The decays $K^+ \rightarrow \pi^+ \nu \bar{\nu}$ and $K^0_L \rightarrow \pi^0 \nu \bar{\nu}$, the latter being studied by the KOTO experiment in Japan \cite{Iwai:2012qya}, are unique probes to test the Standard Model.\\%lepton universality in weak interactions, flavour conservation in neutral currents or CP violation in charged currents.\\

Both decays are \textit{Flavour-Changing Neutral-Current} (FCNC) processes, a kind of transition that is strongly suppressed in the Standard Model and therefore very sensitive to SM extensions or new Physics scenarios, as summarised in \figurename~\ref{fig:bsm}. This plot reports the branching ratios predicted by the Standard Model and by BSM (Beyond the Standard Model) theories such as the \textit{Constrained Minimal Flavour Violation} effective theory, the \textit{Minimal Supersymmetric Standard Model} and the \textit{Four-Generation} model \cite{flavianet}.  A 10\% measurement of $B.R.(K^+ \rightarrow \pi^+ \nu \bar{\nu})$ alone would rule out a large set of models.\\

Moreover, the above processes feature an exceptional cleanliness and precise theoretical prediction: in the NA62 case, the Standard Model prediction is B.R.$\left( K^+ \rightarrow \pi^+ \nu \bar{\nu}\right) = \left( 7.81 {^{+0.80}_{-0.71}}_{\rm CKM} \pm 0.29 \right) \, 10^{-11} $ \cite{PhysRevD.83.034030}, where the first error accounts for the uncertainty on the CKM matrix elements, and the second one is the pure theoretical uncertainty. Any deviation from this expectation would be an evidence for new Physics. These decay channels are thus extremely sensitive, and represent a powerful tool to probe theories beyond the Standard Model (\figurename~\ref{fig:bsm}) \cite{Mescia:2007kn,Hurth:2008jc}.\\
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Images/pdf/Krare_BSM_new}
\caption{The ``Mescia-Smith'' plot \cite{flavianet} illustrating $K^+ \rightarrow \pi^+ \nu \bar{\nu}$ and $K^0_L \rightarrow \pi^0 \nu \bar{\nu}$ Physics sensitivities. Experimental data on the branching ratios of these ultra-rare decays would allow to put SM and BSM theories at a test.}\label{fig:bsm}
\end{figure}

Both processes arise from a $s \rightarrow d \nu \bar{\nu}$ transition at quark level, and can be described by ``penguin'' Feynman diagrams (1-loop diagrams).
%A hint for new particles intervening in the loop would arise if a discrepancy is detected between experimental data and SM predictions.\\
If a discrepancy were detected between experimental data and SM predictions, this would be a hint for the existence of unknown particles intervening in the loop. We will discuss further in Section~\ref{sect:na62:phys} the theoretical framework behind this kind of transition.\\

In case of agreement with the Standard Model, 
%at the same time
instead, a precise measurement of the branching ratio of this process would 
%reflect on an 
improve the accuracy of the current experimental determination of the $\vert V_{td} \vert$ parameter, which is one of the least precisely known elements of the CKM matrix. This particular measurement would be independent from those already obtained in the analysis of the B system \cite{Jaffe:307897,Flynn:680727,Sergi:1523434}.\\

%Moreover, since $V_{td}$ is one of the two CKM matrix elements encoding CP violation, the study of $K \rightarrow \pi \nu \bar{\nu}$ processes allows to test our understanding of the charge-parity symmetry.