New Physics implications and searches at LHCb

The Standard Model of Particle Physics (henceforth, SM) is a very successful theory. Nevertheless, it fails to explain some important questions present in nature. Therefore, a new model beyond the SM is needed. This is often known as New Physics, NP. Some of these alternatives are Supersymmetry (SUSY) and models with extra dimensions. This thesis represents a comprehensive study of NP searches and its implications, performed in a threefold manner. Firstly, a study of NP implications in kaon physics is presented, through the $K_S^0 \rightarrow \pi^0 \mu^+ \mu^-$ sensitivity study at LHCb and phenomenological studies on $K_S^0 \rightarrow \mu^+ \mu^-$. The former serves as important input for the SM prediction of the $K_L^0 \rightarrow \pi^0 \mu^+ \mu^-$ branching fraction, that is a relevant channel in models with extra dimensions. Concerning the latter, it plays a key role in scenarios where flavour violation does not originate from the CKM matrix, \textit{non-minimal Flavour Violating} (non-MFV), where it carries information complementary to $B_{s(d)}^0 \rightarrow \mu^+ \mu^-$. A phenomenological study is shown under the simplest supersymmetrical extension of the SM, the \textit{Minimal Supersymmetric Standard Model} (MSSM). Secondly, a review of other $\Delta F = 2$ transitions is carried out. This is done via the measurement of the weak mixing phase $\phi_s$ in $B_s^0 \rightarrow J/\psi \phi$ at LHCb, accompanied by a phenomenological study of this and more physics observables. The weak mixing phase $\phi_s$ measures the $CP$ violation that happens in the interference between mixing and decay. The SM prediction for the $B_s^0 \rightarrow J/\psi \phi$ channel is very precise, making it a golden mode where to measure this observable. As for the phenomenological study, several physics observables, such as $\phi_s$ and the Electric Dipole Moments (EDMs), are computed under the MSSM paradigm. To finalize, a more general study is done within the \textit{minimal Anomaly Mediated Supersymmetry Breaking}, mAMSB, and the \textit{Constrained Minimal Supersymmetric Standard Model}, CMSSM. In these scenarios, the prospects for the detection of dark matter, as well as for the prediction of the $(g-2)_{\mu}$ anomaly and flavour observables are presented, together with the available parameter space in light of the present experimental constraints.