Asymptotic expansions of Feynman integrals

Feynman integrals are integrals over the momenta of virtual (unobservable) particles, and their evaluation plays a fundamental role in achieving high-precision theoretical predictions in subnuclear physics, as well as in testing the Standard Model. In scattering experiments and particle decays, the main measurable quantities are the cross section and the decay rate. In both quantum mechanics and relativistic quantum theory, these quantities are determined through Fermi’s Golden Rule, which involves the transition amplitude. The evaluation of the latter, in turn, requires the computation of Feynman integrals. Depending on the process under consideration and on the perturbative order, such integrals may depend on the masses and momenta of several particles, making their exact calculation particularly challenging. However, in many situations the mass scales involved differ by several orders of magnitude. In such cases, rather than pursuing an exact evaluation, it is often more convenient to approximate them through suitable asymptotic expansions in the ratio of a small mass scale to a large one. In this thesis, after outlining the physical context and presenting the main methods for solving the simplest Feynman integrals, we focus on the so-called method of regions (developed by Smirnov and Beneke in 1997). This technique is then applied to the study of the self-energy of a scalar field at one loop in different kinematic configurations.