According to the distributions presented in the previous sections, the rejection power of the RICH-based trigger algorithm I developed for the NA62 experiment mostly emerges from a selection of the radius of the \v{C}erenkov rings. Therefore, I performed a study on the rejection achievable with the cut
\begin{align}
&R_{c} \leq R_{\rm th}
\end{align}

\tablename~\ref{tb:r_rejection} shows both the signal acceptance and the background rejection for various threshold values.
The first column lists the threshold values $R_{\rm th}$ used.
The second column shows the percentage of signal events that pass the selection, computed as $n_{\rm pass} / n_{\rm tot}$. The signal is selected according to the requirements listed on \tablename~\ref{tb:sigdef}.
The third column shows the $\pi^+\pi^0$ rejection capability of the RICH-based L0 algorithm alone, which in this study consists in the $R_{c} \leq R_{\rm th}$ cut only. The rejection power is computed as $1 - n_{\rm pass}/n_{\rm tot}$.
The ``RICH~$\vert$~L0'' column is the most significant one. It highlights the rejection power of my algorithm rescaled to the background events that have passed the selection performed by the standard hardware L0 trigger.
Finally, the last column shows the combined rejection level achieved by the two trigger algorithms -- RICH-based and standard L0 -- running in parallel.
The last row shows the $\pi^+ \pi^0$ background rejection achieved by the standard hardware L0 trigger described in \chaptername~\ref{sect:na62l0}.\\%, and is the same for every experiment.\\

\begin{table}
\centering
\textbf{Signal definition:}
\begin{tabular}{|c|c|}
\hline
\hline
Event type & $K^+ \rightarrow \pi^+ \nu \bar{\nu}$\T\B\\
\hline
$\pi^+$ momentum along $z$ & 15 to 35 \si{\giga\electronvolt/\textit{c}}\T\B\\
\hline
Decay vertex $z$ position & 105 to 165 \si{\meter}\T\B\\
\hline
\multirow{2}{*}{Square missing mass to the $K^+$ and $\pi^+$} & 0 to 0.01 \si{\giga\electronvolt\tothe{2}/\textit{c}\tothe{4}}\T\B\\
\hhline{~-}
 & 0.026 to 0.068 \si{\giga\electronvolt\tothe{2}/\textit{c}\tothe{4}}\T\B\\
\hline
\hline
\end{tabular}
\caption{Offline analisys requirements for the selection of $K^+ \rightarrow \pi^+ \nu \bar{\nu}$ events in the NA62 experiment.}\label{tb:sigdef}
\end{table}

\begin{table}
\centering%\footnotesize
\begin{tabular*}{\textwidth}{lcccc}
\hline 
\hline 
%$R_{\rm th}$ (cm) & Signal acceptance & L0 rejection & RICH rejection & RICH $\vert$ LO & L0+RICH\T\B \\ 
$R_{\rm th}$ (cm) & Signal efficiency & RICH rejection & RICH $\vert$ L0 & L0+RICH\T\B \\ 
\hline 
17.60 & 92.6 $ \pm $ 0.8 & 63.4 $ \pm $ 0.5 & 70 $ \pm $ 1 & 93.0 $ \pm $ 0.2\T \\ 
17.70 & 96.0 $ \pm $ 0.6 & 61.3 $ \pm $ 0.5 & 68 $ \pm $ 1 & 92.6 $ \pm $ 0.2 \\ 
17.75 & 97.1 $ \pm $ 0.6 & 59.8 $ \pm $ 0.6 & 68 $ \pm $ 1 & 92.5 $ \pm $ 0.2 \\
17.80 & 97.6 $ \pm $ 0.5 & 58.5 $ \pm $ 0.6 & 67 $ \pm $ 1 & 92.3 $ \pm $ 0.2 \\ 
17.85 & 98.3 $ \pm $ 0.4 & 56.8 $ \pm $ 0.7 & 66 $ \pm $ 1 & 92.1 $ \pm $ 0.2 \\ 
\textbf{17.90} & \textbf{98.6 $ \pm $ 0.4} & \textbf{55.2 $ \pm $ 0.8} & \textbf{65 $ \pm $ 1} & \textbf{91.8 $ \pm $ 0.2} \\ 
17.95 & 99.0 $ \pm $ 0.4 & 53.3 $ \pm $ 0.8 & 64 $ \pm $ 1 & 91.5 $ \pm $ 0.2 \\ 
18.00 & 99.5 $ \pm $ 0.3 & 51.3 $ \pm $ 0.8 & 62 $ \pm $ 1 & 91.2 $ \pm $ 0.2 \\ 
18.10 & 99.6 $ \pm $ 0.3 & 47.0 $ \pm $ 1.0 & 59 $ \pm $ 1 & 90.4 $ \pm $ 0.3\B \\ 
%
% with errors
%
%17.60 & 0.926$ \pm $0.008 & 0.634$ \pm $0.005 & \textbf{0.70$ \pm $0.01} & 0.930$ \pm $0.002\T \\ 
%17.70 & 0.960$ \pm $0.006 & 0.613$ \pm $0.005 & \textbf{0.68$ \pm $0.01} & 0.926$ \pm $0.002 \\ 
%17.75 & 0.971$ \pm $0.006 & 0.598$ \pm $0.006 & \textbf{0.68$ \pm $0.01} & 0.925$ \pm $0.002 \\
%17.80 & 0.976$ \pm $0.005 & 0.585$ \pm $0.006 & \textbf{0.67$ \pm $0.01} & 0.923$ \pm $0.002 \\ 
%17.85 & 0.983$ \pm $0.004 & 0.568$ \pm $0.007 & \textbf{0.66$ \pm $0.01} & 0.921$ \pm $0.002 \\ 
%\textbf{17.90} & \textbf{0.986$ \pm $0.004} & \textbf{0.552$ \pm $0.008} & \textbf{0.65$ \pm $0.01} & \textbf{0.918$ \pm $0.002} \\ 
%17.95 & 0.990$ \pm $0.004 & 0.533$ \pm $0.008 & \textbf{0.64$ \pm $0.01} & 0.915$ \pm $0.002 \\ 
%18.00 & 0.995$ \pm $0.003 & 0.513$ \pm $0.008 & \textbf{0.62$ \pm $0.01} & 0.912$ \pm $0.002 \\ 
%18.10 & 0.996$ \pm $0.003 & 0.470$ \pm $0.010 & \textbf{0.59$ \pm $0.01} & 0.904$ \pm $0.003\B \\ 
%
%
% complete numbers
%17.60 & 0.926$ \pm $0.005 & 0.634$ \pm $0.006 & \textbf{0.70$ \pm $0.01} & 0.930$ \pm $0.002\T \\ 
%17.70 & 0.960 & 0.613 & \textbf{0.683$ \pm $0.01} & 0.926 \\ 
%17.75 & 0.971 & 0.598 & \textbf{0.677$ \pm $0.01} & 0.925 \\
%17.80 & 0.976 & 0.585 & \textbf{0.670$ \pm $0.01} & 0.923 \\ 
%17.85 & 0.983 & 0.568 & \textbf{0.660$ \pm $0.01} & 0.921 \\ 
%\textbf{17.90} & \textbf{0.986} & \textbf{0.552} & \textbf{0.648$ \pm $0.01} & \textbf{0.918} \\ 
%17.95 & 0.990 & 0.533 & \textbf{0.638$ \pm $0.01} & 0.915 \\ 
%18.00 & 0.995 & 0.513 & \textbf{0.620$ \pm $0.01} & 0.912 \\ 
%18.10 & 0.996 & 0.470 & \textbf{0.590$ \pm $0.01} & 0.904\B \\ 
%
\hline 
%\hline 
\end{tabular*}
\begin{tabular*}{\textwidth}{>{\centering}p{0.5\textwidth}<{\centering}>{\centering}p{0.5\textwidth}<{\centering}}
\T\B Standard L0 rejection & 76.7 $\pm$ 0.04\tabularnewline
\hline
\hline
\end{tabular*}
\normalsize
\caption{Rejection power and signal acceptance achievable with a cut on the radius of \v{C}erenkov rings. All values are expressed as percentages, and the errors shown are pure statistical uncertainties due to the size of the data samples. The last row shows the $\pi^+ \pi^0$ background rejection achieved by the standard hardware L0 trigger. For this study I used two separate data sets consisting of 10000 $K^+$ decays each. One data set contained only $\pi^+ \nu \bar{\nu}$ events, while the other one consisted of $K^+ \rightarrow \pi^+ \pi^0$ decays.}\label{tb:r_rejection}
\end{table} 

%The boldface numbers in \tablename~\ref{tb:r_rejection} represent a positive result. 
The selection performed by the $R_c \leq R_{\rm th}$ requirement makes use of different observables than the existing hardware-based L0 trigger, and it makes it possible to cut the data rate fed to the following trigger stage by an additional 60\% to 70\%. Such reduction may in turn allow the L1 stage to perform deeper analisys, without stretching its latency or increasing the amount of resources required.\\

The boldface row in \tablename~\ref{tb:r_rejection} is an attempt to select a starting working point for further studies, as a reasonable compromise between signal efficiency and backround rejection. The efficiency for the signal for the quoted $R_{\rm th} = 17.9$~\si{\centi\meter} is (98.6 $\pm$ 0.4)\%, which means that the signal will remain essentially unchanged. The background rejection achieved over the L0 trigger is (64.8 $\pm$ 1.2)\% and the overall $\pi^+ \pi^0$ rejection adds up to (91.8 $\pm$ 0.2)\%.\\

In a subsequent study I tried to estimate the effect of a cut on the $\delta$ variable introduced in Section~\ref{sect:more-rej-delta}. This selection does not add much to that on the \v{C}erenkov radius, as it operates on correlated variables: the faster a pion crosses the RICH detector, the worse its characteristics can be measured. However, I found out that a cut
\begin{align}
&R_c \leq 17.9\; \si{\centi\meter}\\
&\delta \leq 0.0025
\end{align}
yields a maximum $\pi^+\pi^0$ rejection of (71.5 $\pm$ 0.9)\% on particles that passed the standard L0 selection, while a cut
\begin{align}
&R_c \leq 17.9\; \si{\centi\meter}\\
&\delta \leq 0.002
\end{align}
yields a slightly higher rejection of (72 $\pm$ 1)\%, maintaining a signal acceptance of (98.0 $\pm$ 0.4)\%.\\

Finally, I introduced a ``ring fit quality'' selection. A greater background rejection can be expected if there is a possibility of detecting the presence of more than one \v{C}erenkov ring, for the reasons explained in \chaptername~\ref{ch:2body}.\\

The simplest way to check if the hits detected belong to one or more rings is to perform a cut on a $\chi^2$ variable computed on the result of the ring fit. Montecarlo simulations show that a threshold $\chi^2_{th} = 1$ allows to discriminate between clean 1-ring events and multi-ring or noisy events. Higher values of $\chi^2_{th}$ did not sensibly change the result. The following cuts:
\begin{align}
&R_c \leq 17.9\; \si{\centi\meter}\\
&\delta \leq 0.002\\
&\chi^2_{ring} \leq 1
\end{align}
therefore allow to attain (77 $\pm$ 1)\% additional $\pi^+\pi^0$ rejection after the standard L0 cuts, with (96.2 $\pm$ 0.7)\% signal acceptance.\\

These numbers should be interpreted as the best achievable with a RICH-based online trigger, since both the characteristics of the target processes and the acceptance and resolution of this detector are taken into account.