MUON DECAY ASYMMETRIES FROM $K^0_L \to \pi^0 \mu^+ \mu^-$ DECAYS
MILIND V. DIWAN
HONG MA
T.L. TRUEMAN
{\em Physics Department, Brookhaven National Laboratory, Upton, NY 11973 }
We have examined the decay $K^0_L \to \pi^0 \mu^+ \mu^-$ in which the branching ratio, the muon energy asymmetry and the muon decay asymmetry could be measured. In particular, we find that within the Standard Model the longitudinal polarization ($P_L$) of the muon is proportional to the direct CP violating amplitude. On the other hand the energy asymmetry and the out-of-plane polarization ($P_N$) depend on both indirect and direct CP violating amplitudes. Although the branching ratio is small and difficult to measure because of background, the asymmetries could be large $\cal{O}$(1) in the Standard Model. A combined analysis of the energy asymmetry, $P_L$ and $P_N$ could be used to separate indirect CPV, direct CPV, and CP conserving contributions to the decay.
pmm.tex
%%
%   send to kaon2001@pi.infn.it 
%
%
\documentclass{article}
\usepackage{FPSpro,epsfig}
%

\def\kmu{$K^+\to \mu^+ \pi^0 \nu$}
\def\ke{$K^+\to e^+ \pi^0 \nu$}
\def\kp{$K^+\to \pi^+ \pi^0$}
\def\km{$K^+\to \mu^+ \nu$}
\def\rkm{$K^+\to \mu^+ \nu \gamma$}
\def\kppn{$K^+\to \pi^+ \pi^0 \pi^0$}
\def\kpp{$K^+\to \pi^+ \pi^+ \pi^-$}
\def\pll{$K \to \pi l^+ l^-$}
\def\pllc{$K^+ \to \pi^+ l^+ l^-$}
\def\plll{$K^0_L \to \pi^0 l^+ l^-$}
\def\pmm{$K \to \pi \mu^+ \mu^-$}
\def\pmmc{$K^+\to \pi^+ \mu^+ \mu^-$}
\def\pmml{$K^0_L \to \pi^0 \mu^+ \mu^-$}



%
\begin{document}
%
\title{ 
  MUON DECAY ASYMMETRIES FROM  $K^0_L \to \pi^0 \mu^+ \mu^-$ DECAYS
  }
\author{
 MILIND V. DIWAN, HONG MA,
T.L. TRUEMAN  \\
  {\em 
 Physics Department,
Brookhaven National Laboratory, Upton, NY 11973
} \\
  }
\maketitle
\baselineskip=11.6pt
\begin{abstract}
  We have examined
the decay
$K^0_L \to \pi^0 \mu^+ \mu^-$
 in which  the branching ratio, the muon energy
asymmetry and the muon decay asymmetry
 could be  measured.
In particular, we find that
within the Standard Model
 the longitudinal polarization ($P_L$) of the muon
is proportional to  the direct CP violating amplitude.
On the other hand the energy asymmetry and the out-of-plane
polarization ($P_N$) depend on both indirect and direct CP violating
amplitudes.
Although the branching ratio is small and difficult to
measure because of background, the asymmetries could be large $\cal{O}$(1)
in the Standard Model.
 A combined analysis of
the  energy asymmetry, $P_L$ and $P_N$
 could be used to separate indirect CPV, direct CPV, and CP conserving
contributions to the decay.
\end{abstract}
%
\baselineskip=14pt
%



\section{Introduction}

\noindent

 There are three possible
contributions to the \plll ~decay amplitude: 1) direct
CP-violating contribution from electroweak penguin
and W-box diagrams, 2) indirect CP-violating amplitude
from the $K_1 \to \pi^0 l^+ l^-$ component in $K_L$, and
3) CP-conserving amplitude from the $\pi^0 \gamma \gamma$
intermediate state {\cite{winwolf,rwoj}}.
 The sizes of the three contributions
depend on whether the final-state lepton is an electron or a muon.
The CP-conserving two-photon contribution to the electron mode is expected to
be $(1-4)\times10^{-12}$, based on $K_L\rightarrow \pi^0 \gamma \gamma$ data.
Although suppressed in phase space, this contribution to the muon mode is
comparable to the electron mode because of the scalar form factor
which is  proportional to
lepton mass\cite{hsehgal}.
The interesting direct CP-violating component
must be extracted from any signal found
for $K_L \to \pi^0 l^+ l^-$ in the presence of two formidable obstacles:
 the theoretical uncertainty on contamination from indirect CP-violating and
CP-conserving contributions, and
 the experimental background
from  $l^+ l^- \gamma \gamma$ \cite{greenlee}.


To subtract the indirect CP and CP conserving contributions,
several authors have examined
the use of measurements from
$K_L \to \pi^0 \gamma \gamma$, $K_S \to \pi^0 e^+ e^-$, as well as
 the lepton energy
asymmetry{\cite{rwoj,donoghue,dambrosio,bgeng91,gabval}}.
With better measurements expected in the near future, it is useful
to reexamine \plll decays 
\cite{piggktev, na48ks}. 


In this paper, we examine if the muon decay asymmetries give additional
constraints on CP violation in \pmml.   Indeed, we find that
within the Standard Model the P-odd longitudinal polarization
of the muon is non-zero only if the direct CP violating
amplitudes are non-zero. We also find that other asymmetries
that involve the polarization of both muons can be constructed to
isolate the direct and indirect contributions to CP violation.
We will proceed in analogy to the charged version of the decay,
$K^+\to \pi^+ \mu^+ \mu^-$
which has been analyzed extensively\cite{anbg}$^-$\cite{bgt}. 
The structure of
 $K_L\to \pi^0 l^+ l^-$ decays is more complex than
 that of  $K^+ \rightarrow \pi^+ l^+ l^-$ decays
because of  CP suppression.
In particular,
 the one-photon intermediate state contribution to the
vector form factor ($F_V$), which is expected to be almost
real and dominant for \pllc ~decays, is
CP-suppressed in \plll ~decays.

The \plll decay process can have contributions from
scalar, vector, pseudo-scalar and
axial-vector interactions, with corresponding form factors, $F_S$, $F_V$,
$F_P$, and $F_A${\cite{anbg}}:
\begin{eqnarray}
{\cal M} & = & F_S \bar u(p_l, s)v(\bar p_l, \bar s)
+ F_P \bar u(p_l, s)\gamma_5 v(\bar p_l, \bar s)  \nonumber \\
 & & + F_V p^\mu_k \bar u(p_l, s)\gamma_\mu  v(\bar p_l, \bar s)
    + F_A p^\mu_k \bar u(p_l, s)\gamma_\mu  \gamma_5 v(\bar p_l, \bar s)
\label{amppmm}
\end{eqnarray}
Here $p_k$, $p_\pi$, $p_l$, and $\bar p_l$ are the kaon, pion, lepton, and
antilepton 4-momenta.
Transverse muon polarization effects arise from interference between terms
with non-zero phase differences, while longitudinal polarization results
from interference between terms with the same phase.


 The scalar form factor, $F_S$,  is expected to get
a  contribution from only the two-photon intermediate state,
$K_L \to \pi^0 \gamma \gamma \to \pi^0 \mu^+ \mu^-$.
Pseudo-scalar, $F_P$,  and  axial-vector, $F_A$, get contributions from the
short-distance ``Z-penguin'' and ``W-box'' diagrams only,
 where the dominant term arises
from t-quark exchange;  both form factors therefore  depend on
$V_{ts}V^*_{td}$, with a small contribution from the charm quark.
$F_V=F_V^+ + F_V^-$ where $F_V^+$ is the  CP-even
 contribution   from
the two photon process and is proportional to
$p_k\cdot(p_l-\bar{p_l})$. $F_V^-$ is the CP-odd contribution, the sum of
$F_V^{MM}$, the indirect CP violation in $K_L$ decays, and $F_V^{dir}$, the
short distance, direct CP violating contribution.
Unlike $K^+$,
all the amplitudes in the $K_L$ decay
 are likely to be of  the same order of magnitude, and
 polarization effects should therefore be  large (${\cal O}(1)$), unless
there are strong cancellations \cite{mpla}.


The symmetries of the decay are clearest
 in the $l^-l^{+}$ CM frame. In this reference frame,
$p_l=(E,\vec{p}), \,\bar{p_l}=(E,-\vec{p})$ and $\lambda ,\, \mu$ are the
helicities of the lepton
and anti-lepton, respectively.
Using this notation,  the helicity amplitudes, $M_{\mu \lambda}(\hat{p})$,
 in terms of the four
form factors
$F_S,F_P,F_V, F_A$ are:
\begin{eqnarray}
M_{++}(\hat{p})&=& -F_S \frac{p}{m}+F_P \frac{E}{m} - F_V\, p_k
\cos{\theta} +F_A E_k \nonumber \\
M_{--}(\hat{p}) &=& +F_S \frac{p}{m}+F_P \frac{E}{m} + F_V\, p_k
\cos{\theta} +F_A E_k \nonumber \\
M_{+-}(\hat{p}) &=& +F_V \frac{E}{m}\, p_k\, \sin{\theta}  -F_A
\frac{p}{m}\, p_k
\,\sin{\theta} \nonumber \\
M_{-+}(\hat{p}) &=& +F_V \frac{E}{m}\, p_k \,\sin{\theta} +F_A
\frac{p}{m}\, p_k
\,\sin{\theta}
\end{eqnarray}
Where $p$, $E$, $m$ are the momentum, energy and the mass of the
negative lepton, and $\theta$ is the angle between the negative lepton
and the kaon in the $l^-l^+$ CM frame.
$M_{\mu,\lambda}(\hat{p})$ can depend on various invariants like
$p_k\cdot (p_l+\bar{p_l})$ but
we single out  the unit vector $\hat{p}=\vec{p}\,/|\,\vec{p}\,|$ because it
changes under the $CP$-operation.
Under $CP$ $M_{\mu,\lambda}(\hat{p}) \rightarrow
-M_{-\lambda,-\mu}(-\hat{p})$. As we have already discussed,
$F_V = F_V^+ + F_V^-$
and $F_V^{\pm}(\hat{p}) = \mp F_V^{\pm}(-\hat{p})$. In this set of graphs
the form factors have no
$\theta$ dependence other than an explicit factor of $\cos{\theta}$ in
$F_V^+$.

These helicity amplitudes can be used to create interesting CP-odd
observables such as the following three:
\begin{eqnarray}
|M_{++}(\hat{p})|^2 - |M_{--}(-\hat{p})|^2 & = &  -4 Re\{(F_S \frac{p}{m} +
F_V^+p_k
\cos{\theta})^* \nonumber \\
& & \times (F_P\frac{E}{m}+F_A E_k -F_V^- p_k \cos{\theta})\}  \nonumber \\
|M_{+-}(\hat{p})|^2 -|M_{+-}(-\hat{p})|^2 &=& + 4 Re\{(F_V^+
\frac{E}{m})^* \nonumber \\
& & \times (F_V^-\frac{E}{m} -F_A \frac{p}{m})\} p_k^2 \sin^2{\theta}
\nonumber \\
|M_{-+}(\hat{p})|^2 -|M_{-+}(-\hat{p})|^2 &=& 4 Re\{(F_V^+ \frac{E}{m})^*
\nonumber  \\
& &  \times (F_V^-\frac{E}{m} +F_A \frac{p}{m})\} p_k^2 \sin^2{\theta}
\end{eqnarray}
 The difference between  the last two asymmetries isolates
 the interference between
 the $CP=+1$ part of $F_V$  and purely direct CP violating amplitudes.


The above asymmetries involve the measurement of the polarization of both
leptons in the same event. It is possible to get direct
information about CP
violating amplitudes from measuring asymmetries for one of the leptons
only, even though the
asymmetry is not intrinsically CP-viloating. It is
well-known{\cite{pi0mumu}} that the
out-of-plane polarization of the muon, transverse to the plane formed by the
$\pi$ and the
lepton momenta, gets contributions from CP-violating amplitudes,  but several
effects are
mixed up and it is not possible to give a separation of the direct and the
indirect pieces. It is, however,  
not well-known that more information can be
obtained from the
 parity-violating single lepton 
longitudinal asymmetries, even
though they are not
intrinsically CP-violating. The longitudinal polarizations
 are given in the
lepton-lepton c.m. frame as:
\begin{eqnarray}
P'_L(\hat{p}) &=& \left[|M_{++}|^2 +|M_{-+}|^2 - |M_{+-}|^2 -|M_{--}|^2
\right] /\rho
\nonumber
\\
    &=& [-4 Re\{(F_S\frac{p}{m} +F_V p_k \cos{\theta})^*(F_P \frac{E}{m} +
F_A E_k)\}  \nonumber  \\
& & + 4 \frac{pE}{m^2}\, p^2_k
\sin^2{\theta} Re(F_A^*F_V) ]/\rho, \nonumber \\
\bar{P'}_L(-\hat{p}) &=& \left[|M_{++}|^2 +|M_{+-}|^2 - |M_{-+}|^2
-|M_{--}|^2\right]/\rho
\nonumber \\
    &=& [-4 Re\{(F_S \frac{p}{m} +F_Vp_k \cos{\theta})^*(F_P \frac{E}{m} +
F_A E_k)\}  \nonumber \\
& & - 4 \frac{pE}{m^2}\, p^2_k
\sin^2{\theta} Re(F_A^*F_V) ] /\rho, \nonumber \\
\rho & = &  |M_{++}|^2 +|M_{-+}|^2 + |M_{+-}|^2 + |M_{--}|^2 .
\end{eqnarray}
We use the notation $P'_L$ here to distinguish the polarization components
measures in the
$l \bar{l}$ cm to distinguish them from the components $P_L$ measured in
the kaon rest
frame, to be used later.

These polarizations have the property that  although they are not CP
violating, they are zero if there is no direct CP violation.
The combination
($P'_{L}(\hat{p}) + P'_{L}(-\hat{p})+\bar{P'}_L(\hat{p}) +
\bar{P'}_L(-\hat{p})$) cancels the inteference between pairs of CP violating
amplitudes and leaves a pure CP-violating observable. Similarly
($P'_{L}(\hat{p}) + P'_{L}(-\hat{p})-\bar{P'}_L(\hat{p})-
\bar{P'}_L(-\hat{p})$) is CP-even.
The angular dependence  can be integrated to give a particularly
simple result:
\begin{eqnarray}
<P'_L + \bar{P'}_L >& = & -16 Re\{(F_S \frac{p}{m}
+\frac{1}{3}F_V^+(\theta=0)\,p_k )^*(F_P
\frac{E}{m} + F_A E_k)\}/\rho \nonumber \\
<P'_L - \bar{P'}_L> & = &
\frac{32}{3}\frac{pE}{m^2}\, p^2_k
 Re(F_A^*F_V^-)/\rho
\end{eqnarray}
The first of these is C-even and purely direct, and the second is
C-odd and contains both direct and indirect amplitudes.
It should be noted that for these asymmetries
 the muon and anti-muon polarizations can be measured separately over
the same part of phase space. Indeed, if a complete angular analysis of one
lepton's
polarization can be performed, four out of the 6 independent products the
form factors can be
determined.

We will show our numerical results in the kaon rest frame.
We concentrate on four measurable quantities: the total decay rate,
the energy asymmetry between the muons and the out-of-plane
 and the longitudinal
components of the muon polarization. The in-plane transverse polarization
should also be considered when designing an experiment.

For the purposes of discussing possible experiments, it is useful to have
the lepton
polarization given in a covariant way. The form is the same as for  $K^+
\rightarrow \pi^+l^-
l^+{}$ {\cite{anbg}}:
\begin{eqnarray}
P(s) & = & \{-2 Re(F_SF_P^*) m (s\cdot \bar{p}_l) \nonumber \\
&+& 2 Re(F_VF_A^*) m [2(s\cdot p_k)(\bar{p}_l \cdot p_k) - m_K^2 (s \cdot
\bar{p}_l)]\nonumber
\\
&-& 2Re(F_PF_V^*) [ -\frac{1}{2} q^2(p_k \cdot s) + (p_k \cdot p_l)(\bar
{p}_l \cdot
s)] \nonumber \\
&+& 2 Re(F_S F_A^*)[\frac{1}{2}(q^2 - 4 m^2)(p_k \cdot s) -(\bar{p}_l \cdot
s)(p_l \cdot
 p_k)]\nonumber \\ &+& [2 Im(F_PF_A^*) + 2 Im(F_SF_V^*)] \epsilon^{\mu \nu
\rho \sigma}p_{k\mu}
p_{l\nu} \bar{p}_{l\rho} s_{\sigma}\}/(m^2 \rho) \,,
\end{eqnarray}
where $q^2=(p_l + \bar{p}_l)^2$ and $s$ denotes the covariant spin vector
of the lepton.
The decay rate in the kaon rest frame is given as \cite{anbg}:
\begin{eqnarray}
m^2 \rho_0(E_l, \bar{E}_l) &=&  |F_S|^2\frac{1}{2}(q^2-4m^2)
+|F_P|^2\frac{1}{2}q^2 \nonumber
\\
     &+& |F_V|^2 m_k^2 (2E_l\bar{E}_l - \frac{1}{2}q^2) \nonumber \\
 &+& |F_A|^2 m_k^2 [ 2E_l\bar{E}_l - \frac{1}{2}(q^2-4m^2)] \nonumber \\
 &+& 2Re (F_SF_V^*)m m_k (\bar{E}_l-E_l) \nonumber \\
 &-& 2Re(F_PF_A^*)\frac{m}{2}(m_k^2-M_\pi^2+q^2)].
\end{eqnarray}
The total rate is given by
\begin{eqnarray}
 \Gamma = \frac{m^2}{2 m_k}\int \rho_0(E_l,\bar{E}_l)
\frac{dE_ld\bar{E}_l}{(2\pi)^3}.
\end{eqnarray}

The spin vector $s_L$ in the direction of the $\mu^-$ momentum in the kaon
rest frame is
\begin{equation}
s_L = (p, E_l \sin{\theta_{l \bar{l}}}, 0, E_l \cos{\theta_{l \bar{l}}})/m
\end{equation}
where we take the decay to be in the $x-z$ plane with $\vec{\bar{p}_l}$
pointing in the
$z$-direction. Then the longitudinal polarization of $\mu^-$ 
in the kaon rest frame
(not the same as Eq.4
which is in the $\mu^-
\mu^+$ c.m.) is given as
\def\vp{\vec{p}}
\def\pp{\vp_l\cdot\vec {\bar{p}}_l)}
\begin{eqnarray}
 P_L & = &  [- 2 Re (F_SF_P^*) (\bar{E}_l - \frac{E_l}{p_l^2} \pp )
\nonumber \\
    & + & 2 Re(F_VF_A^*)m_k^2(\bar{E}_l+\frac{E_l}{p^2_l}\pp) \nonumber  \\
    & + & 2Re(F_PF_V^*)m_k(m+\frac{m}{p_l^2}\pp) \nonumber  \\
    & + & 2Re(F_SF_A^*)m_k(-m+\frac{m}{p^2_l}\pp) ]\,p_l/(m^2 \rho_0)
\end{eqnarray}
where $\vp_l(\vec {\bar{p}}_l)$ and $E_l(\bar{E}_L)$ are the momentum and
energy of
$\mu^-(\mu^+)$. Notice that in this frame $F_V^+$ is odd under
$E_l\leftrightarrow  E_{\bar l}$,
while
 $F_V^-$ and the other form factors are even under the same interchange.

The transverse (out of decay plane) polarization perpendicular to the muon
momentum vector in the kaon center of mass frame is given as
\begin{eqnarray}
P_N & =& - 2 [Im(F_SF_V^*)+Im(F_PF_A^*)]\,p_l \, {\bar{p}}_l\, \sin{\theta_{l
\bar{l}}} /(m^2
\rho_0).
\end{eqnarray}
It should be noted that  $F_P$ and $F_A$ are in phase and therefore do not
contribute to $P_N$.

For completeness,  we also give  the expression for the transverse,
in-the-plane polarization $P_T$. For this the spin vector is  $s_T
= (0, \cos{\theta_{l
\bar{l}}}, 0 , -\sin{\theta_{l \bar{l}}})$, and the expression is as follows:
\begin{eqnarray}
P_T &= &-[2 Re(F_SF_P^*)\, m\, \nonumber \\ &+& 2 Re(F_P
F_V^*)\, m_K \,E_l \,
 \nonumber \\ &+& 2 Re(F_SF_A^*)\, m_K\,E_l\,
 \nonumber \\ &+& 2 Re(F_V F_A^*)\, m \,m_K^2 \,
] \,\bar{p}_l \,\sin{\theta_{l \bar{l}}}/(m^2 \rho_0).
\end{eqnarray}
It depends on the same quantities as $P_L$ and, depending on the
experimental configuration,
the measured quantity may be a linear combination of the two. ($P'_L$ is
the linear
combination of $P_L$ and $P_T$ given by the rotation of the lepton spin
through the angle
between the kaon and the lepton momenta as seen in the rest
frame of the
lepton.)


We now consider values of  the form-factors that we will use in estimating
the polarization.

\noindent
$F_V^{MM}$:
The decays $K_S\to \pi^0 l^+ l^-$  have been studied in chiral perturbation
theory extensively.
The same framework of analysis is often applied to the similar decay
of $K^+ \to \pi^+ l^+ l^-$.  In D'Ambrosio {\it et
al}\cite{dambrosio}, the decays are
analyzed beyond the leading order ${\cal O}(p^4)$.
The vector form factor for $K_S$ decays is parametrized as
\begin{eqnarray}
F_V^S = -G_F\frac{\alpha}{4\pi} (a_S+b_S z )
      -\frac{\alpha}{4\pi m_k^2} W^{\pi\pi}(z)
\end{eqnarray}
The function $W^{\pi\pi}$ comes from the pion loop contribution, which is
estimated to
${\cal O}(p^6)$ using $K\to\pi\pi\pi$ data, and the rest of the contributions
are parametrized in the linear term.
Using ideas from VMD a further assumption is sometimes made,
\begin{eqnarray}
b_S & = & \frac{a_S}{M_V^2/m_k^2} = \frac{a_S}{2.5}
\end{eqnarray}
where $M_V$ is the vector meson mass.
The vector form factor for indirect CP violation in $K_L \to \pi^0 \mu^+
\mu^-$
 is then
\begin{eqnarray}
F_V^{MM} & = & \epsilon F_V^S  \label{eqfvm}
\end{eqnarray}
The value of $a_S$ is quite unknown, however it is considered to be ${\cal
O}(1)$.
The study of the similar
decay $K^+ \to \pi^+ l^+ l^-$  gives a value of the equivalent parameter to
be $-0.59\pm0.01$,\cite{e865pee}$^-$\cite{e865pmm} therefore we use the  value, $-0.6$, for our numerical
results.

Most of the other values are taken from from Donoghue and
Gabbiani\cite{donoghue}.

\noindent
$F_V^{dir}$, $F_A$, and $F_P$:
These form factors get contributions from short distance
box and penguin diagrams.
We use the notation from Donoghue and Gabbiani\cite{donoghue}:
\begin{eqnarray}
F^{dir}_V & = & 2\frac{G_F}{\sqrt 2}\, i\, y_{7V} Im{\lambda_t} \nonumber  \\
F_A & = & -2\frac{G_F}{\sqrt 2} \,i \,y_{7A} Im{\lambda_t} \nonumber  \\
  y_{7V} & = & 0.743 \, \alpha,\mbox{at} \, m_t=175  GeV \nonumber \\
   y_{7A} & = & -0.736 \, \alpha, \mbox{at} \, m_t=175  GeV \nonumber
\nonumber \\
 \lambda_t & = & V_{td}V_{ts}^*.
\end{eqnarray}
We have used $\mbox{Im}\lambda_t = 10^{-4}$ in out numerical calculation. 
Similar to the case of $K^+$ decay  $F_P$ is related to
$F_A$ by\cite{anbg}
\begin{eqnarray}
F_P =-m_l (1-\frac{f_-}{f_+}) F_A \label{eqfp}
\end{eqnarray}
where $f_+$ and $f_-$ are charged current semileptonic decay
form factors of $K_L$.

\noindent
$F_S$: Since \cite{donoghue} is concerned only with electron final states,
the scalar
contribution is negligible and they do not calculate it. Therefore we use
the earlier result of
Ecker {\it et al}{\cite{pi0mumu}}. They have calculated $F_S$ to ${\cal
O}(p^4)$ as
\begin{eqnarray}
F_S &=&\frac{iG_8\alpha^2}{4\pi}m\,E(z)\nonumber\\
E(z) &=& \frac{1}{\beta z}log(\frac{1-\beta}{1+\beta})
    [(z-r_\pi^2)F(z/r_\pi^2) - (z-1-r_\pi^2)F(z)]. \label{eqfs}
\end{eqnarray}
$z= (p_l + \bar{p_l})^2/m_k^2$, $r_\pi = m_\pi/m_k$, and
$\beta = \sqrt{1-4\frac{m_l^2}{m_k^2 z}}$. $F(z)$ is a known function
described by Ecker {\it et al}.


\noindent
$F_V^+$: For this we return to \cite{donoghue}.
 Using chiral perturbation theory they obtain
\begin{eqnarray}
F_V^+ &=& 2  G_8\alpha^2 p_k\cdot(p_l-\bar{p_l}) K(s) \\
 K(s) &=& \frac{B(x)}{16\pi^2m_k^2}[ \frac{2}{3}ln(\frac{M_\rho^2}{-s})
          -\frac{1}{4}ln(\frac{-s}{m^2})+\frac{7}{18}] \label{eqfvp}
\end{eqnarray}
where $s=(p_l+ \bar{p_l})^2$, $x=s/4M_\pi^2$, and
$B(x)$ is a complex funtion described in \cite{donoghue}. $G_8$ denotes
the octet coupling
constant in chiral perturbation theory.  We use the value $G_8 =
\frac{G_F}{\sqrt{2}}
|V_{ud}V_{us}^*| g_8$ with $g^{tree}_8 = 5.1$\cite{donoghue}. The value
of $g^{loop}_8=
4.3$ from a higher order calculation does not alter  our conclusions
significantly \cite{g8}. The function  contains  a free parameter,
$\alpha_V = -0.72\pm 0.08$, determined from
experimental $K_L\to \pi^0 \gamma \gamma$ data\cite{piggktev}.
We have used $\alpha_V = -0.70$ for our numerical results.
It should be noted that Heiliger {\it et al} \cite{hsehgal}
have performed an analysis of both $F_S$ and $F_V^+$ in a two component
(pion loop and vector meson dominance (VMD)) model.
This calculation, however, needs to be checked against the
latest data on $K_L \to \pi^0 \gamma \gamma$. 
Recently, Gabbiani and Valencia have pointed out that a more complete
formulation of
$F_V^+$ requires three free parameters which can be obtained from
$K_L \to \pi^0 \gamma \gamma$ data\cite{gabval}. 



Using the above discussed values for the various form factors
the total decay rate is about
$\sim 1.1 \times 10^{-11}$, and it
is dominated by the scalar interaction which makes most of the contribution
for mu-mu mass above 280 MeV.
 This is because of the two pion loop contribution
embodied in the form factor $F_S$.
Unlike  $F_S$, the rest of the form factors are expected
to give contributions that fall with $q^2$.
The contribution from $F_P$ is small because of the suppression due to the
lepton mass.
The region  $\sqrt{q^2} < 280$ MeV will be affected
by interference effects as shown in figure \ref{dalitz}.
It is interesting to note that the destructive interference causes
the decay rate to be almost zero in the region where the $\mu^+$ is at
rest in the kaon rest frame.
\begin{figure}
\begin{center}
\epsfig{figure=dalitz.eps,height=2.4in,width=2.4in}
\end{center}
\caption{Dalitz decay distribution for  for $K_L \to \pi^0 \mu^+ \mu^-$.
 The units for the decay rate contours
 are arbitrary. The total
calculated branching fraction
was $1.1\times 10^{-11 }$. The form factors and the 
values of the parameters used are described
in the text.   }
\label{dalitz}
\end{figure}
The longitudinal and out-of-plane polarizations of the $\mu^-$ are shown
in figure \ref{twopols}.
\begin{figure}
\begin{center}
\mbox{\epsfig{figure=polpltl.eps,height=2.4in,width=2.4in}
\epsfig{figure=polpltt.eps,height=2.4in,width=2.4in}}
\end{center}
\caption{Longitudinal (left) and out-of-plane (right) polarizations  of
$\mu^-$ in
$K_L \to \pi^0 \mu^+ \mu^-$ decay
plotted as a function of $\mu^+$ and $\mu^-$ energy 
 in the kaon rest frame. }
\label{twopols}
\end{figure}
The energy asymmetry and the out-of-plane polarizations are large, but they
also have large dependence on
the parameter $a_S$.
If the out-of-plane polarization is used to constrain $a_S$ then
  the short distance physics could be extracted by fitting
the measured  distribution for the longitudinal polarization which
mainly comes from $Re(F_SF_A^*)$ and $Re(F_AF_V^*)$.



The modes $K_L \to \pi^0 e^+ e^-$ and $K_L \to \pi^0 \mu^+ \mu^-$
  have not as yet been  observed;
the  current  best limits on the branching ratios  for $K_L \to \pi^0 l^+ l^-$
 were obtained by the KTeV experiment
at FNAL;
$B(K_L \to \pi^0 \mu^+ \mu^-) < 3.8 \times 10^{-10}$
and $B(K_L \to \pi^0 e^+ e^-) < 5.1\times 10^{-10}$ \cite{ktevpmm, ktevpee}. 
These limits were based on  2 observed events in each case,
 and expected backgrounds  of
$0.87\pm 0.15$ for the muon mode and $1.06\pm 0.41$ for the electron mode.
The main backgrounds for the muon mode
 were estimated to be from $\mu^+ \mu^- \gamma \gamma$
($0.37\pm 0.03$)  and $\pi^+ \pi^- \pi^0$ ($0.25\pm 0.09$), in which both
charged pions
decay in flight.  Of these, the former background could be irreducible and
therefore of
great concern.
For any future experiment  it seems unlikely that the
background due to $\mu^+ \mu^- \gamma \gamma$
can be lowered.
The signal to background ratio, assuming
that only $\mu^+ \mu^- \gamma \gamma$ will contribute in a future experiment,
will  be around 1/5, if the standard-model signal is taken as
$B(K_L \to \pi^0 \mu^+ \mu^-) \sim 5\times 10^{-12}$.\,{\cite{mpla}}

Measuring the muon polarization asymmetries in $K_L \to \pi^0 \mu^+ \mu^-$,
together with the branching ratio and the lepton energy asymmetry,
could be a good way of defeating  the intrinsic background from
CP-conserving and indirect CP-violating amplitudes and
  the experimental background from $\mu^+ \mu^- \gamma \gamma$.
The large predicted asymmetries should be easy to measure with sufficient
statistics at new intense proton accelerators such as the Brookhaven
National Laboratory
AGS, the Fermilab main injector, or the Japanese Hadron Factory.
An examination of the functional form
of  the form factors $F_S$ and $F_V$ is needed to see if the present function
form in terms of the parameters $a_V$ and $a_S$ is adequate.
 Examination of
 how to measure the different components of the polarization in
the laboratory as well as
the $\mu \mu \gamma \gamma$ background
is  needed to understand the dilution of
the asymmetries
on the Dalitz plot.


\noindent



We  thank Laurence Littenberg, German Valencia, Willian Marciano,
 Steve Kettell for useful discussions.  This work was
supported by DOE grant DE-AC02-98CH10886.





%
\begin{thebibliography}{99}


\bibitem{winwolf} B. Winstein and L. Wolfenstein,
Rev. Mod. Phys. {\bf 65}, 1113 (1993).


\bibitem{rwoj} J. L. Ritchie, and S. G. Wojcicki, Rev. Mod. Phys.
{\bf 65}, 1149 (1993).

\bibitem{hsehgal}
P. Heiliger and L.M. Sehgal, Phys. Rev. D {\bf 47}, 4920 (1993).

\bibitem{greenlee}
H. B. Greenlee, Phys. Rev. D {\bf 42}, 3724 (1990).

\bibitem{donoghue}
J.F.Donoghue and F. Gabbiani, Phys. Rev. D. {\bf 51}, 2187 (1995).



\bibitem{dambrosio}
G. D'Ambrosio, et al., JHEP {\bf 8}, 4 (1998).

\bibitem{bgeng91}
G. Belanger and  C. Q. Geng,  Phys. Rev. D {\bf 43}, 140 (1991).

\bibitem{gabval}
F. Gabbiani and G. Valencia, hep-ph/0105006.



\bibitem{piggktev}
A. Alavi-Harati et al. [KTeV Collaboration], Phys. Rev. Lett. {\bf 83}, 917
(1999)

\bibitem{na48ks}
R. Batley, et al. [NA48 Collaboration], CERN/SPSC 2000-002, 
Dec. 1999. 

\bibitem{anbg}
 Pankaj Agrawal, John N. Ng, G. Belanger, C.Q. Geng,
Phys. Rev. {\bf D45}, 2383 (1992).


\bibitem{wise1}
Ming Lu, Mark B. Wise, and Martin J. Savage, Phys. Rev. {\bf D46},
5026 (1992).

\bibitem{wise2}
Martin J. Savage, Mark B. Wise, Phys. Lett. {\bf B250}, 151 (1990).

\bibitem{buchalla}
G. Buchalla, A.J. Buras,  Phys. Lett. {\bf B336}, 263 (1994).

\bibitem{gourdin}
Michel Gourdin, PAR-LPTHE-93-24, May 1993.


\bibitem{bgt}
 G. Belanger, C.Q. Geng, P. Turcotte,
Nucl. Phys. {\bf B390}, 253 (1993).




\bibitem{mpla}
M. Diwan and Hong Ma,
International Journal of Modern Physics A, Vol. 16, No. 14 (2001), 2449-2471.
 hep-ex-00073738.

\bibitem{pi0mumu}
G. Ecker, A. Pich, E. de Rafael, Nucl. Phys. B{\bf 303}, 665 (1988).
also see G. Ecker and A. Pich, Nucl. Phys. B366, 189 (1991).

\bibitem{e865pee}
R. Appel, et al., Phys. Rev. Lett. {\bf 83}, 4482 (1999).

\bibitem{e787pmm}
S. Adler, et al., Phys. Rev. Lett. {\bf 79}, 4756 (1997).

\bibitem{e865pmm}
H. Ma, et al., Phys. Rev. Lett. {\bf 84}, 2580 (2000).


\bibitem{g8}
J. Kambor, J. Missimer, and D. Wyler, Phys. Lett. B261, 496 (1991).

\bibitem{ktevpmm}
A. Alavi-Harati, et al., Phys. Rev. Lett. {\bf 84}, 5279 (2000).

\bibitem{ktevpee}
A. Alavi-Harati, et al., hep-ex/0009030.

%\bibitem{epichmumu}




%\bibitem{63} \label{ksgg} G. D. Barr, et al., Phys. Lett. B351,  579 (1995).


%\bibitem{67} \label{ecker291}
%G. Ecker, A. Pich, E. de Rafael, Nucl. Phys. B{\bf 291}, 692 (1987).



%\bibitem{71} \label{e787pnn}
%S. Adler, et al., Phys. Rev. Lett. {\bf 84}, 3768 (2000).

%\bibitem{72} \label{littpnn}
%L. Littenberg, Phys. Rev. D{\bf 39}, 3322 (1989).




%\bibitem{76} \label{ktevmmgg}
%A. Alavi-Harati, et al., Phys. Rev. D{\bf 62}, 112001  (2000).


%\bibitem{80} \label{e926}
%KOPIO (E926) proposal, submitted to the BNL-HEP program committee,
%Sep 4, 1996.  See also
%T. Numao (for the BNL-E926 Collaboration),
%Presented  at the 6th Conference on
%the Intersections of Particle and Nuclear Physics (CIPANP 97),
%Big Sky, MT, 27 May - 2 Jun 1997.

%Proceedings ``Big Sky 1997, Intersections between particle and nuclear
%physics'',  807-811.




\end{thebibliography}
%
\end{document}

hep-ex0108025.extracted_bib
\begin{thebibliography}{99}


\bibitem{winwolf} B. Winstein and L. Wolfenstein,
Rev. Mod. Phys. {\bf 65}, 1113 (1993).


\bibitem{rwoj} J. L. Ritchie, and S. G. Wojcicki, Rev. Mod. Phys.
{\bf 65}, 1149 (1993).

\bibitem{hsehgal}
P. Heiliger and L.M. Sehgal, Phys. Rev. D {\bf 47}, 4920 (1993).

\bibitem{greenlee}
H. B. Greenlee, Phys. Rev. D {\bf 42}, 3724 (1990).

\bibitem{donoghue}
J.F.Donoghue and F. Gabbiani, Phys. Rev. D. {\bf 51}, 2187 (1995).



\bibitem{dambrosio}
G. D'Ambrosio, et al., JHEP {\bf 8}, 4 (1998).

\bibitem{bgeng91}
G. Belanger and  C. Q. Geng,  Phys. Rev. D {\bf 43}, 140 (1991).

\bibitem{gabval}
F. Gabbiani and G. Valencia, hep-ph/0105006.



\bibitem{piggktev}
A. Alavi-Harati et al. [KTeV Collaboration], Phys. Rev. Lett. {\bf 83}, 917
(1999)

\bibitem{na48ks}
R. Batley, et al. [NA48 Collaboration], CERN/SPSC 2000-002, 
Dec. 1999. 

\bibitem{anbg}
 Pankaj Agrawal, John N. Ng, G. Belanger, C.Q. Geng,
Phys. Rev. {\bf D45}, 2383 (1992).


\bibitem{wise1}
Ming Lu, Mark B. Wise, and Martin J. Savage, Phys. Rev. {\bf D46},
5026 (1992).

\bibitem{wise2}
Martin J. Savage, Mark B. Wise, Phys. Lett. {\bf B250}, 151 (1990).

\bibitem{buchalla}
G. Buchalla, A.J. Buras,  Phys. Lett. {\bf B336}, 263 (1994).

\bibitem{gourdin}
Michel Gourdin, PAR-LPTHE-93-24, May 1993.


\bibitem{bgt}
 G. Belanger, C.Q. Geng, P. Turcotte,
Nucl. Phys. {\bf B390}, 253 (1993).




\bibitem{mpla}
M. Diwan and Hong Ma,
International Journal of Modern Physics A, Vol. 16, No. 14 (2001), 2449-2471.
 hep-ex-00073738.

\bibitem{pi0mumu}
G. Ecker, A. Pich, E. de Rafael, Nucl. Phys. B{\bf 303}, 665 (1988).
also see G. Ecker and A. Pich, Nucl. Phys. B366, 189 (1991).

\bibitem{e865pee}
R. Appel, et al., Phys. Rev. Lett. {\bf 83}, 4482 (1999).

\bibitem{e787pmm}
S. Adler, et al., Phys. Rev. Lett. {\bf 79}, 4756 (1997).

\bibitem{e865pmm}
H. Ma, et al., Phys. Rev. Lett. {\bf 84}, 2580 (2000).


\bibitem{g8}
J. Kambor, J. Missimer, and D. Wyler, Phys. Lett. B261, 496 (1991).

\bibitem{ktevpmm}
A. Alavi-Harati, et al., Phys. Rev. Lett. {\bf 84}, 5279 (2000).

\bibitem{ktevpee}
A. Alavi-Harati, et al., hep-ex/0009030.

%\bibitem{epichmumu}




%\bibitem{63} \label{ksgg} G. D. Barr, et al., Phys. Lett. B351,  579 (1995).


%\bibitem{67} \label{ecker291}
%G. Ecker, A. Pich, E. de Rafael, Nucl. Phys. B{\bf 291}, 692 (1987).



%\bibitem{71} \label{e787pnn}
%S. Adler, et al., Phys. Rev. Lett. {\bf 84}, 3768 (2000).

%\bibitem{72} \label{littpnn}
%L. Littenberg, Phys. Rev. D{\bf 39}, 3322 (1989).




%\bibitem{76} \label{ktevmmgg}
%A. Alavi-Harati, et al., Phys. Rev. D{\bf 62}, 112001  (2000).


%\bibitem{80} \label{e926}
%KOPIO (E926) proposal, submitted to the BNL-HEP program committee,
%Sep 4, 1996.  See also
%T. Numao (for the BNL-E926 Collaboration),
%Presented  at the 6th Conference on
%the Intersections of Particle and Nuclear Physics (CIPANP 97),
%Big Sky, MT, 27 May - 2 Jun 1997.

%Proceedings ``Big Sky 1997, Intersections between particle and nuclear
%physics'',  807-811.




\end{thebibliography}