In the previous sections I examined the simple analysis chain I have used in order to extract physics information from the RICH data.
The relevant parameters used are: \cite{na62-tdr}
\begin{itemize}
\item $\theta_{\rm rich} = +0.00175571$~\si{\radian}
\item $P_{\rm kick} = -270$~\si{\mega\electronvolt/\textit{c}}
\item $\theta_K = +0.0012$~\si{\radian}
\end{itemize}

Let us summarize the outcome of the analysis.
\begin{itemize}
\item The velocity of the particle is computed from the radius of the detected ring as of Eqn.~\ref{eqn:rich-p}, and does not undergo any further modification.
\item The first adjustment performed consists in a hit-by-hit shift of the PMT coordinates, needed in order to take the mirrors tilt into account.
\item Once the hits are repositioned, we can fit a ring to the data. We then change the $x$ coordinate of the found ring centre according to Eqn.~\ref{eqn:center-shift}.
\item We now use the $x$ and $y$ coordinates to compute the $\theta_x,\theta_y$ components of the particle direction (Eqns.~\ref{eqn:theta-x}, \ref{eqn:theta-y}). Of these, $\theta_x$ is propagated back according to Eqn.~\ref{eqn:theta-x-final}.
\end{itemize}
%The velocity of the particle is computed from the radius of the detected ring as of Eqn.~\ref{eqn:rich-p}, and does not undergo any further modification.\\
%
%The first adjustment performed consists in a hit-by-hit shift of the PMT coordinates, needed in order to take the mirrors tilt into account.\\
%
%Once the hits are repositioned, we can fit a ring to the data. We then change the $x$ coordinate of the found ring centre according to Eqn.~\ref{eqn:center-shift}.\\
%
%We now use the $x$ and $y$ coordinates to compute the $\theta_x,\theta_y$ components of the particle direction (Eqns.~\ref{eqn:theta-x}, \ref{eqn:theta-y}). Of these, $\theta_x$ is propagated back according to Eqn.~\ref{eqn:theta-x-final}.\\

We finally obtain the decay angle:
\begin{align}
&\theta_{K\pi}^2 = {\theta_x}_{(K\pi)}^2 + \theta_y^2
\end{align}

\figurename~\ref{img:reco-res} shows the accuracy achieved in the reconstruction of $\beta_\pi$ and $\theta_{K\pi}$.
% achieved by means of the simple analysis process presented here. 
The decay angle is reconstructed with a resolution $\sigma(\theta_{K\pi}) \approx 650$ \si{\micro\radian} (\figurename~\ref{img:reco-res-angle}). On the other hand, the distribution of $\beta$ (\figurename~\ref{img:reco-res-beta}) shows a slight asymmetry towards values higher than the true speed of the particle. This is probably due to an uneliminable effect of the discrete binning of photomultipliers. The asymmetry arises as the \v{C}erenkov light emission angle is generally overestimated, as it can be inferred from \figurename~\ref{img:reco-res-cherangle}. In fact, the distributions shown here were obtained by use of the Taubin algorithm for circle fitting, described in Appendix~\ref{app:taubin-algo}, which is one of the most robust single-ring fit procedures available with respect to the radius of the circle. Quoting the RMS of the distribution as a measurement of the reconstruction accuracy, we may state that the resolution in $\beta$ is approximately $\sigma(\beta) \approx 5 \times 10^{-6}$.\\

\begin{figure}
\centering
\subfigure[Decay angle resolution]{\includegraphics[width=0.49\textwidth]{Images/pdf/L0algo/angle_res}\label{img:reco-res-angle}}
\subfigure[Particle speed resolution]{\includegraphics[width=0.49\textwidth]{Images/pdf/L0algo/beta_res}\label{img:reco-res-beta}}
\caption{RICH reconstruction resolution.}\label{img:reco-res}
\end{figure}

Let us finally discuss a few critical aspects concerning this analysis. At the beginning of this section, the magnetic kick due to the spectrometer dipole was computed assuming the detected particle is a $\pi^+$, therefore using its mass in Eqn.~\ref{eqn:theta-kick}. In fact, most events without pions will be rejected by the hardware L0 trigger. However, on some occasions we might still wrongly assume a particle is a pion and propagate it with a wrong formula. This would not in any way affect the efficiency for the signal of the trigger presented in this thesis, since events without charged pions would be discarded during the stage of offline analysis. Besides, the Montecarlo study reported in this work does not examine background channels different from $K^+ \rightarrow \pi^+ \pi^0$, as the analysis process involved would be substantially different.

\begin{figure}
\centering
\includegraphics[width=\textwidth]{Images/pdf/L0algo/cherenkov_angle_2}
\caption{Reconstruction resolution on the \v{C}erenkov light emission angle. An asymmetry is clearly visible between RICH reconstructed data and the ``Montecarlo truth''.}\label{img:reco-res-cherangle}
\end{figure}