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\newcommand{\kzpnn}    {\mbox{$K \! \rightarrow \! \pi \nu \overline{\nu}$ }}
\newcommand{\kpnn}    {\mbox{$K^+ \! \rightarrow \! \pi^+ \nu \overline{\nu}$ }}
\newcommand{\klpnn}   {\mbox{$K^\circ_L \! \rightarrow \! \pi^\circ \nu \overline{\nu}$ }}
\newcommand{\kpen}    {\mbox{$K^+ \! \rightarrow \! \pi^\circ e^+ \nu_e$ }}
\newcommand{\vtd}     {\mbox{$V_{td}$ }}
\newcommand{\Vtd}     {\mbox{$| V_{td} |$ }}
\newcommand{\vus}     {\mbox{$V_{us}$ }}
\newcommand{\Vus}     {\mbox{$| V_{us} |$ }}
\newcommand{\vcb}     {\mbox{$V_{cb}$ }}
\newcommand{\Vcb}     {\mbox{$| V_{cb} |$ }}
\newcommand{\vub}     {\mbox{$V_{ub}$ }}
\newcommand{\Vub}     {\mbox{$| V_{ub} |$ }}
\newcommand{\bpsiks}  {\mbox{$B^\circ_d \! \rightarrow \! J/\psi K^\circ$ }}
\newcommand{\bmix}    {\mbox{$B^\circ$---$\overline{B^\circ}$ }}
\newcommand{\bdmix}   {\mbox{$B_d^\circ$---$\overline{B_d^\circ}$ }}
\newcommand{\bsmix}   {\mbox{$B_s^\circ$---$\overline{B_s^\circ}$ }}
\newcommand{\bsbd}    {\mbox{$\Delta M_{B_s}/\Delta M_{B_d}$ }}
\newcommand{\ek}      {\mbox{$\varepsilon_K$ }}
\newcommand{\sinb}    {\mbox{$\sin2\beta$ }}
\newcommand{\lamt}    {\mbox{$\lambda_t$ }}

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\begin{center}

{\large\bf\boldmath{Estimate of $B(\kzpnn)|_{SM}$ Using 
the Kaon Unitarity Triangle }} \\

% version submitted to archive on Dec 20th.

\vskip .2cm

S. H. Kettell$^{1}$, L. G. Landsberg$^{2}$, H. Nguyen$^{3}$ \\

${^1}$  Brookhaven National Lab, Upton, NY, USA \\
${^2}$  Institute of High Energy Physics, Serpukhov, Russia \\
${^3}$  Fermi National Accelerator Lab, Batavia, IL, USA \\

\end{center}

% \begin{abstract}
We estimate $B(\kpnn)$ in the context of the Standard Model using the
`kaon unitarity triangle' relation by fitting data on \ek and \sinb
for \lamt.  Our estimate is independent of the CKM matrix element \vcb
and of B-mixing. This estimate can be compared to current and future direct
measurements of $B(\kpnn)$ and to predictions made from B-mixing.  If
discrepancies arise, this technique will help to resolve various new
physics scenarios.\\
\\
\\
% \end{abstract}

% \maketitle



\twocolumngrid

The ultra-rare FCNC kaon decays \kpnn and \klpnn are of particular
interest as these `gold-plated decays' can be predicted in the
Standard Model framework with very high theoretical accuracy.

The \kzpnn decays are treated in detail in a number of
papers\cite{bb,Inami-Lim,longdistance,adler,BNL949,CKM,8,9,10,11,13,14,marciano,falk,blo,ambrosio,sa1,sa2,laplace,hocker,bk,18,20,6,12,22}.
We list some of the key aspects of these decays.
\begin{itemize}

\item[a)] The main contribution to these FCNC processes arises at
small distances $r\sim 1/m_t, 1/m_Z$; therefore, a very accurate
description for the strong interactions is possible in the framework
of perturbative QCD. This analysis has been carried out in the leading
logarithmic order (LLO) with corrections in the next to leading order
(NLO) approximation\cite{bb}.

\item[b)] The calculation of the matrix element $\langle\pi |
H_w|K\rangle_{\pi\nu\bar{\nu}}$ from quark-level processes involves
long-distance physics. However, these long-distance effects can be
avoided by the renormalization procedure developed by Inami and
Lim\cite{Inami-Lim}, relating the matrix element to that of the well
known decay \kpen through isotopic-spin symmetry. Other possible
long-distance contributions to $B(\kpnn)$have been shown to be
negligble\cite{longdistance}.


\item[c)] Since the effective vertex $Zd\bar{s}$ in the diagrams of
Figure~\ref{kpinn_beach02} is short-distance, these processes are
also sensitive to the contributions from new heavy objects (e.g.,
supersymmetric particles).
\end{itemize} 
\begin{figure}[tb]
\hskip .25in
\psfig{figure=kpinn_beach02.eps,height=2.5in,width=2.5in}
\caption{The dominant contributions to \kzpnn.
\label{kpinn_beach02}}
\end{figure}

A very important step in the study of \kpnn was achieved by the E787
experiment\cite{adler} at BNL in which two clean events were found in
favorable background conditions, indicating a branching ratio of
$B(\kpnn)|_{EXP}$ = $15.7^{+17.5}_{-8.2} \times 10^{-11}$.  This
observation has opened the door for future more precise study of the
\kpnn decay\cite{BNL949,CKM}.

In the Standard Model, the \kpnn decay is described by penguin and box
diagrams presented in Figure~\ref{kpinn_beach02}.  The partial widths
have the form:
\begin{eqnarray}
\Gamma(\kpnn) & = & \kappa^+ \cdot |\lambda_cF(x_c) + 
\lambda_tX(x_t)|^2  \nonumber \\
& = & \kappa^+ \cdot[ ( Re\lambda_cF(x_c) + Re\lambda_tX(x_t) )^2 \nonumber \\
& + &  (Im\lambda_cF(x_c) + Im\lambda_tX(x_t) )^2 ]  \nonumber \\
& \simeq & \kappa^+\cdot[( Re\lambda_cF(x_c)+Re\lambda_tX(x_t) )^2  \nonumber \\
& + & (Im\lambda_tX(x_t) )^2 ] \label{math/1}
\end{eqnarray}
where
\begin{eqnarray}
\kappa^+ = \left(\frac{G_F}{\sqrt{2}}\right)^2
\cdot |\langle\pi^+\nu\bar{\nu}|H_w|K^+\rangle |^2\cdot  
3\left(\frac{\alpha} {2\pi\sin^2\vartheta_w}\right)^2  \nonumber 
\end{eqnarray}
The factor of 3 in the expression for $\kappa^+$ results from the
three flavors of neutrinos $(\nu_e, \nu_\mu, \nu_r)$ participating in
the \kpnn decays. The factors $F(x_c)$ and $X(x_t)$ are the Inami-Lim
functions\cite{Inami-Lim} for the loop diagrams in
Figure~\ref{kpinn_beach02}.  They depend on the variables $x_i =
(m_i/m_W)^2$ with the masses of the $+\frac{2}{3}$ quarks, $m_i:
i=c,t$.  The $\lambda_i \equiv V_{id}V^*_{is}$ are vectors in the
complex plane that satisfy the unitarity relation:
\begin{equation}
\label{math/14}
\lambda_t + \lambda_c + \lambda_u = 0\quad (\lambda_i = V_{id}V^*_{is}\;;\;i=u,c,t).
\end{equation}
This equation describes the `kaon unitarity triangle', which can be
completely determined from measurement of the three kaon decays:
\kpen, \kpnn and \klpnn.  This triangle is highly elongated with a
base to height ratio of $\sim$1000.

The QCD LLO and NLO corrections to the Inami-Lim functions have been
evaluated\cite{bb,8,9,10,11}.  With current data (see
Table~\ref{tab:data}) we fix $F(x_c) = (9.8 \pm 1.4)\times 10^{-4}$
and $X(x_t) = 1.53\pm 0.05$ (the accuracy of QCD calculations improves
with increasing quark mass).  The $c$-quark contribution in
(\ref{math/1}) is smaller than the $t$-quark contribution, but is
non-negligible.  Although $F(x_c)/X(x_t)\sim 10^{-3}$, the
$Re(\lambda_c)$ is much larger than ($Re\lambda_t$ and
$Im\lambda_t$). $Re\lambda_c\sim \lambda$ while $Re\lambda_t$,
$Im\lambda_t$ and $Im\lambda_c$ are less than $\lambda^5$).

For the $CP$-violating\cite{13,14} \klpnn decay
\begin{eqnarray}
\label{math/2}
\Gamma(\klpnn) & \simeq &
\frac{1}{2}|{A(K^0\to\pi^0\nu\bar{\nu}) - A(\bar{K}^0\to\pi^0\nu\bar{\nu})} |^2  \nonumber \\
& = &  \kappa^0 \cdot\frac{1}{2} | \lambda_cF(x_c) + 
 \lambda_tX(x_t) - h.c. |^2 \nonumber \\
& = &  \kappa^0 \cdot 2\left[Im\lambda_cF(x_c) + Im\lambda_tX(x_t)\right]^2 \nonumber \\
& \simeq &  \kappa^0 \cdot 2\left[Im\lambda_tX(x_t)\right]^2
\end{eqnarray}
where
\begin{eqnarray}
\kappa^0 = \left(\frac{G_F}{\sqrt{2}}\right)^2
\cdot |\langle\pi^0\nu\bar{\nu}|H_w|K^0\rangle |^2\cdot 3\left(\frac{\alpha}
{2\pi\sin^2\vartheta_w}\right)^2 \nonumber
\end{eqnarray}
There is no $c$-quark contribution since $Im\lambda_cF(x_c)
\ll Im\lambda_tX(x_t)$. 

The partial width for the well-known decay mode \kpen is
given by:
\begin{eqnarray}
\Gamma(\kpen) = \left(\frac{G_F}{\sqrt{2}}\right)^2
|V_{us}|^2 |\langle\pi^0e^+\nu_e|H_w|K^+\rangle |^2 \nonumber
\end{eqnarray}
As mentioned above, one can relate this to
$\langle\pi^+\nu\bar{\nu}|H_w|K^+ \rangle$ and
$\langle\pi^0\nu\bar{\nu}|H_w|K^0\rangle$ with the help of
isotopic-spin symmetry:
\begin{equation}
\left|\frac{\langle\pi^+\nu\bar{\nu}|H_w|K^+\rangle}{\langle\pi^0e^+\nu_e
|H_w|K^+\rangle}\right|^2 = \left|\frac{
\langle\pi^+|H_w|K^+\rangle}{\langle\pi^0|H_w|K^+\rangle}\right|^2 = 2r_+,
\label{math/3}
\end{equation}
\begin{equation}
\left|\frac{\langle\pi^0\nu\bar{\nu}|H_w|K^0\rangle}{\langle\pi^0e^+\nu_e
|H_w|K^+\rangle}\right|^2 = \left|\frac{
\langle\pi^0|H_w|K^0\rangle}{\langle\pi^0|H_w|K^+\rangle}\right|^2 = r_0.
\label{math/4}
\end{equation}
The factor 2 in (\ref{math/3}) accounts for the pion quark structure
$|\pi^0\rangle = \frac{1}{\sqrt{2}}|u\bar{u} - d\bar{d}\rangle$ and
$|\pi^+\rangle = |u\bar{d}\rangle$. The factors $r_+ = 0.901$ and
$r_0= 0.944$ arise from the phase space corrections and the breaking
of isotopic symmetry\cite{marciano}.

Hence from (\ref{math/1}), (\ref{math/3}) and (\ref{math/4}) the
branching ratio for the \kpnn decay is
\begin{eqnarray}
\label{math/5}
&B(\kpnn)|_{SM} = R_+\cdot \frac{X(x_t)^2}{\lambda^2} 
\nonumber \\
&  \cdot \left\{[ Re\lambda_c f \frac{F(x_c)}{X(x_t)} 
+ Re\lambda_t ]^2 + [Im\lambda_t]^2\right\}
\end{eqnarray}
where
\begin{eqnarray}
\label{math/6}
\left.{\begin{array}{rll}
R_+ & = & B(\kpen) \cdot \frac{3\alpha^2}
{ 2\pi^2\sin^4\vartheta_w}\cdot r_+  \\
 & = & 7.50\times 10^{-6} \\
f \frac{F(x_c)}{X(x_t)}  & = & (6.61 \pm 0.95)\times 10^{-4} \\ 
f & = & 1.03 \pm 0.02  \end{array}} \right\}
\end{eqnarray}
Here, $f$ is an additional correction factor to the c-quark term to
take into account non-perturbative effects of dimension-8
operators\cite{falk}.  The branching ratio for the \klpnn decay is
\begin{equation}
\label{math/7}
B(\klpnn)|_{SM} = R_0\cdot \frac{X(x_t)^2}{\lambda^2}[Im
\lambda_t]^2
\end{equation}
with
\begin{eqnarray}
\label{k0}
R_0  & = & R_+\cdot \displaystyle\frac{\mathstrut r_0}{r_+}\cdot  
\displaystyle\frac{\mathstrut \tau(K_L^0)}{\tau(K^+)} = 3.28
\times 10^{-5} \nonumber \\
\quad r_0/r_+ & = & 1.048 ~~~~~ \tau(K_L^0)/\tau(K^+) = 4.17 \nonumber
\end{eqnarray}

The intrinsic theoretical uncertainty of the SM prediction for
$B(\kpnn)|_{SM}$ is $\sim7\%$ and is limited by the $c$-quark
contribution, whereas for $B(\klpnn)|_{SM}$ the uncertainty is
1--2\%. In practice the uncertainties of numerical evaluations of the
\kzpnn branching ratios are dominated by the current uncertainties in
the CKM matrix parameters.

The parameters $Im\lambda_t$, $Re\lambda_t$, $Re\lambda_c$ can be
estimated within the standard unitarity triangle (UT) framework using
the improved Wolfenstein parameterization\cite{blo}
$\bar{\eta},~\bar{\rho},~A,~\rm{and}~\lambda$ (with
$A\lambda^2=|V_{cb}|, \bar{\rho} \equiv \rho(1-\frac{\lambda^2}{2})$
and $\bar{\eta} \equiv \eta(1-\frac{\lambda^2}{2})$ ).  In these
variables the CKM quark matrix is
\begin{eqnarray}
\label{math/10}
V_{CKM} & = &
\left( \begin{array}{ccc} 
V_{ud} & V_{us} & V_{ub} \\
V_{cd} & V_{cs} & V_{cb} \\
V_{td} & V_{ts} & V_{tb} 
\end{array} \right)  \\ \nonumber \\ 
 & = & \left(
\begin{array}{ccc}
1-\frac{\lambda^2}{2} & \lambda & A\lambda^3(\rho-i\eta) \\
-\lambda & 1-\frac{\lambda^2}{2} & A\lambda^2 \\
 A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1
\end{array}\right) \nonumber \\ 
& + & O(\lambda^4) \nonumber 
\end{eqnarray}
and
\begin{equation}
\label{math/11}
\left.\begin{array}{ccl}
Re\lambda_c & \simeq & -\lambda \left(1-\frac{\lambda^2}{2}\right)+ O(\lambda^5) \\
Re\lambda_t & = & -A^2\lambda^5\left(1-\frac{\lambda^2}{2}\right)(1-\bar{\rho})
 + O(\lambda^7) \\
Im\lambda_t & = & \eta A^2\lambda^5 + O(\lambda^9)
\end{array}\right\}
\end{equation}

The current values of these and other parameters used in this paper
can be found in Table~\ref{tab:data}.  Using (\ref{math/11}) and
Table~\ref{tab:data}, equations (\ref{math/5}) and (\ref{math/7}) can
be naively solved to give the branching ratios for \kpnn and \klpnn:
\begin{widetext}
\begin{eqnarray}
\label{math/12}
B(\kpnn)|_{SM} & = & R_+\cdot A^4\lambda^8X(x_t)^2\cdot 
\left\{\frac{1}{\sigma}[(\rho_0-\bar{\rho})^2 + (\sigma\bar{\eta})^2]\right\}
\nonumber \\
& = &  R_+\cdot |V_{cb}|^4 X(x_t)^2\cdot 
\left\{\frac{1}{\sigma}[(\rho_0-\bar{\rho})^2 + (\sigma\bar{\eta})^2]\right\} 
\nonumber \\
& = & 7.50\times 10^{-6} \cdot [2.88\times 10^{-6}\pm (19.4\%)]
[2.34 \pm (6.5\%)]\{1.43\pm  (19\%)\} \nonumber \\
& = & [7.23\pm (28\%)]\times 10^{-11} = [7.2\pm 2.0]\times 10^{-11}
\end{eqnarray}
\begin{eqnarray}
\label{math/13}
B(\klpnn)|_{SM} & = & R_0\cdot A^2\lambda^8X(x_t)^2\cdot
\left\{\sigma\bar{\eta}^2\right\} \nonumber \\ 
& = &  R_0\cdot |V_{cb}|^4X(x_t)^2\cdot\left\{\sigma\bar{\eta}^2\right\} 
\nonumber \\
& = & 3.28\times 10^{-5}\cdot[2.88\times 10^{-6} 
\pm (19.4\%)][2.34\pm (6.5\%)] \cdot \{ 0.129\pm    (28.6\%)\} \nonumber \\
& = & [2.8\pm (35\%)]\times 10^{-11} = [2.8\pm 1.0]\times 10^{-11}
\end{eqnarray}
\end{widetext}
with $\rho_0 = 1+\Delta = 1+ fF(x_c)/(|V_{cb}|^2X(x_t)) = 1.39\pm 
0.07$ and $\sigma = 1/(1-\frac{1}{2}\lambda^2)^2 = 1.051$.

The uncertainties of $B(\kzpnn)$ in (\ref{math/12}) and
(\ref{math/13}) are dominated by the current uncertainties in the CKM
parameters (see Reference~\citenum{PDG}) and are significantly larger
than the intrinsic theoretical uncertainties.  The uncertainty of \Vcb
is quite significant in the evaluation of $B(\kzpnn)$ due to the
$|\vcb|^4$ dependence.  CLEO has recently measured\cite{21} a somewhat
higher \Vcb value of $(46.9\pm 3.0)\times 10^{-3}$, which would cause
a significant increase to (\ref{math/12}) and (\ref{math/13}).

The numerical solutions of equations (\ref{math/12}) and
(\ref{math/13}) do not include correlations between $\bar{\rho}$,
$\bar{\eta}$, and \vcb. Rather, these calculation are used to
demonstrate the influence of different factors in the calculation of
B(\kzpnn).  An evaluation\cite{11} employing a scanning method and
conservative errors for $V_{CKM}$ obtained the following values:
$B(\kpnn) = (7.5 \pm 2.9) \times 10^{-11}$ and $B(\klpnn) = (2.6 \pm
1.2) \times 10^{-11}$.  A more recent evaluation with similar CKM
inputs, but employing a Gaussian fit obtained $B(\kpnn) = (7.2 \pm
2.1) \times 10^{-11}$\cite{ambrosio}. This value is not very different
from (\ref{math/12}). In some recent
analyses\cite{sa1,sa2,laplace,hocker} the effects of correlations are
clearly seen and increase the precision on $B(\kzpnn)$.

A more aggressive approach for the evaluation of errors of \Vcb and
the other constraints on standard UT\cite{6,12} can significantly
increase the precision for $B(\kzpnn)$. Solving equations
(\ref{math/12}) and (\ref{math/13}) with these values for CKM
parameters\cite{11} gives $B(\kpnn)=(7.6\pm1.2)\times10^{-11}$ and
$B(\klpnn)=(2.9\pm0.5)\times10^{-11}$.  However, given the unclear
situation with \Vcb and $\xi$, the SU(3) breaking
correction to B-mixing, the more conservative approach of
Reference~\citenum{PDG} seems better justified.

Fitting to all data will provide the best precision on $B(\kpnn)$, but
as we will demonstrate, the precision on $B(\kpnn)$ from \ek and \sinb
alone is currently competitive and has the advantage of simplicity
(with no dependence on \Vcb).

In this work we directly evaluate $Re \lambda_t$ and $Im \lambda_t$
to calculate $B(\kzpnn)$ from (\ref{math/5}) and (\ref{math/7}).
This approach has been discussed in the literature\cite{bk,22}, but as
far as we know, no calculations of $B(\kzpnn)$ exist by this
method. In order to minimize uncertainty from \Vcb, it is natural to
consider \ek and \sinb in terms of the kaon UT.  We recall that
$\lambda_u = V_{ud}V_{us}^* \simeq \lambda(1-\frac{1}{2}\lambda^2)$ is
real, and $\lambda_c = V_{cd}V_{cs}^*$ has a very small complex phase
$\varphi(\lambda_c)\simeq \frac{Im\lambda_c}{Re\lambda_c}\simeq\eta
A^2\lambda^4\simeq 6\times 10^{-4}$.  The angle ($\beta_K$) between
$\lambda_t$ and $\lambda_u$ is
\begin{eqnarray}
\label{math/17}
\beta_K & = & \pi  - \varphi(V_{td}V_{ts}^*) = \pi - \varphi(V_{td}) + \varphi(V_{ts}) =
\beta + 1.0^{\circ} \nonumber \\
& = & (24.6\pm 2.3)^{\circ}
\end{eqnarray}
The phase of $V_{ts}$ is $ImV_{ts}/ReV_{ts}\simeq\eta\lambda^2 =
0.0172 = \pi + 1.0^{\circ}$.  The phase of $V_{td}$ is $\varphi
(V_{td}) = -\beta$.  The angle $\beta$ is measured precisely and in a
theoretically-clean fashion in the CP asymmetry of \bpsiks decays:
\sinb = $0.734\pm 0.054$\cite{23}. The preferred solution is $\beta =
(23.6\pm 2.3)^{\circ}$.

In the Standard Model, the apex of the kaon UT ($\lambda_t^a$) is
constrained by the measurements as shown in Figure~\ref{ideal_kut}
(without errors).  The constraint from \ek is expressed
as\cite{8,9,10}
\begin{eqnarray}
|\varepsilon_k| & = & L\cdot \hat{B}_k Im\lambda_t \cdot 
\{ Re\lambda_c[\eta_{cc} S_0(x_c) - \eta_{ct} S_0(x_c;x_t)] \nonumber \\
& & - Re\lambda_t\cdot \eta_{tt}\cdot S_0(x_t)\}
\label{math/18}
\end{eqnarray}
with parameters as shown in Table~\ref{tab:data}.  We can find the
apex of the kaon UT as the intercept of the \ek curve with
the line representing the constraint from \sinb:
\begin{equation}
Im \lambda_t = -{\rm tan}\beta_K\cdot Re \lambda_t = 
(-0.458\pm 0.049) \cdot Re \lambda_t
\label{math/19}
\end{equation}
\begin{figure}
\hskip .25in
\psfig{figure=ideal_kut.eps,height=2.5in,width=2.5in}
\caption{The apex ($\lambda_t^a$) of the kaon unitarity triangle (no
errors are shown).  The circle labeled \vub is described by (\ref{math/25}),
which uses the standard unitarity relation, and has a radius
R$\sim$\vcb\vub. The thick black lines (\ek and \sinb)
illustrate the constraints used in this paper. The dashed lines
illustrate the constraints from \kzpnn.  The inset shows the triangle
not drawn to scale.
\label{ideal_kut}}
\end{figure}

To calculate a probability density function (PDF) for $\lambda_t^a$,
we follow the Bayesian approach of References~\citenum{6}
and~\citenum{12}.  Let $f(\bold{x})$ be the PDF for $\bold{x}$, where
$\bold{x}$ is a point in the space of
$(\beta_K,~\ek,~\hat{B}_K,~m_t,~m_c,~...)$.  Equations (\ref{math/18})
and (\ref{math/19}) define the mapping from $\bold{x}$ to
$\lambda_t^a$.  Through these equations and $f(\bold{x})$, we derive
$f(\lambda_t^a)$, the PDF for $\lambda_t^a$.  We make the usual
assumption that $f(\bold{x})$ is the product of the PDF's for the
components of $\bold{x}$.  The component PDF's are taken from
Table~\ref{tab:data}, and are assumed to be uncorrelated.

Figure~\ref{limit_ek_sin2beta} shows the PDF for $\lambda_t^a$.  We
find the following central values:
\begin{equation}
\label{math/21}
\left.\begin{array}{c}
Re \lambda_t^a = (-2.85 \pm 0.29) \times 10^{-4} \\
Im \lambda_t^a = ( 1.30 \pm 0.12) \times 10^{-4}  
\end{array}\right\}
\end{equation}
\begin{figure}
\hskip .25in
\psfig{figure=limit_ek_sin2beta.eps,height=2.5in,width=2.5in}
\caption{1 $\sigma$ and 2 $\sigma$ C.L. intervals on $\lambda_t^a$,
obtained from the measurements of \ek and \sinb.
\label{limit_ek_sin2beta}}
\end{figure}

For $B(\kpnn)|_{SM}$ we obtain from Equations~(\ref{math/5}),
(\ref{math/7}) and (\ref{math/21}):
\begin{eqnarray}
\label{math/22}
B(\kpnn)|_{SM} & =  R_+ 
\frac{X^2(x_t)}{\lambda^2} \cdot \nonumber \\ 
&  \left\{\left[{Re\lambda_c} f \frac{F(x_c)}{X(x_t)} - Re\lambda_t ^a\right]^2 +
 [Im \lambda_t^a]^2\right\} \nonumber \\
& = (7.21 \pm 1.26)\times 10^{-11}
\end{eqnarray}
The probability distribution for $B(\kpnn)$ is presented in
Figure~\ref{brpdf}.  
\begin{figure}
\hskip .25in
\psfig{figure=brpdf.eps,height=2.5in,width=2.5in}
\caption{The PDF for $B(\kpnn)|_{SM}$, obtained from the measurements
of \ek and \sinb. The 95\% C.L. upper limit is $9.5 \times 10^{-11}$ and
95\% C.L. lower limit is $5.4 \times 10^{-11}$. \label{brpdf}}
\end{figure}
For \klpnn we obtain from (\ref{math/7}),
(\ref{k0}), and (\ref{math/21}):
\begin{eqnarray}
\label{math/23}
B(\klpnn)|_{SM} & = & R_0\frac{X(x_t)^2}{\lambda^2}
\left[Im\lambda_t^a\right]^2 \nonumber \\ 
& = & (2.66\pm 0.54)\times 10^{-11}
\end{eqnarray}
The results of new calculations (\ref{math/22}) and (\ref{math/23}) of
\kzpnn branching ratios with the kaon unitarity triangle variables are
in a good agreement with the calculations in the standard unitarity
triangle variables (\ref{math/12}) and (\ref{math/13}) but are free of
uncertainties in \Vcb. The main source of the uncertainties in
(\ref{math/22}) and (\ref{math/23}) are the lattice calculations of
the value of $\hat{B}_k = 0.86\pm 0.15$ and $m_c = 1.3 \pm 0.1$
GeV/c$^2$.  (We note that some recent lattice calculations using
domain-wall fermions\cite{cppacs,rbc,sa1} find values of $\hat{B}_k$
that are 10--15\% lower than the world average\cite{lellouch,gazzon}
that we use in Table~\ref{tab:data}.)  If future lattice QCD
calculations\cite{24} can significantly reduce the uncertainty in
$\hat{B}_k$, a corresponding improvement in $B(\kzpnn)|_{SM}$ will be
possible.

Given that it is difficult to assign PDF's to theoretical
uncertainties, we explore here the influence of $\hat{B}_k$ and $m_c$
on $B(\kpnn)$ using a scanning technique  while keeping other theoretical uncertainties gaussian.
We take the extremal values for $\hat{B}_k$ to be 0.72 and 1.00, and
$m_c$ to be 1.2 and 1.4 GeV/c$^2$.  For $\hat{B}_k$=0.72 and
$m_c$=1.4, we find $B(\kpnn)<10.4\times10^{-11}$ at 95\% CL. For
$\hat{B}_k$=1.00 and $m_c$=1.2, we find $B(\kpnn)>5.0\times10^{-11}$
at 95\% CL. These limits are not much worse than those derived from
Figure~\ref{brpdf}.

We've emphasized that our estimate uses only \sinb and $|\ek|$.
Nevertheless, it is interesting to consider how the measurements of
$B_d$-mixing and $|V_{ub}|$ would constrain $\lambda_t^a$.  From the
following relations:
\begin{eqnarray}
\Delta m_{B_d} & = & \frac{G_F}{6\pi^2}M^2_W m_{B_d} f^2_{B_d} \hat{B}_{B_d}
\eta_{B_d} S_0(x_t) |V_{td}V^*_{tb}|^2  \nonumber \\
0 & = & V_{ud} V^*_{ub} + V_{cd}V^*_{cb} + V_{td}V^*_{tb}  \nonumber
\end{eqnarray}
and using the approximations of (\ref{math/10}): $V^*_{tb} \approx 1$, $V_{us} =
\lambda$, $V_{ud} \approx (1-\lambda^2/2)$, and $V_{cb} \approx
-V_{ts}$, we convert the equations above into:
\begin{equation}
\Delta m_{B_d} = \frac{G_F}{6\pi^2}M^2_W m_{B_d} f^2_{B_d} \hat{B}_{B_d}
\eta_{B_d} S_0(x_t) \frac{|\lambda_t|^2}{|V_{cb}|^2} \\
\label{math/24}
\end{equation}
\begin{equation}
|\lambda_t| = |V^*_{ub} V^*_{cb}(1-\lambda^2/2) - \lambda (V^*_{cb})^2 |
\label{math/25}
\end{equation}
These two equations describe two circles whose intersections contain
the apex of the kaon UT (see Fig.~\ref{ideal_kut}), and are correlated
somewhat through \vcb.  Similar to the case of $\hat{B}_K$, there are
significant theoretical uncertainties in $B_d$-mixing, and the
extraction of \Vub and \Vcb.  The uncertainty in $B$-mixing will
presumably be significantly improved by the addition of $B_s$-mixing,
once the current confusing situation with $\xi$ is resolved (this will
be further improved once $B_s$-mixing is actually observed).  Using
the Bayesian procedure described earlier and the parameters in
Table~\ref{tab:data}, the PDF for $\lambda^a_t$ derived solely from
$B_d$-mixing and $|V_{ub}|$ is shown in Fig.~\ref{Bconstraints}.  We
see that this PDF does not constrain the kaon UT apex as well as \sinb
and $|\ek|$.  Combining all four constraints, we get the PDF for
$B(\kpnn)$ shown in Fig.~\ref{brcombined}, which is only slightly more
precise than Fig.~\ref{brpdf}.
\begin{figure}
\hskip .25in
\psfig{figure=limit_bd_vub.eps,height=2.5in,width=2.5in}
\caption{1 $\sigma$ and 2 $\sigma$ C.L. intervals on $\lambda_t^a$,
obtained from the measurements of $B_d$-mixing and $|V_{ub}|$.
\label{Bconstraints}}
\end{figure}
\begin{figure}
\hskip .25in
\psfig{figure=brcombined.eps,height=2.5in,width=2.5in}
\caption{The PDF for $B(\kpnn)|_{SM}$,
obtained from the measurements of \ek, \sinb, $B_d$-mixing,
and $|V_{ub}|$.
\label{brcombined}}
\end{figure}

Until recently the only observation of CP-violation had been in the
$K$ system.  With the recent observation of CP-violation in the
$B$-system, and the rather precise values of \sinb =
$0.734\pm0.054$ now available from the B-factories\cite{23} in good
agreement with \sinb = $0.715^{+0.055}_{-0.045}$, obtained by
fits to $\bar\rho$ and $\bar\eta$ using all other experimental
information\cite{6}, the SM picture of CP-violation has received a
significant validation.  This agreement is important because the latter 
value of \sinb includes B-mixing and \ek, which
are sensitive to loop FCNC processes, where the possibility of new
physics is more likely than that of tree processes such as
\bpsiks. 

Thus far, the CKM description appears to be the dominant source of CP
violation.  However, there are new physics scenarios that preserve the
equality between \sinb as measured from \bpsiks and global CKM fits,
but allow for a significantly different $B(\kzpnn)$\cite{aguilaz}.
Therefore, a crucial test of the CKM description will be to compare
$\beta$ derived from $B(\kzpnn)$ to that from \bpsiks.  The most
important new information on the CKM matrix will be measurements of
$B(\kpnn)$\cite{CKM} and $B(\klpnn)$\cite{kopio} to 10\% precision.
The combination of these, in context of the SM, will determine
\sinb to 0.05\cite{bb,6,10,11}, competitive with the current
uncertainty on \sinb.  If the measurement of this angle
obtained from B(\kzpnn) is consistent with that of \bpsiks, this would
provide very strong support for the SM description of CP-violation.

Another critical test of the SM will be comparison of \Vtd from \kpnn
and \bsbd.  Currently, the E787 measurement of $B(\kpnn) =
15.7^{+17.5}_{-8.2} \times 10^{-11}$ is consistent with the SM
expectation, but the central experimental value exceeds it by a factor
of two. To date there is only a limit on $B_s$-mixing, but it is
likely to be observed soon at the Tevatron.  Until $B_s$-mixing is
observed, the limit can be used to set an upper limit on
$B(\kpnn)$\cite{bb}.  A recent calculation of this
limit\cite{ambrosio} gives $B(\kpnn)|_{\rm SM} < 13.2 \times
10^{-11}$, which is below the current experimental value\cite{adler}.
This work used a value of $\xi = 1.15 \pm 0.06$, whereas recent work
would suggest a higher value\cite{sa1,sa2,kronfeldryan}. A value of
$\xi = 1.32\pm 0.10$ as suggested by Reference~\citenum{kronfeldryan}
would raise the upper limit to $15.5 \times 10^{-11}$.  The
theoretical situation with $\xi$ will hopefully be resolved soon.
Reference~\citenum{ambrosio} considered possible new physics scenarios
in the $s \to d\nu\bar\nu$ transition or $B-$mixing.  If a
disagreement persists, we have outlined a procedure using only \ek and
\sinb to ascertain which sector $K$ or $B$ is influenced by new
physics.


Our work is an estimation of $B(\kpnn)|_{\rm SM}$ based solely on \ek
and \sinb and is not dependent on \Vcb or B-mixing.  Our 95\%
C.L. upper limit is $9.5 \times 10^{-11}$ and the single large
systematic error of this approach is $\hat{B}_K$.  The current
experimental value is a little less than 1$\sigma$ above this limit.  The uncertainty
from our prediction is comparable to the expected experimental
uncertainties that might be achieved in the future measurements of
\kpnn~\cite{BNL949,CKM}.  An experimental measurement significantly
larger than our SM limit of $B(\kpnn) \le 10\times10^{-11}$ will be a
clear signal of new physics in either the $s \to d\nu\bar\nu$
transition or $K-$mixing.

\begin{table}
\caption{Some SM and CKM matrix parameters used for evaluation of
$B(\kzpnn)|_{SM}$ (see~\cite {10,11,PDG}).  The subscript G(U) denote
the Gaussian(Uniform) probability density distribution for the errors.
Errors shown without subscripts are assumed to be Gaussian. }
\begin{tabular}[t]{l}\hline
$\left.\begin{array}{l}
\lambda = 0.222\pm 0.002 \\
\bar{\rho} = 0.22\pm 0.10 \\
\bar{\eta} = 0.35\pm 0.05 \\
|V_{cb}| = (41.2\pm 2.0)\cdot 10^{-3} \\
A = |V_{cb}|/\lambda^2 = 0.836\pm 0.044 \\
\end{array}\right\}\begin{array}{l}
${\rm PDG\cite{PDG} }$ \end{array}$ \\
$\left.\begin{array}{l}
\bar{\rho} = 0.173\pm 0.046 \\
\bar{\eta} = 0.357\pm 0.027 \\
|V_{cb}| = (40.9\pm 0.8)\cdot 10^{-3} \hspace{1cm} \\
\end{array}\right\}\begin{array}{l}
${\rm \cite{6,12,stocchi} }$ \end{array}$ \\
$\left.\begin{array}{l}
\beta_K = \beta + 1^\circ = (24.6 \pm 2.3)^\circ \\
|\ek| = (2.282 \pm 0.017) \cdot 10^{-3}  \\
\hat{B}_k = 0.86 \pm 0.06_{\rm G} \pm 0.14_{\rm U} \\
m_t = \bar{m}_t = 166\pm 5 {\rm GeV/c^2} \\
m_c = \bar{m}_c = 1.3\pm 0.1 {\rm GeV/c^2} \\
\end{array}\right\}\begin{array}{l}
{\rm  } \end{array}$ \\
$\left.\begin{array}{l}
X(x_t) = 1.53\pm 0.05 \\
F(x_c) = \frac{2}{3}X^e_{NL}(x_c)+\frac{1}{3}X^\tau_{NL}(x_c) \\
 = (9.82 \pm 1.36) \cdot 10^{-4} \\
f = 1.03 \pm .02 \\
f \cdot F(x_c)/X(x_t) = (6.61 \pm 0.95) \cdot 10^{-4}
\end{array}\right\}\begin{array}{l}
{\rm Inami-Lim } \\
{\rm functions~for } \\
{\rm K\to\pi\nu\bar{\nu}} \end{array}$ \\
$\left.\begin{array}{l}
S_0(x_c) = (2.42\pm 0.39)\cdot 10^{-4} \\
S_0(x_c,x_t) = (2.15\pm 0.31)\cdot 10^{-3} \\
S_0(x_t) = 2.38\pm 0.11 \\
\eta_{cc} = 1.45\pm 0.38 \\
\eta_{ct} = 0.47\pm 0.04 \\
\eta_{tt} = 0.57\pm 0.01 \\
L = 3.837\times10^{-4}$\cite{10}$
\end{array}\right.\left\}\begin{array}{l}
{\rm Inami-Lim}\\
{\rm functions \ and} \\
{\rm QCD \ corrections} \\
{\rm for}\,K^0\rightleftarrows \bar{K}^0\,{\rm and} \\
|\varepsilon_k|\,{\rm evaluation}
\end{array}
\right.$
\\
$\left.\begin{array}{l}
\Vcb({\rm incl.})=(40.4\pm0.7_{\rm G}\pm0.8_{\rm U})\cdot 10^{-3} \\
|V_{cb}|({\rm excl.})=(42.1\pm1.1_{\rm G}\pm1.9_{\rm U})\cdot 10^{-3} \\
|V_{ub}|({\rm incl.})=(40.9\pm4.6_{\rm G}\pm3.6_{\rm U})\cdot 10^{-4} \\
|V_{ub}|({\rm excl.})=(32.5\pm2.9_{\rm G}\pm5.5_{\rm U})\cdot 10^{-4} \\
\Delta m_{B_d} = 0.489 \pm 0.008 ~{\rm ps^{-1}}  \\
f_{B_d} \sqrt{\hat{B}_{B_d}} = 230 \pm 30_{\rm G} \pm 15_{\rm U}~{\rm MeV}
\end{array}\right.$
$\left\}\begin{array}{l}
B_d-{\rm mixing} \\
{\rm and} ~ |V_{ub}| \\
{\rm parameters ~ used}\\
{\rm in ~ evaluating~the}\\
{\rm constraint ~ on ~\lambda_t^a} \\
{\rm in ~ Fig.~\ref{Bconstraints} }
\end{array}
\right.$\\
$\left.\begin{array}{l}
\xi = \frac{f_s}{f_d}\cdot \sqrt{\frac{\hat{B}_s}{\hat{B}_d}} =1.15 \pm 0.06\\
\end{array}\right.
\left\}\begin{array}{l}
{\rm old ~ value}\\
\end{array}
\right.$\\
$\left.\begin{array}{l}
\xi = 1.32\pm0.10$\cite{kronfeldryan}$\\
\xi = 1.18\pm0.04^{+0.12}_{-0.0}$\cite{lellouch}$\\
\xi = 1.22\pm0.07$\cite{becizevic}$\\
\xi = 1.16\pm0.08$\cite{sa1}$\\
\end{array}\right.
\left\}\begin{array}{l}
{\rm new ~ data ~ with}\\
{\rm chiral ~ log ~ extrapolation}\\
\end{array}
\right.$\\

\label{tab:data}
\end{tabular}
\end{table}


\section{Acknowledgments}

We would like to thank P.~Cooper, P.~Mackenzie, U.~Nierste, A.~Soni
and D.~Jaffe for useful discussions and W.~Marciano for the original
stimulation to perform this type of calculation. This work was in part
supported by the U.S. Department of Energy through contract
\#DE-AC02-98CH10886 and in part by the Fermilab Particle Physics
Division.

\section{Note}

During the final preparation of this work for publication we found
that Reference~\citenum{stocchi} considered fitting for the apex of
the UT from the CP-violating data only, as we do in
Figure~\ref{limit_ek_sin2beta}. However, Reference~\citenum{stocchi}
used ($\bar{\rho},\bar{\eta}$), which is dependent on \Vcb and is not
as suitable for analysis of \kzpnn.

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\end{document}

