The physics and the mixed Hodge structure of Feynman integrals

TL;DR

The paper develops a deep link between Feynman integrals and periods through unitarity, world-line methods, and mixed Hodge structures. It formulates a parametric representation with the first and second Symanzik polynomials, interprets them via period matrices and Green functions, and situates Feynman integrals within a relative cohomology framework. A concrete algorithm for the Picard–Fuchs equation of all-equal-mass banana graphs is provided, with explicit results for the one- and two-loop cases and discussions of elliptic and polylogarithmic structures, including Mahler measures and regulator mathematics. Overall, it highlights how Hodge-theoretic and motivic techniques illuminate the analytic structure of multi-loop amplitudes and offer systematic tools for deriving differential equations and special-function solutions.

Abstract

This expository text is an invitation to the relation between quantum field theory Feynman integrals and periods. We first describe the relation between the Feynman parametrization of loop amplitudes and world-line methods, by explaining that the first Symanzik polynomial is the determinant of the period matrix of the graph, and the second Symanzik polynomial is expressed in terms of world-line Green's functions. We then review the relation between Feynman graphs and variations of mixed Hodge structures. Finally, we provide an algorithm for generating the Picard-Fuchs equation satisfied by the all equal mass banana graphs in a two-dimensional space-time to all loop orders.

Figures (5)

• Figure 1: The nested structure of the contours of integration for the variable $v^-_i$ corresponding to the ordering $0<v^+_2<v^+_3<\cdots<v^+_{n-2}<1$ of the $v_+$ variables.
• Figure 2: Examples of $\varphi^3$ vacuum graphs at (a) two-loop order, (b) and (c) at three-loop order.
• Figure 3: Graph for the banana graph with $n$ propagators.
• Figure 4: (a) One-loop $n$-point graph and (b)-(c) two-loop four-point graphs.
• Figure 5: After blowup, the coordinate triangle becomes a hexagon in $P$ with three new divisors $D_i$. The elliptic curve $X_\circleddash=\{\mathcal{F}_2(x,y;t)=0\}$ now meets each of the six divisors in one point.