Magnus series and deflation for Feynman integrals

This thesis aims at presenting the mathematical methods of Magnus Series Expansion and Eigenvalue Deflation, to improve the analytic evaluation of multi-loop Feynman integrals through the Differential Equations approach. In the first part of the work, the formalism of multi-loop integrals in dimensional regularization and the relations among them are introduced, discussing how integration-by-parts identities, Lorentz invariance identities and Euler’s scaling equation can be used to derive first order differential equations for Feynman integrals. The analytic properties of the solutions are then investigated by introducing the concepts of iterated integrals and uniform transcendentality. The central part of the thesis is dedicated to the mathematical systematization of the Magnus Series Expansion and of the Eigenvalue Deflation, employed to addreess the determination of the solution of a system of differential equations by means of algebraic techniques. An original derivation of the Eigenvalue Deflation method, based upon the operations of deflation and balance transformation, is here presented. The final part consists of the detailed application of both Magnus Series and Eigenvalue Deflation methods to the one-loop box diagram, to the two-loop ladder diagram and to the nontrivial three-loop ladder diagram, which enter the evaluation of 2->2 scattering process among massless partons up to the next-to-next-to-next leading order in Quantum Electrodynamics and Quantum Chromodynamics. The analytic expressions of the corresponding integrals, as well as the ones corresponding to their subdiagrams, previously known in the literature, are hereby re-derived one order higher and in a simpler way. The presented approaches can be applied in a wider context, ranging from high-precision collider phenomenology to the study of formal aspects of scattering amplitudes in gauge theories.