The exchange graph and variations of the ratio of the two Symanzik polynomials

TL;DR

This paper investigates how the ratio of Symanzik polynomials in Feynman graphs varies under geometric perturbations, introducing the exchange graph to analyze the boundedness of these variations within the context of Hodge theory.

Contribution

It introduces the exchange graph to study the variation of Symanzik polynomial ratios and proves their boundedness under geometric perturbations, connecting graph theory with Hodge theory.

Findings

The ratio of Symanzik polynomials remains bounded under bounded geometric perturbations.

The exchange graph encodes exchange properties between spanning trees and 2-forests.

Connected components of the exchange graph are characterized, aiding in the boundedness proof.

Abstract

Correlation functions in quantum field theory are calculated using Feynman amplitudes, which are finite dimensional integrals associated to graphs. The integrand is the exponential of the ratio of the first and second Symanzik polynomials associated to the Feynman graph, which are described in terms of the spanning trees and spanning 2-forests of the graph, respectively. In a previous paper with Bloch, Burgos and Fres\'an, we related this ratio to the asymptotic of the Archimedean height pairing between degree zero divisors on degenerating families of Riemann surfaces. Motivated by this, we consider in this paper the variation of the ratio of the two Symanzik polynomials under bounded perturbations of the geometry of the graph. This is a natural problem in connection with the theory of nilpotent and SL2 orbits in Hodge theory. Our main result is the boundedness of variation of the ratio. For this we define the exchange graph of a given graph which encodes the exchange properties between spanning trees and spanning 2-forests in the graph. We provide a description of the connected components of this graph, and use this to prove our result on boundedness of the variations.