\section{\Dz-\Dzbar mixing and \CP violation with Dalitz plot analyses}
\label{sec:Appendix_DalitzAnalyses}

Time-dependent Dalitz plot analyses have been proposed for measuring \Dz-\Dzbar mixing and 
 \CPV~\cite{Asner:2005sz} from CLEO collaboration and extended to a large data sample 
 from BaBar~\cite{delAmoSanchez:2010xz},~\cite{Aubert:2008zh} and 
 Belle collaboration~\cite{Abe:2007rd}. The advantage of performing time-dependent 
 Dalitz plot analyses relies in the possible presence of double Cabibbo suppressed and 
 \CP amplitudes, interfering among each other, in a single 3-body decay mode. 
 This characteristic depends on the dynamics of the specific 3-body final 
 state that we 
 are considering.
 \par 
 In addition, for \Dz self-conjugate states (e.g. \kshh, $\pip\pim\pi^0$) 
 it is possible to 
 extract directly the mixing parameters $x$, $y$ without \Dz-\Dzbar relative strong phase 
 uncertainties by assuming \CP conservation in the \Dz decay. 
As a drawback, the introduction of a phenomenological model for the parameterization 
of the 3-body \Dz decay amplitude introduces a ``model'' uncertainty, 
sometimes called ``Dalitz model'' error.

\subsection{Considerations for 3-body self-conjugate \Dz decay modes}

Self-conjugate 3-body decay modes (e.g. \kshh, $\pip\pim\pi^0$) have the pecularity that 
 the final state $|f\rangle$, and the charge conjugate state 
 $ C |f\rangle = |\bar f\rangle$, can be both identified in a single Dalitz plot 
 $(\sab,\sac)$. 
 In particular if $|f\rangle = (\sab',\sac')$ then  $|\bar f\rangle = (\sac',\sab')$.
 For example, this is not possible for the decay mode $\Dz \to K^+\pim\pi^0$.
 Time-dependent analysis of self-conjugate 3-body decays with flavor tag allows to extract 
 mixing parameters $x$, $y$ directly when assuming \CP conservation in the decay amplitude,
 $\Abarf \equiv \Afbar$. Infact the time-dependence:
%
\begin{eqnarray}
\frac{d\Gamma[\Dz_{\textrm{phys}}\to f]/dt}{e^{-\Gamma t}{\cal N}_f} &=&
\bigg[ (|\Af|^2 + |\frac{q}{p}|^2 |\Abarf|^2) \cosh(\Gamma y t) 
+ ( |\Af|^2- |\frac{q}{p}|^2 |\Abarf|^2) \cos(\Gamma x t) \nonumber \\
& + &  2\re \left( \Af^*\Abarf\frac{q}{p} \right)  \sinh(\Gamma y t)  
- 2\im \left( \Af^*\Abarf\frac{q}{p} \right) \sin (\Gamma x t)  \bigg] 
\label{eq:DalitzFlavorTagTD}
\end{eqnarray}
% 
with \CP assumed to be conserved in the decay becomes:
%
\begin{eqnarray}
\frac{d\Gamma[\Dz_{\textrm{phys}}\to f]/dt}{e^{-\Gamma t}{\cal N}_f} &=&
 \bigg[ (|\Af|^2 + |\frac{q}{p}|^2 |\Afbar|^2) \cosh(\Gamma y t) 
 +  ( |\Af|^2- |\frac{q}{p}|^2 |\Afbar|^2) \cos(\Gamma x t)  \nonumber \\
 & + &  2\re \left( \Af^*\Afbar\frac{q}{p} \right)  \sinh(\Gamma y t)  
-  2\im \left( \Af^*\Afbar\frac{q}{p} \right) \sin (\Gamma x t)  \bigg] 
\label{eq:DalitzFlavorTagTD_CPC}
\end{eqnarray}
% 
which is written in terms of the \Dz decay amplitude and can be fitted directly from 
 the data. In Eq.~\ref{eq:DalitzFlavorTagTD_CPC} is no more present the \Dzbar decay amplitude
 or the relative \Dz-\Dzbar strong phase. 
\par
In general \CP is conserved in the decay of charm meson with a good precision in the Standard Model.
However, for \Dz decays with a $K_S^0$ final particle, 
\CP violation is expected to be observed at  the level of 0.3\% due to \CPV 
in the $K^0-\bar K^0$ mixing. This effect is probably not 
 negligible for the level of precision that we want to achieve at SuperB {\it (need to be more 
 quantitative on this point)}. In addition the Datitz model error will probably limit the 
 precision of these measurements when the high statistics data sample will be available. 


\subsection{Relevance of \Dz-\Dzbar threshold data}
\label{sec:DzDzbar_data}
Measurements at \Dz-\Dzbar threshold will definitely help in reducing the Dalitz model uncertainty
  and also in avoiding the need of the assumption of \CP conservation in the \Dz decay. 
Let's consider for the moment the time-integrated rate, reported in Eq.~\ref{eq:cp_kshh_int}. 
%for 3-body self-conjugate decay modes. 
For each Dalitz plot region or bin, identified with the index $i$, we have the possibility 
to access the information on the relative \Dz-\Dzbar phase as reported below:
%
\begin{eqnarray}
&&\int_i d{\cal P}\int d(\dt) \left[\frac{\Rodd(\Sm,\kshh;\dt)}{\left|A_{\Sm} \right|^2}
-\frac{\Rodd(\Sp,\kshh;\dt)}{\left|A_{\Sp} \right|^2}\right]e^{-\Gamma|\Delta t|} =\nonumber\\
&&\frac{16}{\Gamma} \int_i d{\cal P} \bigg[  \re(\Af^*\Abarf) (1+y^2\cos^2\phi -x^2\sin^2\phi )%\nonumber \\
  -  \im(\Af^*\Abarf) \cos\phi \sin\phi (x^2+ y^2)\bigg] = \nonumber \\
&&\frac{16}{\Gamma}\int_i d{\cal P} \left[\re(\Af^*\Abarf)+ {\cal O}(x^2+ y^2)\right] =
 \frac{16}{\Gamma} c_i \sqrt{T_i\bar T_i}  + {\cal O}(x^2+ y^2)\nonumber \\
\label{eq:cp_kshh_ci_si}
\end{eqnarray}
%
\begin{eqnarray}
\int_i d{\cal P } \int dt \frac{d\Gamma[\Dz_{\textrm{phys}}\to f]/dt}{2{\cal N}_f/\Gamma} =
\frac{1}{2}\int_i d{\cal P } \bigg[ (|\Af|^2 + |\frac{q}{p}|^2 |\Abarf|^2) \frac{1}{1-y^2} 
+ ( |\Af|^2- |\frac{q}{p}|^2 |\Abarf|^2) \frac{1}{1+x^2}  \bigg] \nonumber \\
\simeq \frac{1}{2} \int_i d{\cal P }\bigg[ (2+y^2-x^2) |\Af|^2 + \left(|\frac{q}{p}|^2 |\Abarf|^2\right)(x^2+y^2)  \bigg] =  \nonumber \\
\Ti \left(1+\frac{y^2-x^2}{2}\right)+ \Tbari \left(\absqop^2\frac{x^2+y^2}{2}\right)  \nonumber \\ \\
\int_i d{\cal P } \int dt \frac{d\Gamma[\Dzbar_{\textrm{phys}}\to f]/dt}{2{\cal N}_f/\Gamma} =
\frac{1}{2}\int_i d{\cal P } \bigg[ (|\Abarf|^2 + |\frac{p}{q}|^2 |\Af|^2) \frac{1}{1-y^2} 
+ ( |\Abarf|^2- |\frac{p}{q}|^2 |\Af|^2) \frac{1}{1+x^2}  \bigg] \nonumber \\
\simeq \frac{1}{2} \int_i d{\cal P }\bigg[ (2+y^2-x^2) |\Abarf|^2 + \left(|\frac{p}{q}|^2 |\Af|^2\right)(x^2+y^2)  \bigg] =  \nonumber \\
\Tbari \left(1+\frac{y^2-x^2}{2}\right)+ \Ti \left(\abspoq^2\frac{x^2+y^2}{2}\right) \nonumber \\
\label{eq:cp_kshh_Ti}
\end{eqnarray}
%
where $\int_i d{\cal P}$ indicates the integral over the phase space relative to the 
 Dalitz plot bin $i$ and also
%
\begin{eqnarray}
\Af = |\Af| e^{i\delta_f} \hspace{3cm} \Abarf = |\Abarf| e^{i\bar \delta_f} \nonumber \\
\int_i  |\Af|^2 d{\cal P} = T_i \hspace{3cm} \int_i  |\Abarf|^2 d{\cal P} = \bar T_i \nonumber \\
\frac{\int_i  \re(\Af^*\Abarf)d{\cal P}}{\sqrt{T_i\bar T_i}}  = \frac{\int_i |\Af||\Abarf| \cos(\bar\delta_f-\delta_f)d{\cal P}}{\sqrt{T_i\bar T_i}} = c_i \nonumber \\  
 \frac{\int_i  \im(\Af^*\Abarf) d{\cal P}}{\sqrt{T_i\bar T_i}}  =  \frac{\int_i |\Af||\Abarf| \sin(\bar\delta_f-\delta_f)d{\cal P}}{\sqrt{T_i\bar T_i}}  = s_i \nonumber \\ 
\label{eq:ci_si_def}
\end{eqnarray}
%
For 3-body self-conjugate decay modes the relations 
$c_i \equiv c_{\bar i}$ and $s_i \equiv - s_{\bar i}$ hold in case of \CP 
 conservation in the decay. Here, with $\bar i$ we intend the Dalitz plot region or bin obtained 
 by mirroring the bin $i$ over the axis of symmetry of the Dalitz plot. 
In case bin sizes are chosen in such a way that the amplitude variation within the $i$-th 
bin is relatively small, the following approximation might be used to determine the $s_i$ 
coefficients $| s_i| =  \sqrt{1 - c_i^2}$ and the $s_i$ sign could be determined directly 
from the data as proposed by Bondar {\it et al.}~\cite{Bondar:2008hh,Bondar:2010qs}.
In particular in ~\cite{Bondar:2008hh} it is shown how the double-tag 
 $(\kspipi)_D(\kspipi)_D$ events improve the determination of $s_i$.
% 
\par 
Eq.~\ref{eq:DalitzFlavorTagTD} can be written in a model independent way for 
each bin $i$ of the Dalitz plot in terms of the coefficient 
$c_i$, $s_i$, \Ti, \Tbari:
%
\begin{eqnarray}
\frac{d\Gamma_i[\Dz_{\textrm{phys}}\to f]/dt}{e^{-\Gamma t}{\cal N}_f} &=&
 \bigg[ \left(\Ti + |\frac{q}{p}|^2 \Tbari\right) \cosh(\Gamma y t) 
+ \left( \Ti- |\frac{q}{p}|^2 \Tbari \right) \cos(\Gamma x t) \nonumber \\
& + & 2\left( c_i\sqrt{\Ti\Tbari}\absqop\cos\phi-s_i\sqrt{\Ti\Tbari}\absqop\sin\phi\right)  \sinh(\Gamma y t)  \nonumber \\
&-& 2\left( c_i\sqrt{\Ti\Tbari}\absqop\sin\phi+s_i\sqrt{\Ti\Tbari}\absqop\cos\phi \right) \sin (\Gamma x t)  \bigg] \nonumber \\ 
\label{eq:DalitzFlavorTagTD_MI}
\end{eqnarray}
%
The above time-dependence does not assume \CP conservation in the decay of the \Dz and also 
 does not require any phenomenological parameterization of the \Dz decay amplitude.
The parameters $c_i$, $s_i$ can be obtained in a time-integrated measurement 
 of $(\kspipi)_D$ decays at \Dz-\Dzbar threshold as explained above, 
 and the parameters \Ti, \Tbari can be extracted with a time-integrated measurement 
 of flavor tagged decays both at the $\Upsilon(4S)$ and at \Dz-\Dzbar threshold.
 In both cases, the relative uncertainty on the extraction of the parameters from the 
 time-integrated measurements is about $10^{-4} \simeq {\cal O} (x^2+y^2)$ or lower.


\newpage
We report here eq.~\ref{eq:DDbarphys_rate} again for convenience:
\begin{eqnarray}
&&\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}}=\nonumber\\
&&\left(|a_+|^2+|a_-|^2\right)\cosh(y\Gamma \Delta t)+
\left(|a_+|^2-|a_-|^2\right)\cos(x\Gamma \Delta t)\nonumber\\
&&-2\re((a^*_+a_-)\sinh(y\Gamma \Delta t)+2\im(a^*_+a_-)\sin(x\Gamma \Delta t)\nonumber\\
\label{eq:DDbarphys_rate_2}
\end{eqnarray}

From the definition of $a_+$ and $a_-$ it follows:
\begin{eqnarray}
&&|a_+|^2=|\bar A_{f_1}|^2|A_{f_2}|^2+|A_{f_1}|^2|\bar A_{f_2}|^2-2\re(\bar A_{f_1}A_{f_2}A^*_{f_1}\bar A^*_{f_2})\\\label{eq:ap2}
&&|a_-|^2=|p/q|^2|A_{f_1}|^2|A_{f_2}|^2+|q/p|^2|\bar A_{f_1}|^2|\bar A_{f_2}|^2-2\re(p/q\  p^*/q^* A_{f_1}\bar A^*_{f_1} A_{f_2}\bar A^*_{f_2})\\\label{eq:am2}
&&a^*_+a_-=p/q |A_{f_2}|^2 A_{f_1}\bar A^*_{f_1}-p/q\ |A_{f_1}|^2 A_{f_2}\bar A^*_{f_2}-q/p\ |\bar A_{f_1}|^2 A^*_{f_2}\bar A_{f_2}+q/p\ |\bar A_{f_2}|^2 A^*_{f_1}\bar A_{f_1}\label{eq:apam}
\end{eqnarray}
We define:
\begin{eqnarray}
&&\int_i |A_{f_j}|^2 dP=T_{ji}\\
&&\int_i |\bar A_{f_j}|^2 dP=\bar T_{ji}\\
&&\int_i \bar A^*_{f_j}A_{f_j}dP=\sqrt{T_{ji}\bar T_{ji}}(c_{ji}+is_{ji})
\end{eqnarray}
where $\int_i dP$ is the integral overal the $i$-th bin of the Dalitz plot, $j=1,2$ indicates the two $D$ decays and $c_{ji}$ and $s_{ji}$ are the real and imaginary part of $\int_i dP/ \sqrt{T_{ji}\bar T_{ji}}$, respectively.
\begin{eqnarray}
\int_i\int_j|a_+|^2&=&\bar T_{1i}T_{2j}+T_{1i}\bar T_{2j}-2\sqrt{T_{1i}\bar T_{1i}T_{2j}\bar T_{2j}}(c_{1i}c_{2j}+s_{1i}s_{2j})\label{eq:int_a+}\\
\int_i\int_j|a_-|^2&=&\left|\frac{p}{q}\right|^2 T_{1i}T_{2j}+\left|\frac{q}{p}\right|^2 \bar T_{1i}\bar T_{2j}-\nonumber\\
&&2\sqrt{T_{1i}\bar T_{1i}T_{2j}\bar T_{2j}}\left(\cos 2\phi\ (c_{1i}c_{2j}-s_{1i}s_{2j})-\sin2\phi\ (c_{1i}s_{2j}+s_{1i}c_{2j})\right)\label{eq:int_a-}\\
\int_i\int_j \re(a^*_+a_-)&=&\left(\left|\frac{p}{q}\right| T_{2j}+\left|\frac{q}{p}\right| \bar T_{2j}\right)\sqrt{T_{1i}\bar T_{1i}}(\cos\phi\ c_{1i}+\sin\phi\ s_{1i})-\nonumber\\
&&\left(\left|\frac{p}{q}\right| T_{1i}+\left|\frac{q}{p}\right|\bar T_{1i}\right)\sqrt{T_{2j}\bar T_{2j}}(\cos\phi\ c_{2j}+\sin\phi\ s_{2j})\label{eq:int_re_a+a-}\\
\int_i\int_j \im(a^*_+a_-)&=&\left(\left|\frac{p}{q}\right| T_{2j}+\left|\frac{q}{p}\right| \bar T_{2j}\right)\sqrt{T_{1i}\bar T_{1i}}(\cos\phi\ s_{1i}-\sin\phi\ c_{1i})-\nonumber\\
&&\left(\left|\frac{p}{q}\right| T_{1i}+\left|\frac{q}{p}\right|\bar T_{1i}\right)\sqrt{T_{2j}\bar T_{2j}}(\cos\phi\ s_{2j}-\sin\phi\ c_{2j})\label{eq:int_im_a+a-};
\end{eqnarray}


Relastions~\ref{eq:int_a+}-\ref{eq:int_im_a+a-} put in~\ref{eq:DDbarphys_rate_2} tells what can be gained by performing a double-$K^0_S\pi^+\pi^-$ time-dependent analysis compared to the time-integrated case, where one is basically sensitive to the $|a_+|^2$ term only.
 