Feynman graph polynomials

TL;DR

The paper surveys the two graph polynomials that govern the Feynman parametrization of multi-loop integrals, linking the first and second Symanzik polynomials to rich graph-theoretic structures. It reviews multiple computation methods—from spanning-tree/forest representations and the matrix-tree theorem to deletion-contraction recursions, duality for planar graphs, and matroid theory—and it discusses their roles in analyzing singularities and thresholds via Landau equations. Key contributions include explicit formulae, Dodgson-type identities with factorization properties, and connections to the multivariate Tutte polynomial and matroids, which provide a unifying framework for when different graphs share the same polynomial invariants. Collectively, the work offers both practical algorithms for loop-integral calculations and deeper mathematical understanding of graph polynomials in quantum field theory.

Abstract

The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.

Figures (19)

• Figure 1: The "double box"-graph: A two-loop Feynman diagram with four external lines and seven internal lines. The momenta flowing out along the external lines are labelled $p_1$, ..., $p_4$, the momenta flowing through the internal lines are labelled $q_1$, ..., $q_7$.
• Figure 2: The one-loop two-point function with equal masses. This graph shows a normal threshold for $p^2=4m^2$.
• Figure 3: The left picture shows a spanning tree for the graph of fig. \ref{['fig1']}, the right picture shows a spanning $2$-forest for the same graph. The spanning tree is obtained by deleting edges $4$ and $7$, the spanning $2$-forest is obtained by deleting edges $1$, $4$ and $7$.
• Figure 4: A two-loop two-point graph.
• Figure 5: The set of spanning trees for the two-loop two-point graph of fig. \ref{['fig3']}.