The Time Dependent CP Violation in Charm

A model which describes the time-dependent CP formalism in $D^0$ decays has recently been proposed. There it has been highlighted a possible measurement of the angle $\beta_c$, in the charm unitarity triangle, using the decays $D^0\to K^+ K^-$ and $D^0\to \pi^+ \pi^-$, and a measurement of the mixing phase $\phi_{MIX}$. The same method can be used to measure the value of the parameter $x$, one of the two parameters defining charm mixing. We numerically evaluate the impact of a time-dependent analysis in terms of the possible outcomes from present and future experiments. We consider the scenarios of correlated $D^0$ mesons production at the center of mass energy of the $\Psi(3770)$ at Super$B$, uncorrelated production at the center of mass energy of the $\Upsilon(4S)$ at Super$B$ and Belle II, and LHCb. Recently a hint of direct CP violation in charm decays was reported by the LHCb collaboration, we estimate the rate of time-dependent asymmetry that could be achieved using their available data, and we generalise the result for the full LHCb program. We conclude that LHCb is already able to perform a first measurement of $\beta_{c,eff}$, and slightly improve the present constraints on the parameters $x$ and $\phi_{MIX}$. A more precise determination of $\beta_{c,eff}$, $\phi_{MIX}$ and $x$ will require a larger data sample, and most probably the cleaner environment of the new high luminosities $B$-factories (both Super$B$ and Belle II) will be needed. We show that Super$B$ will be able to measure $\beta_{c,eff}$ and $\phi_{MIX}$ with a precision of $1.4^o$ and improve the precision on $x$ by a factor of two.