figure 1: Triangles with one external massive leg.
  • Figure 2: Triangles with two external massive legs.
  • Figure 3: Triangles with three external massive legs.
  • Figure 4: Example of cuts of $T(0,p_2^2,p_3^2;m_{12}^2,0,0)$: (a) single cut in an external channel; (b) single cut in an internal mass; (c) double cut in external channels; (d) double cut in an external channel and an internal mass. Thin dotted lines correspond to cuts on external channels and imply complex conjugation of a region of the diagram. Thick dashed lines correspond to mass cuts and do not imply any complex conjugation.
  • Figure 5: All possible cut diagrams of $T(0,p_2^2,p_3^2;m_{12}^2,0,0)$. A diagram in the top row and its corresponding diagram in the bottom row are associated to the same momentum channel, but opposite energy flow.