\begin{figure}
\centering
\def\svgwidth{\textwidth}
\input{Images/pdf/na62/tuttona62-tex.pdf_tex}
\caption{Sketch of the full NA62 detector. The CEDAR (KTAG), 8 out of 12 LAV stations, the LKr and SAC calorimeters and the CHOD used for NA48 are already installed on site. The CHANTI, the STRAW spectrometer, the RICH, the IRC calorimeter and the muon vetoes are currently under construction, while the GTK is in a phase of advanced design \cite{newson}. See \chaptername~\ref{ch:exp-setup} for a brief description of the above subdetectors.}\label{img:na62:all-detectors}
\end{figure}

The presence of two neutrinos and of a single charged track in the final state makes NA62's goal a challenging precision measurement, requiring hermetic background rejection as well as an excellent detector system for particle identification, tracking, calorimetry and spectrometry. The signature of a $K^+ \rightarrow \pi^+ \nu \bar{\nu}$ decay consists in one and only one charged track. Any other event for which only one charged track is detected contributes to the background.\\

The $K^+ \rightarrow \pi^+ \nu \bar{\nu}$ process features only three measurable quantities, which are the momenta of the kaon and of the charged decay product, and the laboratory frame angle between the two. It is convenient to construct the squared missing mass to the kaon and the measured decay product, and use it as a discriminating kinematic variable:
\begin{align}
m^2_{\rm miss} &= {\left( P^{\mu}_K - P^{\mu}_{\pi} \right)}^2\\
&= {\left(E_K - E_\pi \right)}^2 - \left(P_K^2 + P_\pi^2 -2|P_K||P_\pi| \cos \theta_{K\pi} \right)\notag
\end{align}
where $P_K$ and $P_\pi$ are the momenta of the decaying kaon and of the pion, $E_K$ and $E_\pi$ are their energies and $\theta_{K\pi}$ is the decay angle in the laboratory frame, i.e. the angle between the kaon and pion tracks.\\

The missing mass is computed under the assumption that the detected decay product is a pion. The $m^2_{\rm miss}$ variable allows to separate the  signal from the most important background processes, as shown in the first panel of \figurename~\ref{img:m2miss_regions}. The ``fiducial'' signal region, i.e. the $m^2_{\rm miss}$ interval allowed for data analisys, is split into two parts, excluding the region dominated by the $K^+ \rightarrow \pi^+ \pi^0$ process. In addition, upper and lower limits are set in order to isolate the $\mu^+ \nu$ and $3\pi$ channels. These three channels account for 92\% of the $K^+$ decay events \cite{PDBook}.\\

%The distributions of \figurename~\ref{img:m2miss_regions} do not take the experimental errors into account, due to the finite resolution of the NA62 detector. 
%Background rejection will therefore rely on both kinematic constraints, and PID/VETO systems.\\

The second panel of \figurename~\ref{img:m2miss_regions} gathers missing mass spectra for background modes which are not kinematically constrained. These modes include radiative versions of the channels described above, and $3-$ and $4-$body semi-leptonic channels. In these cases, the missing mass distributions overlap that of the signal; hence the only way to push background rejection to the needed limits for these processes is to employ reliable particle identification (PID) and VETO systems.\\

\begin{table}
\centering
\begin{tabular}{ccc}
\hline
\hline
Decay mode & B. R. & Rejection\\
\hline
$\mu^+ \nu_\mu$ & 63\% & Kinematics + $\mu$ PID\\
$\pi^+ \pi^0$ & 21\% & Kinematics + $\gamma$ Veto\\
$\pi^+ \pi^+ \pi^-$ & 6\% & Kinematics + $\pi^\pm$ Veto\\
$\pi^+ \pi^0 \pi^0$ & 2\% & Kinematics + $\gamma$ Veto\\
$\pi^0 e^+ \nu_e$ & 5\% & $e$ PID + $\gamma$ Veto\\
$\pi^0 \mu^+ \nu_\mu$ & 3\% & $\mu$ PID + $\gamma$ Veto\\
\hline
\hline
\end{tabular}
\caption{Main background channels for the NA62 experiment.}\label{tab:modes}
\end{table}

\tablename~\ref{tab:modes} lists the most frequent $K^+$ decay modes, together with the respective rejection techniques.

\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Images/pdf/na62/m2miss_regions_bgs}
\caption{Distribution of the recostructed squared missing mass
% reconstructed from the generated kaon and the 
resulting from the decay of the generated kaon into the
detected charged daughter, under the assumption that the latter is a pion. The solid line represents the $K^+ \rightarrow \pi^+ \nu \bar{\nu}$ signal in both plots. The first sketch shows the shape of the kinematically constrained backgrounds, which are also the channels with the largest branching ratios. The second plot shows the other main background processes for which the reconstructed missing mass overlaps that of the signal. The distributions shown here were computed %negleting
without taking into account the errors due to the finite resolution of the NA62 detector \cite{Ceccucci:1070584}. The darkened areas display the current choice for the fiducial signal regions.}\label{img:m2miss_regions}
\end{figure}