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\title{Electromagnetic corrections to $K \rightarrow \pi \pi$ and 
Dispersive Matching}

\author{V. Cirigliano}

\address{Department of Physics, University of Massachusetts, 
Amherst MA 01003 USA}

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\maketitle

\abstracts{I report on work done in collaboration with J.F. Donoghue 
and E. Golowich on electromagnetic corrections to $K \rightarrow \pi \pi$
 decays \cite{em1,em2}.}

In the analysis of $K \rightarrow \pi \pi$ decays, the $\Delta I =
1/2$ enhancement provides an amplification mechanism for isospin
breaking (IB) effects. This circumstance makes IB corrections to 
$\Delta I \geq 3/2$ amplitudes of the natural size of about $10 \%$.
In this contribution I report on some issues arising in the analysis 
of electromagnetic (EM) effects \cite{others}. 
 
In presence of EM interactions, the kaon partial decay
width $\Gamma_{n}$ (where $n = +-,00,+0$) can be written as: 
% 
\begin{equation}
\Gamma_{n} (\omega) \ = \ \frac{\Phi_{n}}{2 \, \sqrt{s_{n}}} \, 
	| \overline{\cal A}_{n} |^2 \, G_{n} (\omega) \ . 
\label{eq1} 
\end{equation} 
% 
We indicate with $\omega$ its dependence on how much of the $n \, +$
soft $\gamma$ channel is included in the measured rate.  $\Phi / (2
\sqrt{s})$ is a kinematical factor, $\overline{A}_{n}$ is the infrared
finite amplitude - defined by standard procedure \cite{F61,Weinberg65}
- and $G_{n} (\omega)$ is the infrared factor associated with the
combined effect of {\em virtual} and {\em real} soft photons. $G_{n}
(\omega)$ has to be calculated with the same prescriptions on soft
photons cuts used in the measurement of $\Gamma_n$.  Since the PDG
\cite{PDG} data come with no reference to this, no precise inclusion
of radiative corrections is possible at present \cite{em2}. The 
data analysis at KLOE will  hopefully fill this gap \cite{franzini}.

The IR finite amplitudes can then be parameterized as follows, 
%
\begin{eqnarray}
\overline{\cal A}_{+-} &=& \left( A_{0} + \delta A_0 \right)
e^{i(\delta_0 + \gamma_0)} + { 1 \over \sqrt{2}} \left( A_{2} +
\delta A_2  \right)
e^{i(\delta_2 + \gamma_2)}
\ , \nonumber \\
\overline{\cal A}_{00} &=& \left( A_{0} + \delta A_0  \right)
e^{i(\delta_0 + \gamma_0)} - \sqrt{2} \left( A_{2} +
\delta A_2  \right)
e^{i(\delta_2 + \gamma_2)} \ \ ,
\label{eq2} \\ 
\overline{\cal A}_{+0} &=& {3 \over 2} \left( A_{2}
+ \delta A_2^{+}  \right)
e^{i (\delta_2 + \gamma_2^+)} \ \ ,
\nonumber
\end{eqnarray}
%
in terms of shifts $\delta A_I$, $\gamma_I$ to the original isospin
amplitudes $A_I$ and strong rescattering phases $\delta_I$, entering
the usual parameterization as required by Watson's theorem. In general
$\delta A_2 \neq \delta A_{2}^{+}$, due to a $\Delta I = 5/2$
amplitude induced by electromagnetism.

We have calculated $\delta A_I$ within an approach inspired by
dispersive theory \cite{em1}, where the EM amplitude is written as
follows:
%
\begin{equation}
A_{f \, i} (p_{\rm ext}) =  \frac{e^2}{2} \,              
\int \,\frac{d^{4} q}{(2 \pi)^4} \, D_{\mu \nu}  (q) 
\ A_{f \, i}^{\mu \nu} \, (q, p_{\rm ext})  = \frac{\alpha}{4 \pi} 
\int \, d Q^2  \  A_{f \, i} \,(Q^2, p_{\rm ext}) . 
\label{eq3}
\end{equation}
% 
$D_{\mu \nu}$ is the photon propagator, $A^{\mu \nu}_{f i}$ is the
amplitude for $\gamma (q) \, i (p_i) \rightarrow \gamma (q) \,
f(p_f)$, and the second form is obtained upon integration over the
angular variables. 
Model independent information on the low $Q^2$ region of the integrand 
is provided by chiral symmetry through the tool of chiral lagrangians. 
The OPE is used to investigate the high $Q^2$ behavior. 
We then interpolate between the two regions using
the assumption that low lying resonances play a dominant role in the
intermediate $Q^2$ region. The matching procedure provides also an
estimate of the theoretical uncertainty. Our results are:
%
\begin{eqnarray}
\delta A_0 / A_{0} &  = & (0.46 \pm 0.13) \, \%   \nonumber \\
\delta A_2 / A_{2} &  = & (6.0 \pm 2.5) \, \%      \label{eq4} \\
\delta A_2^{+} / A_{2} & = & - (3.3 \pm 3.6) \, \%    \nonumber 
\end{eqnarray}
% 
We have also analyzed the dynamical content of the phases $\gamma_I$
through unitarity relations in presence of IB interactions
\cite{em2}.  We find that $\gamma_2$ is enhanced by the large ratio
$A_0 / A_2$ and is related to EM corrections to $\pi \pi$ scattering
\cite{empipi} at energy equal to the kaon mass. A leading order estimate
 of these effects gives $\gamma_2 = 3.1 ^{o}$, which already amounts
to $40 \% $ of the strong phase shift $\delta_2$ at $s = M_K^2$.
Sizeable higher order corrections $\gamma_2^{(e^2 p^2)}$ cannot be
excluded and require a full analysis of $\pi \pi$ scattering at this 
order. \\
The implications of this work on the direct CP violation quantity 
$\epsilon ' / \epsilon$ are discussed in Ref. \cite{em2}.



\begin{thebibliography}{99}

\bibitem{em1} V. Cirigliano, J.F. Donoghue and E. Golowich, 
Phys. Lett. {\bf B450}  (1999) 241;
Phys.Rev. {\bf D61}: 093001, 2000;
Phys.Rev. {\bf D61}, 093002, 2000. 

\bibitem{em2} V. Cirigliano, J.F. Donoghue and E. Golowich, hep-ph/0008290. 

\bibitem{others} For a list of references on strong and EM isospin 
breaking, see Ref. \cite{em2}.

\bibitem{F61} D.R. Yennie, S.C. Frautschi, H. Suura, Ann.Phys. (NY) {\bf 13}
 (1961) 379.

\bibitem{Weinberg65} S. Weinberg, Phys.Rev. {\bf 140} (1965) 516.

\bibitem{PDG} PDG98, C. Caso et al., Eur.Phys.J. {\bf C3} (1998) 1.

\bibitem{franzini} Talk given by P. Franzini at {\em Chiral 2000}, 
JLAB, July 17-22 2000. 

\bibitem{empipi} U. Meissner, G. M\"uller, S. Steininger, Phys.Lett.{\bf B406} 
(1997) 154, Err.{\bf B407} (1997) 454; 
M. Knecht, R. Urech, Nucl.Phys.{\bf B519} (1998) 329. 




\end{thebibliography}

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