\boldmath
\section{Relationship between $\epsilon_K$ and \kkm}\label{app:kappaepsilon}

\subsection{Corrections to the usual $\epsilon_K$ formula}
\unboldmath In Sec.~\ref{ssec:kkmbasics}, we have discussed the
connection between $\epsilon_K$ and \kkm. The resulting equation,
\eq{eq:epsilon0}, has been obtained thanks to several approximations
(concerning the phase $\phi_\epsilon$, neglect of $\xi$, computation
of $M_{12}$ from lowest-dimension operators of the effective
Hamiltonian).  The corrections to these approximations are encoded in
the deviation of the factor $\kappa_\epsilon$ from 1.  A series of
papers~\cite{run2,Andriyash:2003ym,bg,bgi} has assessed more precisely
the value of this factor in order to account for the terms neglected
by the previous approximations.

$\xi\neq0$ and $\phi_\epsilon\neq 45^\circ$ imply $\kappa_\epsilon \neq 1$, 
Ref.~\cite{bg} finds $\kappa_\epsilon = 0.92 \pm 0.02$.
In Ref.~\cite{LLVdW} a lattice QCD
calculation of $\imag A_2$ is combined with the experimental value of
$\epsilon'_K/\epsilon_K$ to compute $\xi$ and finds 
agreement with Ref.~\cite{bg} while quoting
an even smaller uncertainty: $\kappa_\epsilon = 0.92 \pm 0.01$. 
Finally, correcting for $\imag M_{12}^K\neq \imag
M_{12}^{(6)}$ by including higher-order terms of the operator product
expansion leads to $\kappa_\epsilon = 0.94\pm 0.02$.
Actually, in the correction factor $\kappa_\epsilon$, the three
approximations have to be treated in different ways, since they mix
uncertainties from experimental and theoretical origins. The correction
from $\phi_\epsilon$ is of experimental nature and can be treated
easily.

The correction involving $\xi$ could be computed by combining the
experimental value of ${\rm Re}\ A_0$ and the theoretical computation of
${\rm Im}\ A_0$ using the effective Hamiltonian $H^{|\Delta S|=1}$. 
However, the latter is dominated by the matrix element of the
QCD penguin in the $I=0$ channel $\langle (\pi\pi)_0 |Q_6|K\rangle$,
which is poorly known. Here we follow the method of Refs.~\cite{Andriyash:2003ym,bg,LLVdW} 
which uses $\epsilon'_K$ to correlate $\langle (\pi\pi)_0 |Q_6|K\rangle$ with
$A_2$. The latter amplitude involves the matrix element of the
electroweak penguin $Q_8$ in the $I=2$ channel, $\langle
(\pi\pi)_2|Q_8|K\rangle$, which has been computed using lattice
simulations and sum rules. Indeed, one finds \cite{bg}
\begin{equation}\label{eq:Omega}
\frac{\epsilon'_K}{\epsilon_K}= 
         -\omega\frac{\xi}{\sqrt{2}|\epsilon_K|}(1-\Omega),
\qquad\quad \mbox{with}~~~
\omega=\frac{{\rm Re}A_2}{{\rm Re A_0}},\qquad
\Omega=\frac{1}{\omega}\frac{{\rm Im}A_2}{{\rm Im} A_0}.
\end{equation}
$\omega =0.0450$ is obtained from experiment, whereas $\Omega$
describes the weight of the (imaginary part of the) $I=2$ contribution
with respect to the $I=0$ one. 

In practice, a numerical estimate of the contributions from other
(subleading) operators than $Q_6$ and $Q_8$ in $H^{|\Delta S|=1}$ has
been obtained~\cite{BurasLectures,BurasJamin,DG} under the assumptions
that ${\rm Im}\ A_0$ and ${\rm Im}\ A_2$ can be computed accurately
combining the effective Hamiltonian approach and experimental values for
${\rm Re}\ A_0$ and ${\rm Re}\ A_2$:
\begin{equation}\label{eq:epsprimeovereps}
\frac{\epsilon'_K}{\epsilon_K} = N_0 + N_1 \cdot R_6 + N_2 \cdot R_8, 
\qquad \qquad R_6=R_6[(\epsilon'/\epsilon)_{\rm exp},R_8]
\end{equation}
where $N_0, N_1, N_2$ are numbers, coming mainly from $\lambda_t$, the
experimental values for the real parts of $A_0$ and $A_2$, and $R_6$ and
$R_8$ are rescaled bag parameters.  Following the review of
ref.~\cite{Cirigliano}, the authors of ref.~\cite{BurasJamin} proposed 
the following conservative range
$R_8=1.0\pm 0.2$ which we will follow. Using
$\epsilon'_K/\epsilon_K=(1.65\pm 0.26)\cdot 10^{-3}$, one can use
eq.(\ref{eq:epsprimeovereps}) to determine $\Omega$, which
corresponds to the ratio between the $I=2$ and $I=0$ contributions to $\epsilon'_K/\epsilon_K$.
In principle, this would require to split $N_0$ into two pieces coming respectively from $I=0$ and $I=2$, and to assess the size of these contributions following Ref.~\cite{BurasLectures}. A quicker way to obtain a similar estimate consists in computing~\cite{DG}:
\begin{equation}
\qquad
\Omega_1 = \frac{N_2 \cdot R_8}{N_0 + N_1\cdot R_6}
\qquad 
\Omega_2 = \frac{N_0 + N_2 \cdot R_8}{N_1\cdot R_6}
\label{romega12}
\end{equation}
where $\Omega_1$ and $\Omega_2$ correspond to the (extreme) hypothesis that $N_0$ is saturated either by either by $I=0$ or $I=2$ contributions.
Eq.~(\ref{eq:Omega}) can be then used to compute $\xi$ in either hypotheses, and we will take the spread of the obtained values as a (conservative) systematic uncertainty in the determination of $\xi$. A more detailed analysis of the contributions to $N_0$ would allow us to decrease this systematic uncertainty.

The last correction comes from higher-dimension contributions to ${\rm
  Im} M_{12}$. As shown in ref.~\cite{bgi}, there are two different
corrections at $d=8$, corresponding to the $\Delta S=2$ $d=8$ operators
and the double insertion of $\Delta S=1$ operators connected by a $u,c$
loop. The first is expected to be very suppressed compared to the $d=6$
contributions, whereas the second one is essentially dominated by
long-distance pion exchanges estimated in Chiral Perturbation Theory for
weak processes, leading to:
\begin{equation}
\epsilon_K=\sin\phi_\epsilon e^{i\phi_\epsilon} \left[\frac{{\rm Im} M_{12}^{(6)}}{\Delta M} + \rho\xi \right], \qquad\qquad
 \rho=0.6\pm 0.3.
\end{equation}

For our purposes, it will be simpler to combine these corrections with
the experimental input for $\epsilon_K$:
\begin{equation}
  \epsilon_K^{(0)}=\epsilon_K\left(\frac{1}{\sqrt{2}\sin\phi_\epsilon}+\rho\frac{\epsilon'_K}{\epsilon_K}
    \frac{1}{\omega(1-\Omega)}\right)\equiv 
    \frac{\epsilon_{K,exp}}{\kappa_\epsilon}
\end{equation}
where $\epsilon_K^{(0)}$ denote the approximate value of $\epsilon_K$ in
eq.~(\ref{eq:epsilon0}) with $\kappa_\epsilon=1$. Depending on the
choice of $\Omega$ (i.e. the respective part of $I=0$ and $I=2$ contributions in
the formula for $\epsilon'/\epsilon$), we get
\begin{eqnarray}
\kappa_\epsilon^{(1)} &=& 0.943\pm 0.003 (\epsilon'/\epsilon)\pm 0.012 (\phi_\epsilon)\pm 0.004 (R_8)\pm 0.015 (\rho)\\
\kappa_\epsilon^{(2)}
  &=& 0.940 \pm 0.003 (\epsilon'/\epsilon)\pm 0.012 (\phi_\epsilon)\pm 0.004 (R_8)\pm 0.018 (\rho)
\end{eqnarray}
Combining the first two errors in quadrature for the Gaussian part and the last two errors linearly and taking the spread of the values into the Rfit part, we obtain the estimate
\begin{equation}
\kappa_\epsilon = 0.940 \pm 0.013 \pm 0.023
\end{equation}
in good agreement with 
% the fact that $\lambda_\epsilon $ is the inverse of $\kappa_\epsilon$,
% and with the estimate
$\kappa_\epsilon=0.94\pm 0.02$ in ref.~\cite{bgi}. This determination relies on the assumption that $\epsilon'_K$ is not affected by New Physics.

\subsection{Error budget for $\epsilon_K$}
Recently, it has been claimed that there is a discrepancy between the $\epsilon_K$
constraint and its prediction~\cite{bg,smtensions}. With the input values used in our fit 
and with the Rfit treatment of theoretical uncertainties we do not observe any 
sizeable discrepancy when comparing the prediction for $\epsilon_{K}$ 
(Table~\ref{tab:fitResults_SM1}) with its measured value 
(Table~\ref{tab:ExperimentalInputs}). 
This can also be seen in Fig.~\ref{fig-rhoeta_smfit} where the combined fit prefers 
a region in $\rhobar$ and $\etabar$ that is close to the edge of, though still inside
the $95~\%$ CL region of the $\epsilon_{K}$ constraint.

As illustrated in Fig.~\ref{fig-epsK} one can obtain a minor discrepancy 
at 1.2 $\sigma$ if one treats as Gaussian all the parameters involved in \eq{eq:epsilon0} 
(i.e., $|V_{cb}|$, $\Bag_K$, $\kappa_\epsilon$, the QCD correction terms $\eta_{cc,ct,tt}$ 
and the masses $\bar{m}_{c,t}$), but such a treatment of the systematics remains questionable.
Another way of seeing the absence of discrepancy is to compare the prediction for 
$\kappa_\epsilon$ and estimates of these quantity. One can see clearly from 
Fig.~\ref{fig-kappa} that the global fit can cope easily with a value of $\kappa_\epsilon$ 
down to 0.9, and that the  prediction from the global fit agrees well with the recent 
estimates of this quantity.

\begin{nfigure}{!Htb}
  \includegraphics[width=10cm]{epsK_BG_meas}
  \caption{\small Constraint on $|\epsilon_K|$ predicted from the global fit,
 compared to the experimental value. The green curve is obtained with our Rfit 
 treatment of the uncertainties for the parameters entering \eq{eq:epsilon0}.  
 The dashed curve is obtained by treating all the errors as Gaussian.
\label{fig-epsK}}
\end{nfigure}
\begin{nfigure}{!Htb}
  \includegraphics[width=10cm]{kappa}
  \caption{\small Constraint on $\kappa_\epsilon$ predicted from the global fit 
   using \eq{eq:epsilon0}, compared to three different theoretical inputs 
   $\kappa_\epsilon=0.94\pm 0.02$ with either a Rfit or a Gaussian treatment of 
   the error, or $\kappa_\epsilon=0.94\pm 0.013\pm 0.023$ corresponding to our 
   own treatment of the uncertainties involved in its evaluation. 
                 \label{fig-kappa}}
\end{nfigure}

In order to make the comparison of our $|\epsilon_K|$ analysis with
the one of Ref.~\cite{LLVdW} easier we treat all the errors as
Gaussian and calculate the error budget for the fit prediction of
$|\epsilon_K|$. With our inputs we find
\begin{eqnarray}
10^3|\epsilon_K|&=& 1.911 \pm 0.020 
  \large|_A \pm 0.021\large|_\lambda \pm 0.064 
    \large|_{(\bar\rho,\bar\eta)} \nonumber\\
                & &       \ \ \ 
  \pm 0.180\large|_{\Bag_K} \pm 0.019\large|_\mathrm{top} 
  \pm 0.088\large|_\mathrm{charm} \pm 0.054\large|_\mathrm{\kappa_\epsilon}
                \nonumber\\
                &=& 1.91 ^{+0.26}_{-0.24}
\end{eqnarray}
which is $1.2~\sigma$ away from the experimental measurement, while with the inputs of the last reference in~\cite{LLVdW} we find
\begin{eqnarray}
10^3|\epsilon_K|&=& 1.769 \pm 0.019\large|_A \pm 0.019\large|_\lambda \pm 0.061\large|_{(\bar\rho,\bar\eta)} \nonumber\\
                & &       \ \ \ \pm 0.062\large|_{\Bag_K} \pm 0.018\large|_\mathrm{top} \pm 0.067\large|_\mathrm{charm} \pm 0.033\large|_\mathrm{\kappa_\epsilon}
                \nonumber\\
                &=& 1.77 ^{+0.18}_{-0.16}
\end{eqnarray}
which is $2.4~\sigma$ away from the experimental measurement, in agreement with~\cite{LLVdW}. Hence we see that the difference between our analysis and
Ref.~\cite{LLVdW} mainly comes from our input for $\Bag_K$ that has a larger theoretical error, and from our central value for $|\epsilon_K|$ that is larger
because of the different analysis of the CKM parameters. We thus conclude 
that the potential anomaly in $|\epsilon_K|$ cannot yet be
precisely quantified independently of the theoretical inputs and 
therefore deserves further investigations.

\clearpage
