New Physics in the Flavour Sector in the presence of Flavour Changing Neutral Currents

Flavour-Changing-Neutral-Currents (FCNC) play an important r\^ ole in testing the Standard Model (SM) while probing the possibility of having New Physics beyond the SM. In the SM, FCNC are forbidden at three level, but arise through calculable one-loop contributions. We review some of the features of FCNC in two examples of minimal extensions of the SM. In the first example, we consider an extension of the SM consisting of the addition of one vector-like quark either of the up-type ($Q=2/3$) or the down type ($Q=- 1/3$). In this extension there are non-vanishing but naturally suppressed Z-mediated FCNC at tree level. In the second example, we discuss extensions of the SM with two Higgs doublets, without the assumption of natural flavour conservation, giving rise to Higgs mediated FCNC. The existence of strict experimental limits on processes sensitive to Higgs FCNC requires a strong suppression of these currents. We present scenarios resulting from discrete symmetries where all new flavour structures in the quark sector are parametrized by elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, together with the ratio of vacuum expectation values of the Higgs doublets in the Higgs basis defined by the symmetry. We extend these scenarios to the leptonic sector with the Pontecorvo-Maki-Nakagawa-Sakata matrix playing a r\^ ole similar to the CKM matrix in the quark sector.