\section{Mixing and \CP, \T and \CPT violation in the neutral \D system}
\label{sec:MixCPV}
In the following we use the PDG convention\cite{pdg10_cprev} to express the time-dependent decay amplitudes and rates of a neutral pseudoscalar meson $|M^0\rangle$ and its $CP$ conjugate $|\bar{M}^0\rangle$. 
%
\begin{eqnarray}
|\Mphys\rangle=\left(g_+(t)+\z g_-(t)\right)|\M\rangle -\sqrt{1-\z^2}\frac{q}{p} g_-(t)|\Mbar\rangle\\
|\Mbarphys\rangle=\left(g_+(t)-\z g_-(t)\right)|\Mbar\rangle -\sqrt{1-\z^2}\frac{p}{q} g_-(t)|\M\rangle
\end{eqnarray}
where
\begin{equation}
g_\pm = \frac{1}{2}\left(e^{-im_Ht-\frac{1}{2}\Gamma_H t}\pm e^{-im_L t - \frac{1}{2}\Gamma_L t}\right)
\end{equation}
and $q$, $p$ are the parameters regulating indirect \CP violation and 
\z regulates \CPT violation.
\par
Defining $x\equiv\Delta m/\Gamma$ and $y\equiv\Delta\Gamma/(2\Gamma)$, and assuming $z=0$, one obtains the following time-dependent decay rates:
\begin{eqnarray}
&&\frac{d\Gamma[\Mphys\to f]/dt}{e^{-\Gamma t}\mathcal{N}_f}=\nonumber\\
&&\left(|A_f|^2+|(q/p)\bar A_f|^2\right)\cosh(y\Gamma t)+
\left(|A_f|^2-|(q/p)\bar A_f|^2\right)\cos(x\Gamma t)\nonumber\\
&&+2\re((q/p)A^*_f\bar A_f)\sinh(y\Gamma t)-2\im((q/p)A^*_f\bar A_f)\sin(x\Gamma t)\nonumber\\
&&\label{eq:mphys_rate}\\
&&\frac{d\Gamma[\Mbarphys\to f]/dt}{e^{-\Gamma t}\mathcal{N}_f}=\nonumber\\
&&\left(|(p/q)A_f|^2+|\bar A_f|^2\right)\cosh(y\Gamma t)-
\left(|(p/q)A_f|^2-|\bar A_f|^2\right)\cos(x\Gamma t)\nonumber\\
&&+2\re((p/q)A_f\bar A^*_f)\sinh(y\Gamma t)-2\im((p/q)A_f\bar A^*_f)\sin(x\Gamma t)\nonumber\\
&&\label{eq:mbarphys_rate}
\end{eqnarray}

When neutral pseudoscalar mesons are produced coherently in pairs from the decay of a vector resonance, $V\to M^0\bar M^0$, the time-dependence of their subsequent decays to final state $f_1$ and $f_2$ has a similar form to eqs.~(\ref{eq:mphys_rate},\ \ref{eq:mbarphys_rate}):
\begin{eqnarray}
&&\frac{d\Gamma[V_{\text{phys}}(t_1,t_2)\to f_1 f_2]/d\Delta t}{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}
\end{eqnarray}
where $\Delta t\equiv t_2-t_1$ is the difference in the production times, $t_1$ and $t_2$, of $f_1$ and $f_2$, respectively, and the dependence on the average decay time and on decay angles has been integrated out. The coefficients $a_+$ and $a_-$ are defined as
\begin{eqnarray}
&&a_+\equiv \bar A_{f_1} A_{f_2}-A_{f_1}\bar A_{f_2},\nonumber\\
&&a_-\equiv -\sqrt{1-\z^2}\left( \frac{q}{p}\bar A_{f_1}\bar A_{f_2}-\frac{p}{q}A_{f_1}A_{f_2}\right)+\z \left(\bar A_{f_1}A_{f_2}+ A_{f_1}\bar A_{f_2}\right) \nonumber \\
\label{eq:def_aplus_aminus}
\end{eqnarray}
%

\subsection{Comparison of time-dependent measurements at $\Upsilon(4S)$ and at charm threshold}
Time-dependent measurements performed at $\Upsilon(4S)$ use flavor-tagged $D$ decays whose flavor at $t=0$ is determined through the charge of the soft pion in the decay sequence $D^{*+}\to D^0\pi^+$ or $D^{*-}\to \bar D^0\pi^-$. The time-dependent rates are given in Eq.~\ref{eq:mphys_rate} and~\ref{eq:mbarphys_rate}.

The same time-dependent rates can be measured also in $\Psi(3770)\to D\bar D$ decays if one of the $D$ mesons is selected in a semi-leptonic final state. In fact, assuming \CPT conservation ($z=0$) and identifying $\dt\to t$ and $f_2 \to f$, we find that Eqs.~\ref{eq:DDbarphys_rate} and~\ref{eq:def_aplus_aminus} reduce to Eq.~\ref{eq:mphys_rate} if $\A_{f_1}=0$ and $\Abar_{f_1}=1$, or to Eq.~\ref{eq:mbarphys_rate} if $\Abar_{f_1}=0$ and $\A_{f_1}=1$. 

At the charm threshold, however, the quantum coherence of the $D\bar D$ system gives access to a number of additional tag states other than the flavor-eigenstates $D^0$ or $\bar D^0$. The relations between the time-dependent rate and the mixing/$CP(T)$ parameters depend on the final state of the two $D$ mesons in a way described by Eq.~\ref{eq:DDbarphys_rate} and~\ref{eq:def_aplus_aminus}. In appendix~\ref{sec:Appendix_UsefulFormulae} we have written explicitly these relations for several combinations of $D$ final states. The list is also reported in Tab.~\ref{tab:final_states_list}.

%In conclusion, a run at the charm threshold provides additional handles to investigate the mixing and $CP(T)$ violation in the charm sector. %Our goal is to quantify 

%QUI SOTTO TAVOLA GRANDE CON CANALI VS NUM EVENTS E TAU reso
%usare questa per indicare la lista di canali citata nella sezione sopra
%\begin{table}
%\end{table}

\subsection{Comparison of time-dependent and time-integrated measurements at charm threshold}
Before analyzing the specific $D$ final states at charm threshold it is instructive to make some general considerations regarding the comparison between the time-dependent and time-integrated measurements.

Since the mixing parameters $x$, $y$ are small ($<10^{-2}$) the time-dependent rate in Eq.~\ref{eq:DDbarphys_rate} can be expanded by safely neglecting the terms $o(x^2)$, $o(y^2)$:
\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(2+\frac{y^2-x^2}{2}\left(\Gamma\Delta t\right)^2\right)|a_+|^2+\left(\frac{y^2+x^2}{2}\right)|a_-|^2\left(\Gamma\Delta t\right)^2+\nonumber\\
&&2|a_+|^2+\left(\frac{y^2-x^2}{2}|a_+|^2+\frac{y^2+x^2}{2}|a_-|^2\right)\left(\Gamma\Delta t\right)^2+\nonumber\\ 
&&2\left(-y\re(a^*_+a_-)+ x\im(a^*_+a_-)\right)\Gamma \Delta t 
\label{eq:DDbarphys_rate_xy}
\end{eqnarray}
Analogously, after neglecting terms of order $o(x^2)$ and $o(y^2)$ the time-integrated rate can be written as:
\begin{eqnarray}
\Gamma=\int \frac{d\Gamma}{d\Delta t} d\Delta t \propto\frac{2}{\Gamma}\left( (2+y^2-x^2)|a_+|^2+(y^2+x^2)|a_-|^2\right)\label{eq:DDbar_rate_timeint_xy}
\end{eqnarray} 

Let us compare Eq.~\ref{eq:DDbarphys_rate_xy} and~\ref{eq:DDbar_rate_timeint_xy}. Both the time-dependent and time-independent measurements are usually mostly sensitive to the term $|a_+|^2$, except for specific cases as for example the double semileptonic tag where $a_+=0$ (see appendix~\ref{sec:Appendix_UsefulFormulae}). The term $|a_-|^2$ is in both cases suppressed by $x^2+y^2\sim 10^{-4}-10^{-5}$. The time-dependent rate keeps in addition the dependence on $-y\re(a^*_+a_-)+x\im(a^*_+a_-)$. This term constitutes the substantial difference between the time-dependent and the time-integrated measurements.
 
Our next steps can be summarized as follows
\begin{itemize}
\item evaluate what additional information is provided by the quantity $-y\re(a^*_+a_-)+x\im(a^*_+a_-)$, as a function of the selected $D$ final states
\item study how the sensitivity on the physics parameters extracted from $-y\re(a^*_+a_-)+x\im(a^*_+a_-)$, $|a_+|^2$ and $|a_-|^2$ scale as a function of the CM boost
\end{itemize}
Regarding the second item there are two competing factors that must be evaluated. One is how the time resolution improves as the boost increases, and how the improved time resolution affects the precision of the physics parameters extracted from the time-dependent measurement. The other factor is how the reconstruction efficiency reduces as the boost increases, and consequently what is the effect on the precision of the physics parameters.

At this time we have performed preliminary studies based on the selection of double $K^\pm\pi^\mp$ decays to quantify how the time resolution and the reconstruction efficiency scale as a function of the boost. The results are summarized in Fig.~\ref{fig:aveDeltaTErr_vs_bg_kpikpi} and~\ref{fig:eff_vs_bg_kpikpi}. We still have to evaluate how the error of the physics parameters extracted from the time-dependent measurement is affected by the time resolution.

