Mixed Hodge Structures and Renormalization in Physics

TL;DR

The paper develops a rigorous link between renormalization in perturbative quantum field theory and limiting mixed Hodge structures by recasting Feynman amplitudes in parametric form and studying the monodromy of toric and graph-hypersurface degenerations. It builds a comprehensive combinatorial and geometric framework using Hopf algebras, toric blowups, and graph polynomials to isolate log-poles via projective integrals, and then encodes the resulting renormalization as a nilpotent monodromy matrix N acting on a period vector. A central thrust is the MOM renormalization scheme, where homogeneity simplifies the extraction of p1(Γ) from co-graphs and residues, and the leading terms are interpreted as periods of a limiting mixed Hodge structure. The results suggest a deep, motivic structure underlying renormalized amplitudes, with a concrete pathway to understanding their periods and weight filtrations through the monodromy data and LMHS theory.

Abstract

We relate renormalization in perturbative quantum field theory to the theory of limiting mixed Hodge structures using parametric representations of Feynman graphs.

Figures (14)

• Figure 1: Picture of $X$ and $L$
• Figure 2: Dunce's cap. Here and in following figures, external half-edges are often not drawn and are determined by the requirement that all vertices are four-valent.
• Figure 3: This vertex graph has a propagator correction given by edges $4,5,6$. The non-trivial part of the coproduct then delivers on the left the subgraph with internal edges $4,5,6$ amongst other terms. The coproduct on the right has a co-graph on edges $1,2,3$. There is a two-point vertex $u$ between edges $2,3$. Choosing two labels $u=m^2$ or $u=\Box$ allows to distinguish between mass and wave-function renormalization. We remind the reader that the corresponding monomials in the Lagrangian are $m^2\phi^2/2$ and $\phi\Box\phi/2$.
• Figure 4: In Eq.(\ref{['copw3']}), we give the coproduct for this wheel with three spokes in the core Hopf algebra.
• Figure 5: A graph with overlapping subdivergences. The renormalization Hopf algebra gives $\Delta^\prime{123456}=56\otimes 1234+1256\otimes 34+3456\otimes 12$. Note that each edge belongs to some subgraph with $\textrm{sdd}\geq 0$.

Theorems & Definitions (66)

• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• Proposition 3.1
• proof