A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries

TL;DR

The paper introduces the concept of rigidity for Feynman integrals and proves a universal bound of $2(L-1)$ on the rigidity of massless four-dimensional integrals at $L$ loops, with maximal rigidity achieved by marginal graphs having $(L+1)D/2$ propagators. It shows that marginal integrals generically involve Calabi–Yau geometries whose dimension matches the integral's rigidity, providing explicit infinite families (tardigrades, paramecia, amoebas) and finite four-dimensional massless $\phi^4$ examples that saturate the bound. The analysis relies on the Symanzik polynomial formalism, where marginality corresponds to $\mathfrak{F}$ being linear in each Schwinger parameter, leading to integrals that can be viewed as weight-two polylogs over Calabi–Yau manifolds. The work discusses broader implications, conjectures extending the bound, and the relationship between geometry, planarity, and higher-loop structure, proposing a rich geometrical landscape beyond polylogarithms for loop integrals.

Abstract

We define the rigidity of a Feynman integral to be the smallest dimension over which it is non-polylogarithmic. We argue that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show that this bound may be saturated for integrals that we call marginal: those with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman integrals in D dimensions generically involve Calabi-Yau geometries, and we give examples of finite four-dimensional Feynman integrals in massless $φ^4$ theory that saturate our predicted bound in rigidity at all loop orders.