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\newcommand{\beq}{\begin{equation}}
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\begin{document}
\thispagestyle{empty}   % to suppress the page number on the first page
%\begin{flushright}
%Preliminary version\\
% Jun 21, 1995
%\end{flushright}
\begin{flushleft}

ZTF - 95/04\\

\end{flushleft}
\vspace{1.4cm}

\begin{center}

  \begin{Large}

  \begin{bf}

Chiral Anomaly Route to the $K_L\rightarrow \pi^+ \pi^- \gamma $ Decay

  \end{bf}

  \end{Large}

\vspace{1cm}

  \begin{Large}

Kre\v{s}imir Kumeri\v{c}ki  and Ivica Picek
\vspace{0.5cm}
%\end{Large}

{\small Departement of Physics, Faculty of Science, University of Zagreb, 
POB 162, 41000 Zagreb, Croatia\\}    

\end{Large}

\end{center}

\vspace{1.0cm}
\begin{center}
  {\bf Abstract}
\end{center}
\begin{quotation}
\noindent
The increasing body of experimental data motivates us to 
 investigate a new contribution to the radiative $K_L\rightarrow \pi^+ \pi^-
 \gamma $  decay, which is ${\cal{O}}(p^4)$ within
the framework of chiral perturbation theory.
The new contribution refers to the $\Delta S = 1$ WZW action term,
established recently for the direct,  ${\cal{O}}(p^4)$,  
$K_L\rightarrow \gamma \gamma (-)$ decay amplitude.
%This approach represents a viable parametrisation for the whole set of
%anomalous non-leptonic radiative $K$-decays reveals the anatomy of
%the respective short distance amplitudes.
We present additional evidence in support of such flavour-changing generalization
of  the  WZW action term as a viable parametrisation of the  anomalous radiative  
non-leptonic kaon decays.

\end{quotation}

\newpage

\begin{Large}

1. Introduction

\end{Large}

\vspace{0.5cm}

In a search for the origin of CP violation in particle physics, the neutral
kaon system still seems to be the most prosperous laboratory. A distinguished
feature of this system is the fact that it embodies two distant 
scales of energy:
pseudo-Goldstone kaons have a long-distance (LD) appearance,
whereas CP-violating $K$-decay amplitudes probe the short-distance (SD)
scale.Namely, since $K$ contains only quarks of the first and second generation, CP
violation requires the appearance of (heavy) quarks of the third generation
in loops. Thus,  CP-violating $K$-decays add to an eminent class of physical
phenomena which experience many scales of distance.

By exploring nature by this route, one might reveal some of its most profound 
features. An example is the appearance of quantum
mechanical processes on the macroscopic scale (like low-temperature superconductivity) normally
thought to proceed at the microscopic scale only.  The other known example
of a phenomenon spreading over different distance scales (and employed
in the present account) is known as the chiral, triangle, or the WZW anomaly. How
it is  termed depends on the theoretical framework and the corresponding 
distance scale at which this phenomenon is considered. 
Its history started with  a
$VVA$ fermion (proton) triangle loop accounting for the 
$\pi\rightarrow 2\gamma$ decay. 
Effectively, the anomaly leads to the electromagnetic transition in a situation
where one would expect none ($\pi^{0}$ is electrically neutral). In
this way, the anomaly represents a signature of a substructure (at the
level of charged constituents, the electromagnetic transition appears
naturally).
Further direct manifestation of the chiral anomaly one can find in the 
low-energy interactions of the pseudoscalar meson octet.
Once quarks have been established as fundamental fermions, 
the anomalous triangle received its ``chiral'' counterpart towards 
ultraviolet (UV), 
and its ``WZW'' counterpart towards infra-red (IR). 
Then, it is almost inevitable 
to recognize the {\em anomaly matching principle} in a way in which 
't Hooft  \cite{t Hooft} did it:
one requires the anomaly appearing at any scale (with its particular degrees
of freedom), including the eventual preonic (constituents
of the quark) level.
 
  In this paper we exploit the fact that (in the SM) at the electroweak
scale the triangle anomaly originates at the UV/SD scale, 
the same scale at which the CP-violating $K$-decay amplitude
in SM model seem to originate. 
In general, the theoretical 
analysis in the $K$-system is plagued by LD uncertanities, and the separation 
of the SD and the LD parts faces almost insurmountable difficulties.
However, by focusing on the selected anomaly-governed amplitudes at the 
mesonic scale, we could possibly infer features of
some SD (CP-violating) $K$-decay 
amplitudes. In this sense, recognising the anomalous portions of $K$-decay 
amplitudes might be essential in decision making in experimentation: among
the rare processes which are on the border of feasibility, the
anomalous ones have a chance to persist from their SD (short-wavelength)
origin to the LD (long-wavelength) scale of experimentation.  

Following previous investigations   of the potentially  large CP-violating 
$K_S\rightarrow \gamma \gamma (-)$ amplitude \cite{ep93,ep94},
we now focus on the  $K_L\rightarrow \pi^+ \pi^-  \gamma $ amplitude.
Thereby we elaborate in more detail a recent observation \cite{ep94}
that the decay amplitudes for these processes contain an anomalous
piece. In fact, a microscopic treatment of the $K_L\rightarrow \gamma \gamma$
decay (first \cite{ep94} within the chiral quark model, followed by a confirmation 
\cite{k3p} in the bound-state calculation) revealed the previously unknown
{\em direct-decay} amplitude. On basis of such microscopic experience
Eeg and Picek  \cite{ep94}  suggested a generalization
of the Wess-Zumino-Witten \cite{WZ} term to the $\Delta S = 1$
$K_L\rightarrow \gamma \gamma$ transition.
In this paper we investigate  whether such WZW 
description, {\em generalized} to a wider class of anomalous
procesess, could
shed some light onto the nature of the direct amplitude measured
recently in the  $K_L\rightarrow \pi^+ \pi^-  \gamma $ decay \cite{Forty}.

\vspace{0.5cm}

\begin{Large}

2. Direct  CP violation outside $K \rightarrow \pi \pi$

\end{Large}

\vspace{0.2cm}

After thirty years, the evidence for CP violation in particle physics  has
remained restricted to the measurement of the value of the $K_L$ 
to $K_S$ ratios:
%\begin{description}
\[
|\eta_{+-}|\simeq |\eta_{oo}| \sim 2\times 10^{-3}~.
\]
%\end{description}
The ratio for  charged pions
\begin{equation}
\eta_{+-}   =  \frac{A(K_L \rightarrow \pi^+ \pi^-)}{A(K_S \rightarrow 
\pi^+ \pi^-)} 
= \epsilon + \epsilon' 
\label{etapm}
\end{equation}
and for neutral pions
\begin{equation}
\eta_{oo} = \frac{A(K_L \rightarrow \pi^o \pi^o)}{A(K_S \rightarrow 
\pi^o \pi^o)} = \epsilon - 2 \epsilon'  
\label{eta00}
\end{equation}
differ in the parameter $\epsilon'$ . This parameter measures $\Delta S = 1$ (direct)
 CP violation, whereas  $\epsilon$ is a measure of $\Delta S = 2$ (indirect)
 CP violation.
 The present data on the parameter $\epsilon'$ quantifying direct CP violation are
inconclusive.  Whereas the Fermilab \cite{E731} measurement is consistent with zero,
\[
{\rm Re}~\frac{\epsilon^\prime}{\epsilon}=
(7.4 \pm 5.2 \pm 2.9) \times 10^{-4} ~~~\mbox{[E731],}
\]
the  NA31 experiment at CERN  \cite{NA31} gives
$3\sigma$ evidence,
\[
{\rm Re}~\frac{\epsilon^\prime}{\epsilon}=
(23.0 \pm 3.6 \pm 5.4) \times 10^{-4} ~~~\mbox{[NA31].}
\]
The sensitivity  at the $10^{-4}$ level is expected from the forthcoming
facilities at Fermilab (E832), CERN (NA48), and the Frascati $\Phi$-factory
(DA$\Phi$NE).

Therefore, a study of other (rare) K-decay modes might elucidate the
existence of direct CP violation. There are several  promising
candidate decays advocated in recent reviews \cite{Forty,RW,Peccei}.
For the $K_{\rm S}\rightarrow3\pi$ decays among them,
CPLEAR (CERN) has already  
provided first results. The $\eta_{+-o}$
parameter  related to the $K_{\rm S}\rightarrow \pi^+\pi^-\pi^o$ decay
mode (having both the CP-conserving and CP-violating pieces) is
\[
{\rm Re}~(\eta_{+-o})=
(5 \pm 2\pm 7) \times 10^{-3} ~~~\mbox{[CPLEAR]}
\]

and
\[
{\rm Im}~(\eta_{+-o})=
(1.6 \pm 2.4 \pm 1.8) \times 10^{-2} ~~~\mbox{[CPLEAR].}
\]
The $K_{\rm S}\rightarrow 3\pi^o$ decay mode (having only the 
CP-violating piece)
differs from the previous one in the direct-CP-violation piece:
\[
\eta_{ooo}=\epsilon+\epsilon^\prime_{ooo}~; \;\;
\eta_{+-o}=\epsilon+\epsilon^\prime_{+-o}~.
\]
Still, a foreseeable sensitivity both for CPLEAR and for 
the DA$\Phi$NE
will stay at the level  of $\epsilon$, so that the direct CP violation issue 
remains inconclusive.

For similar reasons, one of the authors started to study the
radiative $K_{L,S} \rightarrow 2 \gamma$ decays \cite{ep93}
in the hope of telling
direct CP violation from signals in these processes.

The parameter for $CP= +1$ photons, 
\begin{equation}
\eta_+ \equiv \eta_{\gamma \gamma (+)} =  { A(K_L \rightarrow \gamma \gamma (+))
\over A(K_S \rightarrow \gamma \gamma (+))} 
= \epsilon + \epsilon'_{\gamma \gamma (+)} \simeq \epsilon \quad,
\end{equation}
is essentially determined by indirect CP violation.
This is due to the fact that, in chiral perturbation theory,  $K_S \rightarrow 2
\gamma$ first appears through the pion loop. Such a structure,
\begin{center}

 $K_S \rightarrow
2 \pi \rightarrow
2 \gamma ,$

\end{center}
indicates that the CP-violating
amplitude of this decay has the same origin as in $K_S \rightarrow
2 \pi$, implying
$\epsilon'_{\gamma\gamma(+)}  \simeq  \epsilon'$ 
or $\eta_{\gamma\gamma(+)} \simeq \eta_{+-}$ in (\ref{etapm}), 
where the direct CP-violating 
parameter $\epsilon'$  is inhibited by a factor of 22 coming from the $\Delta I
= 1/2$  rule. 
Thus, direct CP violation will be manifested through $\epsilon'_{\gamma\gamma(-)}$, 
entering the ratio of the physical amplitudes for $K_{L,S} \rightarrow \gamma
\gamma (-)$ decays (where $(-)$ refers to the $CP = -1$ state of the
photons),
\begin{equation}
\eta_- \equiv  \eta_{\gamma \gamma (-)} = { A(K_S \rightarrow \gamma \gamma (-))
\over A(K_L \rightarrow \gamma \gamma (-))} 
= \epsilon + \epsilon'_{\gamma \gamma (-)} .
\end{equation}
Since such a study awaits a dedicated experiment at CPLEAR \cite{Forty},
the rest 
of this paper will be devoted to the $K_L\rightarrow \pi^+ \pi^- \gamma $
channel 
where CP violation has already been measured! It required 
the use of the $\gamma$ energy 
spectrum in order to separate  contributions from internal bremsstrahlung (IB)
from those of direct emission (DE). The first measurement by E731 \cite{E731a}
\[
|\eta_{+-\gamma} | =
(2.15 \pm 0.26 \pm 0.17)  \times 10^{-3} ~~~\mbox{[E731]}\quad,
\]
confirmed  recently by E773 \cite{E773}
\[
|\eta_{+-\gamma} | =
(2.414 \pm 0.065 \pm 0.062)\times 10^{-3}  ~~~\mbox{[E773]} \quad,
\]
refers actualy to the IB contribution to the quantity which in 
total acquires the form \cite{DA94}
\begin{equation}
\eta_{+-\gamma} =  { A(K_L \rightarrow \pi^{+}\pi^{-}\gamma)_{IB+E1}
\over A(K_S \rightarrow \pi^{+}\pi^{-}\gamma)_{IB+E1}}\simeq
\eta_{+-}+\epsilon'_{\pi\pi\gamma}\frac{A(K_S \rightarrow \pi^{+}\pi^{-}\gamma
)_{E1}}{A(K_S \rightarrow \pi^{+}\pi^{-}\gamma)_{IB}} \quad.
\end{equation}
Although the interesting direct CP-violating $\epsilon'_{\pi\pi\gamma}$
is suppresed by a small factor $A_{E1}/A_{IB}$, it is not suppresed
by the factor dictated by the $\Delta I=1/2$ rule, and might be improved
by the time evolution measurement at DA$\Phi$NE.

However, these experimental achievements and prospects
are confronted with rather
poor theorethical predictions \cite{DMS}. Therefore we are trying to attack
this problem form another side, by employing the anomalous nature of the
decay under consideration.


\vspace{0.5cm}

\begin{Large}

3. Three facets of anomaly in non-leptonic $K$-decays

\end{Large}

\vspace{0.2cm}

(i) A major source of the theoretical uncertainty in predicting the
$K_L\rightarrow \pi^+ \pi^- \gamma$ amplitude originates from the
cancellation of the potentially leading contributions. Those are
termed the {\em reducible} contributions, where the photonic
couplings to mesons originate from the WZW term
\begin{equation}
{\cal L}_{WZW}= -{i N_c \over 48\pi^2}\epsilon^{\mu\nu\alpha\beta}~
 Tr [W(U, l, r)_{\mu\nu\alpha\beta} - W(I, l, r)_{\mu\nu\alpha\beta}] \; ,
\label{eq:wz}
\end{equation}
where $N_c$ is the number of colours. 
Such term closes
the gauging procedure imposed on the original 
bosonic lagrangian 
represented non-linearly by the $3 \times 3$ matrix:
\begin{equation}
U\equiv \exp\biggl({{2i\over f}\Pi}\biggr) \; .
\label{eq:uf}
\end{equation}
Here  $\Pi = \sum_a \pi^a\lambda^a/2 \,$ is given by the eight Goldstone fields
$\pi^a (a=1,..,8)$, and 
$f$ can be identified with the pion decay constant
$f = f_\pi = (92.4 \pm 0.2)~{\rm MeV}$ ($=f_K,$ in the chiral limit).

For the purely electromagnetic ($A_\mu$ -- field) gauging 
\begin{equation}
l_\mu= r_\mu= a_\mu \equiv -e A_\mu Q; \quad Q = \frac{1}{3} diag(2,-1,-1),
\end{equation}
the tensor in (\ref{eq:wz}) acquires the form (symmetric under
$U\leftrightarrow U^{\dagger}, \Sigma^{L}_{\mu}=U^{\dagger}
\partial_{\mu}U\leftrightarrow \Sigma^{R}_{\mu}=U\partial_{\mu}U^{\dagger}$)
\begin{eqnarray}
W(U, a)_{\mu\nu\alpha\beta} \, = \,
\Sigma^{L}_{\mu} U^\dagger\partial_\nu a_\alpha U a_\beta
+ \Sigma^{L}_{\mu} a_\nu \partial_\alpha   a_\beta \nonumber \\
+ \Sigma^{L}_{\mu} \partial_\nu a_\alpha  a_\beta
- i \Sigma^{L}_{\mu} \Sigma^{L}_{\nu} \Sigma^{L}_{ \alpha }  a_\beta
- ( L \leftrightarrow R ) \; .
\end{eqnarray}
Then by expanding
the expression in the parenthesis in (\ref{eq:wz})
\begin{equation}
[W(U, a)_{\mu\nu\alpha\beta} - W(I, a)_{\mu\nu\alpha\beta}] 
\; = \; V_{\mu\nu\alpha\beta}  \; ,
\label{eq:wz1}
\end{equation}
to the wanted order, one generates the anomalous meson-photon couplings 
containing
the well-known flavour-diagonal term of order ${\cal O}(p^4)$ 
responsible for  $\pi^0 \rightarrow \gamma \gamma$
\begin{eqnarray}
{\cal L}^{(4)}_{WZW} \, = \,
 -{N_c \alpha \over 24 \pi f_\pi} \epsilon_{\mu\nu\rho\sigma}
F^{\mu\nu}F^{\rho\sigma} \biggl(\pi^0 + {\eta \over \sqrt{3}}
\biggr) \nonumber \\
\, + \, { i  N_c e \over 12 \pi^2f_\pi^3} \epsilon_{\mu\nu\rho\sigma} A^\mu
\partial^\nu \pi^+ \partial^\rho \pi^- \partial^\sigma\biggl(\pi^0 +
{\eta \over \sqrt{3}}\biggr) \quad.
\label{eq:piwz}
\end{eqnarray}
Such couplings, in conjunction with the non-leptonic $\Delta S = 1$ 
transition (\ref{eq:cro}) lead to the LD appearance of  the chiral
anomaly in the radiative non-leptonic $K$-decays.
 There is nonvanishing tree level
contribution to $K_{L}\rightarrow \pi^0 \pi^0 \gamma\gamma$ (Fig. 1a),
the real representative of ${\cal O}(p^4)$ reducible anomalous process.

 However, very similarly to $K_L\rightarrow\gamma\gamma$ (Fig. 1b), the pole
diagrams for $K_L\rightarrow \pi^+ \pi^- \gamma$ (Fig. 2a), generated
by the chiral WZW term (\ref{eq:wz}), vanish in leading order chiral
perturbation theory($\chi$PT). There is nonvanishing tree level
contribution to $K_{L}\rightarrow \pi^0 \pi^0 \gamma\gamma$ (Fig. 1a),
the real representative of ${\cal O}(p^4)$ reducible anomalous process.
The cancellation at this ${\cal O}(p^4)$ order
of the reducible amplitudes for procesess
adresses us to the study of less known direct amplitudes.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{fmffile}{neet}

\begin{figure}
\fmfframe(80,35)(0,35){
\begin{fmfgraph*}(120,70)
\fmfleft{i1}
\fmfright{o1,o2}
\fmf{fermion}{i1,v1}
\fmfv{decor.shape=square,decor.filled=0, label=(a),
label.angle=-110,label.dist=0.63h}{v1}
\fmf{fermion,label=$\noexpand\pi^0,,\noexpand\eta$}{v1,v2}
\fmf{photon}{v2,o1}
\fmf{photon}{v2,o2}
\fmfv{label=$\noexpand\gamma$}{o1,o2}
\fmfdot{v2}
\fmfv{label=$K_{L}$}{i1}
\fmffreeze
\fmftop{t1}
\fmfbottom{b1}
\fmf{fermion}{v1,t1}
\fmf{fermion}{v1,b1}
\fmfv{label=$\noexpand\pi^{0}$}{t1,b1}
\end{fmfgraph*}}
\fmfframe(50,35)(0,35){
\begin{fmfgraph*}(120,70)
\fmfleft{i1}
\fmfv{label=$K_{L}$}{i1}
\fmfright{o1,o2}
\fmf{fermion}{i1,v1}
\fmf{fermion,label=$\noexpand\pi^0,,\noexpand\eta$}{v1,v2}
\fmf{photon}{v2,o1}
\fmf{photon}{v2,o2}
\fmfv{label=$\noexpand\gamma$}{o1,o2}
\fmfdot{v2}
\fmfv{decor.shape=square,decor.filled=0,label=(b),label.angle=-90,
       label.dist=0.6h}{v1}
\end{fmfgraph*}}

\caption{Long distance contributions to (a) $K_{L}\rightarrow\pi^0\pi^0
    \gamma\gamma$ and (b) $K_{L}\rightarrow\gamma
    \gamma$ induced by the leading WZW term.}
\end{figure}


\begin{figure}
\fmfframe(80,35)(0,35){
\begin{fmfgraph*}(120,70)
\fmfleft{i1}
\fmfv{label=$K_{L}$}{i1}
\fmfright{o1,o2,o3}
\fmf{fermion}{i1,v1}
\fmf{fermion,label=$\noexpand\pi^0,,\noexpand\eta$}{v1,v2}
\fmf{fermion}{v2,o1}
\fmfv{label=$\noexpand\pi^{-}$}{o1}
\fmf{photon}{v2,o2}
\fmfv{label=$\noexpand\gamma$}{o2}
\fmf{fermion}{v2,o3}
\fmfv{label=$\noexpand\pi^{+}$}{o3}
\fmfdot{v2}
\fmfv{decor.shape=square,decor.filled=0,label=(a),label.angle=-90,
       label.dist=0.6h}{v1}
\end{fmfgraph*}}
\fmfframe(50,35)(0,35){
\begin{fmfgraph*}(120,70)
\fmfleft{i1}
\fmfv{label=$K_{L}$}{i1}
\fmfright{o1,o2,o3}
\fmf{fermion, tension=2}{i1,v1}
\fmfv{decor.shape=square,decor.filled=.5,label=(b),label.angle=-90,
   label.dist=0.6h}{v1}
\fmf{fermion}{v1,o1}
\fmfv{label=$\noexpand\pi^{-}$}{o1}
\fmf{photon}{v1,o2}
\fmfv{label=$\noexpand\gamma$}{o2}
\fmf{fermion}{v1,o3}
\fmfv{label=$\noexpand\pi^{+}$}{o3}
\end{fmfgraph*}}

\caption{(a) Reducible contribution to the 
  $K_{L}\rightarrow\pi^{+}\pi^{-}\gamma$
      which vanishes in the $SU(3)$ symmetry limit. (b) The same  
     process viewed from the standpoint of an $\Delta S=1$
     extended WZW term, encapsulating contributions from all scales.}
\end{figure}

\end{fmffile}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(ii) In fact, the process $K_L\rightarrow \pi^+ \pi^- \gamma$ we
are focusing to, is accustomed to be a representative of
a class of {\em direct anomalous} process.
The respective odd intrinsic parity
 weak lagrangian at the order ${\cal O}(p^4)$,
offered by Bijnens, Ecker and
Pich (BEP) \cite{bep}, is as follows:
\begin{eqnarray}
{\cal L}^{\Delta S=1}_{\rm An} &=& -ie {G_8 f^2_\pi\over 16\pi^{2} }
\epsilon^{\mu\nu\rho\sigma}{F}_{\rho\sigma}
\biggl[a_2 \mbox{Tr}(\lambda_- [U^{\dagger} Q U,L_{\mu} L_{\nu} ]) 
  + 3a_3 \mbox{Tr}(\lambda_- {L_\mu})\mbox{Tr}((Q+U^{\dagger} Q U) L_{\nu})
\nonumber \\
&+& a_4  \mbox{Tr}(\lambda_-L_{\mu})\mbox{Tr}((Q-U^{\dagger} Q U)L_{\nu}) \biggr] \;+
h.\, c. 
\label{a123}
\end{eqnarray}
where $a_2, a_3, a_4$ are coupling constants, 
$\lambda_{\pm} = (\lambda_6  \pm i\lambda_7)/2$ is the matrix 
that projects out  $\Delta S =  \pm 1$ transitions, 
and $L_{\mu} = i  U^{\dagger} D_{\mu}U$ is expressed through
the covariant derivative $D_{\mu} U=\partial_\mu U-ieA_\mu[Q,~U]$.
The overall coupling constant $|G_8|=9\times10^{-6}$GeV$^{-2}$
is the same as that appearing in the standard lowest-order $\Delta S=1$ 
term displayed later in eq. (\ref{eq:cro}).
The origin of the direct anomalous term (\ref{a123}) can be traced
back to the bosonization of four-quark operators in the odd-intrinsic
parity sector \cite{bep,pder91}, by using the functional derivative
of the action $S=\int d^4 x {\cal{L}}$ as an identification of the
quark current:
\begin{equation}
\bar{q} \gamma^{\mu} L q \sim  \frac{\delta S}{\delta l_{\mu}}  \; .
\end{equation}
 To obtain the direct anomalous terms BEP \cite{bep} started from a four-quark
effective hamiltonian essentially involving products of two weak
currents (in the factorizable limit)
% wrote down an
%expression formally of the form
\begin{equation}
 {\cal{L}}_{\rm An} \, \sim G_F \,
\biggl( \frac{\delta S^{(2)}}{\delta l_{\mu}} \biggr)
\; \biggl( \frac{\delta S_{WZW}}{\delta l^{\mu}} \biggr) \; ,
\end{equation}
where $S^{(2)}$ is the normal action ${\cal{O}}(p^2)$ 
\begin{equation}
{\cal{L}}^{(2)}_{strong/em} \, = \, \frac{f_{\pi}^2}{4} \,
 Tr(D^{\mu} U D_{\mu} U^{\dagger}) \quad.
\end{equation}
The result, written in the form  (\ref{a123}), contributes to
$\bar{K}^0 \rightarrow \pi^+ \pi^- \gamma$ but not to
$\bar{K}^0 \rightarrow \gamma \gamma$, because the commutator
$[\Pi,Q]$ doesn't contain neutral kaons ($Q$ acts as the unit matrix
in the $s, d$ sector, and $U Q U^\dagger = Q + [\Pi,Q] + ....$).
 To find a contribution giving the $\bar{K}^0 \rightarrow \gamma
\gamma$ amplitude in this way, one is then addressed to go further to
the ${\cal{O}}(p^6)$ terms.

(iii)  We shall instead explore another route \cite{ep94} 
which is close to the one used by Cronin 
\cite{cro67} to generalize the ${\cal{O}}(p^2)$ strong/electromagnetic term
(\ref{eq:cro})
to the corresponding non-leptonic $\Delta S = 1$ term \cite{cro67}
\begin{equation}
{\cal{L}}^{(2)}_{\Delta S = 1} \, = \, G_8 \,
 Tr(\lambda_+ D^{\mu} U D_{\mu} U^{\dagger}) \; .
\label{eq:cro}
\end{equation}
The new flavour-changing WZW-like term \cite{ep94}, predicting
uniquely $\bar{K}^{0}\rightarrow 2\gamma$ at the ${\cal O}(p^4)$
was obtained from the
ordinary WZW term by the simple substitution
$ 1 \rightarrow {\cal{G}}_{\rm WZW} \, \lambda_+$
in front of $V_{\mu\nu\alpha\beta}$ in (\ref{eq:wz1}):
\begin{equation}
{\cal{L}}^{\Delta S=1}_{\rm WZW} \, = \,  -{i N_c \over 48\pi^2}
{\cal{G}}_{\rm WZW} \,
 \epsilon^{\mu\nu\alpha\beta}
\, Tr(\lambda_+ V_{\mu\nu\alpha\beta}) \; .
\label{eq:gwz}
\end{equation}
 The effective coupling ${\cal{G}}_{\rm WZW}$ accounts for both
 the CP-conserving ({\em even}) and  CP-violating ({\em odd}) 
transitions, by the respective terms in
\begin{center}
${\cal{G}}_{\rm WZW} \, = \, {\cal{G}} _{\rm WZW}^{even} \, + 
\, i \,  {\cal{G}} _{\rm WZW}^{odd} $. \\
\end{center}

In the present paper we go a step further in exploiting the anomalous
${\cal{L}}^{\Delta S=1}_{\rm WZW}$ terms in order to predict other radiative
non-leptonic processes. We argue that also for a class of flavour-changing
anomalous processes, there should be no new coupling strengths when we
go from one anomalous process to another. This justifies relation
(\ref{eq:gwz}), providing a predictive link among processes involving
an odd number of Goldstone bosons.
The expression (\ref{eq:gwz}) taken at its face value accounts for
the transitions
$\bar{K}^0 \rightarrow  \gamma \gamma$, 
$\pi^{+} \pi^{-} \gamma$, $\pi^{0}\pi^{0}\gamma$,
$\pi^{+} \pi^{-} \gamma \gamma$ and
$\pi^{0} \pi^{0} \gamma \gamma$:
\begin{eqnarray}
{\cal{L}}^{\Delta S=1}_{\rm WZW}{(4)}\, = \,
{\cal{G}} _{\rm WZW} {N_c \alpha \over 24 \pi f_\pi} \frac{2 \sqrt{2}}{3}
 \epsilon_{\mu\nu\rho\sigma}
A^{\nu}F^{\rho\sigma} \partial^{\mu}\bar{K}^{0}
 \nonumber \\
+ {\cal{G}}_{\rm WZW} {i N_c e \over 12 \pi^2f_\pi^3} \frac{\sqrt{2}}{3}
\epsilon_{\mu\nu\rho\sigma}
A^{\sigma} \partial^{\mu}\bar{K}^{0}
[\partial^{\nu}\pi^- \partial^{\rho}\pi^+]
\nonumber \\
+ {\cal{G}}_{\rm WZW} {N_c \alpha \over 128 \sqrt{2} \pi f_\pi^3}
\epsilon_{\mu\nu\rho\sigma}F^{\nu\rho}A^{\sigma}
[\bar{K}^{0}\pi^{0}\partial_{\mu}\pi^{0} - \partial_{\mu}
\bar{K}^{0}\pi^{0}\pi^{0}] + h. c.
\label{eq:kwz}
\end{eqnarray}
Provided that (\ref{eq:kwz}) explains the total  $K_{L} \rightarrow  
\gamma \gamma$  rate, an extraction of
 the coupling ${\cal{G}}_{\rm WZW}$ from the first term in 
Eq. (\ref{eq:kwz}) enables 
one to determine the $K_L \rightarrow 
\pi^+ \pi^- \gamma$ decay  (the second term in (\ref{eq:kwz}) shown
on Fig. 2b) .

In determining the coupling ${\cal{G}}_{\rm WZW}$ from the $K^0$ decay
into two photons, we have to rewrite our  flavour-non-diagonal WZW interaction 
(\ref{eq:gwz}) in the physical (decay-state) basis. We obtain it
by combining the 
%amplitudes 
$A(\bar{K}^{0} \rightarrow \gamma \gamma$)
% arrising from the
%${\cal{G}}_{\rm WZW} \,\lambda_+$  
and  $A(K^0 \rightarrow \gamma \gamma$) 
%given by the analogous
%${\cal{G}}_{\rm WZW}^* \,\lambda_-$  substitution :
amplitudes:
\begin{equation}
{\cal{L}}^{\Delta S=1}_{\gamma \gamma (-)}\, = \,
 {N_c \alpha \over 24 \pi f_\pi} \frac{2}{3}
[ \; {\cal{G}} _{\rm WZW}^{even} \; \Phi_{K_2} 
- i \; {\cal{G}} _{\rm WZW}^{odd} \; \Phi_{K_1} ]
 \epsilon_{\mu\nu\rho\sigma}
F^{\mu \nu} F^{\rho\sigma} \; .
\label{eq:eogwz}
\end{equation}
The two terms in (\ref{eq:eogwz}) correspond to the 
CP-conserving  
$K_L (\simeq K_2) \rightarrow \gamma \gamma(-)$ and
 the CP-violating
$K_S (\simeq K_1) \rightarrow \gamma \gamma(-)$ amplitudes,
respectively. 

Under a plausible assumption that the CP-conserving amplitude
from (\ref{eq:eogwz}) dominates the measured 
$K_L \rightarrow \gamma \gamma$ decay rate, one can determine
${\cal{G}} _{\rm WZW}^{even}$  \cite{ep94} from the 
measured $K_2  \gamma \gamma$ coupling $C_{K_2}$ .
 The rate for $K_2 \rightarrow   \gamma \gamma$   gives
(in the limit $f_K \, = \,  f_\pi$)
\begin{center} 
$|C_{K_2}| \, = \, {2 \over  3} \, |{\cal{G}}_{\rm WZW}^{even}| \, C_{\pi^0 } \, = 
\, 5.9 \times 10^{-11} MeV^{-1}\, ,$
\end{center}
and thus the dimensionless coupling is
\begin{equation}
  {\cal G}_{WZW}\simeq|{\cal{G}}_{\rm WZW}^{even}| \, 
\simeq  \,  2 \times 10^{-7} \; .  
\label{gwzw}
\end{equation}

  Armed with the knowledge of coupling ${\cal G}_{WZW}$ let us analyse
the $\Delta S=1$ direct anomalous term
in the $K_L\rightarrow \pi^+ \pi^- \gamma$ decay .

\vspace{5mm}
{\Large
 4. Anomalous $K_{L}\rightarrow\pi^{+}\pi^{-}\gamma$ decay}
\vspace{2mm}

  Experimentally, the $K_{L}\rightarrow\pi^{+}\pi^{-}\gamma$ amplitude appears
to be suitable to study the DE amplitude. The first step in
this direction is represented by substracting the IB contribution
from the total rate. The kinematical cuts in photon energy allow
one to eliminate the uninteresting IB predicted fully by
gauge invariance.

  On the theoretical side the cancellation of the reducible
amplitudes related to the WZW functional leaves some space
for the study of less known direct amplitudes.
Once we have ${\cal G}_{WZW}$ (\ref{gwzw}) we can predict
the WZW-related process  $K_L \rightarrow \pi^{+} \pi^-\gamma$
explicated in the second term of eq. (\ref{eq:kwz}). Transforming this
term  into the physical K-meson basis,  we obtain the following expression
for the CP-conserving
amplitude which presumably dominates the rate:
\begin{equation}
{\cal{L}}^{\Delta S=1}_{WZW(1\gamma)} \, = \,  {\cal{G}} _{\rm WZW}^{even} 
 { N_c e \over 12 \pi^2 f_\pi^3} \frac{2}{3}
\epsilon_{\mu\nu\rho\sigma}
A^{\mu} \partial_{\nu}\Phi_{K_2}
[\partial_{\rho}\pi^+ \partial_{\sigma}\pi^-] \; .
\label{wzw1g}
\end{equation}
Let us parametrize 
the amplitude for the one-gamma emission 
$K_L (k) \rightarrow  \pi^+ (p_+) \pi^- (p_-) \gamma (q) $ as follows
\begin{equation}
A(K_L \rightarrow  \pi^+ \pi^- \gamma) \; = \;
- i e G_{1 \gamma}
 \epsilon_{\mu \nu \rho \sigma} \epsilon ^{\mu} k^{\nu}
p_+^{\rho}p_-^{\sigma} \, .
\end{equation}
This amplitude reproduces the measurment by
 the E731 experiment at Fermilab  \cite{E731a}
\begin{center}
$Br(K_L \rightarrow  \pi^+ \pi^- \gamma) \; = 
\; (3.19\pm 0.16)\times 10^{-5}$ , 
\end{center}
with the value of the effective coupling $G_{1\gamma}$
\begin{equation}
 |G_{1 \gamma}^{expt}| \; \simeq \;  4.9 \; \times 10^{-6} GeV^{-3} .
\end{equation}
On the other hand, by employing the value (\ref{gwzw}) for 
$|{\cal{G}} _{\rm WZW}^{even}|$ in eq. (\ref{wzw1g}) we predict
\begin{eqnarray}
 |G_{1 \gamma}^{theo}| \; = \;   |{\cal{G}}_{\rm WZW}^{even}| 
 { N_c \over 12 \pi^2 f_\pi^3} \frac{2}{3} 
\nonumber \\
\; \simeq \; 4.28 \times 10^{-6} GeV^{-3} \quad, 
\end{eqnarray}
which agrees well with the upper experimental value.
This indicates that flavour-extended WZW term successfully 
relates non-leptonic radiative kaon decays.

Let us in addition compare our prediction to the direct
(factorizable) contribution \cite{bep} derived from eq. (\ref{a123})
of (ii) part of the preceeding section.
\begin{equation}
{\cal L}^{\Delta S=1}_{BEP}=\frac{G_{8} e}{4\sqrt{2}\pi^{2}
f_{\pi}}(a_{2}+2a_{4})\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}
\partial^{\mu}\bar{K}^{0}\partial^{\nu}\pi^{-}\pi^{+}\quad.
\end{equation}
  A direct comparison of this and our contribution 
to the $K_{L}\rightarrow\pi^{+}\pi^{-}\gamma$ (\ref{eq:kwz}) implies
\[
      a_{2}+2a_{4}\simeq 1 \quad,
\]
which is consistent with a claim \cite{bep} 
\[
	a_2, a_4 \lessoe 1 \quad,
\]
drawn from the analysis of the 
$K^{+}\rightarrow\pi^{+}\pi^{0}\gamma$ decay.
This represents another successful prediction of the flavour-changing
extension of the WZW, described in part (iii) of the previous section.

\vspace{5mm}
{\Large
5. Conclusions
}
\vspace{2mm}

  In a situation that the $K_{L}\rightarrow\pi^{+}\pi^{-}\gamma$
amplitude is claimed to be unpredictable within the customary
analysis within the $\chi PT$, we take an
alternative look at the direct part of this amplitude governed by
the chiral Wess-Zumino term.
In fact, we check a viability of the prescription to generate
the anomalous non-leptonic interaction terms.
We argue that the anomaly mechanism takes care of the uniqueness
of such a prescription, i. e. for the univesal coupling
${\cal G}_{WZW}$ in equation (\ref{eq:gwz}). Thus we arrive at the
predictive relations linking different radiative non-leptonic
$K$-decays.
The viability of such new parametrisation is demonstrated here by
analysing the available body of the experimental and 
theoretical data on the $K_{L}\rightarrow\pi^{+}\pi^{-}\gamma$
process. 
An additional support commes from another proces,
$K_{L}\rightarrow\pi^{0}\pi^{0}\gamma\gamma$, contained in our
relation (\ref{eq:kwz}). Assuming dominant CP contribution, it is
given by
\begin{displaymath}
  {\cal L}^{\Delta S=1}_{WZW(2\gamma)}={\cal G}_{\rm WZW}
\frac{\alpha N_{c}}{54 \pi f_{\pi}^{2}}\epsilon^{\mu\nu\alpha\beta}
F_{\nu\alpha}A_{\beta}[K_{2}\pi^{0}\partial_{\mu}\pi^{0} -
(\partial_{\mu}K_{2})\pi^{0}\pi^{0}]\quad.
\end{displaymath}
The explicit comparison to the existing theoretical analysis of
the (reducible) contribution \cite{DFR} shows that we have
a comparable prediction for the rate of this process too. 

To conclude,
the ``anomaly road'', advocated here, which starts form true-contact
(non-separable) SD mechanism, seems to incorporate a whole body of
the anomalous transitions: reducible, direct (factorizable) and
the contact (direct-nonfactorizable). On the top of the previous
parametrisations, including links among processes, the presented
``anomaly road'' incorporates, at the end of the chain, the
$K$ to vacuum transition. Relying on the anomaly as a
mechanism which has an
imprint on different distance scale, the flavour-changing 
generalization of the WZW term seems to bear the closest
contact with the bosonisation of $\Delta S=1$ non-leptonic radiative
$K$ decays. Simultaneously, WZW flavour-changing term seems to
provide a unified treatment of such processes in the sense that
it incorporates contributions irrespectivelly of the scale from
which they originate. In this sense such parametrisation is compatible
to the $\chi$PT framework.

Finally, such an ``anomaly road'', being able to give a new insight into
SD/contact contributions for selected rare kaon decays, might
improve the thoretical-predictivity power. Isolating the direct
anomalous/contact terms of radiative non-leptonic $K$-decays seems
to be a necessary ingredient in separating direct from indirect
CP violation. In this sense the processes like $K_{L}\rightarrow
\gamma\gamma$ and  $K_{L}\rightarrow\pi^{+}\pi^{-}\gamma$,
where LD (reducible)  anomalous amplitudes cancel out to the
leading order, seem to have enough sensitivity to  the SD part of decay 
amplitude in order to make progress in further study of CP violation. 

%As clear a separation
%as possible of the LD and SD mechanisms is a prerequisite needed
%to face the analysis of forthcoming data from rare-decay
%measurements. By residing firmly in the SD part of decay amplitudes,
%the {\em direct anomalous} terms most directly decouple from the
%offending LD uncertainties. Without a theoretical understanding of 
%SD/contact contributions, one could hardly hope to make progress
%in further study of CP violation. Isolating the direct
%anomalous/contact terms of radiative non-leptonic $K$-decays seems
%to be a necessary ingredient in separating direct from indirect
%CP violation.
\vspace{5mm}
\begin{Large}

Acknowledgment

\end{Large}
\vspace*{2mm}
The authors thank J. O. Eeg for helpful discussions, and 
acknowledge the support of the EU contract CI1$^{\ast}$-CT91-0893.

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\end{document}



