Complexity of Feynman Integrals; The relation between Feynman integrals and the o-minimal structure of sub-Pfaffian sets

Abstract

Dimensionally regulated Feynman integrals are the cornerstone of all perturbative calculations in Quantum Field Theory. They are known to exhibit a rich mathematical structure. Recently it was shown that Feynman integrals, as functions of the masses and the in- and out-going momenta, are tame functions, in the sense of being definable in an o-minimal structure. O-minimal structures are a concept from mathematical logic encapsulating tameness properties. Some o-minimal structures, like the semi-algebraic sets, have a natural notion of complexity, given by a format and a degree. In the mentioned example the number of variables and the sum of degrees of the polynomials involved in defining these sets. Explicit bounds on the topological complexity of these sets can then be given.\\ Another example of a structure admitting a natural notion of complexity is the structure of restricted sub-Pfaffian sets. These are unions of projections of sets defined by polynomial equations and inequalities involving Pfaffian functions. These are functions that are defined as polynomials in the variables $x_1, ..., x_l$ and analytic functions $f_1, ..., f_r$, satisfying a special system of first order differential equations.\\ In this thesis we will explore the connection between Feynman integrals and Pfaffian functions. We present a number of examples of Feynman integrals that are Pfaffian functions. Our main result is that both the real and imaginary parts of one-loop Feynman integrals Pfaffian functions.