Espressioni Analitiche per ampiezze ad un loop che contribuiscono al processo pp → t t_bar j all'ordine O(epsilon^2)

The top quark mass is a very important free parameter of the Standard Model, as it plays a crucial role in Standard Model precision tests. The value can be extracted from high energy collider experiments where a top quark pair is produced. To obtain a precise picture of this process, we must understand the radiation patterns that appear in association with the top quark, and obtain fully differential distributions that may be compared with theoretical predictions. The theoretical description for the production of a top quark pair in association with a QCD jet is a key observable which must be improved in order to match the experimental uncertainties. The technology for the computation of the scattering amplitude ingredients necessary to take theory predictions to next-to-next-to-leading order in perturbative QCD is currently missing. Of particular importance is the understanding of the loop integrals and special function basis that can be used to describe the fixed order amplitudes. A first step in this direction is to study the one-loop integrals and amplitudes to higher order in the dimensional regularisation parameter ϵ where we encounter pentagon integrals for the first time. An analytic description of the special function basis would enable a fast and efficient numerical evaluation. In this thesis we study the analytic form of the pentagon integrals appearing in pp → tt̄j using the method of differential equations. By identifying a ‘dlog’ form of the differential equation for the master integrals, we are able to provide a representation of the expansion in terms of Chen iterated integrals up to transcendental weight four. We are then able to find a Symbol level representation of the one-loop helicity amplitudes in which the universal infrared and ultraviolet pole structure is identified analytically.