\section{Inputs}\label{ssec:inputs}

In this Section, we discuss all relevant experimental and theoretical inputs
entering the fits. The corresponding values and uncertainties are quoted in 
Tables~\ref{tab:ExperimentalInputs} and~\ref{tab:TheoreticalInputs}. In general, 
if there is only one uncertainty quoted, we understand
this error as a statistical 
one. In case of two error contributions, the first one is taken as a statistical
error while the second, theoretical, error is treated as an allowed range for 
the observable or the parameter under consideration. This kind of uncertainty is 
treated in the Rfit scheme described in Refs.~\cite{CKMfitter1,CKMfitter2}.


\subsection{Hadronic parameters and method of averaging}
\label{ssec:MethodofAveraging}

Several hadronic inputs are required for the fits presented, and we mostly rely
on Lattice QCD (LQCD) simulations to estimate these quantities involving strong interactions at low energies. 
The presence of results from different lattice QCD collaborations 
with various statistics and systematics make it all 
the more necessary to combine them in a careful way. The procedure that we have 
chosen to determine these lattice averages is as follows:\\
We collect the relevant calculations of the quantity that we are interested in
and 
we take only unquenched results with 2 or 2+1 dynamical fermions, even those from 
proceedings without a companion article (flagged with a star). In these results, 
we separate the error estimates into a Gaussian part and a flat part that is 
treated \`{a} la Rfit. The Gaussian part collects the uncertainties from purely 
statistical origin, but also the systematics that can be controlled and treated 
in a similar way (e.g., interpolation or fitting in some cases). The remaining 
systematics constitute the Rfit error. If there are several sources of error in 
the Rfit category, we add them linearly \footnote{Keeping in mind that in many 
papers this combination is done in quadrature and the splitting
between different sources is not published.}.
The Rfit model is simple but also very strict. It amounts to assuming that the 
theoretical uncertainty is rigorously constrained by a mathematical bound that 
is our only piece of information. If Rfit is taken stricto sensu and the individual 
likelihoods are combined in the usual way (by multiplication), the final uncertainty 
can be underestimated, in particular in the case of marginally compatible values.
We correct this effect by adopting the following averaging recipe. 
We first combine the Gaussian uncertainties by combining the likelihoods 
restricted to their Gaussian part. Then we assign to this combination the smallest 
of the individual Rfit uncertainties. The underlying idea is twofold:
\begin{itemize}
\item the present state of the art cannot allow us to reach a better theoretical 
      accuracy than the best of all estimates.
\item this best estimate should not be penalized by less precise methods (as it 
      would happen be the case if one took the dispersion of the individual 
      central values as a guess of the combined theoretical uncertainty).
\end{itemize}
It should be stressed that the concept of a theoretical uncertainty is ill-defined, 
and the combination of them even more. Thus our approach is only one among the 
alternatives that can be found in the literature \cite{LubiczTarantino,LLVdW}. 
In contrast to some of the latter, ours is algorithmic and can be reproduced. 
Moreover, we differ from the PDG-like method advocated in Ref.~\cite{LLVdW} on 
two points. We separate systematic and statistic errors, which prevents us from 
assigning a reduced systematics to a combination of several results suffering from 
the same systematic uncertainty. We do not attempt at estimating the (partial) 
correlations between the results from different collaborations, even though we are 
aware of their existence (results from the same gauge configuration, using the same 
procedure to determine the lattice spacing\ldots). Whatever the averaging method 
chosen, one should emphasize that it relies crucially on the quality of the error 
estimation performed by each collaboration.

The following tables show the inputs used and the average obtained by applying the 
procedure described above for the following hadronic parameters: 
the decay constant $f_{B_{s}}$ for the $B_{s}$ meson (Table~\ref{tab:fBs}), 
the ratio of decay constants $f_{B_{s}}/f_{B_{d}}$ (Table~\ref{tab:fBsOverfB}), 
the scheme-invariant bag parameter $\hat{\Bag}_{B_{s}}=1.523 \, \Bag_{B_{s}}\big(m_{b}\big)$
for the $B_{s}$ meson discussed after Eq.~\ref{defb} (Table~\ref{tab:Bshat}),
the ratio of bag parameters $\Bag_{B_{s}}/\Bag_{B_{d}}$ (Table~\ref{tab:BsOverBdhat}),
and the bag parameter $\Bag_{K}(2 \, \mbox{GeV})$ for the neutral kaon (Table~\ref{tab:BKMSbar2GeV}).

\begin{table}[Htp]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{|c|c|c|c|}\hline
Reference    & $N_f$   & $f_{B_{s}} \pm \sigma_{stat} \pm \sigma_{Rfit}$ & Article                      \\
\hline
CP-PACS01    & 2       & $242 \pm 9 ^{+53}_{-34}$                        & \cite{CP-PACS01}             \\
MILC02       & 2       & $217 \pm 6 ^{+58}_{-31}$                        & \cite{MILC02}                \\
JLQCD03      & 2       & $215 \pm 9 ^{+19}_{-15}$                        & \cite{JLQCD03}               \\
HPQCD03      & 2+1     & $260 \pm 7 \pm 39$                              & \cite{HPQCD03}               \\
FNAL-MILC09* & 2+1     & $243 \pm 6 \pm 22$                              & \cite{FNAL-MILC09}           \\
HPQCD09      & 2+1     & $231 \pm 5 \pm 30$                              & \cite{HPQCD09}               \\
\hline
Our average  &         & $228 \pm 3 \pm 17$                              &                              \\
\hline
\end{tabular}
\caption[fBS]
{Calculations and average used for the decay constant $f_{B_{s}}$.
 $N_f$ stands for the number of  dynamical flavours used in the simulation.
 The first uncertainty quotes the statistical uncertainty, the second the Rfit error.}
\label{tab:fBs}
\end{table}

\begin{table}[Htp]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{|c|c|c|c|}\hline
Reference    & $N_f$   & $f_{B_{s}}/f_{B_{d}} \pm \sigma_{stat} \pm \sigma_{Rfit}$ & Article              \\
\hline
CP-PACS01    & 2       & $1.179 \pm 0.018 \pm 0.023$                               & \cite{CP-PACS01}     \\
MILC02       & 2       & $1.16  \pm 0.01 ^{+0.08}_{-0.04}$                         & \cite{MILC02}        \\
JLQCD03      & 2       & $1.13  \pm 0.03 ^{+0.17}_{-0.02}$                         & \cite{JLQCD03}       \\
FNAL-MILC09* & 2+1     & $1.245 \pm 0.028 \pm 0.049$                               & \cite{FNAL-MILC09}   \\
HPQCD09      & 2+1     & $1.226 \pm 0.020 \pm 0.033$                               & \cite{HPQCD09}       \\
\hline
Our average  &         & $1.199 \pm 0.008 \pm 0.023$                               &                      \\
\hline
\end{tabular}
\caption[fBsOverfB]
{Calculations and average used for the ratio of decay constants $f_{B_{s}}/f_{B_{d}}$.
 $N_f$ stands for the number of  dynamical flavours used in the simulation.
 The first uncertainty quotes the statistical uncertainty, the second the Rfit error.
}
\label{tab:fBsOverfB}
\end{table}

%Given the very small (and controversial) systematic error quoted by HPQCD05, 
%we keep it in the average but (arbitrarily) assign the second largest 
%theoretical uncertainty to the final number.

\begin{table}[Htp]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{|c|c|c|c|}\hline
Reference     & $N_f$   & $\hat{\Bag}_{B_{s}} \pm \sigma_{stat} \pm \sigma_{Rfit}$ & Article          \\
\hline
JLQCD03       & 2       & $1.299  \pm 0.034^{+0.122}_{-0.095}$                     & \cite{JLQCD03}       \\
HPQCD06       & 2+1     & $1.168  \pm 0.105 \pm 0.140$                             & \cite{HPQCD06}       \\
RBC/UKQCD07*  & 2+1     & $1.21   \pm 0.05  \pm 0.05$                              & \cite{RBC/UKQCD07*}  \\
HPQCD09       & 2+1     & $1.326  \pm 0.04  \pm 0.03$                              & \cite{HPQCD09}       \\
\hline
Our average   &         & $1.28   \pm 0.02  \pm 0.03$                              &                      \\
\hline
\end{tabular}
\caption[Bshat]
{Calculations and average used for the bag parameter $\hat{\Bag}_{B_{s}}$.
 $N_f$ stands for the number of  dynamical flavours used in the simulation.
 The first uncertainty quotes the statistical uncertainty, the second the Rfit error.
}
\label{tab:Bshat}
\end{table}

\begin{table}[Htp]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{|c|c|c|c|}\hline
Reference     & $N_f$   & $\Bag_{B_{s}}/\Bag_{B_{d}} \pm \sigma_{stat} \pm \sigma_{Rfit}$ & Article          \\
\hline
JLQCD03       & 2       & $1.017 \pm 0.016 ^{+0.076}_{-0.017}$                            & \cite{JLQCD03}   \\
HPQCD09       & 2+1     & $1.053 \pm 0.020 \pm 0.030$                                     & \cite{HPQCD09}   \\
\hline
Our average   &         & $1.05  \pm 0.01 \pm 0.03$                                       &                  \\
\hline
\end{tabular}
\caption[BsOverBdhat]
{Calculations and average used for the bag parameter ratio $\Bag_{B_{s}}/\Bag_{B_{d}}$.
 $N_f$ stands for the number of dynamical flavours used in the simulation.
 The first uncertainty quotes the statistical uncertainty, the second the Rfit error.
}
\label{tab:BsOverBdhat}
\end{table}


\begin{table}[Htp]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{|c|c|c|c|}\hline
Reference     & $N_f$      & $\Bag_{K}(2 \, \mbox{GeV}) \pm \sigma_{stat} \pm \sigma_{Rfit}$ & Article              \\
\hline
JLQCD08       & 2          & $0.537 \pm 0.004  \pm 0.072$                                    & \cite{JLQCD08}       \\
HPQCD/UKQCD06 & 2+1        & $0.618 \pm 0.018  \pm 0.179$                                    & \cite{HPQCD/UKQCD06} \\
RBC/UKQCD07   & 2+1        & $0.524 \pm 0.010  \pm 0.052$                                    & \cite{RBC/UKQCD07}   \\
ALVdW09       & 2+1        & $0.527 \pm 0.006  \pm 0.049$                                    & \cite{ALVdW09}   \\
\hline
Our average   &            & $0.527 \pm 0.0031 \pm 0.049$                                    &                  \\
\hline
\end{tabular}
\caption[BKMSbar2GeV]
{Calculations and average used for the bag parameter $\Bag_{K}(2 \, \mbox{GeV})$.
 $N_f$ stands for the number of  dynamical flavours used in the simulation.
 The first uncertainty quotes the statistical uncertainty, the second the Rfit error.
}
\label{tab:BKMSbar2GeV}
\end{table}

We are not aware of lattice estimates for the power-suppressed matrix elements corresponding to $\Bag_{\tilde{R}_i}$. 
We will assign them a value of $1\pm 0.5$, taking a flat uncertainty for the two 
bag parameters contributing the most to $\Delta\Gamma^q$ and $a_{SL}^q$ 
($\Bag_{\tilde{R}_2}$ and $\Bag_{\tilde{R}_3}$ respectively), and a Gaussian error for the other ones.
These bag parameters vary independently for $B_d$ and $B_s$ mesons, \textit{i.e.} we do not assume the exact SU(3) symmetry for them.

\subsection{Observables not affected by New Physics in mixing}\label{ssec:InputsWithoutNP}
In this section, we first discuss the observables allowing us to establish an 
universal preferred region in the $\bar\rho-\bar\eta$ subspace, independent of any NP 
contributions in mixing.
\begin{itemize}
\item The CKM matrix element $|V_{ud}|$ has been determined from three different 
      methods: superallowed nuclear $\beta$-decays, neutron $\beta$-decay and 
      pion $\beta$-decay.
      Currently, the best determination of $|V_{ud}|$ comes from superallowed 
      $\beta$-decays where the uncertainty is dominated by the theoretical 
      error, see e.g. Ref.~\cite{CKM05Vud,PDG2008}. 
      An analysis by Towner and Hardy~\cite{Hardy:2008gy} focusing on an 
      improvement of the isospin-symmetry-breaking terms finds a central value
      for $|V_{ud}|$ which is larger though still compatible when compared to 
      values quoted in the past, with a slightly reduced uncertainty:
      $|V_{ud}|=0.97425 \pm 0.00022$.

\item The matrix element $|V_{us}|$ can be determined from $K_{e3}$ decays, 
      from hadronic $\tau$ decays, and from semileptonic hyperon decays. 
      We are using the $K_{e3}$ average quoted by Flavianet~\cite{flavianet}.
      The experimental number for $f_{+}(q^{2}=0) \cdot |V_{us}|$ obtained by averaging
      results from ISTRA+, KLOE, KTeV, and NA48(/2), as quoted by Flavianet, is 
      $f_{+}(q^{2}=0) \cdot |V_{us}| = 0.21664 \pm 0.00046$~\cite{flavianet}.
      Taking the %most recent 
      Lattice QCD result for the $K \to \pi$ 
      form factor $f_{+}$ of $0.964 \pm 0.005$~\cite{VusUKQCDRBC}, this
      translates into $|V_{us}| = 0.2246 \pm 0.0012$~\cite{flavianet}.

\item The matrix element $|V_{cb}|$ is obtained from semileptonic decays
      $B \to X_{c} \ell \nu$, where $X_c$ is either a $D^{*}$ meson (exclusive 
      method) or a sum over all hadronic final states containing charm 
      (inclusive method). For several years the most precise value has been 
      provided by the inclusive method where the theoretical uncertainties 
      have been pushed below the $2\percent$ level by determining the relevant 
      non-perturbative Heavy Quark Expansion (HQE) parameters from moment 
      measurements in $B \to X_{c}\ell\nu$ and $B \to X_{s}\gamma$ decays.
      The inclusive $|V_{cb}|$ value used in our analysis is taken from the
      Heavy Flavour Averaging Group (HFAG)~\cite{HFAG10},
      $|V_{cb,\text{incl}}|=(41.85\pm0.43\pm0.59) \times 10^{-3}$, where 
      the first error contains the experimental and HQE uncertainties and 
      the second reflects the theoretical uncertainty on the total rate 
      prediction for $B \to X_{c}\ell\nu$.\\
      The theoretical uncertainty on the exclusive $|V_{cb}|$ determination 
      in the calculation of the form factor value at zero recoil $F(1)$ has 
      not been competitive so far.
      A recent calculation provides a significantly smaller error budget: 
      $F(1)=0.921 \pm 0.013 \pm 0.020$~\cite{Bernard:2008dn}, although the exclusive
      $|V_{cb}|$ determination still gives a larger uncertainty.
      Using the average value for the product $F(1) |V_{cb}|=(36.04 \pm 0.52) \times 10^{-3}$
      from the Heavy Flavour Averaging Group (HFAG)~\cite{HFAG10} and applying
      a $0.7\percent$ QED correction~\cite{Sirlin}, one finds 
      $|V_{cb,\text{excl}}|=(38.85\pm 0.56_\text{exp}\pm 0.55_\text{theostat}\pm 0.84_\text{theosys}) \times 10^{-3} 
                           =(38.85 \pm 0.77 \pm 0.84) \times 10^{-3}$,
      which has a smaller central value than the inclusive result.\\
      We average the two $|V_{cb}|$ values in such a way that the smallest 
      theoretical uncertainty is preserved similarly to our procedure to average lattice  
      inputs and we obtain $|V_{cb}| = (40.89 \pm 0.38 \pm0.59) \times 10^{-3}$, 
      keeping in mind, however, that the inclusive and exclusive numbers
      are not in perfect agreement.
      
    \item The two methods to extract $|V_{ub}|$, the inclusive and the
      exclusive ones (using the theoretically cleanest $B \to \pi \ell \nu$
      decays), both suffer from significant theoretical uncertainties.
      The exclusive measurements prefer values around $3.5 \times
      10^{-3}$~\cite{HFAG10}. The numbers quoted are from partial rates
      measured at large $q^{2}$ ($>16\gev^2$) or at small $q^{2}$
      ($<16\gev^{2}$), using form factor calculations from
      LQCD~\cite{FNAL,HPQCD}, or Light Cone Sum Rules
      (LCSR)~\cite{LCSR:BallZwicky}, respectively. 
      The fit input is the average of these numbers, following the same procedure as
      for the Lattice QCD parameters:
      $|V_{ub,\text{excl}}|=(3.51\pm0.10\pm0.46) \times 10^{-3}$.
      \\
      The average of inclusive results quoted by HFAG using~\cite{HFAG10} 
      the Shape Function (SF) scheme~\cite{BLNP} yields
      $(4.32\pm0.16^{+0.22}_{-0.23}) \times 10^{-3}$, where the first
      uncertainty contains the statistical and experimental systematic
      uncertainty as well as the modelling errors for $b \to u \ell \nu$ 
      and $b \to c \ell \nu$ transitions.
      Compared to HFAG, we 
      modify the assignment of the uncertainties as follows. We add
      the following uncertainties in quadrature: statistical uncertainty,
      experimental systematics, $b \rightarrow c$ and $b \rightarrow u$
      modeling and the error from the HQE parameters ($b$-quark mass $m_b$
      and $\mu_{\pi}^{2}$). 
      Several theoretical uncertainties can only be guestimated: the shape
      function uncertainty, contributions from subleading shape functions,
      weak annihilations and the procedure of scale matching. We assign an 
      additional uncertainty on $m_b$, which reflects higher order corrections 
      not accounted for in the partial rate predictions for 
      $B \rightarrow X_u \ell \nu$~\cite{NeubertPrivateCommunication}.
      We choose $50~{\rm MeV}$ as the additional uncertainty.
      All these 
      uncertainties of a second type are added linearly. As a result, 
      we obtain a significantly larger theoretical uncertainty compared to the 
      uncertainty quoted by the HFAG: $(4.32^{+0.21}_{-0.24} \pm 0.45) \times 10^{-3}$.\\
      The exclusive and the inclusive inputs are then averaged using the
      same recipe as for the Lattice QCD parameters and we obtain: 
      $|V_{ub}|=(3.92\pm0.09\pm0.45) \times 10^{-3}$.

\item A measurement of the branching fraction for $B^{+} \to\tau \nu$
      allows one to constrain the product 
      $|V_{ub}| \cdot f_{B}$ where $f_{B}$
      is the decay constant of the charged $B$ meson. %\\
      The theoretical prediction for this branching fraction is given by
      \begin{equation}
       \BRB{\tau\nu}
       = \frac{G_F^2 m_{B^{+}} m_{\tau}^2 }{ 8 \pi }
       \left( 1 - \frac{m_{\tau}^2}{m_{B^{+}}^2} \right)^2 
           |V_{ub}|^2 f_{B}^2 \tau_{B^{+}}.
      \end{equation}
      We use the experimental value $\tau_{B^{+}}=1.639 \cdot 
          10^{-12} s$ in our analysis. 
      $\BRB{\tau\nu}$ combined with the constraint from the
      oscillation frequency $\Delta m_{d}$ (see
      Sec.~\ref{ssec:InputsBdWithNP}) removes the
      dependence on the decay constant $f_{B}$ (assuming 
      that the decay constant for the charged and neutral $B$
      meson is the same, i.e., neglecting
      isospin-breaking effects of order $1\%$).\\
      First evidence for the decay $B \to\tau\nu$ has been 
      made by the Belle collaboration~\cite{btaunu_belle1} as well as the first
      observation of this decay~\cite{btaunu_belle2}. 
      The world average for $\BRB{\tau\nu}$ 
      that is used in the analysis calculated from the measurements performed by 
      \babar\ and Belle~\cite{btaunu_belle1,btaunu_belle2,btaunu_babar1,btaunu_babar2}
      is $\BRB{\tau\nu}=(1.73 \pm 0.35) \times 10^{-4}$.

\item The input for the CKM angle $\gamma$ 
      ($=\arg [ -V_{ud}^{\phantom{*}}V_{ub}^*/(V_{cd}^{\phantom{*}}V_{cb}^*) ]$) 
      is taken from a combined full frequentist analysis of the CKMfitter 
      group using \CP-violating asymmetries in charged $B$ decays to 
      neutral $D^{(*)}$ mesons plus charged $K^{(*)}$ mesons.
      The data are taken from HFAG using the three different methods proposed 
      by Gronau, London, Wyler (GLW)~\cite{GLW}, and Atwood, Dunietz, Soni (ADS)~\cite{ADS}, and 
      including also the Dalitz plot approach developed by Giri, Grossman, 
      Soffer and Zupan (GGSZ), and independently by the Belle collaboration~\cite{GGSZ}. 
      At the $68.3\percent$ confidence level (CL), the result of this analysis is
      $(70^{+14}_{-21})^{\circ}$ with a second solution
      at $\gamma+\pi$. The constraint is shown in Figure~\ref{fig-gamma}.

      The above determination of $\gamma$ raises interesting statistical issues. 
      The angle $\gamma$ actually appears as the complex phase of a suppressed
      ratio $r_B$ of decay amplitudes (different $r_B$'s appear for different final states);
      in other words in the limit $r_B\to 0$ there is no constraint left on the CKM
      phase, and for finite $r_B$ the error on $\gamma$ is roughly inversely
      proportional to $r_B$ itself. It turns out that the current data do not exclude
      tiny values for the $r_B$'s, one obtains for
      the $DK$ final state $r_B=0.104^{+0.015}_{-0.025}$ at 68.3\% CL,
      the $D^{*}K$ final state $r_B=0.116^{+0.025}_{-0.024}$ at 68.3\% CL,
      and for the final state $DK^{*}$ final state $r_B=0.113^{+0.061}_{-0.047}$ at 68.3\% CL.
      Because the ratio of amplitudes is related to the bare observables non-linearly, 
      its maximum-likelihood estimate is biased, and
      it can be shown that this bias overestimates the value of $r_B$ which in turn
      implies an underestimate of the uncertainty on $\gamma$. In the statistical
      language, this effect yields a significant \textit{undercoverage} of the
      na\"{\i}ve 68.3 \% CL interval for $\gamma$ computed from the log-likelihood
      variation.

      A better estimate of the statistical uncertainty on $\gamma$ can be obtained
      by inspecting the deviation  of the distribution of 
      the log-likelihood among a large number of toy experiments from its asymptotic limit.
      Problems arise because 
      this distribution is not only non asymptotic, but also depends on
      \textit{nuisance parameters}, that is other parameters than $\gamma$ that
      are necessary to compute the toy experiments. In such a situation the most
      conservative approach is called the \textit{supremum} one, since it maximizes
      the uncertainty over all possible values of the nuisance parameters. To date
      this method which guarantees the coverage properties by construction is the
      default one for the treatment of $\gamma$ in CKMfitter, but it must be kept
      in mind that it actually leads to overcoverage in general~\cite{karimCKM08}.

      Another determination of $\gamma$ which is unaffected by new physics 
      in \bbm\ is obtained by combining measurements of $\alpha$ and $\beta$: 
      The corresponding quantities are changed into
      $\alpha-\phi_d^{\Delta}/2$ and $\beta+\phi_d^{\Delta}/2$ in the presence of a new
      physics phase $\phi_d^{\Delta}$~
%      \footnote{potential non standard electroweak
%      penguin $b\to d$ transitions are excluded by our SM4FC hypothesis}, 
      which therefore drops out of $\gamma=\pi-(\alpha-\phi_d^{\Delta}/2)- (\beta+\phi_d^{\Delta})$. 
      This procedure leads to a significantly more precise determination of $\gamma$ 
      than $B\to D^{(*)} K^{(*)}$ decays. The individual measurements of
      $\alpha-\phi_d^{\Delta}/2$ and $\beta+\phi_d^{\Delta}/2$ are described 
      in the next section. 
\end{itemize}

\begin{nfigure}{Htb}
 \includegraphics[width=10cm]{gamma.eps}
 \caption{\small Constraint on the angle $\gamma$ from a combined analysis of
   $B\to D^{(*)}K^{(*)}$ decays.}\label{fig-gamma}\end{nfigure}

%\noindent 
%Figure~\ref{fig-universalUT} displays the profile of the preferred $(\bar\rho,\bar\eta)$ 
%plane when the fit is performed using only the inputs $|V_{ud}|$, $|V_{us}|$,
%$|V_{cb}|$, $|V_{ub}|$, $\gamma$, $\alpha-\phi_d^{\Delta}/2$ and $\beta+\phi_d^{\Delta}/2$.
%Two additional branches from $\alpha-\phi_d^{\Delta}/2$ and $\beta+\phi_d^{\Delta}/2$ exist 
%in the negative $\bar{\eta}$ plane which are further discussed in Sec.~\ref{ssec:bdbs}. 

%\begin{nfigure}{htb}
%  \includegraphics[width=10cm]{rhoeta_small_tree_bothgam.eps}
%  \caption{\small Constraint on the CKM $\bar\rho-\bar\eta$ coordinates, using only the 
%    inputs which are not affected by new physics in $B_d$ mixing: $|V_{ud}|$, $|V_{us}|$, 
%    $|V_{cb}|$, $|V_{ub}|$ from semileptonic decays and from $B \rightarrow \tau \nu$, 
%    $\gamma$ and $\alpha-\phi_d^{\Delta}/2$ and $\beta+\phi_d^{\Delta}/2$.
%    Regions outside the coloured areas have ${\rm CL} > 95.45~\%$. For the combined fit 
%    the yellow area inscribed by the contour line represents points with ${\rm CL} < 95.45~\%$. 
%    The shaded area inside this region represents points with ${\rm CL} < 68.3~\%$.
%    \label{fig-universalUT}}
%\end{nfigure}

\boldmath
\subsection
{Observables in the {\bbduli} system affected by New Physics in mixing}\label{ssec:InputsBdWithNP}
\unboldmath
\noindent 
The following observables  can be affected by $\vert \Delta F\vert =2$ NP 
contributions in the $B_d^0 - {\bar{B}}_d^0$ system:

\begin{itemize}
\item The oscillation frequency $\Delta m_d$ in the $B_{d}$ sector has been measured 
      with $\Order(1\percent)$ precision mainly due to the $B$-factory data~\cite{HFAG08}.
      The translation of the measured value for $\Delta m_d$ into constraints 
      on the CKM parameter combination ${|V_{td}^{\phantom{*}} V_{tb}^{*}|}^{2}$ 
      suffers from significant uncertainties in the theoretical calculation of the 
      product $f_{B_{d}} \sqrt{\Bag_{B_{d}}}$ of hadronic parameters. These hadronic 
      parameters can be obtained from LQCD computation, computing
      $f_{B_{d}} \sqrt{\Bag_{B_d}}$ 
      from $f_{B_{s}}$ and $\Bag_{B_{s}}$, and the flavour-symmetry breaking ratios 
      $f_{B_{s}}/f_{B_{d}}$ and $\Bag_{B_{s}}/\Bag_{B_{d}}$ as quoted in 
      Section~\ref{ssec:MethodofAveraging} and summarized in Table~\ref{tab:TheoreticalInputs}. 
      The value and uncertainty for the perturbative QCD correction ${\hat{\eta}}_{B}$ 
      has been originally estimated in Ref.~\cite{BBL95}. With up-to-date values and 
      uncertainties for $\alpha_{s}$ and the top-quark mass one obtains 
      $\hat{\eta}_{B} = 0.8393 \pm 0.0034$.
      For the top-quark mass we take the average value of the Tevatron Electroweak 
      Working Group~\cite{TevEWWG}, $m_{t}=(172.4 \pm 1.2)~{\rm GeV}$, combining 
      published and also preliminary results from \Dzero and CDF. This mass, 
      interpreted as a pole mass, is translated into 
      $\ov m_{t}(\ov m_{t})=(165.017 \pm 1.156 \pm 0.11)~{\rm GeV}$ in the $\overline{\rm MS}$-scheme
      at one-loop order. It should be noted, however, that the identification of
      the measured mass value to the pole mass is under debate (with potential new systematics coming from this identification),
      see e.g.~\cite{Langenfeld:2010aj}.

\item In the SM, the decay width difference $\Delta\Gamma_d$ is predicted to be small: 
      $\Delta\Gamma_d=(38^{+11}_{-12}) \times 10^{-4}\psinv$~\cite{ln} (with our inputs).
      The average between DELPHI and \babar\ measurements of the ratio 
      $\Delta\Gamma_d/\Gamma_d$ calculated by HFAG~\cite{DeltaGammad,BabarDeltaGammad,HFAG08} 
      is $0.009\pm0.037$.
      The experimental uncertainty is much larger than the size of the SM prediction,
      $\Delta\Gamma_d/\Gamma_d=(46^{+17}_{-18}) \times 10^{-4}$, and the 
      measured value is in good agreement with the SM prediction within experimental 
      uncertainties so that stringent constraints on NP contributions cannot be derived 
      at the present stage of precision. Since a huge amount of statistics will be needed to
      measure $\Delta\Gamma_d$ at the level predicted by the Standard Model,
      this situation will probably
      not change for quite a long time. Even then if a deviation from the Standard Model
      value were observed due to NP contributions in mixing it would show up
      beforehand in other observables like
      $a_\text{fs}^{d}$ or $\sin{2\beta}$. As a consequence, $\Delta\Gamma_d$ has no 
      visible impact in our discussion 
      and it is not used as an input to the fits presented here.

\item \CP violation in $B_{d}$ mixing (i.e., $|q/p| \ne 1$, with $q/p$ defined in
       Eq.~\eqref{eq:qp}) can be measured from the untagged dilepton rate asymmetry
      \begin{equation}
      a_\text{fs}^{d}=\frac{N_{\ell^{+}\ell^{+}}-N_{\ell^{-}\ell^{-}} }{N_{\ell^{+}\ell^{+}}+N_{\ell^{-}\ell^{-}}}
          = 2(1-|q/p|)~.
      \end{equation}
      With a tagged time-dependent decay asymmetry, one measures 
        \begin{align}
              A_\text{SL}^q(t)& \:\equiv\:
              \frac{\Gamma(\bar B^0_q(t)\to l^{+}X)-\Gamma(B^0_q(t)\to l^{-}X)}
                   {\Gamma(\bar B^0_q(t)\to l^{+}X)+\Gamma(B^0_q(t)\to
                     l^{-}X)}\\ &=\frac{1-|q/p|^4}{1+|q/p|^4}= 2 (1-|q/p|) +
                   \Order\big((|q/p| - 1)^2\big),
        \end{align}
        with the time-dependence dropping out.
      A weighted average of \babar, Belle and CLEO measurements~\cite{ASLdBfactories} 
      results in $A_{\text{SL}}^d=-(47\pm 46)\times 10^{-4}$~\cite{HFAG08}, 
      which is about one standard deviation below the SM prediction
      of $a_\text{fs}^d= (-6.4\epmuli{1.6}{1.8})\times 10^{-4}$
      \cite{ln,bbln,dega} (with our inputs).\\

\item Within the SM the measurement of the $S$ coefficient in the time-dependent 
      \CP asymmetry 
      $A_{\CP}(t)=S\, \sin{(\Delta m_{d} \cdot t)} + C \, \cos{(\Delta m_{d} \cdot t)}$ 
      in decays of neutral $B_{d}$ mesons to final states $(c \bar{c}) K^{0}$ provides 
      a measurement of the parameter $\sin{2\beta}$,
      where $\beta =\arg [ -V_{td}^{\phantom{*}}V_{tb}^* / (V_{cd}^{\phantom{*}}V_{cb}^*) ]$,  
      to a very good approximation. 
      The current uncertainty of $0.023$ on $S$ is still dominated by statistics~\cite{HFAG08}. 
      The difference between the measured $S$ coefficient and $\sin{2\beta}$ due to
      penguin contributions has been theoretically estimated in Ref.~\cite{BoosMannelReuter} to be below 
      the $10^{-3}$ level, while phenomenologically less stringent constraints on this difference are quoted in 
      Refs.~\cite{GrossmanKaganLigeti,LiMishima,Ciuchini1,Faller:2008zc}.\\
      When interpreting the measured $S$ coefficient as $\sin{(2\beta+\phi_{d}^{\Delta})}$
      the \emph{SM4FC} hypothesis does not rigourously apply. However, the gluonic penguin 
      is \emph{OZI}-suppressed and the  $Z$-penguin is estimated to be small so that NP 
      in decay is assumed to be negligible with respect to the leading tree amplitude. 
      We neglect the effect from possible NP in $K-\bar{K}$ mixing on $\sin{(2\beta+\phi_{d}^{\Delta})}$, 
      which is justified given the small value of the well-measured \CP-violating parameter 
      $\epsilon_{K}$.

\item The measurement of $\sin{(2\beta+\phi_{d}^{\Delta})}$ results in two solutions
      for $2\beta+\phi_{d}^{\Delta}$ (in $[0,\pi]$). 
      One of these solutions can be excluded by measuring 
      the sign of $\cos{(2\beta+\phi_{d}^{\Delta})}$. 
      For a recent review of \babar\ and 
      Belle measurements see e.g. Ref.~\cite{lackerbeauty06}. 
      The current experimental 
      results from \babar\ and Belle using a time- and 
      angular-dependent analysis 
      of $B^0\to \jpsi K^{*0}$ decays, a time-dependent 
      Dalitz-plot analysis in 
      $\Bz/\Bzb \to D^{(*)0}/\bar{D}^{(*)0} h^{0}$ with 
      $\Bz/\Bzb \to D^{(*)0}/\bar{D}^{(*)0} h^{0}$, and 
      $\Bz/\Bzb \rightarrow D^{*+} D^{*-} K^{0}_{S}$,
      disfavour negative 
      $\cos{(2\beta+\phi_{d}^{\Delta})}$ values but
      HFAG deems it difficult to average the different measurements
      or to determine a reliable confidence level as a function of 
      $\cos{(2\beta+\phi_{d}^{\Delta})}$~\cite{HFAG08}. Here, as a simplification, it 
      is only assumed that $\cos{(2\beta+\phi_{d}^{\Delta})}>0$.
      
\item The constraint on the CKM angle $\alpha=\pi-\beta-\gamma$ 
      is obtained from time-dependent and
      time-independent measurements in the decays $B \to \pi\pi$, $B \to
      \rho\rho$, and $B \to \rho\pi$.  The time-dependent \CP
      asymmetries measured in $B \to \pi\pi$ provide information on the
      effective parameter $\sin{(2\alpha_\text{eff})}$ (which is a function of
      $\alpha$ and the penguin-to-tree ratio~\cite{Charles:1998qx}).  It is possible to
      translate this measurement into a constraint on $\alpha$ by
      exploiting isospin symmetry which allows to pin down 
      the penguin-to-tree ratio and thus to  determine the
      difference $\alpha-\alpha_\text{eff}$ from data~\cite{GronauLondon}.
      Under the assumption of exact isospin symmetry the amplitudes
      $A^{+-} \equiv A(B^{0} \to \pi^{+}\pi^{-})$, $A^{00} \equiv
      A(B^{0} \to \pi^{0}\pi^{0})$, and $A^{+0} \equiv A(B^{+} \to
      \pi^{+}\pi^{0})$ satisfy a triangular relationship:
      $\sqrt{2}A^{+0} - \sqrt{2}A^{00} = A^{+-}$.  A corresponding
      relationship holds for the \CP conjugated decays:
      $\sqrt{2}{\bar{A}}^{+0} - \sqrt{2}{\bar{A}}^{00} =
      {\bar{A}}^{+-}$.  These isospin triangles can be reconstructed by
      measuring the branching fractions and direct \CP asymmetries for
      the final states $B^{0} \to \pi^{+}\pi^{-}$, $B^{0} \to
      \pi^{0}\pi^{0}$, and $B^{+} \to \pi^{+}\pi^{0}$. Since one
      measures $\sin{(2\alpha_\text{eff})}$ and since the triangle has a twofold
      ambiguity for its apex in the complex
      plane, there is an eightfold ambiguity for $\alpha$ in $[0,\pi]$.
      The extraction of $\alpha$ from the isospin analysis is independent 
      of any possible NP contributions in the $\Delta I=1/2$ decay amplitude 
      except for the singular point $\alpha=0$~\cite{isospinprior}. If there
      is no NP contribution in the $\Delta I=3/2$ decay amplitude (as assumed here), 
      the
      extraction provides $\alpha=\pi-\gamma-\beta-\phi_{d}^{\Delta}/2$. As a consequence, 
      $\alpha$ is equivalent to $\gamma$ if $\beta+\phi_{d}^{\Delta}$ is measured e.g. 
      from $B \to \jpsi K_{S}$  as already pointed out above.
      
      Similar in line an isospin analysis can be performed for the $B
      \to \rho\rho$ system. In this case the analysis needs to take into
      account the measured longitudinal polarisation of the $\rho$
      mesons in the different final states $B^{0} \to \rho^{+}\rho^{-}$,
      $B^{0} \to \rho^{0}\rho^{0}$,
      and $B^{+} \to \rho^{+}\rho^{0}$. Finally, the $\rho\pi$ modes provides
      another crucial input for $\alpha$, by using a model for the Dalitz decay 
      into three pions in addition to the isospin symmetry~\cite{snyderquinn}.
      The results of these analyses that are based on the world averages of
      the various \babar\ and Belle measurements for the CP asymmetries and
      branching fractions determined by the Heavy Flavour Averaging Group~\cite{HFAG08} 
      are displayed in Figure~\ref{fig-alpha}.
      The combined analysis results in $\alpha=(89.0^{+4.4}_{-4.2})^{\circ}$ at $68.3\percent$ CL.

      The most stringent constraint on $\alpha$ comes currently from the
      $B^{0} \to \rho \rho$ channel. The uncertainty is driven by the
      rather large branching fraction $BF(B^{+} \to \rho^{+}\rho^{0})$.
      The input value for $BF(B^{+} \to \rho^{+}\rho^{0})$ has changed
      recently when the \babar collaboration presented a new analysis on the 
      final data set~\cite{babar_rho+rho0}. The large branching fraction value 
      leads to an isospin triangle that just closes. The measurement uncertainty 
      is smaller than the expected uncertainty. As a consequence, the current 
      uncertainty quoted for $\alpha$ might be on the optimistic side.
      It should also be stressed that at this level of precision so-far neglected
      uncertainties (electroweak penguins, $\pi-\eta^{(')}$ mixing, $\rho-\omega$ 
      mixing, other isospin violations, finite $\rho$-width, etc.) should be 
      considered in more detail. 
\end{itemize}

\begin{nfigure}{Htb}
 \includegraphics[width=10cm]{alpha_all.eps}
 \caption{\small Constraint on the angle $\alpha$ from the isospin analyses of 
   $B\to \pi\pi$ and $B\to\rho\rho$ decays and a Dalitz plot analysis of 
    $B\to\rho\pi$ decays.  In the presence of new physics in $B_d$ mixing, the
   quantity shown is $\pi-\gamma-\beta-\phi_{d}^{\Delta}$.}\label{fig-alpha}
\end{nfigure}

\begin{table}[Htp]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{|c|c|c|}\hline
Observable                                      & Value and uncertainties                      & Reference              \\
\hline
$|V_{ud}|$                                      & $0.97425 \pm 0.00022$                        &\cite{Hardy:2008gy}     \\
$|V_{us}|$                                      & $0.2246  \pm 0.0012$                         &\cite{flavianet}        \\
$m_{K}$                                         & $(497.614\pm 0.024)\mev$                     &\cite{PDG2008}          \\
$\GF$                                           & $(1.16637\pm 0.00001) \times 10^{-5}~\gev^2$ &\cite{PDG2008}          \\
$m_{W}$                                         & $(80.398 \pm 0.025)\gev$                     &\cite{PDG2008}          \\
$m_{B_{d}}$                                     & $(5.27917\pm 0.00029)\gev$                   &\cite{PDG2008}          \\
$m_{B^{+}}$                                     & $(5.27953\pm 0.00033)\gev$                   &\cite{PDG2008}          \\
$m_{B_{s}}$                                     & $(5.3663 \pm 0.0006)\gev$                    &\cite{PDG2008}          \\
$|V_{cb}|$                                      & $(40.89  \pm 0.38 \pm 0.59) \times 10^{-3}$  & see text               \\
$|V_{ub}|$                                      & $(3.92\pm0.09\pm0.45) \times 10^{-3}$        & see text               \\
$\mathcal{B}(B \rightarrow \tau \nu_{\tau})$     & $(1.73   \pm 0.35) \times 10^{-4}$           & see text               \\
$\gamma$                                        & $73{^{\,+19}_{\,-24}}^{\circ}$               & see text               \\
\hline
$\Delta m_{d}$                                  & $(0.507  \pm 0.005)\psinv$                   &\cite{PDG2008}          \\
$a_\text{fs}^{d}$                               & $(-47 \pm 46)\times 10^{-4}$                         &\cite{HFAG08}           \\
$\sin (2\beta+\phi_d^{\Delta})$                                   & $0.673   \pm 0.023$                          &\cite{HFAG09}           \\
$\cos (2\beta+\phi_d^{\Delta})$                                   & positive                                     &see text                \\
$\alpha$                                        & $89.0{^{\,+4.4}_{\,-4.2}}^{\circ}$           &see text                \\
\hline
$\Delta m_{s}$                                  & $(17.77  \pm 0.12)\psinv$                    &\cite{dmsexpvalue}      \\
$a_\text{fs}$                                   & $(-85 \pm 28)\times 10^{-4}$                       &see text and \cite{dimuon_evidence_d0,ASLCDF} \\
$a_\text{fs}^{s}$                               & $(-17 \pm 93)\times 10^{-4}$                         &\cite{ASLsD0_3}         \\
%$\Delta \Gamma^{\CP\prime}_{s}$                & $(0.090  \pm 0.050)\psinv$                   &\cite{D0DeltaGammasCP',CDFDeltaGammasCP'} \\
$(\phi^\Delta_{s}-2\beta_{s})$ vs. $\Delta \Gamma_{s}$ & see text                                     &\cite{taggedphaseCDF_2,taggedphaseD0,Punzi}                        \\ 
%Was: \phi_{s}
\hline
$\epsilon_{K}$                                  & $(2.229  \pm 0.010) \times 10^{-3}$          &\cite{PDG2008}          \\
$\Delta m_{K}$                                  & $(3.483  \pm 0.006) \times 10^{-12}\mev$     &\cite{PDG2008}          \\
\hline
\end{tabular}
\caption[ExperimentalInputs]
{Experimental inputs used in the fits.}
\label{tab:ExperimentalInputs}
\end{table}


\begin{table}[htp]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{|c|c|c|}\hline
Theoretical Parameter                          & Value and uncertainties                 & Reference                                 \\
\hline
$f_{B_{s}}$                                    & $(228    \pm 3     \pm 17)\mev$         & see Section~\ref{ssec:MethodofAveraging}  \\
$\Bag_{B_{s}}\big(m_{b}\big)$                  & $0.846   \pm 0.013 \pm 0.020$           & see Section~\ref{ssec:MethodofAveraging}  \\
$f_{B_{s}}/f_{B_{d}}$                          & $1.199   \pm 0.008 \pm 0.023$           & see Section~\ref{ssec:MethodofAveraging}  \\
$\Bag_{B_{s}}/\Bag_{B_{d}}$                    & $1.05    \pm 0.01  \pm 0.03$            & see Section~\ref{ssec:MethodofAveraging}  \\
$\hat{\eta}_{B}$                               & $0.8393   \pm 0.0034$                   & see text                                  \\
$\ov m_{t}(\ov m_{t})$                         & $(165.017 \pm 1.156 \pm 0.11)\gev$      & see text and \cite{TevEWWG}               \\
${\hat{\Bag}}_{K}$                             & $(0.724  \pm 0.004 \pm 0.067)$          & see Section~\ref{ssec:MethodofAveraging}  \\
$f_{K}$                                        & $156.1\mev$                             & see text                                  \\
$\kappa_\epsilon$                              & $0.940 \pm 0.013 \pm 0.023$                      & see text                                  \\
$\eta_{tt}$                                    & $0.5765  \pm 0.0065$                    &{\rd \cite{bjw}}                           \\
$\eta_{ct}$                                    & $0.47    \pm 0.04$                      &{\rd \cite{Nierste2}}                      \\
$\eta_{cc}$                                    & $\rd (1.39 \pm 0.35) \lt( \frac{1.29\, \gev}{\ov m_c}\rt)^{1.1}$ &\cite{Nierste1}   \\
$\lambda_{K}$                                  & $(1.25   \pm 0.00 \pm 0.75)\gev$        & see text                                  \\
$\ov m_{c}(\ov m_{c})$                         & $(1.286  \pm 0.013 \pm 0.040)\gev$      & see text                                  \\
$\widetilde{\Bag}_{S,B_s}/\widetilde{\Bag}_{S,B_d}$ & $1.01 \pm 0 \pm 0.03$              &\cite{BGMPR}                               \\
$\widetilde{\Bag}_{S,B_s}(m_b)$                & $0.91 \pm 0.03 \pm 0.12$                &\cite{BGMPR}                               \\
$\rd \Lambda_{\ov{\rm MS}}^{(5)}$              & $(0.222 \pm 0.027)\gev$                 & from $\alpha_s(M_{Z})$ in \cite{PDG2008}  \\
$\overline{m}_{s}(\overline{m}_{b})$           & $(0.085 \pm 0.017)\gev$                 &\cite{ln}                                  \\
$\overline{m}_{b}(\overline{m}_{b})$           & $(4.248 \pm 0.051)\gev$                 &\cite{HFAG10}                              \\
$m_{b}^{pow}$                                  & $(4.7\pm 0 \pm 0.1)\gev$                &\cite{ln}                                  \\
$\Bag_{R_{0}}$                                 & $1.0 \pm 0.5$                           &\cite{ln}                                  \\
$\Bag_{\tilde{R}_{1}}$                         & $1.0 \pm 0.5$                           &\cite{ln}                                  \\
$\Bag_{R_{1}}$                                 & $1.0 \pm 0.5$                           &\cite{ln}                                  \\
$\Bag_{R_{2}}$                                 & $1.0 \pm 0.5$                           &\cite{ln}                                  \\
$\Bag_{\tilde{R}_{2}}$                         & $1.0 \pm 0 \pm 0.5$                     &\cite{ln}                                  \\
$\Bag_{R_{3}}$                                 & $1.0 \pm 0.5$                           &\cite{ln}                                  \\
$\Bag_{\tilde{R}_{3}}$                         & $1.0 \pm 0 \pm 0.5$                     &\cite{ln}                                  \\
\hline
\end{tabular}
\caption[TheoreticalInputs]
{Theoretical inputs used in the fits.}
\label{tab:TheoreticalInputs}
\end{table}


\boldmath
\subsection
{Observables in the \bbsuli system affected by New Physics in mixing}\label{ssec:InputsBsWithNP}
\unboldmath

\noindent 
Observables which can be possibly affected by $\vert \Delta F\vert =2$ NP 
contributions in the \Bs-\Bsb system are:
\begin{itemize}
\item  The experimental input for the mass difference $\Delta m_s$ taken from HFAG is dominated by 
       the measurement of CDF~\cite{dmsexpvalue}. The dependence of the SM prediction
       $\Delta m_s^{\text{SM}}$ on $\bar \rho-\bar \eta$ coordinates appears very 
       weak through the relevant CKM matrix elements term $\vert V^*_{ts}V^{\phantom{*}}_{tb}\vert^2$, 
       but the physical $\Delta m_s$ gives a direct constraint in the $\Delta_s$ plane, 
       by computing  $f_{B_{s}} \sqrt{\hat{\Bag}_{B_{s}}}$
       from $f_{B_{s}}$ and $\Bag_{B_{s}}$, given in Table~\ref{tab:TheoreticalInputs}.
       The hadronic matrix element for the $B_s$ system       
        can be related to the $B_d$ one via the 
       flavour-${\mathit SU}(3)$ breaking correction parameter $\xi$ defined through 
       $f_{B_s}^2 \Bag_{B_{s}} = \xi^2 f_{B_d}^2 \Bag_{B_{d}}$ 
       Measurements of $\Delta m_s$ thus reduce the uncertainties on $f_{B_d}^2 \Bag_{B_{d}}$
       since $\xi$ is better known from lattice QCD than $f_{B_d}^2 \Bag_{B_{d}}$.
       This relation also generates a strong correlation between the NP parameters 
       $|\Delta_d|$ and $|\Delta_s|$ when allowing for new physics contributions to mixing.

\item  The SM prediction \cite{ln} for $a_\text{fs}^{s}$ is at least one order of magnitude 
       smaller than the one for $a_\text{fs}^{d}$, see Table \ref{tab:fitResults_SM1}.
       Compared to the SM 
       prediction the measurement from \Dzero has a quite large uncertainty, 
       $a_\text{fs}^{s}=(-17 \pm 91{^{+12}_{-23}}) \times 10^{-4}$~\cite{ASLsD0_3}, 
       and hence does not have a strong impact on the NP constraints in the $B_{s}$ sector. 
       It is nevertheless included in our analysis.

\item  \Dzero~\cite{dimuonexp} using $1~fb^{-1}$ and CDF~\cite{ASLCDF} have measured inclusive 
       dimuon CP asymmetries. The \Dzero result corresponding to the measurement quoted in 
       Ref.~\cite{dimuonexp} presented as 
       \(
           a_\text{fs}^{d} + \frac{f_{s} Z_{s}}{f_{d} Z_{d}} a_\text{fs}^{s} = -0.0028\pm0.0013\pm0.0009
       \)
       is provided in~\cite{D0CombinedCPasymmetry} whereas CDF~\cite{ASLCDF} quotes the result as
       \begin{equation}
       a_\text{fs}= \frac{f_{d} Z_{d} a_\text{fs}^{d} + f_{s} Z_{s} a_\text{fs}^{s}}{f_{d} Z_{d} + f_{s} Z_{s}}
                  =0.0080\pm0.0090\pm0.0068, 
       \end{equation}
       where $f_{d(s)}$ is the fraction of neutral $B_{d(s)}$ mesons 
       in the fragmentation and $Z_{d(s)}$ is given by 
       \begin{equation}
          Z_{q}= \frac{1}{1-y_{q}^{2}}-\frac{1}{1+x_{q}^{2}}
       \end{equation}
       with $y_{q}=\Delta \Gamma_{q}/2\Gamma_{q}$ and $x_{q}=\Delta m_{q}/\Gamma_{q}$
       (see also Ref.~\cite{nir2006}).
       Very recently \Dzero has presented a new measurement of $a_\text{fs}$~\cite{dimuon_evidence_d0}
       using $6.1~fb^{-1}$ integrated luminosity which shows a $3.2\sigma$  deviation from the 
       (almost zero) SM prediction, and is the first direct evidence against the SM in $B$ meson observables:
       $a_\text{fs}=-0.00957 \pm 0.00251 \pm 0.00146$. This result supersedes the former 
       result in Ref.~\cite{D0CombinedCPasymmetry}. The average between the new \Dzero result and the
       CDF result reads 
       \begin{equation}
       a_\text{fs}=-0.0085 \pm 0.0028,
       \end{equation}
        which is $3.0$ standard deviations away
       from the SM prediction.\\
       For the interpretation of the measured observables we use the following values and 
       uncertainties for $f_{d(s)}$, $y_{q}$ and $x_{q}$: 
       $f_{d}=0.333 \pm 0.030$, $f_{s}=0.121 \pm 0.015$ with a correlation coefficient of
       $+0.439$ are taken from Ref.~\cite{HFAG10}, $x_{d}$ is calculated from 
       $\Delta m_{d} = (0.507 \pm 0.005)\cdot 10^{12} s^{-1}$~\cite{PDG2008} and $\tau_{B_{d}}=(1.525 \pm 0.009) \cdot 10^{-12} s$~\cite{HFAGforPDG2009},
       $y_{d}$ is set to zero since $\Delta \Gamma_{d}$ is expected to be very small
       in the Standard Model and is hardly affected by New Physics, $x_{s}$ is calculated from 
       $\Delta m_{s} = (17.77 \pm 0.12)\cdot 10^{12} s^{-1}$~\cite{PDG2008} and $y_{s}$ from 
       $\tau_{B_{s}}=(1.515 \pm 0.034) \cdot 10^{-12} s$~\cite{HFAGforPDG2009}, the measurement of
       $\Delta \Gamma_{s}$ in $B_s\to J/\psi\, \phi$ (see below)
       and the flavour-specific $B_{s}$
       lifetime $\tau_{B_{s}}^{FS}=(1.417 \pm 0.042) \cdot 10^{-12} s$~\cite{HFAGforPDG2009}. This results in
       $Z_{d}= 0.3741 \pm 0.0054$ and $Z_{s}= 1.0044^{+0.0058}_{-0.0032}$\, hence actually the error on $Z_{d,s}$
       has negligible impact on our global fits.
       
\item  CDF and \Dzero have presented time-dependent tagged analyses\cite{taggedphaseCDF,taggedphaseD0}
       of $B_s\to J/\psi\, \phi$ decays  which provide information on $\Delta_s$ through 
       $\Delta \Gamma_{s}$ and $\phi_s^\Delta-2\beta_s$. 
       These analyses supersede the previous untagged studies of Refs.~\cite{dgexpnew, CDFdg} 
       (which use the same data sample, but without the tagging information)
       and we will not use the untagged results in our analysis. 
       These time-dependent tagged analyses have raised a lot of attention recently, 
       in particular when the UTfit collaboration claimed an evidence of New Physics 
       of at least $3 \sigma$ based on a global fit where these measurements played a 
       central role~\cite{Bona:2008jn}. It has been later shown, though, that this 
       conclusion came from an overinterpretation of the data~\cite{capri}. 
       In the framework of the HFAG~\cite{HFAG08} the CDF and \Dzero have determined 
       a combined constraint based on the measurements in Refs.~\cite{taggedphaseCDF,taggedphaseD0}. 
       In 2009 CDF has updated the analysis on a larger dataset using $2.8 fb^{-1}$ 
       of data~\cite{taggedphaseCDF_2}. The new average between \Dzero~\cite{taggedphaseD0} 
       and  CDF~\cite{taggedphaseCDF_2} has been presented in Summer 2009~\cite{Punzi}.
       Using this new average the deviation of the measured value for $\beta_s$ with 
       respect to the SM value is essentially unchanged and reads $2.3$ standard deviations. 
       This average is our default input for the corresponding observables, supplemented by the 
       constraint on the flavour-specific $B_{s}$
       lifetime $\tau_{B_{s}}^{FS}=(1.417 \pm 0.042) \cdot 10^{-12}$~\cite{HFAGforPDG2009} which can be viewed as an independent measurement
       of $\Delta \Gamma_{s}$.
       New results for $B_s\to J/\psi\, \phi$ have been presented in Summer 2010 by
       CDF (with $5.2 fb^{-1}$)~\cite{taggedphaseCDF_3} and D\O\ (with $6.1 fb^{-1}$)~\cite{taggedphaseD0} 
       collaborations, in closer agreement to the Standard Model expectations, but these measurements    
       have not been combined together yet. They have not be included in the present analysis, 
       but are briefly discussed together with the SM significance tests below.
\end{itemize}

\subsection{The neutral kaon system\label{Sec:InputsKaonWithNP}}

The master formula for $\epsilon_K$ has been given in \eq{cons}, from the relation 
between $\epsilon_K$ and $M_{12}^s$. The translation of $\epsilon_{K}$ into a constraint 
on $\rhobar$ and $\etabar$ suffers from sizeable uncertainties in the Wolfenstein 
parameter $A$ (the determination of which being driven by $|V_{cb}|^{4}$), ${\hat{\Bag}}_{K}$ 
(see Table~\ref{tab:TheoreticalInputs} and Section~\ref{ssec:MethodofAveraging}), 
from the long-distance corrections to the relation between $M_{12}^K$ and $\epsilon_K$ encoded in $\kappa_\epsilon$,  
and, though of less importance, from uncertainties in the QCD corrections coming from 
$\eta_{cc}$~\cite{Nierste1}, from the charm quark mass $\ov m_{c}(\ov m_{c})$ in 
the $\ov {\text{MS}}$ scheme, from $m_{t}$ and the perturbative QCD corrections 
$\eta_{tt}$~\cite{Nierste1} and $\eta_{ct}$~\cite{Nierste1}.


\begin{itemize}
\item From the experimental point of view, the number on the 
 LHS of Eq.~\ref{cons} has shifted substantially over time. For instance in 1995 the 
 corresponding number was $1.21\cdot 10^{-7}$~\cite{Nierste2}. More recently, the 
 numerical value for $\epsilon_{K}$ has shifted by about $2.3~\%$ (a $3.7~\sigma$ effect) 
 between the 2004 and 2006 edition of the  Particle Data Group (PDG) from 
 $(2.284\pm0.014) \times 10^{-3}$~\cite{PDG2004} down to 
 $(2.232\pm0.007) \times 10^{-3}$~\cite{PDG2008}. 
 This shift has been mainly driven by improved measurements of the branching fraction 
 $BF(K_{L} \to \pi^{+}\pi^{-})$ performed by the KTeV, KLOE and NA48 collaborations 
 leading to a reduction of $5.5~\%$ of the semileptonic branching fraction values.
\item As discussed in sec.~\ref{ssec:kkmbasics}, 
 the relation between $\epsilon_K$ and \kkm\ is affected by several corrections
 encoded in $\kappa_\epsilon$. We have combined them with the experimental result for $\epsilon_K$ as indicated in this section.
\item
 The kaon decay constant $f_{K}$ is taken from the review on pseudoscalar decay 
 constants in Ref.~\cite{PDG2008} which is calculated from the measured branching 
 fraction $BF(K^{+} \rightarrow \mu^{+} \nu_{\mu}(\gamma))$ and the measured 
 charged kaon decay time using an external input for $|V_{us}|$. In Ref.~\cite{PDG2008}
 $|V_{us}|=0.2255 \pm 0.0019$ from $K_{\ell 3}$ decays is used as external input
 which leads to $f_{K}=155.5 \pm 0.2 \pm 0.8 \pm 0.2$ MeV where the first error is due
 to the experimental uncertainties, the second due to the uncertainty from $|V_{us}|$
 and the third due to higher order corrections~\cite{PDG2008}.
 With our input of $|V_{us}|=0.2246 \pm 0.0012$ this translates into
 $f_{K}=156.1 \pm 0.2 \pm 0.6 \pm 0.2$ MeV. In the fit, we do not consider the uncertainties
 on $f_{K}$ at this point since they are currently negligible with respect to the other 
 uncertainties. Since the $f_{K}$ value obtained in this way is anticorrelated with
 our $|V_{us}|$ input a consistent treatment would require including the leptonic kaon 
 decay in the fit and constrain $f_{K}$ simultaneously. This would then lead to an even
 smaller uncertainty on $f_{K}$ given the improved uncertainty on $|V_{us}|$ imposed by 
 the global fit. Such a fit is technically possible but has not been performed here since the 
 error from $f_{K}$ on $\epsilon_{K}$ does not play an important role. 
\item
 Various determinations of the charm quark mass are available. For instance,
 the charm quark mass in the kinetic mass scheme can be obtained from fits 
 to data from lepton energy and hadronic mass moments in $B \to X_{c}\ell\nu$ decays 
 combined with photon energy moments measured in $B \to X_{s}\gamma$ decays, see
 e.g. Refs.~\cite{BuchmuellerFlaecher,HFAG08}. 
 The most recent value for the kinetic mass quoted by HFAG is 
 $m_{c}^{kin}=(1.165\pm 0.050)~{\rm GeV}$~\cite{HFAG08}, corresponding to a
 value in the $\overline{\rm MS}$ scheme:
 $\ov m_{c}(\ov m_{c})=(1.265\pm0.060\pm0.050)~{\rm GeV}$.
 The first uncertainty on the charm quark mass is correlated with the b-quark 
 mass uncertainty obtained from the same fits quoted in Table~\ref{tab:ExperimentalInputs}
 with a linear correlation coefficient of order $98~\%$.
 A second uncertainty of $50$ MeV has been added following the discussion in 
 Ref.~\cite{BuchmuellerFlaecher}, to take into account 
 the low renormalisation scale and the size of higher-order perturbative corrections
 when translating the mass from one scheme to another.
 \\
 As an alternative, the charm quark mass can also be determined from $e^{+}e^{-}$
 annihilation data into hadrons created from quark-antiquark pairs. 
 The OPE-based method consists in writing sum rules for moments of the cross 
 section $\sigma(e^{+} e^{-} \rightarrow c \bar{c})$, which are dominated by 
 the perturbative term and the contribution proportional to the gluon condensate. 
 An older analysis of Steinhauser and K\"uhn based on a three-loop calculation
 finds $\ov m_{c}(\ov m_{c})=(1.304\pm0.027)~{\rm GeV}$ with a small uncertainty~\cite{SteinhauserKuehn}. 
 A similar analysis by Jamin and Hoang~\cite{JaminHoang} obtains a consistent 
 result, $\ov m_{c}(\ov m_{c})=(1.290\pm0.070)~{\rm GeV}$, but quotes a 
 significantly larger uncertainty. This can be ascribed to the choice of the 
 OPE scale separating short- and long-distance physics and it can be viewed 
 as the impact of (neglected) higher order terms in perturbation theory on 
 the determination of $m_c$ through such moments. 
 In a more recent calculation at four loops for the perturbative contribution, 
 the uncertainty has been further reduced:
 $\ov m_{c}(\ov m_{c})=(1.286\pm0.013)~{\rm GeV}$~\cite{Kuhn:2007vp}~\footnote{Using an 
 improved routine for the renormalisation group evolution of the computed 
 value $m_c(3 GeV)=(0.986 \pm 0.013)$ GeV the same group finds a slightly shifted central value 
 $\ov m_{c}(\ov m_{c})=1.268$ GeV~\cite{Steinhauser:2008pm}.}. However, in this reference, there 
 is only a limited discussion of the freedom in the choice of the renormalisation 
 scale for the perturbative series and the gluon condensate is varied only in a 
 limited range, even though these two effects were seen as bringing significant 
 systematics in Ref.~\cite{JaminHoang}.
 In the absence of further studies 
 on the systematics discussed above in the case of the four-loop analysis of 
 ref.~\cite{Kuhn:2007vp}, we assign an additional theoretical uncertainty of 
 $0.040~{\rm GeV}$ and use as our input value: 
 $\ov m_{c}(\ov m_{c})=(1.286\pm0.013\pm0.040)~{\rm GeV}$ which is consistent 
 with the values from the moment fits but has smaller uncertainties.
\end{itemize}
