Charm CPV and rare decays at LHCb

Charm physics has been playing all along a role in particle physics, by contributing to the for-mulation of the Standard Model (SM) as it is known nowadays. The level of attention on it has tremendously increased in recent years because of the first experimental observations of the slow mixing rate of the $D^0 − \overline{D}^0$ flavour oscillations, providing definitely a full range of probes, entirely complementary to the $B$ and $K$ mesons, for mixing and $CP$- violation. In fact the charm quark is the only up-type quark that manifests flavour oscillations. Only in recent years it has been possible to collect huge and very clean samples of $D$ meson decays, several orders of magnitude larger in size than in the past, allowing also for the first time approaching the small SM expectations for $CP$-violation below the $10^{−3}$ level. Thus, the dynamics of the charm quark can be probed for the presence of New Physics with negligible SM “background”, since any generic non-SM contribu-tion would naturally carry additional $CP$-violating phases, which could enhance the observable $CP$-violation relative to SM predictions. It is worth to mention that $D^0 − \overline{D}^0$ mixing can proceed in the SM through a double weak boson exchange (short distance contributions) or through inter-mediate states that are accessible to both $D^0$ and $\overline{D}^0$ (long distance contributions). The potentially large long distance contributions are non–perturbative and therefore difficult to estimate from the theory, however the interplay of many and even more precise experimental measurements, that only recently approached the desired precision, along with the foreseen theoretical improvements in calculating such amplitudes, should allow constraining theoretical uncertainties in the next and far future increasing the sensitivity to possible contributions from new processes and particles at mass scales beyond the reach of direct searches. The LHCb experiment at the Large Hadron Collider is currently the main player and this brief write-up covers the most recent LHCb results on ${CP}$-violation in the charm sector and on the search of rare charm decays.