A modern approach to Feynman Integrals and Differential Equations

In this Thesis we discuss recent ideas concerning the evaluation of multi-loop Feynman Integrals in the context of Dimensional Regularization. In the first part we study relations fulfilled by Feynman Integrals, with a particular focus on Integration By Parts Identities (IBPs). We present the latter both in the standard momentum space representation, where we essentially we integrate a set of denominators over the loop momenta, and in Baikov representation, in which denominators are promoted to integration variables, and the Gram determinant of the whole set of loop and external momenta, referred to as Baikov Polynomial, emerges as a leading object. IBPs in Baikov representation naturally lead to the study and the implementation of concepts and algorithms developed in Computational Algebraic Geometry, such as Sygyzies. We present a Mathematica code devoted to IBPs generation in Baikov representation. In the second part we focus on the Method of Differentil Equations for Feynman Integrals, with a particular emphasis on the algorithm based on the Magnus Exponential to achieve the Canonical Form: in both of them, an underlying algebraic structure arises. We present applications relevant to phenomenology: namely we compute the Mis for the 1-loop box which appear in the e ! e scattering and we obtain the Canonical Form for a 2-loops non plaanar 3-points function, which is part of a wider task regarding the calculation of the 2-loops non planar box which is needed for the qq ! tt process. In the last part we analyze the role of Cut Integrals as solutions of homogeneous Differential Equations, and their implementation in Baikov representation. Working on an explicit example, we show how different IBPs-compatible integration regions lead to different solutions for a higher order Differential Equation.