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Cutkosky Rules and Outer Space

Spencer Bloch, Dirk Kreimer

TL;DR

This work provides a rigorous mathematical framework for Cutkosky rules by embedding Feynman amplitudes into a cubical chain complex of graphs and leveraging Pham’s vanishing cycles. It creates a bridge between the analytic structure of amplitudes (thresholds, monodromy, dispersion) and the combinatorics of graph reductions, encoded via a Hopf-algebraic renormalization and a novel Outer Space perspective. Central contributions include a precise Cutkosky theorem in the presence of renormalization, a hierarchical description of leading and anomalous thresholds through graph polynomials, and a matrix calculus (the M_i^Γ) that reconstructs amplitude variations from lower-dimensional data. The framework integrates Thom’s isotopy, Whitney stratifications, and Hodge-theoretic considerations to illuminate the real structure essential for on-shell cuts, with concrete examples such as the one-loop bubble and the triangle illustrating the method and its dispersive interpretation.

Abstract

We derive Cutkosky's theorem starting from Pham's classical work. We emphasize structural relations to Outer Space.

Cutkosky Rules and Outer Space

TL;DR

This work provides a rigorous mathematical framework for Cutkosky rules by embedding Feynman amplitudes into a cubical chain complex of graphs and leveraging Pham’s vanishing cycles. It creates a bridge between the analytic structure of amplitudes (thresholds, monodromy, dispersion) and the combinatorics of graph reductions, encoded via a Hopf-algebraic renormalization and a novel Outer Space perspective. Central contributions include a precise Cutkosky theorem in the presence of renormalization, a hierarchical description of leading and anomalous thresholds through graph polynomials, and a matrix calculus (the M_i^Γ) that reconstructs amplitude variations from lower-dimensional data. The framework integrates Thom’s isotopy, Whitney stratifications, and Hodge-theoretic considerations to illuminate the real structure essential for on-shell cuts, with concrete examples such as the one-loop bubble and the triangle illustrating the method and its dispersive interpretation.

Abstract

We derive Cutkosky's theorem starting from Pham's classical work. We emphasize structural relations to Outer Space.

Paper Structure

This paper contains 37 sections, 15 theorems, 146 equations.

Table of Contents

  1. Introduction and results
  2. Introduction
  3. Results
  4. Graphs and spanning forests
  5. Momenta and masses
  6. Powercounting
  7. Multiple edges
  8. Example
  9. The cubical chain complex and $M_\Gamma^i$
  10. The cubical chain complex
  11. The matrix $M_i^\Gamma$
  12. Graphs and their Hopf algebra structure
  13. Hopf algebra structure
  14. The Hopf algebra and pairs $(\Gamma,F)$
  15. Kinematics
  16. ...and 22 more sections

Key Result

Lemma 2

Let $\Gamma$ be a connected graph with at least $1$ edge and no self-loops. Then $\Gamma$ admits a connected $2$-partition.

Theorems & Definitions (41)

  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • Remark 4
  • Lemma 5
  • proof
  • Remark 6
  • Remark 7
  • Lemma 8
  • ...and 31 more