The aim of this work is to further reduce the rate of $ K^+ \rightarrow \pi^+ \pi^0 $ events fed to the L1 farm.
%The target background for this work is the two-body decay $ K^+ \rightarrow \pi^+ \pi^0 $.
This is the most prominent background after the standard L0 trigger selection, as shown in \tablename~\ref{tab:bg_rej_mc}.\\

Among the kinematically constrained backgrounds (see \chaptername~\ref{ch:na62-strategy}), $ K^+ \rightarrow \pi^+ \pi^0 $ is the only process falling in the same missing mass region as the signal. This $2-$body process features a closed kinematics and could therefore be detected as a Dirac delta in the spectrum of the missing mass in the decay of the beam kaon into $\pi^+$ (neglecting resolution effects). When the distribution of experimental errors is convoluted with the expected spectrum for this process, the squared missing mass $m^2_{\rm miss}$ distributes as a Gaussian around the squared mass of the neutral pion.\\

%Therefore, the signal region is split into two sub-regions on the two sides of the $M^2_{\pi^0}$ line, referred to as $R_1$ and $R_2$ in the following.
%
%At a theoretical level, this process features a closed kinematics and could be detected as a Dirac delta in the spectrum of the missing mass to the charged beam kaon and its decay product $\pi^+$. \figurename~\vref{img:trigintro:mmiss} shows the expected missing mass spectra for the ``signal'' process $K^+ \rightarrow \pi^+ \nu \bar{\nu}$, and for the major backgrounds (see RIFERIMENTO TABELLA MODI DECADIMENTO K+ -- DA FARE).\\

From an experimental point of view, the signature of the $ K^+ \rightarrow \pi^+ \pi^0 $ process consists of one or more charged tracks detected (to account for the charged pion and the eventual conversion of photons from $\pi^0$ decay).\\

This process may yield multiple \v{C}erenkov rings in case of:
\begin{itemize}
\item inelastic scattering on the material of the beam pipe or of the upstream detectors
\item Dalitz decay of the neutral pion: $\pi^0 \rightarrow e^+ e^- \gamma$
\item photon conversion $\gamma \rightarrow e^+ e^-$ in a sufficiently upstream region.
\end{itemize}

The identification of a \v{C}erenkov ring on the RICH detector provides information on both the direction and the velocity of the particle that has crossed the detector volume. In principle, the relationship between these quantities can tell whether a detected pion comes from a 2-body decay or not. Let us go through the kinematics involved in the $K^+ \rightarrow \pi^+ \pi^0$ decay.\\

Let $P_K$, $P_\pi$ and $P_0$ be the four-momenta of the kaon, of the charged pion and of the neutral pion respectively.
In the centre of mass frame, the equality $P_K = P_\pi + P_0$ reads
\begin{align}
&\left\{\begin{array}{l}
m_K = E_{\pi}^* + E_0^*\\
\vec{0} = \vec{P_{\pi}^*} + \vec{P_0^*}
\end{array}\right.\label{eqn:4momenta-sum}
\end{align}
where $E_{\pi}^*$ and $E_0^*$ are the centre of mass energies of the charged and neutral pion respectively. The second relation can be written as
\begin{align}
&{\left( E_{\pi}^* \right)}^2 - m_{\pi}^2 = {\left( E_{0}^* \right)}^2 - m_{0}^2
\end{align}
and therefore, using the first of Eqns.~\ref{eqn:4momenta-sum}, we find the centre of mass energy of the $\pi^+$, that is single-valued:
\begin{align}
&E_{\pi}^* = \frac{m_K^2 + m_0^2 - m_{\pi}^2}{2 m_K}
\end{align}

Now consider the invariant product $P_K^{\mu} {P_{\pi}}_{\mu}$. In the rest frame of the decaying kaon it reads
\begin{align}
&{\left(P_K^{\mu} {P_{\pi}}_{\mu}\right)}^* = m_K E_{\pi}^* - \vec{0} \cdot \vec{P_{\pi}^*} = m_K E_{\pi}^*\label{eqn:pp1}
\end{align}
On the other hand, in the laboratory frame we have to consider the momenta of both particles:
\begin{align}
&P_K^{\mu} {P_{\pi}}_{\mu} = E_K E_\pi - \vec{P_K} \cdot \vec{P_\pi} = E_K E_\pi - |\vec{P_K}|  |\vec{P_\pi}| \cos \theta_{K\pi}\label{eqn:4mom-betatheta}
\end{align}
where $\theta_{K\pi}$ is the angle comprised between the $K^+$ and the $\pi^+$ tracks.\\

The position of the \v{C}erenkov ring of the pion provides a measurement of the direction of the pion, which can be related to $\theta_{K\pi}$ by means of the simple backwards propagation steps described in \chaptername~\ref{ch:reco}, and of the velocity $\beta_\pi$ of the pion. Therefore, if we assume that the detected particle is really a charged pion, we can compute its energy and momentum:
\begin{align}
&E_\pi = \gamma_\pi m_\pi = \frac{m_\pi}{\sqrt{1-\beta_{\pi}^2}}\label{eqn:e_pi}\\
&|\vec{P_\pi}| = \beta_\pi E_\pi = \frac{\beta_\pi m_\pi}{\sqrt{1-\beta_{\pi}^2}}\label{eqn:p_pi}
\end{align}
%\begin{itemize}
%\item DISTRIBUZIONE IMPULSI including very large momentum $\pi^+$
%\item DISTRIBUZIONE VERTICI
%\item one or more charged tracks detected, depending on the detectors geometrical acceptance, to account for the charged pion and the neutral pion decay photons through $\gamma \rightarrow e^+e^-$
%\end{itemize}
%
%\begin{figure}
%\centering
%\includegraphics[width=0.6\textwidth]{Images/pdf/trigIntro/mmiss}
%\caption{boh}\label{img:trigintro:mmiss}
%\end{figure}
If we substitute Eqns.~\ref{eqn:e_pi} and \ref{eqn:p_pi} in Eqn.~\ref{eqn:4mom-betatheta}, and match the two expressions~\ref{eqn:pp1} and~\ref{eqn:4mom-betatheta} for the $P_K^{\mu} {P_{\pi}}_{\mu}$ invariant, we find that the velocity of the pion is constrained by the laboratory-frame decay angle $\theta_{K\pi}$ through the relation
\begin{align}
&\theta_{K\pi} ( \beta_\pi )\bigg|_{K^{+} \rightarrow \pi^+ \pi^0} = \arccos \left[ \frac{1}{\beta_\pi |\vec{P_K}|} \left(E_K - \frac{m_K}{m_{\pi}}  E_{\pi}^* \sqrt{1 - \beta_{\pi}^2} \right)\right]\label{eqn:thetabeta}
\end{align}
that arises from the closed kinematics of 2-body decays.\\

In principle, $E_K$ and $| \vec{P_K} |$ are beam parameters that are known to a precision of 1\%; therefore, a RICH measurement alone would allow to check if Eqn.~\ref{eqn:thetabeta} holds. In case of a positive result, a $K^+ \rightarrow \pi^+ \pi^0$ event could be identified and safely discarded.