This work presents some essential techniques that constitute the modern strategy for computing scattering amplitudes. It begins with an introductory chapter to fill the gap between a standard QFT course and the latest developments in the field. The author then tackles the main bottleneck: the computation of the loop Feynman integrals. The most efficient technique for their computation is the method of the differential equations. This is discussed in detail, with a particular focus on the mathematical aspects involved in the derivation of the differential equations and their solution. Ample space is devoted to the special functions arising from the differential equations, to their analytic properties, and to the mathematical techniques which allow us to handle them systematically. The thesis also addresses the application of these techniques to a cutting-edge problem of importance for the physics programme of the Large Hadron Collider: five-particle amplitudes at two-loop order. It presents the first analytic results for complete two-loop five-particle amplitudes, in supersymmetric theories and QCD. The techniques discussed here open the door to precision phenomenology for processes of phenomenological interest, such as three-photon, three-jet, and di-photon + jet production.sses of phenomenological interest, such as three-photon, three-jet, and di-photon + jet production.

This work presents some essential techniques that constitute the modern strategy for computing scattering amplitudes. It begins with an introductory chapter to fill the gap between a standard QFT course and the latest developments in the field. The author then tackles the main bottleneck: the computation of the loop Feynman integrals. The most efficient technique for their computation is the method of the differential equations. This is discussed in detail, with a particular focus on the mathematical aspects involved in the derivation of the differential equations and their solution. Ample space is devoted to the special functions arising from the differential equations, to their analytic properties, and to the mathematical techniques which allow us to handle them systematically. The thesis also addresses the application of these techniques to a cutting-edge problem of importance for the physics programme of the Large Hadron Collider: five-particle amplitudes at two-loop order. It presents the first analytic results for complete two-loop five-particle amplitudes, in supersymmetric theories and QCD. The techniques discussed here open the door to precision phenomenology for processes of phenomenological interest, such as three-photon, three-jet, and di-photon + jet production.

Lectures in Scattering Theory discusses problems in quantum mechanics and the principles of the non-relativistic theory of potential scattering. This book describes in detail the properties of the scattering matrix and its connection with physically observable quantities. This text presents a stationary formulation of the scattering problem and the wave functions of a particle found in an external field. This book also examines the analytic properties of the scattering matrix, dispersion relations, complex angular moments, as well as the separable representation of the scattering amplitude. The text also explains the method of factorizing the potential and the two-particle scattering amplitude, based on the Hilbert-Schmidt theorem for symmetric integral equations. In investigating the problem of scattering in a three-particle system, this book notes that the inapplicability of the Lippman-Schwinger equations can be fixed by appropriately re-arranging the equations. Faddeev equations are the new equations formed after such re-arrangements. This book also cites, as an example, the scattering of a spin-1/2 particle by a spinless particle (such as the scattering of a nucleon by a spinless nucleus). This text is suitable for students and professors dealing with quantum mechanics, theoretical nuclear physics, or other fields of advanced physics.

Abstract: In this thesis, we develop and apply the D-dimensional numerical unitarity method for the computation of multi-loop scattering amplitudes, which are the key building blocks for precise theoretical predictions for the ongoing physics program of the Large Hadron Collider experiment at CERN. We formulate a framework for consistently implementing dimensional regularization in numerical approaches, which in particular can be applied for efficient numerical evaluations of dimensionally-regularized helicity amplitudes with external fermions. We present the next-to-leading order QCD predictions for the production of W boson in association with a pair of bottom quarks and light jets at the Large Hadron Collider, which is an irreducible background to the the studies of decays of the Higgs boson to a pair of bottom quarks. We include all the effects of the non-vanishing bottom-quark mass. To obtain the one-loop matrix elements required for the process, we implement an extension of the numerical unitarity method to massive particles in the loop. We present the first benchmark values and analytic form of all two-loop five-parton helicity amplitudes in QCD in the leading-color approximation. The analytic expressions are reconstructed from exact numerical evaluations with the numerical unitarity method. Our results provide all two-loop amplitudes required for the calculation of next-to-next-to-leading order QCD corrections to the production of three jets at hadron colliders in the leading-color approximation.

"The computation of massless 5-point 2-loop scattering amplitudes requires the analytical expressions for about 32400 master integrals. In this thesis, we present a method to evaluate all these integrals from a much smaller set corresponding to a fix ordering of the external particles. The integration constants for different leg orderings are determined by using multiple scaling limits of the associated (canonical) differential equations' alphabet letters. The choice of scaling limits is constrained by the requirement the paths they describe in the kinematic space are generated only by harmonic polylogarithms (HPLs). Such constraint ensures using scaling limits makes the analytic evaluation of the solutions fast and simple. Our method is suitable to show the existence of a web of computational highways in the kinematic space relating all evaluated master integrals"--