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Isospin Breaking in $K\to3π$ Decays III: Bremsstrahlung and Fit to Experiment

We complete here our work on isospin violation in the $K\to3π$ system. We first calculate $K\to2π$ to the same order as we did $K\to3π$ in papers I and II of this series. This adds effects of order $G_{27} p^2 (m_u-m_d)$ and $G_{27} p^2 e^2$ to earlier work. We calculate also the lowest order Bremsstrahlung contributions, $K\to2πγ,3πγ$. With these and our earlier results we perform a full fit to all available CP conserving data in the $K\to2π,3π$ system including isospin violation effects. We perform these fits under various input assumptions as well as test the factorization and the vector dominance model for the weak NLO low energy constants.