Bounded Collection of Feynman Integral Calabi-Yau Geometries

We define the rigidity of a Feynman integral to be the smallest dimension over which it is nonpolylogarithmic. We prove that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L - 1) at L loops provided they are in the class 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 phi(4) theory that saturate our predicted bound in rigidity at all loop orders.