Reflexive Polytopes and the Convergence of Feynman Integrals

In the parametric representation, Feynman integrals can be viewed as Euler integrals defined by the Symanzik polynomials of a graph. The convergence properties of these integrals are intimately tied to the combinatorial geometry of their associated Newton polytopes; specifically, finiteness is guaranteed when the polytope contains interior points. We present a classification of Feynman integrals associated with polytopes containing a unique interior point, identifying a subset that are reflexive. Our results show that such reflexive polytopes are surprisingly scarce within the space of Feynman graphs. We conclude by computing several infinite families of these integrals and exploring their connections to mirror symmetry and toric geometry. This is based on joint work with Leonardo de la Cruz and Pavel Novichkov.

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