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%Title page
\begin{titlepage}
\begin{flushright}
LNF-92/xxx \\
CERN--TH.6504/92 \\
BUTP--92/20
\end{flushright}

\vspace*{1cm}
\begin{center} {\Large \bf SEMILEPTONIC KAON DECAYS  \\[5pt]
IN CHIRAL PERTURBATION THEORY}

\vspace{1cm}
{\bf{J. Bijnens and G. Ecker$^{\sharp}$}}

\vspace{.2cm}

 CERN \\
 CH$-$1211 Geneva 23 \\


\vspace{.7cm}

  and

\vspace{.7cm}

{\bf{J. Gasser$^{\star}$}}


\vspace{.2cm}

 Institut f\"ur theoretische Physik\\
Universit\"at Bern, Sidlerstrasse 5 \\
CH$-$3012 Bern\\

\nopagebreak[3]

\end{center}

\vspace{2cm}
\vspace{2cm}
\noindent
{\underline{\hspace{5cm}}}\\

\noindent
$^{\sharp}$ Permanent address:
Institut f\"ur theoretische Physik, Universit\"at  Wien, Boltz-\\
$^{\hspace{1.8ex}}$manngasse 5,
A--1090 Wien, Austria\\
\noindent
$^{\star}$ Research supported in part by Schweizerischer Nationalfonds\\

\noindent
$^{\hspace{1.8ex}}$CERN-TH.6504/92 \\
$^{\hspace{1.8ex}}$May 1992

\end{titlepage}

\begin{titlepage}
\vfill

\begin{abstract}
 We present the matrix elements for the semileptonic kaon decays
$K_{l2\gamma}$, $K_{l2l^+l^-}$, $K_{l2l'^+l'^-}$, $K_{l3}$, $K_{l3\gamma}$ and
$K_{l4}$ at
next-to-leading order in chiral perturbation theory and compare the predictions
with experimental data. Monte Carlo event generators are used to calculate the
corresponding rates at DAFNE. We discuss the possibilities to improve our
knowledge of the low-energy structure of the Standard Model
 at this and similar machines.

\end{abstract}
\vfill
\end{titlepage}

\tableofcontents
\begin{center}
 {\underline{Note:}}
\end{center}
\begin{itemize}
\item
The number of events quoted for DAFNE are based on a
luminosity of $5\cdot 10^{32}~ cm^{-2}s^{-1}$,
which is equivalent \footnote{P. Franzini, private communication.}
to an annual rate of $9\cdot 10^9$ $(1.1\cdot 10^9)$ tagged
$K^{\pm}$ $(K_L)$ (1 year = $10^7~ s$ assumed).
\item
Whenever we quote a branching ratio for a semileptonic
$K^0$ decay, it stands for the branching ratio of the corresponding
$K_L$ decay, e.g.,
\bearr
BR(K^0 \to \pi^- l^+ \nu ) \equiv BR(K_L \to \pi^{\pm} l^{\mp} \nu )~.
\nonumber
\eearr
\item
More  notation is provided in appendix \ref{notation}.
\end{itemize}

\setcounter{chapter}{0}

\chapter{INTRODUCTION TO CHIRAL SYMMETRY}
\label{Intro}
\thispagestyle{empty}

\newpage

Chiral perturbation theory (CHPT)
is a systematic approach to formulate the standard model as a
quantum field theory at the hadronic level. In its general form, it
uses only the symmetries of the standard model, in particular its
spontaneously broken chiral symmetry.
It is characterized by an effective chiral
Lagrangian in terms of pseudoscalar meson fields (and possibly other
low-lying hadronic states) giving rise to a systematic low-energy
expansion of amplitudes \cite{Wein79,GL1}.

In the formulation of Ref.\cite{GL1}, one considers the generating
functional $Z[v,a,s,p]$ of
connected Green functions of quark currents associated
with the fundamental Lagrangian
\beq
{\cal L} = {\cal L}^0_{QCD} + \bar q \gamma^\mu (v_\mu + \gamma_5 a_\mu)q
- \bar q (s - i \gamma_5 p)q . \label{eq:QCD}
\eeq
${\cal L}^0_{QCD}$ is the QCD Lagrangian with the masses of the
three light quarks set to zero. The external fields $v_\mu$, $a_\mu$,
$s$ and $p$ are hermitian $3 \times 3$ matrices in flavour space.
To describe electromagnetic and semileptonic interactions, the
relevant external gauge fields of the standard model are
\footnote{We adopt the present conventions of the Particle Data
Group \cite{PDGR}.}
\beqa
r_\mu  =  v_\mu + a_\mu & = & - eQ  A_\mu
\label{eq:gf} \\*
l_\mu  =  v_\mu - a_\mu & = &  - eQ  A_\mu
        - \dfrac{e}{\sqrt{2}\sin{\theta_W}} (W^+_\mu T_+ + h.c.) \no
\eeqa $$
Q = \dfrac{1}{3} \rm{diag}(2,-1,-1), \qquad
T_+ = \left( \ba{ccc}
0 & V_{ud} & V_{us} \\
0 & 0 & 0 \\
0 & 0 & 0 \ea \right)  $$
where the $V_{ij}$ are Kobayashi--Maskawa matrix elements.
The quark mass matrix
\beq
{\cal M} = \mbox{diag}(m_u,m_d,m_s)
\eeq
is contained in the scalar field $s(x)$.
The Lagrangian (\ref{eq:QCD}) exhibits a local $SU(3)_L \times SU(3)_R$
symmetry
\beqa
q & \ra & g_R \, \dfrac{1}{2} (1 + \gamma_5)q +
                  g_L \, \dfrac{1}{2} (1 - \gamma_5)q \no \\*
r_\mu  & \ra & g_R r_\mu g^\dagger_R +
                             i g_R \partial_\mu g^\dagger_R \no \\*
l_\mu  & \ra & g_L l_\mu g^\dagger_L +
                             i g_L \partial_\mu g^\dagger_L \\*
s + i p & \rightarrow & g_R (s + i p) g^\dagger_L \no \\*
g_{R,L} & \in & SU(3)_{R,L}. \no \eeqa

The generating functional $Z$ admits an expansion in powers of
external momenta and quark masses (CHPT).
In the meson sector at leading
order in CHPT, it is given by the classical action
\beq
Z = \int d^4 x {\cal L}_2 (U,v,a,s,p). \eeq
${\cal L}_2$ is the non-linear $\sigma$ model Lagrangian coupled to the
external fields $v,a,s,p$
\beq
{\cal L}_2 = \frac{F^2}{4} \langle D_\mu U D^\mu U^\dagger +
             \chi U^\dagger + \chi^\dagger U \rangle \label{eq:L2} \eeq
where
\beq
D_\mu U = \partial_\mu U - ir_\mu U + iU l_\mu, \qquad
\chi = 2 B_0(s + ip), \eeq
and $\langle A \rangle$ stands for the trace of the matrix $A$. $U$ is a
unitary $3 \times 3$ matrix
$$
U^\dagger U = {\bf 1}, \qquad \det U = 1,
$$
which transforms as
\beq
U \rightarrow g_R U g^\dagger_L \eeq
under $SU(3)_L \times SU(3)_R$.
$U$ incorporates the fields of the eight pseudoscalar Goldstone
bosons. A convenient parametrization is \footnote{We follow the
Condon-Shortley-de Swart phase conventions.}
\beq
U = \exp{(i\sqrt{2}\Phi/F)},\qquad
\Phi = \left( \ba{ccc}
\dfrac{\pi^0}{\sqrt{2}} + \dfrac{\eta_8}{\sqrt{6}} & -\pi^+ & - K^+ \\*
\pi^- & -\dfrac{\pi^0}{\sqrt{2}} + \dfrac{\eta_8}{\sqrt{6}} & - K^0 \\*
K^- & -\ol{K^0} & - \dfrac{2 \eta_8}{\sqrt{6}} \ea \right). \eeq

The parameters $F$ and $B_0$ are the only free
constants at $O(p^2)$: $F$ is the pion decay constant
in the chiral limit,
\beq
F_\pi =  F(1 + O(m_{quark})) = 93.2 MeV , \eeq
whereas $B_0$ is related to the quark condensate,
\beq
\langle 0|\bar u u |0\rangle = - F^2 B_0(1 + O(m_{quark})). \eeq
$B_0$ always appears multiplied by quark masses. At $O(p^2)$, the
product $B_0 m_q$ can be expressed in terms of meson masses, e.g.
\beq
M^2_{\pi^+} = B_0 (m_u + m_d). \eeq

The Lagrangian (\ref{eq:L2}) is referred to as the effective chiral
Lagrangian of $O(p^2)$. The chiral counting rules are the following:
the field $U$ is of $O(p^0)$, the derivative $\partial_\mu$ and the
external gauge fields $v_\mu, a_\mu$ are terms of $O(p)$, and the
fields $s, p$ count as $O(p^2)$.

At order $p^4$ the generating functional consists of three terms
\cite{GL1} :
\begin{enumerate}
\item[i)] The  one-loop graphs generated by the lowest order Lagrangian
(\ref{eq:L2}).
\item[ii)] An explicit local action of order $p^4$.
\item[iii)] A contribution to account for the chiral anomaly.
\end{enumerate}

We briefly discuss the contributions ii) and iii)
and start with the local action of $O(p^4)$.
It is generated by the Lagrangian ${\cal L}_4$ \cite{GL1}:
\beqa
{\cal L}_4 & = & L_1 \langle D_\mu U^\dagger D^\mu U\rangle^2 +
                 L_2 \langle D_\mu U^\dagger D_\nu U\rangle
                     \langle D^\mu U^\dagger D^\nu U\rangle \no \\*
& & + L_3 \langle D_\mu U^\dagger D^\mu U D_\nu U^\dagger D^\nu U\rangle +
    L_4 \langle D_\mu U^\dagger D^\mu U\rangle \langle \chi^\dagger U +
    \chi U^\dagger\rangle  \no \\*
& & +L_5 \langle D_\mu U^\dagger D^\mu U(\chi^\dagger U + U^\dagger
\chi)\rangle
    +
    L_6 \langle \chi^\dagger U + \chi U^\dagger \rangle^2 +
    L_7 \langle \chi^\dagger U - \chi U^\dagger \rangle^2  \no \\*
& & + L_8 \langle \chi^\dagger U \chi^\dagger U +
 \chi U^\dagger \chi U^\dagger\rangle
    -i L_9 \langle F_R^{\mu\nu} D_\mu U D_\nu U^\dagger +
      F_L^{\mu\nu} D_\mu U^\dagger D_\nu U \rangle \no \\*
& & + L_{10} \langle U^\dagger F_R^{\mu\nu} U F_{L\mu\nu}\rangle +
    L_{11} \langle F_{R\mu\nu} F_R^{\mu\nu} + F_{L\mu\nu} F_L^{\mu\nu}\rangle +
    L_{12} \langle \chi^\dagger \chi \rangle,
\eeqa
where
\beqa
  F_R^{\mu\nu} & = & \partial^\mu r^\nu -
                     \partial^\nu r^\mu -
                     i [r^\mu,r^\nu]   \\*
  F_L^{\mu\nu} & = & \partial^\mu l^\nu -
                     \partial^\nu l^\mu -
                     i [l^\mu,l^\nu] . \no
\eeqa
The twelve new low-energy couplings $L_1, \ldots, L_{12}$ arising
here are in general
divergent (except $L_3,L_7$). They absorb the divergences of the one-loop
graphs via the renormalization
\bearr
L_i&=&L_i^r +\Gamma_i \lambda
\nonumber \\
\lambda &=&(4\pi)^{-2}\mu^{d-4} \left\{ \frac{1}{d-4} - \frac{1}{2} \left(
{\mbox{ln}}(4\pi) +\Gamma'(1) +1 \right) \right\}
\eearr
in the dimensional regularization scheme. The coefficients $\Gamma_i$ are
displayed in table \ref{tab:Li}. They govern the scale dependence of the
renormalized, finite couplings $L_i^r(\mu)$,
\be
L_i^r(\mu_2) =L_i^r(\mu_1) + \frac{\Gamma_i}{16\pi^2} \ln\frac{\mu_1}{\mu_2} \;
\; . \label{eq:scale}
\ee
Observable quantities are independent of the scale $\mu$, once the loop
contributions are included.

The constants $F,B_0$, together with $L_1^r,\ldots,L_{10}^r$,
completely determine the low-energy behaviour of pseudoscalar meson
interactions to $O(p^4)$. $L_{11}^r$ and $L_{12}^r$ are contact terms which
are not directly accessible to experiment.
Similarly to $F$ and $B_0$ discussed above, the constants $L_i^r$ are not
determined
by chiral symmetry $-$ they are fixed by the dynamics of the underlying theory
through the renormalization group invariant scale $\Lambda$ and by the heavy
quark masses $m_c,m_b,\ldots \; .$ With present techniques, it is, however, not
possible to evaluate them directly from the
QCD Lagrangian. In the absence of such
a calculational scheme, they have been determined by comparison with
experimental low-energy information and by using large$-N_C$ arguments.
The result is shown in column 2 of table \ref{tab:Li}, where
$L_1^r,\ldots,L_{10}^r$ are
displayed at the scale $\mu=M_{\rho}$. The experimental information
underlying these values is shown in column 3.
[$L_1,L_2$ and $L_3$ are taken from a recent overall
fit to $K_{e4}$ and $\pi \pi$ data \cite{Kl4Rig}, see also the section
on $K_{l4}$ decays in chapter \ref{semi} and Ref. \cite{Kl4Bij}.
$L_4,\ldots,L_{10}$ are from \cite{GL1}.
For $L_9$ see also \cite{BC88}.
In Refs. \cite{GL1,Res} it was shown that the values for the
$L^r_i(M_\rho)$ can be understood in terms of meson resonance exchange.
For recent attempts to evaluate $L_i$ directly from the QCD
Lagrangian see \cite{rafael}.]


\begin{table}[t]
\begin{center}
\caption{Phenomenological values and source for the renormalized coupling
constants $L^r_i(M_\rho)$.
The quantities $\Gamma_i$
in the fourth column determine the scale dependence of the $L^r_i(\mu)$
according to Eq. (\protect\ref{eq:scale}). $L_{11}^r$ and $L_{12}^r$ are not
directly accessible to experiment.} \label{tab:Li}
\vspace{.5cm}
\begin{tabular}{|c||r|l|r|}  \hline
i & $L^r_i(M_\rho) \times 10^3$ & source & $\Gamma_i$ \\ \hline
  1  & 0.7 $\pm$ 0.5 & $K_{e4},\pi\pi\rightarrow\pi\pi$ & 3/32  \\
  2  & 1.2 $\pm$ 0.4 &  $K_{e4},\pi\pi\rightarrow\pi\pi$&  3/16  \\
  3  & $-$3.6 $\pm$ 1.3 &$K_{e4},\pi\pi\rightarrow\pi\pi$&  0     \\
  4  & $-$0.3 $\pm$ 0.5 & Zweig rule &  1/8  \\
  5  & 1.4 $\pm$ 0.5  & $F_K:F_\pi$ & 3/8  \\
  6  & $-$0.2 $\pm$ 0.3 & Zweig rule &  11/144  \\
  7  & $-$0.4 $\pm$ 0.2 &Gell-Mann-Okubo,$L_5,L_8$ & 0             \\
  8  & 0.9 $\pm$ 0.3 & \small{$M_{K^0}-M_{K^+},L_5,$}&
5/48 \\
     &               &   \small{ $(2m_s-m_u-m_d):(m_d-m_u)$}       & \\
 9  & 6.9 $\pm$ 0.7 & $<r^2>_{em}^\pi$ & 1/4  \\
 10  & $-$5.5 $\pm$ 0.7& $\pi \rightarrow e \nu\gamma$  &  $-$ 1/4  \\
\hline
11   &               &                                & $-$1/8 \\
12   &               &                                & 5/24 \\
\hline
\end{tabular}
\end{center}
\end{table}

Here, it is of interest to know which of the low-energy couplings occur in the
matrix elements for the
semileptonic kaon decays discussed in chapter \ref{semi}.
This information is given in table \ref{tab:semili}.
(There is an ambiguity concerning the bookkeeping of $L_4$ and $L_5$: some of
these contributions may be absorbed into the physical decay constants
$F_\pi,F_K$. Here we have chosen the convention which corresponds to the
amplitudes displayed in chapter \ref{semi}.
 Furthermore, in $K_{\mu4}$ decays, additional constants  may occur via
the form factor $R$ which has not yet been worked out at one-loop level
\cite{colgass}. This channel is therefore omitted in the table.)



\begin{table}[t]
\protect
\begin{center}
\caption{Occurrence
of the low-energy coupling constants $L_1,\ldots,L_{10}$ and of
the anomaly  in the semileptonic decays discussed
in chapter \protect\ref{semi}.\label{tab:semili}
  }
\vspace{1em}
\begin{tabular}{|c||c|c|c|c|c|c|c|}  \hline
 & & & & & $K^+\rightarrow$ & $K^+\rightarrow$ & $K^0\rightarrow$
\\
     &
$K_{l2\gamma}$ &
$K_{l2ll}$    &
$K_{l3}$    &
$K_{l3\gamma}$&
$\pi^+\pi^-e^+\nu_e$&
$\pi^0\pi^0e^+\nu_e$&
$\pi^0\pi^-e^+\nu_e$ \\ \hline
% 1     times2 times3 times4 times5 times6  times7  times8
 $L_1$ &      &      &      &      & $\times$ &$\times$ &
\\
 $L_2$&      &      &      &      & $\times$ &$\times$ &
\\
 $L_3$ &      &      &      &      & $\times$ &$\times$ & $\times$
\\
 $L_4$ &      &      &      &      & $\times$ &$\times$ &
\\
 $L_5$&       &      &      &      & $\times$ &$\times$ &$\times$
\\
 $L_9$&      &$\times$&$\times$&$\times$& $\times$ &$\times$ &$\times$
\\
 $L_9+L_{10}$ &$\times$&$\times$&      &$\times$&       & &
\\
\hline
Anomaly&$\times$&$\times$& &$\times$&$\times$&&$\times$
\\
\hline
\end{tabular}
\end{center}
\end{table}
We now turn to point iii) above.
A functional $Z[U,l,r]$ which reproduces the chiral anomaly
was first constructed by Wess and Zumino \cite{WZ}. For practical
purposes, it is useful to write it in the explicit form given by Witten
 \cite{Witten}:
\beqa
Z [U,l,r]_{WZW} &=&-\dfrac{i N_C}{240 \pi^2}
\int_{M^5} d^5x \epsilon^{ijklm} \langle \Sigma^L_i
\Sigma^L_j \Sigma^L_k \Sigma^L_l \Sigma^L_m \rangle \label{eq:WZW} \\*
 & & - \dfrac{i N_C}{48 \pi^2} \int d^4 x
\varepsilon_{\mu \nu \alpha \beta}\left( W (U,l,r)^{\mu \nu
\alpha \beta} - W ({\bf 1},l,r)^{\mu \nu \alpha \beta} \right)
\no \\
W (U,l,r)_{\mu \nu \alpha \beta} & = &
\langle U l_{\mu} l_{\nu} l_{\alpha}U^{\dg} r_{\beta}
+ \frac{1}{4} U l_{\mu} U^{\dg} r_{\nu} U l_\alpha U^{\dg} r_{\beta}
+ i U \partial_{\mu} l_{\nu} l_{\alpha} U^{\dg} r_{\beta}
\no  \\
& & + i \partial_{\mu} r_{\nu} U l_{\alpha} U^{\dg} r_{\beta}
- i \Sigma^L_{\mu} l_{\nu} U^{\dg} r_{\alpha} U l_{\beta}
+ \Sigma^L_{\mu} U^{\dg} \partial_{\nu} r_{\alpha} U l_\beta
\no \\
& & -\Sigma^L_{\mu} \Sigma^L_{\nu} U^{\dg} r_{\alpha} U l_{\beta}
+ \Sigma^L_{\mu} l_{\nu} \partial_{\alpha} l_{\beta}
+ \Sigma^L_{\mu} \partial_{\nu} l_{\alpha} l_{\beta}  \\
& & - i \Sigma^L_{\mu} l_{\nu} l_{\alpha} l_{\beta}
+ \frac{1}{2} \Sigma^L_{\mu} l_{\nu} \Sigma^L_{\alpha} l_{\beta}
- i \Sigma^L_{\mu} \Sigma^L_{\nu} \Sigma^L_{\alpha} l_{\beta}\rangle
\no \\
& & - \left( L \leftrightarrow R \right) \no \eeqa
$$
\Sigma^L_\mu = U^{\dg} \partial_\mu U \qquad
\Sigma^R_\mu = U \partial_\mu U^{\dg} $$
$$ N_C = 3 \qquad \varepsilon_{0123} = 1  $$
where $\left( L \leftrightarrow R \right)$ stands for the interchange
$$
U \leftrightarrow U^\dg, \qquad l_\mu \leftrightarrow r_\mu,
\qquad \Sigma^L_\mu \leftrightarrow \Sigma^R_\mu . $$
The integration in the first term in Eq. (\ref{eq:WZW}) is over a
five-dimensional
  manifold whose boundary is four-dimensional Minkowski space, such
that
\be
\int_{M^5} d^5x \epsilon^{ijklm}\partial_m T_{ijkl}=\int d^4x \epsilon^{\mu \nu
\rho\sigma} T_{\mu \nu\rho\sigma}
\ee
according to Stoke's theorem. [This term involves at least five pseudoscalar
fields and will not be needed in the following chapter.]   The convention used
in
Eq. (\ref{eq:WZW}) ensures that $Z[U,l,r]_{WZW}$ conserves parity and
reproduces the anomaly under $SU(3)_L\times SU(3)_R$ transformations in
Bardeen's form \cite{bardeen} (in particular, it  is invariant under
 transformations generated by the vector currents).

The Wess-Zumino-Witten functional contains all the
 anomalies which contribute to the semileptonic meson decays considered in the
following chapter. The relevant piece for e.g. $K_{l4}$ decays is
\be
Z[U,l,r]_{WZW}=\frac{i\sqrt{2}}{ 4 \pi^2 F^3}
\int d^4 x \epsilon_{\mu \nu \rho \sigma}<\partial^\mu \Phi \partial^\nu \Phi
\partial^\rho \Phi v^\sigma> + \cdots \; \; .
\ee

This short introduction to CHPT (see Refs.\cite{Georgi,Gasser} for more
extensive treatments with references to the original literature)
contains all the ingredients necessary
for the calculation of semileptonic $K$ decay amplitudes to $O(p^4)$
presented in the next chapter. For the low energies involved in these
decays, the momentum dependence of the $W$ propagator connecting to the
lepton-neutrino pair in the final state can be neglected.
The chiral realization of the
non-leptonic weak interactions is discussed in the corresponding
chapters on non-leptonic $K$ decays.

\newpage
\addcontentsline{toc}{section}{\hspace{1cm}Bibliography}
\begin{thebibliography}{99}

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 S. Weinberg, Physica 96A (1979) 327.

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\bibitem{PDGR}
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\bibitem{Kl4Rig}
 C. Riggenbach, J. Gasser, J.F. Donoghue and
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\bibitem{Kl4Bij}
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\bibitem{BC88}
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\bibitem{Res}
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\bibitem{colgass}
G. Colangelo and J. Gasser, in preparation.

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 J. Wess and B. Zumino, Phys. Lett. B37 (1971) 95.

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 E. Witten, Nucl. Phys. B223 (1983) 422; for further discussions see also\\
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\bibitem{Gasser}
 J. Gasser, The QCD Vacuum and Chiral Symmetry, in:
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dei Laboratori Nazionali di Frascati, P.O. Box, 13-I-00044 Frascati (Italy)) p.
291; \\  H. Leutwyler, Chiral Effective Lagrangians, Lectures given at
the 30. Int. Universit\"{a}tswochen f\"{u}r Kernphysik, Schladming and at the
Advanced Theoretical Study Institute in Elementary Particle Physics,
Boulder, 1991.

\end{thebibliography}

%***********************
%Now Chapter 2
%***********************
\newcommand{\pl}{PL}
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\chapter{SEMILEPTONIC KAON DECAYS}
\label{semi}
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\newpage

\section{Radiative $K_{l2}$ decays}

We consider the $K_{l2\gamma}$ decay
\be
K^+ (p) \rightarrow l^+ (p_l) \nu_l (p_\nu) \gamma (q) \hspace{1cm}
[K_{l2\gamma}] \label{k1}
\ee
where $l$ stands for $e$ or $\mu$, and $\gamma$ is a real photon with $q^2 =
0$. Processes where the (virtual) photon converts into a $e^+ e^-$ or
$\mu^+\mu^-$ pair are considered in the next section. The $K^-$ mode is
obtained from (\ref{k1}) by charge conjugation.

\subsection{Matrix elements and kinematics}

The matrix element for $K^+\rightarrow l^+ \nu_l \gamma$ has the
structure
\be
T = -iG_F eV_{us}^\star \epsilon^\star_\mu \left \{ F_K L^\mu - H^{\mu \nu}
l_\nu \right \}
\label{k3}
\ee
with
\bearr
L^\mu &=& m_l \bar{u}(p_\nu) (1 + \gamma_5) \left ( \frac{2p^\mu}{2pq}
- \frac{2 p^\mu_l + \not \!{q} \gamma^\mu}{2 p_l q} \right ) v (p_l)
\nonumber \\
l^\mu &=& \bar{u} (p_\nu)\gamma^\mu  (1 -\gamma_5) v (p_l)
\nonumber \\
H^{\mu \nu} &=& i V (W^2) \epsilon^{\mu \nu \alpha \beta} q_\alpha p_\beta -
A(W^2) (q W g^{\mu \nu} - W^\mu q^\nu)
\nonumber \\
W^\mu &=& (p-q)^\mu = (p_l + p_\nu)^\mu.
\label{k4}
\eearr

Here, $\epsilon_\mu$ denotes the polarization vector of the photon with
$q^\mu \epsilon_\mu= 0$, whereas $A$,
$V$ stand for two Lorentz
invariant amplitudes which occur in the general decomposition of the
tensors

\be
I^{\mu \nu} =
\int dx e^{iqx+iWy} < 0 \mid T V^\mu_{em} (x) I^\nu_{4-i5}(y) \mid K^+(p)>
\; \;, \; \; I=V,A \; \; .
\label{k5}
\ee
The form factor $A$  $(V)$ is related to the matrix element of
the axial (vector)
current in (\ref{k5}).
 In appendix \ref{kl2g} we display the general
decomposition of $A^{\mu \nu}$,
$V^{\mu\nu}$ for $q^2 \neq 0$ and provide also the link with the notation
used by the PDG \cite{pdg} and in \cite{ke22,km21}.

The  term proportional to $L^\mu$ in (\ref{k3}) does not contain
unknown quantities -- it
is determined by the amplitude of the nonradiative decay $K^+ \rightarrow l^+
\nu_l$. This part of the amplitude is usually referred to as "inner
Bremsstrahlung (IB) contribution", whereas the term proportional to $H^{\mu
\nu}$ is called "structure dependent (SD) part" .

The form factors are analytic functions in the complex $W^2$-plane cut
along the positive real axis. The cut starts at $W^2 = (M_K + 2 M_\pi)^2$ for
$A$ (at $W^2 = (M_K + M_\pi)^2$ for $V$). In our phase convention, $A$ and
$V$ are real in the physical region of $K_{l 2 \gamma}$ decays,
\be
m_l^2 \leq W^2 \leq M^2_{K}.
\label{k6}
\ee

The kinematics of (spin averaged) $K_{l 2 \gamma}$ decays needs two
variables, for which we choose the conventional quantities
\be
x = 2 p q/M^2_{K} \hspace{0.2cm} , \hspace{0.2cm} y = 2 p  p_l/
M_{K}^2 \; \; .
\label{k15}
\ee
In the $K$ rest frame, the variable $x$ ($y$) is proportional to the photon
(charged lepton) energy,
\be
x=2 E_\gamma /M_K \; \; , \; \; y=2 E_l/M_K \; \; ,
\label{k15a}
\ee
and the angle $\theta_{l\gamma}$ between the photon and the charged lepton is
related to $x$ and $y$ by
\be
x=\frac{ (1-y/2+A/2)(1-y/2-A/2)}{1-y/2+A/2 {\mbox{cos}} \theta_{l \gamma}} \;
\; ; A=\sqrt{y^2-4 r_l} \; \; .
\label{k15b}
\ee
In terms of these quantities, one has
\be
W^2 = M^2_{K} (1-x) \hspace{0.2cm} ; \hspace{0.2cm} (q^2 = 0) \hspace{0.2cm} .
\label{k16}
\ee

We write the physical  region for $x$ and $y$ as
$$
2 \sqrt{r_l} \leq y \leq 1 + r_l
$$
\be
1 - \frac{1}{2} (y + A) \leq x \leq 1 - \frac{1}{2} (y - A)
\label{k17}
\ee
or, equivalently, as
\bearr
0 \leq & x &\leq 1-r_l \nonumber \\
1-x +\frac{r_l}{(1-x)} \leq & y & \leq 1+r_l
\label{k17a}
\eearr
where
\be
r_l = m^2_l/ M^2_{K} = \left \{ \begin{array}{ll}
1.1 \cdot 10^{-6} (l = e) \\
4.6 \cdot 10^{-2} (l = \mu) \; \; .
\end{array} \right .
\label{k18}
\ee


  \subsection{Decay rates}

The partial decay rate is
\be
d\Gamma = \frac{1}{2M_K (2\pi)^5} \sum_{spins} |T|^2 d_{LIPS}(p;p_l,p_\nu,q).
\ee
The Dalitz plot density
\be
\rho(x,y) = \frac{d^2\Gamma}{dx dy} = \frac{M_K}{256\pi^3} \sum_{spins} |T|^2
\ee
is a Lorentz invariant function which contains $V$ and $A$ in the following
form \cite{brym},
\bearr
\rho(x,y)&=&
\rho_{\mbox{\tiny{IB}}}(x,y)  +  \rho_{\mbox{\tiny{SD}}}(x,y)
+  \rho_{\mbox{\tiny{INT}}}(x,y)
\nonumber \\
 \rho_{\mbox{\tiny{IB}}}(x,y)& =& A_{\mbox{\tiny{IB}}}
f_{\mbox{\tiny{IB}}}(x,y)
\nonumber \\
 \rho_{\mbox{\tiny{SD}}}(x,y)& =& A_{\mbox{\tiny{SD}}}
      M_K^2   \left[ (V+A)^2
f_{{\mbox{\tiny{SD}}}^+}(x,y) + (V-A)^2 f_{{\mbox{\tiny{SD}}}^-} (x,y) \right]
\nonumber \\
 \rho_{\mbox{\tiny{INT}}}(x,y)& =&
 A_{\mbox{\tiny{INT}}} M_K
\left [ (V+A) f_{{\mbox{\tiny{INT}}}^+} (x,y) + (V-A)
f_{{\mbox{\tiny{INT}}}^-} (x,y) \right]
\label{k19}
\eearr
where
\bearr
f_{\mbox{\tiny{IB}}}(x,y)& =&\left[ \frac{1-y+r_l}{x^2(x+y-1-r_l)}\right]
\left[x^2 +2(1-x)(1-r_l) -\frac{2x r_l (1-r_l)}{x+y-1-r_l} \right]
\nonumber \\
 f_{{\mbox{\tiny{SD}}^+}}(x,y)& =& \left[ x+y-1-r_l\right]
\left[ (x+y-1)(1-x)-r_l \right]
\nonumber \\
 f_{{\mbox{\tiny{SD}}^-}}(x,y)& =& \left[1-y+r_l \right]
 \left[ (1-x) (1-y) +r_l\right]
\nonumber \\
 f_{{\mbox{\tiny{INT}}^+}}(x,y)& =& \left[ \frac{1-y+r_l}{x(x+y-1-r_l)}\right]
\left[ (1-x)(1-x-y)+r_l \right]
\nonumber \\
 f_{{\mbox{\tiny{INT}}^-}}(x,y)& =& \left[ \frac{1-y+r_l}{x(x+y-1-r_l)}\right]
\left[ x^2 -(1-x)(1-x-y)-r_l\right]
\label{k20}
\eearr

and
\bearr
A_{\mbox{\tiny{IB}}}& =& {4r_l} \left ( \frac{F_K}{M_K} \right)^2
A_{\mbox{\tiny{SD}}}
\nonumber \\
A_{\mbox{\tiny{SD}}}& =& \frac{G_F^2 |V_{us}|^2 \alpha}{32 \pi^2} M_K^5
\nonumber \\
A_{\mbox{\tiny{INT}}}& =&{4r_l} \left ( \frac{F_K}{M_K} \right)
A_{\mbox{\tiny{SD}}} \; \; .
\label{k20a}
\eearr
For later convenience, we note that
\be
A_{\mbox{\tiny{SD}}}
                    = \frac{\alpha}{8\pi}
\frac{1}{r_l(1-r_l)^2} {\left( \frac{M_K}{F_K} \right) }^2
\Gamma (K\rightarrow l\nu_l) \; .
\label{k20b}
\ee
The indices IB, SD and INT stand respectively for the contribution from
inner Bremsstrahlung, from the structure dependent part and from the
interference term between the IB and the SD part in the amplitude.

To get a feeling for the magnitude of the various contributions
IB,$\mbox{SD}^\pm$ and ${\mbox{INT}}^\pm$ to the decay rate, we consider
the integrated  rates
\be
\Gamma_I = \int_{R_I} dxdy  \rho_I (x,y) \; \; ; \; \; I =
\mbox{SD}^\pm, \mbox{INT}^\pm ,\mbox{IB} \; \; ,
\label{dr1}
\ee
where $\rho_{SD}=\rho_{SD^+} + \rho_{SD^-}$ etc.
 For the region $R_I$ we take the full phase space
 for $I \neq \mbox{IB}$, and
\be
R_{{\mbox{{\tiny{IB}}}}} =214.5 {\mbox{MeV/c}} \leq p_l \leq 231.5
{\mbox{MeV/c}}\; . \label{dr2}
\ee
for the Bremsstrahlung contribution. Here $p_l$ stands for the modulus of the
lepton three momentum in the kaon rest system \footnote{
This cut has been used in \cite{km21} for $K_{\mu 2\gamma}$, because this
kinematical region is free from $K_{\mu3}$ background.
We apply it here for illustration also to the electron mode $K_{e2\gamma}$.
                                 }.
  We consider constant
form factors
$V$, $A$ and write for the rates and for the corresponding branching ratios
\bearr
\Gamma_I & =& A_{\mbox{\tiny{SD}}} \left \{ M_K (V \pm A) \right \}^{N_I}X_I \;
\;
\nonumber \\
{\mbox{BR}}_I &\doteq& \Gamma_I / \Gamma_{\mbox{\tiny{tot}}} = N \left \{
M_K (V \pm A) \right \}^{N_I} X_I
\label{dr3}
\eearr
with
\be
N = A_{\mbox{\tiny{SD}}}/{\Gamma_{\mbox{\tiny{tot}}}}
 = 8.348 \cdot 10^{-2}.
\label{dr3a}
\ee
The values for $N_I$ and $X_I$ are listed in table \ref{t:pskl2}.

\begin{table}[t]
\protect
\begin{center}
\caption{
 The quantities
 $X_I,N_I$.
 SD$^\pm$ and INT$^\pm$ are evaluated with full phase space,
 IB with
restricted kinematics (\protect\ref{dr2}).
\label{t:pskl2}         }
\vspace{1em}
\begin{tabular}{|c||c|c|c|c|c||c|} \hline
\multicolumn{1}{|c||}{}&{${\mbox{SD}}^+$}&{${\mbox{SD}}^-$}&
{${\mbox{INT}}^+$}&{INT$^-$}
&{${\mbox{IB}}$}& \multicolumn{1}{|c|}{}\\ \hline
{$X_I$} & {$1.67\cdot 10^{-2}$}&{$1.67\cdot 10^{-2}$}&
{$-8.22 \cdot 10^{-8}$}&{$3.67\cdot 10^{-6}$}&{$3.58\cdot 10^{-6}$}&
{$K_{e2\gamma}$} \\ \hline
{$X_I$} & {$1.18\cdot 10^{-2}$}&{$1.18\cdot 10^{-2}$}&
{$-1.78\cdot 10^{-3}$}&{$1.23\cdot 10^{-2}$}&{$3.68\cdot 10^{-2}$}&
{$K_{\mu 2 \gamma}$} \\ \hline
{$N_I$} &2&2&1&1&0&\multicolumn{1}{|c|}{} \\ \hline
  \end{tabular}
\end{center}
\end{table}

To estimate $\Gamma_I$ and ${\mbox{BR}}_I$, we note that the form factors $V,A$
 are of
 order
\be
M_K(V+A) \simeq -10^{-1} \; \; , \; \; M_K(V-A) \simeq -4 \cdot 10^{-2} \; \; .
\ee
{}From this and from the entries in the table
one concludes that for the above regions $R_I$, the interference terms
$\mbox{INT}^\pm$ are negligible in
$K_{e2\gamma}$, whereas they are important in $K_{\mu 2 \gamma}$. Furthermore,
IB is negligible for $K_{e2\gamma}$, because it is helicity suppressed as can
be seen from the factor $m_l^2$ in $A_{\mbox{\tiny{IB}}}$. This term
dominates however in $K_{\mu 2 \gamma}$.


\subsection{Determination of $A(W^2)$ and $V(W^2)$}

The decay rate  contains two real functions
\be
F^\pm(W^2) = V(W^2) \pm A(W^2)
\label{k21}
\ee
as the only unknowns. In Figs. (\ref{psp1},\ref{psp2}) we display
contour plots for the density distributions
$f_{\mbox{\tiny{IB}}}, \ldots,$ $f_{{\mbox{\tiny{INT}}^\pm}}$ for $l = \mu,
e$. These
five terms have obviously very different Dalitz plots. Therefore, in principle,
one can determine the strength of each term by choosing a suitable kinematical
region of observation. To pin down $F^\pm$, it would   be sufficient to
measure
at each photon energy the interference term INT$^\pm$. This has not yet
\begin{figure}[p]
\vspace{7.5in}
\caption{Contour plots for
 $f_{{\mbox{\protect\tiny{IB}}}}$,\ldots,
 $f_{{\mbox{\protect\tiny{INT}}}^\pm}$
 [$K_{\mu 2 \gamma}$]. The numbering on the
lines points towards increasing modulus. The normalization is arbitrary.
\label{psp1} }
\end{figure}
\begin{figure}[p]
\vspace{7.5in}
\caption{Contour plots for
 $f_{{\mbox{\protect\tiny{IB}}}}$,\ldots,
 $f_{{\mbox{\protect\tiny{INT}}}^\pm}$
 [$K_{e 2 \gamma}$]. The numbering on the
lines points towards increasing modulus. The normalization is arbitrary.
 \label{psp2}}
\end{figure}
been achieved so far, either because the contribution of INT$^\pm$
is too small (in $K_{e 2 \gamma}$)
, or because too few events have been collected  (in
$K_{\mu 2 \gamma}$). On the other hand, from a
measurement
of {\mbox{SD}}$^\pm$ alone one can determine $A, V$ only up to a fourfold
 ambiguity:
\be
\mbox{SD}^\pm \rightarrow \left \{ (V,A); - (V,A); (A,V); - (A,V) \right \}.
\label{k22}
\ee
In terms of the ratio
\be
\gamma_K = A/V
\label{k23}
\ee
this ambiguity  amounts to
\be
\mbox{SD}^\pm \rightarrow \left \{ \gamma_K ; 1/ \gamma_K \right \}.
\label{k24}
\ee
Therefore, in order to pin down the amplitudes $A$ and $V$ uniquely, one must
measure the interference terms INT$^\pm$ as well.

\subsection{Previous experiments}

 \vspace{.5cm}

{\underline{{\bf{$K^+ \rightarrow e^+ \nu_e  \gamma$}}}}


 \vspace{.5cm}

The PDG uses data from two experiments \cite{ke22,ke21}, both of which have
been
sensitive mainly to the {\mbox{SD}}$^+$ term in (\ref{k19}). In \cite{ke21}, 56
events
with $E_\gamma > 100$ MeV, $E_{e^+} > 236$ MeV and $\theta_{e^+\gamma}> 120^0$
have been identified, whereas the later experiment \cite{ke22} has
collected 51 events with $E_\gamma > 48$ MeV, $E_{e^+} > 235$ MeV and
$\theta_{e^+ \gamma} > 140^0$. In these kinematical regions, background from
$K^+ \rightarrow e^+ \nu_e \pi^0$ is absent because $E_{e}^{\mbox{\tiny{max}}}
(K_{e3}) = 228$ MeV. The combined result of both experiments is
\footnote{In all four experiments \cite{ke21,ke22,km21,km22} discussed here
and below , the form factors $A$ and $V$ have been treated as constants.}
\cite{ke22}
\be
\Gamma(\mbox{SD}^+) / \Gamma(K_{\mu2}) = (2.4 \pm 0.36) \cdot 10^{-5}.
\label{k25}
\ee
For {\mbox{SD}}$^-$, the bound
\be
\Gamma(\mbox{SD}^-) / \Gamma_{\mbox{\tiny{total}}} < 1.6 \cdot 10^{-4}
\label{k26}
\ee
has been obtained from a sample of electrons with energies 220 MeV $\leq
E_{e} \leq 230$ MeV \cite{ke22}. Using (\ref{dr3},\ref{dr3a}), the
result (\ref{k25}) leads to
\be
 M_K \mid V+A \mid = 0.105 \pm 0.008 \; \; .
\label{k27}
\ee

The bound (\ref{k26}) on the other hand implies \cite{ke22}
\be
\mid V -A \mid / \mid V+A \mid < \sqrt{11},
\label{k28}
\ee
from where one concludes \cite{ke22} that $\gamma_K$ is outside the range
$-1.86$ to $-0.54$,
\be
\gamma_K  \not \in [-1.86, - 0.54] \; \; .
\label{k29}
\ee
As we already mentioned, the interference terms INT$^\pm$ in $K \rightarrow
e \nu_e \gamma$ are small and can hardly ever be measured.
As a result of this,
the amplitudes $A,V$ and the ratio $\gamma_K$ determined from $K_{e2\gamma}$
are
subject to the ambiguities (\ref{k22}), (\ref{k24}).

\vspace{.5cm}

{\underline{{\bf{$K^+ \rightarrow \mu^+ \nu_\mu \gamma$}}}}

\vspace{.5cm}

Here, the interference terms INT$^\pm$ are nonnegligible in appropriate regions
of phase space (see Figs. (\ref{psp1},\ref{psp2})). Therefore, this
decay allows one in principle to
pin down $V$ and $A$. The PDG uses data from two experiments
\cite{km21,km22}. In \cite{km21},
the momentum spectrum of the muon was measured in the region (\ref{dr2}).
 In total $2 \pm 3.44$ {\mbox{SD}}$^+$ events
have been found with 216 MeV/c  $< p_\mu< $  230 MeV/c and $E_\gamma > 100$
 MeV, which leads to
\be
M_K \mid V+ A \mid < 0.16\; \;.
\label{k30}
\ee
In order to identify
the effect of the {\mbox{SD}}$^-$ terms, the region 120 MeV/c
$<p_\mu < $150 MeV/c was searched. Here, the background from $K_{\mu3}$ decays
was very serious. The authors found 142 $K_{\mu \nu \gamma}$ candidates and
conclude that
\be
- 1.77 < M_K (V-A) < 0.21.
\label{k31}
\ee


The result (\ref{k30}) is consistent with
(\ref{k27}), and the bound  (\ref{k31}) is worse than the
result (\ref{k28}) obtained from   $K_{e 2 \gamma}$. The
branching ratios which follow  \cite{km21} from
(\ref{k30},\ref{k31}) are displayed in table
\ref{t:erkl2}, where we also show the $K_{e2\gamma}$ results \cite{ke21,ke22}.
The entry SD$^-$+INT$^-$
for $K_{\mu 2 \gamma}$ is based on additional
 constraints from $K_{e2\gamma}$ \cite{km21}.


\begin{table}[t]
\begin{center}
\protect
\caption{ Measured
 branching ratios $\Gamma (K\rightarrow l \nu_l \gamma)
/\Gamma_{\mbox{{\protect\tiny{total}}}}$. The $K_{e2\gamma}$ data are from
\protect\cite{ke21,ke22}, the $K_{\mu 2\gamma}$ data from
\protect\cite{km21,km22}. The last column corresponds \protect\cite{km21}
 to the cut
(\protect\ref{dr2}).
\label{t:erkl2}         }
\vspace{1em}
{\footnotesize{
\begin{tabular}{|c|c|c|c|c|c|} \hline
&{${\mbox{SD}}^+$}&{${\mbox{SD}}^-$}&{${\mbox{INT}}^+$}&
{${\mbox{SD}}^- + {\mbox{INT}}^-$}&{total}
\\ \hline
{$K_{e2\gamma}$} & {$(1.52 \pm 0.23)\cdot 10^{-5}$}&{$<1.6\cdot 10^{-4}$}&
{}&{}&{ }
\\ \hline
{$K_{\mu 2 \gamma}$} & {$< 3\cdot 10^{-5}$}&{}&
{$<2.7\cdot 10^{-5}$}& {$<2.6\cdot 10^{-4}$}&
{$(3.02\pm 0.10)\cdot 10^{-3}$}
\\
\multicolumn{1}{|c|}{}&{}&{}&{(modulus)}&{(modulus)}&{}
\\ \hline
\end{tabular}
}}
\end{center}
\end{table}




\subsection{Theory}

The amplitudes $A(W^2)$ and $V(W^2)$ have been worked out in the framework of
various approaches, viz., current algebra, PCAC, resonance exchange,
dispersion relations, \ldots . For a rather detailed review together with an
extensive list of references up to 1976 see \cite{BARDIN}. Here, we concentrate
on the predictions of $V, A$ in the framework of CHPT.

\vspace{.5cm}

{\bf{A) Chiral expansion to one loop}}

\vspace{.5cm}

The amplitudes $A$ and $V$ have been evaluated \cite{donog,beg} in the
framework
of CHPT to one loop. At leading order in the low-energy expansion, one has
\be
A=V=0 .
\label{k36}
\ee
As a consequence of this, the rate is
entirely
given by the IB contribution
at leading order. At the one-loop level, one finds
\bearr
A &=& -\frac{4}{F} (L_9^{r} + L_{10}^{r})
\nonumber \\
V &=& - \frac{1}{8 \pi^2} \frac{1}{F}
\nonumber \\
\gamma_K &=& 32 \pi^2 (L_9^r + L_{10}^r)\; ,
\label{k37}
\eearr
where $L_9^r$ and $L_{10}^r$ are the renormalized low-energy couplings
evaluated at the scale $\mu$ (the combination $L_9^r + L_{10}^r$ is scale
independent). The vector form factor stems from the Wess-Zumino term
\cite{wessz} which enters the low-energy expansion at order $p^4$,
 see chapter \ref{Intro}.

{\underline {Remarks}:}

\begin{description}

\item(i)
 At this order in the low-energy expansion, the form factors $A,V$ do
not
exhibit any $W^2$-dependence. A nontrivial $W^2$-dependence only occurs at the
next order in the energy expansion (two-loop effect, see  the discussion
below). Note that the available analyses of experimental data of $K \rightarrow
l \nu_l \gamma$ decays \cite{ke21,ke22,km21,km22} use constant form factors
throughout.

\item (ii) Once the combination $L_9 + L_{10}$ has been pinned down
from other processes, Eq. (\ref{k37}) allows one to
evaluate
$A,V$ unambiguously at this order in the low-energy expansion. Using $L_9 +
L_{10} = 1.4 \cdot 10^{-3}$  and $F=F_\pi$, one has
\bearr
 M_K (A+V) &=& -0.097
\nonumber \\
M_K(V-A) &=& -0.037
\nonumber \\
\gamma_K &=& 0.45 \; \; .
\label{k38}
\eearr
The result for the combination $(A+V)$ agrees with (\ref{k27}) within the
errors,
while $\gamma_K$ is consistent with (\ref{k29}).

\end{description}

We display in table \ref{t:chpr} the branching ratios ${\mbox{BR}}_I$
 (\ref{dr3})
 which follow from
the prediction (\ref{k38}). These predictions satisfy of course the
inequalities found from experimental data (see table \ref{t:erkl2}).

\begin{table}[t]
\protect
\begin{center}
\caption{Chiral
prediction at order $p^4$ for the branching ratios
$\Gamma(K\rightarrow l \nu_l \gamma)/\Gamma_{\mbox{\protect\tiny{total}}}$. The
cut used in the last column is  given in Eq. (\protect\ref{dr2}).
\label{t:chpr}
}
\vspace{1em}
\begin{tabular}{|c|c|c|c|c|c|} \hline
{}&{{\mbox{SD}}$^+$}&{{\mbox{SD}}$^-$}&{{\mbox{INT}}$^+$}&{{\mbox{INT}}$^-$}&{
total} \\ \hline
{$K_{e2\gamma}$} & {$1.30\cdot 10^{-5}$}&{$1.95\cdot 10^{-6}$}&
{$6.64\cdot 10^{-10}$}&{$-1.15\cdot 10^{-8}$}&{$2.34 \cdot 10^{-6}$}
\\ \hline
{$K_{\mu 2\gamma}$} & {$9.24\cdot 10^{-6}$}&{$1.38\cdot 10^{-6}$}&
{$1.44\cdot 10^{-5}$}&{$-3.83\cdot 10^{-5}$}&{$3.08 \cdot 10^{-3}$}
\\ \hline
\end{tabular}
\end{center}
\end{table}

\newpage

\vspace{.5cm}
{\bf{B) $W^2$-dependence of the form factors}}

\vspace{.5cm}

The chiral prediction gives constant form factors at order $p^4$. Terms of
order $p^6$ have not yet been calculated. They would, however, generate a
nontrivial $W^2$ - dependence both in $V$ and $A$. In order to estimate the
magnitude of these corrections, we consider one class of $p^6$ - contributions:
terms which are generated
by vector and axial vector resonance exchange with
strangeness \cite{BARDIN,egpr},
\be
V(W^2) = \frac{V}{1-W^2/{M_{K^\star}}^2}\; \; ,\; \; A(W^2) =
\frac{A}{1-W^2/{M_{K_1}}^2} \label{k40}
\ee
where $V,A$ are given in (\ref{k37}). We now examine the effect of the
denominators in (\ref{k40}) in the region $y \geq 0.95, x \geq 0.2$
which has been explored in $K^+ \rightarrow e^+ \nu_e \gamma$ \cite{ke22}.
We put $m_e = 0$ and evaluate the rate
\be
\frac{dP(x)}{dx} = \frac{N_{\mbox{\tiny{tot}}}}{\Gamma_{\mbox{\tiny{tot}}}}
\int^1_{y=0.95}  \rho_{{\mbox{\tiny{SD}}}^+}(x,y) dy
\label{k41}
\ee
where $N_{\mbox{\tiny{tot}}}$ denotes the total number of $K^+$ decays
considered, and $\Gamma^{-1}_{\mbox{\tiny{tot}}} = 1.24 \cdot 10^{-8}$
sec.

\begin{figure}[t]
\vspace{9cm}
\caption{The rate $dP(x)/dx$ in (\protect\ref{k41}), evaluated with the form
factors
(\protect\ref{k40}) and
$N_{\mbox{\protect\tiny{tot}}} = 9
\cdot 10^9$.
The solid line corresponds to $M_{K^\star}
= 890$ MeV, $M_{K_1} = 1.3$ GeV. The dashed  line is evaluated with
$M_{K^\star} = 890$ MeV, $ M_{K_1} = \infty$ and the dotted line corresponds to
$M_{K^\star} = M_{K_1} = \infty$.
 The total number of events is also indicated in each case.
 \label{figform}
        }
\end{figure}

The function $\frac{dP(x)}{dx}$ is displayed in Fig. (\ref{figform}) for
three different values of $M_{K^\star}$ and $M_{K_1}$, with
 $N_{\mbox{\protect\tiny{tot}}} = 9
\cdot 10^9$.
 The total number of events
\be
N_P = \int^{1}_{x= 0.2} dP(x)
\label{k42}
\ee
is also indicated in each case. The difference between the dashed and the
dotted line shows  that
the nearby singularity in the anomaly form factor influences the decay rate
substantially at low photon energies.
The effect disappears at $x \rightarrow 1$, where $W^2 = M_K^2 (1-x)
\rightarrow 0$. To minimize the effect of resonance exchange, the large
$x$-region should thus be
considered. The low $x$-region, on the other hand, may be used to explore the
$W^2$-dependence of $V$ and of $A$. For a rather exhaustive discussion
of the relevance of this $W^2$ - dependence
for the analysis of $K_{l2\gamma}$ decays we refer the reader to Ref.
\cite{BARDIN}.



\subsection{Improvements at DAFNE}

Previous experiments have used various cuts in phase space in order (i)
to identify the individual contributions IB, {\mbox{SD}}$^\pm$, INT$^\pm$ as
far
 as
possible, and (ii) to reduce the background from $K_{l3}$ decays. This
background has in fact forced so severe cuts that only the upper end of the
lepton spectrum remained.

The experimental possibilities to reduce background from $K_{l3}$ decays are
presumably more favourable with today's techniques. Furthermore, the
annual yield of $9\cdot 10^9 K^+$ decays at DAFNE is more than two orders of
magnitude higher than the samples which were available in
 \cite{ke22,km21,ke21,km22}.
This allows for a big improvement in the determination
of the amplitudes $A$ and $V$, in particular in $K_{\mu 2 \gamma}$ decays.
 It would  be very interesting to pin
down the
combination $L_9 + L_{10}$ of the low-energy constants which occur
in the chiral representation of the amplitude $A$  and to
investigate the $W^2$-dependence of the form factors.

\newpage

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\section{The decays $K^{\pm} \rightarrow l^{\pm} \nu l'^{+}l'^{-}$}

Here we consider decays where the photon turns into a
lepton-anti-lepton pair,
\begin{eqnarray}
K^+ &\to&e^+ \nu \mu^+\mu^- \label{S21}\\
K^+ &\to&\mu^+ \nu e^+ e^- \label{S22}\\
K^+ &\to&e^+ \nu e^+ e^- \label{S23}\\
K^+ &\to&\mu^+ \nu \mu^+ \mu^- \ .\label{S24}
\end{eqnarray}

\subsection{Matrix elements}

We start with the processes (\ref{S21}) and (\ref{S22}),
\begin{eqnarray}
K^+(p) &\to&l^+(p_l) \nu(p_\nu) l'^{+}(p_1) l'^{-}(p_2)\nonumber\\
(l,l')&=& \ (e,\mu)\mbox{ or } (\mu,e) .
\end{eqnarray}
The matrix element is
\begin{equation}
\label{S27}
T = -i G_F e V_{us}^* \overline{\epsilon}_\rho   \left\{
F_K \overline{L}^\rho - \overline{H}^{\rho\mu}l_{\mu} \right\}
\end{equation}
where
\begin{eqnarray}
\overline{L}^\mu & = & m_l \overline{u}(p_\nu) (1+\gamma_5)
\left\{
\frac{2 p^\mu - q^\mu}{2 p q - q^2} -\frac{2p^\mu_l +\not\!q\gamma^\mu}
{2 p_l q + q^2}\right\}
v(p_l)
\nonumber\\
l^\mu &=& \overline{u}(p_\nu)\gamma^\mu (1-\gamma_5) v(p_l)
\nonumber\\
\overline{H}^{\rho\mu} & = &i V_1 \epsilon^{\rho\mu\alpha\beta}
 q_\alpha p_\beta - A_1 ( qW g^{\rho\mu} - W^\rho q^\mu )
\nonumber\\&&
-A_2 (q^2 g^{\rho\mu} - q^\rho q^\mu )
-A_4 (qW q^\rho - q^2 W^\rho) W^\mu
\end{eqnarray}
with
\begin{equation}
A_4 = \frac{2 F_K}{M_K^2 - W^2} \frac{F_V^{K}   (q^2) - 1 }{q^2} + A_3
\ .
\end{equation}
The form factors $A_i(q^2,W^2),\ V_1(q^2,W^2)$ are the ones defined
in appendix \ref{kl2g}.
                $F_V^{K}  (q^2)$ is the electromagnetic form factor
of the $K^+$. Finally the quantity $\overline{\epsilon}^{ \mu}$ stands
for
\begin{equation}
\overline{\epsilon}^{\mu} =   \frac{e}{q^2} \overline{u}(p_2)
\gamma^\mu v(p_1)\ ,
\end{equation}
and the four-momenta are
\begin{equation}
q = p_1 + p_2 ,\ W = p_l + p_\nu = p - q
\end{equation}
such that $q_\mu \overline{\epsilon}^{\mu} = 0$.

In order to obtain the matrix element for (\ref{S23}) and (\ref{S24}),
\begin{equation}
K^+ (p) \to l^+(p_l) \nu(p_\nu) l^+(p_1) l^-(p_2)\ ,
\end{equation}
one identifies $m_l$ and $m_l'$ in (\ref{S27}) and subtracts the
contribution obtained from interchanging $p_1 \leftrightarrow p_l$ :
\begin{eqnarray}
(p_1 , p_l) &\to& (p_l,p_1) \nonumber\\
q &\to& p_l + p_2\nonumber\\
W &\to& p- q=p_\nu + p_1\ .
\end{eqnarray}

\subsection{Decay distributions}

The decay width is given by
\begin{equation}
d\Gamma  =  \frac{1}{2M_K (2\pi)^8 }
\sum_{spins} |T^2| d_{LIPS}(p;p_l,p_\nu,p_1,p_2)
\end{equation}
and the total rate is the integral over this for the case $l\ne l'$.
For the case $l = l'$ the integral has to be divided by the factor 2
for two identical particles in the final state.

We first consider the case where
$l\ne l'^{}$ and introduce the dimensionless variables
\begin{eqnarray}
x &=& \frac{2 pq}{M_K^2} \nonumber\\
y &=& \frac{2p_l p}{M_K^2}\nonumber\\
z &=& \frac{q^2}{M_K^2}\nonumber\\
r_l &=& \frac{m_l^2}{M_K^2}\nonumber\\
r_l' &=&\frac{m_{l'}^2}{M_K^2} \; \;.
\end{eqnarray}
Then one obtains, after integrating over $p_1$ and $p_2$ at fixed $q^2$
\cite{KRISHNA},
\begin{eqnarray}
\label{S16}
d\Gamma_{K^+ \to l^+ \nu l'^{+}l'^{-}} &=&
\alpha^2 G_F^2 |V_{us}|^2 M_K^5 F(z,r'_l) \left\{
   -\sum_{spins} \overline{T}_\mu^* \overline{T}^\mu \right\}
    dx dy dz
\nonumber\\
F(z,r'_l)&=& \frac{1}{192 \pi^3 z} \left\{
     1 + \frac{2r'_{l}}{z}\right\}
  \sqrt{1 - \frac{4r'_{l}}{z}}
\nonumber\\
{\overline{T}}^\mu &=& M_K^{-2} \left\{
F_K \overline{L}^\mu -
\overline{H}^{\mu\nu}l_\nu\right\} .
\end{eqnarray}
The quantity $\left\{ -
\sum_{spins}\overline{T}_\mu^* \overline{T}^\mu \right\}$
is displayed in appendix \ref{TRACES}.
This result allows one to evaluate, e.g.,
the distribution $d\Gamma/dz$ of produced $l'^+ l'^-$ pairs rather
easily. The kinematically allowed region is
\begin{eqnarray}
4 r'_l \le&z&\le 1 + r_l - 2 \sqrt{r_l}
\nonumber\\
2 \sqrt{z} \le &x& \le 1 + z - r_l
\nonumber\\
A - B \le & y & \le A+B
\end{eqnarray}
with
\begin{eqnarray}
A&=&\frac{(2-x)(1+z+r_l -x)}{2 ( 1+z-x)}\nonumber\\
B&=&\frac{(1+z-x-r_l)\sqrt{x^2 - 4 z}}{2(1+z-x)} \; \; .
\end{eqnarray}
The case $l = l'$ is slightly more elaborate. We feel that it does not
make sense to display the term $\sum_{spins} |T|^2$ because it is of
considerable complexity in the general case when all the form factors
$A_i$, $V_1$ and $F_V^{K}$ are $q^2$ and $W^2$ dependent. The expression
together with the Monte Carlo program to do the phase space integrals
is available on request from the authors.

\subsection{Theory}

The form factors $A_i,\ V_1$ and $F_V^K$ have been discussed in all
kinds of models, Vector Meson Dominance, hard meson, etc.. For a
discussion see Ref. \cite{BARDIN}. We will restrict ourselves to the
predictions in the framework of CHPT.

To leading order we have
\begin{eqnarray}
V_1&=&0 \nonumber\\
A_1\ =\ A_2&=&A_3\ = 0\ .
\end{eqnarray}
We also have $F_V^K = 1$. The rate here is entirely given by the
inner Bremsstrahlung contribution. At the one-loop level
several form factors get non-zero values \cite{beg}
\begin{eqnarray}
V_1&=&- \frac{1}{8\pi^2 F}\nonumber\\
A_1&=&- \frac{4}{F}\left( L_9^r + L_{10}^r \right) \nonumber\\
A_2&=&- \frac{2 F_K ( F_V^K(q^2) -1 ) }{q^2} \nonumber\\
A_3&=&0\nonumber\\
F_V^K(q^2)&=&1+H_{\pi\pi}(q^2)+2 H_{KK}(q^2) \; \; .
\label{ourresult}
\end{eqnarray}
These results obey the current algebra relation of Ref. \cite{BARDIN}.
The function $F_V^K(q^2)$ does, however, deviate somewhat from the
linear parametrization often used.
The function $H(t)$ is defined in appendix \ref{loop}.

The fact that the form factors at
next-to-leading order could be written in terms of the kaon
electromagnetic form factor
       in a simple way is          not true anymore at the $p^6$
level.
The Lagrangian at order $p^6$ contains a term of the form
\begin{equation}
\mbox{\rm tr}\left\{D_\alpha F_L^{\alpha\mu} U^{\dag} D^{\beta}
F_{R\beta\mu} U \right\}
\end{equation}
that contributes to $A_2$ and $A_3$ but not to the kaon electromagnetic
form factor,  $F_V^K (q^2)$.

\subsection{Numerical results}

Using the formulas of the previous subsections and appendix \ref{TRACES}
we have
calculated the rates for a few cuts, including those
given in the literature.
For the case of \underline{unequal} leptons, the results are given
in table \ref{KLLL1} for the decay
$K^+ \to \mu^+ \nu e^+ e^-$. These include the cuts used in Refs.
\cite{KRISHNA} and \cite{DIAMANT},
$x \ge 40\ MeV /M_K$ and $ z \ge (140\ MeV/M_K)^2$,
respectively.
\begin{table}
\caption{\label{KLLL1}
         Theoretical values for the branching ratios for the decay
$K^+ \to \mu^+ \nu e^+ e^-$ for various cuts.}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
               &  tree level & form factors as given by CHPT\\
\hline
full phase space &$2.49 \cdot 10^{-5}$  &         $2.49 \cdot 10^{-5}$\\
\hline
$z\le 10^{-3}$ & $ 2.07\cdot 10^{-5} $ & $ 2.07 \cdot 10^{-5}$\\
\hline
$z\ge 10^{-3}$ &$4.12\cdot 10^{-6}$& $4.20\cdot 10^{-6}$\\
\hline
$z \ge ( 20\ MeV/M_K)^2$ & $ 3.15\cdot 10^{-6}$ & $3.23\cdot 10^{-6}$\\
\hline
$z \ge (140\ MeV/M_K)^2$ & $ 4.98\cdot 10^{-8}$ & $8.51\cdot 10^{-8}$\\
\hline
$ x\ge 40\ MeV/M_K$ & $ 1.58\cdot 10^{-5} $ & $1.58\cdot 10^{-5}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
It can be seen that for this decay most of the branching ratio is
generated at very low electron-positron invariant masses. As can be seen
from the result for the cuts used in Ref. \cite{DIAMANT}, the effect
of the structure dependent terms is most visible at high invariant
electron-positron invariant mass. Our calculation, including the effect
of the form factors agrees well with their data. We disagree, however,
with the numerical result obtained by Ref. \cite{KRISHNA} by about an
order of magnitude.

For the decay $K^+ \to e^+ \nu \mu^+\mu^-$, we obtain for the tree level
or IB contribution a branching ratio
\begin{equation}
BR_{IB}(K^+ \to e^+ \nu \mu^+ \mu^- ) =
     3.06\cdot 10^{-12}
\end{equation}
      and,
including the form factors,
\begin{equation}
BR_{total}(K^+ \to e^+ \nu \mu^+ \mu^- ) =
 1.12\cdot 10^{-8}.
\end{equation}
 Here the structure
dependent terms are the leading contribution since the inner
Bremsstrahlung  contribution is helicity suppressed as can be seen
from the factor $m_l$ in $\overline{L_\mu}$.

For the decays with \underline{identical}
leptons we obtain for the muon case a
branching ratio  of
\begin{equation}
BR_{total}(
K^+ \to \mu^+\nu\mu^+\mu^-) =
1.35\cdot10^{-8}
\label{predicted}
\end{equation}
for the full phase space including the effects of the
form factors.
The inner Bremsstrahlung or the tree level branching
ratio for this decay is
\begin{equation}
BR_{IB}(
K^+ \to \mu^+\nu\mu^+\mu^-) =
3.79\cdot10^{-9}.
\end{equation}
For the decay with two positrons and one electron the integration
over full phase space
for the tree level results
is very sensitive to the behaviour for small
pair masses.
 We have given the tree level and the full prediction,
including form factor effects in table \ref{KLLLP}. The cuts are
always on both invariant masses :
\begin{eqnarray}
z&=&(p_1 + p_2)^2 / M_K^2 \nonumber\\
z_1 &=& (p_l + p_2)^2 / M_K^2\ .
\end{eqnarray}
\begin{table}
\caption{\label{KLLLP}
         Theoretical values for the branching ratios for the decay
$K^+ \to   e^+ \nu e^+ e^-$ for various cuts.}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
               &  tree level & form factors as given by CHPT\\
\hline
full phase space & $\approx 4 \cdot 10^{-9}$ & $ 1.8\cdot 10^{-7}$\\
\hline
$z,\ z_1 \ge 10^{-3}$&  $3.0\cdot 10^{-10}$&      $1.22 \cdot 10^{-7}$\\
\hline
$z,\ z_1 \ge (50\ MeV/M_K)^2 $
&  $5.2\cdot 10^{-11}$&      $8.88 \cdot 10^{-8}$\\
\hline
$z,\ z_1 \ge (140\ MeV/M_K)^2 $
&  $2.1\cdot 10^{-12}$&      $3.39 \cdot 10^{-8}$\\
\hline
\end{tabular}
\end{center}
\end{table}
The values for the masses used are those of     $K^+$ and $\pi^+$.
For $L_9$ and $L_{10}$ we used the values given in chapter \ref{Intro},
\begin{eqnarray}
L_9^r (M_\rho ) & = & 6.9 \cdot 10^{-3} \nonumber\\
L_{10}^r (M_\rho ) &=& -5.5 \cdot 10^{-3} \; \; .
\end{eqnarray}

\subsection{Present experimental status}

Only decays with an electron positron pair in the final state,
decays (\ref{S22}) and (\ref{S23}),
have been observed.

 Both have been measured
in the same experiment       \cite{DIAMANT}.
The decay $K^+ \to \mu^+ \nu e^+ e^-$ was measured with a branching
ratio of $(1.23 \pm 0.32)\cdot 10^{-7}$ with a lower cut on the
electron positron invariant mass of $140\ MeV$.
The measurement is compatible with our calculation including the
form factor effects for the relevant region of phase space.
This measurement
was then  extrapolated \cite{DIAMANT}
using the result of \cite{KRISHNA} to the
full phase space. Since we  disagree with that calculation,
we also disagree with the extrapolation.

In the same experiment,  4 events of the type
                                            $K^+ \to e^+\nu e^+ e^-$
were
observed where both electron positron pair invariant masses were
above 140 $MeV$. This corresponds to a branching ratio for this
region of phase space of $(2.8^{+2.8}_{-1.4})\cdot 10^{-8}$.
This result is compatible within errors  with our calculation, see
table \ref{KLLLP}.
The matrix element of Ref. \cite{KRISHNA} was again used for the
extrapolation to full phase space\cite{DIAMANT}.
Apart from our numerical disagreement,
the calculation of Ref. \cite{KRISHNA} was for the case of
non-identical leptons and cannot be applied here.

For the decay $K^+ \to \mu^+\nu\mu^+\mu^-$ an upper limit
of $4.1 \cdot 10^{-7}$ exists \cite{ATIYA}. This upper limit is
compatible with our theoretical result, Eq. (\ref{predicted}).

The decay $K^+ \to e^+\nu\mu^+\mu^-$ has not been looked for so far
and should be within the capabilities of DAFNE given the
branching ratio predicted in the previous subsection.
This decay proceeds
almost entirely through the structure dependent terms and is as such a
good test of our calculation.

\subsection{Improvements at DAFNE}

The decays discussed in this section, $K^+ \to l^+\nu
l^+l^-$, are
complementary to the decays $K^+ \to l^+\nu\gamma$. As was the case
for the analogous decay, $\pi^+ \to e^+\nu e^+ e^-$ \cite{egli},
 it may be
possible to explore phase space
more easily with this process than with
$K^+ \to l^+ \nu \gamma$ to resolve ambiguities in the form factors.

As can be seen from our predictions, tables \ref{KLLL1} and
\ref{KLLLP}, all the decays considered in this section should be
observable at DAFNE. Large improvements in statistics are
possible since less severe cuts than those used in the past
experiments should be possible. In the decays with a $\mu^+ \mu^-$
pair and the decay $K^+ \to e^+ \nu e^+ e^-$
the effects of the form factors are already large in the total
rates and should be easily visible at DAFNE. In the decay
$K^+ \to \mu^+ \nu e^+ e^-$
most of the total rate is for small invariant
mass of the pair and is given by the inner Bremsstrahlung
contribution.  There are, however, regions of phase space where the
form factor effects are large and DAFNE should have enough
statistics to be able to study these regions.


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\section{$K_{l3}$ decays}

The decay channels considered in this section are
\bearr
 K^+(p) &\rightarrow& \pi^0 (p') l^+ (p_l) \nu _l (p_\nu) \hspace{1cm}
[K_{l3}^+] \label{s31}
\\
K^0(p) &\rightarrow &\pi^- (p') l^+ (p_l) \nu_l (p_\nu) \hspace{1cm} [K_{l3}^0]
\label{s32}
\eearr
and their charge conjugate modes. The symbol  $l$ stands for $\mu$ or $e$. We
do
not consider electromagnetic corrections and correspondingly set $\alpha = 0$
throughout this section.

\subsection{Matrix elements and kinematics}

The matrix element for $K_{l3}^+$
has the general structure
\bearr
T& =& \frac{G_F} {\sqrt{2}} V_{us}^\star l^\mu { F_\mu}^+ (p',p)
\label{s34}
\eearr
with
\bearr
l^\mu& =& \bar{u} (p_\nu)\gamma^\mu  (1- \gamma_5) v (p_l)
\nonumber \\
{ F_\mu}^+ (p',p)& =& < \pi^0 (p') \mid V_\mu^{4-i5} (0)
\mid K^+(p)>
\nonumber \\
&=& \frac{1}{\sqrt{2}} [(p'+p)_\mu f^{K^+\pi^0}_+ (t) + (p-p')_\mu
f_-^{K^+\pi^0} (t)]. \label{s35}
\eearr
To obtain the matrix element for $K_{l3}^0$ , one replaces $F^+_\mu$ by
\bearr
{ F_\mu }^0 (p',p)& =& < \pi^- (p') \mid V_\mu^{4-i5} (0)
\mid K^0(p)>
\nonumber \\
&=&  (p'+p)_\mu f^{K^0\pi^-}_+ (t) + (p-p')_\mu f_-^{K^0\pi^-} (t).
\label{s35a}
\eearr
 The processes (\ref{s31})
and (\ref{s32}) thus involve the four $K_{l3}$ form factors
$f^{K^+\pi^0}_\pm (t)$, $f^{K^0 \pi^-}_\pm (t)$ which depend on
\be
t = (p'-p)^2 = (p_l + p_\nu)^2,
\label{s36}
\ee
the square of the four momentum transfer to the leptons.


Let $f_\pm^{K\pi}=f_\pm^{K^+\pi^0}$ or $f_\pm^{K^0\pi^-}$.
$f_+^{K\pi}$ is referred to as the vector form factor, because
it specifies the $P$-wave  projection of the crossed channel matrix elements
 $< 0 \mid V^{4-i5}_\mu(0) \mid K^+, \pi^0 \;\mbox{in} >$.
 The $S$-wave projection is described by the scalar form factor
\be
f^{K\pi}_0 (t) = f^{K\pi}_+ (t) + \frac{t}{M^2_K - M^2_\pi} f^{K\pi}_-(t) \; \;
{}.
 \label{s37}
\ee
Analyses of $K_{l3}$ data frequently assume a linear dependence
\be
f^{K\pi}_{+,0} (t) = f^{K\pi}_+ (0) \left[ 1 + \lambda_{+,0}
\frac{t}{M^2_{\pi^+}} \right] \; \; .
\label{s38}
\ee
For a discussion of the validity of this approximation see \cite{GL2,pdg}
and references cited therein. Eq. (\ref{s38}) leads to a constant
$f^{K\pi}_- (t)$ ,
\be
f^{K\pi}_- (t) = f^{K\pi}_- (0) = f^{K\pi}_+ (0) (\lambda_0 -
\lambda_+) \frac{M_K^2 - M_\pi^2}{M_{\pi^+}^2}.
\label{s381}
\ee
The form factors $f^{K\pi}_{\pm,0} (t)$ are analytic functions in the complex
$t$-plane cut along the positive real axis. The cut starts at $t=(M_K +
M_\pi)^2$. In our phase convention, the form factors are real in the physical
region
\be
m^2_l \leq t \leq (M_K - M_\pi)^2.
\label{s39}
\ee
The kinematics of (spin averaged) $K_{l3}$ decays needs two variables, for
which we choose
\be
y = 2 p p_l/M_K^2\;,\; \; z = 2 p p'/ M_K^2 =(-t+M_\pi^2 +M_K^2)/M_K^2 \; \;.
\label{s311}
\ee
In the $K$ rest frame, $y$ ($z$) is proportional to the charged lepton (pion)
energy,
\be
y=2E_l/M_K \; \; , \; \; z=2E_\pi/M_K \; \; .
\label{s311a}
\ee

The physical region for $y$ and $z$ is
\bearr
2 \sqrt{r_l} \leq & y& \leq 1 + r_l - r_\pi
\nonumber \\
A(y)-B(y) \leq& z& \leq A(y)+B(y)
\nonumber\\
A(y)&=&(2-y)(1+r_l+r_\pi-y)/[2(1+r_l-y)]
\nonumber \\
B(y)&=& \sqrt{y^2-4r_l} (1+r_l-r_\pi-y)/[2(1+r_l-y)]
\nonumber \\
r_l &=& m^2_l/M_K^2, r_\pi = M^2_\pi/ M_K^2.
\label{s3121}
\eearr
or, equivalently,
\bearr
2\sqrt{r_\pi} \leq &z& \leq 1+r_\pi -r_l
\nonumber\\
C(z)-D(z) \leq &y& \leq C(z)+D(z)
\nonumber \\
C(z)&=& (2-z)(1+r_\pi +r_l -z)/[2(1+r_\pi-z)]
\nonumber \\
D(z)&=&\sqrt{z^2-4r_\pi}(1+r_\pi -r_l -z)/[2(1+r_\pi -z)] \; \; .
\nonumber \\
\label{s312}
\eearr


  \subsection{Decay rates}

The differential decay rate for $K_{l3}^+$ is given by
\bearr
d\Gamma & =& \frac{1}{2M_K(2\pi)^5} \sum_{spins}|T|^2 d_{LIPS}(p;p_l,p_\nu,p')
\; \; .
 \label{s312a}
\eearr
The Dalitz plot density
\be
\rho(y,z)=\frac{d^2\Gamma}{dy dz} = \frac{M_K}{256\pi^3} \sum_{spins}|T|^2
\label{s312b}
\ee
is a Lorentz invariant function which contains $f_{\pm}^{K^+\pi^0}$ in the
following form,
\bearr
\rho(y,z) = \frac{M_K^5G^2_F \mid V_{us}\mid^2 }{256 \pi^3}
 \left [ A (f^{K^+\pi^0}_+)^2 + B f^{K^+\pi^0}_+ f^{K^+\pi^0}_- + C
(f^{K^+\pi^0}_-)^2 \right]
 \label{s313}
\eearr
with
\bearr
A(y,z)& =& 4 (z+y-1) (1-y) + r_l [4 y+ 3 z-3] - 4 r_\pi + r_l (r_\pi - r_l)
\nonumber \\
B(y,z)& =& 2 r_l (3-2y - z + r_l - r_\pi)
\nonumber \\
C(y,z)& =& r_l (1+ r_\pi - z - r_l).
\label{s314}
\eearr
The quantities $(A,B,C)$ are related to the ones quoted by the PDG \cite{pdg}
by
\be
(A,B,C) = \frac{8}{M_K^3} (A,B,C)_{\mbox{\tiny{PDG}}} \; \; .
\label{s315}
\ee
To obtain the rate for $K_{l3}^0$, one replaces in (\ref{s313})
$f^{K^+\pi^0}_\pm $ by $\sqrt{2} f^{K^0\pi^-}_\pm$.

For convenience we also display the $K_{\mu3}/K_{e3}$ rates evaluated in the
approximation (\ref{s38}) for the form factors,
\bearr
\Gamma (K^+_{\mu3}) / \Gamma (K^+_{e3}) &=&
\frac{
0.645 +2.087 \lambda_+ +1.464 \lambda_0 +3.375 \lambda_+^2 +2.573 \lambda_0^2}
{1 +3.457 \lambda_+ +4.783 \lambda_+^2}
\nonumber  \\
\Gamma (K^0_{\mu3}) / \Gamma (K^0_{e3})& = &
\frac{
0.645 +2.086 \lambda_+ +1.459 \lambda_0 +3.369 \lambda_+^2 +2.560\lambda_0^2}
{1+3.456\lambda_+ +4.776 \lambda_+^2} \; .
\label{s316b}
\eearr


We have used the physical masses \cite{pdg} in evaluating these ratios and
$M_{\pi^+}$ to scale the slope in both cases. The
terms linear and quadratic in $\lambda_0$ are proportional to $m_l^2$ and
 therefore strongly suppressed in the
electron case. We do not include them in the denominators,
 because these coefficients
are smaller than $10^{-4}$. The interference term $\lambda_0\lambda_+$ is
absent by angular momentum conservation.
Furthermore, one has
\be
\int dy\;dz A(y,z) = \left\{ \begin{array}{ll}
                             0.0623 & [K_{\mu 3}^+] \nonumber \\
                             0.0606 & [K_{\mu 3}^0] \; \; .\\
                             \end{array}
                         \right.
\ee
\subsection{Determination of
the $K_{l3}$ form factors}

Measurements of the Dalitz plot distribution (\ref{s313}) of $K_{\mu3}$ data
allow one in principle to pin down the form factors
(up to a sign) in the range
 $m_\mu^2 \leq t \leq
(M_K - M_\pi)^2$.
 Measuring the $K_{\mu3}/K_{e3}$ branching ratio and
then using
(\ref{s316b}) gives a relationship between $\lambda_+$ and
$\lambda_0$  which is valid
in the approximation (\ref{s38}). Furthermore, muon polarization experiments
measure the weighted average of the ratio $f_-^{K\pi}(t)/f_+^{K\pi}(t)$ over
the $t$
range of the experiment \cite{pdg,mupol}. On the other hand, the electron modes
$K_{e3}$ are sensitive to $f_+^{K\pi}$ only, because the other contributions
are suppressed by the factor $(m_e/M_K)^2 \simeq 10^{-6}$, see eqs.
(\ref{s313}), (\ref{s314}).

Isospin breaking effects in $f^{K^+\pi^0}_+ (0)$ and $f_+^{K^0 \pi^-}(0)$
play
a central role in the determination of the Kobayashi-Maskawa matrix element
$V_{us}$ from $K_{e3}$ data, see \cite{lroos} for a detailed discussion of this
point. In the following we concentrate on the measurement of the slopes
$\lambda_{+,0}$.

\subsection{Previous measurements}

We refer the reader to the 1982 version of the PDG \cite{pdg82}
\footnote{Please note that the most recent measurements of $\lambda_{+,0}$ go
back to 1981 \cite{pdg}!}
for a critical discussion of the wealth of experimental information on
$\lambda_{+,0}^{K\pi}$. Here we content ourselves with a short summary.

\underline{$K_{e3}$-experiments}

The $\lambda_+$ values obtained are fairly consistent. The average values are
\bearr
K^+_{e3} \;:\; \lambda_+ &=& 0.028 \pm 0.004 \; \; \; \; \mbox{Ref.} \cite{pdg}
\nonumber \\
K_{e3}^0 \ : \, \lambda_+& =& 0.030 \pm 0.0016 \; \; \mbox{Ref.} \cite{pdg} \;
\; .
\label{s318}
\eearr

\underline{$K_{\mu 3}$-experiments}

The result by Donaldson et al. \cite{donaldson}
\bearr
\lambda_+& =& 0.030 \pm 0.003
\nonumber \\
\lambda_0& =& 0.019 \pm 0.004
\label{s319}
\eearr
dominates the statistics in the $K^0_{\mu3}$ case. The $\lambda_+$ value
(\ref{s319}) is consistent with the $K_{e3}$ value (\ref{s318}).
 However, the situation concerning the slope $\lambda_0$
is rather unsatisfactory, as the following (chronological) list illustrates.
\be
\lambda_0 = \left\{\begin{array}{llll}
              0.0341 &\pm &0.0067 & \cite{birulev} \\
              0.050 &\pm &0.008  & \cite{cho} \\
              0.039 &\pm &0.010  &  \cite{hill} \\
              0.047 &\pm &0.009  & \cite{clark} \\
              0.025 &\pm &0.019  & \cite{buchanan} \\
              0.019 &\pm &0.004  & \cite{donaldson} \; \; .
                \end{array}
              \right.
\label{l:s324}
\ee
The $\chi^2$ fit to the $K^0_{\mu3}$ data yields $\lambda_+ = 0.034 \pm 0.005$,
$\lambda_0 = 0.025 \pm 0.006$ with a $\chi^2/DF = 88/16$ \cite[p.76]{pdg82}!
The situation in the charged mode $K^+_{\mu3}$ is slightly better \cite{pdg82}.


\subsection{Theory}

The theoretical prediction of $K_{l3}$ form factors has a long history,
starting
in the sixties with the current algebra evaluation of $f^{K^+\pi^0}_\pm$. For
an
early review of the subject and for references to work prior to CHPT
evaluations
of $f_\pm$ we refer the reader to \cite{chounet} (see also Ref.\cite
{shabkl4}). Here we concentrate on the
evaluation of the form factors in the framework of CHPT. We restrict our
consideration to the isospin symmetry limit $m_u =m_d$, as a result of which
one has
\be
f^{K^0 \pi^-}_{\pm,0} (t) =  f^{K^+\pi^0}_{\pm,0} (t) \equiv f_{\pm,0}(t) \; ;
\; m_u = m_d \; \; . \label{s321}
\ee

\vspace{.5cm}

{\bf{A) Chiral prediction at one-loop order}}

\vspace{.5cm}

In Ref. \cite{GL2}, the vector current matrix elements $< M' \mid q
\gamma^\mu \frac{\lambda^a}{2} q \mid M>$ have been calculated up to and
including terms of order $t = (p'-p)^2$ and of order $m_u, m_d$ and $m_s$ in
the invariant form factors. For reasons which will become evident below, we
consider here, in addition to the $K_{l3}$ form factors, also
the electromagnetic form factor of the pion
\be
< \pi^+ (p') \mid V^\mu_{em} (0) \mid \pi^+ (p) > = (p'
+p)^\mu F^\pi_V (t).
\label{s322}
\ee
The low-energy representation for $F_V^\pi (t)$ \cite{GL2,bijnens} and
 $f_+ (t)$ \cite{GL2} reads

$$
F_V^\pi (t) = 1+ 2 H_{\pi \pi} (t) + H_{KK} (t)
$$
\be
f_+ (t) = 1 + \frac{3}{2} H_{K\pi} (t) + \frac{3}{2} H_{K\eta} (t).
\label{s323}
\ee
The quantity $H(t)$ is a loop function
displayed  in appendix \ref{loop}. It contains the low-energy constant $L_9$.
 The indices attached to $H(t)$ denote the masses running in the loop.

Since $L_9$ is  the only unknown occurring in $F_V^\pi (t)$ and
in $f_+
(t)$, we need experimental information on the {\em slope} of one of these two
form factors to obtain a parameter-free low-energy representation of the
other.

The analogous low-energy representation of the scalar form factor is
\bearr
f_0 (t) &=& 1 + \frac{1}{8F^2} \left( 5 t - 2 \Sigma_{K\pi} - 3
\frac{\Delta^2_{K\pi}}{t}  \right) \bar{J}_{K\pi} (t)
\nonumber \\
&+& \frac{1}{24F^2} \left( 3 t - 2 \Sigma_{K\pi} -
\frac{\Delta_{K\pi}^2}{t} \right) \bar{J}_{K\eta} (t)
\nonumber \\
&+& \frac{t}{\Delta_{K\pi}} \left( \frac{F_K}{F_\pi} - 1 \right).
\label{s325}
\eearr
The function $\bar{J} (t)$ is listed in appendix \ref{loop}, and
$\Sigma_{K\pi}$ and $\Delta_{K\pi}$ stand for
\bearr
\Sigma_{K\pi}& =& M_K^2 + M_\pi^2
\nonumber \\
\Delta_{K \pi}& =& M_K^2 - M_\pi^2 \; \; .
\label{s326}
\eearr
The measured value \cite{lroos} $F_K/F_\pi = 1.22 \pm 0.01$ may be used to
obtain a parameter-free prediction of the scalar form factor $f_0 (t)$.

\vspace{.5cm}

{\bf{B) Momentum dependence of the vector form factor}}

\vspace{.5cm}

In the spacelike interval $\sqrt{-t} < 350$ MeV the low-energy representation
(\ref{s323}) for the electromagnetic form factor $F_V^\pi (t)$ is very well
approximated by the first two terms in the Taylor series expansion
around $t=0$,
\be
F_V^\pi (t) = 1 + \frac{1}{6} <r^2>^\pi_V t + \cdots \; \; .
\label{s324}
\ee
Likewise, the linear approximation
\be
f_+ (t) = f_+ (0) \left \{ 1 + \frac{1}{6} <r^2>_V^{K\pi}  t
+ \cdots \right\}
\label{s325a}
\ee
reproduces the low-energy representation (\ref{s323}) very
well, see Fig. \ref{fkl31}.
\begin{figure}[t]
\vspace{8cm}
\caption{The vector and scalar form factors $f_+(t)$ and $f_0(t)$.
\label{fkl31}
         }
\end{figure}
 This is  in agreement with the observed Dalitz plot distribution, which is
consistent with a form factor linear in $t$. The charge radii are
$$
<r^2>^\pi_V = \frac{12 L_9^{r}}{F^2}- \frac{1}{32 \pi^2 F^2} \left \{
2 \ln \frac{M_\pi^2}{\mu^2} + \ln \frac{M_K^2}{\mu^2} + 3 \right \}
$$
$$
< r^2>^{K\pi}_V = < r^2>_V^\pi - \frac{1}{64 \pi^2 F^2} \left \{ 3 h_1
\left (\frac{M_\pi^2}{M_K^2} \right) + 3 h_1 \left (\frac{M_\eta^2}{M_K^2}
\right ) \right.
$$
\be
\hspace{-1cm} \left. + \frac{5}{2} \ln \frac{M_K^2}{M_\pi^2} + \frac{3}{2} \ln
\frac{M_\eta^2}{M_K^2} - 6 \right \}
\label{s327}
\ee
where
\be
h_1(x) = \frac{1}{2} \frac{(x^3 - 3x^2 - 3x +1)}{(x-1)^3} \ln x + \frac{1}{2}
\left( \frac{x+1}{x-1} \right)^2 - \frac{1}{3} \; \; .
\label{s328}
\ee
To evaluate these relations numerically, we use the measured charge radius of
the pion:
\be
<r^2>_V^\pi = 0.439 \pm 0.008 \mbox{fm}^2 \; \; {\mbox{Ref.}} \cite{r2pion}
\label{s329}
\ee
as input and obtain the prediction
\bearr
\lambda_+& =& \frac{1}{6} M^2_{\pi^+} <r^2>_V^{K\pi} = 0.031
\label{s330}
\eearr
in agreement with
the experimental results (\ref{s318}), (\ref{s319})
\footnote{
We do not quote an error for the result (\ref{s330}), because one should
estimate  higher order chiral corrections for this purpose.}
                                .
{}From this (and from the considerably more detailed discussion in Ref.
\cite{GL2}), one concludes, in agreement with other
theoretical investigations \cite{kl3other}, that the measured charge radii
$<r^2>_V^{\pi}$ and $<r^2>_V^{K \pi}$ are consistent with the low-energy
prediction.

\vspace{.5cm}

{\bf{C) Momentum dependence of $f_0 (t)$. Dashen-Weinstein and
Callan-Treiman relations}}

\vspace{.5cm}

In the physical region of $K_{l3}$ decay the low-energy representation
(\ref{s325}) for the scalar form factor is approximated by the linear
formula
\be
f_0 (t) = f_+ (0) \left \{ 1 + \frac{1}{6} <r^2>^{K\pi}_S t +
\cdots \right \}
\label{s331}
\ee
to within an accuracy of 1 \%. (See Fig. \ref{fkl31}). The curvature generated
by higher
order terms is also expected to be negligible in the physical region of the
decay \cite{GL2}. For the slope $<r^2>^{K\pi}_S$ one obtains
 \bearr
<r^2>^{K\pi}_S& = &\frac{6}{M_K^2 - M_\pi^2} \left ( \frac{F_K}{F_\pi}-1
\right) + \delta_2 + O(\hat{m},m_s)
\nonumber \\
\delta_2 &=& - \frac{1}{192 \pi^2F^2} \left \{ 15 h_2 \left(
\frac{M_\pi^2}{M_K^2} \right) + \frac{19 M_K^2 + 3 M^2_\eta}{M_K^2 + M_\eta^2}
h_2 \left( \frac{M_\eta^2}{M_K^2} \right) - 18 \right\}
\nonumber \\
\label{s332}
\eearr
where
\bearr
h_2(x)& =& \frac{3}{2} \left( \frac{1+x}{1-x}\right)^2 +
\frac{3x(1+x)}{(1-x)^3} \ln x,
\nonumber \\
h_2 (x) &=& h_2 \left(\frac{1}{x} \right)\; ,\;h_2(1) = 1,
\nonumber \\
\hat{m}&=&(m_u+m_d)/2\; \; .
\eearr
This (parameter-free) predic\-tion is a modi\-fied vers\-ion of the
Dashen-Wein\-stein
relation \cite{dawein}, which results if the nonanalytic contribution
$\delta_2$ is dropped. Dashen, Li, Pagels and Weinstein \cite{dalipa} were the
first to point out that the low-energy singularities generated by the Goldstone
bosons affect this relation. The modified relation is formulated as a
prediction for the slope of $f_0(t)$ at the unphysical point $t_1 = M^2_K +
M^2_\pi$. Their expression for this slope however has two shortcomings: (i) it
does not account for all corrections of order ${\cal M}$; (ii) The slope at
$t_1$ differs substantially from the slope in the physical region of the decay
\cite{GL2,slope}, see Fig. \ref{fkl32}.
\begin{figure}[t]
\vspace{8cm}
\caption{ The normalized slopes of the vector and the scalar form factors.
Curve 1: the normalized slope $M^2_{\pi^+} df_+(t)/dt$. Curve 2: the
normalized
slope $M^2_{\pi^+} df_0(t)/dt$. Near the $\pi K$ threshold $t_0=(M_K+M_\pi)^2$,
the vector form factor behaves as $f_+(t)=f_+(t_0) + O[(t-t_0)]$, whereas
$f_0(t)=f_0(t_0) + O[(\protect\sqrt{t-t_0})]$. The slope of the scalar form
factor is therefore singular at $t=(M_K+M_\pi)^2$.
\label{fkl32}
          }
\end{figure}

Algebraically, the correction $\delta_2$ is of the same order in the low-energy
expansion as the term involving $F_K/F_\pi -1$. Numerically, the correction is
however small: $\delta_2$ reduces the prediction by 11 \%. With $F_K/F_\pi =
1.22 \pm 0.01$ the low-energy theorem (\ref{s332}) implies
\bearr
<r^2>^{K\pi}_S &=& 0.20 \pm 0.05 \mbox{fm}^2
\nonumber \\
\lambda_0 &=& \frac{1}{6} M_{\pi^+}^2 <r^2>^{K\pi}_S = 0.017 \pm 0.004
\label{s333}
\eearr
where the error is an estimate of the uncertainties due to higher order
contributions. The prediction (\ref{s333}) is in agreement with the
high-statistics experiment \cite{donaldson} quoted in (\ref{s319}) but in
flat disagreement with some of the more recent data listed in
(\ref{l:s324}).

In the formulation of Dashen and Weinstein \cite{dawein}, the Callan-Treiman
relation \cite{caltrei} states
that the scalar form factor evaluated at $t= M_K^2 - M^2_\pi$ differs from
$F_K/F_\pi$ only by terms of order $m_u, m_d$: the quantity
\be
\Delta_{\mbox{\tiny{CT}}} = f_0 (M_K^2 - M^2_\pi) -
\frac{F_K}{F_\pi} \label{s334}
\ee
is of order $\hat{m}$. Indeed,  the low-energy
representation (\ref{s325})  leads to
\be
\Delta_{\mbox{\tiny{CT}}} = - \frac{M^2_\pi}{2 F^2} \left \{ \bar{J}_{K\pi}
(M_K^2 - M_\pi^2) + \frac{1}{3} \bar{J}_{K\eta} (M^2_K - M_\pi^2) \right\} +
O(\hat{m}m_s) \; \; .
\label{s335}
\ee
Numerically, $\Delta_{\mbox{\tiny{CT}}} = - 3.5 \cdot 10^{-3}$. The
Callan-Treiman relation should therefore hold to a very high degree of
accuracy. If the form factor is linear from $t=0$ to $t = M_K^2 - M^2_\pi$ then
the slope must be very close to
\be
\lambda_0^{\tiny{CT}} = \frac{ {{M_{\pi^+}}^{2}} }{M_K^2-M^2_\pi}
 \left( \frac{F_K}{F_\pi}-1 \right) = 0.019
,
\label{s336}
\ee
in agreement with (\ref{s333}) and with the experimental result of Ref.
\cite{donaldson}, but in disagreement with, e.g., the value found in Ref.
\cite{cho}. We see no way to reconcile the value $\lambda_0 = 0.050$ with
chiral symmetry.


\subsection{Improvements at DAFNE}

DAFNE provides the opportunity to improve our knowledge of $K_{l3}$ decays in a
very substantial manner - in particular, it should be possible to clarify the
issue of the slope $\lambda_0$ of the scalar form factor $f_0$. To illustrate,
we compare in table \ref{tkl3} the hitherto
obtained
number of events (third column)  with the expected ones at DAFNE
(fourth column). The last column displays the remarkable increase in statistics
obtainable at DAFNE.


\begin{table}
\protect
\begin{center}
\caption{Rates of $K_{l3}$ decays.
The number of events in the
third column
 corresponds to  those data which are of relevance for the
determination of the  slope $\lambda_0$ of the scalar
form factor.
\label{tkl3}
}
\vspace{1em}
\begin{tabular}{|c|c|c|c|c|} \hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{$\sharp$ events} & \\ \hline
  & branching  & Particle Data
  & DAFNE  & improve-                  \\
  & ratio & Group
  & {1 year} & ment \\ \hline
$K^+ \rightarrow \pi^0 \mu^+ \nu_\mu $ &
 $3.18 \cdot 10^{-2} $ & $ 10^5 $ & $3\cdot 10^8$ & $3\cdot 10^3$      \\
\hline  $K_L \rightarrow \pi^{\pm} \mu^{\mp} \nu $ &
 $27 \cdot 10^{-2} $ & $ 4\cdot 10^6 $ & $3\cdot 10^8$ & $70$  \\ \hline
\end{tabular}
\end{center}
\end{table}


%before each section

\setcounter{equation}{0}
\setcounter{subsection}{0}
\setcounter{table}{0}
\setcounter{figure}{0}
\clearpage
%New version of Sect. 4: c3I, dI expressed in terms of c3I
%Vhat instead of V_{irr} : date June 4, 1992
\section{Radiative $K_{l3}$ decays} \label{section4}
The decay channels considered in this section are
\beqa
K^+(p) & \ra & \pi^0(p') l^+(p_l) \nu_l(p_{\nu}) \gamma(q) \qquad
[K^+_{l3\gamma}] \no \\*
K^0(p) & \ra & \pi^-(p') l^+(p_l) \nu_l(p_{\nu}) \gamma(q) \qquad
[K^0_{l3\gamma}] \no
\eeqa
and the charge conjugate modes. We only consider real photons ($q^2 = 0$).

\subsection{Matrix elements}
The matrix element for $K^+_{l3\gamma}$ has the general structure
\beqa
T & = & \left.\dfrac{G_F}{\sqrt{2}} e V^*_{us} \ve^{\mu}(q)^*
\right\{(V^+_{\mu\nu} -
A^+_{\mu\nu}) \ol{u}(p_{\nu}) \gamma^{\nu} (1 - \gamma_5)
v(p_l) \label{eq:T}  \\*
  &   & + \left. \dfrac{F^+_{\nu}}{2 p_lq} \ol{u}(p_{\nu}) \gamma^{\nu}
(1 - \gamma_5) (m_l - \fsl p_l - \fsl q) \gamma_{\mu} v(p_l)\right\}
\equiv \ve^{\mu *} A^+_{\mu}. \no
\eeqa
The diagram of Fig. \ref{fig41}.a corresponding to the first part of
Eq. (\ref{eq:T}) includes Bremsstrahlung off the $K^+$.
The lepton Bremsstrahlung diagram of Fig. \ref{fig41}.b is represented
by the second part of Eq. (\ref{eq:T}).
The hadronic tensors $V^+_{\mu\nu}, A^+_{\mu\nu}$ are defined as
\beq
I^+_{\mu\nu} = i \int d^4x e^{i q x} \langle \pi^0(p') \mid
T\{V^{em}_\mu(x) I^{4-i 5}_\nu(0)\} \mid K^+(p) \rangle ,
\qquad I = V,A .
\eeq
$F^+_\nu$ is the $K^+_{l3}$ matrix element
\beq
F^+_\nu = \langle \pi^0(p') \mid V^{4-i 5}_\nu(0) \mid K^+(p) \rangle.
\eeq
The tensors $V^+_{\mu\nu}$ and $A^+_{\mu\nu}$ satisfy the Ward identities
\beqa
q^\mu V^+_{\mu\nu} & = & F^+_\nu \label{eq:Ward} \\*
q^\mu A^+_{\mu\nu} & = & 0  \no
\eeqa
leading in turn to
\beq
q^\mu A^+_\mu = 0~,\label{eq:WI}
\eeq
as is required by gauge invariance.

For $K^0_{l3\gamma}$, one obtains the corresponding amplitudes and
hadronic tensors by making the replacements
\beqa
K^+ & \ra & K^0,\qquad \pi^0 \ra \pi^- \no \\*
V^+_{\mu\nu} & \ra & V^0_{\mu\nu},\qquad A^+_{\mu\nu} \ra A^0_{\mu\nu}\\*
F^+_\nu & \ra & F^0_\nu,\qquad A^+_\mu \ra A^0_\mu. \no
\eeqa

To make the infrared behaviour transparent,
it is convenient to separate the tensors $V^+_{\mu\nu}, V^0_{\mu\nu}$
into two parts:
\beqa
V^+_{\mu\nu} & = & \hat{V}^+_{\mu\nu} + \dfrac{p_\mu}{pq} F^+_\nu
\label{eq:Low} \\*
V^0_{\mu\nu} & = & \hat{V}^0_{\mu\nu} + \dfrac{p'_\mu}{p'q} F^0_\nu. \no
\eeqa
Due to Low's theorem, the amplitudes $\hat{V}^{+,0}_{\mu\nu}$
are finite for $q \ra 0$. The axial amplitudes
$A^{+,0}_{\mu\nu}$ are automatically infrared finite.
The Ward identity (\ref{eq:Ward}) implies that the vector amplitudes
$\hat{V}^{+,0}_{\mu\nu}$ are transverse:
\beq
q^\mu \hat{V}^{+,0}_{\mu\nu} = 0. \eeq

For on-shell photons, Lorentz and parity invariance together with gauge
invariance allow the general
decomposition (dropping the superscripts +,0 and terms that vanish
upon contraction with the photon polarization vector)
\beqa
\hat{V}_{\mu\nu} & = & V_1 \left(g_{\mu\nu} - \dfrac{W_\mu q_\nu}
{qW}\right) + V_2 \left(p'_\mu q_\nu - \dfrac{p'q}{qW} W_\mu q_\nu
\right) \no \\*
& & + V_3 \left( p'_\mu W_\nu - \dfrac{p'q}{qW} W_\mu W_\nu \right)
+ V_4\left( p'_\mu p'_\nu - \dfrac{p'q}{qW} W_\mu p'_\nu \right)
\label{eq:tensor} \\
A_{\mu\nu} & = & i \ve_{\mu\nu\rho\sigma} (A_1 p'^\rho q^\sigma +
A_2 q^\rho W^\sigma) + i \ve_{\mu\lambda\rho\sigma} p'^\lambda
q^\rho W^\sigma (A_3 W_\nu + A_4 p'_\nu) \no \\
F_\nu & = & C_1 p'_\nu + C_2 (p - p')_\nu  \no \\*
W & = &  p_l + p_\nu. \no
\eeqa
With the decomposition (\ref{eq:Low}) we can write the matrix element
for $K^+_{l3\gamma}$ in (\ref{eq:T}) in a form analogous to Eq.
(\ref{k3}) for $K_{l2\gamma}$:
\beqa
T & = & \left.\dfrac{G_F}{\sqrt{2}} e V^*_{us} \ve^{\mu}(q)^*
\right\{(\hat{V}^+_{\mu\nu} -
A^+_{\mu\nu}) \ol{u}(p_{\nu}) \gamma^{\nu} (1 - \gamma_5) v(p_l)
\label{eq:Tnew}  \\*
  &   & + \left. F^+_{\nu} \ol{u}(p_{\nu}) \gamma^{\nu}
(1 - \gamma_5) \left[ \dfrac{p_\mu}{pq} -
\dfrac{(\fsl p_l + \fsl q - m_l) \gamma_\mu}
{2 p_lq} \right] v(p_l)\right\}~.  \no
\eeqa

The four invariant vector amplitudes $V_1,\ldots,V_4$ and
the four axial amplitudes $A_1,\ldots,A_4$ are functions of three scalar
variables. A convenient choice for these variables is
\beq
E_\gamma = pq/M_K ,\; E_\pi = pp'/M_K ,\; W = \sqrt{W^2}
\label{eq:kin1} \eeq
where $W$ is the invariant mass of the lepton pair. The amplitudes
$C_1, C_2$ can be expressed in terms of the $K_{l3}$ form factors
and depend only on the variable
\begin{figure}[t]
\vspace{7cm}
\caption{Diagrammatic representation of the $K^+_{l3\gamma}$
amplitude.} \label{fig41}
\end{figure}
$ (p - p')^2 = M^2_K + M^2_\pi - 2 M_K E_\pi$.
For the full kinematics of $K_{l3\gamma}$ two more variables are
needed, e.g.
\beq E_l = pp_l/M_K ,\; x = p_lq/M^2_K. \label{eq:kin2} \eeq
The variable $x$ is related to the angle $\theta_{l\gamma}$ between the
photon and the charged lepton in the $K$ rest frame:
\beq
x M^2_K = E_\gamma (E_l - \sqrt{E^2_l - m^2_l} \cos{\theta_{l\gamma}}).
\eeq
T invariance implies that the vector amplitudes $V_1,\ldots,V_4$, the
axial amplitudes $A_1,\ldots,A_4$ and the $K_{l3}$ form factors
$C_1, C_2$ are (separately) relatively real in the physical region.
We choose the standard
phase convention in which all amplitudes are real.

For $\theta_{l\gamma} \ra 0$ (collinear lepton and photon), there is
a lepton mass singularity in (\ref{eq:T}) which is numerically
relevant for $l = e$.
The region of small $E_\gamma, \theta_{l\gamma}$ is dominated by the
$K_{l3}$ matrix elements. The new theoretical information of
$K_{l3\gamma}$ decays resides in the tensor amplitudes
$\hat{V}_{\mu\nu}$ and $A_{\mu\nu}$. The relative importance of these
contributions can be enhanced by cutting away the region of low
$E_\gamma, \theta_{l\gamma}$. It may turn out to be of advantage to
reduce the statistics by applying more severe cuts than necessary from
a purely experimental point of view.

\subsection{Decay rates} \label{subsec-tree}
The total decay rate is given by \beqa
\Gamma(K \ra \pi l \nu \gamma) & = & \dfrac{1}{2 M_K (2 \pi)^8}
\int d_{LIPS}(p;p',p_l,p_\nu,q)
\sum_{spins}\mid T \mid^2 \label{eq:PS}
\eeqa
in terms of the amplitude $T$ in (\ref{eq:T}). The square of the matrix
element, summed over photon and lepton polarizations, is a bilinear form
in the invariant amplitudes $V_1,\ldots,V_4$, $A_1,\ldots,A_4$,
$C_1,C_2$.
Pulling out common factors, we write (\ref{eq:PS}) in the form
\beq
\Gamma(K \ra \pi l \nu \gamma) = \dfrac{4 \alpha G^2_F \mid V_{us} \mid^2}
{(2 \pi)^7 M_K} \int d_{LIPS}(p;p',p_l,p_\nu,q)\ SM. \label{eq:rate} \eeq
$SM$, the reduced matrix element squared, is given in
App. \ref{KL3G} as a function
of scalar products and invariant amplitudes. For the actual numerical
calculations, we have found it useful to employ a tensor decomposition
different from the one in Eqs. (\ref{eq:Low}) and (\ref{eq:tensor})
\beqa
V_{\mu\nu} & = & B_1 g_{\mu\nu} + B_2 W_\mu q_\nu + B_3 p'_\mu q_\nu
+ B_4 W_\mu p'_\nu \no \\
& & + B_5 W_\mu W_\nu + B_6 p'_\mu W_\nu + B_7 p'_\mu p'_\nu~.
\label{eq:basis} \eeqa
One advantage is that (\ref{eq:basis}) applies equally well to both
charge modes while (\ref{eq:Low}) does not. Moreover, the expression for
$SM$ in App. \ref{KL3G} is more compact when written in terms of the
$B_i$. In the numerical evaluation of the amplitudes, gauge invariance
can of course be used to express three of the $B_i$ in terms of the
remaining ones and of $C_1, C_2$.

To get some feeling for the magnitude of the various decay rates, let
us first consider the tree level amplitudes to lowest order $p^2$ in
CHPT. With the sign conventions of Ref.\cite{galenp1} exhibited
in chapter \ref{Intro}, these amplitudes are
\cite{beg,Hol} : \begin{flushleft} $\ul{K^+_{l3\gamma}:}$ \end{flushleft}
\beqa
V^+_{\mu\nu} & = & \dfrac{1}{\sqrt{2}} \left [ g_{\mu\nu} + \dfrac{
(p'+W)_\mu(2 p'+W)_\nu}{pq} \right ] \no \\*
A^+_{\mu\nu} & = & 0 \label{eq:tree+} \\*
F^+_\nu & = & \dfrac{1}{\sqrt{2}} (p+p')_\nu \no \eeqa
\begin{flushleft} $\ul{K^0_{l3\gamma}:}$ \end{flushleft}
\beqa
V^0_{\mu\nu} & = & - g_{\mu\nu} + \dfrac{p'_\mu(2 p'+2 q+W)_\nu}
{p'q} \no \\*
A^0_{\mu\nu} & = & 0 \label{eq:tree0} \\*
F^0_\nu & = & (p+p')_\nu. \no \eeqa

In table~\ref{tab:tree} the corresponding branching ratios
are presented for the four decay modes for
$E_\gamma \geq 30 MeV$ and $\theta_{l\gamma} \geq 20^\circ$.
For $K^0_{l3\gamma}$, the rates are to be understood as
$\Gamma(K_L \ra \pi^{\pm} l^{\mp} \nu \gamma)$. The
number of events correspond to the design values for DAFNE (cf.
App. \ref{notation} ).
\begin{table}[t]
\begin{center}
\caption{Branching ratios for tree level amplitudes for $E_\gamma
\geq 30 MeV$ and $\theta_{l\gamma} \geq 20^\circ$ in the $K$ rest
frame.} \label{tab:tree} \vspace{.5cm}
$\begin{array}{|c|c|c|}
\hline
$decay$  & $BR(tree)$
& \# $events/yr$ \\ \hline
K^+_{e3\gamma} & 2.8 \times 10^{-4} & 2.5 \times 10^6 \\
K^+_{\mu 3\gamma} & 1.9 \times 10^{-5} & 1.7 \times 10^5 \\
K^0_{e3\gamma} & 3.6 \times 10^{-3} & 4.0 \times 10^6 \\
K^0_{\mu 3\gamma} & 5.2 \times 10^{-4} & 5.7 \times 10^5 \\
\hline  \ea $
\end{center}
\end{table}

\subsection{Previous experiments}
The data sample for $K_{l3\gamma}$ decays is very limited and it
is obvious that DAFNE will be able to make significant improvements.
The present experimental status is summarized in table~\ref{tab:exp}.
\begin{table} \centering
\caption{Experimental results for $K_{l3\gamma}$ decays}
\label{tab:exp} \vspace{.5cm}
$\ba{|c|c|c|r|r|c|}
\hline
$decay$ & $exp.$ & E_{\gamma,min} & $\# events$
 & \multicolumn{1}{|c|}{$BR$} & \\ \hline
K^+_{e3\gamma} & \cite{Bol} & 10 MeV & 192 & (2.7 \pm 0.2)
\times 10^{-4} & 0.6 < \cos{\theta_{e\gamma}} < 0.9 \\
K^+_{e3\gamma} & \cite{Rom} & 10 MeV & 13 & (3.7 \pm 1.3)
\times 10^{-4} &  - " - \\
K^+_{e3\gamma} & \cite{Ljung} & 30 MeV & 16 & (2.3 \pm 1.0)
\times 10^{-4} & \cos{\theta_{e\gamma}} < 0.9 \\
K^+_{\mu 3\gamma} & \cite{Ljung} & 30 MeV & 0 & < 6.1
\times 10^{-5} & 90 \% \; c.l. \\
K^0_{e3\gamma} & \cite{Peach} & 15 MeV & 10 & (1.3 \pm 0.8)
\times 10^{-2} & \\ \hline
\ea $ \end{table}

A comparison between tables \ref{tab:tree} and \ref{tab:exp} shows
the tremendous improvement in statistics to be expected at DAFNE.
We shall come back to the question whether this improvement will
be sufficient to test the standard model at the next-to-leading order,
$O(p^4)$, in CHPT.

\subsection{Theory}
Prior to CHPT, the most detailed calculations of $K_{l3\gamma}$
amplitudes were performed by Fearing, Fischbach and Smith
\cite{Fischbach} using current algebra techniques.

In the framework of CHPT, the amplitudes are
given by (\ref{eq:tree+}) and (\ref{eq:tree0})
to leading order in the chiral expansion.

\vspace{.5cm}

{\bf{A) CHPT to $O(p^4)$}}

\vspace{.5cm}

There are in general three types of contributions~\cite{galenp1}:
anomaly, local contributions due to ${\cal L}_4$ and loop amplitudes.

\begin{figure}
\vspace{7.5cm}
\caption{Loop diagrams (without tadpoles) for $K_{l3}$ at $O(p^4)$. For
$K_{l3\gamma}$, the photon must be appended on all charged lines and
on all vertices.} \label{fig42}
\end{figure}

The anomaly contributes to the axial amplitudes \beqa
A^+_{\mu\nu} & = & \dfrac{i \sqrt{2}}{16 \pi^2 F^2} \left\{\ve_
{\mu\nu\rho\sigma}q^\rho(4 p'+W)^\sigma + \dfrac{4}{W^2-M^2_K}
\ve_{\mu\lambda\rho\sigma}W_\nu p'^\lambda q^\rho W^\sigma \right\}
\label{eq:anom} \\*
A^0_{\mu\nu} & = & - \dfrac{i}{8 \pi^2 F^2}\ve_{\mu\nu\rho\sigma}
q^\rho W^\sigma.\no \eeqa

The loop diagrams for $K_{l3\gamma}$ are shown in
Fig. \ref{fig42}. We first write the $K^+_{l3}$ matrix element
in terms of three functions $f^+_1, f^+_2,f^+_3$ which will also appear
in the invariant amplitudes $V^+_i$. Including the contributions
from the low-energy constants $L_5,L_9$ in ${\cal L}_4$, the
$K_{l3}$ matrix element $F^+_\nu$ is given by \beqa
F^+_\nu & = & f^+_1(t) p'_\nu + \left[\dfrac{1}{2}(M^2_K - M^2_\pi - t)
f^+_2(t) + f^+_3(t)\right](p - p')_\nu \no \\
f^+_1(t) & = & \sqrt{2} + \dfrac{4 \ol{L_9}}{\sqrt{2} F^2} t +
2 \sum_{I=1}^{3} (c^I_2 - c^I_1) B^I_2(t) \no \\*
f^+_2(t) & = & - \dfrac{4 \ol{L_9}}{\sqrt{2} F^2} + \dfrac{1}{t}
\sum_{I=1}^{3} \left\{ (c^I_1 - c^I_2)\left [2 B^I_2(t)
- \dfrac{(t+\Delta_I)
\Delta_I J_I(t)}{2 t}\right ] - c^I_2 \Delta_I J_I(t)
\right\} \no \\*
f^+_3(t) & = & \dfrac{F_K}{\sqrt{2} F_\pi} + \dfrac{1}{2 t}
\sum_{I=1}^{3} \left\{(c^I_1+c^I_2)(t+\Delta_I)
- 2 c^I_3\right\}\Delta_I J_I(t) \label{eq:fi} \\
\ol{L_9} & = & L^r_9(\mu) - \dfrac{1}{256 \pi^2} \ln{\dfrac
{M_\pi M^2_K M_\eta}{\mu^4}}  \no \\
\Delta_I & = & M^2_I-m^2_I\enskip ,\enskip t \enskip = \enskip (p-p')^2~.
\no \eeqa
$\ol{L_9}$ is a scale independent coupling constant and we have traded
the tadpole contribution together with $L_5$ for $F_K/F_\pi$ in
$f^+_3(t)$. The sum over I corresponds to the three loop diagrams
of Fig. \ref{fig42} with coefficients $c^I_1,c^I_2,c^I_3$ displayed in
table~\ref{tab:loop+}.
\begin{table} \centering
\caption{Coefficients for the $K^+_{l3\gamma}$ loop amplitudes
corresponding to the diagrams $I=1,2,3$ in Fig. \protect\ref{fig42}.
All coefficients $c^I_i$ must be divided by
$6 \protect\sqrt{2} F^2$.}
\label{tab:loop+} \vspace{.5cm}
$\ba{|l|l|l|r|r|c|}
\hline
I & M_I & m_I & c^I_1
& c^I_2  & c^I_3 \\ \hline
1 & M_K & M_\pi & 1 & -2 & - M^2_K - 2 M^2_\pi \\
2 & M_K & M_\eta & 3 & -6 & - M^2_K - 2 M^2_\pi \\
3 & M_\pi & M_K & 0 & -6 & -6 M^2_\pi \\
\hline
\ea$ \end{table}
We use the Gell-Mann--Okubo mass formula
throughout to express $M^2_{\eta}$ in terms of $M^2_K, M^2_\pi$.
The functions $J_I(t)$ and $B^I_2(t)$ can be found in App. \ref{loop}.

The standard $K_{l3}$ form factors $f_+(t), f_-(t)$
as given in the previous section~\cite{GL2} are
\beqa
f_+(t) & = & \dfrac{1}{\sqrt{2}}f^+_1(t) \\*
f_-(t) & = & \dfrac{1}{\sqrt{2}}\left[(M^2_K-M^2_\pi-t)f^+_2(t)
+2 f^+_3(t)-f^+_1(t)\right]. \no \eeqa

It remains to calculate the infrared finite tensor amplitude
$\hat{V}^+_{\mu\nu}$. The invariant amplitudes $V^+_i$ can be
expressed in terms of the previously defined functions $f^+_i$ and of
additional amplitudes $I_1,I_2,I_3$. Diagrammatically, the latter
amplitudes arise from those diagrams in Fig. \ref{fig42} where the
photon is not appended on the incoming $K^+$ (non-Bremsstrahlung
diagrams). The final expressions are
\beqa
V^+_1 & = & I_1 + p'W_q f^+_2(W^2_q) + f^+_3(W^2_q)
\no \\*
V^+_2 & = & I_2 -\dfrac{1}{pq}\left[p'W_q f^+_2(W^2_q)
+ f^+_3(W^2_q)\right]
\no \\*
V^+_3 & = & I_3 +\dfrac{1}{pq}\left[p'W f^+_2(W^2)+f^+_3(W^2)
-p'W_q f^+_2(W^2_q) - f^+_3(W^2_q)\right]  \label{eq:Vhat} \\*
V^+_4 & = & \dfrac{f^+_1(W^2)-f^+_1(W^2_q)}{pq}  \no \\*
W_q & = & W + q \enskip = \enskip p - p'~.
\no \eeqa

The amplitudes $I_1, I_2, I_3$ in Eq.(\ref{eq:Vhat}) are given by
\beqa
I_1 & = & \dfrac{4 qW}{\sqrt{2}F^2}(\ol{L_9}+\ol{L_{10}}) + \dfrac
{8 p'q}{\sqrt{2}F^2}\ol{L_9}  \no \\*
&  & +\sum_{I=1}^{3}\left\{\left [(W^2_q+\Delta_I)(c^I_1+c^I_2)-
2(c^I_2 p'W_q+c^I_3)\right ]
\left[\dfrac{(W^2_q-\Delta_I)\hat{J}_I}
{2 W^2_q} -2 G_I) \right] \right.\no \\*
& & +\dfrac{(c^I_2-c^I_1)}{2}
\left [\dfrac{p'W_q}{W^2_q}(\dfrac{(W^4_q-\Delta^2_I)
\hat{J}_I}{W^2_q}+4\hat{B}^I_2)+p'(W-q)L^I_m\right ] \no \\*
& & +\dfrac{2(c^I_2-c^I_1)}{qW}\left.\left [p'q(F_I-
(W^2_q+\Delta_I)G_I)+p'W(\hat{B}^I_2-B^I_2)\right ] \right\}
\no \\
I_2 & = & -\dfrac{8\ol{L_9}}{\sqrt{2}F^2}+\dfrac{2}{qW}\sum_{I=1}
^{3}(c^I_2-c^I_1)\left [F_I-(W^2+\Delta_I)G_I\right ]  \no \\
I_3 & = & -\dfrac{4\ol{L_9}}{\sqrt{2}F^2}+\sum_{I=1}^{3}
\left\{2(c^I_2-c^I_1)
\left [G_I+\dfrac{L^I_m}{4}+\dfrac{\hat{B}^I_2-B^I_2}{qW}
\right ] -c^I_1 \dfrac{\Delta_I J_I}{W^2} \right\} \label{eq:loop} \\
\ol{L_{10}} & = & L^r_{10}(\mu)+\dfrac{1}{256\pi^2}\ln{\dfrac{M_\pi
M^2_K M_\eta}{\mu^4}}  \no \\
L^I_m & = & \dfrac{\Sigma_I}{32\pi^2\Delta_I}
\ln{\dfrac{m^2_I}{M^2_I}} \no \\
F_I & = & \hat{B}^I_2-\dfrac{W^2}{4}L^I_m+\dfrac{1}{qW}
\left(W^2 B^I_2-W^2_q \hat{B}^I_2\right) \no \\
G_I & = & \dfrac{M^2_I}{2}C(W^2_q,W^2,M^2_I,m^2_I)+\dfrac{1}{8 qW}
\left[(W^2_q+\Delta_I)\hat{J}_I-(W^2+\Delta_I)J_I\right]
+\dfrac{1}{64\pi^2} \no \\
J_I & \equiv & J_I(W^2), \; \hat{J}_I \equiv J_I(W^2_q) \no \\
B^I_2 & \equiv & B^I_2(W^2), \; \hat{B}^I_2 \equiv B^I_2(W^2_q).
 \no \eeqa

The function $C(W^2_q,W^2,M^2_I,m^2_I)$ is given in App. \ref{loop}.
All the invariant amplitudes $V^+_1,\ldots,V^+_4$ are
real in the physical region. Of course, the same is true for the
$K_{l3}$ matrix element $F^+_\nu$.

The $K^0_{l3\gamma}$ amplitude has a very similar structure. Both the
$K^0_{l3}$ matrix element $F^0_\nu$ and the infrared finite
vector amplitude $\hat{V}^0_{\mu\nu}$ can be obtained from the
corresponding quantities $F^+_\nu$  and
$\hat{V}^+_{\mu\nu}$ by the following steps:
\bit
\item interchange $p'$ and $-p \:$ ;
\item replace $\dfrac{F_K}{F_\pi}$ by $\dfrac{F_\pi}{F_K}$ in $f^+_3$;
\item insert the appropriate coefficients $c^I_i$ for
$K^0_{l3\gamma}$ listed in table~\ref{tab:loop0};
\item multiply $F^+_\nu$ and $\hat{V}^+_{\mu\nu}$ by a
factor $-\sqrt{2}$.
\eit
\begin{table} \centering
\caption{Coefficients for the $K^0_{l3\gamma}$ loop amplitudes
corresponding to the diagrams $I=1,2,3$ in Fig. \protect\ref{fig42}. All
coefficients $c^I_i$ must be divided by $6 \protect\sqrt{2}F^2$.}
\label{tab:loop0} \vspace{.5cm}
$\ba{|l|l|l|r|r|c|}
\hline
I & M_I & m_I  & c^I_1
& c^I_2 & c^I_3 \\ \hline
1 & M_K & M_\pi & 0 & -3 & -3 M^2_K \\
2 & M_K & M_\eta & 6 & -3 &  M^2_K+2 M^2_\pi \\
3 & M_\pi & M_K & 4 & -2 & -2 M^2_K + 2 M^2_\pi \\
\hline \ea $
\end{table}

\vspace{.5cm}

{\bf{B) Numerical results}}

\vspace{.5cm}

In calculating the rates with the complete amplitudes of the previous
subsection, we use the same cuts as for the tree level rates in
Sect.~\ref{subsec-tree}:
\beqa
E_\gamma & \geq & 30 MeV \\*
\theta_{l\gamma} & \geq & 20^\circ. \no \eeqa
The physical values of $M_\pi$ and $M_K$ are used in the amplitudes.
$M_\eta$ is calculated from the Gell-Mann--Okubo mass formula. The
values of the other parameters can be found in chapter \ref{Intro}
 and in appendix \ref{notation}.

The results for $K^+_{l3\gamma}$ and $K^0_{l3\gamma}$ are displayed in
tables \ref{tab:+} and \ref{tab:0}, respectively. For comparison, the
tree level branching ratios of table \ref{tab:tree} and the rates for
the amplitudes without the loop contributions are also shown. The
separation between loop and counter\-term contributions is of course
scale dependent. This scale dependence is absorbed in the scale invariant
constants $\ol{L_9}, \ol{L_{10}}$ defined in Eqs.(\ref{eq:fi}),
(\ref{eq:loop}). In other words, the entries in tables \ref{tab:+},
\ref{tab:0} for the amplitudes without loops correspond to setting
all coefficients $c^I_i$ in tables \ref{tab:loop+},
\ref{tab:loop0} equal to zero.

\begin{table} \centering
\caption{Branching ratios and expected number of events at DAFNE
for $K^+_{l3\gamma}$.} \label{tab:+} \vspace{.5cm}
$\ba{|l|c|c|}
\hline
\multicolumn{1}{|c|}{K^+_{e3\gamma}} &
$BR$  & \# $events/yr$ \\ \hline
$full $ O(p^4) $ amplitude$ & 3.0\times 10^{-4} & 2.7\times 10^6 \\
$tree level$ & 2.8\times 10^{-4} & 2.5\times 10^6 \\
O(p^4) $ without loops$ & 3.2\times 10^{-4} & 2.9\times 10^6 \\
\hline \ea$ \\*[.5cm] $\ba{|l|c|c|} \hline
\multicolumn{1}{|c|}{K^+_{\mu 3\gamma}}  &
$BR$ & \# $events/yr$ \\ \hline
$full $ O(p^4) $ amplitude$ & 2.0\times 10^{-5} & 1.8\times 10^5 \\
$tree level$ & 1.9\times 10^{-5} & 1.7\times 10^5 \\
O(p^4) $ without loops$ & 2.1\times 10^{-5} & 1.9\times 10^5 \\
\hline \ea$ \end{table}
\nopagebreak[2]
\begin{table} \centering
\caption{Branching ratios and expected number of events at DAFNE
for $K^0_{l3\gamma}$.} \label{tab:0} \vspace{.5cm}
$\ba{|l|c|c|}
\hline
\multicolumn{1}{|c|}{K^0_{e3\gamma}}  &
$BR$  & \# $events/yr$ \\ \hline
$full $ O(p^4) $ amplitude$ & 3.8\times10^{-3} & 4.2\times 10^6 \\
$tree level$ & 3.6\times 10^{-3} & 4.0\times 10^6 \\
O(p^4) $ without loops$ & 4.0\times 10^{-3} & 4.4\times 10^6 \\
\hline \ea$ \\*[.5cm] $\ba{|l|c|c|} \hline
\multicolumn{1}{|c|}{K^0_{\mu 3\gamma}}  &
$BR$ & \#$events/yr$ \\ \hline
$full $ O(p^4) $ amplitude$ & 5.6\times 10^{-4} & 6.1\times 10^5 \\
$tree level$ & 5.2\times 10^{-4} & 5.7\times 10^5 \\
O(p^4) $ without loops$ & 5.9\times 10^{-4} & 6.5\times 10^5 \\
\hline \ea$ \end{table}

\subsection{Improvements at DAFNE}
The numerical results given above
demonstrate very clearly that the non-trivial
CHPT effects of $O(p^4)$ can be detected at DAFNE in all four channels
without any problem of statistics. Of course, the rates are bigger for
the electronic modes. On the other hand, the relative size of the
structure dependent terms is somewhat bigger in the muonic channels
(around 8\% for the chosen cuts). We observe that there is negative
interference between the loop and counterterm amplitudes.

The sensitivity
 to the counterterm coupling constants $L_9, L_{10}$ and to the
chiral anomaly can be expressed as the difference in the number of events
between the tree level and the $O(p^4)$ amplitudes (without loops).
In the optimal case of $K^0_{e3\gamma}$, this amounts to more than
$4 \times 10^5$ events/yr at DAFNE. Almost all of this difference is
due to $L_9$. It will be very difficult to extract the coupling constant
$L_{10}$ from the total rates. A more detailed study is needed to
determine whether $L_{10}$ can be extracted from differential
distributions.

The chiral anomaly is more important for $K^+_{l3\gamma}$, but even there
it influences the total rates rather little. Once again, a dedicated
study of differential rates is necessary to locate the chiral anomaly,
if possible at all.

On the other hand, taking
into account that $L_9$ is already known to good accuracy (see
chapter \ref{Intro}), $K_{l3\gamma}$
decays will certainly allow for precise and unambiguous tests
of the one-loop effects in CHPT \cite{beg}.


%before each section
\newpage
\setcounter{equation}{0}
\setcounter{subsection}{0}
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         \section{$K_{l4}$ decays}
         In this section we discuss the decays
         \bearr
          K^+(p) & \rightarrow & \pi^+(p_1) \;\pi^-(p_2) \;l^+(p_l) \; \nu_l
(p_{\nu})          \label{h1} \\
          K^+(p) & \rightarrow & \pi^0(p_1)\; \pi^0(p_2) \;l^+(p_l) \;\nu_l
(p_{\nu})         \label{k2}\\
          K^0(p) & \rightarrow & \pi^0(p_1)\; \pi^-(p_2)\; l^+(p_l) \;\nu_l
(p_{\nu})         \label{h3}
        \eearr
        and their charge conjugate modes. The letter $l$ stands for
        $e$ or $\mu$. We do
        not consider isospin violating contributions and
        correspondingly set $m_u = m_d$, $\alpha = 0$.

        \subsection{Kinematics}

        We start with the process (\ref{h1}).
       The full kinematics of this decay requires five variables.
       We will use the
       ones introduced by Cabibbo and Maksymowicz \cite{cabmak}. It is
       convenient to
       consider three reference frames, namely the $K^+$ rest system
       $(\Sigma_K)$, the
       $\pi^+
       \pi^-$ center-of-mass system $(\Sigma_{2 \pi})$ and the
       $l^+\nu_l$ center-of-mass system $(\Sigma_{l \nu})$. Then
       the variables are

       \begin{enumerate}
       \item $s_\pi$, the effective mass squared of the dipion system,

       \item $s_l$, the effective mass squared of the dilepton system,

       \item $\theta_\pi$, the angle of the $\pi^+$ in $\Sigma_{2\pi}$
       with respect to the dipion line of flight in $\Sigma_K$,

       \item $\theta_l$, the angle of the $l^+$ in $\Sigma_{l\nu}$
       with  respect to the dilepton line of flight in
       $\Sigma_K$, and

       \item $\phi$, the angle between the plane formed by the pions
       in
       $\Sigma_K$ and the corresponding plane formed by the dileptons.

       \end{enumerate}

       The angles $\theta_\pi$, $\theta_l$ and $\phi$ are displayed in
       Fig.
       \ref{figkin}. In order to specify these variables more
\begin{figure}[t]
\vspace{6cm}
\caption{Kinematic variables for $K_{l4}$ decays. The angle $\theta_\pi$ is
defined in $\Sigma_{2\pi}, \theta_l$ in $\Sigma_{l\nu}$ and $\phi$ in
$\Sigma_K$.
\label{figkin}
         }
\end{figure}
       precisely, let
       $\vec{p}_1$ be the three-momentum of the $\pi^+$ in
       $\Sigma_{2\pi}$ and
       $\vec{p}_l$ the three-momentum of the $l^+$ in
       $\Sigma_{l\nu}$.
       Furthermore, let $\vec{v}$ be a unit vector along the direction
       of flight of the dipion in
       $\Sigma_K$, and $\vec{c}\,(\vec{d}\,)$ a unit vector along
       the  projection
       of $\vec{p}_1(\vec{p_l})$ perpendicular to $\vec{v}(-\vec{v})$,
       \bearr
       \vec{c} & = & (\vec{p}_1 - \vec{v} \vec{v} \cdot \vec{p}_1) /
       [\vec{p}^{\;2}_1 - (\vec{p}_1 \cdot \vec{v})^2]^{1/2}
       \nonumber \\
       \vec{d} & = & (\vec{p}_l - \vec{v} \vec{v} \cdot \vec{p}_l) /
       [\vec{p}^{\;2}_l - (\vec{p}_l \cdot \vec{v})^2]^{1/2} \; \; .
       \nonumber
       \eearr
       The vectors  $\vec{v}$, $\vec{c}$ and $\vec{d}$
       are indicated in Fig. \ref{figkin}. Then, one has
      \bearr
      s_\pi & = & (p_1 + p_2)^2 \; \;, \;\; s_l = (p_l + p_\nu)^2
      \nonumber \\
      \cos \theta_\pi & = & \vec{v} \cdot \vec{p}_1 / \mid \vec{p}_1
      \mid,\; \;  \cos \theta_l = - \vec{v} \cdot \vec{p}_l / \mid \vec{p}_l
\mid       \nonumber \\
      \cos \phi & = & \vec{c} \cdot \vec{d}, \; \; \sin \phi = (\vec{c}
      \times \vec{v}) \cdot \vec{d}.
      \label{h5}
      \eearr
      The range of the variables is
      \bearr
      4 M^2_\pi & \leq & s_\pi \leq (M_K - m_l)^2
      \nonumber \\
      m^2_l & \leq & s_l \leq (M_K - \sqrt{s_\pi})^2
      \nonumber \\
      0 & \leq & \theta_\pi, \theta_l \leq \pi, 0 \leq \phi \leq 2
      \pi. \label{h6}
      \eearr
      It is useful to furthermore introduce the following combinations
      of four vectors
      \be
      P = p_1 + p_2, \; \; Q = p_1 - p_2 \;,\; L = p_l + p_\nu \; , \;
      N = p_l - p_\nu
      \label{k6a}
      \ee
      together with the corresponding Lorentz invariant scalar
      products
      \bearr
      P^2 & = & s_\pi, \; \; Q^2 = 4 M^2_\pi - s_\pi,\; \;  L^2 = s_l, \; \;N^2
      = 2 m_l^2 - s_l,
      \nonumber \\
      PQ & = & 0, \nonumber \\
      PL & =& \frac{1}{2} (M_K^2 - s_\pi - s_l),
      \nonumber \\
      PN & = & z_l PL + (1-z_l) X \cos \theta_l,
      \nonumber \\
      QL & = & \sigma_\pi X \cos \theta_\pi,
      \nonumber \\
      QN & = & z_l QL + \sigma_\pi (1-z_l) \left[ PL \cos \theta_\pi \cos
      \theta_l \right.
      \nonumber \\
    & & -    \left. (s_\pi s_l)^{1/2} \sin \theta_\pi\sin \theta_l \cos
      \phi \right]
      \nonumber \\
      LN & = & m^2_l
      \nonumber \\
      <LNPQ> & \equiv & \epsilon_{\mu \nu \rho \sigma} L^\mu N^\nu
      P^\rho Q^\sigma
      \nonumber \\
      & = & - (s_\pi s_l)^{1/2}
      \sigma_\pi (1-z_l) X \sin \theta_\pi \sin \theta_l \sin
      \phi
      \label{k7}
      \eearr
      with
      \bearr
      X & = & ((PL)^2 -s_\pi s_l)^{1/2} = \frac{1}{2} \lambda^{1/2} (M_K^2,
s_\pi, s_l)       \nonumber \\
      z_l & = & m^2_l /s_l
      \nonumber \\
      \sigma_\pi & = & (1-4 M^2_\pi/s_\pi)^{1/2}.
      \label{k8}
      \eearr
      Below we will also use the variables
      \bearr
      t & = & (p_1 -p)^2
      \nonumber \\
      u & = & (p_2 - p)^2.
      \label{k9}
      \eearr
      These are related to $s_\pi, s_l$ and $\theta_\pi$ by
      \bearr
      t  +  u &=& 2 M^2_\pi + M^2_K + s_l - s_\pi
      \nonumber \\
      t  -  u& =& - 2 \sigma_\pi X \cos \theta_\pi \; \; .
      \label{k10}
      \eearr

      \subsection{Matrix elements}

      The matrix element for $K^+ \rightarrow \pi^+ \pi^- l^+ \nu_l$
      is
      \be
      T = \frac{G_F}{\sqrt{2}} V^\star_{us} \bar{u} (p_\nu) \gamma_\mu
      (1-\gamma_5) \nu(p_l) (V^\mu - A^\mu)
      \label{k11}
      \ee
      where
      \bearr
      I_\mu & = & < \pi^+ (p_1) \pi^- (p_2) \mbox{out}\mid
      I_\mu^{4-i5} (0) \mid K^+ (p)  >;\; I = V,A
      \nonumber \\
      V_\mu & = & - \frac{H}{M^3_K} \epsilon_{\mu \nu \rho \sigma} L^\nu
      P^\rho Q^\sigma
      \nonumber \\
      A_\mu & = & -i\frac{1}{M_K} \left [ P_\mu F +
      Q_\mu G + L_\mu R \right ]
      \label{k12}
      \eearr
and $\epsilon_{0123}=1$.
 The matrix elements for the other channels
(\ref{k2},\ref{h3}) may be obtained from (\ref{k11},\ref{k12}) by isospin
symmetry, see below.

      The form factors $F,G,R$ and $H$ are real analytic functions of the three
      variables $p_1 p_2$, $p_1 p$ and $p_2 p$. Below,
      we will use instead the variables $\{ s_\pi, s_l, \theta_\pi\}$ or
      $\{s_\pi, t, u \}$.

{\underline{Remark:}} In order to agree with the notation used by Pais and
Treiman \cite{paistr} and by Rosselet et al. \cite{ross}, we have changed our
previous  convention \cite{bijnenskl4,riggen} in the definition of the anomaly
form factor $H$. See also the comments after Eq. (\ref{h18}).



      \subsection{Decay rates}

      The partial decay rate for (\ref{h1}) is given by

      \be
      d\Gamma = \frac{1}{2M_K(2\pi)^8} \sum_{spins} \mid
      T \mid^2 d_{LIPS} (p; p_l, p_\nu, p_1, p_2).
      \label{k13}
      \ee

The quantity $\sum_{spins} \mid T \mid ^2$ is a Lorentz invariant quadratic
form in $F,G,R$ and $H$. All scalar products can be expressed
 in the 5 independent variables $s_\pi,s_l,\theta_\pi,\theta_l$ and $\phi$,
 such that
\be
\sum_{spins} \mid T \mid ^2 = {2G_F^2 \mid V_{us} \mid ^2}{M_K^{-2}}
J_5(s_\pi,s_\l,\theta_\pi,\theta_l,\phi) \; \; .
\label{k13a}
\ee
Carrying out the integrations over the remaining $4 \cdot 3 - 5 = 7$ variables
in (\ref{k13})       gives \cite{cabmak}
 \be
      d \Gamma_5 = G^2_F \mid V_{us} \mid^2  N(s_\pi, s_l) J_5
      (s_\pi, s_l, \theta_\pi, \theta_l, \phi) ds_\pi ds_l d (\cos
      \theta_\pi) d(\cos \theta_l) d\phi
      \label{k14}
      \ee
      where
      \be
      N(s_\pi, s_l) = (1-z_l) \sigma_\pi X
      /(2^{13} \pi^6 M_K^5) \; \; .
      \label{h15}
      \ee
      The form factors $F,G,R$ and $H$ are independent of $\phi$ and
$\theta_l$. It is       therefore possible to carry out two more
      integrations in
      (\ref{k14}) with the result
      \be
      d\Gamma_3 = G_F^2 \mid V_{us} \mid^2  N(s_\pi, s_l)
      J_3 (s_\pi, s_l, \theta_\pi) ds_\pi ds_l d (\cos \theta_\pi).
      \label{k14a}
      \ee
      The explicit form of $J_5$ is
      \bearr
J_5 &=& |F|^2 \left[ (\pl)^2 - (\pn)^2 - \spi s_l + m_l^2 \spi \right]
\nonumber \\
&+& |G|^2 \left[ (\ql)^2 - (\qn)^2 - Q^2 s_l + m_l^2 Q^2 \right]
\nonumber \\
&+& |R|^2 \; m_l^2 \left[ s_l - m_l^2 \right] \frac{ }{ }
\nonumber \\
&+& \frac{1}{M_K^4} |H|^2 \left[ (m_l^2 - s_l)\left[
Q^2 X^2 +s_\pi (QL)^2 \right] -<LNPQ>^2 \right]
\nonumber \\
&+& (F^* G+F G^*)  \left[ (\pl)  (\ql)^{ } - (\pn) (\qn) \right]
\nonumber \\
&+& (F^*R+F R^*) \; m_l^2 \left[(\pl) -  (\pn) \right] \frac{ }{ }
\nonumber \\
&+& \frac{1}{M_K^2} (F^*H+FH^*) \left[ (\qn) (\pl)^2
- (\ql) (\pl) (\pn) -  \spi s_l(\qn)   + \mee \spi  (\ql) \right]
\nonumber \\
&+& (G^*R+G R^*) \; m_l^2 \left[ (\ql) - (\qn) \right]  \frac{ }{ }
\nonumber \\
&+& \frac{1}{M_K^2} (G^*H+GH^*) \left[  (\pl) (\ql) (\qn)
 -(\pn)  (\ql)^2 +  s_l (\pn)  \qq
- \mee (\pl) \qq \right]
\nonumber \\
&+&  \frac{i}{M_K^2}<LNPQ>\left[-(F^*G-FG^*) M_K^2
+ (F^*H-FH^*)  (\pn) \right.
\nonumber \\
&+&\left. (G^*H-GH^*)  (\qn)
+ (R^*H-RH^*)  m_l^2 \right]  \; \; .
      \label{h15a}
      \eearr
      For data analysis it is useful to represent this result in a still
      different form which displays the $\theta_l$ and $\phi$
      dependence more clearly \cite{paistr}:
      \bearr
J_5 &=&2(1-z_l)\left[ I_1 + I_2 \cos 2 \theta_l + I_3 \sin^2 \theta_l \cdot
\cos 2
\phi + I_4 \sin 2 \theta_l \cdot \cos \phi + I_5 \sin \theta_l \cdot \cos \phi
\right. \nonumber \\
&+& \left. I_6 \cos \theta_l + I_7 \sin \theta_l \cdot \sin \phi + I_8 \sin 2
\theta_l \cdot \sin \phi + I_9 \sin^2 \theta_l \cdot \sin 2 \phi \right] .
      \label{h16}
      \eearr
      One obtains
      \bearr
I_1 &=& \frac{1}{4}\left\{ (1 + z_l) |F_1|^2  + \frac{1}{2}
(3+z_l)\left(|F_2|^2 +
|F_3|^2 \right) \sin^2 \thp   + 2z_l  |F_4|^2 \right\}
\nonumber \\
I_2 &=& - \frac{1}{4} (1-z_l)\left\{ |F_1|^2 - \frac{1}{2}
\left( |F_2|^2 + |F_3|^2 \right)  \sin^2 \thp  \right\}
\nonumber \\
I_3 &=& - \frac{1}{4} (1-z_l) \left\{ |F_2|^2 - |F_3|^2 \right\}\sin^2 \thp
\nonumber \\
 I_4 &=& \frac{1}{2}(1-z_l)\mbox{ Re} (F_1^* F_2)  \sin \thp
\nonumber \\
I_5 &=& -\left\{ \mbox{ Re} (F_1^* F_3) +
z_l  \mbox{ Re} (F_4^* F_2) \right\}  \sin \thp
\nonumber \\
I_6 &=&
- \left\{\mbox{ Re} (F_2^* F_3)\sin^2 \thp - z_l \mbox{ Re} (F_1^* F_4)\right\}
\nonumber \\
I_7 &=& - \left\{ \mbox{ Im} (F_1^* F_2) +
z_l  \mbox{ Im} (F_4^* F_3) \right\} \sin \thp
\nonumber \\
I_8 &=& \frac{1}{2} (1-z_l) \mbox{ Im} (F_1^* F_3) \sin \thp
\nonumber \\
I_9 &=& -\frac{1}{2}(1-z_l) \mbox{ Im} (F_2^* F_3) \sin^2 \thp  \;,
      \label{h17}
      \eearr
      where
      \bearr
 F_1& =& X \cdot F + \sigma_\pi (P L) \cos \thp \cdot G
\nonumber \\
 F_2& =& \sigma_\pi \left( s_\pi s_l \right)^{1/2} G
\nonumber \\
 F_3& =&  \sigma_\pi X \left( s_\pi s_l \right)^{1/2}  \frac{H}{M_K^2}
\nonumber \\
 F_4& =&- (P L)  F - s_l R - \sigma_\pi X \cos \thp \cdot G \; \; .
      \label{h18}
      \eearr
      The definition of $F_1, \ldots, F_4$ in (\ref{h18}) corresponds
      to the combinations used by Pais and Treiman \cite{paistr} (the
      different sign in the terms $\sim PL$
 is due to our use of the  metric
      $\mbox{diag} (+ ---)$). The form factors $I_1, \ldots, I_9$
      agree with the expressions given in \cite{paistr}. We conclude that our
convention for the relative phase in the definition of the form factors
 in Eq. (\ref{k12}) agrees with the one used by Pais and Treiman.
 The comparison of
(\ref{h15a}) with
\cite[table II]{ross} shows furthermore that it also agrees with
this reference.

      The quantity $J_3$ can now easily be obtained from (\ref{h16})
      by integrating over $\phi$ and $\theta_l$,
      \be
      J_3 = \int d\phi \; d({\mbox{cos}}\theta_l) J_5 =8 \pi (1 - z_l) \left [
I_1 - \frac{1}{3} I_2 \right].       \label{h19}
      \ee

      \subsection{Isospin decomposition}

      The $K_{l4}$ decays (\ref{k2}) and (\ref{h3}) involve the same
      form factors as displayed in Eq. (\ref{k12}). We denote by
      $A_{+-}$, $A_{00}$ and $A_{0-}$ the current matrix elements of
      the processes (\ref{h1})-(\ref{h3}). These are related by
      isospin symmetry \footnote{We use the  Condon-Shortley phase
conventions.},       \be
      A_{+-} =\frac{A_{0-}}{\sqrt{2}} - A_{00} \; \; .
      \label{i1}
      \ee
      This relation also holds for the individual form factors, which
      may be decomposed into a symmetric and an antisymmetric part under
      $t \leftrightarrow u \; (p_1 \leftrightarrow p_2)$. Because of Bose
      symmetry and of the $\Delta I = \frac{1}{2}$ rule of the
      relevant  weak currents, one has
      \bearr
      (F,G,R,H)_{00} & = & - (F^+, G^-, R^+, H^-)_{+-}
      \nonumber \\
      (F,G,R,H)_{0-} & = & \sqrt{2} (F^-, G^+, R^-, H^
      +)_{+-}
      \label{i2}
      \eearr
      where
      \be
      F^\pm_{+-} = \frac{1}{2} [F(s_\pi, t, u) \pm F(s_\pi, u, t)]
      \label{i3}
      \ee
      and $F(s_\pi, t, u)$ is defined in Eq. (\ref{k12}).

The isospin relation for the decay rates is
\be
\Gamma(K^+\rightarrow \pi^+\pi^-l^+\nu_l)=\frac{1}{2}\Gamma(K_L\rightarrow
\pi^0\pi^\pm l^\mp\nu) +2 \Gamma(K^+\rightarrow \pi^0\pi^0l^+\nu_l) \; \; .
\label{iso}
\ee
Isospin violating contributions affect the matrix elements and phase
space, as a result of which this relation  is modified. In
order to illustrate the (substantial) effects from asymmetries in
phase space, we take
 constant form factors
$F,G$ and set $R=0,H=0$. Eq.  (\ref{iso}) then reads (with
physical
masses for $K^+\rightarrow \pi^+\pi^-l^+\nu_l, \pi^0\pi^0 l^+\nu_l$ and with
$M_{\pi^0}=M_{\pi^\pm}=137$ MeV in  $K_L\rightarrow \pi^0\pi^\mp l^\pm \nu$)
\bearr
(16.0 F^2 +3.1 G^2)\Gamma_0&=& (20.1F^2 +2.0 G^2)\Gamma_0
\nonumber \\
\Gamma_0 &=& V_{us}^2 \cdot 10^2 {\mbox{sec}}^{-1}
\eearr
in the electron mode and
\bearr
(1.79 F^2 +0.25 G^2)\Gamma_0 &=& (2.64 F^2 +0.20 G^2)\Gamma_0
\eearr
in the muon mode.

      \subsection{Partial wave expansion}

      The form factors may be written in a partial wave expansion in
      the variable $\theta_\pi$.
      We consider a definite isospin $\pi \pi$ state. Suppressing isospin
indices, one has \cite{berends,cecile}
      \bearr
      F &= & \sum^{\infty}_{l=0} P_l (\cos \theta_\pi) f_l - \frac{
      \sigma_\pi PL}{X} \cos
      \theta_\pi G
      \nonumber\\
      G & = & \sum^{\infty}_{l=1} P_l' (\cos \theta_\pi) g_l
      \nonumber \\
      R & = & \sum^{\infty}_{l=0} P_l (\cos \theta_\pi) r_l +
      \frac{ \sigma_\pi s_\pi}{X}
      \cos \theta_\pi G
      \nonumber \\
      H & = & \sum^{\infty}_{l= 0} P_l' (\cos \theta_\pi) h_l
      \label{i4}
      \eearr
      where
      \be
      P_l'(z) = \frac{d}{dz} P_l(z) \; \; .
      \label{i10}
      \ee

      The partial wave amplitudes $f_l, g_l, r_l$ and $h_l$ depend on
      $s_\pi$ and $s_l$. Their phase coincides with the phase shifts
      $\delta^I_l$ in elastic $\pi \pi$ scattering (angular momentum
      $l$, isospin $I$).
      More precisely, the quantities
      \bearr
     && e^{-i \delta^0_{2l}} X_{2l}
      \nonumber \\
     && e^{-i\delta^1_{2l+1}} X_{2l+1}\; ; \;l=0,1, \ldots \; ;
      \; X = f,g,r,h
      \label{i11}
      \eearr
      are real in the physical region of $K_{l4}$ decay. The form
      factors $F_1$ and $F_4$ therefore have a simple expansion
      \bearr
      F_1 &=& X \sum_{l} P_l
      ( \cos \theta_\pi) f_l
      \nonumber \\
      F_4& =& - \sum_{l} P_l (\cos \theta_\pi) (PL f_l + s_l r_l).
      \label{i12}
      \eearr


      On the other hand, the phase of the projected amplitudes
      \be
      F_{2l} = \int P_l (\cos \theta_\pi) F_2 d (\cos \theta_\pi) \; ;
      \; l=0,1,2,\ldots
      \label{i13}
      \ee
      is not given by $\delta_l^I$, e.g., $e^{-i\delta^1_1} F_{20}$ is
      \underline{not} real in the isospin one case. A similar remark applies to
$F_3$.

      \subsection{Previous experiments}


\begin{table}[t]
\protect
\begin{center}
\caption{\label{kl441} Rates of $K_{e4}$ decays.}

\vspace{1em}
\begin{tabular}{|c|c|c|c|c|} \hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{$\sharp$ events} & \\ \hline
  & branching  & Particle Data
  & DAFNE  & improve-                  \\
  & ratio & Group
  & 1 yr & ment \\ \hline
  &&&& \\
$ K^+ \rightarrow \pi^+\pi^- e^+ \nu_e $ &
 $3.91 \cdot 10^{-5} $ & $ 3 \cdot 10^4 $ & $3\cdot 10^5$ & $10$      \\
$ K^+ \rightarrow \pi^0\pi^0 e^+ \nu_e $ &
 $ 2.1 \cdot 10^{-5} $ & $ <50 $ & $2\cdot 10^5$ & $>4\cdot 10^3$      \\
${K_L} \rightarrow \pi^0 \pi^{\pm} e^{\mp} \nu $ &
$6.2 \cdot 10^{-5} $ & $ 16 $ & $ 7\cdot 10^4$ & $4 \cdot 10^3$      \\
 &&&& \\ \hline
\end{tabular}
\end{center}
\end{table}



      We display in table \ref{kl441} the number of events collected
      so far.
      The data are obviously dominated by the work of Rosselet et al.
      \cite{ross}, which measures the $\pi^+\pi^-$ final state with
      good statistics. The authors parametrize the form factors as
      \bearr
      F & = & f_s e^{i\delta^0_0} + f_p e^{i \delta^1_1} \cos \theta_\pi
      + \mbox{D-wave}
      \nonumber \\
      G & = & g e^{i\delta^1_1} + \mbox{D-wave}
      \nonumber \\
      H & = & h e^{i\delta^1_1} + \mbox{D-wave}
      \label{e1}
      \eearr
      with $f_s, f_p, g$ and $h$ assumed to be real
      \footnote{Note that, according to what is said in the previous
subsection, the terms denoted
      by ''D-wave'' in Eq. (\ref{e1}) all contain (complex) contributions which
      are proportional to $P_l (\cos \theta_\pi), l \geq 0$.
                }
              .
      Furthermore, they put $m_e = 0$, such that
      the form factors $R$ and $F_4$ drop out in the decay distribution.
Despite       the good statistics, the experiment has not been able to
      separate out the full kinematic behaviour of the matrix
      elements. Therefore certain approximations/assumptions had to
      be made. For example, no dependence on $s_l$ was seen within the
      limits of the data, so that the results were quoted assuming
      that such a dependence is absent. Similarly, $f_p$ was found to
      be compatible with zero, and hence put equal to zero when the
      final result for $g$ was derived. A dependence on $s_\pi$ was
      seen, and found to be compatible with
      \bearr
      f_s (q^2) & = & f_s(0) [1+ \lambda_f q^2]
      \nonumber\\
      g (q^2) & = & g(0) [1+ \lambda_g q^2]
      \nonumber \\
      h (q^2) & = & h(0) [1+ \lambda_h q^2] \nonumber\\
      q^2 & = &  (s_\pi-4 M_\pi^2)/4M_\pi^2
      \label{e2}
      \eearr
      with
      \be
      \lambda_f = \lambda_g = \lambda_h = \lambda.
      \label{e3}
      \ee

      These approximations to the form factors do not agree completely
      with what is found in the theoretical predictions. Dependence on
      $s_l$ and non-zero values for higher partial waves all occur in
      the theoretical results.

      The experimental results for the threshold values and the slopes
      of the form factors are \cite{ross}
      \bearr
      f_s(0) & = & 5.59 \pm 0.14
      \nonumber \\
      g(0) & = & 4.77 \pm 0.27
      \nonumber\\
      h(0) & = & - 2.68 \pm 0.68
      \nonumber \\
      \lambda & = & 0.08 \pm 0.02.
      \label{e4}
      \eearr
      We have used \cite{pdg} $\mid V_{us} \mid = 0.22$ in
      transcribing these results. (We note that from Eqs. (\ref{e1} - \ref{e4})
and $f_p=0$ we
obtain  $\Gamma_{K_{e4}} = (2.94 \pm 0.16)\cdot 10^3$ sec$^{-1}$. This value
must
be compared with $\Gamma_{K_{e4}} = (3.26 \pm 0.15)\cdot 10^3$ sec$^{-1}$
obtained in the same experiment.)  In addition to the threshold
values       (\ref{e4}) of the form factors, the phase shift difference
      $\delta = \delta^0_0 - \delta^1_1$ was determined \cite{ross} in
      five energy bins. The S-wave scattering length $a^0_0$ was then
      extracted by using a model of Basdevant, Froggatt and Petersen
      \cite{bafropet}. This model is based on solutions to Roy
      equations. The result for the scattering length is
      \be
      a^0_0 = 0.28 \pm 0.05.
      \label{e5}
      \ee
      A study by \cite{pipet}, based on a more recent solution to Roy
      equations, gives
      \be
      a^0_0 = 0.26 \pm 0.05.
      \label{e6}
      \ee


Turning now to the other channels, we consider the measured branching ratios
\bearr
 \hspace{-1cm} BR(K^+\rightarrow\pi^0\pi^0e^+\nu_e) &=&\left(
2.0^{+0.5}_{-0.4}\right) \cdot 10^{-5} \;\; \; [25\; \mbox{events}]\; \; \;
\cite{Bol} \label{k50}
\eearr
and
\bearr
BR(K_L\rightarrow\pi^0\pi^\mp e^\pm \nu)&=&\left\{
\begin{array}{lll}
(6.2\pm2.0)\cdot 10^{-5} & [16\;\mbox{events}] & \cite{carroll} \\
(5.8 \pm 0.2 \pm 0.4) \cdot 10^{-5} & [780 \pm 40\;\mbox{events}] &
\cite{barker} \; \; . \end{array}
\right.
\nonumber
\\
\label{k51}
\eearr
The kinematic dependence of the form factors on
the variables $s_\pi,s_l$ and $\theta_\pi$ has not yet been resolved
experimentally in these decays. In order to proceed, we assume that the
$A_{00}$ and $A_{0-}$ form factors are independent of $\theta_\pi$, e.g.,
$F_{00}=F_{00}(s_\pi,t+u)$ etc.
As a result of this assumption, $G_{00},H_{00},F_{0-}$ and $R_{0-}$ all vanish
by Bose statistics.  The contribution from $R_{00}$ is completely
negligible in the electron mode, and the contribution from the anomaly form
factor to the decay (\ref{k51}) is tiny. We neglect it altogether,
as a result of which the above decays are fully determined by $F_{00}$ and
$G_{0-}$.  We write
\be
F_{00}=F_0(1+\lambda q^2) \;, \; \; G_{0-} = G_0(1+\lambda q^2)
\label{k52}
\ee
and obtain for the rate
\bearr
2\Gamma_{K^+\rightarrow\pi^0\pi^0e^+\nu_e} &=&|F_0 V_{us}|^2(
2.01 +1.7 \lambda +O(\lambda^2)) \cdot 10^3 {\mbox{sec}^{-1}}
\label{k53} \\
\Gamma_{K_L\rightarrow\pi^0\pi^\mp e^\pm \nu}&=&
|G_0 V_{us}|^2(0.406 +0.47  \lambda +O(\lambda^2)) \cdot 10^3
{\mbox{sec}^{-1}}
\label{k54}
\eearr
where we have used physical phase space in (\ref{k53}) and
$M_\pi^0=M_\pi^\pm=137$ MeV in (\ref{k54}). This finally gives with
$\lambda=0.08$ from (\ref{e4})
 \bearr
|F_0|&=& 5.58 ^{+0.7}_{-0.6} \; \; \cite{Bol}
\nonumber \\
|G{_0}| &=& \left\{ \begin{array}{ll}
7.5 \pm 1.2 & \cite{carroll} \\
7.3 \pm 0.3 & \cite{barker}
\end{array}
\right.
\eearr
which compares rather well with the isospin predictions (\ref{i2})
\bearr
|F_0|&=& |f_s(0)| = 5.59 \pm 0.14
\nonumber \\
|G_{0}|&=& \sqrt{2}|g(0)| = 6.75 \pm 0.38 \; \; .
\eearr

      \subsection{Theory}

      The theoretical predictions of $K_{l4}$ form factors have a long
      history which started in the sixties with the current algebra
      evaluation of $F$, $G$, $R$ and $H$. For an early review of the
      subject and for references to work prior to CHPT we refer the
      reader to \cite{chounet} (see also \cite{shabkl4}).
      Here we concentrate
on the evaluation of the form factors in the framework of CHPT
      \cite{bijnenskl4,riggen}. We restrict our consideration to the
      isospin symmetry limit $m_u = m_d, \alpha = 0$.

\vspace{.5cm}

     {\bf{A) The one-loop result}}

\vspace{.5cm}

      In Ref. \cite{bijnenskl4,riggen}, the form factors $F$, $G$ and
      $H$ have been evaluated in CHPT at order $p^4$. The analytic
      expression for $R$ has not yet been worked out to this accuracy
      \cite{colang}. This form factor only contributes significantly
      to $K_{\mu 4}$ decays.

      The chiral representation of the form factors at order $E^2$ was
      originally given by Weinberg \cite{weinkl4},
      \bearr
      F & = & G  =  \frac{M_K}{\sqrt{2} F_\pi} = 3.74
      \nonumber \\
      H&  = & 0.
      \label{T1}
      \eearr
      We write the result for $F$ at next-to-leading order in the form
      \bearr
      F(s_\pi, t, u) & = & \frac{M_K}{\sqrt{2} F_\pi} \left\{ 1 +
      F^+(s_\pi, t, u) + F^-(s_\pi, t, u) + O(E^4) \right\}
      \nonumber \\
      F^\pm(s_\pi, t, u) & = & U^\pm_F(s_\pi, t, u) + P^\pm_F (s_\pi,
      t, u) + C^\pm_F
      \label{T2}
      \eearr
      and will use below an analogous expression for the form factor
      $G$. The superscript +($-$) denotes a term which is even (odd)
      under crossing $t \leftrightarrow u$. The contributions
      $U^\pm_F(s_\pi, t, u)$ denote the unitarity corrections generated
      by the one-loop graphs which appear at order $E^4$. They have
      the form
      \bearr
      U^+_F (s_\pi, t, u) & = & F_\pi^{-2} \left[ \Delta_0 (s_\pi) +
      a_F(t) + a_F (u)\right]
      \nonumber \\
      U^-_F (s_\pi, t, u) & = & F_\pi^{-2} \left[b_F (t) - b_F (u)\right]
      \label{T3}
      \eearr
      with
      \bearr
      \Delta_0(s_\pi) & = & \frac{1}{2} (2 s_\pi - M^2_\pi) J^r_{\pi
      \pi} (s_\pi) + \frac{3s_\pi}{4} J^r_{KK} (s_\pi) +
      \frac{M^2_\pi}{2} J^r_{\eta \eta} (s_\pi)
      \nonumber \\
      a_F (t) & = & \frac{1}{32} \left[(14 M^2_K + 14 M^2_\pi - 19t)
      J^r_{K\pi} (t) + (2M^2_K + 2 M^2_\pi - 3t)
      J^r_{\eta K} (t) \right]
      \nonumber \\
      & + & \frac{1}{16} \left[ (3M^2_K - 7 M^2_\pi + 5 t) K_{K \pi}
      (t) + (M^2_K - 5 M^2_\pi + 3t) K_{\eta K} (t) \right]
      \nonumber \\
      & - & \frac{1}{8} \left[ 9 (L_{K\pi} (t) + L_{\eta K} (t)) +
      (3M^2_K - 3 M^2_\pi - 9 t) (M^r_{K\pi} (t) + M^r_{\eta K}
      (t))\right]
      \nonumber \\
      b_F(t) & = & a_F (t) - \frac{1}{2} (M^2_K + M^2_\pi -t)
      J^r_{K\pi} (t).
      \label{T4}
      \eearr
      The loop integrals $J^r_{\pi \pi} (s_\pi), \ldots$ which occur
      in these expressions are listed in the appendix \ref{loop}. The
functions
      $J^r_{PQ}$ and $M^r_{PQ}$ depend on the scale $\mu$ at which the
      loops are renormalized. The scale drops out in the expression
      for the full amplitude (see below).

      The imaginary part of  $F_\pi^{-2} \Delta_0 (s_\pi)$ contains
      the $I = 0$, $S$-wave $\pi \pi$ phase shift
      \be
      \delta^0_0 (s_\pi) = (32 \pi F^2_\pi)^{-1} (2s_\pi - M^2_\pi) \sigma_\pi
     + O(E^4)
      \label{T5}
      \ee
      as well as contributions from $K \bar{K}$ and $\eta \eta$
      intermediate states. The functions $a_F(t)$ and $b_F(t)$ are
      real in the physical region.

      The contribution $P^\pm_F(s_\pi, t, u)$ is a polynomial in
      $s_\pi, t, u$ obtained from the tree graphs at order $E^4$. We
      find
      \be
      P^\pm_F (s_\pi, t, u) = \frac{1}{F^2_\pi} \sum^{9}_{i=1}
      p^\pm_{i,F} (s_\pi, t, u) L^r_i
      \label{T6}
      \ee
      where
      \bearr
      p^+_{1,F} & = & 32 (s_\pi - 2M^2_\pi )
      \nonumber \\
      p^+_{2,F} & = &8 (M_K^2 + s_\pi
      - s_l)
      \nonumber \\
      p^+_{3,F} & = & 2(M^2_K - 8
      M^2_\pi + 5 s_\pi - s_l)
      \nonumber \\
      p^+_{4,F} & = & 32 M^2_\pi
      \nonumber \\
      p^+_{5,F} & = & 4 M^2_\pi
      \nonumber \\
      p^+_{9,F} & = & 2 s_l
      \nonumber \\
      p^-_{3,F} & = & - 2 (t - u) .
      \label{T7}
      \eearr
      The remaining coefficients $p^\pm_{i,F}$ are zero. The symbols
      $L^r_i$ denote the renormalized coupling constants discussed
      in chapter \ref{Intro}.

      Finally we come to the contributions $C^\pm_F$ which contain
      logarithmic terms, independent of $s_\pi, t$ and $u$:
      \bearr
      C^+_F & = & (256 \pi^2 F^2_\pi)^{-1} \left[ 5 M^2_\pi \ln
      \frac{M^2_\pi}{\mu^2} - 2 M^2_K \ln \frac{M^2_K}{\mu^2} - 3
      M^2_\eta \ln \frac{M^2_\eta}{\mu^2} \right]
      \nonumber \\
      C^-_F & = & 0 \; \; .
      \label{T8}
      \eearr
      The corresponding decomposition of the form factor $G$,
      \be
      G^\pm = U^\pm_G + P^\pm_G + C^\pm_G,
      \label{T9}
      \ee
      has the following explicit form:
      \bearr
      U^+_G (s_\pi, t, u) & = & F_\pi^{-2} \left[ \Delta_1 (s_\pi) +
      a_G(t) + a_G(u) \right]
      \nonumber \\
      U_G^- (s_\pi, t, u) & = & F^{-2}_\pi \left[ b_G(t) - b_G(u)\right]
      \label{T10}
      \eearr
      with
      \bearr
      \Delta_1 (s_\pi) & = & 2 s_\pi \left\{ M^r_{\pi \pi} (s_\pi)+
      \frac{1}{2} M^r_{KK}(s_\pi) \right\}
      \nonumber \\
      a_G(t) & = & \frac{1}{32} \left[ (2M^2_K + 2M^2_\pi + 3t)
      J^r_{K\pi}(t) - (2M^2_K + 2M^2_\pi -3 t) J^r_{\eta
      K}(t) \right]
      \nonumber \\
      & + & \frac{1}{16} \left[ (-3 M^2_K + 7 M^2_\pi - 5 t) K_{K \pi}
      (t) + (-M^2_K + 5 M^2_\pi - 3 t) K_{\eta K} (t) \right]
      \nonumber \\
      & - & \frac{3}{8} \left[ L_{K\pi}(t) + L_{\eta K} (t) - (M^2_K -
      M^2_\pi + t) (M^r_{K\pi} (t) + M^r_{\eta K} (t)) \right]
      \nonumber \\
      b_G(t) & = & a_G(t) - \frac{1}{2} (M^2_K + M^2_\pi - t)
      J^r_{K\pi} (t).
      \label{T11}
      \eearr
      The imaginary part of $F_\pi^{-2} \Delta_1(s_\pi)$ contains the
      $I=1$, $P$-wave phase shift
      \be
      \delta^1_1 (s_\pi) = (96 \pi F_\pi^2)^{-1} s_\pi
      \sigma_\pi^{3/2} +  O(E^4)
      \label{T12}
      \ee
      as well as contributions from $K\bar{K}$ intermediate states.
      The functions $a_G, b_G$ are real in the physical region.

      The polynomials
      \be
      P^\pm_G = \frac{1}{F^2_\pi} \sum^{9}_{i=1} p^\pm_{i,G} (s_\pi,
      t, u) L^r_i
      \label{T13}
      \ee
      are
      \bearr
      p^+_{3,G} & = &  - 2 (M^2_K +
      s_\pi - s_l)
      \nonumber \\
      p^+_{5,G} & = & 4M^2_\pi
      \nonumber \\
      p^+_{9,G} & = &  2 s_l
      \nonumber \\
      p^-_{2,G} & = & 8(t - u)
      \nonumber \\
      p^-_{3,G} & = & \frac{1}{4} p^-_{2,G}.
      \label{T14}
      \eearr
      The remaining $p^\pm_{i,G}$ vanish. The logarithms contained in
      $C^\pm_G$ are
      \be
      C^\pm_G = - C^\pm_F.
      \label{T15}
      \ee
      The form factor $H$ starts only at $O(E^4)$. The prediction is
      \be
      H = -\frac{\sqrt{2} M^3_K}{8 \pi^2 F^3_\pi} = -2.66
      \label{T16}
      \ee
      in excellent agreement with the experimental value.

      The results for $F$ and $G$ must satisfy two nontrivial
      constraints: i) Unitarity requires that $F$ and $G$ contain, in
      the physical region $4M^2_\pi \leq s_\pi \leq (M_K-m_l)^2$, imaginary
      parts governed by $S$- and $P$-wave $\pi \pi$ scattering [these
      imaginary parts are contained in the functions
      $\Delta_0(s_\pi), \Delta_1(s_\pi)]$. ii) The scale dependence
      of the low-energy constants $L^r_i$ must be compensated  by
      the scale dependence of $U_{F,G}$ and $C_{F,G}$ for all values
      of
      $s_\pi, t, u, M^2_\pi, M^2_K$. [Since we work at order $E^4$, the
      meson masses appearing in the above expressions satisfy the
      Gell-Mann-Okubo mass formula.] We have checked that these
      constraints are satisfied.

\vspace{.5cm}

 {\bf{ B) Comparison with experiment}}

\vspace{.5cm}

      One striking feature of the chiral prediction for the form
      factors is that the only important dependence on the low-energy
      constants is through $L_1, L_2$ and $L_3$. We proceed by fixing
      $L_4, L_5$ and $L_9$ at the values found in other processes
      (see chapter \ref{Intro}, in particular table 1).

      A rather extensive analysis of the chiral prediction and the
      data on $K^+ \rightarrow \pi^+\pi^- e^+ \nu$ and elastic $\pi
      \pi$ scattering has been given in Refs. \cite{bijnenskl4,riggen}. We
      refer the reader to these articles for details. Here we
      mention the following points.



\begin{table}[t]
\protect
\begin{center}
\caption{ \label{chpred}
 Predictions of chiral symmetry following from the fit to
the $K_{e4}$ data \protect\cite{ross} alone
(column 3) and the combined determination from $\pi \pi$ \protect\cite{nagels}
and $K_{e4}$ data \protect\cite{ross}
(last column). The first column gives the prediction of
the leading order term
in the low-energy expansion of the $\pi \pi$ amplitude.
 }
\vspace{1em}
\begin{tabular}{|l|c|c|c|c|}  \hline
     &
\raisebox{-0.6ex}{leading} &
    &
    &
      \\
     &
order  &
\raisebox{0.8ex}{experiment}  &
\raisebox{0.8ex}{$K_{e4}$ alone}     &
\raisebox{0.8ex}{$K_{e4} + \pi \pi$} \\       \hline
$\lambda_g$  &
    &
$0.08 \pm 0.02$  &
$0.06 \pm 0.02$  &
$0.06 \pm 0.02$  \\
$a_0^0$  &
$0.16$   &
$0.26 \pm 0.05$  &
$0.20$ &
$0.20$  \\
$b_0^0$  &
$0.18$    &
$0.25 \pm 0.03$   &
$0.26$  &
$0.26$  \\
$a_0^2$  &
$-0.045$  &
$-0.028 \pm 0.012$   &
$-0.040$  &
$-0.041$   \\
$b_0^2$  &
$-0.089$  &
$-0.082 \pm 0.008$  &
$-0.069$  &
$-0.070$  \\
$a_1^1$  &
$0.030$    &
$0.038 \pm 0.002$  &
$0.037$  &
$0.036$  \\
$b_1^1$  &
      &
      &
$0.045$   &
$0.043$  \\
$a_2^0$  &
     &
$(17 \pm 3) \cdot 10^{-4}$  &
$21 \cdot 10^{-4}$   &
$20 \cdot 10^{-4}$  \\
$a_2^2$  &
     &
$(1.3 \pm 3) \cdot 10^{-4}$  &
$3.5 \cdot 10^{-4}$   &
$3.5 \cdot 10^{-4}$  \\   \hline
\end{tabular}
\end{center}
\end{table}


      \begin{enumerate}

      \item  Fixing $L_1, L_2$ and $L_3$ from $f_s(0)$, $g(0)$ and
      the slope $\lambda_f$ of the form factor $f_s$ gives
      \bearr
      L_1^{r}(M_\rho) & = & [0.5 \pm 0.3] \cdot 10^{-3}
      \nonumber \\
      L_2^{r}(M_\rho) & = & [1.6 \pm 0.3] \cdot 10^{-3}
      \nonumber \\
      L_3 & = & [-3.2 \pm 1.1] \cdot 10^{-3} \; \; .
      \label{C1}
      \eearr
      The error bar corresponds \cite{riggen} to an increase of
      $\chi^2$ by one. It does not include the error due to unknown
      higher order corrections in the chiral expansion of the form
      factor. Having determined $L_1,L_2$ and $L_3$, one may then work
out the       form factors from the representation (\ref{T2}-\ref{T15}). The
      result is shown in
Fig. \ref{fkl4}, where we plot for the electron mode the quantity
\be
f_s(s_\pi)=\left\{ \frac{1}{(s_l^{\tiny{max}}-s_l^{\tiny{min}})}
\int_{s_l^{\tiny{min}}}^{s_l^{\tiny{max}}} ds_l \left|
{\small{\frac{1}{2}}}\int_{-1}^1
d({\mbox{cos}}\theta_\pi) F(s_\pi,s_l,{\mbox{cos}}\theta_\pi)\right|^2
\right\}^{1/2}
\label{C1a}
\ee
and similarly for $g(s_\pi)$.
  The lowest
      order results Eq. (\ref{T1}) (labelled "tree") plus the experimental
central values and the central values
corresponding to Eq. (\ref{C1}) are displayed.
 Note
      that the slope of the $g$ form factor has not been included in
      the fit and is thus a prediction. It matches very well with the
      experimental data.
\begin{figure}[t]
\vspace{8cm}
\caption{ The form factors $f_s(s_\pi)$ and $g(s_\pi)$
(Eq. \protect\ref{C1a}) according
to the chiral representation (electron mode). The dotted lines show the lowest
order result (\protect\ref{T1}),
and the  dashed lines correspond to $L_1,L_2$ and $L_3$ from
(\protect\ref{C1}).
The experimental result (\protect\ref{e4}) is displayed by a solid line.
\label{fkl4}
           }
\end{figure}

      \item  The decay $K^+ \rightarrow \pi^+ \pi^- e^+ \nu_e$
      allows one to test the large $N_C$ prediction
      \be
      (L_2^r - 2 L_1^r) / L_3 = 0 \hspace{2cm} (\mbox{large }N_C).
      \label{C2}
      \ee
      From the values in Eq. (\ref{C1}), we see that a small non-zero
      result for this combination is preferred, but that it is
      consistent with zero within the errors. The fit was also done
      \cite{riggen} using the variables
      \bearr
      X_1 & = & L_2^r - 2 L_1^r - L_3
      \nonumber \\
      X_2 & = & L_2^r
      \nonumber \\
      X_3 & = & (L_2^r - 2 L_1^r)/L_3
      \label{C3}
      \eearr
      with the result
      \bearr
      X_1 & = & (3.8 \pm 0.9) \cdot 10^{-3}
      \nonumber \\
      X_2 & = & (1.6 \pm 0.3) \cdot 10^{-3}
      \nonumber \\
      X_3 & = & - 0.19_{-0.27}^{+0.16}   \; \; .
      \label{C4}
      \eearr
     ($X_1$ and $X_3$ are scale independent, $X_2$ is evaluated at the rho
mass.)
      The result is that the large $N_C$ prediction works remarkably
      well.

      \item  Having determined the low-energy constants, one is in
      a position to study the predictions. The coefficients $L_1, L_2$
      and $L_3$ also govern elastic $\pi \pi$-scattering, and the real
      test of the theory is that these coefficients are simultaneously
      compatible with the elastic $\pi \pi$ amplitude. The most
      straightforward way to check this is to predict the $\pi \pi$
      threshold parameters. The chiral predictions were worked out in
      Ref. \cite{galean,galepl}. If we use the determination (\ref{C1}), we
      obtain the prediction in  table \ref{chpred}, third column. (For
      $\bar{l}_3, \bar{l}_4$ which occur in $a^I_l, b^I_l$ we have
      used the central value $\bar{l}_3 = 2.9, \bar{l}_4 = 4.3$ from
      Ref. \cite{galean}). The predictions are within $1 \frac{1}{2}$
      standard deviations of the data in all cases. Note, in
      particular, the nice agreement for the $I= 0,2$ $D$-wave
      scattering lengths $a^0_2, a^2_2$. Furthermore, it is comforting
      to see that the SU(2)$\times$SU(2) prediction \cite{galean,galepl}
      \be
      a^0_0 = 0.20 \pm 0.01
      \ee
      survives the $K_{e4}$ test unharmed.

      \item  It is of interest to provide the best determination of
      the low-energy constants by including the maximum amount of
      data. This includes the $K_{e4}$ form factors $f_s(0), g(0)$ and
      $\lambda_f$, as well as the direct determination of $\delta^0_0
      - \delta^1_1$ in $K_{e4}$ decay. We take the other information as the
      $\pi \pi$ threshold parameters $a^1_1, a^0_2, a^2_2, b^2_0$ as
      well as the universal curve \cite{universal}
      \bearr
      X(a^0_0, a^2_0) & = & 2a^0_0 - 5a^2_0 - 0.96 (a^0_0 - 0.3)
      \nonumber \\
      & & -  0.7 (a^0_0 - 0.3)^2
      \nonumber \\
      & = & 0.69 \pm 0.04 \; \; .
      \label{universal}
      \eearr
      The results of the fit are shown in the last column of table
\ref{chpred}.       The corresponding values for $L_1, L_2$ and $L_3$ are
      \bearr
      L^r_1 (M_\rho) & = & (0.7 \pm 0.5) \cdot 10^{-3}
      \nonumber \\
      L^r_2 (M_\rho) & = & (1.2 \pm 0.4) \cdot 10^{-3}
      \nonumber \\
      L_3 & = & (- 3.6 \pm 1.3) \cdot 10^{-3}\; \; .
      \label{C5}
      \eearr
      The error includes the theoretical error bar, see Ref.
      \cite{riggen}.
[The one-loop representation (\ref{T2}-\ref{T16}) of the form
factors $F,G$ and
$H$, evaluated at the central values (\ref{C5}), gives $\Gamma_{K_{e4}}
= 2.5 \cdot 10^3$ sec$^{-1}$. This is somewhat lower than the experimental
\cite{ross} width $\Gamma_{K_{e4}} = 3.26 \cdot 10^3$ sec$^{-1}$ and
the value  $\Gamma_{K_{e4}} = 2.94 \cdot 10^3$ sec$^{-1}$ which follows from
(\ref{e4}). The reason for considering nevertheless (\ref{C5})
as the present best
estimate for $L_1,L_2$ and $L_3$ is discussed at some length in
\cite{riggen}:
since the chiral corrections to the tree level result $F=G=3.74$ are large,
one should not expect that the one-loop corrections already do the complete job
- rather, higher order terms have any right to also contribute accordingly. The
above result for $L_1,L_2$ and $L_3$ includes an estimate for these additional
terms. (The experimental width for $K_{e4}$ is within the
uncertainties $\Delta L_1,\Delta L_2$ and $\triangle L_3$ quoted.)]

      \end{enumerate}

      \subsection{Improvements at DAFNE}

      The chiral analysis of $K_{l4}$ decays has been used so far for
      three purposes:

      \begin{enumerate}

      \item  The $K_{e4}$ data from Ref. \cite{ross} make
      predictions for the slope of the $G$ form factor and for the
      $\pi \pi$ scattering lengths. These are given in table \ref{chpred}.

      \item  The same $K_{e4}$ data allow one to test the large $N_C$
      prediction, see Eqs. (\ref{C2}-\ref{C4}).

      \item The full set of $K_{e4}$ and $\pi \pi$ scattering data
      allows the best determination  of the coefficients $L_1, L_2$
and       $L_3$ in the chiral Lagrangian, see (\ref{C5}).

      \end{enumerate}

      In the next generation of $K_{l4}$ decay experiments, there is the
      opportunity to improve the phenomenology of $K_{l4}$ (see table
      \ref{kl441}):

      \begin{enumerate}

      \item The present experimental uncertainty on $G$ is still too
      large to provide a precise value for the large $N_C$ parameter
      $(L_2^r - 2 L_1^r)/L_3$. $(K^0 \rightarrow \pi^0 \pi^- e^+
 \nu_e$ decays are mainly sensitive to $G^+_{+-}$ which in turn can be
      used to pin down $L_3$. $K^+ \rightarrow \pi^0 \pi^0 e^+ \nu_e$
      is mainly sensitive to $F^+_{+-}$ which contains $L_1, L_2$ and
      $L_3$.)

      \item The observation of all $K_{l4}$ reactions with high
      statistics could provide a cleaner separation of the various
      isospin amplitudes.

      \item A very useful innovation would be to analyze the
      experimental data directly using the framework of chiral
      perturbation theory. Rather than making assumptions about the
      absence of $P$-waves, $D$-waves etc., one could parametrize the
      data using the full chiral perturbation formulas, and directly
      decide the quality of the fit and the favoured values of the
      low-energy constants.

      \item Finally, we come to a most important point. As we
      mentioned already, $K^+ \rightarrow \pi^+ \pi^- e^+ \nu_e$ has
      been used \cite{pipet} to determine the isoscalar $S$-wave
      scattering length with the result $a^0_0 = 0.26 \pm 0.05$. This
      value must be compared with the SU(2)$\times$SU(2) prediction
      \cite{galean,galepl} $a^0_0 = 0.20 \pm 0.01$. Low-energy $\pi \pi$
      scattering is one of the few places where chiral symmetry allows
      one to make a precise prediction within the framework of QCD. In
      their article, Rosselet et al. comment about the discrepancy
      between $a^0_0 = 0.26 \pm 0.05$ and the leading order result
      \cite{weinpi} $a^0_0 = 0.16$ in the following manner: ''... it
      appears that this prediction can be revised without any
      fundamental change in current algebra or in the partial
      conservation of axial-vector current \cite{bonnier,franklin}.''
      Today, we know that this is not the case: It would be a major
      difficulty for QCD, should the central value $a^0_0 = 0.26$ be
      confirmed with a substantially smaller error.

      $K_{l4}$ decays are -- at present \cite{nemenov} -- the only
      available source of clean information on $\pi \pi$ $S$-wave
      scattering near threshold. We therefore feel that it would be
      rather appropriate to clarify this issue.

      \end{enumerate}


 \vspace{2cm}
{\bf{Acknowledgements}}



We thank G. Pancheri for the perfect organization of the DAFNE Workshops
 and the INFN for the hospitality in Frascati.
 We have enjoyed numerous interesting discussions
with the members of the working groups, and we are grateful to
L. Maiani and N. Paver for reading the manuscript and for valuable
suggestions at various stages of this work. We thank J. Beringer for
checking traces in $K_{l2\gamma}$, M. Candusso for doing the contour plots
and C. Riggenbach for providing us with numerical routines for $K_{l4}$.
 E.A. Ivanov has kindly made available to us
unpublished work by D.Yu. Bardin and collaborators on radiative pion and
kaon decays.


\appendix

%before appendix
\newcounter{zahler}
\renewcommand{\thesection}{\Alph{zahler}}
\renewcommand{\theequation}{\Alph{zahler}.\arabic{equation}}
\setcounter{zahler}{0}

%before each section in appendix
\newpage
\setcounter{equation}{0}
\addtocounter{zahler}{1}

\section{Notation}
\label{notation}

The notation for phase space is the one without the factors of $2\pi$.
For the decay rate of a
particle with four momentum $p$ into $n$ particles
with momenta $p_1,\ldots,p_n$ this is
\begin{equation}
d_{LIPS}(p;p_1,\ldots,p_n) = \delta^4(p-\sum_{i=1}^n p_i )
\prod_{i=1}^n \frac{d^3 p_i}{2 p_i^0} ~.
\end{equation}
We use a covariant normalization of one-particle states,
\be
<{\vec{p}} \; '|\vec{p}> = (2\pi)^3 2p^0 \delta^3({\vec{p}} \; '-\vec{p})
 \;
\; , \ee
together with the spinor normalization
\be
\bar{u}(p,r)u(p,s) = 2m\delta_{rs} \; \; .
\ee
The kinematical function $\lambda(x,y,z)$ is defined as
\begin{equation}
\lambda(x,y,z) = x^2 + y^2 + z^2 - 2 (xy + yz + zx)~.
\end{equation}
We take the standard model in the current $\times$ current form, i.e.,
we neglect the momentum dependence of the $W$-propagator. The currents
used in the text are :
\begin{eqnarray}
V_\mu^{4-i5} &=& \bar{q} \gamma_\mu \frac{1}{2}(\lambda_4 - i\lambda_5)q
  ~=~ \overline{s}\gamma_\mu u
\nonumber\\
A_\mu^{4-i5} &=& \bar{q} \gamma_\mu \gamma_5 \frac{1}{2}
(\lambda_4 - i\lambda_5) q
  ~=~ \overline{s}\gamma_\mu \gamma_5 u
\nonumber\\
V^{em}_\mu &=& \bar{q}\gamma_\mu Q q
\nonumber\\
Q&=&\mbox{diag}(2/3, -1/3 ,-1/3)~.
\end{eqnarray}
The numerical values used in the programs are the physical masses
for the particles as given by the Particle Data Group \cite{pdg}.
In addition we have used the values for the decay constants derived
from the most recent measured charged pion and kaon
semileptonic decay rates\cite{pdg,lroos} :
\begin{eqnarray}
F_\pi &=& 93.2 ~MeV\nonumber\\
F_K &=& 113.6~MeV.
\end{eqnarray}
We do not need values for the quark masses. For the processes considered
in this report we can always use the lowest order relations to rewrite
them in   terms of the
pseudoscalar meson masses (see chapter \ref{Intro}). For the
KM matrix element $|V_{us}|$
we used the central value, 0.220, of Ref. \cite{pdg}.
The numerical values for the $L_i^r(M_\rho)$ are those given in
chapter \ref{Intro}.

The number of events quoted for DAFNE are based on a
luminosity of $5\cdot 10^{32}~ cm^{-2}s^{-1}$,
which is equivalent \cite{franzini}
to an annual rate of $9\cdot 10^9$ $(1.1\cdot 10^9)$ tagged
$K^{\pm}$ $(K_L)$ (1 year = $10^7~ s$ assumed).

Whenever we quote a branching ratio for a semileptonic
$K^0$ decay, it stands for the branching ratio of the corresponding
$K_L$ decay, e.g.,
\begin{equation}
BR(K^0 \to \pi^- l^+ \nu ) \equiv BR(K_L \to \pi^{\pm} l^{\mp} \nu )~.
\end{equation}
We use the Condon-Shortley phase conventions throughout.


%before each section in appendix
\newpage
\setcounter{equation}{0}
\addtocounter{zahler}{1}
\section{Loop integrals}
\label{loop}

In this appendix we define the functions appearing in the loop integrals
used in the text.
First we define the functions needed for loops with two propagators,
mainly in the form given in Ref. \cite{galenp1}.
We consider a loop with two masses, $M$ and $m$.
 All needed functions can be given
in terms of the subtracted scalar integral $\bar{J}(t) = J(t) - J(0)$,
\begin{equation}
J(t) = ~  -i
\int \frac{d^4p}{(2\pi)^d} \frac{1}{((p+k)^2 - M^2)(p^2 - m^2)}
\end{equation}
with $t = k^2$.
The functions used in the text are then :
\begin{eqnarray}
\bar{J}(t)&=&-\frac{1}{16\pi^2}\int_0^1 dx~
\log\frac{M^2 - t x(1-x) - \Delta x}{M^2 - \Delta x}
\nonumber\\&=&
\frac{1}{32\pi^2}\left\{
2 + \frac{\Delta}{t}\log\frac{m^2}{M^2} -\frac{\Sigma}{\Delta}
\log\frac{m^2}{M^2} - \frac{\sqrt{\lambda}}{t}
\log\frac{(t+\sqrt{\lambda})^2 -
\Delta^2}{(t-\sqrt{\lambda})^2-\Delta^2}\right\}
{}~,
\nonumber\\
J^r(t) &=& \bar{J}(t) - 2k~,
\nonumber\\
M^r(t) &=& \frac{1}{12t}\left\{ t - 2 \Sigma \right\} \bar{J}(t)
+ \frac{\Delta^2}{3 t^2} \bar{J}(t)
+ \frac{1}{288\pi^2} -\frac{k}{6}
\nonumber\\&&
                               - \frac{1}{96\pi^2 t} \left\{
      \Sigma + 2 \frac{M^2 m^2}{\Delta}
     \log\frac{m^2}{M^2} \right\} ~,
\nonumber\\
L(t)&=& \frac{\Delta^2}{4t} \bar{J}(t)~,
\nonumber\\
K(t)&=&\frac{\Delta}{2t}\bar{J}(t)    ~,
\nonumber\\
H(t)&=&\frac{2}{3} \frac{L_9^r}{F^2} t + \frac{1}{F^2}[t M^r(t) - L(t)],
\nonumber\\
\Delta &=& M^2 - m^2~,
\nonumber\\
\Sigma &=& M^2 + m^2  ~,
\nonumber\\
\lambda&=&\lambda(t,M^2,m^2) ~=~ (t+\Delta)^2 - 4tM^2  ~.
\end{eqnarray}
In the text these are used with subscripts,
\begin{equation}
\bar{J}_{ij}(t)  =  \bar{J}(t)~~~\mbox{with}~~~M = M_i , m = M_j~
\end{equation}
and similarly for the other symbols.
The subtraction point
dependent part
is contained in the constant $k$
\begin{equation}
k = \frac{1}{32\pi^2} \frac{M^2 \log \left( \frac{M^2}{\mu^2} \right)
                       - m^2 \log\left(\frac{m^2}{\mu^2}\right)}
   {M^2 - m^2},
\end{equation}
where $\mu$ is the subtraction scale.

In addition, in section \ref{section4} these functions and symbols
appear in a  summation
over loops $I$
with
\begin{eqnarray}
J_I(t) &=& \bar{J}(t) ~~~\mbox{with}~~~M = M_I , m = m_I       ~;
\nonumber\\
\Sigma_I&=&M_I^2 + m_I^2
\end{eqnarray}
and again similarly for the others.
There the combination $B_2$ appears  as well :
\begin{eqnarray}
B_2(t,M^2,m^2)&=&B_2 (t,m^2,M^2)
\\
&=&\frac{1}{288\pi^2} \left( 3\Sigma - t\right)
-\frac{\lambda(t,M^2,m^2) \bar{J}(t)}{12t}
+\frac{t\Sigma - 8M^2 m^2}{384\pi^2 \Delta} \log\frac{M^2}{m^2} ~.
\nonumber
\end{eqnarray}

The last formula to be defined is the three propagator loop integral
function $C(t_1,t_2,M^2, m^2)$ where one of the three external momenta
has zero mass and two of the propagators have the same mass $M$.
Here $t_1 = ( q_1 + q_2)^2$, $t_2 = q^2_2$ and $q_1^2 = 0$.
\begin{eqnarray}
C(t_1,t_2,M^2,m^2)&=& - i \int \frac{d^4p}{(2\pi)^d}
\frac{1}{(p^2 - M^2 ) ((p+q_1)^2 - M^2) ((p+q_1+q_2)^2 - m^2)}
\nonumber\\ &=&
-\frac{1}{16\pi^2}\int_0^1 dx \int_0^{1-x} dy
\frac{1}{M^2 - y ( \Delta + t_1 ) +  xy (t_1 - t_2) + y^2 t_1}
\nonumber\\
&=& \frac{1}{(4\pi)^2 (t_1 - t_2)}
\left\{ Li_2 \left( \frac{1}{y_+(t_2)}\right) +
  Li_2\left(\frac{1}{y_- (t_2)}\right)\right.
\nonumber\\& &  \left.
  - Li_2\left(\frac{1}{y_+(t_1)}\right)
  - Li_2\left(\frac{1}{y_-(t_2)}\right)\right\}~,
\nonumber\\
y_{\pm}(t)&=&\frac{1}{2t}\left\{ t + \Delta \pm
\sqrt{\lambda(t,M^2,m^2)}\right\}
\end{eqnarray}
where $Li_2$ is the dilogarithm
\begin{equation}
Li_2(x) = - \int_0^1    \frac{dy}{y}\log(1-xy)   ~.
\end{equation}

%before each section in appendix
\newpage
\setcounter{equation}{0}
\addtocounter{zahler}{1}

\section{Decomposition of the hadronic tensors $I^{\mu \nu}$}
\label{kl2g}
Here we consider the tensors
\be
I^{\mu \nu} =
\int dx e^{iqx+iWy} < 0 \mid T V^\mu_{em} (x) I^\nu_{4-i5}(y) \mid K^+(p)>
\; \;, \; \; I=V,A \; \;
\ee
and detail its connection with the matrix element (\ref{k3}).

The general decomposition of $A^{\mu\nu}, V^{\mu\nu}$
 in terms of Lorentz invariant amplitudes reads \cite{BARDIN,beg} for $q^2
\neq 0$
\bearr
\frac{1}{\sqrt{2}} A^{\mu\nu} &=& - F_K \left \{ \frac{(2W^\mu + q^\mu)
W^\nu}{M_K^2 - W^2} + g^{\mu\nu} \right \}
\nonumber \\
&+& A_1 (q W g^{\mu\nu} - W^\mu q^\nu) + A_2 (q^2 g^{\mu\nu} - q^\mu
q^\nu)
\nonumber \\
&+& \left \{ \frac{2 F_K (F_V^K(q^2)-1)}{(M_K^2-W^2)q^2}
+ A_3 \right\}
 (qWq^\mu - q^2 W^\mu) W^\nu
\label{A6}
\eearr
and
\be
\frac{1}{\sqrt{2}} V^{\mu \nu} = iV_1 \epsilon^{\mu \nu \alpha \beta}
q_\alpha p_\beta
\label{A7}
\ee
where the form factors $A_i(q^2,W^2)$ and $V_1(q^2,W^2)$ are analytic functions
of $q^2$ and $W^2$. $F^K_V (q^2)$ denotes the electromagnetic form factor of
the kaon $(F_V^K (0) = 1)$. $A^{\mu\nu}$ satisfies the Ward identity
\be
q_\mu A^{\mu \nu} = - \sqrt{2} F_K p^\nu.
\label{A8}
\ee

In the process (\ref{k1}) the photon is real. As a consequence of this, only
the two form factors $A_1(0,W^2)$ and $ V_1(0,W^2)$ contribute. We set
\bearr
A(W^2)& = &A_1(0,W^2)
\nonumber \\
V(W^2) &=& V_1 (0, W^2)
\label{A10}
\eearr
and obtain for the matrix element (\ref{k3})
\be
T= -iG_F/\sqrt{2} e {V_{us}}^\star \epsilon^\star_\mu \left \{ \sqrt{2} F_K
l_1^\mu - (V^{\mu\nu} - A^{\mu\nu}) l_\nu \right \}_{ \mid_{q^2=0}} \; \;
, \label{A2}
\ee
with
\bearr
l^{\mu} & =& \bar{u} (p_\nu)\gamma^\mu  (1-\gamma_5) v(p_l)
\nonumber \\
l_1^\mu &=& l^\mu + m_l \bar{u} (p_\nu) (1+\gamma_5) \frac{2p_l^\mu + \not
\!{q} \gamma^\mu}{m_l^2 - (p_l + q)^2} v(p_l) \; \; .
\label{A3}
\eearr
Grouping terms into an IB and a SD piece gives  (\ref{k3},\ref{k4}).
As a consequence of (\ref{A8}), $T$  is
 invariant
under the gauge transformation $\epsilon_\mu \rightarrow \epsilon_\mu +
q_\mu$.

The amplitudes $A_1, A_2$ and $V_1$ are related to the
corresponding quantities $F_A,R$ and $F_V$ used by the PDG \cite{pdg} by
\be
- \sqrt{2} M_K (A_1, A_2, V_1) = (F_A, R, F_V).
\label{A9}
\ee
The last term in (\ref{A6}) is omitted in \cite{pdg}. It
contributes to
processes with a virtual photon, $K^\pm \rightarrow l^\pm \nu_l l'^+ l'^-$.

Finally, the relation to  the notation used in
\cite{ke22,km21} is
\bearr
2 (A \pm V)^2 &=& (a_k \pm v_k)^2 \; \; \; \cite{ke22} \nonumber \\
\sqrt{2} (A,V)& =& (F_A,F_V) \; \; \; \cite{km21} \; \; .
\label{A11}
\eearr


%before each section in appendix
\newpage
\setcounter{equation}{0}
\addtocounter{zahler}{1}

\section{Formulas for the traces in terms of $x,y$ and $z$
for the decays $K^+ \to l^+ \nu l'^+ l'^-$}
\label{TRACES}

This is the FORTRAN program used to evaluate the differential
decay rate    in terms of the kinematic variables used in the text.
The formfactors are A1,A2,A4 and V1 with their complex conjugates
A1C, A2C, A4C and V1C, all made dimensionless by multiplying with the
relevant power of $M_K$. T is the quantity
$\left\{ - \sum_{spins}\overline{T_\mu}^* \overline{T}^\mu\right\}$.
The matrix element squared, before the integration over the lepton
pair kinematic variables, is available on request from the authors.
\begin{verbatim}
C  ALL QUANTITIES ARE IN UNITS OF MK TO THE RELEVANT POWER
      W2   = 1.0 - X + Z
      P$W  = 1.0 - X/2.0
      P$PN = P$W - Y/2.0
      PL$PN= (W2 - RL)/2.0
      PL$W = RL + PL$PN
      PN$W = W2 - PL$W
      Q$W  = X/2.0 - Z
      Q$PL = Y/2.0 - PL$W
      Q$PN = P$PN - PN$W
      DENOM1 = 1.0/(2.0*Q$PL + Z)
      DENOM2 = 1.0/(X - Z)
      A11 = REAL ( A1 + A1C)
      A22 = REAL ( A2 + A2C)
      A44 = REAL ( A4 + A4C)
      VV  = REAL ( V1 + V1C)
      A12 = REAL (A1*A2C + A1C*A2)
      A14 = REAL (A1*A4C + A1C*A4)
      A1V = REAL (A1*V1C + A1C*V1)
      A24 = REAL (A2*A4C + A2C*A4)
      A2V = REAL (A2*V1C + A2C*V1)
      A4V = REAL (A4*V1C + A4C*V1)
      A1A1C = REAL (A1*A1C)
      A2A2C = REAL (A2*A2C)
      A4A4C = REAL (A4*A4C)
      V1V1C = REAL (V1*V1C)
      T = 0.0
      T = T + A1A1C * ( 16*Q$PL*Q$PN*W2 - 16*Q$PL*Q$W*PN$W - 16*Q$PN*
     +    Q$W*PL$W - 8*PL$PN*Z*W2 )
      T = T + A2A2C * (  - 16*Q$PL*Q$PN*Z - 8*PL$PN*Z**2 )
      T = T + A4A4C * ( 8*Q$W**2*PL$PN*Z*W2 - 16*Q$W**2*PL$W*PN$W*Z -
     +    8*PL$PN*Z**2*W2**2 + 16*PL$W*PN$W*Z**2*W2 )
      T = T + V1V1C * (  - 8*P$PN*Q$PL*X + 8*P$PN*Y*Z + 16*Q$PL*Q$PN
     +     - 4*Q$PN*X*Y )
      T = T + FK**2*DENOM1**2 * (  - 32*Q$PL**2*PL$PN*Z**(-1)*RL - 32*
     +    Q$PL*Q$PN*RL - 32*Q$PL*PL$PN*RL + 32*Q$PN*RL**2 + 8*PL$PN*Z*
     +    RL + 32*PL$PN*RL**2 )
      T = T + FK**2*DENOM1*DENOM2 * ( 32*P$PN*Q$PL*RL + 32*Q$PL*PL$PN*X
     +    *Z**(-1)*RL - 16*Q$PN*Y*RL - 32*PL$PN*Y*RL )
      T = T + FK**2*DENOM2**2 * (  - 8*PL$PN*X**2*Z**(-1)*RL + 32*PL$PN
     +    *RL )
      T = T + FK*DENOM1*A11 * ( 16*Q$PN*Q$W*RL - 16*Q$PN*PL$W*RL + 16*
     +    Q$W*PL$PN*RL + 8*PN$W*Z*RL )
      T = T + FK*DENOM1*A22 * (  - 16*Q$PL*Q$PN*RL + 24*Q$PN*Z*RL + 16*
     +    PL$PN*Z*RL )
      T = T + FK*DENOM1*A44 * ( 16*Q$PL*Q$W*PN$W*RL - 8*Q$PN*Z*W2*RL +
     +    8*Q$W*PN$W*Z*RL - 16*PL$W*PN$W*Z*RL )
      T = T + FK*DENOM1*VV * ( 16*P$PN*Z*RL - 8*Q$PN*X*RL )
      T = T + FK*DENOM2*A11 * (  - 16*P$PN*Q$W*RL + 16*P$W*Q$PN*RL )
      T = T + FK*DENOM2*A22 * (  - 16*P$PN*Z*RL + 8*Q$PN*X*RL )
      T = T + FK*DENOM2*A44 * ( 16*P$W*PN$W*Z*RL - 8*Q$W*PN$W*X*RL )
      T = T + A12 * (  - 8*Q$PL*PN$W*Z - 8*Q$PN*PL$W*Z - 8*Q$W*PL$PN*Z
     +     )
      T = T + A14 * ( 8*Q$PL*PN$W*Z*W2 + 8*Q$PN*PL$W*Z*W2 - 16*Q$W*PL$W
     +    *PN$W*Z )
      T = T + A1V * (  - 8*P$PN*Q$PL*Q$W - 8*P$PN*PL$W*Z - 4*Q$PL*PN$W*
     +    X + 4*Q$PN*Q$W*Y + 4*Q$PN*PL$W*X + 4*PN$W*Y*Z )
      T = T + A24 * ( 8*Q$PL*Q$W*PN$W*Z + 8*Q$PN*Q$W*PL$W*Z - 8*Q$W**2*
     +    PL$PN*Z + 8*PL$PN*Z**2*W2 - 16*PL$W*PN$W*Z**2 )
      T = T + A2V * (  - 16*P$PN*Q$PL*Z + 8*Q$PN*Y*Z )
      T = T + A4V * ( 8*P$PN*Q$PL*Z*W2 - 8*P$PN*Q$W*PL$W*Z - 8*P$W*Q$PL
     +    *PN$W*Z + 8*P$W*Q$PN*PL$W*Z - 4*Q$PN*Y*Z*W2 + 4*Q$W*PN$W*Y*Z
     +     )
      T = -T
\end{verbatim}
%before each section in appendix
\newpage
\setcounter{equation}{0}
\addtocounter{zahler}{1}

\section{FORTRAN routine for the calculation of the reduced square of
the $K_{l3\gamma}$ matrix element $SM$}  \label{KL3G}

\begin{verbatim}

      FUNCTION SM(EG,EP,W2,EL,X)
C THE FUNCTION SM CALCULATES THE REDUCED SQUARE OF THE MATRIX
C ELEMENT OF SECT. 4 IN TERMS OF THE SCALAR VARIABLES EG(PHOTON
C ENERGY), EP(PION ENERGY), W2(INVARIANT MASS SQUARED OF LEPTON
C PAIR), EL(ENERGY OF CHARGED LEPTON) AND X=PL.Q/MK^2 AND IN
C TERMS OF THE VECTOR AMPLITUDES B1,..,B7, AXIAL AMPLITUDES
C A1, A2, A3  (A4=0 TO O(P^4)) AND C1, C2 DEFINED IN SECT. 4.
C ALL DIMENSIONFUL QUANTITIES ARE NORMALIZED TO THE KAON MASS:
C MP=M(PION)/M(KAON), ML=M(LEPTON)/M(KAON), ETC.
      REAL ML,MP,ML2,MP2,ML4
      COMMON/MASSES/ML,MP
      ML2=ML**2
      MP2=MP**2
      ML4=ML**4
C SCALAR PRODUCTS: QPP=Q.P', PLPP=P(LEPTON).P', WPP=W.P',
C QW=Q.W, ALL SCALED TO M(KAON)=1; W=P(LEPTON)+P(NEUTRINO)
      QPP=EP+EG+(W2-MP2-1.)/2.
      PLPP=EL-X-(W2+ML2)/2.
      WPP=-EG+(1.-MP2-W2)/2.
      QW=-EP+(1.+MP2-W2)/2.
C FOR ILLUSTRATION, THE TREE LEVEL AMPLITUDES B1,...,B7,
C A1,A2,A3,C1,C2 FOR K0(L3GAMMA) ARE LISTED BELOW
      B1=-1.
      B2=0.
      B3=2./QPP
      B4=0.
      B5=0.
      B6=1./QPP
      B7=B3
      A1=0.
      A2=0.
      A3=0.
      C1=2.
      C2=1.
C IN THE FOLLOWING, SM IS CALCULATED IN TERMS OF SCALAR
C PRODUCTS, MASSES AND INVARIANT AMPLITUDES.
C THIS PART IS INDEPENDENT OF THE CHOICE OF SCALAR VARIABLES
C TO SPECIFY THE KINEMATICS (P(LEPTON).Q IS DENOTED X)
      R1=B1*(B1-B5*ML2)*(-ML2 + W2)
      R1=R1 + B1*B2*2*ML2*(X-QW)
      R11=B1*B3+B2*B4*W2+(B2*B7+B3*B4)*WPP+B3*B7*MP2
      R1=R1+R11*(4*X*PLPP-2*X*WPP-2*PLPP*QW-
     +    QPP*ML2 + QPP*W2)
      R1=R1 + B1*(B4+B6)*2*ML2*(PLPP-WPP)
      R1=R1 + B1*A1*(-4*X*WPP + 4*PLPP*QW)
      R1=R1 + B1*A2*(4*X*W2-2*QW*ML2-2*QW*W2)
      R1=R1 + B2*(B2*W2+2*B3*WPP)*2*X*(X-QW)
      R1=R1 + B2*(B5*W2+B6*WPP)*2*ML2*(X-QW)
      R12=B2*A1+B3*A2+A3*ML2*(A2-C2/2./X)
      R1=R1+R12*(-2*X*QW*WPP+2*X*QPP*W2+2*PLPP
     +    *QW**2-QW*QPP*ML2-QW*QPP*W2)
      R1=R1 + B3**2*2*MP2*X*(X-QW)
      R1=R1 + B3*(B5*WPP+B6*MP2)*2*ML2*(X-QW)
      R1=R1+(B4**2*W2+B7**2*MP2+2*B1*B7)*(-1./2.*ML2*MP2+1./2.*MP2*W2
     +     + 2*PLPP**2-2*PLPP*WPP)
      R1=R1+(B4*(B5*W2+B6*WPP)+B7*(B5*WPP+B6*MP2))*2*ML2*(PLPP-WPP)
      R1=R1+B4*B7*WPP*(4*PLPP**2-4*PLPP*WPP-
     +    ML2*MP2+MP2*W2)
      R1=R1+(B4*A1+B7*A2)*(-2*X*WPP**2+2*X*MP2*W2+2*PLPP
     +    *QW*WPP-2*PLPP*QPP*W2-QW*ML2*MP2-QW*MP2*W2
     +     + QPP*WPP*ML2 + QPP*WPP*W2)
      R1=R1+(B5*(B5*W2/2+B6*WPP)+B6**2*MP2/2)*ML2*(ML2-W2)
      R1=R1 + A1**2*(2*X**2*MP2-4*X*PLPP*QPP-2*X*
     +    QW*MP2 + 2*X*QPP*WPP + 2*PLPP*QW*QPP)
      R1=R1 + A1*A2*(-4*X**2*WPP + 4*X*PLPP*QW + 2*X
     +    *QW*WPP + 2*X*QPP*ML2-2*PLPP*QW**2-QW*QPP*
     +    ML2-QW*QPP*W2)
      R1=R1 + (A1+C1/2./X)*A3*ML2*(2*X*QW*MP2-2*X*QPP*WPP
     +    -2*PLPP*QW*QPP-2*QW**2*MP2 + 4*QW
     +    *QPP*WPP + QPP**2*ML2-QPP**2*W2)
      R1=R1 + A2**2*(2*X**2*W2-2*X*QW*ML2-2*X*QW*
     +    W2 + QW**2*ML2 + QW**2*W2)
      R1=R1+A3**2*ML2*(1./2.*QW**2*ML2*MP2-1./2.*QW**2
     +    *MP2*W2-QW*QPP*WPP*ML2+QW*QPP*WPP*W2 + 1./
     +    2.*QPP**2*ML2*W2-1./2.*QPP**2*W2**2)
      R2=C1**2*(-1./2.*ML4*MP2 + 1./2.*ML2*MP2*W2
     +     + X**2*MP2-2*X*PLPP*QPP-X*QW*MP2 + 2*
     +    X*QPP*WPP-X*ML2*MP2 + 2*PLPP**2*ML2 + 2*
     +    PLPP*QPP*ML2-2*PLPP*WPP*ML2 + QW*ML2*MP2-2*
     +    QPP*WPP*ML2)
      R2=R2 + C1*C2*(2*X**2*WPP-2*X*PLPP*QW + 4*X*
     +    PLPP*ML2-X*QPP*ML2 + X*QPP*W2-4*X*WPP*
     +    ML2 + 2*PLPP*ML4-2*WPP*ML4)
      R2=R2 + C2**2*(1./2.*ML**6-1./2.*ML4*W2 + 2*X**2*
     +    ML2 + X**2*W2-3*X*QW*ML2 + 2*X*ML4-X*
     +    ML2*W2-QW*ML4)
      RI=B1*C1*(-2*X*WPP + 2*PLPP*QW-2*PLPP*ML2
     +     + QPP*ML2-QPP*W2 + 2*WPP*ML2)
      RI=RI + B1*C2*(-ML4 + ML2*W2-2*X*ML2-2*X*
     +    W2 + 4*QW*ML2)
      RI=RI + B2*C1*(2*X**2*WPP-2*X*PLPP*QW-2*X*
     +    PLPP*ML2-2*X*PLPP*W2-2*X*QW*WPP-X*QPP*
     +    ML2 + X*QPP*W2 + X*WPP*ML2 + X*WPP*W2 + 2*
     +    PLPP*QW**2 + PLPP*QW*ML2 + PLPP*QW*W2 + QW*QPP*
     +    ML2-QW*QPP*W2 + 1./2.*QPP*ML4-1./2.*QPP*W2**2)
      RI=RI+B2*C2*ML2*(-2*X**2-X*ML2-X*W2
     +    +2*QW**2+QW*ML2+QW*W2)
      RI=RI + B3*C1*(2*X**2*MP2-4*X*PLPP**2-4*X*
     +    PLPP*QPP + 2*X*PLPP*WPP-2*X*QW*MP2 + 2*X*
     +    QPP*WPP + 2*PLPP**2*QW + 2*PLPP*QW*QPP + PLPP*QPP*
     +    ML2-PLPP*QPP*W2 + QPP**2*ML2-QPP**2*W2)
      RI=RI + B3*C2*(-4*X**2*PLPP + 2*X**2*WPP + 2*X*
     +    PLPP*QW-2*X*PLPP*ML2-X*QPP*ML2-X*QPP*
     +    W2 + 2*PLPP*QW*ML2 + 2*QW*QPP*ML2)
      RI=RI + B4*C1*(1./2.*ML4*MP2-1./2.*MP2*W2**2 + 2*
     +    X*PLPP*WPP-2*X*WPP**2 + X*MP2*W2-2*PLPP**2
     +    *QW-2*PLPP**2*ML2-2*PLPP**2*W2 + 2*PLPP*QW*WPP-
     +    PLPP*QPP*ML2-PLPP*QPP*W2 + 2*PLPP*WPP*ML2 + 2*
     +    PLPP*WPP*W2-QW*MP2*W2 + QPP*WPP*ML2 + QPP*WPP*W2)
      RI=RI + B4*C2*ML2*(-2*X*PLPP+X*WPP-
     +    PLPP*QW-PLPP*ML2-PLPP*W2 + 2*QW*WPP
     +    +1./2.*QPP*ML2-1./2.*QPP*W2 + WPP*ML2
     +     + WPP*W2)
      RI=RI + B5*C1*ML2*(X*WPP-PLPP*QW-PLPP*
     +    ML2-PLPP*W2-1./2.*QPP*ML2 + 1./2.*QPP*W2
     +    +WPP*ML2 + WPP*W2)
      RI=RI + B5*C2*ML2*(-1./2.*ML4+1./2.*W2**2-X*
     +    ML2+QW*W2)
      RI=RI + B6*C1*ML2*(X*MP2-2*PLPP**2-2*PLPP
     +    *QPP+2*PLPP*WPP-QW*MP2 + 2*QPP*WPP)
      RI=RI + B6*C2*ML2*(-2*X*PLPP+X*WPP+
     +    PLPP*QW-PLPP*ML2 + PLPP*W2-1./2.*QPP*
     +    ML2 + 1./2.*QPP*W2)
      RI=RI + B7*C1*(2*X*PLPP*MP2-X*WPP*MP2-4*
     +    PLPP**3-4*PLPP**2*QPP + 4*PLPP**2*WPP-PLPP*QW*
     +    MP2 + 4*PLPP*QPP*WPP + PLPP*ML2*MP2-PLPP*MP2*
     +    W2 + 1./2.*QPP*ML2*MP2-1./2.*QPP*MP2*W2)
      RI=RI + B7*C2*(-4*X*PLPP**2 + 4*X*PLPP*WPP-X
     +    *MP2*W2-2*PLPP**2*ML2 + 2*PLPP*WPP*ML2 + QW*ML2
     +    *MP2)
      RI=RI + A1*C1*(2*X**2*MP2-4*X*PLPP*QPP + 2*X*
     +    PLPP*WPP-2*X*QW*MP2 + 4*X*QPP*WPP-X*ML2
     +    *MP2-X*MP2*W2-2*PLPP**2*QW + PLPP*QPP*ML2 +
     +    PLPP*QPP*W2 + 2*QW*ML2*MP2-2*QPP*WPP*ML2)
      RI=RI + A1*C2*(4*X**2*WPP-4*X*PLPP*QW-2*X*
     +    QPP*ML2 + 2*QW*QPP*ML2)
      RI=RI + A2*C1*(-2*X**2*WPP + 2*X*PLPP*QW-2*X
     +    *PLPP*W2 + X*QPP*ML2-X*QPP*W2 + X*WPP*ML2
     +     + X*WPP*W2 + PLPP*QW*ML2 + PLPP*QW*W2-2*QW*WPP
     +    *ML2-1./2.*QPP*ML4 + QPP*ML2*W2-1./2.*QPP*W2**2)
      RI=RI + A2*C2*(-4*X**2*W2 + 4*X*QW*ML2 + 2*X*
     +    QW*W2-2*QW**2*ML2)
      SM=R1+RI/X+R2/X**2
      RETURN
      END
\end{verbatim}

\newpage
\addcontentsline{toc}{section}{\hspace{1cm}List of Tables}
\listoftables
\newpage
\addcontentsline{toc}{section}{\hspace{1cm}List of Figures}
\listoffigures
\newpage
\addcontentsline{toc}{section}{\hspace{1cm}Bibliography}
\begin{thebibliography}{99}

\newcommand{\PL}[3]{{Phys. Lett.}        {#1} {(19#2)} {#3}}
\newcommand{\PRL}[3]{{Phys. Rev. Lett.} {#1} {(19#2)} {#3}}
\newcommand{\PR}[3]{{Phys. Rev.}        {#1} {(19#2)} {#3}}
\newcommand{\NP}[3]{{Nucl. Phys.}        {#1} {(19#2)} {#3}}



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\end{thebibliography}

\end{document}

