text:
\section{Introduction}

Rare $K$ decays, directly sensitive to short-distance FCNC processes, offer an
invaluable window into the physics at play at high-energy scales. Besides the
two $K\rightarrow\pi\nu\bar{\nu}$ golden modes, the decays $K_{L}%
\rightarrow\pi^{0}e^{+}e^{-}$ and $K_{L}\rightarrow\pi^{0}\mu^{+}\mu^{-}$ also
exhibit good sensitivities, thanks to the theoretical control achieved over
their long-distance components\cite{DEIP,BDI03,IsidoriSmithUnter}.
Importantly, these modes are sensitive to different combinations of
short-distance FCNC currents, and thus allow in principle to discriminate
among possible New Physics scenarios.

In this respect, the pair of $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ decays
is unique since, though their dynamics is similar, the very different lepton
masses allow to probe helicity-suppressed effects in a particularly clean way.
Only the $K_{L}\rightarrow\ell^{+}\ell^{-}$ modes share this characteristic,
but the dominance of the long-distance two-photon contribution unfortunately
prevents from acceding to the short-distance physics with a good degree of
precision\cite{IsidoriU03}. On the contrary, the corresponding two-photon
contribution to $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ is under control. It
represents only $30\%$ of the total rate for the muonic mode, and is
negligible for the electronic one\cite{BDI03,IsidoriSmithUnter}.

The main purpose of the paper is to illustrate how this fact can be used to
constrain or identify the nature of possible New Physics effects. More
precisely, our goals are:

\begin{enumerate}
\item To analyze the impacts arising from all possible $\Delta S=1$
four-fermion operators in a model-independent way, i.e. operators of the form
$\left(  \bar{s}\Gamma d\right)  \left(  \bar{\ell}\Gamma\ell\right)  $,
$\Gamma=1,\gamma_{5},\gamma^{\mu},\gamma^{\mu}\gamma_{5},\sigma^{\mu\nu}$, and
to show how combined measurements of $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$
can disentangle them. An important distinction is made between
helicity-suppressed operators, like the $\Gamma=1,\gamma_{5}$ ones arising for
example in the MSSM at large $\tan\beta$\cite{TanBeta,Foster05}, and
helicity-allowed operators like in SUSY without R parity\cite{Rparity} or from
leptoquark interactions\cite{DavidsonBC94}. Also, the electromagnetic tensor
operator $\bar{s}\sigma_{\mu\nu}dF^{\mu\nu}$ will be briefly
considered\cite{BurasCIRS99,DAmbrosioGao}. Finally, constraints from
$K_{L}\rightarrow\ell^{+}\ell^{-}$ (for scalar/pseudoscalar operators) and
$K\rightarrow\pi\nu\bar{\nu}$ (for helicity-allowed tensor/pseudotensor
interactions) will be analyzed.

\item To improve the control over the long-distance two-photon contribution.
Indeed, this is needed to estimate interference effects with New Physics
short-distance contributions. In addition, once achieved, the $K_{L}%
\rightarrow\pi^{0}\mu^{+}\mu^{-}$ forward-backward (or lepton energy)
CP-asymmetry\cite{2pp,AFB} will be computed reliably for the first time, both
in the Standard Model and beyond, and will prove to be an interesting
complementary source of information on the New Physics at play.
\end{enumerate}

The paper is organized as follows. In Section 2, we present the ingredients
needed to deal with the long-distance dominated contributions, and, in the
spirit of Ref.\cite{IsidoriSmithUnter}, analyze the possible signals of New
Physics in the vector or axial-vector operators. Then, in Section 3, we
analyze all other operators, both in the helicity-suppressed and
helicity-allowed cases. The corresponding analysis of $K_{L,S}\rightarrow
\ell^{+}\ell^{-}$ is in appendix. Finally, our results are summarized in the Conclusion.

\section{$K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ with standard
short-distance operators}

The $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ decays receive essentially three
types of contributions, depicted in Fig.\ref{Fig1}.

A first class of effects, purely sensitive to short-distance physics, results
from heavy particle FCNC loops (the $W$ and $Z$ bosons and the $t$ and $c$
quarks in the SM, see Fig.\ref{Fig1}a), and can be parametrized by a set of
local effective operators. In the SM, the leading relevant effective
Hamiltonian induced by these effects reads \cite{BuchallaBL96}:%
\begin{equation}
\mathcal{H}_{eff}^{\mathrm{V,A}}=-\frac{G_{F}\alpha}{\sqrt{2}}\lambda
_{t}\left[  y_{7V}\,\left(  \bar{s}\gamma_{\mu}d\right)  \left(  \bar{\ell
}\gamma^{\mu}\ell\right)  +y_{7A}\left(  \bar{s}\gamma_{\mu}d\right)  \left(
\bar{\ell}\gamma^{\mu}\gamma_{5}\ell\right)  \right]  +h.c.\;, \label{Eq2}%
\end{equation}
with $\alpha\equiv\alpha\left(  M_{Z}\right)  $ and $\lambda_{q}=V_{qs}^{\ast
}V_{qd}$. Of course, in the presence of New Physics, other types of effective
interactions could be produced. This will be the subject of Section 3. For
now, we will assume that New Physics affects only the values of the
coefficients $y_{7A,7V}$, and leave them as free
parameters\cite{IsidoriSmithUnter}.

Beyond SM scenarios leading to such modifications of the vector and
axial-vector couplings are numerous. Examples are the MSSM for moderate values
of $\tan\beta$ (see e.g. Ref.\cite{MSSM}, and references therein), for large
$\tan\beta$ (from charged Higgs penguins, see e.g. Ref.\cite{isidoriP06}), or
the Enhanced Electroweak Penguins of Refs.\cite{BurasS99,BFRS04}. Of course,
in specific models, New Physics also affects operators with different flavor
quantum numbers. We have opted here for a decoupled, model-independent
analysis, considering thus only the operators relevant for $K_{L}%
\rightarrow\pi^{0}\ell^{+}\ell^{-}$.

Also, the four-quark operators $Q_{1,...,6}$ have not been explicitly included
in Eq.(\ref{Eq2}). This is because their impact on the direct CP-violating
(DCPV) contribution to $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$, associated
to the local effective Hamiltonian $\mathcal{H}_{eff}^{\mathrm{V,A}}$ in
standard terminology, can be safely neglected in the SM
\cite{BuchallaBL96,BDI03}. New Physics cannot change this picture as new
sources of CP-violation from $Q_{1,...,6}$ are bounded from purely hadronic
$K$ decay observables.

A second class of contributions, dominated by long-distance dynamics, is
driven by the coupling of leptons to photons, via $K^{0}-\bar{K}^{0}$ mixing
(Fig.\ref{Fig1}b) or via a two photon loop (Fig.\ref{Fig1}c). Let us now
analyze these in more detail.

\subsection{Long-distance dominated contributions}

The remarkable point with the contributions depicted in Fig.1b and 1c is that
they can be entirely determined from experimental data. Their estimation is
thus not affected by possible New Physics effects. As they remain as an
unavoidable background to the interesting short-distance contributions, we
briefly recall (and partly improve, in the case of two-photon amplitudes) the
way they are dealt with.%

\begin{figure}
[t]
\begin{center}
\includegraphics[
height=2.2632in,
width=4.6103in
]%
{Fig1.eps}%
\caption{(a) Short-distance penguin and box diagrams for the initial
conditions of $y_{7A,7V}$ in the SM. (b) $K^{0}-\bar{K}^{0}$ mixing-induced
contribution, with a long-distance dominated, CP-conserving effective
$K_{S}\rightarrow\pi^{0}\gamma^{\ast}$ vertex. (c) Long-distance two-photon
induced CP-conserving contributions.}%
\label{Fig1}%
\end{center}
\end{figure}

\paragraph{\textit{The indirect CP-violating contribution}}

(ICPV, Fig.\ref{Fig1}b) originates from $K^{0}-\bar{K}^{0}$ mixing. The
subsequent CP-conserving $K_{S}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ decay is
dominated by the long-distance process $K_{S}\rightarrow\pi^{0}\left(
\gamma^{\ast}\rightarrow\ell^{+}\ell^{-}\right)  $, producing the lepton pair
in a $1^{--}$ state. The corresponding amplitude reads:%
\begin{equation}
\mathcal{M}_{\mathrm{ICPV}}=\varepsilon\,\mathcal{M}\left(  K_{1}%
\rightarrow\pi^{0}\ell^{+}\ell^{-}\right)  =-\varepsilon\frac{G_{F}\alpha
_{em}}{4\pi}W_{S}\left(  z\right)  \left(  P+K\right)  ^{\mu}\left\{
\bar{u}_{p}\gamma_{\mu}v_{p^{\prime}}\right\}  \;, \label{Eq7}%
\end{equation}
where $P,K,p$ and $p^{\prime}$ denote the momenta of the $K,\pi,\ell^{-}$ and
$\ell^{+}$ states, respectively, $z=T^{2}/M_{K^{0}}^{2}$ with $T=p+p^{\prime}$
and $\alpha_{em}\approx1/137$. The $W_{S}$ function has been analyzed in
detail in Chiral Perturbation Theory (ChPT) in Ref.\cite{DEIP}, and can be
parametrized as follows:
\begin{equation}
W_{S}\left(  z\right)  =a_{S}+b_{S}z+W_{S}^{\pi\pi}\left(  z\right)
+W_{S}^{KK}\left(  z\right)  \;. \label{Eq8}%
\end{equation}
The pion and kaon loops ($W_{S}^{\pi\pi},W_{S}^{KK}$) were found small and, to
a good approximation, a single (real) counterterm dominates: $W_{S}\left(
z\right)  \approx a_{S}$. This counterterm can then be extracted from the
experimental $K_{S}\rightarrow\pi^{0}e^{+}e^{-}$ and $K_{S}\rightarrow\pi
^{0}\mu^{+}\mu^{-}$ branching fractions: $\left|  a_{S}\right|  =1.20\pm0.20$
\cite{KSpill}. Eq.(\ref{Eq7}) is thus indeed entirely determined in terms of
measured quantities ($|a_{S}|$, $\varepsilon$).

\paragraph{\textit{The two-photon CP-conserving contribution}}

(Fig.\ref{Fig1}c), $K_{L}\rightarrow\pi^{0}\left(  \gamma^{\ast}\gamma^{\ast
}\rightarrow\ell^{+}\ell^{-}\right)  $, produces the lepton pair in either a
phase-space suppressed tensor state $2^{++}$ or a helicity-suppressed scalar
state $0^{++}$. The former is found negligible from experimental constraints
on $K_{L}\rightarrow\pi^{0}\left(  \gamma\gamma\right)  _{2^{++}}$%
\cite{BDI03}, while for the latter the amplitude reads:%
\begin{equation}
\mathcal{M}_{\mathrm{\gamma\gamma}}=\frac{G_{8}\alpha_{em}^{2}}{2\pi^{2}}%
M_{K}\frac{r_{\ell}}{z}\mathcal{F}_{\ell}\left(  z\right)  \left\{
\bar{u}_{p}v_{p^{\prime}}\right\}  \;, \label{Eq8b}%
\end{equation}
with $r_{\ell}=m_{\ell}/M_{K}$ and $\left|  G_{8}\right|  =9.1\cdot10^{-12}$
MeV$^{-2}$. It is dominated by the two-loop process $K_{L}\rightarrow\pi
^{0}\left(  P^{+}P^{-}\rightarrow\gamma^{\ast}\gamma^{\ast}\rightarrow\ell
^{+}\ell^{-}\right)  _{0^{++}}$ with $P=\pi,K$, computed in ChPT, and closely
related to $K_{S}\rightarrow\pi^{+}\pi^{-}\rightarrow\gamma^{\ast}\gamma
^{\ast}\rightarrow\ell^{+}\ell^{-}$. With the parametrization $\mathcal{M}%
\left(  K_{L}\rightarrow\pi^{0}P^{+}P^{-}\right)  \sim a_{1}^{P}\left(
z\right)  $ for the momentum distribution entering the subprocess $P^{+}%
P^{-}\rightarrow\gamma^{\ast}\gamma^{\ast}\rightarrow\ell^{+}\ell^{-}$, the
two-loop form-factor $\mathcal{F}_{\ell}\left(  z\right)  $ can be expressed
as ($r_{\pi}=M_{\pi}/M_{K}$)%
\begin{equation}
\mathcal{F}_{\ell}\left(  z\right)  =a_{1}^{\pi}\left(  z\right)
\mathcal{I}\left(  \frac{r_{\ell}^{2}}{z},\frac{r_{\pi}^{2}}{z}\right)
-a_{1}^{K}\left(  z\right)  \mathcal{I}\left(  \frac{r_{\ell}^{2}}{z}%
,\frac{1}{z}\right)  \;, \label{Eq9a}%
\end{equation}
with $\mathcal{I}\left(  a,b\right)  $ given in
Refs.\cite{EckerPich91,IsidoriSmithUnter} (for practical purposes, a numerical
representation is given in Appendix B).

A reliable estimation of $\Gamma\left(  K_{L}\rightarrow\pi^{0}\ell^{+}%
\ell^{-}\right)  _{\mathrm{\gamma\gamma}}$ can then be obtained from the
measured $K_{L}\rightarrow\pi^{0}\gamma\gamma$ rate\cite{ggrate} thanks to the
stability of the ratio%
\begin{equation}
R_{\mathrm{\gamma\gamma}}=\frac{\Gamma\left(  K_{L}\rightarrow\pi^{0}\ell
^{+}\ell^{-}\right)  _{\mathrm{\gamma\gamma}}}{\Gamma\left(  K_{L}%
\rightarrow\pi^{0}\gamma\gamma\right)  }\;, \label{Eq9}%
\end{equation}
with respect to changes in $a_{1}^{P}\left(  z\right)  $
\cite{IsidoriSmithUnter}. Note that some $\mathcal{O}\left(  p^{6}\right)  $
ChPT effects are thus included in the estimated $\Gamma\left(  K_{L}%
\rightarrow\pi^{0}\ell^{+}\ell^{-}\right)  _{\mathrm{\gamma\gamma}}$, most
notably those responsible for the large observed $K_{L}\rightarrow\pi
^{0}\gamma\gamma$ rate compared to the $\mathcal{O}\left(  p^{4}\right)  $
ChPT prediction.

However, theoretical control over $R_{\mathrm{\gamma\gamma}}$ only is not
sufficient to deal with New Physics interactions that produce the lepton pair
in a $0^{++}$ state (generating interference effects) or compute
forward-backward asymmetries. For this, we need to control $\mathcal{M}%
_{\mathrm{\gamma\gamma}}$ too. To fill this gap, the key is to use the
similarity of behavior of the $K_{L}\rightarrow\pi^{0}\gamma\gamma$ and
$K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ spectra at the origin of the
stability of $R_{\mathrm{\gamma\gamma}}$. First consider the fact that the
$K_{L}\rightarrow\pi^{0}\gamma\gamma$ normalized spectrum is rather
well-described with the parametrizations
\begin{subequations}
\label{Eq10}%
\begin{align}
\mathcal{O}\left(  p^{4}\right)  \;\text{ChPT}  &  :a_{1}^{\pi}\left(
z\right)  _{ChPT}=z-r_{\pi}^{2},\\
K_{L}\overset{}{\rightarrow}\pi^{0}\pi^{+}\pi^{-}\text{ Dalitz}  &
:a_{1}^{\pi}\left(  z\right)  _{Dalitz}=-0.46+2.44z-0.95z^{2}\,,
\end{align}
and $a_{1}^{K}\left(  z\right)  =z-1-r_{\pi}^{2}$ in both cases. For
$a_{1}^{\pi}\left(  z\right)  _{Dalitz}$, only the $z$-dependent part of the
$K_{L}\rightarrow\pi^{0}\pi^{+}\pi^{-}$ effective vertex is kept. Now, to
account for the rescaling of $\mathcal{B}\left(  K_{L}\rightarrow\pi^{0}%
\ell^{+}\ell^{-}\right)  _{\mathrm{\gamma\gamma}}$ by $\mathcal{B}\left(
K_{L}\rightarrow\pi^{0}\gamma\gamma\right)  ^{\exp}$, we rescale $a_{1}%
^{P}\left(  z\right)  _{ChPT}\rightarrow1.45a_{1}^{P}\left(  z\right)
_{ChPT}$ and $a_{1}^{P}\left(  z\right)  _{Dalitz}\rightarrow1.36a_{1}%
^{P}\left(  z\right)  _{Dalitz}$, such that the rate computed from
Eq.(\ref{Eq8b}) is kept frozen. Clearly, the theoretical control on the
resulting $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ differential rate is not
as good as on the total rate, but is nevertheless satisfactory. In practice,
the error inherent to the procedure can be probed by comparing the predictions
obtained using either $a_{1}^{\pi}\left(  z\right)  _{ChPT}$ or $a_{1}^{\pi
}\left(  z\right)  _{Dalitz}$.

Finally, as far as the sign of $\mathcal{M}_{\mathrm{\gamma\gamma}}$ is
concerned, we checked that Eq.(\ref{Eq8b}) is consistent with the conventions
used in the rest of the paper for hadronic matrix elements. Furthermore, under
the reasonable assumption that the sign of $G_{8}$ as fixed by the
factorization approximation is not changed by the non-perturbative evolution
down to $M_{K}$ (see for example Ref.\cite{GerardST}), our conventions
correspond to $G_{8}<0$.

\subsection{Vector and axial-vector short-distance contributions}

Let us now consider the DCPV piece induced by the vector and axial-vector
operators of Eq.(\ref{Eq2}):%
\end{subequations}
\begin{equation}
\mathcal{M}_{\mathrm{V,A}}=\langle\pi^{0}\ell^{+}\ell^{-}|-\mathcal{H}%
_{eff}^{\mathrm{V,A}}|K_{L}\rangle=iG_{F}\alpha\langle\pi^{0}|\bar{s}%
\gamma_{\mu}d|K^{0}\rangle\left\{  \bar{u}_{p}\gamma^{\mu}\left(
\operatorname{Im}\left(  \lambda_{t}y_{7V}\right)  +\operatorname{Im}\left(
\lambda_{t}y_{7A}\right)  \gamma_{5}\right)  v_{p^{\prime}}\right\}
\label{Eq3b}%
\end{equation}
(the sizeable $c$-quark contribution, known at NLO\cite{BuchallaBL96}, is
understood in $y_{7V}$), with the matrix element%
\begin{gather}
\langle\pi^{0}\left(  K\right)  |\bar{s}\gamma^{\mu}d|K^{0}\left(  P\right)
\rangle=\frac{1}{\sqrt{2}}\left(  \left(  P+K\right)  ^{\mu}f_{+}^{K^{0}%
\pi^{0}}\left(  z\right)  +\left(  P-K\right)  ^{\mu}f_{-}^{K^{0}\pi^{0}%
}\left(  z\right)  \right)  \;,\label{Eq4}\\
f_{-}^{K^{0}\pi^{0}}\left(  z\right)  =\frac{1-r_{\pi}^{2}}{z}\left(
f_{0}^{K^{0}\pi^{0}}\left(  z\right)  -f_{+}^{K^{0}\pi^{0}}\left(  z\right)
\right)  \;.\nonumber
\end{gather}
The slopes are extracted from $K_{\ell3}$ decays \cite{KL3,PDG}, neglecting
isospin breaking:%
\begin{equation}
f_{0,+}\left(  z\right)  \equiv f_{0,+}^{K^{0}\pi^{0}}\left(  z\right)
=\frac{f_{+}\left(  0\right)  }{1-\lambda_{0,+}z},\;\;\lambda_{0}=0.18\text{,
}\lambda_{+}=0.32\;, \label{Eq5}%
\end{equation}
in the pole parametrization. Accounting for isospin breaking in $\pi^{0}-\eta$
mixing for the value at zero momentum transfer\cite{MarcianoParsa}, one gets:%
\begin{equation}
f_{+}\left(  0\right)  =\left(  1.0231\right)  ^{-1}f_{+}^{K^{0}\pi^{+}%
}\left(  0\right)  \approx0.939\;, \label{Eq6}%
\end{equation}
with the Leutwyler-Ross prediction $f_{+}^{K^{0}\pi^{+}}\left(  0\right)
=0.961\left(  8\right)  $\cite{LeutwylerRoos}, confirmed by lattice
studies\cite{LatticeF0}. This quite precise value, together with the knowledge
of the form-factor slopes, renders the theoretical prediction of the vector
and axial-vector contributions to $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$
remarkably clean.

Inserting Eq.(\ref{Eq4}) in $\mathcal{M}_{\mathrm{V,A}}$, the vector current
is seen to produce the lepton pair in a vector state $1^{--}$, while the
axial-vector part produces it in an axial-vector $1^{++}$ or pseudoscalar
$0^{-+}$ state. This latter component is helicity suppressed, and enters thus
only the muon mode. Besides, the vector current and ICPV amplitudes produce
the same final state, they thus interfere in the rate. Recent theoretical
analyses point towards a constructive interference, i.e., $a_{S}<0$
\cite{BDI03,FGD04}.%

\begin{figure}
[t]
\begin{center}
\includegraphics[
height=2.808in,
width=3.8009in
]%
{Fig2.eps}%
\caption{Behavior of $\mathcal{B}\left(  K_{L}\rightarrow\pi^{0}\mu^{+}\mu
^{-}\right)  $ against $\mathcal{B}\left(  K_{L}\rightarrow\pi^{0}e^{+}%
e^{-}\right)  $, in units of $10^{-11}$, as $y_{7A,7V}$ are rescaled by a
common factor (dots), or allowed to take arbitrary values (red sector). The
ellipses denote $25,50,75\%$ confidence regions in the SM, assuming
constructive DCPV -- ICPV interference (destructive interference is around the
$-1$ dot), or for Ref.\cite{BFRS04}, in which $y_{7A}\approx-3.2$ and
$y_{7V}\approx0.9$.}%
\label{Fig2}%
\end{center}
\end{figure}

\paragraph{\textit{Total rates}:}

Altogether, the branching ratios are predicted to be%
\begin{gather}
\mathcal{B}_{\mathrm{V,A}}^{\ell^{+}\ell^{-}}=\left(  C_{dir}^{\ell}\pm
C_{int}^{\ell}\left|  a_{S}\right|  +C_{mix}^{\ell}\left|  a_{S}\right|
^{2}+C_{\gamma\gamma}^{\ell}\right)  \cdot10^{-12}\;,\smallskip\label{Eq11}\\%
\begin{tabular}
[c]{ll}%
$C_{dir}^{e}=\left(  4.62\pm0.24\right)  \;\left(  w_{7V}^{2}+w_{7A}%
^{2}\right)  ,$ & $C_{dir}^{\mu}=\left(  1.09\pm0.05\right)  \left(
w_{7V}^{2}+2.32w_{7A}^{2}\right)  ,$\\
$C_{int}^{e}=\left(  11.3\pm0.3\right)  \;w_{7V},$ & $C_{int}^{\mu}=\left(
2.63\pm0.06\right)  \;w_{7V},$\\
$C_{mix}^{e}=14.5\pm0.5,$ & $C_{mix}^{\mu}=3.36\pm0.20,$\\
$C_{\gamma\gamma}^{e}\approx0,$ & $C_{\gamma\gamma}^{\mu}=5.2\pm1.6,$%
\end{tabular}
\nonumber
\end{gather}
with $w_{7A,7V}=\operatorname{Im}\left(  \lambda_{t}y_{7A,7V}\right)
/\operatorname{Im}\lambda_{t}$. In the Standard Model, the coefficients
$y_{7A,7V}$ are real \cite{BuchallaBL96}:%
\begin{equation}
y_{7A}\left(  M_{W}\right)  =-0.68\pm0.03,\;\;y_{7V}\left(  \mu\approx
1\;\text{GeV}\right)  =0.73\pm0.04\;, \label{Eq3}%
\end{equation}
and, with $\operatorname{Im}\lambda_{t}=\left(  1.407\pm0.098\right)
\cdot10^{-4}$\cite{CKMfitter}, one gets%
\begin{equation}
\mathcal{B}_{\mathrm{SM}}^{e^{+}e^{-}}=3.54_{-0.85}^{+0.98}\;\left(
1.56_{-0.49}^{+0.62}\right)  \cdot10^{-11},\;\;\mathcal{B}_{\mathrm{SM}}%
^{\mu^{+}\mu^{-}}=1.41_{-0.26}^{+0.28}\;\left(  0.95_{-0.21}^{+0.22}\right)
\cdot10^{-11}\;, \label{Eq11b}%
\end{equation}
for constructive (destructive) interference. The present experimental bounds
are one order of magnitude above these predictions:%
\begin{equation}
\mathcal{B}_{\mathrm{\exp}}^{e^{+}e^{-}}<28\cdot10^{-11}\text{\cite{KTeVelec}%
},\;\;\mathcal{B}_{\mathrm{\exp}}^{\mu^{+}\mu^{-}}<38\cdot10^{-11}%
\text{\cite{KTeVmuon}}\;. \label{Eq11c}%
\end{equation}
The differential rate for the electronic mode is trivial (no CPC piece), while
for the muonic mode it can be found in Ref.\cite{IsidoriSmithUnter}.

The Standard Model confidence region on the $\mathcal{B}^{e^{+}e^{-}}%
$--$\mathcal{B}^{\mu^{+}\mu^{-}}$ plane is shown in Fig.\ref{Fig2}. As
discussed in Ref.\cite{IsidoriSmithUnter}, this plane is particularly
well-suited to search for New Physics signal, and identify its specific
nature. Indeed, the fact that helicity suppression is rather inefficient for
the muonic mode introduces a genuine difference of sensitivity to the
short-distance V,A currents, i.e. to $y_{7A,7V}$, for the two $K_{L}%
\rightarrow\pi^{0}\ell^{+}\ell^{-}$ modes. These two types of contributions
can thus be disentangled by measuring both rates. Suffices to note that the
coefficients approximately obey $C_{i}^{\mu}/C_{i}^{e}\approx0.23$,
$i=dir,int,mix$, which is simply the phase-space suppression, except for the
enhanced $y_{7A}$ contribution to $C_{dir}^{\mu}$, which comes from the
production of helicity-suppressed pseudoscalar $0^{-+}$ states. Without this,
no matter the New Physics contributions to $y_{7A,7V}$, the rates would always
fall on a trivial straight line. Thanks to this effect, on the contrary, for
arbitrary values of $y_{7A,7V}$, the two modes can lie anywhere inside the red
sector in Fig.\ref{Fig2} for the $C_{i}^{\ell}$ at their central values.
Accounting for theoretical errors at $1\sigma$, this area is mathematically
expressed as%
\begin{equation}
0.1\cdot10^{-11}+0.24\mathcal{B}_{\mathrm{V,A}}^{e^{+}e^{-}}<\mathcal{B}%
_{\mathrm{V,A}}^{\mu^{+}\mu^{-}}<0.6\cdot10^{-11}+0.58\mathcal{B}%
_{\mathrm{V,A}}^{e^{+}e^{-}}\;. \label{Eq11d}%
\end{equation}
For definiteness, we have also indicated the curve corresponding to a common
rescaling of $y_{7A}$ and $y_{7V}$ from New Physics, and drawn the confidence
region for the enhanced electroweak penguins of Ref.\cite{BFRS04}, to
illustrate the opposite situation in which $y_{7A}=-3.2$ is strongly enhanced,
while $y_{7V}=0.9$ stays roughly the same as in the SM. Finally, note that the
extent of the confidence regions essentially reflects the uncertainty on
$a_{S}$ (whose effect is included in the bounds Eq.(\ref{Eq11d})), and could
be reduced by more precise measurements of the $K_{S}\rightarrow\pi^{0}%
\ell^{+}\ell^{-}$ branching fractions.%

\begin{figure}
[t]
\begin{center}
\includegraphics[
height=1.8455in,
width=5.9205in
]%
{Fig3.eps}%
\caption{Left: $A_{FB}^{\mu}$ (in \%) as a function of $z$, in the SM. Right:
Integrated $A_{FB}^{\mu}$ (in \%) as a function of $w_{7V}$, with $w_{7A}$
fixed at its SM value. Red (blue) lines correspond to constructive
(destructive) interference between ICPV and vector current contributions,
while plain\ (dashed) lines correspond to $a_{1}^{P}\left(  z\right)
_{Dalitz}$ ($a_{1}^{P}\left(  z\right)  _{ChPT}$), respectively.}%
\label{Fig3}%
\end{center}
\end{figure}

\paragraph{\textit{Forward-backward asymmetry}:}

The differential forward-backward asymmetry\cite{2pp,AFB} is defined by%
\begin{equation}
A_{FB}^{\ell}\left(  z\right)  =\frac{%
{\displaystyle\int_{0}^{y_{0}}}
\dfrac{d^{2}\Gamma}{dydz}dy-%
{\displaystyle\int_{-y_{0}}^{0}}
\dfrac{d^{2}\Gamma}{dydz}dy}{%
{\displaystyle\int_{0}^{y_{0}}}
\dfrac{d^{2}\Gamma}{dydz}dy+%
{\displaystyle\int_{-y_{0}}^{0}}
\dfrac{d^{2}\Gamma}{dydz}dy}\;, \label{Eq12}%
\end{equation}
with%
\begin{equation}
y=\frac{P\cdot(p-p^{\prime})}{M_{K}^{2}},\;\;y_{0}=\frac{1}{2}\beta_{\ell
}\beta_{\pi}\;,
\end{equation}
and $\beta_{\ell}^{2}=1-4r_{\ell}^{2}/z$, $\beta_{\pi}^{2}=\lambda\left(
1,r_{\pi}^{2},z\right)  $, $\lambda\left(  a,b,c\right)  =a^{2}+b^{2}%
+c^{2}-2\left(  ab+bc+ac\right)  .$ The variable $y$ is related to the angle
between the $K$ and $\ell^{-}$ momenta in the dilepton rest-frame (hence the
name forward-backward), and also, by definition, to the energy difference
$E_{\ell^{-}}-E_{\ell^{+}}$ in the $K$ rest-frame (one then speaks of lepton
energy asymmetry).

This observable requires CP-violation, and arises from the $\operatorname{Re}%
[\mathcal{M}_{\mathrm{1}^{--}}^{\ast}\left(  \mathcal{M}_{\mathrm{0}^{++}%
}+\mathcal{M}_{\mathrm{2}^{++}}\right)  ]$ interference term. Let us start by
assuming that $\mathcal{M}_{\mathrm{2}^{++}}$ is negligible, as for the total
rates. Then:%
\begin{equation}
A_{FB}^{\ell}\left(  z\right)  =\frac{A_{1}^{\ell}\left(  z\right)  }%
{d\Gamma/dz}\operatorname{Re}\left(  \varepsilon^{\ast}W_{S}^{\ast}\left(
z\right)  \mathcal{F}_{\ell}\left(  z\right)  \right)  +\frac{A_{2}^{\ell
}\left(  z\right)  }{d\Gamma/dz}\operatorname{Im}\left(  \lambda_{t}%
y_{7V}\right)  \operatorname{Im}\mathcal{F}_{\ell}\left(  z\right)  \;,
\label{Eq12b}%
\end{equation}
with $A_{1,2}^{\ell}\left(  z\right)  $ some combinations of constants and
form-factors. The electronic asymmetry $A_{FB}^{e}\left(  z\right)  $ is
negligible in this case since $\mathcal{F}_{e}\left(  z\right)  $ is helicity
suppressed, while for the muon it is shown in Fig.\ref{Fig3} in the case of
the Standard Model. Note that the theoretical control gained over this
quantity would be difficult to improve since it relies on the specific
parametrization of $\mathcal{M}_{\mathrm{\gamma\gamma}}$. In addition, the
experimental sensitivity required to measure $A_{FB}^{\ell}\left(  z\right)  $
is unlikely to be achieved soon. We will therefore not consider $A_{FB}^{\ell
}\left(  z\right)  $ anymore, but rather concentrate on the integrated
asymmetry:%
\begin{equation}
A_{FB}^{\ell}=\frac{%
{\displaystyle\int_{4r_{\ell}^{2}}^{(1-r_{\pi})^{2}}}
dz\left(
{\displaystyle\int_{0}^{y_{0}}}
\dfrac{d^{2}\Gamma}{dydz}dy-%
{\displaystyle\int_{-y_{0}}^{0}}
\dfrac{d^{2}\Gamma}{dydz}dy\right)  }{%
{\displaystyle\int_{4r_{\ell}^{2}}^{(1-r_{\pi})^{2}}}
dz\left(
{\displaystyle\int_{0}^{y_{0}}}
\dfrac{d^{2}\Gamma}{dydz}dy+%
{\displaystyle\int_{-y_{0}}^{0}}
\dfrac{d^{2}\Gamma}{dydz}dy\right)  }=\frac{N\left(  E_{\ell^{-}}>E_{\ell^{+}%
}\right)  -N\left(  E_{\ell^{-}}<E_{\ell^{+}}\right)  }{N\left(  E_{\ell^{-}%
}>E_{\ell^{+}}\right)  +N\left(  E_{\ell^{-}}<E_{\ell^{+}}\right)  }\;.
\label{Eq12c}%
\end{equation}
Compared to $A_{FB}^{\mu}\left(  z\right)  $, it is more stable:%
\begin{equation}
A_{FB}^{\mu}=\left(  1.3\left(  1\right)  w_{7V}\pm1.7(2)\left|  a_{S}\right|
\right)  \cdot10^{-12}\,\text{\thinspace}/\,\mathcal{B}_{\mathrm{V,A}}%
^{\mu^{+}\mu^{-}}\;. \label{Eq13}%
\end{equation}
In the Standard Model, $A_{FB}^{\mu}=\left(  20\pm4\right)  \%$ for
constructive and $\left(  -12\pm4\right)  \%$ for destructive interference,
with the error coming from varying $a_{1}^{\pi}\left(  z\right)  $ between
$\mathcal{O}\left(  p^{4}\right)  $ ChPT and Dalitz accounting for $\pm2\%$.

For general axial currents, it is clear that $A_{FB}^{\mu}$ decreases when
$w_{7A}$ increases since it does not contribute to the interference
Eq.(\ref{Eq12b}). For $w_{7V}$, the interference is linear while the rate is
quadratic, and thus $|A_{FB}^{\mu}|$ reaches a maximum, around $23\%$ for
$w_{7A}$ at its SM value, before decreasing again, see Fig.\ref{Fig3}. The
absolute maximum for $A_{FB}^{\mu}$ is around $\pm25\%$ for $w_{7A}\approx0$
and $w_{7V}\approx\pm1$ or $\pm4$, depending on the direct - indirect CPV
interference sign.

In the SM, even if $A_{FB}^{\mu}$ is polluted by the theoretical error on the
two-photon amplitude, its measurement could fix the sign of $a_{S}$. As can be
seen in Fig.\ref{Fig3}, this remains true if New Physics is found from the
measurements of $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ total rates with
$|w_{7V}|\lesssim2$.

Let us now consider the interference term $\operatorname{Re}[\mathcal{M}%
_{\mathrm{1}^{--}}^{\ast}\mathcal{M}_{\mathrm{2}^{++}}]$. The amplitude
$\mathcal{M}_{\mathrm{2}^{++}}$ is discussed in detail in Ref.\cite{BDI03}. It
leads to a helicity-allowed, but phase-space suppressed contribution to
$A_{FB}^{\ell}$, hence contributes mostly for the electronic mode.
Unfortunately, the theoretical control on the $\mathcal{M}_{\mathrm{2}^{++}}$
amplitude is not good as it depends on unknown phenomenological parameters.
Though sufficient for deriving, from $K_{L}\rightarrow\pi^{0}\gamma\gamma$, a
tight upper bound on the $\mathrm{2}^{++}$ contribution to $C_{\gamma\gamma
}^{\ell}$ in Eq.(\ref{Eq11})\cite{BDI03}, $A_{FB}^{e}$ remains largely
unconstrained. Indeed, both $\mathcal{M}_{\mathrm{1}^{--}}^{\ast}$ and
$\mathcal{M}_{\mathrm{2}^{++}}$ contribute mostly at low $z$, and their
sizeable interference can generate $A_{FB}^{e}$ anywhere between $0\%$ and
about $\pm60\%$\cite{2pp}, depending on the phenomenological parameters of
Ref.\cite{BDI03}.

Concerning $A_{FB}^{\mu}$, the situation is better. The impact of
$\mathcal{M}_{\mathrm{2}^{++}}$ corresponds to an additional $\pm3\%$
uncertainty, and therefore does not affect the potential of $A_{FB}^{\mu}$ in
determining the sign of $a_{S}$.

\section{$K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ with generic new physics operators}

In addition to the modification of vector and axial-vector couplings
considered in the previous section, new four-fermion effective interactions
could be generated by the integration of New Physics heavy degrees of freedom.
The effective Hamiltonian comprising all the possible dimension-six
semi-leptonic four-fermion structures\cite{Michel} relevant for $K_{L}%
\rightarrow\pi^{0}\ell^{+}\ell^{-}$ (as well as quark bilinear electromagnetic
couplings) reads%
\begin{equation}
\mathcal{H}_{eff}=\mathcal{H}_{eff}^{\mathrm{V,A}}+\mathcal{H}_{eff}%
^{\mathrm{P,S}}+\mathcal{H}_{eff}^{\mathrm{T,\tilde{T}}}+\mathcal{H}%
_{eff}^{\mathrm{EMO}}\;, \label{NP0}%
\end{equation}
with $\mathcal{H}_{eff}^{\mathrm{V,A}}$ given in Eq.(\ref{Eq2}) and%
\begin{align}
\mathcal{H}_{eff}^{\mathrm{P,S}}  &  =\frac{G_{F}^{2}M_{W}^{2}}{\pi^{2}%
}\frac{m_{s}m_{\ell}}{M_{W}^{2}}\left[  y_{P}\left(  \bar{s}d\right)  \left(
\bar{\ell}\gamma_{5}\ell\right)  +y_{S}\left(  \bar{s}d\right)  \left(
\bar{\ell}\ell\right)  \right]  +h.c.\label{NP1}\\
\mathcal{H}_{eff}^{\mathrm{T,\tilde{T}}}  &  =\frac{G_{F}^{2}M_{W}^{2}}%
{\pi^{2}}\frac{m_{s}m_{\ell}}{M_{W}^{2}}\left[  y_{T}\left(  \bar{s}%
\sigma_{\mu\nu}d\right)  \left(  \bar{\ell}\sigma^{\mu\nu}\ell\right)
+y_{\tilde{T}}\left(  \bar{s}\sigma_{\mu\nu}d\right)  \left(  \bar{\ell}%
\sigma^{\mu\nu}\gamma_{5}\ell\right)  \right]  +h.c.\label{NP2}\\
\mathcal{H}_{eff}^{\mathrm{EMO}}  &  =\frac{G_{F}}{\sqrt{2}}M_{K}\frac{Q_{d}%
e}{16\pi^{2}}\left[  y_{\gamma}^{\pm}\left(  \bar{s}\sigma_{\mu\nu}\left(
1\pm\gamma_{5}\right)  d\right)  F^{\mu\nu}\right]  +h.c.\;\;. \label{NP3}%
\end{align}
A low scale ($\mu\lesssim m_{c}$) is understood for the evaluation of the
Wilson coefficients, quark masses and matrix elements of the operators in the
above equation. Dimension-eight operators, containing two powers of the
external momenta, are not considered as they are very small in the SM and are
expected to remain so in the presence of New Physics (see discussion in
\cite{BBKU02}).

Our goal is to analyze the impact of these new operators on the $K_{L}%
\rightarrow\pi^{0}\ell^{+}\ell^{-}$ branching fractions and asymmetries in a
model-independent way. Still, some comments on specific scenarios behind the
various operators are in order:

\begin{enumerate}
\item For the scalar and pseudoscalar operators $Q_{S}=\left(  \bar
{s}d\right)  \left(  \bar{\ell}\ell\right)  $ and $Q_{P}=\left(  \bar
{s}d\right)  \left(  \bar{\ell}\gamma_{5}\ell\right)  $, we have explicitly
included the $m_{s}m_{\ell}/M_{W}^{2}$ helicity suppression factor to give a
realistic description of models where these operators are generated from an
extended Higgs sector. For example, large $y_{P,S}$ can arise in the MSSM with
large $\tan\beta$ (see e.g. \cite{TanBeta,Foster05}) and sizeable trilinear
soft-breaking couplings. This kind of scenarios has been analyzed in many
works, but usually focuses on other decay modes. See e.g.
Refs.\cite{IsidoriRetico,BBKU02} for a MSSM analysis with emphasis on the
$K,B\rightarrow\ell^{+}\ell^{-}$ decays.

\item The tensor and pseudotensor operators $Q_{T}=\left(  \bar{s}\sigma
_{\mu\nu}d\right)  \left(  \bar{\ell}\sigma^{\mu\nu}\ell\right)  $ and
$Q_{\tilde{T}}=\left(  \bar{s}\sigma_{\mu\nu}d\right)  \left(  \bar{\ell
}\sigma^{\mu\nu}\gamma_{5}\ell\right)  =i\varepsilon^{\mu\nu\rho\sigma}\left(
\bar{s}\sigma_{\mu\nu}d\right)  \left(  \bar{\ell}\sigma_{\rho\sigma}%
\gamma_{5}\ell\right)  $, to our knowledge, have not been included in studies
of $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ so far. These modes are however
the most promising source of information on $Q_{T,\tilde{T}}$, since these
cannot contribute to $K\rightarrow\ell^{+}\ell^{-}$. Though they do not arise
in the SM, they do in the MSSM but, in addition to being helicity suppressed,
they are usually suppressed by loop factors\cite{BBKU02}. Similar operators
have been considered in $\Delta F=2$ processes (see e.g.
Refs.\cite{Ciuchini98,DF2}) and for the $B\rightarrow X\ell^{+}\ell^{-}$ rate
and asymmetries (see e.g. Ref.\cite{Btensor}).

\item For completeness, we have included the dimension-five electromagnetic
tensor operators $Q_{\gamma}^{\pm}=\left(  \bar{s}\sigma_{\mu\nu}\left(
1\pm\gamma_{5}\right)  d\right)  F^{\mu\nu}$. These were considered for
example in Refs.\cite{BurasCIRS99,DAmbrosioGao}. Since the $\sigma_{\mu\nu
}\gamma_{5}$ part does not contribute to $K_{L}\rightarrow\pi^{0}\ell^{+}%
\ell^{-}$, only $y_{\gamma}\equiv y_{\gamma}^{+}+y_{\gamma}^{-}$ is accessible
here. Note that in principle these operators already arise in the Standard
Model, however they are too small to affect $K_{L}\rightarrow\pi^{0}\ell
^{+}\ell^{-}$. In the MSSM, they are correlated with the chromomagnetic tensor
operators and thus strongly constrained by other
observables\cite{BurasCIRS99,DAmbrosioGao}.

\item In the last section, we consider the general framework in which neither
$Q_{S,P}$ nor $Q_{T,\tilde{T}}$ are helicity-suppressed, i.e. we remove the
$m_{s}m_{\ell}/M_{W}^{2}$ factors in Eqs.(\ref{NP1},\ref{NP2}). A large class
of models with such helicity-allowed FCNC operators are theories with
leptoquark interactions (for a review, see \cite{DavidsonBC94}), among which
specific GUT models. Alternatively, SUSY without R-parity can also induce
helicity-allowed $Q_{S,P}$ interactions through tree-level sneutrino exchanges
(see e.g. \cite{Rparity}).
\end{enumerate}

The distinction between helicity-suppressed and helicity-allowed scenarios is
necessary as the corresponding signatures, i.e. impacts on $K_{L}%
\rightarrow\pi^{0}e^{+}e^{-}$ and $K_{L}\rightarrow\pi^{0}\mu^{+}\mu^{-}$,
will obviously be very different. Let us now analyze these impacts systematically.

\subsection{Scalar and pseudoscalar operators}

The relevant matrix element reads%
\begin{equation}
\langle\pi^{0}|\bar{s}d|K^{0}\rangle=-\frac{M_{K}^{2}-M_{\pi}^{2}}{\sqrt
{2}\left(  m_{s}-m_{d}\right)  }f_{0}\left(  z\right)  \; \label{SP1}%
\end{equation}
in the sign convention of Eq.(\ref{Eq4}). This matrix element is enhanced
compared to its vector counterpart due to the large value of the quark
condensate (i.e., the large ratio of meson masses over quark masses).

The scalar (pseudoscalar) operator produces the lepton pair in a CP-even
$0^{++}$ (CP-odd $0^{-+}$) state, therefore it is the real (imaginary) part of
its Wilson coefficient that contributes to $K_{L}\rightarrow\pi^{0}\ell
^{+}\ell^{-}$:%
\begin{align}
\mathcal{M}_{\mathrm{P}}  &  =\frac{G_{F}^{2}M_{W}^{2}}{\pi^{2}}\frac{m_{\ell
}}{M_{W}^{2}}\left(  M_{K}^{2}-M_{\pi}^{2}\right)  \;i\operatorname{Im}%
y_{P}\;f_{0}\left(  z\right)  \;\left\{  \bar{u}_{p}\gamma_{5}v_{p^{\prime}%
}\right\}  \;,\label{SP2}\\
\mathcal{M}_{\mathrm{S}}  &  =\frac{G_{F}^{2}M_{W}^{2}}{\pi^{2}}\frac{m_{\ell
}}{M_{W}^{2}}\left(  M_{K}^{2}-M_{\pi}^{2}\right)  \;\operatorname{Re}%
y_{S}\;f_{0}\left(  z\right)  \;\left\{  \bar{u}_{p}v_{p^{\prime}}\right\}
\;. \label{SP3}%
\end{align}
To reach these expressions, $m_{d}$ has been neglected against $m_{s}$ in
Eq.(\ref{SP1}). The pseudoscalar current interferes with the
helicity-suppressed pseudoscalar part of the axial-vector current, while the
scalar current interferes with the helicity-suppressed two--photon 0$^{++}$
contribution. Since in addition $Q_{S}$ and $Q_{P}$ are themselves
helicity-suppressed, only the muon mode can be affected by $y_{S,P}$.

\paragraph{\textit{For the pseudoscalar operator,}}

including also the $\mathrm{V}$ and $\mathrm{A}$ contributions, the
differential rate reads:%
\begin{gather}
\frac{d\Gamma_{\mathrm{V,A\&P}}}{dz}=\frac{G_{F}^{2}M_{K}^{5}\alpha^{2}}%
{64\pi^{3}}\left(  \operatorname{Im}\lambda_{t}\right)  ^{2}\beta_{\ell}%
\beta_{\pi}\left(  \frac{\beta_{\pi}^{2}}{6}A_{+}\left(  f_{+}\left(
z\right)  \right)  ^{2}+A_{0}\left(  f_{0}\left(  z\right)  \right)
^{2}\right)  ,\label{SP4}\\
A_{+}=w_{7A}^{2}\beta_{\ell}^{2}+w_{7V}^{2}\frac{3-\beta_{\ell}^{2}}%
{2},\;\;A_{0}=\frac{r_{\ell}^{2}}{z}\left(  w_{7A}\left(  1-r_{\pi}%
^{2}\right)  +z\rho_{P}\operatorname{Im}y_{P}\right)  ^{2},\nonumber
\end{gather}
with the prefactor%
\begin{equation}
\rho_{P}=\frac{1}{\operatorname{Im}\lambda_{t}}\frac{1}{2\pi\sin^{2}\theta
_{W}}\frac{M_{K}^{2}-M_{\pi}^{2}}{M_{W}^{2}}\approx0.18\;. \label{SP5}%
\end{equation}
$\operatorname{Im}y_{P}\sim\mathcal{O}\left(  10\right)  $ is thus required to
get $\mathcal{O}\left(  1\right)  $ effects. Numerically, the contributions to
the total rates, to be added to Eq.(\ref{Eq11}), are
\begin{subequations}
\label{SP6}%
\begin{align}
\mathcal{B}_{\mathrm{P}}^{e^{+}e^{-}}  &  =\left(  1.9\,w_{7A}%
\,\operatorname{Im}y_{P}+0.038\left(  \operatorname{Im}y_{P}\right)
^{2}\right)  \cdot10^{-17}\;,\\
\mathcal{B}_{\mathrm{P}}^{\mu^{+}\mu^{-}}  &  =\left(  0.26\,w_{7A}%
\,\operatorname{Im}y_{P}+0.0085\left(  \operatorname{Im}y_{P}\right)
^{2}\right)  \cdot10^{-12}\;,
\end{align}
showing the very strong helicity suppression at play for the electron mode.

\paragraph{\textit{For the scalar operator,}}

from Eqs.(\ref{Eq8b}) and (\ref{SP3}), one immediately gets for the total
$0^{++}$ contribution:%
\end{subequations}
\begin{equation}
\frac{d\Gamma_{\mathrm{\gamma\gamma\&S}}}{dz}=\frac{G_{8}^{2}M_{K}^{5}%
\alpha_{em}^{4}}{512\pi^{7}}\beta_{\pi}\beta_{\ell}^{3}\frac{r_{\ell}^{2}}%
{z}\left|  \mathcal{F}_{\ell}\left(  z\right)  -\rho_{S}\operatorname{Re}%
y_{S}\,z\,f_{0}\left(  z\right)  \right|  ^{2}\;, \label{SP8}%
\end{equation}
with the suppression factor%
\begin{equation}
\rho_{S}=-\frac{\sqrt{2}\pi}{\sin^{2}\theta_{W}}\frac{\alpha G_{F}}%
{\alpha_{em}^{2}G_{8}}\frac{M_{K}^{2}-M_{\pi}^{2}}{M_{W}^{2}}\approx0.13\;.
\label{SP9}%
\end{equation}
Since $\mathcal{F}_{\mu}\left(  z\right)  \sim\mathcal{O}\left(  1\right)  $,
the helicity suppression $M_{W}^{-2}$ turns out to be nearly compensated.
Performing the $z$ integral, we find
\begin{subequations}
\label{SP10}%
\begin{align}
\mathcal{B}_{\mathrm{S}}^{e^{+}e^{-}}  &  =(1.5\left(  3\right)
\operatorname{Re}y_{S}+0.0039\left(  \operatorname{Re}y_{S}\right)  ^{2}%
)\cdot10^{-16}\;,\\
\mathcal{B}_{\mathrm{S}}^{\mu^{+}\mu^{-}}  &  =(0.04\left(  1\right)
\operatorname{Re}y_{S}+0.0041\left(  \operatorname{Re}y_{S}\right)  ^{2}%
)\cdot10^{-12}\;.
\end{align}
The error on the interference term is estimated by varying the distribution
$a_{1}^{\pi}\left(  z\right)  $ between $\mathcal{O}\left(  p^{4}\right)  $
ChPT and Dalitz (giving respectively $0.034$ and $0.053$ for the muonic mode).

\paragraph{\textit{Total rates:}}

The $Q_{S}$ and $Q_{P}$ operators do not affect the electronic mode due to
their strong helicity suppression. For the muonic mode, combining
Eqs.(\ref{SP6},\ref{SP10}) with Eq.(\ref{Eq11}) and fixing $w_{7A,7V}$ at
their SM values Eq.(\ref{Eq3}), the impacts on the total rate are summarized
in the first three columns of Table \ref{TableSP}.%

\begin{table}[t] \centering
$%
\begin{tabular}
[c]{cccccc}\hline
& Enhancement & Enhancement & Maximal & Bound from & Experimental\\
& of 50\% & of 100\% & suppression & $K_{L}\rightarrow\mu^{+}\mu^{-}$ & bound
(Eq.(\ref{Eq11c}))\\\hline
$\operatorname{Im}y_{P}$ & $-20,40$ & $-30,50$ & 7\% for $\operatorname{Im}%
y_{P}\approx10\;\,\,$ & $\left|  \operatorname{Im}y_{P}\right|  \lesssim8$ &
$|\operatorname{Im}y_{P}|\lesssim220$\\
$\operatorname{Re}y_{S}$ & $-45,35$ & $-65,55$ & 1\% for $\operatorname{Re}%
y_{S}\approx-5$ & $\left|  \operatorname{Re}y_{S}\right|  \lesssim50$ &
$|\operatorname{Re}y_{S}|\lesssim300$\\\hline
\end{tabular}
\ $%
\caption{Numerical analysis of scalar and pseudoscalar
operator impacts on $\mathcal{B}%
^{\mu^+\mu^-}$.
\label{TableSP}}
\end{table}%

A well-motivated scenario in which $\operatorname{Re}y_{S}$ and
$\operatorname{Im}y_{P}$ can be large is the MSSM for large values of
$\tan\beta$\cite{TanBeta,Foster05}. In that context, the contributions of
$Q_{S,P}$ are related to those of $Q_{P}^{\prime}=\left(  \bar{s}\gamma
_{5}d\right)  \left(  \bar{\ell}\gamma_{5}\ell\right)  $ and $Q_{S}^{\prime
}=(\bar{s}\gamma_{5}d)(\bar{\ell}\ell)$: $y_{S,P}=y_{P,S}^{\prime}$, with
$y_{P}^{\prime}$ and $y_{S}^{\prime}$ further correlated. The contributions of
$Q_{P,S}^{\prime}$ to $K_{L}\rightarrow\mu^{+}\mu^{-}$ were analyzed for
example in Ref.\cite{IsidoriRetico}, with the result that values of a few tens
for $y_{S,P}^{\prime}$ are compatible with $\Delta S=2$ and $B$-physics
data\footnote{Assuming a degenerate SUSY spectrum $M_{\tilde{d}}\sim\left|
\mu\right|  \sim M_{A}\sim M_{\tilde{g}}$, the gluino contributions to
$y_{S,P}$ scale as\cite{IsidoriRetico}%
\[
y_{S,P}\sim\left(  M_{W}^{2}/M_{A}^{2}\right)  \tan^{3}\beta\left(
1+0.01\tan\beta\operatorname{sign}\mu\right)  ^{-2}\left(  \left(  \delta
_{LL}^{D}\right)  _{12}+18\left(  \delta_{RR}^{D}\right)  _{13}\left(
\delta_{LL}^{D}\right)  _{32}\right)  \;.
\]
With $\tan\beta\sim50$, $\mu>0$ (as favored from the muon $g-2$) and $M_{A}$
in the range $300-500$ GeV\cite{Foster05}, one gets $y_{S,P}\sim
\mathcal{O}\left(  15\right)  $ with the single mass-insertion $(\delta
_{LL(RR)}^{D})_{12}\sim10^{-2}$ (compatible with $\varepsilon_{K}$ constraints
\cite{Ciuchini98}) and $y_{S,P}\sim\mathcal{O}\left(  30\right)  $ with the
double one $\left(  \delta_{RR}^{D}\right)  _{13}\left(  \delta_{LL}%
^{D}\right)  _{32}\sim10^{-3}$ (constrained by $\Delta M_{s,d}$
\cite{Foster05}).}.

Without restricting ourselves to the MSSM, we can investigate the constraints
on $y_{S,P}$ derived from the experimental $K_{L}\rightarrow\mu^{+}\mu^{-}$
rate under the assumption that $y_{S,P}=y_{P,S}^{\prime}$ approximately holds.
This analysis is presented in Appendix A, leading to $\left|
\operatorname{Im}y_{S}^{\prime}\right|  \lesssim8$ and $\left|
\operatorname{Re}y_{P}^{\prime}\right|  \lesssim35$ for $y_{7A}^{\prime
}=y_{7A}=-0.68$ (for the $Q_{7A}^{\prime}=\left(  \bar{s}\gamma_{\mu}%
\gamma_{5}d\right)  \left(  \bar{\ell}\gamma^{\mu}\gamma_{5}\ell\right)  $
operator). Allowing for New Physics in the axial-vector current with the bound
$\left|  y_{7A}^{\prime}\right|  \lesssim3$ corresponds to $\left|
\operatorname{Re}y_{P}^{\prime}\right|  \lesssim70$, a much larger range since
these two contributions interfere in the rate.

Our numerical analysis can be summarized drawing the allowed regions on the
$\mathcal{B}^{e^{+}e^{-}}$--$\mathcal{B}^{\mu^{+}\mu^{-}}$ plane
(Fig.\ref{Fig4}). Compared to the region spanned for general values of
$y_{7A,7V}$ (Fig.\ref{Fig2}), turning on $y_{S,P}$ basically extends the
vertical spread since only the muon mode is affected, and this mostly in the
upward direction (i.e., enhancements). This is illustrated in Fig.\ref{Fig4}
by the light-blue/dark-blue region, corresponding to $|\operatorname{Re}%
y_{S}|<90$ / $|\operatorname{Im}y_{P}|<35$, respectively. Imposing further the
constraints from $K_{L}\rightarrow\mu^{+}\mu^{-}$ under the assumption
$y_{7A,S,P}=y_{7A,P,S}^{\prime}$ gives the yellow region. Obviously, the
sensitivity of $K_{L}\rightarrow\pi^{0}\mu^{+}\mu^{-}$ to $Q_{S,P}$ is quite good.

It should also be noted that $Q_{S,P}$ contribute mostly for large $z$. While,
as explained in Ref.\cite{IsidoriSmithUnter}, introducing a cut off at
$z\approx4r_{\ell}^{2}$ can reduce the contribution of the two-photon
amplitude with respect to the $Q_{7A,7V}$ ones, thereby reducing the
theoretical error, such a procedure would also reduce the sensitivity to
$\operatorname{Im}y_{P}$ and $\operatorname{Re}y_{S}$ significantly, and may
thus not be desirable.%

\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.8228in,
width=6.4835in
]%
{Fig4.eps}%
\caption{Left: Impacts of scalar and pseudoscalar operators in the
$\mathcal{B}^{e^{+}e^{-}}-\mathcal{B}^{\mu^{+}\mu^{-}}$ plane of
Fig.\ref{Fig2}. Light blue (dark blue) corresponds to arbitrary $y_{7A,7V}$
together with $\operatorname{Re}y_{S}<90$ ( $\operatorname{Im}y_{P}<35$),
resp., while the yellow region corresponds to $y_{7V,7A,S,P}$ arbitrary but
compatible with $\mathcal{B}\left(  K_{L}\rightarrow\mu^{+}\mu^{-}\right)
^{\exp}$ (see text). The dashed, light-blue line indicates the lower extent of
the corresponding region. Right: The asymmetry $A_{FB}^{\mu}$ as a function of
$w_{7V}$ and $\operatorname{Re}y_{S}$, assuming constructive interference
($a_{S}<0$), $y_{7A}=-0.68$ and $\operatorname{Im}y_{P}=0$.}%
\label{Fig4}%
\end{center}
\end{figure}

\paragraph{\textit{Forward-backward asymmetry}:}

$\operatorname{Re}y_{S}$ enters the numerator of $A_{FB}^{\mu}$ through the
interference term $\mathcal{M}_{\mathrm{ICPV}}^{\ast}\mathcal{M}_{\mathrm{S}}$
only:
\end{subequations}
\begin{equation}
A_{FB}^{\mu}=(1.3\left(  1\right)  w_{7V}\pm1.7(2)\left|  a_{S}\right|
\mp0.057\left|  a_{S}\right|  \operatorname{Re}y_{S})\cdot10^{-12}%
\,\text{\thinspace}/\,\mathcal{B}_{\mathrm{V,A,S,P}}^{\mu^{+}\mu^{-}}\;.
\label{SP11}%
\end{equation}
There is no interference between $\mathcal{M}_{\mathrm{S}}$ and $\mathcal{M}%
_{\mathrm{V}}$ because of their 90$%
{{}^\circ}%
$ relative phase. For this reason, $A_{FB}^{\mu}$ goes to zero when $y_{7V}$
and/or $\operatorname{Re}y_{S}$ becomes large, and reaches its maximum for
moderate values, as shown in Fig.\ref{Fig4} for $y_{7A}=-0.68$ and
$\operatorname{Im}y_{P}=0$. If these latter two values are enhanced, since
they contribute only to $\mathcal{B}_{\mathrm{V,A,S,P}}^{\mu^{+}\mu^{-}}$,
$A_{FB}^{\mu}$ decreases, i.e. the figure remains the same but the absolute
size of $A_{FB}^{\mu}$ is reduced. Finally, the figure for destructive
interference is readily obtained by performing a vertical axis reflexion
($w_{7V}\rightarrow-w_{7V}$) followed by an overall sign change for
$A_{FB}^{\mu}$.

No matter the New Physics behind $y_{7A,7V,S,P}$, $|A_{FB}^{\mu}|$ is always
smaller than $30\%$, i.e., not very far from its SM value Eq.(\ref{Eq13}).
Given the theoretical errors, $A_{FB}^{\mu}$ does not appear very promising to
get a clear signal of New Physics. Nevertheless, as said before, it offers a
very interesting possibility of constraining the relative signs of
$\operatorname{Re}y_{S}$, $y_{7V}$ and $a_{S}$ when considered in conjunction
with the $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ total rates.

\subsection{Tensor and pseudo-tensor operators}

Let us now turn to the tensor operators of Eqs.(\ref{NP2}) and (\ref{NP3}).
The relevant matrix element assumes the form%
\begin{equation}
\langle\pi^{0}\left(  K\right)  |\bar{s}\sigma^{\mu\nu}d|K^{0}\left(
P\right)  \rangle=i\frac{P^{\mu}K^{\nu}-P^{\nu}K^{\mu}}{\sqrt{2}M_{K}}%
B_{T}\left(  z\right)  , \label{NPT1}%
\end{equation}
with $\langle\pi^{0}|\bar{s}\sigma^{\mu\nu}\gamma_{5}d|K^{0}\rangle$ obtained
through $\sigma^{\mu\nu}\gamma_{5}=i\varepsilon^{\mu\nu\rho\sigma}\sigma
_{\rho\sigma}$. The tensor form-factor was studied on the lattice
\cite{Mescia00}, with the result $B_{T}\left(  z\right)  \approx
1.2f_{+}\left(  0\right)  /(1-0.29z)$ at $\mu\simeq m_{c}$ in the
$\overline{MS}$ scheme (an earlier order-of-magnitude estimate may be found in
Ref.\cite{OldBT}).

\paragraph{\textit{The electromagnetic tensor operator}}

produces the lepton pair in a $1^{--}$ state and the transition is
CP-violating:
\begin{equation}
\mathcal{M}_{\mathrm{EMO}}=i\frac{G_{F}\alpha}{\sqrt{2}}\frac{Q_{d}}{2\pi
}\operatorname{Im}y_{\gamma}B_{T}\left(  z\right)  \left\{  \bar{u}_{p}%
\!\not\!%
Pv_{p^{\prime}}\right\}  \;. \label{NPT2}%
\end{equation}
Let us take $\lambda_{T}=\lambda_{+}$, which is good enough for our purpose,
and can be justified in a pole model through the fact that $z<<M_{T,V}%
^{2}/M_{K}^{2}$ with $M_{T,V}$ the nearest vector and tensor resonances. The
effect of $Q_{\gamma}^{\pm}$ can then be absorbed into the vector-current
Wilson coefficient $w_{7V}$\cite{BurasCIRS99}%
\begin{equation}
w_{7V}\rightarrow w_{7V}^{\prime}=w_{7V}+\frac{\operatorname{Im}y_{\gamma}%
}{\operatorname{Im}\lambda_{t}}\frac{Q_{d}}{4\pi}\frac{B_{T}\left(  0\right)
}{f_{+}\left(  0\right)  }\;. \label{NPT3}%
\end{equation}
The above redefinition is independent of the lepton flavor, hence the
$K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ modes cannot disentangle possible
New Physics effects arising from $Q_{\gamma}^{\pm}$ from those arising in the
vector current electroweak penguins and boxes. Such New Physics effects were
analyzed in Section 2, see Fig.\ref{Fig2}.

\paragraph{\textit{The tensor operator}}

also induces a CP-violating contribution:%
\begin{equation}
\mathcal{M}_{\mathrm{T}}=i\frac{G_{F}\alpha}{\sqrt{2}}\operatorname{Im}%
\lambda_{t}B_{T}\left(  z\right)  r_{\ell}\rho_{T}\operatorname{Im}%
y_{T}\left\{  \bar{u}_{p}\left(  2r_{\ell}%
\!\not\!%
P-\frac{P\cdot(p-p^{\prime})}{M_{K}}\right)  v_{p^{\prime}}\right\}  \;,
\label{NPT4}%
\end{equation}
where we have defined%
\begin{equation}
\rho_{T}=\frac{1}{\operatorname{Im}\lambda_{t}}\frac{2}{\pi\sin^{2}\theta_{W}%
}\frac{m_{s}M_{K}}{M_{W}^{2}}\approx\frac{1}{4} \label{NPT5}%
\end{equation}
for $m_{s}\approx150$ MeV. As for the magnetic operator, the $%
\!\not\!%
P$ part can be absorbed into $w_{7V}$, but now the $m_{\ell}$ dependence
introduces an effective breaking of $\mu-e$ universality in the vector
current:%
\begin{equation}
w_{7V}\rightarrow w_{7V}^{\ell}=w_{7V}+\rho_{T}\operatorname{Im}y_{T}r_{\ell
}^{2}\frac{B_{T}\left(  0\right)  }{f_{+}\left(  0\right)  }\;. \label{NPT6}%
\end{equation}
The second term in Eq.(\ref{NPT4}) is also CP-violating because $P\cdot
(p-p^{\prime})=yM_{K}^{2}$ is CP-odd. It produces the lepton pair again in a
$1^{--}$ state and can thus interfere with both the ICPV and vector operator
contributions, producing an extra contribution to the $A_{+}$ factor defined
in Eq.(\ref{SP4}):%
\begin{equation}
\left(  A_{+}\right)  _{\mathrm{T}}=r_{\ell}^{2}\frac{B_{T}\left(  z\right)
^{2}}{f_{+}\left(  z\right)  ^{2}}\left(  \rho_{T}\operatorname{Im}%
y_{T}\right)  ^{2}\frac{z\beta_{\ell}^{4}}{8}+r_{\ell}^{2}\beta_{\ell}%
^{2}\frac{B_{T}\left(  z\right)  }{f_{+}\left(  z\right)  }\rho_{T}%
\operatorname{Im}y_{T}\left(  w_{7V}^{\ell}-\frac{\alpha_{em}\operatorname{Im}%
\left(  \varepsilon W_{S}\left(  z\right)  \right)  }{\sqrt{8}\pi\alpha
f_{+}\left(  z\right)  \operatorname{Im}\lambda_{t}}\right)  \,. \label{NPT7}%
\end{equation}
In the muonic case, the overall $\beta_{\mu}^{5}$ factor for the first term,
corresponding to an orbital angular momentum of two between the muons ($y$
counts as one unit of angular momentum), makes this contribution significantly
phase-space suppressed compared to the one shifting $w_{7V}$, Eq.(\ref{NPT6}).
Numerically,
\begin{subequations}
\label{NPT8}%
\begin{align}
\mathcal{B}_{\mathrm{T}}^{e^{+}e^{-}}  &  =\left(  \left(  10^{-5}%
+0.08\right)  \left(  \operatorname{Im}y_{T}\right)  ^{2}+\left(  \left(
29+15\right)  w_{7V}\pm\left(  36+18\right)  \left|  a_{S}\right|  \right)
\operatorname{Im}y_{T}\right)  \cdot10^{-19}\;,\\
\mathcal{B}_{\mathrm{T}}^{\mu^{+}\mu^{-}}  &  =\left(  \left(
0.25+0.02\right)  \left(  \operatorname{Im}y_{T}\right)  ^{2}+\left(  \left(
29+4\right)  w_{7V}\pm\left(  36+5\right)  \left|  a_{S}\right|  \right)
\operatorname{Im}y_{T}\right)  \cdot10^{-15}\;,
\end{align}
where the first numbers in each parenthesis come from Eq.(\ref{NPT6}), the
second from Eq.(\ref{NPT7}), and the $\pm$ sign corresponds to $a_{S}%
=\mp\left|  a_{S}\right|  $.

\paragraph{\textit{The pseudotensor operator}}

produces the lepton pair in a $1^{+-}$ state, and is thus CP-conserving%
\end{subequations}
\begin{equation}
\mathcal{M}_{\mathrm{\tilde{T}}}=-\frac{G_{F}\alpha}{\sqrt{2}}%
\operatorname{Im}\lambda_{t}B_{T}\left(  z\right)  r_{\ell}\rho_{T}%
\operatorname{Re}y_{\tilde{T}}\frac{P\cdot(p-p^{\prime})}{M_{K}}\left\{
\bar{u}_{p}\gamma_{5}v_{p^{\prime}}\right\}  \;. \label{NPT10}%
\end{equation}
As none of the other CP-conserving contributions produces such a final state,
it represents a distinct contribution to the rate:%
\begin{equation}
\frac{d\Gamma_{\mathrm{\tilde{T}}}}{dz}=\frac{G_{F}^{2}M_{K}^{5}\alpha^{2}%
}{3072\pi^{3}}\left(  \operatorname{Im}\lambda_{t}\right)  ^{2}\beta_{\ell
}^{3}\beta_{\pi}^{3}r_{\ell}^{2}\left(  \rho_{T}\operatorname{Re}y_{\tilde{T}%
}\right)  ^{2}\left(  B_{T}\left(  z\right)  \right)  ^{2}z\;. \label{NPT11}%
\end{equation}
Numerically, this contribution is phase-space suppressed and very small:
\begin{subequations}
\label{NPT12}%
\begin{align}
\mathcal{B}_{\mathrm{\tilde{T}}}^{e^{+}e^{-}}  &  =7.9\left(
\operatorname{Re}y_{\tilde{T}}\right)  ^{2}\cdot10^{-21}\;,\\
\mathcal{B}_{\mathrm{\tilde{T}}}^{\mu^{+}\mu^{-}}  &  =4.9\left(
\operatorname{Re}y_{\tilde{T}}\right)  ^{2}\cdot10^{-17}\;.
\end{align}

\paragraph{\textit{Total rate and forward-backward asymmetry: }}

Overall, the effects of the $Q_{T,\tilde{T}}$ operators are smaller than those
of $Q_{P,S}$ because of the smaller matrix elements and the phase-space
suppression. In addition, $y_{T,\tilde{T}}<y_{S,P}$ in realistic scenarios,
and tensor operators seem beyond reach. The only exception is the effective
breaking of $\mu-e$ universality in the vector current induced by
$\operatorname{Im}y_{T}$, which slightly extends downwards the ''V,A
only''\ region of Fig.\ref{Fig2} when $\left|  \operatorname{Im}y_{T}\right|
\gtrsim25$.

For $A_{FB}^{\mu}$, $\operatorname{Im}y_{T}$ enters through interference with
$\operatorname{Im}\mathcal{F}_{\mu}\left(  z\right)  $ (not with
$\operatorname{Re}y_{S}$ since they are out of phase by 90$%
{{}^\circ}%
$):%
\end{subequations}
\begin{equation}
A_{FB}^{\mu}=(1.3\left(  1\right)  w_{7V}\pm1.7(2)\left|  a_{S}\right|
\mp0.057\left|  a_{S}\right|  \operatorname{Re}y_{S}+0.033\left(  4\right)
\operatorname{Im}y_{T})\cdot10^{-12}\,\text{\thinspace}/\,\mathcal{B}%
_{\mathrm{V,A,S,P,T,\tilde{T}}}^{\mu^{+}\mu^{-}}. \label{NPT13}%
\end{equation}
$Q_{\tilde{T}}$ does not contribute directly to $A_{FB}^{\mu}$ as it leads to
a real amplitude, out of phase from the $Q_{7A}$ one by 90$%
{{}^\circ}%
$. Therefore, assuming $\operatorname{Im}y_{T}<\operatorname{Re}y_{S}$, no
impact can arise for $A_{FB}^{\mu}$. Similarly, the electronic asymmetry is
not affected, even taking into account interferences with the $\mathcal{M}%
_{\mathrm{2}^{++}}$ piece.

\subsection{Helicity-allowed effective interactions}

We now lift the constraint of helicity suppression in Eqs.(\ref{NP1}) and
(\ref{NP2}), symbolically as:%
\begin{equation}
\frac{G_{F}^{2}M_{W}^{2}}{\pi^{2}}\frac{m_{s}m_{\ell}}{M_{W}^{2}}%
y_{i}\rightarrow\frac{g_{NP}^{2}}{\Lambda_{i}^{2}}\;,\;i=S,P,T,\tilde{T}\;.
\label{EI1}%
\end{equation}
More precisely, all former expressions remain valid provided one makes the
substitutions:
\begin{equation}
\operatorname{Re}y_{S,\tilde{T}},\operatorname{Im}y_{P,T}\rightarrow32\pi
^{2}\times\frac{g_{NP}^{2}}{g^{4}}\times\frac{M_{W}^{2}}{\Lambda_{i}^{2}%
}\times\frac{M_{W}^{2}}{m_{s}m_{\ell}}\rightarrow\left\{
\begin{array}
[c]{l}%
\ell=e:1.7\ 10^{4}/\bar{\Lambda}_{i}^{2}\\
\ell=\mu:83/\bar{\Lambda}_{i}^{2}%
\end{array}
\right.  \;, \label{EI2}%
\end{equation}
where the normalization $\bar{\Lambda}_{i}=(\Lambda_{i}/100\,$TeV$)$
corresponds to typical lower bounds obtained from lepton-number violating
processes (see e.g. \cite{LFV}), and we have assumed $g_{NP}/g^{2}%
\sim\mathcal{O}\left(  1\right)  $. Plugging these expressions in
Eqs.(\ref{SP6},\ref{SP10},\ref{NPT8},\ref{NPT12}), we obtain:
\begin{subequations}
\label{EI3}%
\begin{align}
\mathcal{B}_{\mathrm{S,P,T,\tilde{T}}}^{e^{+}e^{-}}  &  =\left(
\frac{115}{\bar{\Lambda}_{S}^{4}}+\frac{2.6}{\bar{\Lambda}_{S}^{2}}%
+\frac{112}{\bar{\Lambda}_{P}^{4}}+\frac{0.3w_{7A}}{\bar{\Lambda}_{P}^{2}%
}+\frac{2.3}{\bar{\Lambda}_{T}^{4}}+\frac{0.07w_{7V}}{\bar{\Lambda}_{T}^{2}%
}\pm\frac{0.09\left|  a_{S}\right|  }{\bar{\Lambda}_{T}^{2}}+\frac{2.3}%
{\bar{\Lambda}_{\tilde{T}}^{4}}\right)  \cdot10^{-12}\;,\\
\mathcal{B}_{\mathrm{S,P,T,\tilde{T}}}^{\mu^{+}\mu^{-}}  &  =\left(
\frac{29}{\bar{\Lambda}_{S}^{4}}+\frac{4.5}{\bar{\Lambda}_{S}^{2}}%
+\frac{60}{\bar{\Lambda}_{P}^{4}}+\frac{22w_{7A}}{\bar{\Lambda}_{P}^{2}%
}+\frac{1.9}{\bar{\Lambda}_{T}^{4}}+\frac{2.8w_{7V}}{\bar{\Lambda}_{T}^{2}}%
\pm\frac{3.4\left|  a_{S}\right|  }{\bar{\Lambda}_{T}^{2}}+\frac{0.34}%
{\bar{\Lambda}_{\tilde{T}}^{4}}\right)  \cdot10^{-12}\;.
\end{align}
Without helicity-suppression, it is the electronic mode that is the most
sensitive to these operators, simply because of the phase-space suppression in
the muonic mode. Therefore, allowing for these interactions typically produces
total rates in the lower part of the $\mathcal{B}^{e^{+}e^{-}}$--$\mathcal{B}%
^{\mu^{+}\mu^{-}}$ plane, a region which cannot be attained by the New Physics
interactions studied up to now (see Figs.\ref{Fig2} and \ref{Fig4}). Let us
further investigate what signals could be expected.

\paragraph{\textit{Scalar/pseudoscalar operators}:}

As before, assuming that the Wilson coefficients for the scalar and
pseudoscalar operators contributing to $K_{L}\rightarrow\ell^{+}\ell^{-}$ and
$K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ have approximately the same values,
$\bar{\Lambda}_{S^{\prime},P^{\prime}}\approx\bar{\Lambda}_{P,S}$, the
constraints from $K_{L}\rightarrow\mu^{+}\mu^{-}$ still leave the possibility
of sizeable effects. However, now that helicity suppression is no longer
effective, one gets a very tough constraint from $K_{L}\rightarrow e^{+}e^{-}%
$. From the measurement $\mathcal{B}\left(  K_{L}\rightarrow e^{+}%
e^{-}\right)  ^{\exp}=9_{-4}^{+6}\cdot10^{-12}$\cite{PDG}, and since%
\end{subequations}
\begin{equation}
\mathcal{B}\left(  K_{L}\rightarrow e^{+}e^{-}\right)  _{\mathrm{S,P}}%
\approx6.8\cdot10^{-8}\left(  \frac{1}{\bar{\Lambda}_{S^{\prime}}^{4}%
}+\frac{1}{\bar{\Lambda}_{P^{\prime}}^{4}}\right)  \;, \label{EI4}%
\end{equation}
one gets $\bar{\Lambda}_{S^{\prime}},\bar{\Lambda}_{P^{\prime}}\gtrsim8$. For
such large values, the effect on both $\mathcal{B}^{\ell^{+}\ell^{-}}$ is of a
few percents for $y_{7A}$ at its SM value, hence well beyond reach. Though
$\bar{\Lambda}_{S^{\prime},P^{\prime}}$ and $\bar{\Lambda}_{P,S}$ can be
different in specific models, the large difference needed to get observable
effects on $\mathcal{B}^{\ell^{+}\ell^{-}}$ would require a somewhat
fine-tuned scenario. We can therefore reasonably rule out this possibility.

\paragraph{\textit{Tensor/pseudotensor operators}:}

To get a somehow realistic estimate, we bound them from $K^{+}\rightarrow
\pi^{+}\nu\bar{\nu}$ assuming that the operators $(\bar{s}\sigma^{\mu\nu
}d)(\bar{\nu}\sigma_{\mu\nu}\nu)$ and $(\bar{s}\sigma^{\mu\nu}d)(\bar{\nu
}\sigma_{\mu\nu}\gamma_{5}\nu)$ are governed by the same scale factors
$(g_{NP}/\Lambda_{T,\tilde{T}})^{2}$. Their contribution reads%
\begin{equation}
\mathcal{B}\left(  K^{+}\rightarrow\pi^{+}\nu\bar{\nu}\right)
_{\mathrm{T,\tilde{T}}}\approx4.4\cdot10^{-12}\left(  \frac{1}{\bar{\Lambda
}_{T}^{4}}+\frac{1}{\bar{\Lambda}_{\tilde{T}}^{4}}\right)  \;, \label{EI5}%
\end{equation}
and comparing with Eq.(\ref{EI3}) shows that the sensitivities of
$K\rightarrow\pi\nu\bar{\nu}$ and $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ to
tensor interactions are similar.

Let us assume that $\bar{\Lambda}_{T}\approx\bar{\Lambda}_{\tilde{T}}$, and
given the SM prediction of $\left(  8.0\pm1.1\right)  \cdot10^{-11}%
$\cite{Buras05} and the current measurement of $14.7_{-8.9}^{+13.0}%
\cdot10^{-11}$\cite{E787E949} (V,A currents do not interfere with tensor
operators for massless neutrinos), one finds $\bar{\Lambda}_{T,\tilde{T}%
}\gtrsim0.35$. Interestingly, for such small values, the charged lepton modes
are quite large, $\mathcal{B}\left(  K_{L}\rightarrow\pi^{0}e^{+}e^{-}\right)
\approx3\cdot10^{-10}$ and $\mathcal{B}\left(  K_{L}\rightarrow\pi^{0}\mu
^{+}\mu^{-}\right)  \approx2\cdot10^{-10}$, i.e., around their current upper
limits Eq.(\ref{Eq11c}). The conclusion of this order-of-magnitude estimate is
thus that there is still room for large effects from tensor operators. The
corresponding allowed region in the $\mathcal{B}^{e^{+}e^{-}}$--$\mathcal{B}%
^{\mu^{+}\mu^{-}}$ plane is shown in Fig.\ref{Fig5}.%

\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.3166in,
width=3.4013in
]%
{Fig5.eps}%
\caption{Impact of helicity-allowed tensor and pseudotensor operators in the
$\mathcal{B}^{e^{+}e^{-}}$ -- $\mathcal{B}^{\mu^{+}\mu^{-}}$ plane of
Fig.\ref{Fig4}. The green area (up to the green dashed line) corresponds to
arbitrary $y_{7A,7V}$ together with $\Lambda_{T,\tilde{T}}$ compatible with
$\mathcal{B}\left(  K^{+}\rightarrow\pi^{+}\nu\bar{\nu}\right)  $, as
explained in the text.}%
\label{Fig5}%
\end{center}
\end{figure}

\paragraph{\textit{The forward-backward asymmetry}:}

The muonic asymmetry $A_{FB}^{\mu}$ can be enhanced only by $Q_{S}$ and
$Q_{T}$ contributions, as explained in earlier sections, and this through
interferences with $\mathcal{M}_{\mathrm{ICPV}}^{\ast}$ and $\mathcal{M}%
_{\mathrm{\gamma\gamma-0}^{++}}$, respectively, see Eq.(\ref{NPT13}). Tuning
$y_{7V}$ and the helicity-allowed $1/\bar{\Lambda}_{T}^{2}$ term, a maximum of
about $60\%$ can be reached. Still, this requires large contributions from
$Q_{T}$ with both $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ rates above
$10^{-10}$. With $\bar{\Lambda}_{T}\gtrsim1$, this maximum falls to about
$30\%$, i.e. close to the SM value Eq.(\ref{Eq13}).

For the electronic mode, one could think that a small $\bar{\Lambda}_{S}$
could generate a significant asymmetry through the helicity-allowed
interference with $\mathcal{M}_{\mathrm{ICPV}}^{\ast}$. However, this is not
the case because the impact of $Q_{S}$ on $\mathcal{B}^{e^{+}e^{-}}$ is then
much more pronounced. Tensor interactions, for their parts, have a small
impact on $A_{FB}^{e}$ because interference effects with $\mathcal{M}%
_{\mathrm{2}^{++}}$ are helicity-suppressed.

\section{Summary and Conclusion}

The $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ modes offer a unique opportunity
to probe a large range of New Physics $\Delta S=1$ effective operators, see
Table \ref{Table1}. They are therefore an essential tool in the investigation
of flavor structures beyond the Standard Model. Our work was to analyze, in a
model-independent way, the possible experimental signatures of these New
Physics interactions.%

\begin{table}[t] \centering
$%
\begin{tabular}
[c]{lcccc}\hline
& $%
\begin{array}
[c]{c}%
\text{Short-distance}\\
\text{operator}%
\end{array}
$ & $%
\begin{array}
[c]{c}%
\text{CP-property}\\
\text{\& }J^{PC}\left(  \bar{\ell}\ell\right)
\end{array}
$ & $%
\begin{array}
[c]{c}%
\text{Helicity-}\\
\text{suppressed}%
\end{array}
$ & $%
\begin{array}
[c]{c}%
\text{Helicity-}\\
\text{allowed}%
\end{array}
$\\\hline
ICPV ($K^{0}$-$\bar{K}^{0}$) & -- & CPV ($1^{--}$) &
\multicolumn{2}{c}{Eq.(\ref{Eq11})}\\
Two-photon & -- & CPC ($0^{++}$) & \multicolumn{2}{c}{Eq.(\ref{Eq11})}\\\hline
Vector & $(\bar{s}\gamma_{\mu}d)(\bar{\ell}\gamma^{\mu}\ell)$ & CPV ($1^{--}%
$) & \multicolumn{2}{c}{Eq.(\ref{Eq11})}\\
Axial-vector & $(\bar{s}\gamma_{\mu}d)(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell)$
& CPV ($0^{-+},1^{++}$) & \multicolumn{2}{c}{Eq.(\ref{Eq11})}\\\hline
Pseudoscalar & $\left(  \bar{s}d\right)  \left(  \bar{\ell}\gamma_{5}%
\ell\right)  $ & CPV ($0^{-+}$) & Eq.(\ref{SP6}) & Eq.(\ref{EI3})\\
Scalar & $\left(  \bar{s}d\right)  \left(  \bar{\ell}\ell\right)  $ & CPC
($0^{++}$) & Eq.(\ref{SP10}) & Eq.(\ref{EI3})\\
Tensor & $\left(  \bar{s}\sigma_{\mu\nu}d\right)  \left(  \bar{\ell}%
\sigma^{\mu\nu}\ell\right)  $ & CPV ($1^{--}$) & Eq.(\ref{NPT8}) &
Eq.(\ref{EI3})\\
Pseudotensor & $\left(  \bar{s}\sigma_{\mu\nu}d\right)  \left(  \bar{\ell
}\sigma^{\mu\nu}\gamma_{5}\ell\right)  $ & CPC ($1^{+-}$) & Eq.(\ref{NPT12}) &
Eq.(\ref{EI3})\\\hline
\end{tabular}
$%
\caption{Summary of the contributions entering the $K_L\rightarrow\pi^0\ell^+\ell^-$ rate,
and references to the relevant formulas in the text. The CP-property indicates which of the real or imaginary part of
the respective Wilson coefficient contributes. Interferences occur whenever the $\bar{\ell}%
\ell$
pair is produced in the same state. `Helicity-suppressed/allowed' refers to the last four operators.
\label{Table1}%
}
\end{table}%

In the presence of vector and axial-vector interactions only, the two rates
are bounded in the $\mathcal{B}^{e^{+}e^{-}}$--$\mathcal{B}^{\mu^{+}\mu^{-}}$
plane (see Fig.\ref{Fig5}), i.e., at $1\sigma$,
\begin{equation}
0.1\cdot10^{-11}+0.24\mathcal{B}_{\mathrm{V,A}}^{e^{+}e^{-}}<\mathcal{B}%
_{\mathrm{V,A}}^{\mu^{+}\mu^{-}}<0.6\cdot10^{-11}+0.58\mathcal{B}%
_{\mathrm{V,A}}^{e^{+}e^{-}}\;.
\end{equation}
Any signal outside this region is an indication of New Physics FCNC operators
of different structures (baring the possibility of large $\mu-e$ universality
breaking in the V,A currents). We have identified two possible mechanisms.

The first is from helicity-suppressed (pseudo-)scalar operators, as arising in
the MSSM at large $\tan\beta$. These enhance the muonic mode without affecting
the electronic mode. Such interactions would be revealed by measuring the two
rates above the V,A region. Also, comparing with $K_{L}\rightarrow\mu^{+}%
\mu^{-}$ shows, model-independently, that the $K_{L}\rightarrow\pi^{0}\mu
^{+}\mu^{-}$ is more sensitive (besides being cleaner) to these types of
interactions (see the yellow region in Fig.\ref{Fig5}).

The second possibility is from helicity-allowed (pseudo-)tensor interactions,
which could arise for example from tree-level leptoquark interactions. Because
of the phase-space suppression, it is now the electronic mode which is more
affected. These interactions would manifest themselves in a signal
significantly below the V,A region (see the green region in Fig.\ref{Fig5}).
Even assuming the presence of similar contributions for the $K\rightarrow
\pi\nu\bar{\nu}$ modes, there is at present no severe constraint on these effects.

On the other hand, we found that both helicity-suppressed (pseudo-)tensor
interactions and helicity-allowed (pseudo-)scalar interactions should not lead
to observable effects. For the former, this is so because of significant
phase-space suppression, while the latter are already strongly bounded by the
very rare $K_{L}\rightarrow e^{+}e^{-}$ decay. Also, concerning the
electromagnetic tensor operator, $\left(  \bar{s}\sigma_{\mu\nu}d\right)
F^{\mu\nu}$, measurements of the $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$
rates cannot disentangle it from vector operator contributions.

For the (integrated) forward-backward asymmetry $A_{FB}^{\mu}$, the first
reliable estimate has been obtained. It is typically of a few tens of
percents, and can give important information, complementary to the total
rates. In the SM, it could fix the sign of the interference between the vector
operator and ICPV contributions. Similarly, beyond the SM, it can be used to
discriminate among various solutions once both $K_{L}\rightarrow\pi^{0}%
\ell^{+}\ell^{-}$ rates are measured. On the other hand, the electronic
asymmetry $A_{FB}^{e}$ is found either completely dominated by its (unknown)
SM value, or too small to be of any use to constrain either New Physics or
$a_{S}$.

We have not included differential rates or differential asymmetries in the
present study (though they can be trivially computed from our analyses). New
physics does affect these observables, but they require a higher experimental
sensitivity, so total rates and integrated asymmetries are more promising.

With the general expressions for both $K_{L}\rightarrow\pi^{0}\ell^{+}\ell
^{-}$ rates and asymmetries computed in Sections 2 and 3, the way is now paved
for more model-dependent analyses. In this context, the $\mathcal{B}%
^{e^{+}e^{-}}$--$\mathcal{B}^{\mu^{+}\mu^{-}}$ plane remains as a particularly
convenient phenomenological tool to display the correlations among operators
specific to a given model.

In conclusion, the $K_{L}\rightarrow\pi^{0}\ell^{+}\ell^{-}$ system, together
with the neutrino modes $K\rightarrow\pi\nu\bar{\nu}$, has a considerable
potential for unveiling/constraining the nature of possible New Physics in
$\Delta S=1$ FCNC, therefore playing an important role in the quest for a
better understanding of the quark flavor sector.

\subsection*{Acknowledgements}

We are pleased to thank Gino Isidori for stimulating discussions and for
reading the manuscript, and Paride Paradisi for useful comments. This work is
partially supported by IHP-RTN, EC contract No.\ HPRN-CT-2002-00311
(EURIDICE). The work of C.S. is also supported by the Schweizerischer
Nationalfonds; S.T.~acknowledges the support of the DFG grant No.~NI 1105/1-1.

redpajama_set_name: RedPajamaArXiv


Every once in awhile, the Disney fan in me meets up with a real life Disney star. During the summer, I go to a few car events in my area. A couple of weeks ago, I came across a beautiful 1951 Hudson Hornet. Yes, like many cars in the Disney animated movie Cars, "Doc" Hudson … Continue reading "Meet the Real Doc Hudson from the Disney Movie, Cars"
New artwork was released last week for Mickey and Minnie's Runaway Railway, opening in 2019 in Disney's Hollywood Studios' Chinese Theatre. The first ride-through attraction themed to Mickey Mouse and Minnie Mouse will feature a new original story and lovable theme song as it takes Disney's Hollywood Studios guests on a journey inside the wacky and unpredictable world of Disney Channel's Emmy Award-winning Mickey Mouse cartoon shorts. New technologies will transform the two-dimensional cartoon world into an astounding multi-dimensional experience.


The story of how we are dealing with climate change has moved on quite a way since sustainability organisations sprang up over ten years ago in Redland, Bishopston and Westbury-on-Trym.
The story of how we are dealing with climate change has moved on quite a way since sustainability organisations sprang up over ten years ago in Redland, Bishopston and Westbury-on-Trym. Their overarching themes were to learn about the issues and bring them to the attention of their communities through different actions. Both Redland and Westbury-on-Trym raised awareness of the need for locally produced food through starting Farmer's Markets, and all three organised public talks and events about things like transport, recycling, energy, housing, self sufficiency and the importance of community, from which we still benefit.
Most people in our areas are now aware of the language of climate change. Thanks to help from David Attenborough our communities know how bad the plastic marine pollution we have caused is, use their own bags for shopping, and pressure supermarkets to stop using it. The waste recycling system we have did not exist ten years ago, we have a better understanding of the need for and benefits of slower driving, using public transport, walking or cycling, and reclaiming the streets has become a widely understood concept. We know about the need for water conservation, and city wide we have the Bristol Food Network, the Energy Cooperative and the Bristol Pound, all of which are designed to make our communities more resilient, self sufficient, and less energy intense.
With that achieved in ten years, what might we do in the next ten? Looking through the lenses of food, transport, energy and education the choice is almost unlimited. How can we cut food miles to 90% of 2005 through eating more locally grown food and growing our own? How can we seriously reduce traffic congestion and make our air clean and safe? How can we make it commonly accepted practice that living more sustainably with less energy enhances, not diminishes, the quality of our lives. Then it would only be a small step to understanding things like the health benefits of not using cars all the time, the disadvantages of supermarkets and the advantages of shopping diversity and growing food in public and private spaces. How can we build on what we have done so far to reclaim our streets, build a greater sense of community, and work cooperatively to make our building stock less energy intensive? What technologies do we need to develop and bygone skills resurrect? What is the role of education?
If you want to become active in using a skill you have to help us live more sustainably, why not use the structures of Sustainable Redland, Bishopston or Westbury-on-Trym to get going? The experience and contacts of those organisations could push you a long way on your journey, and all three have good websites to start you off.


Micro-Cap 11 is an integrated schematic editor and mixed analog/digital simulator that provides an interactive sketch and simulate environment for electronics engineers. Since its original release in 1982, Micro-Cap has been steadily expanded and improved. Micro-Cap 11, the eleventh generation, blends a modern, intuitive interface with robust numerical algorithms to produce unparalleled levels of simulation power and ease of use. Nothing else comes close.
Algorithmic improvements, optimized code, and an integrated, seamless, analog/digital simulation interface contribute to the stunning speed of Micro-Cap 11.
Numerous features contribute to Micro-Cap 11's power.
• Integrated schematic editor and simulator.
• Integral circuit optimizer with multiple optimization methods.
• Device library with over 32,000 parts.
• Smith charts / Polar plots.
• Extensive mathematical operators and variables.
The graphical, user-friendly interface is simple to learn and use. Familiar SPICE models, plus extensions, are easy to apply. Over 500 warnings and messages help you through problems, when the error occurs, not later in a text file. Micro-Cap 11 plots waveforms during the run, not after the run, when it is too late to change critical values.
You can easily spend twice the cost of Micro-Cap 11 for other simulators without matching its power, speed, and ease of use.
Micro-Cap 11 is fully guaranteed. Our full, 30 day money-back guarantee makes it easy to try.


Taking this since I'm familiar with the topic.
According to https://bugzilla.mozilla.org/show_bug.cgi?id=977588#c118 the mergeable/switchable flags from TelephonyCall should be used instead and these checks removed.
I can't test this patch since I don't have a CDMA device available and the emulator doesn't support it yet; but if the telephony code is doing its job properly on real devices this should work.
I've removed tests that didn't make sense anymore and updated the remaining ones.
M4, could you please test the patch provided by Gabriele.
See my comments on the PR. Would also like to get the results of partner testing before we land this.
Addressed the review comments. I've moved around some tests and added more to better capture the changes I made. I hope this is good enough; I have to ask you to land this for me once it's r+'d and handle the upflit too.
Same code as attachment 8585765 [details] [review], just squashed.
Needinfo to request uplift. We should wait for M4's test results first, though.
Doug, please uplift this patch to 2.2.
[Bug caused by] (feature/regressing bug #): Incomplete functionality from bug 977588.
[User impact] if declined: CDMA users will be unable to merge conference calls into 3-way calls.
[Testing completed]: m4 tried this. See comment 13.
[Risk to taking this patch] (and alternatives if risky): Medium as we can't test this ourselves since it's CDMA-only, and it's a fairly big patch, but there's no real alternative.
Bhavana, please get this uplifted to 2.2.


text:
Juan L. Mercado
Beggars By Choice
by Juan L. Mercado June 24, 2014
written by Juan L. Mercado June 24, 2014
Remember the ancient folk tale of "Juan Tamad"? Today, it is the motif of local government units (LGUs) that prefer the begging bowl to self-reliance.
The Local Government Code (LGC) empowered cities and towns to generate their own income two decades back. No, thank you, many LGUs responded. They opted to panhandle internal revenue allotments (IRA) instead, the Department of Finance notes in the latest issue of "Tax Watch."
Seven out of every eight provinces collect less than 15 centavos of every peso they spend. They patch that gap by siphoning from IRAs. As a result, local-government spending is hobbled.
Instead of collecting taxes, towns and provinces scramble for IRA slabs from a stretched national treasury, Prof. Benjamin Diokno of the UP School of Economics noted earlier. Their delivery of service has stagnated. So has their governance.
Diokno ticked off these points at an earlier Institute of Public Economics and Regulation forum. The LGC sought to spur autonomy by decentralization. Instead, it spawned fiscal beggars. "LGUs no longer seek other sources of funds." There—that's the Juan Tamad syndrome.
Back in 1919, an unknown author printed the folk tale into a book titled: "Buhay na Pinagdaanan ni Juan Tamad."
The 1957 compilation of "Philippine Tales and Fables" by Manuel E. Arguilla included the story of Juan Tamad with illustrations by Romeo V. Tabuena. A 1965 edition was illustrated with woodcuts by the late J. Elizalde Navarro, posthumously named National Artist for the Visual Arts. The out-of-print book is a collector's item today.
It spins the tales of an indolent Juan Tamad who bucks climbing a guava tree. Instead, he lies down below the fruits, waiting with open mouth for gravity to work. Buy mud crabs at the market, his mother ordered. Juan Tamad frees them in a ditch, instructing them to go ahead while he snoozed.
Land or real property taxes (RPT) are the latest focus of the finance department's Tax Watch campaign. Obsolete valuations, in 62 of 80 provinces, curry favor with land-owning elites but cripple delivery of services—from health to education.
Given the festering insurgency and bandit gangs, Sulu province is the tailender, Tax Watch reports. Jolo's local income made up a tenth of a percent of its total 2012 regular income. The rest were national doles. In contrast, the best performer was Aklan. Almost half (45.3 percent) was gleaned from local taxes and other forms of revenue.
The Juan Tamad syndrome spurs many LGUs to dodge the requirement to update taxes on real property valuation every three years. Of the 80 provinces more than half (50) dither on long-overdue mandatory chore.Maguindanao, for instance, still uses 1985 as base year. Bureau of Local Government Finance data pinpoints "deadbeats" deploying obsolete base years: Basilan (1992), Marinduque (1993), Pampanga (1994), Camarines Sur (1997), Negros Oriental (1998), Guimaras (1999), Tarlac (1999), and Zambales (1999).
In contrast, 30 provinces were "generally compliant" with the required update. But 12 of these did not use the latest base valuation year.
The 18 that used the most recent base valuations were: Agusan del Sur, Camarines Norte, Capiz, Catanduanes, Cebu, Compostela Valley, Davao del Sur, Kalinga, Laguna, Lanao del Norte, Negros Occidental, Northern Samar, Nueva Ecija, Oriental Mindoro, Siquijor, Sorsogon, South Cotabato and Surigao del Norte.
The tailenders among the "compliant" provinces were: Abra, Agusan del Norte, Benguet, Cavite, Iloilo, Mountain Province, North Cotabato, Nueva Vizcaya, Pangasinan, Rizal, Romblon and Tawi-Tawi.
The IRA share of the LGUs is 40 centavos out of every peso of national internal revenue taxes. The municipalities get the biggest IRA slab: 34 percent. The provinces and cities bag 23 percent. The barangays are at the end of the begging-bowl queue: 20 percent. In 2012, the LGUs got a hefty P273 billion in IRA. Diokno said this spurs the stampede to convert ill-prepared towns into cities. Cities receive larger IRA chunks because of populations swollen by the flood of migrants and their sprawling land areas.
There were 60 cities in 1960. By 2014, their numbers had more than doubled to 144, many of them infected by the Juan Tamad virus, as seen in local officials' "sense of entitlement" to doles.
Take the 20-percent Local Development Fund (LDF). It had been crafted after the 1972 UN Environment Conference in Stockholm. Delegates from 113 countries, including the Philippines, adopted an action plan that proposed a "20-20 Pact." Governments earmark 20 percent of resources for the poorest. The fund would address the needs of the most deprived: nutrition, healthcare, medicine, potable water, sanitation, primary schooling, etc.
Instead, the LDF is the "most abused" budget item, Interior Secretary Jesse Robredo told local officials at a Cebu conference, shortly before his death in a plane crash. Six out of every 10 LGUs flunk the "full disclosure" criterion on tax spending in both the appropriations law and local government code, the Magsaysay awardee said.
He issued guidelines in 2011 that directed: no underwriting of "salaries, wages or overtime pay"; no cash gifts, bonuses, food allowances, medical assistance, uniforms, no junkets or lakbay aral out-of-town jaunts; no dipping into the fund for "registration fees in training, seminars, conferences or conventions."
That policy remains in force today. But, the policy is honored more in the breach than in practice. The begging bowl syndrome persists.
(E-mail: juanlmercado@gmail.com)
Juan L. Mercado is a columnist for Philippine Daily Inquirer and Cebu Daily News. He writes a Sunday column for Sun-Star Cebu and its syndicated members, Bohol Chronicle and other community papers. He started his career in journalism in early 1950s and served as correspondent for foreign publications including London's Financial Times and Honolulu-Star Bulletin. Mercado became the Philippine Press Institute's founding-director. He also edited DEPTHNews, published by the Magsaysay Award-winning Press Foundation of Asia. Following the "People Power Uprising," and after 19 years of United Nations service, Mercado returned to the Philippines to resume journalism work. He was cited for excellence in opinion writing by Society of Publishers in Asia. In 2005 and 2007, he received the "best columnist award" from the Catholic Archdiocesan Commission for Mass Media.
Ancient Curse
Lawmaker Faults U.S. Army At Congressional Hearing On FilVets
Marcos Guerilla Claims 'Fraudulent' And 'Absurd'
Hanging Up A Star
Replay Buttons
An Asian Francis?
Bottom-line Question: 'Will it last?'
Buzzwords Rehashed
Tightening Water Spigots
Writing '-30-'
Nitwits' Ride

redpajama_set_name: RedPajamaCommonCrawl


After a nine-month slowdown, home sales are picking up again, according to NEFAR data released last week. The May 2018 Housing Stats Jacksonville FL saw a 6.1 percent drop in the number of homes sold compared to May 2017, but the 2,748 homes sold this past month was the largest number of sales since August. Pending sales were up 10.8 percent compared to a year ago.
About every other indicator shows that the housing market is continuing its hot streak. Try and find a house in a great location that does not have multiple offers or a cash buyer. Its a great market for anyone in real estate.
On average, homes in northeast Florida are spending 60 days on the market until they're sold, a number that drops to 53 days in Duval County. In May 0f 2017, homes in northeast Florida were selling in an average of 73 days. Long-term data shows that homes are selling faster than they have since 2006.
Just like last year at this time, prospective home buyers should expect a competitive housing market for the next several months. With payrolls trending upward and unemployment trending downward month after month in an extensive string of positive economic news, demand remains quite strong.
The supply of inventory, a measure of how many months it would take for the current number of homes on the market to sell, was at 3.4 months in the May 2018 Housing Stats Jacksonville FL. This despite 3,797 new listings in May, homes are selling almost as fast as they reach the local markets. A balanced housing market has between five and six months of supply. The sellers' market is being felt more acutely in Duval County, which had 2.8 months of supply in May.
May's median sales price of $225,000, a more than $100,000 increase since 2012, increased 5.8 percent increase since last year. Likewise, the housing affordability index dropped by 11.7 percent since last year. Look at this data as compared to last month here.
Apparently not. And with the Federal Reserve on track to raise rates another 2 more times this year, it is not slowing down people looking for a home. Inventory levels decreased 9.1 percent to 8,398 units in the May 2018 Housing Stats Jacksonville FL. Although home sales may actually drop in year-over-year comparisons over the next few months, that has more to do with low inventory than a lack of buyer interest.
As lower days on market and higher prices persist year after year, one might rationally expect a change in the outlook for residential real estate, yet the current situation has proven to be remarkably sustainable likely due to stronger fundamentals in home loan approvals than were in place a decade ago.
A great local employment track record. New technology jobs in Jacksonville for the Financial and Mortgage Industries. Jacksonville appears poised to lead the state in growth and in new jobs. Now is a great time to be in Jacksonville.
As I close in on $100 million sold in real estate during my career, I can tell you not all Realtors are alike. Look at my listing here, I have worked my tail off to get to list properties like this one. You need to have great people behind you and a great team in the office. You need technology, a great website and you need to be responsive. I have been in real estate as a Realtor since 2005 and I can say that I want to be YOUR representative in the largest transaction you may have in your lifetime. Call me today to discuss how I can change your life, (904) 210-6399. Please let me know you saw my May 2018 Housing Stats Jacksonville FL blog post. Would you like to know the value of your home?


text:
Drug Action Core
Pharmaceutical Sciences Division
Drug Action focuses on areas related to pharmacology, toxicology, cellular differentiation, development, and disease. Interests include the impact of drugs and toxins on biological systems, mechanisms of normal biology, and mechanisms of disease. These are studied at the cellular, genetic, molecular, and biochemical levels using diverse model systems.
Faculty research interests include:
Use of inductively heated, antibody targeted, core-shell nanoparticles for selective removal of targeted cells in vitro and in vivo. Development of high resolution nanoparticle labeling systems for correlative microscopy (Albrecht)
Intercellular communication: uncovering mechanisms that coordinate the development of multicellular organisms (Bashirullah)
Genetic and pharmacologic approaches to study disease development and treatment, focusing on cancer and central nervous system diseases (Collier)
Understanding the transcriptional networks that control keratinocyte proliferation/differentiation and epidermal barrier functions during development and under pathological conditions; developing novel therapeutic strategies for the treatment of inflammatory skin diseases and skin cancer (Dai)
Bioactivation and detoxification of drugs, industrial chemicals, and environmental toxicants; biochemical basis for target organ selectivity; biomarkers of toxicant exposure (Elfarra)
Signal transduction, transcriptional control of neuroprotective genes and neurotoxicity in Parkinson's, Alzheimer's, Huntington's and Neuromuscular disease (Johnson)
Analytical neurochemistry; neuropeptides; proteomics and peptidomics; glycomics and glycoproteomics; biomarker discovery in neurodegenerative diseases; quantitative system biology; metabolomics; microseparations; imaging mass spectrometry and its application to drug delivery and biodistribution; biological mass spectrometry (Li)
Molecular basis of prostate development, prostate cancer progression, and benign prostatic hyperplasia: roles of cell-cell signaling pathways and the use of mouse genetics to discover novel pathways that underlie prostatic diseases (Marker)
CRISPR-based genetic screens in pathogenic bacteria to understand gene function and antibiotic mode of action (Peters)
Understanding the molecular mechanisms involved with hormone therapy in the prevention and treatment of urologic cancers and benign diseases. Focus areas include: translational research, steroids and small molecules, stromal-epithelial interactions, endocrine disrupting chemicals, mouse models of disease progression (Ricke)
Preclinical model systems to investigate the role of the growth hormone/IGF-I axis in prostate carcinogenesis (Swanson)
Development and function of the blood-brain barrier; regulation of major histocompatibilty complex; modulation of neuroinflammation (Taylor)
Pharmacogenomics of xenobiotic toxicity, including both therapeutic drugs and environmental carcinogens. Mechanisms of familial and acquired risk for sulfamethoxazole drug hypersensitivity ("sulfa allergy"). Genetic variability in phase II detoxification pathways (especially GSTs and cytochrome b5 reductase) and cancer risk both in humans and dogs. (Trepanier)
Molecular basis of prostate and urinary tract development, physiology, and toxicology (Vezina)
Extensive communication occurs between the Discovery, Action, and Delivery Cores because of the central importance of drug recognition processes.
In some measure the Action Core draws its identity from the traditional fields of pharmacology, toxicology, cell biology, and genetics. Research in the Drug Action Core is funded among others by the NIH, NSF, the ALS Society of America, the Packard Center for ALS Research at Johns Hopkins, the Hereditary Disease Foundation, and SEA Grant.
Drug Action Faculty
Ralph Albrecht, Arash Bashirullah, Lara Collier, Jun Dai, Adnan Elfarra, Jeffrey Johnson, Lingjun Li, Paul Marker, Jason Peters, William Ricke, Steven Swanson, Michael Taylor, Lauren Trepanier, and Chad Vezina
This page: https://pharmacy.wisc.edu/about-us/divisions/pharmaceutical-sciences/drug-action-core/


With the advent of huge amounts of data now available, thanks to online interactions and shopping, digital marketing has become hugely reliant on this data-type of information for decision-making. The data has many sources like a company's CRM or ERP system or its website and social media presence.
Data-driven marketing is now the framework that is guiding the development of more targeted campaigns, that are linked to what a company has learned about its customer base and audience, in the hopes that more leads and deals will be achieved. This data-driven marketing is also telling CMOs about how to build better customer relationships and enhance the service experience level.
By all accounts, using this information appears to be working quite well as long as marketers understand how to separate the data and pull out the data that can provide them with the deepest insights. However, maybe you are not yet quite sure how to use this customer data to acquire and retain more customers.
Start with how to measure the ROI of your marketing efforts. Make sure the right type of data is available for use including data from channels like mobile, social media, email, web, SMS and print, and is comparable across those channels (such as using custom links, landing pages, etc). Having this consistency will provide a clear picture of patterns or insights in the data rather than trying to find answers in data that has been collected for different reasons. When we started our invoicing company we saw that there were patterns but didn't correlate them to data. Now we've matured and we correlate our data and study it. Had we not been tracking and more importantly paying attention, we wouldn't have seen ROI coming from conferences, something we spend a lot of time marketing.
Determine the campaign's objective as part of a full evaluation plan. In order to truly get results from this data, you need to create campaigns around a specific objective. Do you want to raise brand awareness, retain more customers, or increase revenue? Each one of these goals has needs for different data to shape the campaign and check the campaign's success once it has finished. Each campaign objective also has a different ROI timeframe, so be sure to accommodate that in the measurements you are using with the data from each campaign.
Continually assess what you are doing with data-driven marketing to determine where improvements can be made. One of the best things about data-driven marketing is it's constantly giving you a performance appraisal on what's working and what needs to change. This means you will see if a channel, message, or call-to-action should be altered. The changes can then be justified because the data is the proof in the pudding.
Increase investment once you have the right formula. Once you know your formula is working, then you want to invest more into the campaign to expand it for greater results and a higher rate of ROI over the life of the campaigns associated with this marketing mix. This also can tell you it may be worth it to try other platforms so you can start deploying marketing campaigns for a wider reach.
Ensure the departments work together to maximize the success of data-driven marketing. No matter how beneficial data-driven marketing is you won't really tap into its potential until you get all aspects of your company to work together, bringing in data from marketing, customer service, financial systems, CRM and more. Having the ability to create a single view that is developed between the departments can further drive the benefits from data-driven marketing because succinct strategic decisions can be made on the type of customer to go after. With all departments working together, there will be a consistency such that the customer experiences pay off in even bigger dividends. This can be done by having an organization-wide data strategy in place so the data can be shared across the departments, by reorganizing functions and units to make collaboration easier, and by creating a cross-functional team to further support collaboration.
With a consistent and collaborative approach to what you are implementing with data-driven marketing, you will gain a better understanding of what your audience and customers think, want, and do in relation to the various ways you are communicating with them across platforms. In turn, this will improve your campaigns and help you get closer to succeeding at every marketing tactic you are implementing and achieve a greater ROI in the process.
With the benefits of data-driven marketing at the forefront of your mind, don't forget you'll have to introduce it into your data management strategy. There is help. Download the ebook Maximize Your Marketing today.


Film Review - Barbecue - Does My Bomb Look Big In This?
Casula Powerhouse is really onto a winner with their food'n'film offerings, popping up regularly in collaboration with an array of different film festivals. I visited earlier this month, during the 2017 Sydney Film Festival, to see Barbecue (2017).
As well as the opportunity to immerse yourself in 102 minutes of barbequed meats from across the globe, this event afforded guests with the opportunity to meet director Matthew Salleh, along with producers Daniel Joyce and Rose Tucker, and settle in for a Venezuelan barbecue meal with them.
Barbecue is a very fitting documentary to view in the Liverpool area where residents speak more than 140 languages, as it takes viewers in language to 12 different geographic locations. While each location has different rituals and traditions surrounding the way they flame-grill meat, they all coalesce around bringing people together. "We kept finding these commonalities between cultures," filmmaker Matthew Salleh explained.
Whether they're having a braii in South Africa; grilling yakitori over binchō-tan (white charcoal) in Japan; eating asado in Uruguay; or doing the "craploads of work" to make a hangi in New Zealand, there's a palpable excitement in each setting. It's instantly clear that there's more at stake than just a well-cooked piece of meat. Participants in these barbecues are continuing longstanding traditions, sharing intergenerational knowledge, building friendships and making community.
By jumping from shawarma in a refugee camp on the border between Syria and Jordan straight to Americans firing weapons for fun in Texas, the filmmaker gently highlights that underneath other tensions, we all come together in very similar ways to grill meat. Excellent cinematography ensures your mouth is watering as you watch rotisserie lechón (roast suckling pig) turned by hand (and foot) in Cebu. Clouds of smoke drifting across the screen from the various styles of barbeque have you almost believing you can smell it. It's a feast for all the senses with a lush orchestral soundtrack by Christopher Larkin, performed by the Budapest Scoring Orchestra. It should bring a smile to the face of anyone who likes Philip Glass.
Pathos comes from watching the sun-starved Swedes approximate tradition on engangsgrill - disposable barbeques - which summon bad memories of being trapped in Ikea with a trolley full of items you won't ever use. You also might cringe when you see contemporary Australian values reflected back at you, with one pub-goer explaining pub barbeques go well "until someone throws some kebabs on there" or something else nobody can cook. If barbeque is about bringing people together, surely we can can put in the effort to get better at cooking multicultural meats?
Barbecue left us with plenty to ponder with fellow movie-goers over a Venezualan-inspired barbeque meal. The meal was put together by Bellbird Chef Federico Rekowski who introduced me to his outdoor smoker earlier in the evening, explaining: "It's like my roots meeting technology." Keep your eyes peeled for smoked items, like pork belly, currently popping up on his regular menu.
Guests were treated to a long table experience down the turbine hall with shared platters of barbequed meats.
The plentiful spread ranged from the brisket I saw smoking earlier, to chorizos, black puddings, and chicken cooked sous vide with lemon and thyme, before being finished on the grill.
Our heaving plates were accompanied by chimmichurri and guasacaca sauce, made from pureed avocado and spices; with salad, bread rolls, roasted potatoes and onions all passed separately.
The ticket price of $47 included film, meal and a glass of Eden Road wine. I accompanied my meal with an intriguing Eden Road P3 chalkboard blend of pinot noir, meunier and gris. It's one of their experimental, limited edition wines that I'd be ever so happy to drink again, with or without barbeque.
NOTE: Find more information about upcoming films and film festivals on the Casula Powerhouse website.