How to Resum Feynman Graphs

TL;DR

This paper introduces a simplified combinatorial approach to quantum field theory by defining universal weights for graphs and spanning trees, enabling the transformation of divergent series into convergent ones via the Loop Vertex Expansion.

Contribution

It presents a novel combinatorial reformulation using rational weights and the Loop Vertex Expansion to improve convergence in quantum field theory series.

Findings

Weights correspond to Hepp's sector percentages in UV analysis

Series reshuffling leads to convergence

Uses intermediate field representation graphs

Abstract

In this paper we reformulate in a simpler way the combinatoric core of constructive quantum field theory We define universal rational combinatoric weights for pairs made of a graph and one of its spanning trees. These weights are nothing but the percentage of Hepp's sectors in which the tree is leading the ultraviolet analysis. We explain how they allow to reshuffle the divergent series formulated in terms of Feynman graphs into convergent series indexed by the trees that these graphs contain. The Feynman graphs to be used are not the ordinary ones but those of the intermediate field representation, and the result of the reshuffling is called the Loop Vertex Expansion.