\section{Quantitative Studies}\label{sec:quant} 

\subsection{Standard Model fit}\label{ssec:SMfit} 

In this section we present the current status of the Standard Model CKM fit.
Fig.~\ref{fig-rhoeta_smfit} shows the fit results in the $\bar\rho-\bar\eta$ 
\begin{nfigure}{Htb}
  \includegraphics[width=12cm]{rhoeta_large_global_Vub.eps}
  \caption{\small Constraint on the CKM $(\bar\rho,\bar\eta)$ coordinates from
                  the global Standard Model CKM-fit.
                  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-rhoeta_smfit}}
\end{nfigure}
plane. Table~\ref{tab:fitResults_SM1} shows the 
fit results for various parameters. 
\begin{table}[Htb]
\begin{center}\begin{tabular}{llll}  \hline
{ Quantity} & { central $\pm$ ${\rm CL}\equiv1\sigma$} & { $\pm$ ${\rm CL}\equiv2\sigma$  } & { $\pm$ ${\rm CL}\equiv3\sigma$ }  \\ 
 \hline  &&& \\[-0.3cm]
$A$ & $0.8184^{\,+0.0094}_{\,-0.0311}$ & $0.818^{\,+0.019}_{\,-0.040}$ & $0.818^{\,+0.028}_{\,-0.048}$
\\[0.15cm]
$\lambda$ & $0.22512^{\,+0.00075}_{\,-0.00075}$ & $0.2251^{\,+0.0015}_{\,-0.0015}$ & $0.2251^{\,+0.0022}_{\,-0.0022}$
\\[0.15cm]
$\bar\rho$ & $0.139^{\,+0.027}_{\,-0.023}$ & $0.139^{\,+0.051}_{\,-0.033}$ & $0.139^{\,+0.063}_{\,-0.042}$
\\[0.15cm]
$\bar\eta$ & $0.342^{\,+0.016}_{\,-0.015}$ & $0.342^{\,+0.031}_{\,-0.027}$ & $0.342^{\,+0.046}_{\,-0.037}$
\\[0.15cm]
\hline &&& \\[-0.3cm]
$\widehat B_K$ (!) & $0.82^{\,+0.26}_{\,-0.14}$ & $0.82^{\,+0.33}_{\,-0.18}$ & $0.82^{\,+0.40}_{\,-0.21}$
\\[0.15cm]
$f_{B_s}$ [MeV] (!) & $234.7^{\,+9.8}_{\,-8.3}$ & $235^{\,+13}_{\,-12}$ & $235^{\,+16}_{\,-15}$
\\[0.15cm]
$\widehat\Bag_{B_{s}}$ (!)                   
                 & $ 1.136 ^{\,+ 0.103}_{\,-  0.043}$   &  1.136 $^{\,+  0.457}_{\,-  0.081}$   & 1.14 $^{\,+  0.61}_{\,-  0.11}$ \\[0.15cm]
$f_{B_s}/f_{B_d}$ (!) & $1.199^{\,+0.058}_{\,-0.044}$ & $1.199^{\,+0.109}_{\,-0.096}$ & $1.20^{\,+0.15}_{\,-0.14}$
\\[0.15cm]
$\Bag_{B_{s}}/\Bag_{B_{d}}$ (!)     
                 & $1.156 ^{\,+   0.075}_{\,- 0.104}$   & $ 1.16 ^{\,+  0.15}_{\,-  0.21}$     & $1.16 ^{\,+  0.23}_{\,-  0.30}$ \\[0.15cm]
\hline &&& \\[-0.3cm]
$m_c\unit{GeV}$ (!) & $1.49^{\,+0.56}_{\,-0.52}$ & $1.49^{\,+0.64}_{\,-0.68}$ & $1.49^{\,+0.70}_{\,-0.85}$
\\[0.15cm]
$m_t\unit{GeV}$ (!) & $153.5^{\,+10.5}_{\,-6.9}$ & $154^{\,+40}_{\,-10}$ & $154^{\,+53}_{\,-13}$
\\[0.15cm]
 \hline &&&      \\[-0.3cm]
$J$~~$[10^{-5}]$ & $2.98^{\,+0.15}_{\,-0.19}$ & $2.98^{\,+0.30}_{\,-0.23}$ & $2.98^{\,+0.45}_{\,-0.28}$
\\[0.15cm]
\hline &&& \\[-0.3cm]
$\alpha\unit{deg}$ (!) & $97.4^{\,+1.5}_{\,-9.5}$ & $97.4^{\,+2.9}_{\,-12.1}$ & $97.4^{\,+4.3}_{\,-13.9}$
\\[0.15cm]
$\beta\unit{deg}$ (!) & $28.11^{\,+0.69}_{\,-1.84}$ & $28.1^{\,+1.4}_{\,-5.0}$ & $28.1^{\,+2.1}_{\,-7.7}$
\\[0.15cm]
$\gamma\unit{deg}$ (!) & $67.7^{\,+3.6}_{\,-4.1}$ & $67.7^{\,+5.2}_{\,-7.7}$ & $67.7^{\,+6.7}_{\,-9.5}$
\\[0.15cm]
$\beta_s$ [deg] & $1.038^{\,+0.048}_{\,-0.047}$ & $1.038^{\,+0.097}_{\,-0.080}$ & $1.04^{\,+0.14}_{\,-0.11}$
\\[0.15cm]
\hline\\
 \vspace{-0.5cm}
\end{tabular}\end{center}
 \caption[.]{\label{tab:fitResults_SM1}%\em
 Fit results of the Standard Model fit. The notation `(!)' means that the fit output represents the indirect constraint, \textit{i.e.}
              the corresponding direct input has been removed from the analysis.}
\end{table}
In Tables~\ref{tab:fitResults_SM1} and~\ref{tab:fitResults_SM2}
we also show the result of the fit for observables that have been individually 
excluded from the fit in order to quantify possible deviations between the 
individual input values and their fit predictions.
The $\chi^{2}/N_\mathrm{dof}$ of the combined Standard Model fit is approximately 
$7.2/7$ and hence corresponds to a good overall agreement. However this global minimum
$\chi^2$ mixes quantities that are in perfect agreement with their fit prediction, with others that
are individually at odds. Possible deviations
between a selection of measured observables and their Standard Model predictions are discussed
in more detail in the following.
%
\begin{table}[Htb]
\begin{center}\begin{tabular}{llll} \hline
{ Quantity} & { central $\pm$ ${\rm CL}\equiv1\sigma$} & { $\pm$ ${\rm CL}\equiv2\sigma$  } & { $\pm$ ${\rm CL}\equiv3\sigma$ }  \\
 \hline  &&& \\[-0.3cm]
$|V_{ud}|$ (!) & $0.97444^{\,+0.00028}_{\,-0.00028}$ & $0.97444^{\,+0.00055}_{\,-0.00056}$ & $0.97444^{\,+0.00082}_{\,-0.00084}$
\\[0.15cm]
$|V_{us}|$ (!) & $0.22544^{\,+0.00095}_{\,-0.00095}$ & $0.2254^{\,+0.0019}_{\,-0.0019}$ & $0.2254^{\,+0.0028}_{\,-0.0029}$
\\[0.15cm]
$|V_{ub}|$ (!) & $0.00353^{\,+0.00015}_{\,-0.00021}$ & $0.00353^{\,+0.00030}_{\,-0.00032}$ & $0.00353^{\,+0.00045}_{\,-0.00042}$
\\[0.15cm]
$|V_{cd}|$ & $0.22498^{\,+0.00075}_{\,-0.00075}$ & $0.2250^{\,+0.0015}_{\,-0.0015}$ & $0.2250^{\,+0.0022}_{\,-0.0022}$
\\[0.15cm]
$|V_{cs}|$ & $0.97348^{\,+0.00021}_{\,-0.00017}$ & $0.97348^{\,+0.00038}_{\,-0.00035}$ & $0.97348^{\,+0.00055}_{\,-0.00052}$
\\[0.15cm]
$|V_{cb}|$ (!) & $0.04548^{\,+0.00078}_{\,-0.00574}$ & $0.0455^{\,+0.0014}_{\,-0.0062}$ & $0.0455^{\,+0.0020}_{\,-0.0067}$
\\[0.15cm]
$|V_{td}|$ & $0.00865^{\,+0.00024}_{\,-0.00039}$ & $0.00865^{\,+0.00036}_{\,-0.00056}$ & $0.00865^{\,+0.00048}_{\,-0.00069}$
\\[0.15cm]
$|V_{ts}|$ & $0.04072^{\,+0.00038}_{\,-0.00146}$ & $0.04072^{\,+0.00076}_{\,-0.00180}$ & $0.0407^{\,+0.0011}_{\,-0.0021}$
\\[0.15cm]
$|V_{tb}|$ & $0.999133^{\,+0.000060}_{\,-0.000016}$ & $0.999133^{\,+0.000074}_{\,-0.000032}$ & $0.999133^{\,+0.000086}_{\,-0.000048}$
\\[0.15cm]
\hline &&& \\[-0.3cm]
$\phi_d$ [deg]  & $-5.5 ^{\,+ 2.5}_{\,-    3.3}$  & $-5.5 ^{\,+    2.7}_{\,-     4.3}$  & $-5.5 ^{\,+    2.9}_{\,-     5.6}$  \\[0.15cm]
$\phi_s$ [deg]  & $0.27 ^{\,+ 0.20}_{\,-    0.18}$  & $0.27 ^{\,+    0.28}_{\,-     0.25}$  & $0.27 ^{\,+    0.35}_{\,-     0.31}$  \\[0.15cm]
\hline &&& \\[-0.3cm]
$|\epsilon_K|$~~$[10^{-3}]$ (!) & $2.01^{\,+0.58}_{\,-0.66}$ & $2.01^{\,+0.73}_{\,-0.74}$ & $2.01^{\,+0.88}_{\,-0.81}$
\\[0.15cm]
$\Delta m_d \unit{ps^{-1}}$ (!) & $0.558^{\,+0.053}_{\,-0.065}$ & $0.558^{\,+0.091}_{\,-0.116}$ & $0.56^{\,+0.13}_{\,-0.16}$
\\[0.15cm]
$\Delta m_s \unit{ps^{-1}}$ (!) & $17.8^{\,+2.0}_{\,-2.2}$ & $17.8^{\,+3.3}_{\,-3.4}$ & $17.8^{\,+4.5}_{\,-4.3}$
\\[0.15cm]
$a_\text{fs}$~~$[10^{-4}]$ (!)            
                & $-3.10 ^{\,+0.83}_{\,-0.98}$  & $-3.1^{\,+ 1.1}_{\,-1.4}$   & $-3.1^{\,+ 1.2}_{\,- 1.9}$  \\[0.15cm]
$a_\text{fs}^{d}$~~$[10^{-4}]$ (!)            
                & $-6.4^{\,+ 1.6}_{\,-1.8}$ & $-6.4^{\,+1.9}_{\,- 2.4}$   & $-6.4^{\,+ 2.2}_{\,- 3.1}$  \\[0.15cm]
$a_\text{fs}^{s}$~~$[10^{-4}]$ (!)            
                & $0.30^{\,+ 0.12}_{\,-0.13}$  & $0.30^{\,+0.19}_{\,- 0.20}$   & $0.30^{\,+ 0.24}_{\,- 0.27}$  \\[0.15cm]
$\Delta\Gamma_d \unit{ps^{-1}}$ (!)            
                & $0.0038^{\,+ 0.0011}_{\,-0.0012}$  & $0.0038^{\,+ 0.0013}_{\,- 0.0014}$   & $0.0038^{\,+ 0.0014}_{\,- 0.0016}$  \\[0.15cm]
$\Delta\Gamma_s \unit{ps^{-1}}$ (!)            
                & $0.124^{\,+ 0.038}_{\,-0.042}$  & $0.124^{\,+ 0.044}_{\,- 0.050}$   & $0.124^{\,+ 0.049}_{\,- 0.058}$  \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\BRB{\tau\nu}$~~$[10^{-4}]$ (!) & $0.763^{\,+0.113}_{\,-0.061}$ & $0.763^{\,+0.214}_{\,-0.097}$ & $0.76^{\,+0.30}_{\,-0.13}$
\\[0.15cm]
$\BRB{\mu\nu}$~~$[10^{-6}]$ & $0.380^{\,+0.046}_{\,-0.043}$ & $0.380^{\,+0.085}_{\,-0.071}$ & $0.380^{\,+0.123}_{\,-0.088}$
\\[0.15cm]
$\BRB{e\nu}$~~$[10^{-11}]$ & $0.89^{\,+0.11}_{\,-0.10}$ & $0.89^{\,+0.20}_{\,-0.17}$ & $0.89^{\,+0.29}_{\,-0.21}$
\\[0.15cm]
\hline
 \vspace{-0.5cm}
\end{tabular}\end{center}
\caption[.]{\label{tab:fitResults_SM2} %\em 
Fit results of the Standard Model fit. The notation `(!)' means that the fit output represents the indirect constraint, \textit{i.e.}
              the corresponding direct input has been removed from the analysis.}
\end{table}

One observes a sizeable discrepancy between the input value of
$\BRB{\tau\nu}$ (see Table~\ref{tab:ExperimentalInputs}) and its fit
prediction (see Table~\ref{tab:fitResults_SM2}) which is mainly driven
by the measured value of $\sin{2\beta}$, and was first discussed in
Ref.~\cite{CKMfitter08}. Removing either $\BRB{\tau\nu}$
or $\sin{2\beta}$ from the list of inputs results in a $\chi^{2}$ change
that corresponds to $2.6$ standard deviations.  This discrepancy could
arise either from a statistical fluctuation in the measured
$\BRB{\tau\nu}$ value, from too small (large) a value of $f_{B_{d}}$
($\HatBag_{B_{d}}$), or from NP in $B \rightarrow \tau \nu$ and/or the $\sin{2\beta}$
measurements in $B\to J/\psi K_S$. 
There is a specific correlation between the $\sin{2\beta}$ and
$\BRB{\tau\nu}$ in the global fit that is a bit at odds with the direct
experimental determination.  This is best viewed in the
$(\sin{2\beta}$, $\BRB{\tau\nu})$ plane (see
Fig.~\ref{fig-sinbeta-Btotaunu}), regarding the prediction from the
global fit without using these measurements.
%
\begin{nfigure}{Htb}
  \includegraphics[width=10cm]{tension_sin2b_Btaunu.eps}
  \caption{\small Constraint in the $(\sin{2\beta}$, $\BRB{\tau\nu})$
    plane. The coloured constraint represents the prediction for these
    quantities from the global fit when these inputs are removed
    while the cross respresents the measurements with the corresponding
    $1~\sigma$ uncertainty.\label{fig-sinbeta-Btotaunu}}
\end{nfigure}
%
The shape of the correlation can be understood more deeply by considering the ratio 
$\BRB{\tau\nu}/\Delta m_d$, where the decay constant $f_{B_{d}}$ cancels, leaving limited 
theoretical uncertainties restricted to the bag parameter $\HatBag_{B_{d}}$. 
The formula for the ratio displays explicitly 
that the correlation between $\BRB{\tau\nu}$ and the angle $\beta$ is controlled by 
the values of $\HatBag_{B_{d}}$, and the angles $\gamma$ and  $\alpha=\pi-\beta-\gamma$:
\begin{eqnarray}
\frac{\BRB{\tau\nu}}{\Delta m_{d}} = \frac{3 \pi}{4} \frac{m_{\tau}^{2}}{m_{W}^{2} S(x_{t})} \left( 1 - \frac{m_{\tau}^{2}}{m_{B^{+}}^{2}} \right)^{2} \tau_{B^{+}} \frac{1}{\HatBag_{B_{d}} {\eta}_B} \frac{1}{{|V_{ud}|}^{2}} \left( \frac{\sin{\beta}}{\sin{\gamma}} \right)^{2}. 
 \label{BRBoverDeltamd}
\end{eqnarray}
%
The comparison of the indirect prediction of
$\HatBag_{B_{d}}$ from the above analytical formula (having only
$\BRB{\tau\nu}$, $\Delta m_{d}$, $\sin{2\beta}$, $\alpha$, $\gamma$ and
$|V_{ud}|$ as inputs, that is an almost completely theory-free
determination of $\HatBag_{B_{d}}$) with the current direct lattice
determination $\HatBag_{B_{d}} = 1.221^{+0.087}_{-0.085}$ is given in
Fig.~\ref{fig-BBd}. For this test the deviation is $2.7~\sigma$,
dominated first by the experimental error on $\BRB{\tau\nu}$, $\alpha$, $\gamma$ and
second by the theoretical uncertainty on $\HatBag_{B_{d}}$. This tests clearly shows that
the semileptonic extraction of $|V_{ub}|$ has little to do with
the $\BRB{\tau\nu}$ anomaly.
%
Further insight is provided by Fig.~\ref{fig-BBd_fBd} where the constraints
on the decay constant $f_{B_{d}}$ and $f_{B_{d}} \sqrt{\HatBag_{B_{d}}}$ are
shown. We compare the fit inputs $f_{B_{d}}$ and $f_{B_{d}} \sqrt{\HatBag_{B_{d}}}$ 
taken from LQCD calculations with their predictions from the fit. The measured
$\BRB{\tau\nu}$ value leads to the constraint on $f_{B_{d}}$ represented by the 
green band. The orange band represents the constraint on $f_{B_{d}} \sqrt{\HatBag_{B_{d}}}$
thanks to the $\Delta m_{d}$ measurement. The combined prediction for both
quantities (red and yellow regions) reveals that the predicted value for
$f_{B_{d}} \sqrt{\HatBag_{B_{d}}}$ is in very good agreement with the LQCD 
input. Therefore, if the discrepancy is driven by too small a
$f_{B_{d}}$ value, the lattice artefact responsible for this underestimation
should not affect the more complicated determination
of the $\Delta B=2$ matrix element proportional to $f_{B_{d}} \sqrt{\HatBag_{B_{d}}}$,
as already demonstrated in Fig.~\ref{fig-BBd} in order 
to preserve the  good agreement between the predicted and calculated 
values for $f_{B_{d}} \sqrt{\HatBag_{B_{d}}}$.

\begin{nfigure}{H}
  \includegraphics[width=10cm]{Bd_from_Deltamd_Btaunu_Angles.eps}
  \caption{\small Comparison between the $\HatBag_{B_{d}}$ parameter used in the global fit
                  and its prediction.
                  The coloured constraint represents the prediction of $\HatBag_{B_{d}}$
                  quantities from the global fit while the blue point represents the 
                  input value with the corresponding
                  $1~\sigma$ uncertainty as used in the global fit.\label{fig-BBd}}
  \includegraphics[width=10cm]{fsqBd-fBd.eps}
  \caption{\small Constraint on $f_{B_{d}}$ and $f_{B_{d}} \sqrt{\HatBag_{B_{d}}}$. 
                  The green band shows the $95.45\%$ CL constraint on $f_{B_{d}}$ when 
                  $\BRB{\tau\nu}$ is included in the fit. The orange band represents 
                  the $95.45\%$ CL constraint on $f_{B_{d}} \sqrt{\HatBag_{B_{d}}}$
                  thanks to the $\Delta m_{d}$ measurement.
                  The combined $68.3\%$ CL and $95.45\%$ CL constraint is shown in red, 
                  respectively, in yellow. The black point shows the $1~\sigma$ 
                  uncertainty on $f_{B_{d}}$ and $f_{B_{d}} \sqrt{\HatBag_{B_{d}}}$ 
                  as used in the fit.\label{fig-BBd_fBd}}
\end{nfigure}

\clearpage

Another potential anomaly related to the $|\epsilon_K|$ asymmetry has been widely discussed 
in the literature~\cite{bg,smtensions}, but does not show up with our choice of 
inputs and statistical treatment. More details
can be found in the Appendix. Other interesting outcomes of the SM global fit concern the prediction
of the recently measured CP-asymmetries  by the TeVatron experiments, namely in $B_s\to J/\psi\phi$ 
and in dimuonic inclusive decays (see Section~\ref{ssec:inputs}). The discrepancy of these measurements with respect to their 
SM fit prediction, together with the $\BRB{\tau\nu}$ anomaly, are summarized in Table~\ref{SMpvalues}.
\begin{table}[Htb]
\begin{center}\begin{tabular}{ll} \hline
Observable & deviation \\ & wrt the SM fit  \\
 \hline  & \\[-0.3cm]
$\BRB{\tau\nu}$ & $2.6~\sigma$ \\[0.15cm]
$-2\beta_s$ & $2.1~\sigma$ \\[0.15cm]
$a_\text{fs}$ & $3.0~\sigma$ \\[0.15cm]
 \hline  & \\[-0.3cm]
$-2\beta_s$ and $a_\text{fs}$ & $3.1~\sigma$ \\[0.15cm]
$\BRB{\tau\nu}$, $-2\beta_s$ and $a_\text{fs}$ & $3.7~\sigma$ \\[0.15cm]
\hline \end{tabular} \end{center}
\caption{Pull values for the most anomalous observables in the SM global fit, in terms of the number of 
equivalent standard deviations between the direct measurement (Table~\ref{tab:ExperimentalInputs}) and the full indirect
fit prediction (Table~\ref{tab:fitResults_SM2}). These numbers are computed from the $\chi^2$ difference with and without 
the input, interpreted with the appropriate number of degrees of freedom.}\label{SMpvalues}
\end{table}
It is worth noting that the SM does not correlate these anomalies between each other, because the standard prediction
for CP-violation ($-2\beta_s$ and $a_\text{fs}$) in the $B_s$ system is essentially zero, and hence at leading 
order has no common parameter with the $\BRB{\tau\nu}$ anomaly.


\boldmath
\subsection[Scenario I]{Scenario I: New Physics in \bbmd\ and \bbms}\label{ssec:bdbs} 
\unboldmath

In this section we present the CKM fit for scenario I where New Physics in mixing is 
independently allowed in the $B_d$ and $B_s$ system (\textit{i.e.} $\Delta_d$ and $\Delta_s$ 
are independent). 
These fits exclude the constraint from $\epsilon_K$ since it is not possible to obtain 
non-trivial constraints for the three NP parameters in the $K$ sector. The first study 
of this kind using only $B$-factory data has been performed in~\cite{Laplace:2002ik} 
followed by a complete quantitative analysis~\cite{CKMfitter2} profiting from the large 
data sets of \babar and Belle. Analyses taking also into account the $B_s$ system have
been performed by the UTfit collaboration~\cite{UTfit2,UTfitDeltaF2}.

In Fig.~\ref{rhobar-etabar-NP} we show the $\bar\rho-\bar\eta$ plane for this fit, 
allowing us to constrain the parameters of the CKM matrix using observables not affected by New Physics
according to our hypothesis.
There are two allowed solutions in $\bar\rho-\bar\eta$ which can not be distinguished 
when using only $|V_{ud}|$, $|V_{us}|$, $|V_{cb}|$, $|V_{ub}|$ from semileptonic decays 
and from $B \rightarrow \tau \nu$, $\gamma$ and $\alpha-\phi_d^\Delta/2$ and $\beta+\phi_d^\Delta /2$.
Once $a_\text{fs}^{d}$ is added, the second solution at negative $\bar\rho$ and 
$\bar\eta$ values is clearly disfavoured leaving as the only solution the one
with positive $\bar\rho$ and $\bar\eta$ values.
\begin{figure}[!Htb]
  \begin{center}
  \mbox{\epsfig{file=NPmix_rhoeta.eps,width=12cm}}
  \caption{\small 
Constraint on the CKM $(\bar\rho,\bar\eta)$ coordinates, using only the 
    inputs which are not affected by new physics in mixing: $|V_{ud}|$, $|V_{us}|$, 
    $|V_{cb}|$, $|V_{ub}|$ from semileptonic decays and from $B \rightarrow \tau \nu$, 
    $\gamma$ (directly and from the combination of $\alpha-\phi_d^{\Delta}/2$ and $\beta+\phi_d^{\Delta}/2$).
    Regions outside the coloured areas have ${\rm CL} > 95.5~\%$. For the combined fit,
    two solutions are available: the usual solution corresponds to
    the yellow area (points with ${\rm CL} < 95.4~\%$, the 
    shaded region corresponding to points with ${\rm CL} < 68.3~\%$), and the second solution
    corresponds to the purple region.}\label{rhobar-etabar-NP}.
  \end{center}
\end{figure}
%
Figures~\ref{fig-Deltad_scenario1} and~\ref{fig-Deltas_scenario1} show
\begin{nfigure}{!Htb}
  \includegraphics[width=12cm]{NPmix_ReImDeltad_FPCP10preliminary_3contours.eps}
  \caption{\small Constraint on the complex parameter $\Delta_d$ 
                  from the fit in scenario I.
                  For the individual constraints the coloured areas represent regions
                  with ${\rm CL} < 68.3~\%$. For the combined fit the red area shows
                  the region with ${\rm CL} < 68.3~\%$ while the two additional contour 
                  line inscribe the regions with ${\rm CL} < 95.45~\%$, 
                  and  ${\rm CL} < 99.73~\%$, respectively.
                  \label{fig-Deltad_scenario1}}
\end{nfigure}
the results of the global fit for scenario I in the
complex $\Delta_d$ and $\Delta_s$ planes, respectively.
We emphasize that we assume that $\Delta_d$ and $\Delta_s$ are
taken as independent in this scenario, but that some of the
constraints correlate them (such that $a_{fs}$ from the inclusive
dimuon asymmetry, and the ratio $\Delta m_d/\Delta m_s$). Therefore
the figures should be understood as two-dimensional projections of a
single multidimensional fit, and not as independent computations.
%
The constraint from $\Delta m_{d}$ in the
$\mbox{Re}{\Delta_d}-\mbox{Im}{\Delta_d}$ plane shows two allowed
ring-like regions. They correspond to the two allowed solutions in the
$\bar\rho-\bar\eta$ plane when $a_\text{fs}^{d}$ is excluded from the
list of inputs. Indeed, in this NP scenario, $\Delta m_{d}$ is
proportional to the product $|\Delta_d|^2 \cdot |V_{td}V_{tb}^*|^2$,
where the second factor is different for the two allowed solutions
since it is the side of the unitarity triangle relating $(1,0)$ and
$(\ov{\rho},\ov{\eta})$.  The impact of $a_\text{fs}^{d}$ highlights
the power of this measurement to exclude a large region of the possible
NP parameter space even with a measurement precision of $O(5 \cdot
10^{-3})$. In the combined fit the inner ring (which corresponds to
the solution for $\phi_d^\Delta$ in the first quadrant near the 
$\imag \Delta_d$ axis) in the complex $\Delta_d$ plane is disfavoured. 
This  leaves us with an allowed region for $|\Delta_d|$ which is in agreement with the
SM value $\Delta_d=1$, albeit with possible deviations up to
$40~\%$.  The New Physics phase $\phi^\Delta_d$, mainly driven by the
$\BRB{\tau\nu}$ vs. $\sin2\beta$ correlation, can be as large as
$-12^{\circ}$ and shows currently a deviation from the Standard Model
of $2.5~\sigma$. It is interesting to note that the combined
individual constraint from $a_\text{fs}^{d}$, $a_\text{fs}^{s}$ and
$a_\text{fs}$ \textit{also} favours a negative New Physics phase
$\phi_{\Delta_d}$, mainly due to the measured negative
$a_\text{fs}^{d}$ value.  When $\BRB{\tau\nu}$ is excluded from the
inputs $\mbox{Im}{\Delta_d}$ and hence $\phi_{\Delta_d}$ is in good
agreement with the Standard Model value (see
Fig.~\ref{fig-Deltad_woBtaunu_scenario1}). At the same time the
allowed range for $|\Delta_d|$ is significantly enlarged since
$\BRB{\tau\nu}$ helps to reduce the uncertainty on $\Delta m_{d}$:
the two rings are enlarged and merge into a single one.

\begin{nfigure}{!Htb}
  \includegraphics[width=12cm]{NPmix_ReImDeltas_FPCP10preliminary_3contours.eps}
  \caption{\small Constraint on the complex parameter $\Delta_s$ 
                  from the fit in scenario I.
                  For the individual constraints the coloured areas represent regions
                  with ${\rm CL} < 68.3~\%$. For the combined fit the red area shows
                  the region with ${\rm CL} < 68.3~\%$ while the two additional contour 
                  line inscribe the regions with ${\rm CL} < 95.45~\%$, and 
                  ${\rm CL} < 99.73~\%$, respectively. 
                  \label{fig-Deltas_scenario1}}
\end{nfigure}
The constraint on $|\Delta_s|$ from $\Delta m_{s}$ is more stringent than that for
$|\Delta_d|$ - thanks to the smaller theoretical uncertainty in its prediction 
compared to $\Delta m_{d}$ - and in good agreement with the Standard Model point.
It is interesting to note that also for the $B_{s}$ system the constraint from 
$\BRB{\tau\nu}$ plays a non-negligible role: when removing this measurement from 
the list of inputs the constraint on $|\Delta_s|$ becomes weaker since this 
measurement improves the input on the decay constant $f_{B_{s}}$ through the 
SU(3)-breaking parameter $\xi$ (compare Fig.~\ref{fig-Deltas_woBtaunu_scenario1} 
with Fig.~\ref{fig-Deltas_scenario1} ).
There is evidence for a non-zero NP phase $\phi^\Delta_s$ at the $3.1~\sigma$ 
level. This discrepancy is driven by the $a_\text{fs}$  from D0
and by the $\phi^\Delta_{s}-2\beta_s$ analyses from Tevatron, but is expected to be somewhat relaxed by the 
updated measurements~\cite{taggedphaseCDF_3,taggedphaseD0_2}.

We note also that there is an interesting difference in the allowed size of the 
NP contribution when comparing the $B_{d}$ and the $B_{s}$ systems. While in the 
$B_{s}$ system the size of the NP contribution is essentially constrained by the 
$\Delta m_{s}$ measurement alone this is not the case in the $B_{d}$ system. 
Indeed, 
%This is due to the fact that 
the theoretical prediction of $\Delta m_{d}$ strongly 
depends on the Wolfenstein parameters $\rhobar$ and $\etabar$ whereas this 
dependence is very weak for  $\Delta m_{s}$. The constraint on NP in $B_{d}$-mixing
relies thus on $|V_{ub}|$ on one hand and $\gamma$ on the other hand, the latter 
being currently dominated by the combination of the $\sin{2\beta}$ and $\alpha$ 
measurements which is independent of NP contributions in $B$-mixing.
The theory prediction for the oscillation frequency $\Delta m_{d}$
depends on the quantity
$|\Delta_d|\equiv\sqrt{(\real \Delta_d)^{2}+(\imag\Delta_d)^{2}}$. Without
a good constraint on $|\Delta_d|$ from other observables it can
only be predicted with a very large uncertainty as observed in
Table~\ref{tab:fitResults_SM2}. The only other observables that are
sensitive to the modulus of $\Delta_d$
are $a_\text{fs}^{d}$ and 
$a_\text{fs}$ but those are measured with a precision that is significantly above 
the SM prediction and thus do no constrain very much the range of $\Delta m_{d}$ 
(even though they proved powerful in eliminating the negative $(\ov\rho,\ov\eta)$ 
solution) (the same statement holds for $\Delta m_{s}$).

Tables~\ref{tab:fitResults_SM1} and~\ref{tab:fitResults_SM2} show the fit results 
for various parameters. We also show the result of the fit for observables 
that have been individually excluded from the fit in order to quantify possible 
deviations between the individual input values and their fit predictions. Among other things it is interesting
to note that the indirect fit prediction for the dimuonic asymmetry $a_\text{fs}=(-42 ^{\,+ 19}_{\,- 18})\times 10^{-4}$ is a bit
smaller in magnitude than the \Dzero\ measurement $(-95.7 \pm 25.1 \pm 14.6)\times 10^{-4}$, and 
remain more precise in spite of the uncertainties on the theoretical and New Physics parameters. Hence future improvements of this measurement
are  expected to give crucial information on the underlying physics.
\begin{table}[!Htb]
 \setlength{\tabcolsep}{0.8pc}
 \begin{center}\begin{tabular}{llll} \hline
 Quantity & central  ${\pm {\rm CL}}\equiv1\sigma$       &  ${\pm {\rm CL}}\equiv2\sigma$    &  ${\pm {\rm CL}}\equiv3\sigma$  \\
 \hline  &&& \\[-0.3cm]
$A$        & $    0.814 ^{\,+    0.014}_{\,-    0.028}$  & $0.814^{\,+    0.023}_{\,-    0.037}$ & $0.814^{\,+    0.033}_{\,-    0.046}$ \\[0.15cm]
$\lambda$  & $  0.22511 ^{\,+  0.00074}_{\,-  0.00075}$  & $0.2251^{\,+  0.0015}_{\,-  0.0015}$   & $0.2251^{\,+  0.0022}_{\,-  0.0022}$ \\[0.15cm]
$\bar\rho$ & $    0.157 ^{\,+    0.037}_{\,-    0.032}$  & $0.157^{\,+    0.071}_{\,-    0.063}$ & $0.157^{\,+    0.135}_{\,-    0.096}$ \\[0.15cm]
$\bar\eta$ & $    0.433 ^{\,+    0.025}_{\,-    0.025}$  & $0.433^{\,+    0.039}_{\,-    0.067}$ & $0.433^{\,+    0.052}_{\,-    0.112}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\mbox{Re}{(\Delta_d)}$  & $ 0.742 ^{\,+ 0.206}_{\,-    0.066}$  & $0.74^{\,+    0.42}_{\,-     0.11}$  & $0.74^{\,+    0.71}_{\,-    0.16}$ \\[0.15cm]
$\mbox{Im}{(\Delta_d)}$  & $-0.161 ^{\,+ 0.045}_{\,-    0.081}$  & $-0.16^{\,+    0.11}_{\,-     0.15}$  & $-0.16^{\,+    0.20}_{\,-     0.25}$ \\[0.15cm]
$\phi^\Delta_d$ [deg]  & $-12.3 ^{\,+ 3.3}_{\,-    3.4}$  & $-12.3 ^{\,+    8.7}_{\,-     5.6}$  & $-12.3 ^{\,+    14.9}_{\,-     7.7}$ \\[0.15cm]
$\mbox{Re}{(\Delta_s)}$  & $ -0.55  ^{\,+ 0.16}_{\,-     0.19}$   & $-0.55^{\,+    0.37}_{\,-    0.43}$ & $-0.55^{\,+    1.84}_{\,-    -0.70}$ \\[0.15cm]
                       & \textit{or} $ 0.55  ^{\,+ 0.21}_{\,-     0.13}$   & \textit{or} $0.55^{\,+    0.46}_{\,-    0.35}$ &  \\[0.15cm]
$\mbox{Im}{(\Delta_s)}$  & $-0.68  ^{\,+ 0.14}_{\,-     0.15}$   & $-0.68^{\,+    0.37}_{\,-    0.36}$ & $-0.68^{\,+    0.63}_{\,-    0.62}$ \\[0.15cm]
$\phi^\Delta_s$ [deg]  & $-129 ^{\,+ 12}_{\,-    12}$  & $-129 ^{\,+    28}_{\,-     27}$  & $-129 ^{\,+    127}_{\,-     44}$ \\[0.15cm]
                       & \textit{or} $ -51.6  ^{\,+ 14.1}_{\,-     9.4}$   & \textit{or} $-52^{\,+    32}_{\,-    25}$ &  \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$f_{B_s}$ [MeV] (!)                   
                 & $ 280 ^{\,+   82}_{\,-      36}$     & $280^{\,+     127}_{\,-  61}$     & $280^{\,+    159}_{\,-     88}$ \\[0.15cm]
$f_{B_s}/f_{B_d}$ (!)     
                 & $1.06 ^{\,+    0.12}_{\,-   0.22}$   & $1.06^{\,+    0.30}_{\,- 0.41}$    & $1.06^{\,+    0.57}_{\,-    0.48}$ \\[0.15cm]
$\widehat\Bag_{B_{s}}$ (!)                   
                 & $ 2.17 ^{\,+   0.71}_{\,-   1.00}$   & $2.2^{\,+    1.8}_{\,-  1.5}$     & $2.2^{\,+    3.1}_{\,-     2.1}$ \\[0.15cm]
$\Bag_{B_{s}}/\Bag_{B_{d}}$ (!)     
                 & $0.48 ^{\,+    0.66}_{\,-   0.24}$   & $>0.10$     & $>-0.01$ \\[0.15cm]
$\widetilde{\Bag}_{S,B_s}(m_b)$ (!)     
                 & $3.4 ^{\,+  2.0}_{\,- 2.9}$   & $ 3.4 ^{\,+  5.2}_{\,-  4.4}$     & $ 3.4 ^{\,+  9.1}_{\,-  6.1}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$J$~~$[10^{-5}]$ 
           & $  3.73 ^{\,+ 0.15 }_{\,-    0.27}$         & $3.73^{\,+     0.27}_{\,-     0.63}$ & $3.73^{\,+     0.38}_{\,-     1.01}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\alpha\unit{deg}$  
           & $  82 ^{\,+ 18}_{\,-  12}$            & $82^{\,+     33}_{\,-     22}$ & $82^{\,+     47}_{\,-     31}$ \\[0.15cm]
$\beta\unit{deg}$   
           & $  26.9 ^{\,+ 1.4}_{\,-   2.0}$        & $26.9^{\,+    2.2}_{\,-    5.1}$ & $26.9^{\,+    3.0}_{\,-    8.6}$ \\[0.15cm]
$\gamma\unit{deg}$ (!) & $70.0^{\,+4.3}_{\,-4.5}$ & $70.0^{\,+8.5}_{\,-9.2}$ & $70^{\,+13}_{\,-20}$
\\[0.15cm]
 \hline
 \end{tabular}
 \end{center}
 \vspace{-0.5cm}
 \caption[.]{\label{tab:fitResults_NPBDBS}%\em
              Fit results in the New Physics scenario I. The notation `(!)' means that the fit output represents the indirect constraint, \textit{i.e.}
              the corresponding direct input has been removed from the analysis.}
\end{table}
\begin{table}[Htb]
 \setlength{\tabcolsep}{0.8pc}
\begin{center}\begin{tabular}{llll} \hline
 Quantity & central  ${\pm \mathrm{CL}}\equiv1\sigma$       &  ${\pm \mathrm{CL}}\equiv2\sigma$    &  ${\pm \mathrm{CL}}\equiv3\sigma$  \\[0.15cm]
 \hline  &&& \\[-0.3cm]
$|V_{ud}|$(!) 
           & $  0.97444 ^{\,+  0.00028}_{\,-  0.00028}$  & $0.97444^{\,+  0.00055}_{\,- 0.00056}$  & $0.97444^{\,+  0.00082}_{\,- 0.00084}$ \\[0.15cm]
$|V_{us}|$(!) 
           & $  0.22542 ^{\,+   0.00095}_{\,-  0.00095}$    & $0.2254^{\,+  0.0019}_{\,-  0.0019}$   & $0.2254^{\,+  0.0028}_{\,-  0.0029}$ \\[0.15cm]
$|V_{ub}|$(!) 
           & $  0.00505 ^{\,+0.00117}_{\,-  0.00065}$  & $0.0051^{\,+0.0018}_{\,- 0.0012}$  & $0.0051^{\,+0.0023}_{\,- 0.0017}$ \\[0.15cm]
$|V_{cd}|$ & $  0.22498 ^{\,+  0.00075}_{\,-  0.00075}$  & $0.2250^{\,+  0.0015}_{\,-  0.0015}$   & $0.2250^{\,+  0.0022}_{\,-  0.0022}$ \\[0.15cm]
$|V_{cs}|$ & $  0.97349 ^{\,+  0.00021}_{\,-  0.00018}$  & $0.97349^{\,+  0.00038}_{\,- 0.00036}$  & $0.97349^{\,+  0.00056}_{\,- 0.00053}$ \\[0.15cm]
$|V_{cb}|$(!) 
            & $  0.0412 ^{\,+  0.0122}_{\,-  0.0076}$  & $0.041^{\,+  0.031}_{\,- 0.019}$  & $0.041^{\,+  0.045}_{\,-  0.032}$ \\[0.15cm]
$|V_{td}|$ & $  0.00880 ^{\,+  0.00036}_{\,-  0.00051}$  & $0.00880^{\,+  0.00066}_{\,- 0.00087}$  & $0.00880^{\,+  0.00097}_{\,- 0.00155}$ \\[0.15cm]
$|V_{ts}|$ & $  0.04053 ^{\,+  0.00061}_{\,-  0.00129}$  & $0.04053^{\,+  0.00098}_{\,- 0.00167}$  & $0.0405^{\,+  0.0014}_{\,-  0.0020}$ \\[0.15cm]
$|V_{tb}|$ & $  0.999140 ^{\,+ 0.000053}_{\,- 0.000026}$ & $0.999140^{\,+ 0.000069}_{\,- 0.000042}$ & $0.999140^{\,+ 0.000084}_{\,- 0.000058}$ \\[0.15cm]
\hline &&& \\[-0.3cm]
$\phi^\Delta_d+2\beta$ [deg]  (!)& $31 ^{\,+ 15}_{\,-    20}$  & $31 ^{\,+    34}_{\,-     44}$  & $31 ^{\,+    67}_{\,-     67}$  \\[0.15cm]
                &  & \textit{or}  $-149 ^{\,+    57}_{\,-     25}$  & \textit{or} $-149 ^{\,+    96}_{\,-     31}$  \\[0.15cm]
$\phi_d$ [deg]  & $-17.2 ^{\,+ 4.3}_{\,-    5.9}$  & $-17.2 ^{\,+    10.1}_{\,-     9.6}$  & $-17 ^{\,+    17}_{\,-     13}$  \\[0.15cm]
$\phi^\Delta_s-2\beta_s$ [deg]  (!)& $-128 ^{\,+ 13}_{\,-    17}$  & $-128 ^{\,+    34}_{\,-     51}$  & $<40$  \\[0.15cm]
                & \textit{or} $-58 ^{\,+ 17}_{\,-    14}$  & \textit{or} $-58 ^{\,+    50}_{\,-     34}$  &  $>137$ \\[0.15cm]
$\phi_s$ [deg]  & $-129 ^{\,+ 12}_{\,-    12}$  & $-129 ^{\,+    28}_{\,-     27}$  & $-129 ^{\,+    127}_{\,-    44}$  \\[0.15cm]
                & \textit{or} $-51.0 ^{\,+ 13.9}_{\,-   9.8}$  & \textit{or} $-51 ^{\,+    32}_{\,-     25}$  &   \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\Delta m_d$ $[{\rm ps}^{-1}]$ (!)            
                & $ 0.25 ^{\,+     0.30}_{\,- 0.15}$    &  $>0.03$                              &  \\[0.15cm]
$\Delta m_s$ $[{\rm ps}^{-1}]$ (!)            
                & $ 10.5 ^{\,+12.0}_{\,- 7.0}$    &  $>0.8$          &  \\[0.15cm]
$a_\text{fs}$ $[10^{-4}]$ (!)            
                & $-42 ^{\,+ 19}_{\,- 18}$ & $-42^{\,+ 27}_{\,- 26}$   & $-42^{\,+ 39}_{\,- 32}$  \\[0.15cm]
$a_\text{fs}^{d}$ $[10^{-4}]$ (!)            
                & $-34.2^{\,+ 11.9}_{\,-6.4}$ & $-34^{\,+ 23}_{\,- 12}$ & $-34^{\,+ 35}_{\,- 17}$  \\[0.15cm]
$a_\text{fs}^{s}$ $[10^{-4}]$ (!)            
                 & $-82 ^{\,+ 31}_{\,- 10}$  & $-82^{\,+ 60}_{\,- 21}$   & $-82^{\,+ 79}_{\,- 28}$ \\[0.15cm]
$\Delta\Gamma_d$ $[{\rm ps}^{-1}]$ (!)            
                & $0.00569^{\,+ 0.00077}_{\,-0.00230}$  & $0.0057^{\,+ 0.0013}_{\,- 0.0033}$   & $0.0057^{\,+ 0.0019}_{\,- 0.0039}$  \\[0.15cm]
$\Delta\Gamma_s$ $[{\rm ps}^{-1}]$ (!)            
                & $-0.114^{\,+ 0.066}_{\,-0.034}$  &    &   \\[0.15cm]
                & \textit{or} $0.126^{\,+ 0.026}_{\,-0.062}$ & $0.126^{\,+ 0.051}_{\,- 0.298}$   & $0.126^{\,+ 0.067}_{\,- 0.191}$  \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\mathcal{B}(B\to\tau\nu)$ $[10^{-4}]$ (!) & $ 1.470 ^{\,+ 0.075}_{\,- 0.404}$ & $ 1.47 ^{\,+ 0.15}_{\,- 0.86}$ & $ 1.47 ^{\,+ 0.23}_{\,- 0.92}$ \\[0.15cm]
 \hline
 \end{tabular}
 \end{center}
 \vspace{-0.5cm}
 \caption[.]{\label{tab:fitResults_NPBDBS2}%\em
              Fit results in the New Physics scenario I. The notation `(!)' means that the fit output represents the indirect constraint, \textit{i.e.}
              the corresponding direct input has been removed from the analysis. }
\end{table}


\begin{table}[!Htb]
\begin{center}\begin{tabular}{ll} \hline
Hypothesis & p-value \\ 
 \hline  & \\[-0.3cm]
$\mathrm{Im}(\Delta_d)=0$ (1D) & $2.5~\sigma$ \\[0.15cm]
$\mathrm{Im}(\Delta_s)=0$ (1D) & $3.1~\sigma$ \\[0.15cm]
 \hline  & \\[-0.3cm]
$\Delta_d=1$ (2D) & $2.5~\sigma$ \\[0.15cm]
$\Delta_s=1$ (2D) & $2.7~\sigma$ \\[0.15cm]
$\mathrm{Im}(\Delta_d)=\mathrm{Im}(\Delta_s)=0$ (2D) & $3.8~\sigma$ \\[0.15cm]
 \hline  & \\[-0.3cm]
$\Delta_d=\Delta_s$ (2D) & $2.1~\sigma$ \\[0.15cm]
 \hline  & \\[-0.3cm]
$\Delta_d=\Delta_s=1$ (4D) & $3.4~\sigma$ \\[0.15cm]
\hline \end{tabular} \end{center}
\caption{p-values for various SM hypotheses in the framework of New Physics scenario I, in terms of the number of 
equivalent standard deviations. These numbers are computed from the $\chi^2$ difference with and without 
the hypothesis constraint, interpreted with the appropriate number of degrees of freedom.}\label{ScIpvalues}
\end{table}
In contrast to the SM fit, our Scenario I relates the $B_d$ and $B_s$ anomalies through the correlated
determination of the $\Delta$ parameters. Hence it is particularly interesting to compute the p-values associated 
with the hypothesis that some specific combination of the $\Delta$ parameters take their SM value. This is shown in Table~\ref{ScIpvalues}.
We also learn from this table that the Scenario III to be discussed below,
is slightly disfavoured by the data when one considers it as a subcase of Scenario I ($\Delta_d=\Delta_s$), in agreement with
the third paper of Ref.~\cite{paperafterD0}.
Finally as already stressed above, the various evidences against the SM shown in Table~\ref{ScIpvalues} will be relaxed when
the new TeVatron average of the $B_s\to J/\psi\phi$ tagged analysis is available~\cite{taggedphaseCDF_3,taggedphaseD0_2}. However 
a very rough estimate allows us to predict that at least the $\mathrm{Im}(\Delta_d)=\mathrm{Im}(\Delta_s)=0$ hypothesis (\textit{i.e.} no CP-violating
phase in neither $B_d$ nor $B_s$ mixing amplitudes) will
remain disfavoured by more than three standard deviations. Indeed although the mixing CP-phase is expected to come back closer
to the SM value~\cite{taggedphaseCDF_3,taggedphaseD0_2}, it remains well compatible with the indirect constraint from the dimuonic asymmetry,
as shown by Fig.~\ref{fig-phis_comparison}.
\begin{nfigure}{Htb}
 \includegraphics[width=10cm]{phis.eps}
   \caption{\small New Physics scenario I. 
   Indirect constraint on the CP phase $\phi^\Delta_s-2\beta_s$ as measured in $B_s\to J/\psi\phi$,
   compared with the direct measurement by the TeVatron measurements: previous world average~\cite{Punzi} (in black) and
   the Summer 2010 CDF and \Dzero\ updates~\cite{taggedphaseCDF_3,taggedphaseD0_2} (in blue). The dotted line represents the full
   scenario I fit with the previous world average for $a_\mathrm{fs}$, while the green curve is the update after the \Dzero\ evidence
   for a non zero dimuonic asymmetry.
                  \label{fig-phis_comparison}}
\end{nfigure}


\begin{nfigure}{htb}
 \includegraphics[width=10cm]{NPmix_ReImDeltas_FPCP10preliminary_woBtaunu_3contours.eps}
   \caption{\small Fit result for the complex parameter $\Delta_d$ 
                   in scenario I when excluding $\BRB{\tau\nu}$
                   from the list of inputs.
                   For the individual constraints the coloured areas represent regions
                   with ${\rm CL} < 68.3~\%$. For the combined fit the red area shows
                   the region with ${\rm CL} < 68.3~\%$ while the contour line inscribes
                   the region with ${\rm CL} < 95.45~\%$.
                  \label{fig-Deltad_woBtaunu_scenario1}}
%\end{nfigure}
%\begin{nfigure}{htb}
 \includegraphics[width=10cm]{NPmix_ReImDeltas_FPCP10preliminary_woBtaunu_3contours.eps}
   \caption{\small Fit result for the complex parameter $\Delta_s$ 
                   in scenario I when excluding $\BRB{\tau\nu}$
                   from the list of inputs.
                   For the individual constraints the coloured areas represent regions
                   with ${\rm CL} < 68.3~\%$. For the combined fit the red area shows
                   the region with ${\rm CL} < 68.3~\%$ while the contour line inscribes
                   the region with ${\rm CL} < 95.45~\%$.
\label{fig-Deltas_woBtaunu_scenario1}}
\end{nfigure}

\clearpage
\subsection[Scenario II]{Scenario II: MFV with small bottom--Yukawa
coupling}\label{ssec:mfv1} 

In this section, we discuss the MFV scenario which allows to connect the $B$
and Kaon sectors. Such a kind of numerical analysis has been first presented 
in~\cite{UTfit1}. In this scenario, there is only one additional real NP parameter
$\Delta$, see \eq{defsc2}. 
As a consequence, this scenario has difficulties to describe a situation 
where the data prefer a non-zero NP phase in $B$-mixing.
Indeed the scenario II hypothesis embedded in scenario I, that is $\Delta_d=\Delta_s=\Delta$
with Im$(\Delta)=0$, is disfavoured by 3.5 standard deviations.
The quality of the fit does not change when $\epsilon_{K}$ is removed 
from the list of inputs. 

Fig.~\ref{fig-Delta_scenario2} shows the impact of the various constraints
on the parameter $\Delta$. The constraint is only slightly changed when
adding to the $B$-meson observables - where $\BRB{\tau\nu}$
has been excluded - the $\epsilon_{K}$ constraint. On the other hand, 
when adding $\BRB{\tau\nu}$, the constraint gets significantly stronger
at the $1\sigma$ level while at $2\sigma$ the reduction in the allowed
parameter space is modest.
\begin{nfigure}{Htb}
  \includegraphics[width=10cm]{MFV_II.eps}
  \caption{\small Constraint on the real parameter $\Delta$ from
                  the fit in scenario II. The red dashed line represents
                  the constraint if only $B_{d}$ and $B_{s}$ observables
                  are used excluding $\BRB{\tau\nu}$ and $\epsilon_{K}$.
                  When adding $\epsilon_{K}$ the constraint is essentially
                  unchanged (blue dotted-dashed line). A significantly 
                  stronger constraint is obtained when adding $\BRB{\tau\nu}$ 
                  (dotted green). The full constrain when adding both, 
                  $\BRB{\tau\nu}$ and $\epsilon_{K}$, is shown in green.
                  \label{fig-Delta_scenario2}}
\end{nfigure}
%
In Tables~\ref{tab:fitResults_MFV_II} and~\ref{tab:fitResults_MFV_II2}
we further provide
the constraints on individual parameters obtained from the combined fit in scenario II 
as well as predictions for the parameters not used as fit inputs. The compatibility of
the parameter $\Delta$ with one is good, meaning that this New Physics scenario does not describe
the data better than the SM, as expected since all discussed anomalies seem to require new CP phases.

\begin{table}[Htb]
 \setlength{\tabcolsep}{0.8pc}
 \begin{center}\begin{tabular}{llll} \hline
 Quantity & central  ${\pm \mathrm{CL}}\equiv1\sigma$       &  ${\pm \mathrm{CL}}\equiv2\sigma$    &  ${\pm \mathrm{CL}}\equiv3\sigma$  \\[0.15cm]
 \hline  &&& \\[-0.3cm]
$A$        & $  0.8192 ^{\,+   0.0093}_{\,-   0.0228}$   & $0.819^{\,+   
0.019}_{\,-    0.038}$ & $0.819^{\,+    0.028}_{\,-    0.048}$ \\[0.15cm]
$\lambda$  & $  0.22512 ^{\,+  0.00075}_{\,-  0.00075}$  & $0.2251^{\,+ 
0.0015}_{\,- 0.0015}$   & $0.2251^{\,+  0.0022}_{\,-  0.0022}$ \\[0.15cm]
$\bar\rho$ & $  0.140 ^{\,+    0.025}_{\,-    0.024}$    & $0.140^{\,+   
0.048}_{\,-   0.034}$ & $0.140^{\,+    0.062}_{\,-    0.043}$ \\[0.15cm]
$\bar\eta$ & $  0.342 ^{\,+    0.016}_{\,-    0.012}$    & $0.342^{\,+   
0.031}_{\,-   0.024}$ & $0.342^{\,+    0.047}_{\,-    0.036}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\Delta$   & $ 0.907 ^{\,+ 0.091}_{\,-    0.067}$        & $0.907^{\,+   
0.394}_{\,-     0.099}$  & $0.91^{\,+    0.51}_{\,-    0.13}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$B_K$ (!)                              
                 & $ 0.88 ^{\,+    0.23}_{\,-     0.16}$ & $0.88^{\,+     0.33}_{\,-
    0.29}$ & $0.88^{\,+     0.42}_{\,-     0.34}$ \\[0.15cm]
$\kappa_{\epsilon}$ (!)                              
                 & $ 1.16 ^{\,+    0.39}_{\,-     0.28}$ & $1.16^{\,+0.53}_{\,-     0.44}$ & $1.16^{\,+0.65}_{\,-     0.50}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm] 
$f_{B_s}$ [MeV] (!)                   
                 & $ 252 ^{\,+   10}_{\,-      14}$     & $252^{\,+    19}_{\,-43}$       & $252^{\,+    29}_{\,-     75}$ \\[0.15cm]
$\widehat\Bag_{B_{s}}$ (!)               
   
                 & $ 0.777 ^{\,+ 0.525}_{\,-  0.078}$   & $0.78^{\,+0.83}_{\,- 0.14}$   & $0.78^{\,+    1.20}_{\,-   0.18}$ \\[0.15cm]
$f_{B_s}/f_{B_d}$ (!)     
                 & $1.198 ^{\,+    0.057}_{\,-   0.036}$   & $1.198^{\,+0.108}_{\,- 0.077}$    & $1.20^{\,+    0.15}_{\,-    0.19}$ \\[0.15cm]
$\Bag_{B_{s}}/\Bag_{B_{d}}$ (!)     
                 & $1.153 ^{\,+   0.075}_{\,- 0.105}$   & $1.15^{\,+0.15}_{\,- 0.20}$     & $1.15^{\,+  0.23}_{\,-  0.29}$ \\[0.15cm]
$\widetilde{\Bag}_{S,B_s}(m_b)$ (!)     
                 & $-1.1 ^{\,+  1.4}_{\,- 2.0}$   & $-1.1 ^{\,+  2.7}_{\,-  3.5}$     & $-1.1 ^{\,+  4.3}_{\,-  4.8}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$J$~~$[10^{-5}]$ 
           & $  2.98 ^{\,+ 0.15 }_{\,-    0.14}$         & $2.98^{\,+     0.30}_{\,-     0.29}$ & $2.98^{\,+     0.45}_{\,-     0.41}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\alpha$ [deg] (!)
           & $ 93.1 ^{\,+ 4.9}_{\,-  5.8}$            & $93.1^{\,+     7.3}_{\,-     8.1}$ & $93.1^{\,+     9.4}_{\,-     10.0}$ \\[0.15cm]
$\beta$  [deg] (!)
           & $  28.09 ^{\,+ 0.70}_{\,-   2.43}$        & $28.1^{\,+    1.4}_{\,-    5.4}$ & $28.1^{\,+    2.1}_{\,-    7.7}$ \\[0.15cm]
$\gamma$ [deg] (!)   
           & $  67.6 ^{\,+ 3.8}_{\,-   3.6}$        & $67.6^{\,+    5.5}_{\,-
   7.1}$ & $67.6^{\,+    7.2}_{\,-    9.3}$ \\[0.15cm]
 \hline
 \end{tabular}
 \end{center}
 \vspace{-0.5cm}
 \caption[.]{\label{tab:fitResults_MFV_II}%\em
              Fit results in the New Physics scenario II. The notation `(!)' means that the fit output represents the indirect constraint, \textit{i.e.}
              the corresponding direct input has been removed from the analysis. }
\end{table}
\begin{table}[Htb]
 \setlength{\tabcolsep}{0.8pc}
\begin{center}\begin{tabular}{llll} \hline
 Quantity & central  ${\pm \mathrm{CL}}\equiv1\sigma$       &  ${\pm
\mathrm{CL}}\equiv2\sigma$   &  ${\pm \mathrm{CL}}\equiv3\sigma$  \\[0.15cm]
 \hline  &&& \\[-0.3cm]
$|V_{ud}|$(!) 
           & $  0.97444 ^{\,+  0.00028}_{\,-  0.00028}$  & $0.97444^{\,+ 
0.00055}_{\,- 0.00056}$  & $0.97444^{\,+  0.00082}_{\,- 0.00084}$ \\[0.15cm]
$|V_{us}|$(!) 
           & $  0.22545 ^{\,+   0.00095}_{\,-  0.00095}$    & $0.2254^{\,+ 
0.0019}_{\,-  0.0019}$   & $0.2254^{\,+  0.0028}_{\,-  0.0029}$ \\[0.15cm]
$|V_{ub}|$(!) 
           & $  0.00354 ^{\,+0.00015}_{\,-  0.00014}$  &
$0.00354^{\,+0.00030}_{\,- 0.00031}$  & $0.00354^{\,+0.00046}_{\,- 0.00046}$
\\[0.15cm]
$|V_{cd}|$ & $  0.22498 ^{\,+  0.00075}_{\,-  0.00075}$  & $0.2250^{\,+  0.0015}_{\,-  0.0015}$   & $0.2250^{\,+  0.0022}_{\,-  0.0022}$ \\[0.15cm]
$|V_{cs}|$ & $  0.97348 ^{\,+  0.00017}_{\,-  0.00017}$  & $0.97348^{\,+  0.00035}_{\,- 0.00035}$  & $0.97348^{\,+  0.00052}_{\,- 0.00052}$ \\[0.15cm]
$|V_{cb}|$(!) 
           & $  0.0490 ^{\,+  0.0030}_{\,-  0.0075}$  & $0.0490^{\,+ 
0.0051}_{\,- 0.0099}$  & $0.0490^{\,+  0.0071}_{\,-  0.0119}$ \\[0.15cm]
$|V_{td}|$ & $  0.00865 ^{\,+  0.00024}_{\,-  0.00025}$  & $0.00865^{\,+  0.00037}_{\,- 0.00059}$  & $0.00865^{\,+  0.00049}_{\,- 0.00081}$ \\[0.15cm]
$|V_{ts}|$ & $  0.04076 ^{\,+  0.00038}_{\,-  0.00111}$  & $0.04076^{\,+  0.00076}_{\,- 0.00175}$  & $0.0408^{\,+  0.0011}_{\,-  0.0022}$ \\[0.15cm]
$|V_{tb}|$ & $  0.999132 ^{\,+ 0.000046}_{\,- 0.000016}$ & $0.999312^{\,+ 0.000072}_{\,- 0.000032}$ & $0.999132^{\,+ 0.000088}_{\,- 0.000048}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\phi_d$ [deg]  & $-7.7 ^{\,+ 3.5}_{\,-    1.1}$  & $-7.7 ^{\,+    4.6}_{\,-     2.2}$  & $-7.7 ^{\,+    5.3}_{\,-     3.5}$  \\[0.15cm]
$\phi_s$ [deg] & $0.39 ^{\,+ 0.90}_{\,-    1.22}$  & $0.4 ^{\,+    2.4}_{\,-     2.4}$  & $0.4 ^{\,+    3.2}_{\,-     3.5}$  \\[0.15cm]
 \hline  &&& \\[-0.3cm]
%$\epsilon_K$~~$[10^{-3}]$ (!)         & $ 1.99 ^{\,+    0.55}_{\,-     0.55}$ & $>1.32}$ &  \\[0.15cm]
$\Delta m_d$ $[{\rm ps}^{-1}]$ (!)            
                & $ 0.541 ^{\,+     0.082}_{\,- 0.057}$  & $0.54^{\,+0.13}_{\,-    0.12}$  & $0.54^{\,+    0.18}_{\,-     0.17}$ \\[0.15cm]
$\Delta m_s$ $[{\rm ps}^{-1}]$ (!)            
                & $ 16.6 ^{\,+     1.7}_{\,- 1.6}$       & $16.6^{\,+3.8}_{\,-   2.7}$  & $16.6^{\,+    8.0}_{\,-     3.7}$ \\[0.15cm]
$a_\text{fs}$ $[10^{-4}]$ (!)            
                & $-4.0 ^{\,+1.3}_{\,-1.1}$  & $-4.0^{\,+3.4}_{\,-2.8}$   & $-4.0^{\,+ 5.0}_{\,- 3.9}$  \\[0.15cm]
$a_\text{fs}^{d}$ $[10^{-4}]$ (!)            
                & $-8.3^{\,+ 2.6}_{\,-1.2}$ & $-8.3^{\,+4.9}_{\,- 3.2}$   & $-8.3^{\,+ 7.1}_{\,- 4.6}$  \\[0.15cm]
$a_\text{fs}^{s}$ $[10^{-4}]$ (!)            
                & $0.4^{\,+ 1.4}_{\,-1.2}$  & $0.4^{\,+2.7}_{\,- 2.6}$   &    $0.4^{\,+ 3.6}_{\,- 5.0}$  \\[0.15cm]
$\Delta\Gamma_d$ $[{\rm ps}^{-1}]$ (!)            
                & $0.00311^{\,+ 0.00150}_{\,-0.00037}$  & $0.0031^{\,+ 0.0021}_{\,- 0.0010}$   & $0.0031^{\,+ 0.0026}_{\,- 0.0014}$  \\[0.15cm]
$\Delta\Gamma_s$ $[{\rm ps}^{-1}]$ (!)            
                & $0.162^{\,+ 0.020}_{\,-0.067}$  & $0.162^{\,+ 0.030}_{\,- 0.091}$   & $0.162^{\,+ 0.040}_{\,- 0.108}$  \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\mathcal{B}(B\to\tau\nu)$ $[10^{-4}]$ (!) & $ 0.639 ^{\,+ 0.310}_{\,- 0.044}$ & $ 0.639 ^{\,+ 0.434}_{\,- 0.081}$ & $ 0.64 ^{\,+ 0.54}_{\,- 0.11}$ \\[0.15cm]
 \hline
 \end{tabular}
 \end{center}
 \vspace{-0.5cm}
 \caption[.]{\label{tab:fitResults_MFV_II2}%\em
              Fit results in the New Physics scenario II. The notation `(!)' means that the fit output represents the indirect constraint, \textit{i.e.}
              the corresponding direct input has been removed from the analysis. }
 \end{table}



 
\clearpage
\subsection[Scenario III]{Scenario III: Generic MFV}\label{ssec:mfv2} 

In Fig.~\ref{fig-Delta_scenario3} we present the result from the
combined fit to $B_{d}$ and $B_{s}$ observables in the complex plane
$\Delta=\Delta_s=\Delta_d$ for the MFV hypothesis allowing for a
large bottom Yukawa coupling $y_b$ (scenario III).  In scenario
III $\Delta$ can have a sizable complex component proportional to
$y_b^2$. One expects that the constraint $\Delta_s=\Delta_d$
reduces the size of the allowed NP contributions significantly with
respect to the general case studied in scenario I.
However, our fit result in \fig{fig-Delta_scenario3} still allows for a NP 
contribution of order $-20~\%$ to $+40~\%$.
The NP phase in this scenario shows evidence for a deviation from the Standard Model
since both $a_\mathrm{fs}$  and the $\phi^\Delta_{s}$ measurement on the one hand
and  the discrepancy between $\sin 2\beta$
measurement and $\BRB{\tau\nu}$ on the other hand, prefer a negative NP phase in $B_{d,s}$ mixing. 
%
\begin{nfigure}{Htb}
\includegraphics[width=12cm]{NPmixScIII_ReImDelta_FPCP10preliminary.eps}
 \caption{\small Constraint on the complex parameter $\Delta \equiv \Delta_d=\Delta_s$ from 
                  the fit in scenario III. 
                  For the individual constraints the coloured areas represent regions
                  with ${\rm CL} < 68.3~\%$. In addition, a ${\rm CL} < 95.45~\%$ contour
                  is shown for the individual constraints obtained from
                  $\Delta m_{d}$ 
                  and $\Delta m_{s}$,
                  from $a_{SL}$, $a_{SL}^{d}$, and $a_{SL}^{s}$, and from $\phi_{d}^{\psi K}$
                  and $\phi_{s}^{\psi \phi}$.
                  For the combined fit the red area shows
                  the region with ${\rm CL} < 68.3~\%$ while the two additional contour 
                  lines inscribe the regions with ${\rm CL} < 95.45~\%$ and 
                  ${\rm CL} < 99.73~\%$, respectively.\label{fig-Delta_scenario3}}
\end{nfigure}

%\begin{nfigure}{htb}
% \includegraphics[width=10cm]{globalNP_newMFV_BdBsEpsK_ChangeOfVariables_woBtaunu_all.eps}
%  \caption{\small Constraint on the $\mbox{Re}{\Delta}-\mbox{Im}{\Delta}$ coordinates from the fit
%                 in scenario III when excluding $\BRB{\tau\nu}$
%                  from the list of inputs.
%                  The shaded area shows the allowed region with ${\rm CL} < 68~\%$.
%                  The allowed region with ${\rm CL} < 95~\%$ is inscribed by the contour line
%                  shown.\label{fig-Delta_woBtaunu_scenario3}}
%\end{nfigure}

Compared to scenario II this model is in much better agreement with the
data regardless of whether one takes into account
$\BRB{\tau\nu}$ as an input or not. While the $B_{d}$ sector constraints do not
allow too large a NP phase, the $B_{s}$ sector prefers a large NP phase,
though with large errors. As a consequence already discussed above,
the quality of the fit in scenario III is
significantly less good ($2.1~\sigma$) than in scenario I.

\begin{table}[Htb]
 \setlength{\tabcolsep}{0.8pc}
 \begin{center}\begin{tabular}{llll} \hline
 Quantity & central  ${\pm \mathrm{CL}}\equiv1\sigma$       &  ${\pm \mathrm{CL}}\equiv2\sigma$    &  ${\pm \mathrm{CL}}\equiv3\sigma$  \\[0.15cm]
 \hline  &&& \\[-0.3cm]
$A$        & $  0.7948 ^{\,+   0.0286}_{\,-   0.0091}$   & $ 0.7948 ^{\,+    0.040}_{\,-    0.018}$ & $0.795 ^{\,+    0.050}_{\,-    0.027}$ \\[0.15cm]
$\lambda$  & $  0.22506 ^{\,+  0.00075}_{\,-  0.00075}$  & $ 0.2251 ^{\,+  0.0015}_{\,-  0.0015}$   & $ 0.2251 ^{\,+  0.0022}_{\,-  0.0022}$ \\[0.15cm]
$\bar\rho$ & $  0.175 ^{\,+    0.031}_{\,-    0.019}$    & $ 0.175 ^{\,+    0.055}_{\,-    0.033}$ & $ 0.175 ^{\,+    0.067}_{\,-    0.051}$ \\[0.15cm]
$\bar\eta$ & $  0.441 ^{\,+    0.013}_{\,-    0.021}$    & $ 0.441 ^{\,+    0.024}_{\,-    0.051}$ & $ 0.441^{\,+    0.034}_{\,-    0.090}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\mbox{Re}{(\Delta)}$  & $ 0.886 ^{\,+ 0.141}_{\,-    0.062}$  & $ 0.886 ^{\,+    0.399}_{\,-     0.099}$  & 0.886 $^{\,+    0.49}_{\,-    0.13}$ \\[0.15cm]
$\mbox{Im}{(\Delta)}$  & $-0.228 ^{\,+ 0.055}_{\,-    0.051}$  & $-0.23 ^{\,+    0.11}_{\,-     0.13}$  & $-0.23 ^{\,+    0.19}_{\,-     0.21}$ \\[0.15cm]
$\mbox{Arg}{(\Delta)}$ [deg]  & $-14.4 ^{\,+ 2.9}_{\,-    2.1}$  & $-14.4 ^{\,+    6.7}_{\,-     4.2}$  & $-14.4 ^{\,+    11.9}_{\,-     6.4}$ \\[0.15cm]
$\Delta_K^{tt}$ & $1.01 ^{\,+ 0.40}_{\,-    0.38}$  & $1.01^{\,+    0.52}_{\,-     0.43}$  & $1.01^{\,+    0.69}_{\,-     0.47}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$f_{B_s}$ [MeV] (!)                   
                 & $ 270 ^{\,+   24}_{\,-      25}$     & $ 270 ^{\,+    50}_{\,-  54}$       & $ 270 ^{\,+    80}_{\,-     87}$ \\[0.15cm]
$\widehat\Bag_{B_{s}}$ (!)                   
                 & $ 1.37 ^{\,+ 0.34}_{\,-  0.63}$   &  1.4 $^{\,+  1.1}_{\,-  1.3}$   & $<3.1$ \\[0.15cm]
$f_{B_s}/f_{B_d}$ (!)     
                 & $1.210 ^{\,+    0.067}_{\,-   0.048}$   & $1.21 ^{\,+    0.12}_{\,- 0.10}$    & $1.21 ^{\,+    0.17}_{\,-    0.20}$ \\[0.15cm]
$\Bag_{B_{s}}/\Bag_{B_{d}}$ (!)     
                 & $1.171 ^{\,+   0.091}_{\,- 0.142}$   & $ 1.17 ^{\,+  0.18}_{\,-  0.25}$     & $1.17 ^{\,+  0.28}_{\,-  0.40}$ \\[0.15cm]
$\widetilde{\Bag}_{S,B_s}(m_b)$ (!)     
                 & $0.9 ^{\,+  1.0}_{\,- 1.7}$   & $ 0.9 ^{\,+  3.0}_{\,-  3.7}$     & $ 0.9 ^{\,+  5.0}_{\,-  5.1}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$J$~~$[10^{-5}]$ 
           & $  3.62 ^{\,+ 0.18 }_{\,-    0.14}$  & $3.62^{\,+     0.29}_{\,-     0.38}$ & $3.62^{\,+     0.39}_{\,-     0.70}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\alpha$   [deg] (!)
           & $  87.6 ^{\,+ 3.4}_{\,-  5.8}$         & $87.6^{\,+    5.9}_{\,-     7.5}$ & $87.6^{\,+     9.4}_{\,-     8.9}$ \\[0.15cm]
$\beta$    [deg] (!)
           & $ 27.94 ^{\,+ 0.66}_{\,-   1.56}$        & $27.9^{\,+    1.3}_{\,-    3.7}$ & $27.9^{\,+    2.1}_{\,-    8.1}$ \\[0.15cm]
$\gamma$ [deg] (!)& $ 68.4 ^{\,+ 2.3}_{\,- 4.1}$         & $ 68.4^{\,+    3.6}_{\,-     7.3}$ & $ 68.4 ^{\,+     5.0}_{\,-     8.9}$ \\[0.15cm]
 \hline
 \end{tabular}
 \end{center}
 \vspace{-0.5cm}
 \caption[.]{\label{tab:fitResults_MFV_III} %\em
              Fit results in the New Physics scenario III. The notation `(!)' means that the fit output represents the indirect constraint, \textit{i.e.}
              the corresponding direct input has been removed from the analysis. }
 \end{table}

\begin{table}[Htb]
 \setlength{\tabcolsep}{0.8pc}
\begin{center}\begin{tabular}{llll} \hline
 Quantity & central  ${\pm \mathrm{CL}}\equiv1\sigma$       &  ${\pm \mathrm{CL}}\equiv2\sigma$    &  ${\pm \mathrm{CL}}\equiv3\sigma$  \\[0.15cm]
 \hline  &&& \\[-0.3cm]
$|V_{ud}|$(!) 
           & $  0.97447 ^{\,+  0.00028}_{\,-  0.00028}$  & $ 0.97447 ^{\,+  0.00055}_{\,- 0.00056}$  & $ 0.97447 ^{\,+  0.00082}_{\,- 0.00084}$ \\[0.15cm]
$|V_{us}|$(!) 
           & $  0.22535 ^{\,+   0.00095}_{\,-  0.00095}$    & $ 0.22535 ^{\,+  0.0019}_{\,-  0.0019}$   & $ 0.22535 ^{\,+  0.0028}_{\,-  0.0029}$ \\[0.15cm]
$|V_{ub}|$(!) 
           & $  0.00596 ^{\,+0.00049}_{\,-  0.00073}$  & $ 0.00596 ^{\,+0.00090}_{\,- 0.00137}$  & $ 0.00596 ^{\,+0.0013}_{\,- 0.0019}$ \\[0.15cm]
$|V_{cd}|$ & $  0.22494 ^{\,+  0.00075}_{\,-  0.00075}$  & $0.2249^{\,+  0.0015}_{\,-  0.0015}$   & $0.2249^{\,+  0.0022}_{\,-  0.0022}$ \\[0.15cm]
$|V_{cs}|$ & $  0.97354 ^{\,+  0.00017}_{\,-  0.00017}$  & $0.97354^{\,+  0.00035}_{\,- 0.00036}$  & $0.97354^{\,+  0.00052}_{\,- 0.00055}$ \\[0.15cm]
$|V_{cb}|$(!) 
           & $  0.0325 ^{\,+  0.0035}_{\,-  0.0038}$  & $ 0.0325 ^{\,+  0.0092}_{\,- 0.0080}$  & $ 0.033 ^{\,+  0.016}_{\,-  0.012}$ \\[0.15cm]
$|V_{td}|$ & $  0.00848 ^{\,+  0.00038}_{\,-  0.00031}$  & $ 0.00848 ^{\,+  0.00053}_{\,- 0.00057}$  & $ 0.00848 ^{\,+  0.00066}_{\,- 0.00071}$ \\[0.15cm]
$|V_{ts}|$ & $  0.03960 ^{\,+  0.00136}_{\,-  0.00038}$  & $0.03960 ^{\,+  0.00184}_{\,- 0.00075}$  & $0.0396^{\,+  0.0022}_{\,-  0.0011}$ \\[0.15cm]
$|V_{tb}|$ & $  0.999180 ^{\,+ 0.000015}_{\,- 0.000058}$ & $0.999180 ^{\,+ 0.000030}_{\,- 0.000078}$ & $0.999180 ^{\,+ 0.000045}_{\,- 0.000095}$ \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\mbox{Arg}(\Delta)+2\beta$ [deg]  (!)& $32.1 ^{\,+ 6.3}_{\,-    8.8}$  & $32 ^{\,+    15}_{\,-     29}$  & $32 ^{\,+    26}_{\,-     44}$  \\[0.15cm]
$\phi_d$ [deg]  & $-21.5 ^{\,+ 3.9}_{\,-    4.0}$  & $-21.5 ^{\,+    8.1}_{\,-     7.0}$  & $-21.5 ^{\,+    13.7}_{\,-     9.8}$  \\[0.15cm]
&&&\textit{or} $-101 ^{\,+    21}_{\,-     18}$\\[0.15cm]
$\mbox{Arg}(\Delta)-2\beta_s$ [deg]  (!)& $-16.6 ^{\,+ 3.9}_{\,-    2.5}$  & $-16.6 ^{\,+    8.7}_{\,-     5.0}$  & $-16.6 ^{\,+    15.2}_{\,-     7.4}$  \\[0.15cm]
$\phi_s$ [deg] & $-14.0 ^{\,+ 3.0}_{\,-    2.4}$  & $-14.0 ^{\,+    7.0}_{\,-     4.9}$  & $-14.0 ^{\,+    12.2}_{\,-     7.3}$  \\[0.15cm]
 \hline &&&      \\[-0.3cm]
$\Delta m_d$ $[{\rm ps}^{-1}]$ (!)            
                & $ 0.549 ^{\,+     0.088}_{\,- 0.076}$  & $ 0.55 ^{\,+    0.14}_{\,-     0.14}$  & $ 0.55 ^{\,+    0.20}_{\,-     0.20}$ \\[0.15cm]
$\Delta m_s$ $[{\rm ps}^{-1}]$ (!)            
                & $ 15.3 ^{\,+     2.4}_{\,- 1.1}$       & $ 15.3 ^{\,+    4.9}_{\,-     2.2}$  & $ 15.3 ^{\,+    8.0}_{\,-     3.1}$ \\[0.15cm]
$a_\text{fs}$ $[10^{-4}]$ (!)            
                & $-21.5 ^{\,+6.4}_{\,-4.0}$  & $-21.5 ^{\,+ 11.8}_{\,- 8.9}$   & $-22 ^{\,+ 19}_{\,- 15}$  \\[0.15cm]
$a_\text{fs}^{d}$ $[10^{-4}]$ (!)            
                & $-30.7^{\,+ 9.4}_{\,-4.4}$ & $-30.7^{\,+ 14.4}_{\,- 8.8}$   & $-31^{\,+ 21}_{\,-13}$  \\[0.15cm]
$a_\text{fs}^{s}$ $[10^{-4}]$(!)            
                & $-14.2^{\,+ 3.5}_{\,-4.4}$  & $-14.2^{\,+ 7.6}_{\,- 13.3}$   & $-14^{\,+ 13}_{\,- 19}$  \\[0.15cm]
$\Delta\Gamma_d$ $[{\rm ps}^{-1}]$ (!)            
                & $0.00398^{\,+ 0.00064}_{\,-0.00138}$  & $0.0040^{\,+ 0.0014}_{\,- 0.0020}$   & $0.0040^{\,+ 0.0017}_{\,- 0.0023}$  \\[0.15cm]
$\Delta\Gamma_s$ $[{\rm ps}^{-1}]$ (!)            
                & $0.160^{\,+ 0.021}_{\,-0.065}$  & $0.160^{\,+ 0.032}_{\,- 0.089}$   & $0.160^{\,+ 0.042}_{\,- 0.107}$  \\[0.15cm]
 \hline &&&      \\[-0.3cm] 
$\mathcal{B}(B\to\tau\nu)$ $[10^{-4}]$ (!) & $ 1.37 ^{\,+ 0.13}_{\,- 0.39}$ & $ 1.37 ^{\,+ 0.24}_{\,- 0.63}$ & $ 1.37 ^{\,+ 0.33}_{\,- 0.80}$ \\[0.15cm]
 \hline
 \end{tabular}
 \end{center}
 \vspace{-0.5cm}
 \caption[.]{\label{tab:fitResults_MFV_III2} %\em
              Fit results in the New Physics scenario III. The notation `(!)' means that the fit output represents the indirect constraint, \textit{i.e.}
              the corresponding direct input has been removed from the analysis. }
 \end{table}

\clearpage

In this scenario, the $B_{d}$- and $B_{s}$-systems decouple from the kaon system 
and hence the constraint on $\mbox{Re}{\Delta}$ and $\mbox{Im}{\Delta}$ can not be 
improved by adding $\epsilon_{K}$ as this introduces the additional NP parameter 
$\Delta_K^{tt}$ in the theoretical prediction for $\epsilon_{K}$. However, when 
including $\epsilon_{K}$ in the fit one is able to constrain $\Delta_K^{tt}$ 
from the fitted values of the CKM parameters.
The parameter is in agreement with the SM expectation but could differ by more than 
$40~\%$ from $\Delta_K^{tt}=1$. Similarly to the previous scenarios,
one-dimensional results from the fit in scenario III are shown in Tables~\ref{tab:fitResults_MFV_III}
and~\ref{tab:fitResults_MFV_III2}. Again one sees that in particular the semileptonic asymmetries 
$a_\text{fs}^{d}$ and $a_\text{fs}$ are more precisely predicted than the measurements, so that
improvements of the data will be extremely instructive.

Finally two tests of the SM are interesting to study within scenario III, and are shown in Table~\ref{ScIIIpvalues}.
\begin{table}[!Htb]
\begin{center}\begin{tabular}{ll} \hline
Hypothesis & p-value \\ 
 \hline  & \\[-0.3cm]
$\mathrm{Im}(\Delta)=0$ (1D) & $3.4~\sigma$ \\[0.15cm]
 \hline  & \\[-0.3cm]
$\Delta=1$ (2D) & $3.1~\sigma$ \\[0.15cm]
\hline \end{tabular} \end{center}
\caption{p-values for various SM hypotheses in the framework of New Physics scenario III, in terms of the number of 
equivalent standard deviations. These numbers are computed from the $\chi^2$ difference with and without 
the hypothesis constraint, interpreted with the appropriate number of degrees of freedom.}\label{ScIIIpvalues}
\end{table}
As in scenario I, they show evidence for New Physics, and that scenario III describes the data significantly better than either the SM or
scenario II, assuming as before that the improved data on $B_s\to J/\psi\phi$ do not change the overall picture dramatically.

    
