\section{Time dependence of correlated decays with \CP violation}
\label{sec:Appendix_UsefulFormulae}

Here we report the expressions for the time dependence of correlated decays at 
\DzDzbar threshold accounting for possible indirect \CP violation ( i.e. $|q/p| \ne 1$, $\phi \ne 0$).
%
 For the sake of simplicity we assume \CPT is conserved and also 
\CP is conserved in the decay and reserve the possibility of considering 
\CP violation in the decay and \CPT violation for more general sensitivity 
studies. 
%
The phase convention that we use is:
%
\begin{equation}
\CP |\Dz\rangle = |\Dzbar\rangle
\label{eq:CP-PhaseConvention}
\end{equation}
%
so that:
%
\begin{equation}
\lambda_{f_{\CP}}=\frac{q}{p}\frac{\overline{A}_{f_{\CP}}}{A_{f_{\CP}}}=\eta\left|\frac{q}{p}\right|e^{i\phi}
\label{eq:lambdaCP}
\end{equation}
%

 
\par
The final states that we consider are:

\begin{enumerate}

\item \Seta, a \CP eigenstate of eigenvalue $\eta=\pm 1$; 

\item a flavor specific semi-leptonic decays to a final state containing \lpm$\X^{\mp}$ is an inclusive set of all final states;

\item \G, hadronic final state such that both \Dz and
 \Dzbar can decay to it. For example, in charmed mesons, $D^0\to
G^+$ is Cabibbo Favored (CF)  and $D^0\to G^-$ is doubly Cabibbo
suppressed, as in $D^0 \to K^-\pi^+$. This implies that the ratio
of DCS to CA decay rates $R$ is small and the results can be expanded in
terms of this ratio.

\end{enumerate}


The time-dependent decay rates $R_{odd}(f_1,f_2,\Delta t)$ for antisymmetric initial 
state ($C$=odd) into final states $f_1$, $f_2$ are reported below and can be 
derived from Eq.~\ref{eq:DDbarphys_rate} with the substitution
%
\begin{equation}
R_{odd}(f_1,f_2,\Delta t)= \frac{d\Gamma[V_{\text{phys}}(t_1,t_2)\to f_1 f_2]/dt}{e^{-\Gamma |\Delta t|}\mathcal{N}_{f_1 f_2}}
\end{equation}
In the next sections we consider several combinations of final states $(f_1,f_2)$ and we write explicitly the time-dependent and time-integrated rates.  
% 
%
\subsection{Semileptonic decays with $CP$ tag}
\begin{eqnarray}  
\Rodd(\lm\Xp,\Seta;\dt) = \left|\bar A_{l^-X^+}A_{S_\eta}\right|^2 &&\hspace{-0.5cm}\Bigg\{ \left(1+ \left|\frac{q}{p} \right|^2 \right)\cosh(y\Gamma\dt)+\left(1- \left|\frac{q}{p} \right|^2 \right) \cos(x\Gamma\dt) \nonumber \\
&& + 2\eta\left|\frac{q}{p}\right|\cos\phi\sinh(y\Gamma\dt)-2\eta\left|\frac{q}{p}\right|\sin\phi\sin(x\Gamma\dt)\Bigg\}\nonumber \\ 
\Rodd(\lp\Xm,\Seta;\dt) = \left|A_{l^+X^-}A_{S_\eta}\right|^2 &&\hspace{-0.5cm}\Bigg\{ \left(1+ \left|\frac{p}{q} \right|^2 \right)\cosh(y\Gamma\dt)+\left(1- \left|\frac{p}{q} \right|^2 \right) \cos(x\Gamma\dt) \nonumber \\ 
&& + 2\eta\left|\frac{p}{q}\right|\cos\phi\sinh(y\Gamma\dt)+2\eta\left|\frac{p}{q}\right|\sin\phi\sin(x\Gamma\dt)\Bigg\}\nonumber \\ \label{eq:semilep_cp_1}
\end{eqnarray}
%
%
In the limit of $|x\Gamma \dt|\ll 1$, $|y\Gamma \dt| \ll 1$ we obtain:
%
%
\begin{eqnarray}
&&\hspace{-1cm}\Rodd(\lm\Xp,\Seta;\dt)=2\left|\bar A_{l^-X^+}A_{S_\eta}\right|^2\bigg\{\nonumber\\
&& 1+ \eta \left|\frac{q}{p} \right| \left(y\cos\phi-x\sin\phi  \right)(\Gamma\dt) +\left(\frac{y^2}{4}\left(1+\left|\frac{q}{p}\right|^2\right)-\frac{x^2}{4}\left(1-\left|\frac{q}{p}\right|^2\right)\right)(\Gamma\dt)^2\, \bigg\}\nonumber \\
&&\hspace{-1cm}\Rodd(\lp\Xm,\Seta;\dt)=2\left|A_{l^+X^-}A_{S_\eta}\right|^2\bigg\{  \nonumber\\
&&1+ \eta \left|\frac{p}{q} \right| \left(y\cos\phi+x\sin\phi  \right)(\Gamma\dt) +\left(\frac{y^2}{4}\left(1+\left|\frac{p}{q}\right|^2\right)-\frac{x^2}{4}\left(1-\left|\frac{p}{q}\right|^2\right)\right)(\Gamma\dt)^2\, \bigg\}\nonumber \\\label{eq:semilep_cp_2}
\end{eqnarray}
%
%
identical time-dependent decay rate as for the case of $\Dz \to f_{\CP}$ at $\Upsilon(4S)$.\\ 
Integrating~\ref{eq:semilep_cp_1} or \ref{eq:semilep_cp_2} over time we obtain:
\begin{eqnarray}  
\int\Rodd(\lm\Xp,\Seta;\dt)e^{-\Gamma|\Delta t|} d(\dt) &=&\nonumber\\
&&\hspace{-2cm}\frac4\Gamma\left|\bar A_{l^-X^+}A_{S_\eta}\right|^2\left[1+\frac{y^2}{2}\left(1+\absqop^2\right)-\frac{x^2}{2}\left(1-\absqop^2\right)\right]\nonumber\\
\int\Rodd(\lp\Xm,\Seta;\dt)e^{-\Gamma|\Delta t|} d(\dt) &=&\nonumber\\
&&\hspace{-2cm}\frac4\Gamma\left|\bar A_{l^-X^+}A_{S_\eta}\right|^2\left[1+\frac{y^2}{2}\left(1+\abspoq^2\right)-\frac{x^2}{2}\left(1-\abspoq^2\right)\right]\label{eq:semilep_cp_3}
\end{eqnarray}
%\\{\it [Matteo]: la~\ref{eq:semilep_cp_1} mi torna. Nella~\ref{eq:semilep_cp_2} ho aggiunto i termini quadratici in $x$ e $y$ che potrebbero essere non trascurabili rispetti agli effetti che in teoria uno vorrebbe vedere.}
%
\subsection{Double $K^\mp\pi^\pm$ decays}
%
For the case of hadronic final state \G, it is interesting to consider the 
 time-dependent decay rate for the final state $(K^+\pi^-,K^+\pi^-)$ 
(and for charge conjugate). It is useful to introduce the following quantities 
and phase conventions:
%
%
\begin{eqnarray}
&&\frac{\bar{A}_{\Kmpip}}{\A_{\Kmpip}}\equiv \frac{A_{\Kppim}}{\Abar_{\Kppim}}=-\sqrt{R_D}e^{-i\delta_{\Kpi}} \\
&&\lambda_{K^-\pi^+}=\frac{q}{p} \frac{\bar{A}_{\Kmpip}}{\A_{\Kmpip}} = 
- \left|\frac{q}{p} \right| \sqrt{R_D}e^{-i(\delta_{\Kpi}-\phi)} \\
&&\bar{\lambda}_{K^+\pi^-}=\frac{p}{q} \frac{A_{\Kppim}}{\bar{\A}_{\Kppim}} = 
- \left|\frac{p}{q} \right| \sqrt{R_D}e^{-i(\delta_{\Kpi}+\phi)} 
\label{eq:lambda_Kpi}
\end{eqnarray}

%
\begin{eqnarray}
\Rodd(K^-\pi^+,K^-\pi^+;\dt) = \left|A_{K^-\pi^+}\right|^4\left|\frac{p}{q}\right|^2 |1-\lambda_{K^-\pi^+}^2|^2\left[ \cosh(y\Gamma\dt)-\cos(x\Gamma\dt)\right]\nonumber \\ 
\Rodd(K^+\pi^-,K^+\pi^-;\dt) = \left|\bar{A}_{K^+\pi^-}\right|^4\left|\frac{q}{p}\right|^2 |1-\bar{\lambda}_{K^+\pi^-}^2|^2\left[ \cosh(y\Gamma\dt)-\cos(x\Gamma\dt)\right]\nonumber \\ \label{eq:k-pi+_k-pi+_1}
\end{eqnarray}

%{\it[Matteo] A me viene:
%\begin{eqnarray}
%\Rodd(K^-\pi^+,K^-\pi^+;\dt) = \left|A_{K^-\pi^+}\right|^4\left|\frac{p}{q}\right|^2 |1-\lambda_{K^-\pi^+}^2|^2\left[ \cosh(y\Gamma\dt)-\cos(x\Gamma\dt)\right]\nonumber \\ 
%\Rodd(K^+\pi^-,K^+\pi^-;\dt) = \left|A_{K^+\pi^-}\right|^4\left|\frac{q}{p}\right|^2 |1-\bar\lambda_{K^+\pi^-}^2|^2\left[ \cosh(y\Gamma\dt)-\cos(x\Gamma\dt)\right]\nonumber \\ 
%\end{eqnarray}
%
%Da ricontrollare. Per\`o la~\ref{eq:k-pi+_k-pi+_2} mi torna.}\\
%
%
%
%
In the limit of $|x\Gamma \dt|\ll 1$, $|y\Gamma \dt| \ll 1$ we obtain:
%
%
\begin{eqnarray}
\Rodd(\Kmpip,\Kmpip;\dt) = \left|A_{K^-\pi^+}\right|^4\abspoq^2 \left[ 1+ \absqop^4R_D^2-2R_D\absqop^2\cos[2(\deltaKpi-\phi)] \right] \frac{x^2+y^2}{2}(\Gamma \dt)^2\nonumber \\ 
\Rodd(K^+\pi^-,K^+\pi^-;\dt) = \left|A_{K^+\pi^-}\right|^4\left|\frac{q}{p}\right|^2 \left[ 1+ \abspoq^4R_D^2-2R_D\abspoq^2\cos[2(\deltaKpi+\phi)] \right] \frac{x^2+y^2}{2}(\Gamma \dt)^2\nonumber \\\label{eq:k-pi+_k-pi+_2} 
\end{eqnarray}

Integrating \ref{eq:k-pi+_k-pi+_1} or \ref{eq:k-pi+_k-pi+_2} over time we obtain:
\begin{eqnarray}
\int\Rodd(K^-\pi^+,K^-\pi^+;\dt)e^{-\Gamma|\Delta t|} d(\dt) = &&\nonumber\\
&&\hspace{-3cm}\frac2\Gamma\left|A_{K^-\pi^+}\right|^4\abspoq^2 \left[ 1+ \absqop^4R_D^2-2R_D\absqop^2\cos[2(\deltaKpi-\phi)] \right](x^2+y^2)\nonumber\\
\int\Rodd(K^+\pi^-,K^+\pi^-;\dt)e^{-\Gamma|\Delta t|} d(\dt) = &&\nonumber\\
&&\hspace{-3cm}\frac2\Gamma\left|A_{K^+\pi^-}\right|^4\left|\frac{q}{p}\right|^2 \left[ 1+ \abspoq^4R_D^2-2R_D\abspoq^2\cos[2(\deltaKpi+\phi)] \right](x^2+y^2)\nonumber\\\label{eq:k-pi+_k-pi+_3} 
\end{eqnarray}

% 
\subsection{Double semileptonic decays}
For the case of double flavor specific final states (\lp\Xm, \lp\Xm) (and
 charge conjugate) we have:
%
\begin{eqnarray}
\Rodd(\lp\Xm,\lp\Xm;\dt) = |A_{l^+X^-}|^4\left|\frac{p}{q}\right|^2 \left[ \cosh(y\Gamma\dt)-\cos(x\Gamma\dt)\right]\nonumber \\ 
\Rodd(\lm\Xp,\lm\Xp;\dt) = |A_{l^-X^+}|^4\left|\frac{q}{p}\right|^2 \left[ \cosh(y\Gamma\dt)-\cos(x\Gamma\dt)\right]\nonumber \\\label{eq:lX_lX_1} 
\end{eqnarray}

%
%
In the limit of $|x\Gamma \dt|\ll 1$, $|y\Gamma \dt| \ll 1$ we obtain:
%
%
\begin{eqnarray}
\Rodd(\lp\Xm,\lp\Xm;\dt) = |A_{l^+X^-}|^4\abspoq^2  \frac{x^2+y^2}{2}(\Gamma \dt)^2\nonumber \\ 
\Rodd(\lm\Xp,\lm\Xp;\dt) = |A_{l^-X^+}|^4\absqop^2  \frac{x^2+y^2}{2}(\Gamma \dt)^2\nonumber \\\label{eq:lX_lX_2}
\end{eqnarray}
%{\it [Matteo]\ref{eq:lX_lX_1} e \ref{eq:lX_lX_2} mi tornano.}\\
Integrating \ref{eq:lX_lX_1} and \ref{eq:lX_lX_2} over time we obtain:
\begin{eqnarray}
\int\Rodd(\lp\Xm,\lp\Xm;\dt)e^{-\Gamma|\Delta t|} d(\dt) = \frac2\Gamma|A_{l^+X^-}|^4\abspoq^2 (x^2+y^2)\nonumber \\ 
\int\Rodd(\lm\Xp,\lm\Xp;\dt)e^{-\Gamma|\Delta t|} d(\dt) = \frac2\Gamma|A_{l^-X^+}|^4\absqop^2 (x^2+y^2)\nonumber \\\label{eq:lX_lX_3}
\end{eqnarray}
%
%
%
\subsection{$K^\mp\pi^\pm$ decays with $CP$ tag}
%
%
%
For the case of \CP tag we consider for example (\Seta, \Kmpip). We have:
%
%
%
\begin{eqnarray}
&&\Rodd(\Seta,\Kmpip;\dt) = \left|A_{S_\eta}A_{\Kmpip}\right|^2 \bigg\{\nonumber\\
&&  \left[ \left(1+ \abspoq^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi+\cos(\deltaKpi-2\phi) \right) + R_D \left(1+\absqop^2\right)\right]\cosh(y\Gamma\dt)\nonumber \\
&& +  \left[ \left( 1 - \abspoq^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi-\cos(\deltaKpi-2\phi) \right) + R_D \left(1-\absqop^2\right) \right] \cos(x\Gamma\dt) \nonumber \\
&& -2 \left[ \eta \abspoq \cos\phi+ \sqrt{R_D} \cos(\deltaKpi-\phi)\left( \absqop + \abspoq\right) + R_D\absqop\cos\phi \right] \sinh(y\Gamma\dt) \nonumber \\
&& +2 \left[ -\eta \abspoq \sin\phi+ \sqrt{R_D} \sin(\deltaKpi-\phi) \left( \absqop - \abspoq\right) + R_D\absqop\sin\phi \right] \sin(y\Gamma\dt) \bigg\}\label{eq:Seta_k-pi+_1}
\end{eqnarray}
%\begin{eqnarray}
%&&\Rodd(\Seta,\Kppim;\dt) = \left|A_{S_\eta}\bar{A}_{\Kppim}\right|^2 \bigg\{\\
%&&  \left[ \left(1+ \absqop^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi+\cos(\deltaKpi+2\phi) \right) + R_D \left(1+\abspoq^2\right)\right]\cosh(y\Gamma\dt)\nonumber \\
%&& +  \left[ \left( 1 - \absqop^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi-\cos(\deltaKpi+2\phi) \right) + R_D \left(1-\abspoq^2\right) \right] \cos(x\Gamma\dt) \nonumber \\
%&& -2 \left[ \eta \absqop \cos\phi+ \sqrt{R_D} \cos(\deltaKpi+\phi)\left( \abspoq + \absqop\right) + R_D\abspoq\cos\phi \right] \sinh(y\Gamma\dt) \nonumber \\
%&& +2 \left[\eta \absqop \sin\phi+ \sqrt{R_D} \sin(\deltaKpi+\phi) \left( \abspoq - \absqop\right) - R_D\abspoq\sin\phi \right] \sin(y\Gamma\dt) \bigg\} \nonumber \\  \\ \nonumber
%\end{eqnarray}
%
%
%
%
In the limit of $|x\Gamma \dt|\ll 1$, $|y\Gamma \dt| \ll 1$ we obtain:
%
%
\begin{eqnarray}
&& \Rodd(\Seta,\Kmpip;\dt) = \left|A_{S_\eta}A_{\Kmpip}\right|^2  \bigg\{ 2 \left(1+ 2\eta\sqrt{R_D}\cos\deltaKpi+R_D \right)  \nonumber \\
&&+ \bigg[ \left(  \eta \abspoq \cos\phi+ \sqrt{R_D} \cos(\deltaKpi-\phi)\left( \absqop + \abspoq\right) + R_D\absqop\cos\phi \right)y \nonumber \\
&&+ \left( -\eta \abspoq \sin\phi+ \sqrt{R_D} \sin(\deltaKpi-\phi) \left( \absqop - \abspoq\right) + R_D\absqop\sin\phi \right) x \bigg] (\Gamma \dt) \nonumber \\
&& +\frac12 \bigg[ \left( \left(1+ \abspoq^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi+\cos(\deltaKpi-2\phi) \right) + R_D \left(1+\absqop^2\right) \right) y^2 \nonumber \\
&& - \left( \left( 1 - \abspoq^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi-\cos(\deltaKpi-2\phi) \right) + R_D \left(1-\absqop^2\right) \right) x^2\bigg]  (\Gamma \dt)^2 \bigg\} \nonumber \\\label{eq:Seta_k-pi+_2}
\end{eqnarray}
%\begin{eqnarray}
%&& \Rodd(\Seta,\Kppim;\dt) = \left|A_{S_\eta}\bar{A}_{\Kppim}\right|^2  \bigg\{ 2 \left(1+ 2\eta\sqrt{R_D}\cos\deltaKpi+R_D \right)  \nonumber \\
%&&+\bigg[ \left(  \eta \absqop \cos\phi+ \sqrt{R_D} \cos(\deltaKpi+\phi)\left( \abspoq + \absqop\right) + R_D\abspoq\cos\phi \right)y \nonumber \\
%&&+ \left( +\eta \absqop \sin\phi+ \sqrt{R_D} \sin(\deltaKpi+\phi) \left( \abspoq - \absqop\right) - R_D\absqop\sin\phi \right) x \bigg] (\Gamma \dt) \nonumber \\
%&& +\frac12 \bigg[ \left( \left(1+ \absqop^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi+\cos(\deltaKpi+2\phi) \right) + R_D \left(1+\abspoq^2\right) \right) y^2 \nonumber \\
%&& - \left( \left( 1 - \absqop^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi-\cos(\deltaKpi+2\phi) \right) + R_D \left(1-\abspoq^2\right) \right) x^2\bigg]  (\Gamma \dt)^2 \bigg\} \nonumber \\ \\ \nonumber
%\end{eqnarray}
Integrating \ref{eq:Seta_k-pi+_1} and \ref{eq:Seta_k-pi+_2} we obtain:
\begin{eqnarray}
&& \int\Rodd(\Seta,\Kmpip;\dt)e^{-\Gamma|\Delta t|} d(\dt) = \frac2\Gamma\left|A_{S_\eta}A_{\Kmpip}\right|^2  \bigg\{ 2 \left(1+ 2\eta\sqrt{R_D}\cos\deltaKpi+R_D \right)  \nonumber \\
&& + \left( \left(1+ \abspoq^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi+\cos(\deltaKpi-2\phi) \right) + R_D \left(1+\absqop^2\right) \right) y^2 \nonumber \\
&& - \left( \left( 1 - \abspoq^2\right) +2\eta\sqrt{R_D} \left( \cos\deltaKpi-\cos(\deltaKpi-2\phi) \right) + R_D \left(1-\absqop^2\right) \right) x^2 \bigg\} \nonumber \\\label{eq:Seta_k-pi+_3}
\end{eqnarray}

\subsection{Double $CP$ tag}
%
%
For the case of double \CP eigenstates (\Sp, \Sm)  we have:
%\\{\it[Matteo]: forse leverei ``(and charge conjugate)'' perch\`e in quel caso c'\`e un segno meno nei termini proporzionali a $\sin$ e $\sinh$, quindi il commento potrebbe confondere. Mi tornano sia la~\ref{eq:cp+_cp-_1}
%che la \ref{eq:cp+_cp-_2}.}\\
% 
%
%
\begin{eqnarray}
\Rodd(\Sp,\Sm;\dt) = |A_{S_+}|^2|A_{S_-}|^2 &&\hspace{-0.5cm}\Bigg\{\left( 4 + \left|\frac{p}{q}\right|^2 +\left|\frac{q}{p}\right|^2+2\cos(2\phi) \right)\cosh(y\Gamma\dt) \nonumber\\
&&+\left( 4 - \left|\frac{p}{q}\right|^2-\left|\frac{q}{p}\right|^2-2\cos(2\phi) \right)\cos(x\Gamma\dt)\nonumber\\
&&-4\cos\phi\left( \left|\frac{q}{p}\right|+ \left|\frac{p}{q}\right| \right)\sinh(y\Gamma\dt)\nonumber\\
&&+4\sin\phi\left( \left|\frac{q}{p}\right|- \left|\frac{p}{q}\right| \right)\sin(x\Gamma\dt)\Bigg\} \nonumber\\ \label{eq:cp+_cp-_1}
\end{eqnarray}
%
%
%
%
In the limit of $|x\Gamma \dt|\ll 1$, $|y\Gamma \dt| \ll 1$ we obtain:
%
%
\begin{eqnarray}
\Rodd(\Sp,\Sm;\dt)=|A_{S_+}|^2|A_{S_-}|^2\bigg\{2+ \bigg[\left(\absqop-\abspoq\right)x\sin\phi - \left(\absqop+\abspoq\right)y\cos\phi  \bigg](\Gamma\dt)\bigg\}\nonumber \\\label{eq:cp+_cp-_2}
\end{eqnarray}
%
%
\par
It is worth noting that in the above time-dependent rates the charge conjugate
 expressions can be otained with the substitutions $|q/p| \leftrightarrow |p/q|$ and $\phi \leftrightarrow -\phi$.

Integrating \ref{eq:cp+_cp-_1} and \ref{eq:cp+_cp-_2} we obtain:
\begin{eqnarray}
\int\Rodd(\Sp,\Sm;\dt)e^{-\Gamma|\Delta t|} d(\dt)=&&\nonumber\\
&&\hspace{-6cm}\frac4\Gamma|A_{S_+}|^2|A_{S_-}|^2\left[4+ \frac{y^2}{2}\left( 4 + \left|\frac{p}{q}\right|^2 +\left|\frac{q}{p}\right|^2+2\cos(2\phi) \right)-\frac{x^2}{2}\left( 4 - \left|\frac{p}{q}\right|^2-\left|\frac{q}{p}\right|^2-2\cos(2\phi) \right)\right]\nonumber \\\label{eq:cp+_cp-_2}
\end{eqnarray}

\subsection{Time-dependent Dalitz plot decay rates with \CP violation}
%
It is useful to define the Dalitz plot (\sab,\sac) in terms of kinematical quantities:
\begin{eqnarray}
\sab\equiv \msqp \equiv m^2(\ks h^+)\nonumber \\ 
\sac\equiv \msqm \equiv m^2(\ks h^-)\nonumber \\
\label{eq:Dalitz_notations}
\end{eqnarray}
%
%
%
The \Dz decay amplitude varies across the Dalitz plot:
%
%
\begin{eqnarray}
\Af \equiv \langle f | {\cal H} | \Dz \rangle \equiv A(\sab,\sac) \equiv A_{\sab,\sac} \nonumber \\
\Abarf \equiv \langle f | {\cal H} | \Dzbar \rangle \equiv \Abar(\sab,\sac) \equiv \AbarDal \nonumber \\
\label{eq:Dalitz_amp}
\end{eqnarray}
%
%

%
%
\begin{eqnarray}
&&\Rodd(\Seta,\kshh;\dt)=|A_{S_\eta}|^2\Bigg\{ \nonumber \\
&&\left[ |\Af|^2\left( 1+\abspoq^2\right)+|\Abarf|^2\left( 1+\absqop^2\right) -4 \eta \cos\phi \left( \cos\phi \re(\Af^*\Abarf) - \sin\phi \im(\Af^*\Abarf)\right) \right] \cosh(y\Gamma \dt) \nonumber \\
&& + \left[ |\Af|^2\left( 1-\abspoq^2\right)+|\Abarf|^2\left( 1-\absqop^2\right) -4 \eta \sin\phi \left( \sin\phi \re(\Af^*\Abarf) + \cos\phi \im(\Af^*\Abarf)\right) \right] \cos(x\Gamma \dt) \nonumber \\
&& -2 \bigg[\eta\cos\phi\left(\left|\frac{p}{q}\right||A_f|^2+\left|\frac{q}{p}\right||\bar A_f|^2\right)-\left(\absqop+\abspoq\right)\left(\cos\phi\re(\Af^*\Abarf)-\sin\phi\im(\Af^*\Abarf)\right) \bigg]\sinh(y\Gamma \dt) \nonumber \\ 
&& -2 \bigg[\eta\sin\phi\left(\left|\frac{p}{q}\right||A_f|^2-\left|\frac{q}{p}\right||\bar A_f|^2\right)-\left(\abspoq-\absqop\right)\left(\sin\phi\re(\Af^*\Abarf)+\cos\phi\im(\Af^*\Abarf)\right) \bigg]\sin(x\Gamma \dt)\Bigg\}\nonumber\\\label{eq:cp_kshh_1}
\end{eqnarray}
%
%
%
In the limit of $|x\Gamma \dt|\ll 1$, $|y\Gamma \dt| \ll 1$ we obtain:
%
%
\begin{eqnarray}
&&\Rodd(\Seta,\kshh;\dt)=\left|A_{\Seta} \right|^2 \bigg\{ 2\left( |\Af|^2 +|\Abarf|^2 -2\eta \re(\Af^*\Abarf)\right) \nonumber \\
&&+2 \bigg[\bigg( \abspoq \left( \cos\phi\re(\Af^*\Abarf) - \sin\phi \im(\Af^*\Abarf) -\eta \cos\phi |\Af|^2)\right) + \nonumber \\
&& \hspace{+5.5cm}+ \absqop \left( \cos\phi \re(\Af^* \Abar_f)-\sin\phi\im(\Af^* \Abar_f) -\eta \cos\phi|\Abarf|^2 \right) \bigg) y \nonumber \\
&& - \bigg( \abspoq \left( -\cos\phi\im(\Af^*\Abarf) - \sin\phi \re(\Af^*\Abarf +\eta \sin\phi |\Af|^2)\right) + \nonumber \\
&& \hspace{+5.5cm}+\absqop \left( \cos\phi \im(\Af^* \Abar_f)+\sin\phi\re(\Af^* \Abar_f) -\eta \sin\phi|\Abarf|^2 \right) \bigg)x \bigg] (\Gamma\dt) \nonumber \\ 
&& +\frac12 \bigg[ \left(|\Af|^2\left( 1+\abspoq^2\right)+|\Abarf|^2\left( 1+\absqop^2\right) -4 \eta \cos\phi \left( \cos\phi \re(\Af^*\Abarf) - \sin\phi \im(\Af^*\Abarf)\right) \right)y^2 \nonumber \\
&&- \left( |\Af|^2\left( 1-\abspoq^2\right)+|\Abarf|^2\left( 1-\absqop^2\right) -4 \eta \sin\phi \left( \sin\phi \re(\Af^*\Abarf) + \cos\phi \im(\Af^*\Abarf)\right) \right) x^2 \bigg] (\Gamma \dt)^2 \bigg\} \nonumber \\\label{eq:cp_kshh_2}
\end{eqnarray}
%
%
Integrating \ref{eq:cp_kshh_1} and \ref{eq:cp_kshh_2} we obtain:
%
%
\begin{eqnarray}
\int\Rodd(\Seta,\kshh;\dt)e^{-\Gamma|\Delta t|} d(\dt)=&&\nonumber\\
&&\hspace{-6cm}\frac2\Gamma \left|A_{\Seta} \right|^2 \bigg\{ 2\left( |\Af|^2 +|\Abarf|^2 -2\eta \re(\Af^*\Abarf)\right) \nonumber \\
&&\hspace{-6cm} + \bigg[ \left(|\Af|^2\left( 1+\abspoq^2\right)+|\Abarf|^2\left( 1+\absqop^2\right) -4 \eta \cos\phi \left( \cos\phi \re(\Af^*\Abarf) - \sin\phi \im(\Af^*\Abarf)\right) \right)y^2 \nonumber \\
&&\hspace{-6cm}- \left( |\Af|^2\left( 1-\abspoq^2\right)+|\Abarf|^2\left( 1-\absqop^2\right) -4 \eta \sin\phi \left( \sin\phi \re(\Af^*\Abarf) + \cos\phi \im(\Af^*\Abarf)\right) \right) x^2 \bigg] \bigg\} \nonumber \\
\label{eq:cp_kshh_int}
\end{eqnarray}
%
that in the limit of \CP conservation is:
%
%
\begin{eqnarray}
\int\Rodd(\Seta,\kshh;\dt)e^{-\Gamma|\Delta t|} d(\dt)=
\frac4\Gamma \left|A_{\Seta} \right|^2 \bigg[ \left( |\Af|^2 +|\Abarf|^2 -2\eta \re(\Af^*\Abarf)\right) \bigg]\left(1+y^2 \right)\nonumber \\
\end{eqnarray}

%
%
Let's consider now a double Dalitz decay where the final state is identical for the two cases $f_1=f_2=\kshh$. The 
 position, (\sab,\sac), in the \Dz Dalitz plot for the two final states $f_1$ and $f_2$ will be different in general,
 as well as the respective decay amplitudes:
%
%
%
\begin{eqnarray}
A_{f_1} \equiv \langle\kshh|{\cal H}|\Dz\rangle \equiv \A(\sab,\sac) \equiv A_1 \nonumber \\
A_{f_2} \equiv \langle\kshh|{\cal H}|\Dz\rangle \equiv \A(\spab,\spac) \equiv A_2 \nonumber \\
\end{eqnarray}
%
%
%
\begin{eqnarray}
&&\Rodd(\kshh,\kshh;\dt) = \nonumber \\
&&\bigg[ |\Aonebar\Atwo|^2+|\Aone\Atwobar|^2-2\re(\Aonebar^*\Atwo^*\Aone\Atwobar) \nonumber \\
&&\hspace{0.5cm}+ \abspoq^2|\Aone\Atwo|^2+\absqop^2|\Aonebar\Atwobar|^2-2\left( \cos(2\phi)\re(\Aone^*\Atwo^*\Aonebar\Atwobar)-\sin(2\phi)\im(\Aone^*\Atwo^*\Aonebar\Atwobar)\right) \bigg] \cosh(y\Gamma \dt)\nonumber \\
&&\bigg[ |\Aonebar\Atwo|^2+|\Aone\Atwobar|^2-2\re(\Aonebar^*\Atwo^*\Aone\Atwobar) \nonumber \\
&&\hspace{0.5cm}- \abspoq^2|\Aone\Atwo|^2-\absqop^2|\Aonebar\Atwobar|^2+2\left( \cos(2\phi)\re(\Aone^*\Atwo^*\Aonebar\Atwobar)-\sin(2\phi)\im(\Aone^*\Atwo^*\Aonebar\Atwobar)\right) \bigg] \cos(x\Gamma \dt)\nonumber \\
&&-2 \bigg[ |\Atwo|^2\left( \abspoq \left( \cos\phi \re(\Aone\Aonebar^*)+\sin\phi\im(\Aone\Aonebar^*)-\re(\Aone\Aonebar^*)\right)\right) \nonumber\\
&&\hspace{7.0cm}-|\Aone|^2\left( \abspoq \left( \cos\phi \re(\Atwo\Atwobar^*)+\sin\phi\im(\Atwo\Atwobar^*) \right)\right) \nonumber \\
&&\hspace{7.0cm}+|\Atwobar|^2\left( \absqop \left( \cos\phi \re(\Aonebar\Aone^*)-\sin\phi\im(\Aonebar\Aone^*) \right)\right) \bigg] \sinh(y\Gamma \dt)\nonumber \\
&&+2 \bigg[ |\Atwo|^2\left( \abspoq \left( \cos\phi \im(\Aone\Aonebar^*)-\sin\phi\re(\Aone\Aonebar^*)-\im(\Aone\Aonebar^*)\right)\right) \nonumber\\
&&\hspace{7.0cm}-|\Aone|^2\left( \abspoq \left( \cos\phi \im(\Atwo\Atwobar^*)-\sin\phi\re(\Atwo\Atwobar^*) \right)\right) \nonumber \\
&&\hspace{7.0cm}+|\Atwobar|^2\left( \absqop \left( \cos\phi \im(\Aonebar\Aone^*)+\sin\phi\re(\Aonebar\Aone^*) \right)\right) \bigg] \sin(x\Gamma \dt)\nonumber \\
\end{eqnarray}
%
%
%
In the limit of $|x\Gamma \dt|\ll 1$, $|y\Gamma \dt| \ll 1$ we obtain:
%
%
%
\begin{eqnarray}
&&\Rodd(\kshh,\kshh;\dt) = \nonumber \\
&&2 \bigg[ |\Aonebar\Atwo|^2+|\Aone\Atwobar|^2-2\re(\Aonebar^*\Atwo^*\Aone\Atwobar)\bigg] \nonumber \\
&&-2 \bigg\{ \bigg[ |\Atwo|^2\left( \abspoq \left( \cos\phi \re(\Aone\Aonebar^*)+\sin\phi\im(\Aone\Aonebar^*)-\re(\Aone\Aonebar^*)\right)\right) \nonumber\\
&&\hspace{7.0cm}-|\Aone|^2\left( \abspoq \left( \cos\phi \re(\Atwo\Atwobar^*)+\sin\phi\im(\Atwo\Atwobar^*) \right)\right) \nonumber \\
&&\hspace{7.0cm}+|\Atwobar|^2\left( \absqop \left( \cos\phi \re(\Aonebar\Aone^*)-\sin\phi\im(\Aonebar\Aone^*) \right)\right) \bigg] y\nonumber \\
&&\hspace{0.5cm}- \bigg[ |\Atwo|^2\left( \abspoq \left( \cos\phi \im(\Aone\Aonebar^*)-\sin\phi\re(\Aone\Aonebar^*)-\im(\Aone\Aonebar^*)\right)\right) \nonumber\\
&&\hspace{7.0cm}-|\Aone|^2\left( \abspoq \left( \cos\phi \im(\Atwo\Atwobar^*)-\sin\phi\re(\Atwo\Atwobar^*) \right)\right) \nonumber \\
&&\hspace{7.0cm}+|\Atwobar|^2\left( \absqop \left( \cos\phi \im(\Aonebar\Aone^*)+\sin\phi\re(\Aonebar\Aone^*) \right)\right) \bigg] x \bigg\} (\Gamma \dt)\nonumber \\
&&+\frac12\bigg\{ \bigg[ |\Aonebar\Atwo|^2+|\Aone\Atwobar|^2-2\re(\Aonebar^*\Atwo^*\Aone\Atwobar) \bigg] (y^2-x^2) \nonumber \\
&&\hspace{0.5cm}+ \bigg[ \abspoq^2|\Aone\Atwo|^2+\absqop^2|\Aonebar\Atwobar|^2-2\left( \cos(2\phi)\re(\Aone^*\Atwo^*\Aonebar\Atwobar)-\sin(2\phi)\im(\Aone^*\Atwo^*\Aonebar\Atwobar)\right) \bigg] (x^2+y^2) \nonumber \\
&&\hspace{16.5cm}\bigg\}(\Gamma \dt)^2\nonumber \\
%&&\hspace{0.5cm}-\bigg[ |\Aonebar\Atwo|^2+|\Aone\Atwobar|^2-2\re(\Aonebar^*\Atwo^*\Aone\Atwobar) \nonumber \\
%&&\hspace{0.5cm}- \abspoq^2|\Aone\Atwo|^2-\absqop^2|\Aonebar\Atwobar|^2+2\left( \cos(2\phi)\re(\Aone^*\Atwo^*\Aonebar\Atwobar)-\sin(2\phi)\im(\Aone^*\Atwo^*\Aonebar\Atwobar)\right) \bigg] x^2 \bigg\}(\Gamma \dt)^2\nonumber \\
\end{eqnarray}






