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\fancyhead[C]{\it {
SuperB Workshop VI Proceedings (Valencia, Jan 7-15, 2008)
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\begin{document}

%Title of paper
\title{Sharpening the physics case for charm physics at SuperB}

%\author{R Faccini, AD Polosa, S Tosi}
%\affiliation{INFN Roma, Piazzale A Moro 2, Roma, I-00185, Italy}
%
%

\begin{abstract}
%Ikaros
New Physics in general generates flavour changing neutral currents (FCNC). Those could be much less suppressed in the up-type than  
the down-type quark sectors. Among the up-type quarks only charm  allows the {\em full} range of probes for FCNC and thus 
New Physics in oscillation phenomena, in particular those involving CP violation. The SM makes nontrivial predictions for CP violation 
in charm transitions: {\em direct} CP violation should occur 
only  in Cabibbo suppressed 
modes on an observable level $\sim$ ${\cal O}(10^{-3})$. 

The recent evidence for $D^0 - \bar D^0$ oscillations -- with $x_D$, $y_D$ $\simeq 0.005 \div 0.01$ --  does not prove the presence of 
New Physics. However it greatly widens the stage, 
on which CP violation can appear and establish itself as a manifestation of New Physics. 
Within the SM time dependent CP asymmetries might reach the $10^{-5}$ 
[$10^{-4}$] level in Cabibbo allowed and 
once [doubly] suppressed modes, whereas New Physics could enhance them by almost three orders of magnitude. A search for New Physics 
should then aim at sensitivity levels 
of ${\cal O}(10^{-3})$ or better and ${\cal O}(10^{-2})$ or better 
in Cabibbo allowed or once suppressed nonleptonic channels and in doubly Cabibbo suppressed or 
wrong-sign semileptonic modes, respectively. Signals for New Physics might actually be clearer in 
$D$ than in $B$ decays: for while {\em conventional} New Physics scenarios tend to create larger effects in the latter than the former, 
those signals would also have to contend with a much larger SM `background' in the latter than the former.  

The required searches can be undertaken very profitably in runs at the $\Upsilon (4S)$ using $D^*$ tagging and tracking of the $D$ production 
and decay vertices. Relatively short runs in the charm threshold region can provide unique and important information on strong phases needed 
for a proper interpretation of results obtained in $\Upsilon (4S)$ runs. 
They might reveal significantly enhanced effects that can be seen only in $e^+e^- \to D^0\bar D^0$ 
running. 
%They might also reveal significantly enhanced effects that can be seen only there. 
\end{abstract}
%\maketitle must follow title, authors, abstract
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Beginning of Ikaros, part 1
\section{Searching for New Physics in charm decays -- mainly CP violation}
\label{sec:charm-intro}

\subsection{The landscape}

New Physics in general generates flavour changing neutral currents (FCNC). The SM had to be crafted carefully to suppress them in the 
strangeness sector down to the observed level. Those FCNC could actually be much less suppressed in the up-type than the down-type quark 
sectors. Among the up-type quarks it is only charm that allows the {\em full} range of probes for FCNC and thus New Physics in oscillation 
phenomena, in particular those involving CP violation: (i) Top quarks decay before they can hadronize, 
and without top hadrons $T^0$ oscillations cannot occur. Furthermore the sheer size of phase space in top decays greatly reduces 
the coherence between different amplitudes needed to make 
direct CP violation observable. (ii) Hadrons built with $u$ and $\bar u$ quarks like the $\pi^0$ and 
$\eta$ are their own antiparticle; thus there can be no $\pi^0 - \pi^0$ etc. oscillations as a matter 
of principle. They also decay very rapidly. In addition they possess so few decay channels that 
CPT invariance largely rules out CP asymmetries in their decays.  

Strong evidence for $D^0 - \bar D^0$ oscillations has been recently found pointing to~\cite{ref:hfag} 
\begin{eqnarray}
x_D &\equiv& \frac{\Delta M_D}{\bar \Gamma_D} = 0.0081 \pm 0.0033~, \\
y_D &\equiv& \frac{\Delta \Gamma_D}{2\bar \Gamma_D} = 0.0031 \pm 0.0028~.
\end{eqnarray} 
According to our present understanding -- or lack thereof -- these quantities could be produced by SM dynamics, yet $x_D$ could also 
contain substantial contributions from New Physics. It will require a 
theoretical breakthrough to resolve this ambiguity in the interpretation of the data. 

One will be on much firmer ground in interpreting CP asymmetries. For on one hand $D^0 - \bar D^0$ 
oscillations greatly widen the stage, on which CP violation can appear and establish itself as a manifestation of New Physics; on the 
other hand the SM makes very nontrivial predictions for CP violation on charm transitions. In CKM dynamics there is a weak 
phase in $\Delta C =1$ transitions 
entering (in the Wolfenstein representation) through $V_{cs}$, yet it is a highly diluted one: 
\beq 
V_{cs} \simeq 1 - \frac{1}{2} \lambda ^2 - i \eta A^2 \lambda ^4 \simeq 0.97 - 6\cdot 10^{-4} i~.
\eeq 
Furthermore one needs two different, yet coherent amplitudes contribute to the same channel to obtain a  direct CP asymmetry. 
Within the SM this can happen only in Cabibbo suppressed 
modes on an observable level, namely no more than ${\cal O}(10^{-3})$. That means that any 
observation of a direct CP violation in Cabibbo allowed or doubly suppressed channel establishes the 
intervention of New Physics. The only exception to this general rule is provided by modes like 
$D^{\pm} \to K_S\pi^{\pm}$, where one becomes sensitive to (i) the interference between 
$D^+ \to \bar K^0 \pi^+$ and $D^+ \to K^0 \pi^+$ and (ii) the slight CP impurity in the $K_S$ state. The latter effect dominates 
inducing a CP asymmetry of $3.3 \cdot 10^{-3}$. 

With $x_D$, $y_D \sim 0.005 \div 0.01$ the possibilities for CP asymmetries proliferate. In addition 
to the aforementioned direct CP violation one can encounter time {\em dependent} CP asymmetries. 
The latter can be induced by CP violation in $\Delta C=2$ dynamics or even by CP conserving 
contributions to the latter that can make the weak phase in an $\Delta C=1$ amplitude observable. 
In both cases an educated SM guess points to time dependent CP asymmetries of order 
$10^{-3}x_D \sim 10^{-5}$. 

%%%%%%%%%%%%%
\subsection{The menu}
%%%%%%%%%%%%

There are three classes of CP asymmetries: 
\begin{enumerate}
\item 
Direct CP violation can lead to a difference in the rates for $D \to f$ and $\bar D \to \bar f$:  
\beq 
|A_f| \equiv |A(D \to f)| \neq |\bar A_{\bar f}| \equiv |A(\bar D \to \bar f)|~.
\label{ANEQBARA}
\eeq 
Such asymmetries in partial widths require strong phase shifts due to final state 
interactions. Since charm decays proceed in an environment populated by many resonances, 
this requirement will hardly represent a limiting factor in general; it might make the interpretation of signals a more complex task though. 
\item 
{\em Indirect} CP violation -- i.e., one that resides purely in $\Delta C=2$ transitions. One measure for it 
is provided by 
\beq
|q/p| \sim 1 + \frac{\Delta \Gamma_D}{\Delta M_D}{\rm sin}\phi_{\rm weak}  \neq 1 ~.
\label{QOVERP} 
\eeq 
The  same educated SM guess mentioned before points to $|1 - |q/p|| \sim {\rm several}\times 10^{-4}$. 
One should note here that the factor $\Delta \Gamma_D/\Delta M_D$ apparently is close to unity and thus provides no suppression 
to this observable unlike in the case of $B^0$ mesons. Thus one has practically undiluted access to a weak phase due to the 
intervention of New Physics in $D^0 - \bar D^0$ 
oscillations. As discussed later 
such an asymmetry can be searched for cleanly in semileptonic decays of neutral $D$ mesons. While we already know the ratio of 
wrong-sign leptons is small, their CP asymmetry could conceivably be as large as several percent! While the rate of wrong-sign 
leptons oscillates with time, the CP asymmetry does not. 

\item
In qualitative analogy to $B_d \to \psi K_S$ a time dependent CP asymmetry can arise due to an interference between an oscillation 
and decay amplitude:
\beq 
\phi_f = {\rm arg} \left( \frac{q}{p}\frac{\overline{A}_{\overline{f}}}{A_f} \right) \ne 0 \; . 
\label{INTER}
\eeq
A CP asymmetry generated by $\phi_f \neq 0$ is also proportional to sin$\Delta M_D t 
\simeq x_D (t/\tau _D)$ and thus effectively bounded by $x_D$; i.e., the present lack of a signal for a 
time dependent CP asymmetry in $D^0 \to K^+K^-$ on about the 1\% level is not telling at all in view 
of $x_D \leq 1\%$. Yet any improvement in experimental sensitivity could reveal a genuine signal. 
\end{enumerate}

Searching for CP violation in charm decays is not a `wild goose chase'. For we know that baryogenesis requires the presence of 
CP violating New Physics. Signals for such New Physics might actually be clearer in 
$D$ than in $B$ decays: for while {\em conventional} New Physics scenarios tend to create larger effects in the latter than the former, 
those signals would also have to contend with a much larger SM `background' in the latter than the former; i.e., the 
theoretical `signal-to-noise' ratio might be better in charm decays.   

The required searches can be undertaken very profitably in runs at the $\Upsilon (4S)$ 
tagging the \Dz flavor at production time using $D^{*+} \to D^0 \pi^+$ decays
and reconstructing the proper decay time and its error tracking the $D$ production and decay vertices with 
contraints provided by the position and size of the tight $e^+ e^-$ interaction region.
Relatively short runs in the charm threshold region, e.g. \psiprpr, can provide unique and important information on strong phases needed 
for a proper interpretation of results obtained in $\Upsilon (4S)$ runs. In the latter \Dz flavor tagging exploits the
quantum correlations at \psiprpr, while the poor proper time resolution (about the \Dz lifetime) 
will make time-dependent measurements challenging. 

{\em In summary}: Comprehensive and precise studies of CP invariance in charm decays provide sensitive probes for the presence of New Physics. 
\begin{itemize}
\item 
`Comprehensive' means that one analyses nonleptonic as well as semileptonic channels on all 
Cabibbo levels in as many modes as possible; i.e., also in final states containing neutrals. 
\item 
`Precise' means that one achieves sensitivity levels of $10^{-3}$ or even better. 

\end{itemize}

Charm decays provide another highly promising avenue towards finding CP violation, namely in final state distributions rather 
than in partial widths considered so far. This issue will be addressed separately 
below. 


\subsection{Side remarks on rare decays}

The obvious motivation for measuring $D^+/D^+_s \to \mu^+ \nu$, $\tau^+\nu$ is to extract the decay 
constants $f_D$ and $f_{D_s}$ to compare them with the findings of lattice QCD and hopefully validate 
the latter's findings with high accuracy. A more ambitious goal is to probe for contributions from a charged Higgs field. 

The mode $D^0 \to \mu^+\mu^-$ arises within the SM mainly through a two photon intermediate 
state -- $D^0 \to "\gamma \gamma" \to \mu^+\mu^-$ -- and can reach the $10^{-12}$ level. With the present 
experimental upper bound of $1.3 \cdot 10^{-6}$ there is a search window for New Physics of six orders 
of magnitude. Multi-Higgs models or SUSY models with R parity breaking could conceivably induce a signal in 
a range as "large" as few$\times 10^{-8}$ and $10^{-6}$. 

Channels like $D\to \gamma h$, $l^+l^-h$, $l^+l^-h_1h_2$ with $h$ denoting a hadron receive 
relatively sizable contributions within the SM from long distance dynamics, making a search for New Physics contributions 
not very promising there, at least unless one can measure 
precisely the lepton spectra in the final states. 

One can probe a rather exotic variant of New Physics by searching for two-body modes 
$D^+ \to K^+/\pi^+ f$; the charge neutral $f$ denotes a `familon', which could arise as the Nambu-Goldstone boson resulting 
from the spontaneous breakdown of a global family symmetry. It has been searched for in $K^+$ and $B^+$ decays, 
but apparently never in $D^+$ decays. 

% end of Ikaros, part 1


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\Dz\Dzb mixing running at \FourS and \psiprpr energies}
\label{sec:charm-mixing}


The parameters describing charm mixing can be measured in time-dependent 
studies of $D$ mesons or with time-integrated observables of $D$'s produced
coherently near charm threhsold.

The time-dependent $\DZ\DZB$ mixing formalism and a summary of recent 
experimental results can be found in Ref.~\cite{pdg2008}.
Many different charm decay modes can be used to search for charm mixing.
\begin{itemize}
\item The appearance of ``wrong-sign'' kaons in semileptonic decays would provide
direct evidence for $\DZ\DZB$ oscillations (or another process of beyond Standard
Model origin).
\item The most precise limits are obtained by exploiting the time-dependence of
$D$ decays produced in $e^+e-$ collision near 10~GeV.
\begin{itemize}
\item The wrong-sign hadronic decay $\DZ\to K^+\pi^-$ is sensitive to linear combinations of the mass and lifetime differences, denoted $x^{\prime 2}$ and $y^{\prime}$.
The relation of these parameters to $x_D$ and $y_D$ is controlled by a strong
phase difference $\delta_{K\pi}$. 
\item Direct measurements of $x_D$ and $y_D$ independent of unknown
strong interaction phases can also be made using time-dependent studies of 
amplitudes present in multi-body decays of the $\DZ$, for example. 
$\DZ \to K^0_S\pi^+\pi^-$. 
\item Direct evidence of $y_D$ can also appear through 
lifetime differences between decays to $CP$ eigenstates. 
The measured quantity in 
this case, $y_{CP}$, is equivalent to $y_D$ in the absense $CP$ violation.
\end{itemize}
\item Another approach is to study quantum correlations near charm threshold
\cite{Asner:2005wf} in
$e^+e^- \to \DZ\DZB(\pi^0)$ and $e^+e^- \to \DZ\DZB\gamma(\pi^0)$ which yield
$C$-odd and $C$-even $\DZ\DZB$ pairs, respectively. Taken together, the 
time-integrated decay rate to semileptonic, $K\pi$, and $CP$ eigenstates provide
sensitivity to $x_D$, $y_D$, and $\cos\delta_{K\pi}$. 
\end{itemize}

Recently, several new results provide evidence that charm mixing is at the upper
end of the range of Standard Model predictions.

BABAR\cite{Aubert:2007wf} and CDF\cite{Aaltonen:2007uc} 
find evidence for oscillations in $\DZ \to K^+\pi^-$ with
3.9$\sigma$ $(\Delta {\rm Log} {\cal L})$ and 3.8$\sigma$ (Bayesian), respectively.
The most precise measurement is from Belle which 
excludes $x^{\prime 2}=y^\prime=0$ at 2.1$\sigma$\cite{Zhang:2006dp} 
(Feldman-Cousins).

Belle\cite{Staric:2007dt} and BABAR\cite{Aubert:2007en} 
see a 3.2$\sigma$ and 3$\sigma$ effect, respectively, 
for $y_{cp}$ in $\DZ\to K^+K^-$. 
The most precise measurement of $y_D$ is in $\DZ \to K^0_S\pi^+\pi^-$ 
from Belle\cite{Abe:2007rd} and is only
1.2$\sigma$ significant. From the same analysis, Belle also reports a 
2.4$\sigma$ significant result for $x_D$. 
The current situation is far from clear.

All mixing measurements can be combined to obtain
world average (WA) values for $x$ and $y$. The Heavy Flavor
Averaging Group (HFAG) has done such a combination 
%in two ways
\cite{Schwartz:2007fw}\cite{Schwartz:2008}
%: {\it (a)}\ 
%adding together log-likelihood functions for $x$, $y$, and $\delta$ as
%obtained from measurements of relevant observables;
%%analyses of 
%%resonance~\cite{hfag-charm}. 
%{(b)}\ doing a global fit to the measured observables for $x$, $y$, $\delta$,
%$\delta^{}_2$, and $R_D$, being careful to account for the correlations among
%observables by using the corresponding error matrices as provided by the 
%experimental collaborations. For both methods the observables used are obtained
%from measurements of $D^0\rightarrow K^+\ell\nu$, $D^0\rightarrow K^+K^-, \pi^+\pi^-$, $D^0\rightarrow K^+\pi^-$,
%$D^0\rightarrow K^+\pi^-\pi^0$, $D^0\rightarrow K^+\pi^-\pi^+\pi^-$, and
%$D^0\rightarrow K^0_S\pi^+\pi^-$ decays, as well as CLEO-c results for 
%double-tagged 
%branching fractions measured at the $\psi(3770)$ resonance (for details see 
%previous sections of this review).
%The first method has the advantage that non-Gaussian errors are accounted for,
%whereas the second method has the advantage that it is readily expanded to
%allow for $CP$ violation. In that case three additional parameters are included
%in the fit:
%$A^{}_D\equiv (R^+_D-R^-_D)/(R^+_D+R^-_D)$, $|q/p|$, and
%${\rm Arg}(q/p)\equiv\phi$. When both methods are applied 
%to the same set of observables, almost identical results 
%are obtained.

%%The likelihood function is condensed to one dimension by
%%letting, for any value of a single fit parameter (column 1 of \Table{tab:HFAG}), 
%%the other fit parameters take the preferred values. The resulting likelihoods 
%%give central values, 68.3\% C.L. intervals and 95\% C.L. intervals are reported in
%%\Table{tab:HFAG}. 
%%%The results of the CLEO-c extended fit\cite{cleoc-tqca} are
%%%compared with the HFAG $CP$ conserving fit are also shown in \Table{tab:HFAG}.
%%%The CLEO-c and HFAG results are comparable.
%%The HFAG likelihood function is condensed to two dimensions in
%%a similar manner. \Figure{fig:hfag_mixing_contour} and \Figure{fig:hfag_cpv_contour} show the two-dimensional $1\sigma$-$5\sigma$ contours for $(x,y)$ and
%%$(|q/p|=1+A_M,Arg(q/p)=\phi)$. The HFAG fit excluded the no mixing scenario 
%%($x=y=0$) at $5.7\sigma$ - somewhat more restrictive than the CLEO-c WA.
%%The CLEO-c determination of mixing averages 
%%%employs a different methodology and 
%%does not yet incorporate the most recent results from BABAR\cite{Aubert:2007en} and 
%%CDF\cite{Aaltonen:2007uc}. 
%%The HFAG average\cite{hfag-charm} includes the
%%results of a fit to the CLEO-c data without external input. 

%For the global fit, confidence contours in the two dimensions $(x,y)$ 
%and $(|q/p|,\phi)$ are obtained by letting, for any point in the
%two-dimensional plane, all other fit parameters take their 
%preferred values. 
The resulting $1\sigma$-$5\sigma$ contours 
are shown in Fig.~\ref{fig:hfag_mixing_contour} and Fig.~\ref{fig:hfag_cpv_contour}. 
The fits exclude the no-mixing 
point ($x\!=\!y\!=\!0$) at $6.7\sigma$ for both the no $CP$ violation scenario
and the case allowing for $CP$ violation. One-dimensional 
likelihood functions for parameters are obtained by allowing, for 
any value of the parameter, all other fit parameters to take their 
preferred values. The resulting likelihood functions give central 
values, 68.3\% C.L. intervals, and 95\% C.L. intervals as listed 
in Table~\ref{tab:HFAG}.

From the results of the HFAG averaging, the following can
be concluded:

\begin{itemize}
\item
The experimental data consistently indicates
that $D^0$'s undergo mixing. The effect is presumably dominated 
by long-distance processes, and unless $|x|\gg |y|$, it may
be difficult to identify new physics from mixing alone.
\item
Since $y_{CP}$ is positive, the $CP$-even state is shorter-lived,
as in the $K^0$-$\overline K{^0}$ system. However, since $x$ appears to be 
positive, the $CP$-even state is heavier, 
unlike in the $K^0$-$\overline K{^0}$ system.
\item
There is no evidence yet for $CP$-violation in the $D^0$-$\overline D{^0}$ 
system.
\end{itemize}
%BEGIN TABLE 4
\begin{table}
\label{tab:HFAG}
\caption{
HFAG Charm Mixing Averages.}
%\vglue -18pt
\begin{tabular}{cccc} \hline
Fit & Parameter & HFAG Average & 95\% C.L. Interval\\
%\hline
%No & $x(\%)$ & $0.98^{+0.26}_{-0.27}$  & \\
%CPV & $y(\%)$ & $0.75\pm 0.18$  & \\
%  & $R_D(\%)$ & $0.335\pm 0.009$  & \\
%  & $\delta_{K\pi}(^\circ)$ & $21.6^{+11.6}_{-12.6}$  & \\
% & $\delta_{K\pi\pi^0}(^\circ)$ & $30.8^{+25.8}_{-25.0}$  & \\
%%\tableheaddoublerule
%Fit & Parameter & HFAG Average & 95\% C.L. Interval \\
\hline
CPV & $x(\%)$ & $0.97^{+0.27}_{-0.29}$ & (0.39:1.48) \\
 & $y(\%)$ & $0.78^{+0.18}_{-0.19}$ (0.41:1.13) &  \\
 & $R_D(\%)$ &  $0.335\pm 0.009$ & (0.316:0.353) \\
  & $\delta_{K\pi}(^\circ)$ & $21.9^{+11.5}_{-12.5}$ & (-6.3:44.6) \\
 & $\delta_{K\pi\pi^0}(^\circ)$ &  $32.4^{+25.1}_{-25.8}$ & (-20.3:82.7) \\
 & $A_D(\%)$ &  $-2.2\pm 2.5$ & (-7.10:2.67) \\
 & $|p/q|$ & $0.86^{+0.18}_{-0.15}$ & (0.59:1.23) \\
 & $\phi(^\circ)$ & $-9.6^{+8.3}_{-9.5}$ & (-30.3:6.5) \\ \hline
\end{tabular}
\end{table}


%\midfigure{fig:hfag_ycp}
%\FigureInsert{hfag_ycp.eps,width=3.25in}{center}
%\Caption
%World average value for $y_{CP}$, from HFAG~\cite{hfag-charm}.
%\endCaption
%\endfigure

\begin{figure}
%\FigureInsert{hfag_contour_mixingcpv.eps,width=3.25in}{center}
\includegraphics[width=0.7\linewidth]{hfag_contour_mixingcpv.eps}
\caption{
Two-dimensional $1\sigma$-$5\sigma$ contours
for $(x,y)$, obtained from a global fit to the measured observables
for $x$, $y$, $\delta_{K\pi}$, $\delta_{K\pi\pi^0}$, and $R_D$
from 
measurements of $D^0\rightarrow K^+\ell\nu$, $D^0\rightarrow h^+h^-$, 
$D^0\rightarrow K^+\pi^-$, $D^0\rightarrow K^+\pi^-\pi^0$, 
$D^0\rightarrow K^+\pi^-\pi^+\pi^-$, and $D^0\rightarrow K^0_S\pi^+\pi^-$ decays,
and double-tagged branching fractions measured 
at the $\psi(3770)$ resonance (from HFAG~\cite{hfag-charm}).}
\label{fig:hfag_mixing_contour}
\end{figure}


\begin{figure}
%\FigureInsert{hfag_contour_cpv.eps,width=3.25in}{center}
\includegraphics[width=0.7\linewidth]{hfag_contour_cpv.eps}
\caption{
Two-dimensional $1\sigma$-$5\sigma$ contours
for $(|q/p|$,Arg$(q/p))$, obtained from a global fit to the measured observables
for $x$, $y$, $\delta_{K\pi}$, $\delta_{K\pi\pi^0}$, and $R_D$
from 
measurements of $D^0\rightarrow K^+\ell\nu$, $D^0\rightarrow h^+h^-$, 
$D^0\rightarrow K^+\pi^-$, $D^0\rightarrow K^+\pi^-\pi^0$, 
$D^0\rightarrow K^+\pi^-\pi^+\pi^-$, and $D^0\rightarrow K^0_S\pi^+\pi^-$ decays,
and double-tagged branching fractions measured 
at the $\psi(3770)$ resonance (from HFAG~\cite{hfag-charm}).}
\label{fig:hfag_cpv_contour}
\end{figure}

{ The interpretation of the new results in terms of New Physics is inconclusive. 
It is not yet clear whether the effect is caused by $x=0$ or $y=0$ or both, although the latter is favored,
as shown in Table~\ref{tab:HFAG}. Furthermore, there is no single 5$\sigma$
observation of charm mixing nor is one anticipated from the current B-factories.
This situation will be remedied by results anticipated from SuperB.
Table~\ref{mixtab2} shows the senstivity to mixing in $\DZ\to K^+\pi^-$, $K^+K^-$, and $K^0_S\pi^+\pi^-$ channels from the $\FourS$ data 
is in excess of 5$\sigma$ if the lifetime and
mass differences in the $\DZ$ system lie at the upper end of the
range of Standard Model predictions. 

Table~\ref{mixtab2} also shows the sensitivity to mixing from two months of running at charm threshold. The sensitivity to the mixing parameters is comparable to
five years at $\FourS$, with distinct systematic uncertainties.
The $\psiprpr$ data provides unique sensitivity to $\cos\delta$. Although
$\cos\delta$ can be determined from a global fit to $\FourS$ results, the
direct measurement from $\psiprpr$ data allow $y^\prime$ and $x^{\prime 2}$
determined from $\DZ\to K^+\pi^-$ to contribute to the precision determination
of $x$ and $y$. 

Although the $D$'s from $\psiprpr$ are produced nearly at rest in the 
center-of-mass frame, the asymmetric $e^+e^-$ collisions make time-dependent 
mixing analyses are possible.
However, since the production rate of 
charm during threshold running and $\Upsilon$(4S) running is comparable, the
statistical power of the time-dependent analyses near threshold is small.



\begin{table}
\caption{Approximate expected precision ($\sigma$) on the measured
quantities using methods described in the text for the integrated
luminosity of 75~ab$^{-1}$ at SuperB at 10~GeV and 300~fb$^{-1}$
($\sim$ two months)
running at charm threshold.
%% GW: add below plus numbers
%The upgraded LHCb numbers are merely the results from Table~\ref{mixtab1} scaled to the
%new integrated luminosity.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{tabular}{|cccc|}
%\noalign{\vspace{-8pt}}
\hline\hline
Mode & Observable & $\FourS$  & $\psi(3770)$ \\
     &            & (75~ab$^{-1}$) &  (300~fb$^{-1}$) \\
$\DZ\!\to\! K^+\pi^-$ & $x^{\prime 2}$  & $3\times 10^{-5}$ & \\
                  & $y^{\prime}$   & $7\times 10^{-4}$ & \\
$\DZ\!\to\! K^+K^-$ & $y_{CP}$ & $5\times 10^{-4}$ & \\
$\DZ\!\to\! K^0_S\pi^+\pi^-$ & $x$ & $4.9\times 10^{-4}$ & \\
                        & $y$  &  $3.5\times 10^{-4}$ & \\
                        & $|q/p|$  &  $3\times 10^{-2}$ & \\
                        & $\phi$  &  $2^\circ$ & \\
$\psi(3770) \!\to\! \DZ\DZB$ & $x^2$  & &  $(1\!-\!2)\times 10^{-5}$ \\
                         & $y$ & & $(1\!-\!2)\times 10^{-3}$ \\
                & $\cos\delta$ & & $(0.01\!-\!0.02)$ \\
\hline\hline
\end{tabular}
\label{mixtab2}
\end{table}

A serious limitation in the interpretation of charm oscillations in terms of New Physics is the theoretical uncertainty on the Standard Model prediction. 
However, the recent evidence for oscillations opens the window to 
searches for $CP$ asymmetries that do provide unequivocal New Physics 
signals. The sensitivity to these New Physics signals is shown in 
Fig.~\ref{fig:hfag_cpv_projection}. SuperB will probe $CP$ violation in charm mixing
at the ${\cal O}(10^{-3})$ level.


\begin{figure}
%\FigureInsert{hfag_contour_cpv.eps,width=3.25in}{center}
%\includegraphics[width=0.7\linewidth]{hfag_contour_cpv.eps}
\vskip 0.7\linewidth
\caption{
Two-dimensional $1\sigma$-$5\sigma$ contours
for $(|q/p|$,Arg$(q/p))$, 
for $(|q/p|$,Arg$(q/p))$, obtained from a global fit to the measured observables
for $x$, $y$, $\delta_{K\pi}$, $\delta_{K\pi\pi^0}$, and $R_D$
from the sensitivity estimates 
%measurements of $D^0\rightarrow K^+\ell\nu$, 
of $D^0\rightarrow h^+h^-$, 
$D^0\rightarrow K^+\pi^-$, 
%$D^0\rightarrow K^+\pi^-\pi^0$, $D^0\rightarrow K^+\pi^-\pi^+\pi^-$, 
and $D^0\rightarrow K^0_S\pi^+\pi^-$ decays,
and double-tagged branching fractions measured 
at the $\psi(3770)$ resonance.}
\label{fig:hfag_cpv_projection}
\end{figure}



%(In progress)

% David

%\begin{itemize}
%\item
%  Recap of sensitivities which can be reached at SuperB and comparison with other charm physics experiments 
%  (CLEO-c, BESIII, SuperKEK, LHCb, Novosibirk tau-charm). Emphasize that mixing
%  will mainly be used for validation, even if better precisions will be reached than any other experiment before.
%\item
%  Discussion of interplay between running at \FourS and at \psiprpr. Emphasis on model independent way to determine \Dz\Dzb relative
%  strong phase. Results against months of running at \psiprpr.
%\item
%  Brief discussion on our estimates showing that at \psiprpr time dependent CPV measurement could also be done since $\sigma (t) ~ \tau_D$).
%However, this is not the case for running at \psiprpr.
%\end{itemize}

%(David)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{CP violation}
\label{sec:charm-CPV}

%CP violation can only be implemente trough a complex phase in some effective couplings.
%To become CP violation observable, two different and coherent amplitudes have to contribute to an observable. 
%This requirement can be implemented through two scenarios: two different $\Delta C=1$ amplitudes of fixed ratio 
%leading to the same final state (direct CP violation), 
%\begin{equation}
%\left|\frac{\overline{A}_f}{A_f}\right| \ne 1~, 
%\end{equation}
%and $\Dz-\Dzb$ oscillations driven by $\Delta C=2$ dynamics providing the second amplitude, the weight of which varies with time.
%In processes involving oscillations there are two gateways through which CP violation can enter. The first, 
%through
%\begin{equation}
%|q/p| \ne 1~,
%\end{equation}
%which unequivocally constitutes CP violation in $\Delta C=2$ dynamics (direct CP violation). $p$ and $q$ are the usual
%complex parameters obtained from diagonalizing the $\Dz - \Dzb$ mass matrix with $|p|^2 + |q|^2 = 1$.
%The second, via
%\begin{equation}
%\phi_f = {\rm arg} \left( \frac{q}{p}\frac{\overline{A}_f}{A_f} \right) \ne 0~, 
%\end{equation}
%which reflects the interference of CP violation in $\Delta C = 1,2$ dynamics (CP violation in interference between mixing and decay).

%CP violation can also be searched for in transitions forbidden by CP invariance, where both the initial and final
%states can be prepared as CP-eigenstates. An example is the transition
%$\psiprpr \to \Dz\Dzb \to f_1 f_2$, where $f_1$ and $f_2$ are different final CP-eigenstates of the same parity.
%However, since the process involve CP-violating decay rates, the observables are of second order in the CP-violating
%parameters, making this strategy probably out of the required sensitivity reach.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Direct CP violation}
\label{sec:charm-directCPV}

Searches for CP violation in $\Delta C=1$ transitions can be performed measuring asymmetries in the partial widths 
or in final state distributions. 

Golden modes for the former are the Cabibbo suppressed decays $D^0 \to h^+ h^-$, $h=K,\pi$, 
and the doubly Cabibbo suppressed $D^0 \to K^+ \pi^-$. 
These studies can be performed either time integrated or analyzing the time dependence
of the \Dz and \Dzb decay rates, although in both cases time integrated asymmetries are measured. 
Data at \FourS provides the largest data sample
with excellent purities (as large as $\sim 99\%$). The contamination from \BB decays can be virtually eliminated
imposing a $2.5$~\gevc cut on the $D$ momentum in center-of-mass frame which keeps more than 85$\%$ of signal events.

The most precise analysis to date~\cite{ref:babarDtoKKandDtoPiPi} compares time integrated 
$\Dz \to h^+ h^-$ and $\Dzb \to h^+ h^-$ rates, 
$a_{CP}^{hh} = [N_{\Dz} - N_{\Dzb}]/[N_{\Dz} + N_{\Dzb}]$, where $N_{\Dz}$ ($N_{\Dzb}$) is the 
number of \Dz (\Dzb) mesons decaying into $h^+ h^-$ final state.
In this construction all CP violation contributions, direct and
indirect are present. The presence of direct CP violation in one or both modes would be signaled by a non-vanishing
difference between the asymmetries for $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$, $a_{CP}^{KK} \neq a_{CP}^{\pi\pi}$ . 
%CP-violating asymmetries in these modes are predicted to be $\sim$ ${\cal O}(10^{-5}) - {\cal O}(10^{-4})$. 
Two are the main experimental challenges of these measurements. 
Firstly, the experimental asymmetry in the \Dz flavor tagging.
To correct for this asymmetry it is measured the relative detection efficiency for soft pions in data using
the Cabibbo allowed decay $D^0 \to K^- \pi^-$ with (tagged) and without (non-tagged) soft-pion flavor tagging, 
as a function of the pion-momentum magnitude abd polar angle in the lab frame. For the azimuth dependence an
integrated scale factor is enough, since charm production is azimuthally uniform.
Since the reconstructed modes are CP-even, this in the only detector asymmetry.
Secondly, the forward-backward (FB) asymmetry in $c\bar{c}$ production at \FourS,
consequence of the $\gamma/Z^0$ interference and higher order QED corrections 
(both at percent level at this energy), coupled with the asymmetric acceptance of the detector,
which produces a difference in the numbers of reconstructed $D^0$ and $\bar{D}^0$ events.
This effect is directly measured by determining the number of \Dz and \Dzb events
(after soft pion asymmetry correction) as a function of $\cos\theta_D^{CM}$ and decomposing
these into an even (that represents the CP asymmetry and should be independent of $|\cos\theta_D^{CM}|$) and odd 
(that represents the FB production asymmetry) parts. The associated systematic uncertainties are therefore
not a limiting factor and have mostly an statistical nature. Other potential sources are highly suppressed because the 
final states are reconstructed identically for $D^0$ and $\bar{D}^0$. 
With a SuperB luminosity of 75 $ab^{-1}$, sensitivities at $3\times10^{-4}$ and $4\times10^{-4}$ level, 
for $a_{CP}^{KK}$ and $a_{CP}^{\pi\pi}$ respectively, are foreseen.

The time-dependent D-mixing analysis of DCS (wrong sign) $\Dz \to K^+ \pi^-$ and $\Dzb \to K^- \pi^+$ decays can be
used to separate the contributions from DCS decays from $\Dz - \Dzb$ mixing, separately for \Dz and \Dzb. 
A direct CP asymmetry can then be constructed from the difference of DCS \Dz and \Dzb decays, 
$A_D = (R_{\Dz} - R_{\Dzb})/(R_{\Dz} + R_{\Dzb})$, where $R_{\Dz}(R_{\Dzb})$  is the \Dz(\Dzb) DCS rate.
%The main advantage of this strategy is that it provides a distinction between direct and indirect CP violation.
The main experimental difficulties in this analysis are the accurate proper time reconstruction and calibration,
together with asymmetry in the \Dz flavor tagging and the modeling of the differences between $K^+$ and $K^-$ 
absorption in the detector. At SuperB, the much smaller luminous region and the 
significantly enhanced vertexing capabilities, will provide proper time significances at $10\sigma$ level 
(3-4 times better than in BaBar~\cite{ref:babarDtoKPi}, with decay length resolution of about 80~$\mu$m, $\sim 3\sigma$),
%In the BaBar analysis~\cite{ref:babarDtoKPi}, the most probable value of the proper time resolution is 0.16 ps
%(decay length resolution of about 80~$\mu$m), providing a $3\sigma$ separation.
%At SuperB, with a much smaller luminous region and the significantly enhanced vertexing capabilities, 
%the proper time significance will significantly increase (about a factor 3 or 4), 
significantly helping in reducing the systematic uncertainties associated to the modelling of the 
long decay time component and possible biases.
%from other D decays in the signal region (outliers of the resolution function model) and possible biases. 
Systematic uncertainties related to the asymmetry in the soft-pion tagging can be keep under
control using a similar procedure to that outlined above. Corrections %and systematic uncertainties
due to the FB production asymmetry and kaon hadronic interactions can be performed
relying mainly on data, through untagged $\Dz \to K^- \pi^+$ and $\Dzb \to K^+ \pi^-$ 
decays measured as a function of $\cos\theta_D^{CM}$.
Scaling the statistical uncertainty from the BaBar analysis to 75 $ab^{-1}$ we obtain a 
sensitivity on $A_D$ of $4\times10^{-3}$. To reach or improve this sensitivie level, systematic uncertainties,
currently $15\times 10^{-3}$, will have to be reduced by a factor 5 or better, something feasible taking into
accout that systematic corrections rely mainly on data.

For asymmetries in final state distributions the simplest way is to compare CP conjugate Dalitz plots for 3-body decays.
Different regions of the Dalitz plot may exhibit CP asymmetries of varying signs that largely cancel out when one
integrates over the whole phase space, therefore subdomains of the Dalitz plot could contain significantly larger 
CP asymmetries than the whole phase space. Since understanding the dynamics is not an easy goal to achive,
one could try up to four strategies, three of which are model independent. First, quantify differences
between the \Dz and \Dzb Dalitz plots in two dimensions. Secondly, look for differences in the angular moments
of \Dz and \Dzb intensity distributions. Thirdly, in a model-dependent approach, look for CP asymmetries
in the amplitudes describing intermediate states in the \Dz and \Dzb decays. Finally, look for the phase-space
integrated asymmetry. Asymmetries in the \Dz flavor asignment and FB production asymetries only affect the last
method, and can be kept under control as previously discussed. From the pionnering Babar analysis using
$\Dz \to \pi^- \pi^+ \pi^0$ and $\Dz \to K^- K^+ \pi^0$~\cite{ref:babarDtoKKpi0andDtoPiPiPi0}, 
sensitivities at $3\times10^{-4}$ and $9\times10^{-4}$ level, respectively, are anticipated.


For more complex final states other probes have to be employed, a golden example discussed in Sec.~\ref{sec:Todd}.


%  Double cabibbo suppressed decays, $\Dp \to \KS \pip$.
%
% Fernando. Contact Brian Meadows; Jonathon Coleman.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Indirect CP violation at \FourS and \psiprpr}
\label{sec:charm-indirectCPV}
% Arantza

CP violation in mixing can be investigated from the data taken at the \FourS and at the \psiprpr resonances
in semi-leptonic transitions.
%following different analysis strategies. 
In both cases one measures an asymmetry from events in which the $D^0$ or $\bar D^0$, previously flavour tagged, 
has oscillated (signaled as a wrong sign decay),%~\cite{ref:charm_cicerone},
\begin{equation} 
a_{SL} = \frac{N^{--}(t)-N^{++}(t)}{N^{--}(t)+N^{++}(t)} = \frac{|q|^4 - |p|^4}{|q|^4 + |p|^4}~, 
\end{equation}
where $N^{--}$ ($N^{--}$) represents the number of 
$\Dz \to \ell^- \nu X$ ($\Dzb \to \ell^+ \nu X$) decays when the other $D$ meson was tagged as \Dz (\Dzb) at production time.  
Data at the \psiprpr largely benefits from a very clean environment with almost no background. Several decay 
channels can be exclusively reconstructed and combined to increase the assymetry sensitivity. Considering  
the $D^0$ and $\bar D^0$ both decaying into $K^-\pi^+$, $K^-\pi^+\pi^0$, $K^-\pi^+\pi^+\pi^-$, $K^-e^+\nu$, 
$K^-\mu^+\nu$, $K^{*-}e^+\nu$, $K^{*-}\mu^+\nu$, $K^{*-}e^+\nu$,  $\pi^- e^+\nu$, $\pi^-\mu^+\nu$, $K^-K^+$ and $\pi^-\pi^+$, 
and using recent results for the $D^0-\bar D^0$ mixing parameters $x$ and $y$ \cite{ref:hfag}, it is expected a sensitivity to CP 
violation of 2.5$\%$ in one month of running at threshold. The quantum correlation insures that the same sign combinations
can only be due to mixing, and hadronic modes can be treated like the semileptonic decays (no DCS contribution).
Control on systematic uncertainties are expected likewise at the percent 
level, dominated by channels with $\pi^0$ and $\nu$ particles \cite{ref:cleoc_hbr,ref:cleoc_slbr}. Missing mass techniques with fully reconstruction of 
\psiprpr $\to D \bar D$ events omitting one of the product particles can be used to evaluate the accuracy in the reconstruction. 
Large control samples of decay channels with unequivocal particle content like $D^0\to K^0_s\pi^+\pi^-$ and $D^+\to K^-\pi^+\pi^+$ will reduce 
the uncertainty on PID efficiencies. Other sources of systematic uncertainties will also beneficit from the precise measurement of the beam energy 
and improved detector performance. 
% {\bf (1m = 150/fb)}.

At the \FourS the soft pion coming from $D^*$ decays ($D^{*+} \to D^0 \pi^+$) 
can be used to tag the flavour of the $D^0$. The measurement of wrong sign leptons in 
semileptonic decays gives then a clear signature of a mixed event. Data are taken from the continuum. 
%Background events from $B$ decays can be reduced imposing a 2.5~\gevc cut on the $D$ 
%momentum in center-of-mass frame which keeps more than 85$\%$ of signal events and suppress almost all of the $B \bar B$ 
%events. %{\bf (other bkg?) }. 
Background events from $B$ decays can be reduced imposing a 2.5~\gevc cut on the $D$ momentum. %{\bf (other bkg?) }.
With this method the statistical sensitivity in the decay assymetries would reach the $1\%$ level in one 
year of data taking. Systematic uncertainties are foreseen to come from the background control and PID management 
(mainly lepton identification), which will benefit from the vertex capabilities to suppress the background 
and large control samples to study the PID. 
%{\bf (1y = 15/ab)}. 
%The main issue here is to be able to control systematic uncertainty to this extend.  
%{\bf (develope, try to improve and evaluate, ask K. Flood)} 

  %Semileptonic and hadronic decays, running at \FourS and at \psiprpr. Quote reach at \psiprpr per month of running.
  %Fernando, Arantza.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{CP violation in interference between mixing and decay}
\label{sec:charm-interferenceCPV}

% Gabriele

%Several experiments have now shown evidence for mixing in D mesons at
%the level of $1\%$~\cite{Aubert:2007wf,Staric:2007dt,Abe:2007rd,dmixcp} 
%which is consistent with Standard Model predictions~\cite{foundations}.  

%Unless there is new physics, CP violation in D decays is expected to be
%unobservably small and an observation it would be a clear sign of NP.

%Such CP violation effects can be
%searched in D decays into CP eigenstates, e.g.  $D \to \Kp\Km$, $D \to
%\pip\pim$, $D \to \KS\rho$, $D \to \KS\phi$.

CP violation in the interplay of $\Delta C = 1,2$ dynamics can be searched for through time-dependent analyses
of $\Dz \to K^+ K^-$ and $\Dz \to \pi^+ \pi^-$ decays. CP violation and $D^0-\Dzb$ mixing alters the decay time
distribution of $\Dz$ and $\Dzb$ mesons that decay into final states of specific \CP, and a time-dependent
analysis of the tagged \Dz and \Dzb intensities allows a measurement of the $\phi_f$.
To a good approximation, these decay time distributions can be treated as exponential with effective 
lifetimes \tauhhp and \tauhhm.
%
%%, given by
%% \begin{equation}
%% \begin{array}{rcl}
%% \tauhhp   &=&
%% \tauKpi\left[ 1 + r_m \left( y\cos\varphi_f -
%%                                  x\sin\varphi_f \right) \right]^{-1} \\
%% \tauhhm &=&
%% \tauKpi\left[ 1 + r_m^{-1} \left( y\cos\varphi_f +
%%                                       x\sin\varphi_f \right) \right]^{-1} \;,
%% \end{array}
%% \end{equation}
%% where $\tauKpi$ is the lifetime for the Cabibbo-favored decays
%% $D^0 \to K^-\pi^+$ and $\Dzb \to K^+\pi^-$, and $\tauhhp$ ($\tauhhm$)
%% is the lifetime for the Cabibbo-suppressed decays
%% of the $D^0$ ($\Dzb$) into \CP-even final states (such as $K^-K^+$
%% and $\pi^-\pi^+$).

The effective lifetimes can be combined into the quantities $\yCP$ and $\Delta Y$:
\begin{displaymath}
\begin{array}{rcl}
\displaystyle\yCP = \frac{\tauKpi}{\langle\tauhh\rangle} - 1\;,%%\\[0.5cm]
&&
\displaystyle\Delta Y = \frac{\tauKpi}{\langle\tauhh\rangle} A_\tau \;,
\label{eq:yCPcalc}
\end{array}
\end{displaymath}
where $\langle \tauhh \rangle = (\tauhhp+\tauhhm)/2$ and
$A_\tau = (\tauhhp-\tauhhm)/(\tauhhp+\tauhhm)$. 
%% Both $\yCP$ and $\deltaY$
%% are zero if there is no $D^0$-$\Dzb$ mixing. 
%% In the limit of \CP conservation, $\yCP = y$ and $\deltaY = 0$,
%% with the convention that $\cos\varphi_f > 0$.
%
The golden mode is $\Dz \to \Kp\Km$ since the combinatorial background
is $\sim 10\times$ smaller than in the $\pi^+\pi^-$ channel and the
selected sample is $\sim 2\times$ larger.  $\Dz \to \KS\phi$ instead has a
large ($\sim10\%$) contribution from S-wave, so it is better analyzed
using the Dalitz plot technique (see Sec.~\ref{sec:charm-3body}).

The sensitivity to $\yCP$ and $\deltaY$ with the SuperB in the $KK$
and $\pi\pi$ mode can be extrapolated from the current BaBar
analysis~\cite{ref:babarDtoKKandDtoPiPi}, assuming that the systematic errors can be kept
under control. Provided that the CP violation in mixing is small
the sensitivity to the CP violating phase is dominated by
the first term in the expression for $\yCP$ and $\deltaY$.
\begin{eqnarray*}
2 \yCP    = (|q/p|+|p/q|) y \cos\phi - (|q/p|-|p/q|) x \sin\phi, \\
2 \deltaY = (|q/p|-|p/q|) y \cos\phi - (|q/p|+|p/q|) x \sin\phi, 
\end{eqnarray*} 
therefore we can estimate the sensitivity as
$\delta(\cos \phi)\simeq \delta (y_{CP}) /y \simeq  3\times 10^{-4} / y$, 
$\delta(\sin \phi)\simeq \delta (\deltaY) / x \simeq  3\times 10^{-4} / x$.  
%The results are summarized in Table~\ref{tab:kkpipi}.

%\begin{table}[htb]
%\centering{
%\begin{tabular}{ccccc}
%decay channel& $y_{cp}$         & $\Delta Y$      & $\cos \phi$ & $\sin \phi$\\
%\hline
%$K^+K^-$     & $3\times 10^{-4}$ & $3\times 10^{-4}$ & $3\times 10^{-4}/y $ & $3\times 10^{-4}/x $ \\
%%$\pi^+\pi^-$   &  xxx              &   xxx  \\
%\hline
%\end{tabular}
%}
%\caption{Sensitivity to CP violation parameters with 75 $ab^{-1}$
%  using $D^0\to f_{CP}$ decays.}
%\label{tab:kkpipi}
%\end{table}

Most of the systematic errors affecting the signal cancel in the
lifetime ratio. The errors associated to the background are instead
unrelated between \Dz and \Dzb and don't cancel, however these improve
with the statistics. In addition the expected superior resolution of
the vertex detector will further reduce the systematic errors
associated with the position measurement. Therefore we expect that the
systematic erros can be keept under control.
%% In the Babar systematic studies the main
%% remaining one is the cut on the error on the proper time which is
%% related to the fraction of outliers for which $\sim 2/3$ is statistical
%% and $\sim 1/3$ is systematic.%\footnote{from private communication}. 

One underlying assumption in the recent babar
analysis~\cite{ref:babarDtoKKandDtoPiPi} is that the resolution bias
is the same for all the channels ($K\pi$, $KK$, $\pi\pi$) and it does
not depend on the polar angle $\theta$. This could introduce a bias in
the measurements because of the different polar angle acceptance in
the various channels. However with a higher statistics sample this
systematic effect can be overcome by splitting  it into polar angle
(or other variable) intervals.  The production asymmetry is not
important with the BaBar statistics while it could become significant
at sensitivities of the order of few $\times 10^{-4}$, however this
can be handled using control samples such as the untagged \Dz which
has about 5 times more events (assuming \Dz and \Dstar have the same
asymmetry), as discussed in Sec.~\ref{sec:charm-directCPV}.

%% The total systematic error on the BaBar analysis is presently
%% $17\times 10^{-4}$. We estimate the total error with the SuperB sample
%% by adding in quadrature the statistical and systematic errors
%% (rescaled appropriately for the luminosity) with an estimate for the
%% irreducible part of the systematic error. The sensitivity as a
%% function of the luminosity is shown in Fig~\ref{fig:kkerr}.
%
%% \vskip0.5cm
%% \begin{figure}[htb] \begin{center}
%% \includegraphics[width=0.6\linewidth]{ycperr.eps}
%% \includegraphics[width=0.6\linewidth]{dyerr.eps}
%% \caption{Total and statistical error on $y_{cp}$ (top) and $\Delta Y$
%%   (left) extracted from $\Dz \to K^+K^-$ as a function of the integrated luminosity.}
%% \label{fig:kkerr}
%% \end{center} \end{figure}

  %Use decays into CP eigenstate, e.g. $D \to \Kp\Km$, $D \to \pip\pim$, $D \to \KS\rho$, $D \to \KS\phi$.
  %$D \to \Kp\Km$ is the golden mode. $D \to \KS\phi$ has large ($\sim10\%$ contribution from S-wave, so better if analyzed 
  %using Dalitz plot tecnique). Measurement exclusively to be done at \FourS.

  %Contact BaBar authors: Chavez, Carlos A.; Coleman, Jonathon; Cowan, Ray; Flood, Kevin; Lockman, William; Petersen, Brian). 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{T odd correlations}
\label{sec:Todd}

% beginning of Ikaros, part 2

All CP asymmetries observed so far have surfaced in partial widths -- with one notable exception, 
namely the forward-backward asymmetry $\langle A \rangle$ in the $\pi^+\pi^-$ and $e^+e^-$ planes in 
$K_L \to \pi^+\pi^-e^+e^-$. $\langle A \rangle \simeq 14\%$ had been 
predicted -- and confirmed by experiment -- as being driven by the indirect CP impurity $|\eta_{+-}| \simeq 0.23 \%$. 
The reason for this magnification by two orders of magnitude is well understand: 
$\langle A \rangle$ is induced by the interference between a CP violating and a CP conserving amplitude, 
both of which are suppressed, albeit for different reasons. This explains why the enhancement of the CP asymmetry 
comes at the expense of the branching ratio, which is about 
$3\cdot 10^{-7}$; i.e., one has traded in branching ratio for the size of the asymmetry. 

It is possible that a similar effect and enhancement occurs in the analogous mode 
$D_L \to K^+K^-\mu^+\mu^-$, where $D_L$ denotes the `long'-lived neutral $D$ meson. This mode 
could be studied uniquely at SuperB operating at the $\psi (3770)$ by CP tagging the other neutral $D$ meson produced as a $D_S$: 
\beq 
e^+e^- \to \gamma^* \to D^0 \overline D^0 \to [K^+K^-]_D D_L
\eeq

There is a more general lesson from the $K_L \to \pi^+\pi^-e^+e^-$ example, namely that 
CP violation could surface in an enhanced  fashion in multi-body final states. This could turn 
an apparent vice in charm decays -- the preponderance of multi-body final states -- into a virtue. 
This issue will be addressed in detail in Sec.~\ref{sec:charm-3body}.
%The one drawback is the initial `overhead' needed for setting up the analysis. Yet such an investment is likely to bring great 
%dividend in the long run: not only can one hope for numerically larger asymmetries, but also for better control 
%over systematics due to nontrivial constraints and cross checks inherent in a Dalitz plot analysis. 

These considerations apply also to four-body modes, although less experience with such studies has been accumulated so far. 
Some intriguing pilot studies have been performed on a comparison of 
$D^0 \to f$ and $\overline D^0 \to f$, $f=K ^+K^- \pi^+\pi^-$ channels. Denoting by $\phi$ the angle between 
the $\pi^+\pi^-$ and $K^+K^-$ planes one has 

\beq
\frac{d\Gamma}{d\phi}(D^0 \to f) = 
\Gamma_1 {\rm cos}^2 \phi + \Gamma_2 {\rm sin}^2 \phi + \Gamma_3 {\rm cos} \phi {\rm sin}\phi~, 
\eeq
%
\beq
\frac{d\Gamma}{d\phi}(\overline D^0 \to f) = 
\overline \Gamma_1 {\rm cos}^2 \phi + \overline \Gamma_2 {\rm sin}^2 \phi - \overline \Gamma_3 {\rm cos} \phi {\rm sin}\phi~. 
\eeq
%
%$$ 
%\frac{d\Gamma}{d\phi}(D^0 \to K^+K^- \pi^+ \pi^-) = 
%$$
%\beq \Gamma_1 {\rm cos}^2 \phi + 
%\Gamma_2 {\rm sin}^2 \phi + \Gamma_3 {\rm cos} \phi {\rm sin}\phi 
%\eeq
%$$
%\frac{d\Gamma}{d\phi}(\overline D^0 \to K^+K^- \pi^+ \pi^-) = 
%$$
%\beq
%\overline \Gamma_1 {\rm cos}^2 \phi + 
%\overline \Gamma_2 {\rm sin}^2 \phi - \overline \Gamma_3 {\rm cos} \phi {\rm sin}\phi 
%\eeq
Upon integrating over $\phi$ the $\Gamma_3$ and $\overline \Gamma_3$ terms out. 
$(\Gamma_1, \Gamma_2) \neq (\overline \Gamma_1, \overline \Gamma_2)$ thus represents a 
CP asymmetry in the partial widths. The $\Gamma_3$ and $\overline \Gamma_3$ terms can be projected out by integrating over two quadrants: 
\begin{eqnarray}
\langle A \rangle = \frac{\int _0^{\pi/2}d\phi \frac{d\Gamma}{d\phi} - \int _{\pi/2}^{\pi}d\phi \frac{d\Gamma}{d\phi}}
{\int _0^{\pi}d\phi \frac{d\Gamma}{d\phi}}= \frac{2\Gamma_3}{\pi (\Gamma_1 + \Gamma_2)}~,\\
\langle \overline A \rangle = 
\frac{\int _0^{\pi/2}d\phi \frac{d\overline \Gamma}{d\phi} - 
\int _{\pi/2}^{\pi}d\phi \frac{d\overline \Gamma}{d\phi}}
{\int _0^{\pi}d\phi \frac{d\overline \Gamma}{d\phi}}= 
\frac{2\overline \Gamma_3}{\pi (\overline \Gamma_1 + \overline \Gamma_2)}~.
\end{eqnarray}
While $\Gamma_3$ and $\overline \Gamma_3$ represent T odd moments, they do not necessarily 
signal T violation, since they could be induced by strong final state interactions. Yet 
\beq 
\Gamma_3 \neq \overline \Gamma_3 \; \; \Longrightarrow \; \; {\rm CP \; violation}.
\eeq
Such an analysis is theoretically clean, since the dependence on the angle $\phi$ is specifically predicted, which in 
turn allows cross checks to control experimental systematics. 

Alternatively one can define another T odd correlation among the pion and kaon momenta, namely 
$C_T \equiv \vec p_{K^+} \cdot (\vec p_{\pi^+} \times \vec p_{\pi^-})$ for $D^0$  and 
$\overline C_T \equiv \vec p_{K^-} \cdot (\vec p_{\pi^-} \times \vec p_{\pi^+})$ for $\overline D^0$. 
Similar to the previous case one has: $C_T \neq - \overline C_T \; \; \Longrightarrow$  CP violation. 
One can then construct T odd moments
\begin{eqnarray}
A_T &=& \frac{\Gamma(C_T > 0 ) - \Gamma(C_T > 0 )}{\Gamma(C_T > 0 ) + \Gamma(C_T > 0 )}~, \\
\overline A_T &=& \frac{\Gamma(\overline C_T > 0 ) - 
\Gamma(\overline C_T > 0 )}{\Gamma(\overline C_T > 0 ) + \Gamma(\overline C_T > 0 )} \;~, 
\end{eqnarray}
and therefore 
\beq 
A_{\not T} = \frac{1}{2} (A_T - \overline A_T) \neq 0 \; \; \Longrightarrow \; \; 
{\rm CP \; violation}. 
\eeq
A preliminary study based on 380 $fb^{-1}$ of BaBar data suggests a sensitivity of $5.3 \times 10^{-3}$ 
in $A_{\not T}$ that 
%would extrapolate to $1\cdot 10^{-3}$ for 10 $ab^{-1}$ and presumably significantly less for 75 $ab^{-1}$. 
%With a sample of 10 $ab^{-1}$ one can analyze even time slices of $A_{\not T}$. 
would extrapolate to $4\cdot 10^{-4}$ for 75 $ab^{-1}$. With such a sample one can analyze even time slices of $A_{\not T}$. 
These are very promising sensitivities. 

Similar CP studies can be performed for other four-body modes, and one can also compare 
$Y^0_L$ moments and even full amplitude analyses. 


\subsection{Charm baryon decays}

Charm baryons are of course sensitive only to direct CP violation. Yet if 
longitudinally polarized beams were available -- motivated by CP studies in $\tau$ production and decays -- they would 
provide an intriguing handle for CP studies in charm baryon decays. For charm baryons would be produced with a net 
longitudinal polarization that would allow to form novel CP odd correlations with the momenta of the particles in 
the final state. Since the polarization could be controlled, one would have a new handle to deal with systematics. 

%% end of Ikaros, part 2. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mixing and CP violation in 3-body decays}
\label{sec:charm-3body}

A golden method for the mixing and \CP-violation in mixing/decay/interference is the Dalitz analysis of
 $\Dz \to \KS \pip \pim$ events. If Dalitz model systematics could be kept under control, direct \CP-violation 
 could be investigated too.
Present \babar\ data~\cite{Aubert:2005iz} show that at \FourS, signal events from the decay chain $\Dstarp \to \Dz \pip$ with 
 $\Dz \to \KS \pip \pim$ can be selected at a rate close to 1000/\invfb with a purity of 97.0\% and a mistag probability 
 of 0.1\%. \KS are reconstructed in $\pip\pim$ final state, an upper cut on \KS proper time ($\le 8 \tau_S$) 
ensure us to push \KL contamination at a level of $10^{-5}$ level.
Reconstructing the $\Dz \to \KS \pip\pim$ decay vertex, the \Dz proper time ($\tau_D$) can be measured with 
  an average error of $\pm 0.2\ps$ in \babar and $\pm0.1\ps$ at a SuperB, to be compared with 0.4\ps, the \Dz 
 lifetime. 
\par
We use the invariant mass of $K\pi$ pairs: $m_+^2=m^2(\KS,\pip)$ and $m_-^2=m^2(\KS,\pim)$, and we define 
the following Dalitz plot amplitudes ($f_D$) and probabilities ($p_D$), which, 
for us, also depend on $t$:
%
%
%
\begin{eqnarray}
p_D(m_+^2, m_{-}^2,t) &\equiv& | f_D(m_+^2,m_{-}^2,t) |^2  \quad \textrm{\Dz tag}\\
\overline{p}_D(m_+^2, m_{-}^2,t) &\equiv& | \overline{f}_D(m_+^2,m_{-}^2,t) |^2 \quad \textrm{\Dzb tag}
\end{eqnarray}
%
%
%
The signatures for interesting processes are the following ones:
\begin{itemize}

\item  Mixing without \CP-violation 
\begin{eqnarray}
 p_D(m_+^2, m_{-}^2,t) & = & \overline{p}_D(m_-^2, m_{+}^2,t) \quad  \forall t \quad  \textrm{but}\\ 
 p_D(m_+^2, m_{-}^2,0) & \neq & p_D(m_+^2, m_{-}^2,t)
\end{eqnarray}
\item \CP-violation in mixing
\begin{eqnarray} 
 p_D(m_+^2, m_{-}^2,0) & = & \overline{p}_D(m_-^2, m_{+}^2,0)  \quad  \textrm{and}\\ 
 p_D(m_+^2, m_{-}^2,t) & \neq & \overline{p}_D(m_-^2, m_{+}^2,t)
\end{eqnarray}
\item Direct \CP-violation 
\begin{equation}
 p_D(m_+^2, m_{-}^2,0)  \neq  \overline{p}_D(m_-^2, m_{+}^2,0)
\end{equation}
\end{itemize}
%
%
%
and the quantities, to be measured, that enter in the previous Dalitz plot distribution functions, are:
$x$, $y$ (mixing parameters), $|q/p|$ or $\epsilon=\frac{1-|q/p|}{1+|q/p|}$ and $\phi=\arg(\frac{q\bar{A_f}}{pA_f})$ 
(\CP-violation parameters).
\par 
$x$, $y$,$\epsilon$ and $\phi$ can be extracted in a Dalitz model dependent analysis with the isobar or K-matrix 
 approach, making global fits. Examples are described in references~\cite{Asner:2005sz,Aubert:2005iz}. 
For the model dependent approach, 
 we estimate the SuperB sensitivity (at 75 \invab), extrapolating in conservative way the present data.
 Statistical errors can be scaled with the square root of luminosity. We get a result which is much better than
 the desired goal of $10^{-3}$, not reachable by BESIII. 
The second source is from systematicc errors due to the experiment. They are mainly due t background paremeterization,
 efficiency variation through the Dalitz plot, experimental resolution biases on Dalitz plot variables, decay time 
 parameterization, and mistag fractions. Background paramterization is checked with sidebands (according to the 
 Monte Carlo the background does not have any bump in the \Dz mass signal region), and it scales with statistics.
 Efficiency variation is studied with Monte Carlo events and it scales with the Monte Carlo statistics.
Biases on Dalitz plot variable mass resolution are negligible. Decay time paramterization improves with data and the 
 time resolution will be immproved at the SuperB. Mistag fractions can be checked with other final $D$ states and 
 their contribution is negligible. There is the chance that these errors from experimental source could be reduced 
scaling to statistics, too, but we prefer to be conservative and avalute them putting a safety factor of two.
They are reported in Table~\ref{tab:dalitz_belle}, and we can see that they are less than the statistical ones.
%
%
%
\begin{table}[htb]
\centering{
\begin{tabular}{ccccc}
Par. & Stat.         &  Exp. Syst.      &  Model Syst. &  Total   \\
\hline \hline
$x$ ($10^{-4}$)        & 30.0 & 8.0  & 12.0 & 33.3 \\
$y$ ($10^{-4}$)        & 24.0 & 10.0 &  7.0 & 26.9 \\
$\epsilon$ ($10^{-4}$) & 15.0 & 2.5  &  4.0 & 15.7 \\
$\phi$ (deg)           & 17.0 & 4.0  &  3.0 & 17.7 \\
\hline
\end{tabular}
}
\caption{Belle present errors on 0.54 \invab on relevant mixing and \CP-violation parameters.}
\label{tab:dalitz_belle}
\end{table}
%
%
%
\begin{table}[htb]
\centering{
\begin{tabular}{ccccc}
Par. & Stat.         &  Exp. Syst.      &  Model Syst. &  Total   \\
\hline \hline
$x$ ($10^{-4}$)        & 2.5  & 1.4   & 4.0 & 4.9  \\
$y$ ($10^{-4}$)        & 2.0 & 1.7  &  2.3 &  3.5  \\
$\epsilon$ ($10^{-4}$) & 1.3 & 0.4   &  1.3 & 1.9  \\
$\phi$ (deg)           & 1.4 & 0.7   &  1.0 & 1.9  \\
\hline
\end{tabular}
}
\caption{SuperB errors on 75 \invab on relevant mixing and \CP-violation parameters.}
\label{tab:dalitz_sens}
\end{table}


The last, but not least, source of systematic errors, is due to the decay models: the isobar, K-matrix, partial-waves 
ones. Uncertaities come from radius paramters, masses and widths of the resonances, the choice of resonances included 
in the fit.
Recent results from CLEO and Belle~\cite{Asner:2005sz,Abe:2007rd} have demonstrated that the mixing and \CP-violation parameters 
 are not very sensitive to Dalitz model variations.
Dalitz model will be checked using two model independent approaches:
\begin{itemize}
\item with very large data sample a partial-wave analysis is capable to determine the amplitude and phase 
 variation over the phase space directly from data.
\item Data collected at charm threshold will make accessible the \Dz-\Dzb relative phase~\cite{White:2007br}.
\end{itemize}
%
%
%
Even if it is extremely difficult to make predictions on the Dalitz model systematics at SuperB it is 
 reasonable to assume that it will be reduced with respect to the present one from Belle~\cite{Abe:2007rd}
 by a sizeble factor. By comparing the Cleo analysis based on 9.0~\invfb with Belle one based on 
 540~\invfb we realize an improvement of the Dalitz model systematic error better than a factor 4 on 
 average. This improvement is mainly due to the fact that the larger statistic data sample allows 
 a better determination of the Dalitz model parameters. 
Considering a factor 3 improvement for the model error 
 at SuperB seems conservative since it is not taking into account the benefits of partial-wave analysis 
 and of data collected at charm threshold.  
 Sensitivity predictions for mixing and \CP-violation parameters at SuperB are reported in 
Table~\ref{tab:dalitz_sens}.


%Golden method for mixing and CP violation from interference between mixing and decay (and may be also direct CP violation
%if Dalitz model systematics can be kept under control. 
%Also, systematic errors in this analysis are different from other analysis (like lietime ratio from $D \to \Kp\Km$).
%Must evaluate how the \psiprpr data (per month of running) impact the Dalitz systematics from the time-dependent analysis 
%using \FourS data. Could take the current BaBar Dalitz model (not yet published, to be done very soon), generate toy MC event, 
%count number of events in the various bins, then estimate the sensitivity on strong phase.

%David, Nicola, Giovanni.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{thebibliography}{99}   

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%

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  %%CITATION = PRLTA,95,121802;%%


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\end{thebibliography}
\end{document}






