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Spanning forest polynomials and the transcendental weight of Feynman graphs

Francis Brown, Karen Yeats

TL;DR

The paper develops a combinatorial framework to predict the transcendental weight of primitive φ^4 Feynman graphs by linking weight behavior to spanning forest polynomials and Dodgson polynomials. It introduces a universal formula for 3-vertex-connected graphs, analyzes weight via hyperlogarithms and denominator reduction, and classifies weight-preserving versus weight-dropping graph operations, yielding infinite families of both types. Key tools include the spanning forest polynomials $\Phi^P_G$, signs in Dodgson polynomials, and a 3-connected graph analysis with a universal polynomial $\rho_L$. The results illuminate which diagrams contribute maximal weight and provide practical criteria for weight drops, advancing understanding of perturbative expansions and their motivic structure.

Abstract

We give combinatorial criteria for predicting the transcendental weight of Feynman integrals of certain graphs in $φ^4$ theory. By studying spanning forest polynomials, we obtain operations on graphs which are weight-preserving, and a list of subgraphs which induce a drop in the transcendental weight.

Spanning forest polynomials and the transcendental weight of Feynman graphs

TL;DR

The paper develops a combinatorial framework to predict the transcendental weight of primitive φ^4 Feynman graphs by linking weight behavior to spanning forest polynomials and Dodgson polynomials. It introduces a universal formula for 3-vertex-connected graphs, analyzes weight via hyperlogarithms and denominator reduction, and classifies weight-preserving versus weight-dropping graph operations, yielding infinite families of both types. Key tools include the spanning forest polynomials , signs in Dodgson polynomials, and a 3-connected graph analysis with a universal polynomial . The results illuminate which diagrams contribute maximal weight and provide practical criteria for weight drops, advancing understanding of perturbative expansions and their motivic structure.

Abstract

We give combinatorial criteria for predicting the transcendental weight of Feynman integrals of certain graphs in theory. By studying spanning forest polynomials, we obtain operations on graphs which are weight-preserving, and a list of subgraphs which induce a drop in the transcendental weight.

Paper Structure

This paper contains 13 sections, 22 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Background
  4. Signs in Dodgson polynomials
  5. Identities
  6. Weight drop in Feynman graphs
  7. Hyperlogarithms
  8. Initial integrations
  9. Denominator reduction
  10. Weight-preserving operations
  11. Families of weight drop graphs
  12. Operations on 3-connected graphs
  13. Examples of 3-connected operations

Key Result

Proposition 6

Let $e\in E(G)$ such that $e\notin I \cup J \cup K$. Let $G\backslash e$ denote the graph obtained by deleting the edge $e$ (and removing any isolated vertices) and let $G/ e$ denote the graph obtained by contracting $e$ (i.e., deleting $e$ and identifying its endpoints).We require that the contract Since $\Psi_{G,K}^{I,J}$ is linear in the Schwinger parameters this implies that

Theorems & Definitions (58)

  • Remark 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • Corollary 7
  • Proposition 8
  • Definition 9
  • Proposition 10
  • ...and 48 more