%%UNIX --- UPDATED ON 13/8/97  
%====================================================================%
%                  sprocl.tex     27-Feb-1995                        %
% This latex file rewritten from various sources for use in the      %
% preparation of the standard proceedings Volume, latest version     %
% by Susan Hezlet with acknowledgments to Lukas Nellen.              %
% Some changes are due to David Cassel.                              %
%====================================================================%

\documentstyle[sprocl,epsfig]{article}

\font\eightrm=cmr8

%\input{psfig}

\bibliographystyle{unsrt} %for BibTeX - sorted numerical labels by
                          %order of first citation.

\arraycolsep1.5pt

% A useful Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}

% Some useful journal names
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.}  B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}

% Some other macros used in the sample text
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}%temp replacemt due to no font

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%BEGINNING OF TEXT                           
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{MEASUREMENT OF DIRECT EMISSION 
$K^+ \! \rightarrow \! \pi^+ \pi^0 \gamma$ DECAY}

\author{T. Nakano}

\address{Research Center for Nuclear Physics, Osaka University,\\
10-1 Mihogaoka, Ibaraki, Osaka 567--0047, Japan
\\E-mail: nakano@rcnp.osaka-u.ac.jp} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle

The direct emission (DE) component of the $K^+ \! \rightarrow \! \pi^+
\pi^0 \gamma$ decay consists of magnetic and electric transitions. At
$O(p^4)$ in ChPT, the magnetic amplitude
consists of a reducible amplitude~\cite{Ecker} which is unambiguously
determined by the WZW functional~\cite{WZW} with a
chiral anomaly and of a direct anomalous amplitude~\cite{Bijnens},
arising from odd-parity octet operators. There is
no definite prediction from ChPT of the electric transition amplitude,
which depends on undetermined constants. Since the electric amplitude
interferes (INT) with the dominant inner bremsstrahlung (IB)
amplitude, it may be distinguished from the magnetic amplitude, which
does not.

The differential decay width for the decay $K^+(p) \! \rightarrow \!
\pi^+ (p_{+})\pi^0(p_{0})\gamma(q)$ is expressed in terms
of a Lorentz invariant variable $W^{2}\equiv (p \cdot q)(p_{+} \cdot
q)/(m_{\pi^+}^{2} m_{K}^{2})$.
The differential width can be written in terms of the IB
differential width as
\begin{eqnarray*}
\frac{\partial^2 \Gamma}{\partial T_+ \partial W} =
\frac{\partial^2 \Gamma_{IB}}{\partial T_+ \partial W}
\left[ 1 + 2\frac{m^2_{\pi^+}}{m_K} Re \left( \frac{E}{e A}\right) W^2
+ \frac{m^2_{\pi^+}}{m_K^2} \left( \left|\frac{E}{e A}\right|^2 +
\left|\frac{M}{e A}\right|^2 \right) W^4 \right],
\end{eqnarray*}
where $T_+$ is the kinetic energy of the charged pion, 
$A$ is the decay amplitude for $K^+\to\pi^+\pi^0$, and $E$ and
$M$ are electric and magnetic DE amplitudes, respectively.  The direct
emission amplitudes $E$ and $M$ are independent of $W$, while the IB
decay amplitude is proportional to $1/W^2$, and thus dominates in the
small $W$ region.

The E787 detector~\cite{E787det} at the Alternating Gradient
Synchrotron of Brookhaven National Laboratory was used to study
$K_{\pi2\gamma}$.  Kaons of 790 MeV/$c$ were incident on a solenoidal
spectrometer with a 1.0 Tesla field.  Measurements of momentum,
kinetic energy and range of charged decay products were performed. A
hermetic calorimeter system consisting of barrel and endcap detectors
measured the positions and energies of photons and other particles
from $K^+$ decays. A total of 19836 events survived all off-line
selection cuts, eight times more than the previous 
experiments~\cite{oldexp}.

Figure~\ref{figure:Wspctl}a shows the $W$ projection of the signal
events in the $T_+$ signal region 55 MeV $<$ $T_+$ $<$ 90 MeV.  
The dashed curve is a best fit to
IB alone with a $\chi^2$ of 61 for 7 degrees of freedom. The solid
curve is the best fit to the sum of IB and DE and gives a $\chi^2$ of
7.9 for 6 degrees of freedom.  Normalized to the IB spectrum the
deviation of the measured spectrum at large $W$ is shown in
Figure~\ref{figure:Wspctl}b.  The branching ratio for DE is determined
to be $B^{T_+}(DE) = (4.7\pm0.8)\times 10^{-6}$ in the signal $T_+$
region. The best fit to the sum of IB, DE and INT
gives the INT component to be $(-0.4\pm 1.6)$\% of the IB component
with a $\chi^2$ of 7.8 for 5 degrees of freedom. We conclude that no
interference term is required for the fit to this spectrum.  Assuming
a pure magnetic transition for the DE amplitude, the result for
$B^{T_+}(DE)$ gives $|M| = ( 2.1 \pm 0.2 ) \times 10^{-7}$ for the
dimensionless magnetic amplitude in Eq. (1). The reducible anomalous
amplitude $|- e G_8 m_K^3 / 2
\pi^2 f |$ in Ref.~\cite{Ecker} is $|M| = 1.8 \times 10^{-7}$ with standard
$O(p^2)$ ChPT coupling constants, $f = 93 \; {\rm MeV}$ and $|G_8| =
9\times10^{-6} \; {\rm GeV}^{-2}$. This result supports the
hypothesis that the dominant contribution to DE is due to the reducible
anomalous amplitude and the other magnetic contributions are small or
cancel.

\begin{figure}[t]
\centerline{
\hbox{\psfig{figure=nakano.wfit_a.eps,width=5cm}}
\hbox{\psfig{figure=nakano.wfit_b.eps,width=5cm}}
}
\caption{a) $W$ spectrum of the observed events
and best fits to IB+DE (solid curve) and IB alone (dashed curve);
b) Measured $W$ spectrum normalized to the IB spectrum.}
\label{figure:Wspctl}
\end{figure}


\section*{References}
\begin{thebibliography}{99}

\bibitem{Ecker}G. Ecker, H. Neufeld and A. Pich,
\Journal{\PLB}{278}{337}{1992}.

\bibitem{WZW}J. Wess and B. Zumino,
\Journal{\PLB}{37}{95}{1971};
E. Witten,
\Journal{\NPB}{223}{422}{1983};

\bibitem{Bijnens}J.~Bijnens, G.~Ecker, and A.~Pich,
\Journal{\PLB}{286}{341}{1992}.

\bibitem{oldexp}
R.J.~Abrams $et~al.$,
\Journal{\PRL}{29}{1118}{1972};
K.M.~Smith  $et~al.$,
\Journal{\NPB}{109}{173}{1976};
V.N.~Bolotov $et~al.$,
Sov. J. Nucl. Phys. {\bf 45}, 1023 (1987).

\bibitem{E787det}
M.S.~Atiya $et~al.$, 
\Journal{\NIM}{321}{129}{1992}.

\end{thebibliography}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% End of sprocl.tex  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

