ETD

• amplitude model
• FCNC transition
• SM phenomenology

The purpose of this thesis is to study the B⁺ → K⁺π⁺π⁻ℓ⁻ℓ⁺ decay and provide the
theoretical model for a full amplitude analysis of the differential decay rate of the process. The reason that makes this decay interesting is the quark-level transition that induces the process: a b → sℓ⁻ℓ⁺ Flavour-Changing Neutral Current transition. In the Standard Model such a rare decay can only occur through one-loop electroweak diagrams, rendering it extremely suppressed. Flavour-Changing Neutral Current transitions like this are among the most valuable probes for studying Flavour Physics, due to the comparable sizes of the effects of the Standard Model and New Physics. Recent measurements carried out by LHCb have highlighted anomalies in decay channels mediated by the b → sℓ⁻ℓ⁺ transition. For this reason, the theoretical model for studying this transition in a five-body decay channel will be provided. In fact, it is the first time such a high multiplicity b → sℓ⁻ℓ⁺ decay is studied in an inclusive amplitude analysis. By investigating the numerous multi-body intermediate states, it is possible to classify the decay channels into three topologies, involving one or more resonances, to which a non-resonant channel is added. After computing the five-body phase space, the main effective theories necessary for describing the FCNC transition have been introduced. The parameterization for the matrix elements of the hadronic transition is provided, both in the full theory and in the Heavy-Quark and Large-Energy Effective Theory limit. At the level of the mesons involved, the hadronic transition is described using helicity amplitudes, which allow the decay amplitude to be constructed in the helicity formalism. The differential decay rate for the B⁺ → K⁺π⁺π⁻ℓ⁻ℓ⁺ decay is presented in terms of five angular variables and three invariant masses, so as to directly connect the theoretical predictions with experimental distributions, thus paving the way for future full amplitude analysis of the decay on real data collected by the LHCb experiment.