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Subdivergence-free gluings of trees

Xinle Dai, Jordan Long, Karen Yeats

TL;DR

The paper investigates subdivergence-free gluings of two rooted trees with equal leaf counts, a combinatorial problem motivated by primitive Feynman graphs in quantum field theory. It establishes two complementary approaches: (i) a permutation-based method that yields closed forms for counting gluings in three infinite tree families via connected permutations, and (ii) two recursive algorithms (root-decomposition and cut-preprocessing) that compute subdivergence-free gluings for arbitrary tree pairs, with practical implementations. Key contributions include explicit closed-form counts for the Line, Two-ended, and Extra-Fan line families, a generalized framework linking subdivergence-free gluings to connected permutations, and scalable algorithms with potential uses in tanglegram enumeration and Feynman-graph analysis. These results illuminate the structure of subdivergence patterns, connect combinatorics to Hopf-algebra concepts, and provide tools for enumerating primitive diagrams in perturbative quantum field theory.

Abstract

A gluing of two rooted trees is an identification of their leaves and un-subdivision of the resulting 2-valent vertices. A gluing of two rooted trees is subdivergence free if it has no 2-edge cuts with both roots on the same side of the cut. The problem and language is motivated by quantum field theory. We enumerate subdivergence-free gluings for certain families of trees, showing a connection with connected permutations, and we give algorithms to compute subdivergence-free gluings.

Subdivergence-free gluings of trees

TL;DR

The paper investigates subdivergence-free gluings of two rooted trees with equal leaf counts, a combinatorial problem motivated by primitive Feynman graphs in quantum field theory. It establishes two complementary approaches: (i) a permutation-based method that yields closed forms for counting gluings in three infinite tree families via connected permutations, and (ii) two recursive algorithms (root-decomposition and cut-preprocessing) that compute subdivergence-free gluings for arbitrary tree pairs, with practical implementations. Key contributions include explicit closed-form counts for the Line, Two-ended, and Extra-Fan line families, a generalized framework linking subdivergence-free gluings to connected permutations, and scalable algorithms with potential uses in tanglegram enumeration and Feynman-graph analysis. These results illuminate the structure of subdivergence patterns, connect combinatorics to Hopf-algebra concepts, and provide tools for enumerating primitive diagrams in perturbative quantum field theory.

Abstract

A gluing of two rooted trees is an identification of their leaves and un-subdivision of the resulting 2-valent vertices. A gluing of two rooted trees is subdivergence free if it has no 2-edge cuts with both roots on the same side of the cut. The problem and language is motivated by quantum field theory. We enumerate subdivergence-free gluings for certain families of trees, showing a connection with connected permutations, and we give algorithms to compute subdivergence-free gluings.

Paper Structure

This paper contains 14 sections, 11 theorems, 13 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Set up
  3. Physics motivation
  4. Initial observations
  5. Outline
  6. Connected permutations and generalizations
  7. Counting subdivergence-free gluings in special families
  8. Line Family
  9. Two-ended family
  10. Lines with an extra fan family
  11. Algorithms
  12. Eating away children of the root
  13. Cut Preprocessing Algorithm
  14. Discussion

Key Result

Lemma 2.3

$c_n = n! - \sum_{i=1}^{n-1} i! c_{n-i}$

Figures (13)

  • Figure 1: A gluing of two rooted trees. (A) shows rooted trees $t_1$ and $t_2$. (B) shows the leaves of $t_1$ and $t_2$ identified with each other. (C) shows the gluing: the result of contracting an edge incident to each leaf pairing.
  • Figure 2: A gluing with a subdivergence from cutting the edges indicated with dashed lines.
  • Figure 3: An example of a Feynman graph, in this case the directed edges represent a fermion propagating, while the dashed edges represent mesons. There are two external edges, one at the left and one at the right.
  • Figure 4: Examples of trees $\ell_k$ in the line family.
  • Figure 5: Labelling of $\ell_4$. Any gluing that fixes a prefix of length 1,2 or 3 creates a subdivergence.
  • ...and 8 more figures

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 3.1
  • Theorem 3.2
  • ...and 22 more