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Enumerative Methods in Quantum Electrodynamics

Ali Assem Mahmoud

TL;DR

The work establishes a direct combinatorial bridge between 1PI Feynman diagrams in QED-type theories (quenched QED and Yukawa theory) and rooted chord diagrams, enabling purely diagrammatic generation and asymptotic analysis of Green functions. By proving bijections and deriving generating-function identities that express diagram classes in terms of connected chord diagrams (via $C(x)$ and $C_{\ge 2}(x)$), the authors replace traditional singularity-based methods with tractable combinatorial tools. In QQED, 2-connected chord diagrams enumerate the renormalization counterterms, while in Yukawa theory multiple diagram classes (tadpoles, vacua, and multi-external-leg graphs) correspond to structured chord-diagram constructions, including a detailed explicit bijection for tadpoles. These results yield straightforward asymptotics for the considered Green functions using the known transforms of connected chord diagrams, highlighting a deep correspondence between physical perturbation theory and combinatorial map-like structures.

Abstract

We show that observables in QED-type theories can be realized in terms of a combinatorial structure called chord diagrams. One advantage of this combinatorial representation is that it simplifies the study of the asymptotic behavior of corresponding Green functions. Particularly, using the new representation, there is no need to use the standard approach of singularity analysis. This relation also reveals the unexplained correlation between the number of Feynman diagrams in Yukawa theory and the diagrams in quenched QED.

Enumerative Methods in Quantum Electrodynamics

TL;DR

The work establishes a direct combinatorial bridge between 1PI Feynman diagrams in QED-type theories (quenched QED and Yukawa theory) and rooted chord diagrams, enabling purely diagrammatic generation and asymptotic analysis of Green functions. By proving bijections and deriving generating-function identities that express diagram classes in terms of connected chord diagrams (via and ), the authors replace traditional singularity-based methods with tractable combinatorial tools. In QQED, 2-connected chord diagrams enumerate the renormalization counterterms, while in Yukawa theory multiple diagram classes (tadpoles, vacua, and multi-external-leg graphs) correspond to structured chord-diagram constructions, including a detailed explicit bijection for tadpoles. These results yield straightforward asymptotics for the considered Green functions using the known transforms of connected chord diagrams, highlighting a deep correspondence between physical perturbation theory and combinatorial map-like structures.

Abstract

We show that observables in QED-type theories can be realized in terms of a combinatorial structure called chord diagrams. One advantage of this combinatorial representation is that it simplifies the study of the asymptotic behavior of corresponding Green functions. Particularly, using the new representation, there is no need to use the standard approach of singularity analysis. This relation also reveals the unexplained correlation between the number of Feynman diagrams in Yukawa theory and the diagrams in quenched QED.

Paper Structure

This paper contains 13 sections, 11 theorems, 38 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction and Setup
  2. Zero-Dimensional Scalar Theories with Interaction
  3. $k$-Connected Chord Diagrams
  4. QED Theories, Quenched QED, and Yukawa Theory
  5. The Partition Function
  6. Quenched QED
  7. Yukawa Theory
  8. Yukawa Tadpole Graphs: $\left.\partial_{\phi_c}^1(\partial_{\psi_c}\partial_{\bar{\psi}_c})^0G^{\text{Yuk}}(\hbar,\phi_c,\psi_c)\right|_{\phi_c=\psi_c=0}$
  9. An Explicit Bijection
  10. Yukawa Vacuum Graphs: $\left.\partial_{\phi_c}^0(\partial_{\psi_c}\partial_{\bar{\psi}_c})^0G^{\text{Yuk}}(\hbar,\phi_c,\psi_c)\right|_{\phi_c=\psi_c=0}$
  11. Yukawa Graphs from $\left.\partial_{\phi_c}^2(\partial_{\psi_c}\partial_{\bar{\psi}_c})^0G^{\text{Yuk}}(\hbar,\phi_c,\psi_c)\right|_{\phi_c=\psi_c=0}$
  12. Yukawa Graphs from $\left.\partial_{\phi_c}^1(\partial_{\psi_c}\partial_{\bar{\psi}_c})^1G^{\text{Yuk}}(\hbar,\phi_c,\psi_c)\right|_{\phi_c=\psi_c=0}$
  13. Yukawa Graphs from $\left.\partial_{\phi_c}^0(\partial_{\psi_c}\partial_{\bar{\psi}_c})^1G^{\text{Yuk}}(\hbar,\phi_c,\psi_c)\right|_{\phi_c=\psi_c=0}$

Key Result

Lemma 2.1

If $D(x), C(x)$ are the generating series for chord diagrams and connected chord diagrams respectively, then

Figures (17)

  • Figure 3: A rooted chord diagram and its linear representation
  • Figure 4: Two tadpoles may differ due to the relative orientation of fermion loops.
  • Figure 5: All loops are now assumed oriented counter-clockwise and the attached boson edges have to be twisted accordingly; Fermion edges are drawn thicker than boson edges.
  • Figure 6: The 27 1PI tadpole graphs with loop number 4
  • Figure 7: Notation for $(T_1,(T_2,d))$
  • ...and 12 more figures

Theorems & Definitions (35)

  • Definition 2.1: Chord diagrams
  • Definition 2.2: Rooted chord diagrams
  • Definition 2.3: Intervals
  • Definition 2.4: The Intersection Graph
  • Remark 2.1
  • Definition 2.5: Connected Chord Diagrams
  • Example 2.1
  • Lemma 2.1: flajoletchordsalipaper2con
  • Definition 2.6: $k$-Connected Chord Diagrams
  • Proposition 2.2: alipaper2con
  • ...and 25 more