Leonardo de la Cruz (IPhT, Saclay)

Polytopal aspects of Feynman Integrals

In this talk, I will introduce Newton polytopes and discuss their relevance in analyzing properties of Feynman integrals, whose analytic evaluation is central to many areas of theoretical physics.  I will discuss how Newton polytopes of the integral's Symanzik polynomials can be used to construct IR and UV finite Feynman integrals. We conjecture that a necessary and sufficient condition for finiteness is that all parameter-space monomial exponents lie in the polytope's interior and present an algorithm for finding all such numerators. In the latter part of the talk, I will introduce reflexive polytopes, which arise in the context of mirror symmetry, and show how they emerge as a special subclass within the set of finite Feynman integrals. I will conclude by presenting a direct search for reflexive polytopes associated with Feynman integrals.