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\begin{document}
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\title{$K^+ \ra \pi^+ \pi^0 \gamma$ in the Standard Model and Beyond}
\author{ P. MERTENS }
\address{Centre for Cosmology, Particle Physics and Phenomenology (CP3)\\
Universit\'e catholique de Louvain,
Chemin du Cyclotron, 2\\
B-1348 Louvain-la-Neuve, Belgium}
\maketitle\abstracts{In this note we show how improved theoretical analysis combined with recent experimental data coming from NA48/2 concerning $K^+ \to \pi^+ \pi^0 \gamma$ decay shed light on the dynamics of the $s \rightarrow d \gamma$ transition. Consequences on NP analysis are also presented.}
\section{Introduction}
In the search for New Physics (NP) the $s\rightarrow d\gamma$ process is complementary to $b\rightarrow s\gamma$ and $\mu\rightarrow e\gamma$, as the relative strength of these transitions is a powerful tool to investigate the NP dynamics. However, since $s\rightarrow d\gamma$ takes place deep within the non-perturbative regime of QCD we have to control hadronic effects and find observables sensitive to the short-distance dynamics, and thereby to possible NP contributions. The purpose of this note is to show how this can be achieved using the $K^+ \to \pi^+ \pi^0 \gamma$ observable \cite{CDI99}.
In section 2, the anatomy of the $s\rightarrow d\gamma$ process in the Standard Model (SM) is shortly detailed. In section 3, we analyse the $K^+ \ra \pi^+ \pi^0 \gamma$ decay in the SM whereas section 4 is devoted to show how, in the MSSM, rare and $K^+ \ra \pi^+ \pi^0 \gamma$ decays, as well as $\operatorname{Re}(\varepsilon_K^{\prime}/\varepsilon_K)$ can be exploited to constrain NP.
\section{The $s \ra d \gamma$ anatomy}
In the SM, the flavour changing electromagnetic process $s \ra d \gamma$ is a loop effects which at low energy scale is described by the effective $\Delta S =1$ Hamiltonian~\cite{BuchallaBL96}%
\begin{equation}
\mathcal{H}_{eff}(\mu \approx 1\text{ GeV})=\sum_{i=1}^{10}C_{i}\left( \mu\right) Q_{i}\left( \mu\right) +C_{\gamma^{\ast}}^{\pm}Q_{\gamma^{\ast}}^{\pm}+C_{\gamma}^{\pm}Q_{\gamma}^{\pm}+h.c.\;, \label{OPE}
\end{equation}
where the $Q_{i}$ are effective four-quarks operators whereas the quark-bilinear electric $Q_{\gamma^{\ast}}^{\pm}$ and magnetic $Q_{\gamma}^{\pm}$ operators are respectively given by\footnote{By definition : $2\sigma^{\mu\nu}=i[\gamma^{\mu},\gamma^{\nu}]$.}
$
Q_{\gamma^{\ast}}^{\pm}=(\bar{s}_{L}\gamma^{\nu}d_{L}\pm\bar{s}_{R}\gamma^{\nu}d_{R})\,\partial^{\mu}F_{\mu\nu}$ and $Q_{\gamma}^{\pm}=(\bar{s}_{L}\sigma^{\mu\nu}d_{R}\pm\bar{s}_{R}\sigma^{\mu\nu}d_{L})\,F_{\mu\nu}$.
In the non perturbative regime of QCD this Hamiltonian is hadronized into an effective weak Lagrangian that shares the chiral properties of the operators contained in $\mathcal{H}_{eff}$. The chiral structures of $Q_{i}$ and $Q_{\gamma^{\ast}}^{\pm}$ allow the usual $\mathcal{O}(p^2)$ weak Lagrangian $\mathcal{L}_{W}=G_8 O_{8} +G_{27}O_{27} +G_{ew}O_{ew}$ (detailed in \cite{us}) whereas the chirality flipping $Q_{\gamma}^{\pm}$ operators induce more involved $\mathcal{O}(p^4)$ local interactions (detailed in \cite{CDI99,us}). The non-trivial dynamics corresponding to the low-energy tails of the photon penguins arise at $\mathcal{O}(p^{4})$ (the $\mathcal{O}(p^{2})$ dynamics being completely predicted by Low's theorem~\cite{Low}) where they are represented in terms of non-local meson loops, as well as additional $\mathcal{O}(p^{4})$ local effective interactions, in particular the $\Delta I=1/2$ enhanced $N_{14},...,N_{18}$ octet counterterms~\cite{CT1,CT2}.
\section{$K^+ \ra \pi^+ \pi^0 \gamma$ in the SM}
For the $K^{+}\rightarrow\pi^{+}\pi^{0}\gamma$ decay,
the standard phase-space variables are chosen as the $\pi^{+}$ kinetic energy
$T_{c}^{\ast}$ and $W^{2}\equiv(q_\gamma \cdot P_K)(q_\gamma\cdot P_{\pi^+})/m_{\pi^{+}}^{2}%
m_{K}^{2}$~\cite{Christ67}. Indeed, pulling out the dominant bremsstrahlung contribution, the differential rate can be written%
\begin{equation}
\frac{\partial^{2}\Gamma}{\partial T_{c}^{\ast}\partial W^{2}}=\frac{\partial
^{2}\Gamma_{IB}}{\partial T_{c}^{\ast}\partial W^{2}}\left( 1-2\frac{m_{\pi
^{+}}^{2}}{m_{K}}\operatorname{Re}\left( \frac{E_{DE}}{eA_{IB}}\right)
W^{2}+\frac{m_{\pi^{+}}^{4}}{m_{K}^{2}}\left( \left| \frac{E_{DE}}{eA_{IB}%
}\right| ^{2}+\left| \frac{M_{DE}}{eA_{IB}}\right| ^{2}\right)
W^{4}\right) \;. \label{DiffRate}%
\end{equation}
In this expression both electric $E_{DE}$ and magnetic $M_{DE}$ direct emission amplitudes are functions of $W^{2}$ and $T_{c}^{\ast}$ and appear at $\mathcal{O}(p^4)$. To a very good approximation we can identify these direct emission amplitudes with their first multipole for which the $\pi^+ \pi^0$ state is in a P wave.
The main interest of $K^{+}\rightarrow\pi^{+}\pi^{0}\gamma$ is that its bremsstrahlung component $A_{IB}=A(K^+ \ra \pi^+ \pi^0)$ is pure $\Delta I=3/2$ hence suppressed, making the direct
emission amplitudes easier to access. The magnetic amplitude $M_{DE}$ is dominated by the
QED anomaly and will not concern us here.
\subsection{Differential rate}
Given its smallness, we can assume the absence of CP-violation when discussing
this observable. Experimentally, the electric and magnetic amplitudes (taken
as constant) have been fitted in the range $T_{c}^{\ast}\leq80$ MeV and
$0.2