UTfit page of D-Dbar mixing

Under construction

\begin{table}[htb] \centering \begin{tabular}{|ccccccc|} \hline Observable & Value & \multicolumn{4}{c}{Correlation Coeff.} & Reference \\ \hline $y_{CP}$ & $(1.064 \pm 0.208)\%$ & & & & & \cite{hep-ex/0004034, hep-ex/0111024, hep-ex/0111026, hep-ex/0703036, 0712.2249, 0908.0761, 0905.4185, Aaij:2011ad} \\ \hline $A_\Gamma$ & $(0.026 \pm 0.231)\%$ & & & & & \cite{hep-ex/9903012, hep-ex/0703036, 0712.2249, Aaij:2011ad} \\ \hline $x$ & $(0.811 \pm 0.334)\%$ & 1 & -0.007 & -0.255$\alpha$ & 0.216 & \cite{0704.1000} \ $y$ & $(0.309 \pm 0.281)\%$ & -0.007 & 1 & -0.019$\alpha$ & -0.280 & \cite{0704.1000} \ $\vert q/p \vert$ & $(0.95 \pm 0.22 \pm 0.10)\%$ & -0.255$\alpha$ & -0.019$\alpha$ & 1 & -0.128 $\alpha$ & \cite{0704.1000} \ $\phi$ & $(-0.035 \pm 0.19 \pm 0.09)$ & 0.216 & -0.280 & -0.128 $\alpha$ & 1 & \cite{0704.1000} \\\hline $x$ & $(0.16 \pm 0.23 \pm 0.12 \pm 0.08)\%$ & 1 & 0.0615 & & & \cite{1004.5053} \ $y$ & $(0.57 \pm 0.20 \pm 0.13 \pm 0.07)\%$ & 0.0615 & 1 & & & \cite{1004.5053} \ \hline $R_M$ & $(0.0130 \pm 0.0269)\%$ & & & & & \cite{hep-ex/9606016, hep-ex/0502012, hep-ex/0408066, 0705.0704, 0802.2952} \\\hline $(x^\prime_+)_{K\pi\pi}$ & $(2.48 \pm 0.59 \pm 0.39)\%$ & 1 & -0.69 & & & \cite{0807.4544} \ $(y^\prime_+)_{K\pi\pi}$ & $(-0.07 \pm 0.65 \pm 0.50)\%$ & -0.69 & 1 & & & \cite{0807.4544} \ $(x^\prime_-)_{K\pi\pi}$ & $(3.50 \pm 0.78 \pm 0.65)\%$ & 1 & -0.66 & & & \cite{0807.4544} \ $(y^\prime_-)_{K\pi\pi}$ & $(-0.82 \pm 0.68 \pm 0.41)\%$ & -0.66 & 1 & & & \cite{0807.4544} \\ \hline $R_D$ & $(0.3030 \pm 0.0189)\%$ & 1 & 0.77 & -0.87 & & \cite{hep-ex/0703020} \ $(x^\prime_+)^2$ & $(-0.024 \pm 0.052)\%$ & 0.77 & 1 & -0.94 & & \cite{hep-ex/0703020} \ $y^\prime_+$ & $(0.98 \pm 0.78)\%$ & -0.87 & -0.94 & 1 & & \cite{hep-ex/0703020} \\\hline $A_D$ & $(-2.1 \pm 5.4)\%$ & 1 & 0.77 & -0.87 & & \cite{hep-ex/0703020} \ $(x^\prime_-)^2$ & $(-0.020 \pm 0.050)\%$ & 0.77 & 1 & -0.94 & & \cite{hep-ex/0703020} \ $y^\prime_-$ & $(0.96 \pm 0.75)\%$ & -0.87 & -0.94 & 1 & & \cite{hep-ex/0703020} \\\hline $R_D$ & $(0.364 \pm 0.018)\%$ & 1 & 0.655 & -0.834 & & \cite{hep-ex/0601029} \ $(x^\prime_+)^2$ & $(0.032 \pm 0.037)\%$ & 0.655 & 1 & -0.909 & & \cite{hep-ex/0601029} \ $y^\prime_+$ & $(-0.12 \pm 0.58)\%$ & -0.834 & -0.909 & 1 & & \cite{hep-ex/0601029} \\\hline $A_D$ & $(2.3 \pm 4.7)\%$ & 1 & 0.655 & -0.834 & & \cite{hep-ex/0601029} \ $(x^\prime_-)^2$ & $(0.006 \pm 0.034)\%$ & 0.655 & 1 & -0.909 & & \cite{hep-ex/0601029} \ $y^\prime_-$ & $(0.20 \pm 0.54)\%$ & -0.834 & -0.909 & 1 & & \cite{hep-ex/0601029} \\\hline \end{tabular} \caption{Experimental data used in the analysis of $D$ mixing, from ref.~\cite{[{}][{and online updates at \url{http://www.slac.stanford.edu/xorg/hfag/}}]1010.1589}. $\alpha = (1 + \vert q/p \vert)^2/2$. Asymmetric errors have been symmetrized. We do not use measurements that do not allow for CP violation in mixing, except for ref.~\cite{1004.5053}.\footnote{As shown in ref.~\cite{0704.1000}, the results for $x$ and $y$ from the Dalitz analysis of $D \to K_s \pi \pi$ are not sensitive to the assumptions about CP violation in mixing.}} \label{tab:dmixexp} \end{table}

We perform a fit to the experimental data in Table~\ref{tab:dmixexp}. We assume that all relevant decay amplitudes in the phase convention $\mathrm{CP}\vert D\rangle = \vert \bar D \rangle$ and $\mathrm{CP}\vert f\rangle = \eta_{\mathrm{CP}}^f \vert \bar f \rangle$ satisfy the relation $\mathcal{A}(D \to f) = \eta_{\mathrm{CP}}^f\mathcal{A}(\bar D \to \bar f)$, which is expected to hold in the SM (in the standard CKM phase convention) with an accuracy much better than present experimental errors. In the same approximation this implies $\Gamma_{12}$ real. We assume flat priors for $x = \Delta m_D/\Gamma_D$, $y = \Delta \Gamma_D/(2 \Gamma_D)$ and $\vert q/p\vert$, with $\vert D_{H,L} \rangle = p \vert D^0 \rangle \pm q \vert \bar D^0 \rangle$ and $\vert p \vert^2+\vert q \vert^2 = 1$. We can then express all observables in terms of $x$, $y$ and $\vert q/p\vert$ using the following formul{\ae}~\cite{Branco:1999fs, hep-ph/0205113, hep-ph/0703204, 0907.3917, 0904.0305}: \begin{eqnarray} \label{eq:xyandco} &&\delta = \frac{1 - \vert q/p \vert^2}{1+\vert q/p \vert^2} \,,\quad \phi = \arg(q/p) = \arg (y+i \delta x)\,,\quad A_M = \frac{\vert q/p \vert^4 -1}{\vert q/p \vert^4+1}\quad R_M =\frac{x^2+y^2}{2}\,, \ && \left( \begin{array}{c} x^\prime \ y^\prime \end{array} \right) = \left( \begin{array}{cc} \cos \delta_{K\pi} & \sin \delta_{K\pi} \ -\sin \delta_{K\pi} & \cos \delta_{K\pi} \end{array} \right) \left( \begin{array}{c} x \ y \end{array} \right) \,,\quad x^{\prime}_\pm = \left\vert \frac{q}{p} \right\vert^{\pm}(x^\prime\cos \phi \pm y^\prime \sin \phi)\,, \quad y^\prime_\pm = \left\vert \frac{q}{p} \right\vert^{\pm 1}(y^\prime\cos \phi \mp x^\prime \sin \phi)\,,\nonumber \\ && y_\mathrm{CP} = \left( \left\vert \frac{q}{p} \right\vert + \left\vert \frac{p}{q} \right\vert \right) \frac{y}{2} \cos \phi- \left( \left\vert \frac{q}{p} \right\vert - \left\vert \frac{p}{q} \right\vert \right) \frac{x}{2}\sin \phi\,,\quad A_\Gamma = \left( \left\vert \frac{q}{p} \right\vert - \left\vert \frac{p}{q} \right\vert \right) \frac{y}{2} \cos \phi- \left( \left\vert \frac{q}{p} \right\vert + \left\vert \frac{p}{q} \right\vert \right) \frac{x}{2}\sin \phi\,, \nonumber % \% && R_D = \frac{\Gamma(D^0 \to K^+\pi^-)+\Gamma(\bar D^0 \to % K^-\pi^+)}{\Gamma(D^0 \to K^-\pi^+)+\Gamma(\bar D^0 \to % K^+\pi^-)}\,, % \quad A_{D} = \frac{\Gamma(D^0 \to K^+\pi^-)-\Gamma(\bar D^0 \to % K^-\pi^+)}{\Gamma(D^0 \to K^+\pi^-)+\Gamma(\bar D^0 \to K^-\pi^+)} \,,\nonumber \end{eqnarray} with $\delta_{K\pi}$ a strong phase. For the purpose of constraining NP, it is useful to express the fit results in terms of the $\Delta D=2$ effective Hamiltonian matrix elements $M_{12}$ and $\Gamma_{12}$: \begin{equation} \vert M_{12} \vert = \frac{1}{\tau_D } \sqrt{\frac{x^2+\delta^2 y^2}{4(1-\delta^2)}}\,,\quad \vert \Gamma_{12} \vert= \frac{1}{\tau_D }\sqrt{\frac{y^2+\delta^2 x^2}{1-\delta^2}}\,, \quad \sin \Phi_{12} = \frac{\vert \Gamma_{12}\vert^2 + 4 \vert M_{12}\vert^2 - (x^2+y^2)\vert q/p\vert^2/\tau_D^2}{4 \vert M_{12} \Gamma_{12}\vert}\,, \label{eq:m12g12} \end{equation} with $\Phi_{12}=\arg \Gamma_{12}/M_{12}$. Consistently with the assumption $\mathcal{A}(D \to f) = \mathcal{A}(\bar D \to \bar f)$, $\Gamma_{12}$ can be taken real with negligible NP contributions, and a nonvanishing $\Phi_{12}$ can be interpreted as a signal of new sources of CP violation in $M_{12}$. For the sake of completeness, we report here also the formul\ae to compute the observables $x$, $y$ and $\delta$ from $M_{12}$ and $\Gamma_{12}$: \begin{eqnarray} \sqrt{2}\, \Delta m &=& \sqrt{4 \vert M_{12} \vert^2 - \vert \Gamma_{12} \vert^2 + \sqrt{(4\vert M_{12} \vert^2+ \vert \Gamma_{12} \vert^2)^2-16 \vert M_{12} \vert^2 \vert \Gamma_{12} \vert^2 \sin^2\Phi_{12}} }\,,\nonumber \ \sqrt{2}\, \Delta \Gamma&=& \mathrm{sign}(\cos\Phi_{12}) \sqrt{\vert \Gamma_{12} \vert^2 - 4\vert M_{12} \vert^2 + \sqrt{(4\vert M_{12} \vert^2+ \vert \Gamma_{12} \vert^2)^2-16 \vert M_{12} \vert^2 \vert \Gamma_{12} \vert^2 \sin^2\Phi_{12}} }\,,\nonumber \ \delta &=&\frac{ 2 \vert M_{12} \vert \vert \Gamma_{12} \vert \sin\Phi_{12}}{(\Delta m)^2 + \vert \Gamma_{12}\vert^2} \,, \label{eq:m12g12inv} \end{eqnarray} in agreement with \cite{0907.3917} up to a factor of $\sqrt{2}$.

\begin{table}[t] \centering \begin{tabular}{|ccc|} \hline parameter & result @ $68\%$ prob. & $95\%$ prob. range \ \hline $\vert M_{12}\vert$ [1/ps] & $(6.5 \pm 2.3) \cdot 10^{-3}$ & $[2.0,11.0] \cdot 10^{-3}$ \ $\vert G_{12}\vert$ [1/ps] & $(19.7 \pm 3.2) \cdot 10^{-3}$ & $[13.3,26.1] \cdot 10^{-3}$ \ $\Phi_{12}$ [$^\circ$] & $(-12 \pm 15)$ & $[-52,17]$ \ \hline $x$ & $(5.2 \pm 2.0) \cdot 10^{-3}$ & $[1.1,9.1] \cdot 10^{-3}$ \ $y$ & $(8.0 \pm 1.3) \cdot 10^{-3}$ & $[5.3,10.6] \cdot 10^{-3}$ \ $\vert q/p \vert -1$ & $(9 \pm 12) \cdot 10^{-2}$ & $[-11,39] \cdot 10^{-2}$ \ $\phi$ [$^\circ$] & $(-3.0 \pm 3.7)$ & $[-12.6,4.4]$ \ \hline \end{tabular} \caption{Results of the fit to $D$ mixing data.} \label{tab:ddmix_res} \end{table}

\begin{figure}[htb] \centering \includegraphics[width=.3\textwidth]{figs/ddmix_new_M12} \includegraphics[width=.3\textwidth]{figs/ddmix_new_G12} \includegraphics[width=.3\textwidth]{figs/ddmix_new_phi12} \caption{One-dimensional p.d.f. for the parameters $\vert M_{12} \vert$, $\vert \Gamma_{12} \vert$ and $\Phi_{12}$.} \label{fig:ddmix_1d} \end{figure}

\begin{figure}[htb] \centering \includegraphics[width=.4\textwidth]{figs/ddmix_new_x} \includegraphics[width=.4\textwidth]{figs/ddmix_new_y} \includegraphics[width=.4\textwidth]{figs/ddmix_new_qopm1} \includegraphics[width=.4\textwidth]{figs/ddmix_new_phizoom} \caption{One-dimensional p.d.f. for the parameters $x$, $y$, $\vert q/p \vert -1$ and $\phi$.} \label{fig:ddmix_1d_2} \end{figure}

\begin{figure}[htb] \centering \includegraphics[width=.4\textwidth]{figs/G12vsM12} \includegraphics[width=.4\textwidth]{figs/Phi12vsM12} \includegraphics[width=.4\textwidth]{figs/yvsx} \includegraphics[width=.4\textwidth]{figs/phivsqopm1}

\caption{Two-dimensional p.d.f. for $\vert \Gamma_{12} \vert$ vs $\vert M_{12} \vert$ (top left), $\Phi_{12}$ vs $\vert M_{12} \vert$ (top right), $y$ vs $x$ (bottom left) and $\phi$ vs $\vert q/p \vert -1$ (bottom right).} \label{fig:ddmix_2d} \end{figure}

The results of the fit are reported in Table \ref{tab:ddmix_res}. The corresponding p.d.f are shown in Figs. \ref{fig:ddmix_1d} and \ref{fig:ddmix_1d_2}. Some two-dimensional correlations are displayed in Fig. \ref{fig:ddmix_2d}.

% \section{Standard Model Analysis} % \subsection{Fit Results} % \subsection{Predictions And Compatibility} % \subsection{DISCUSSION OF VARIOUS ISSUES}

% \section{New Physics Analysis} % \subsection{Fit Results} % \subsection{Predictions And Compatibility} % \subsection{DISCUSSION OF VARIOUS ISSUES}

% \section{Effective Field Theory Analysis} % \subsection{Fit Results} % \subsection{Constraints On The New Physics Scale}

% \section{Conclusions And Outlook}

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