Impact of|ΔI|=5/2transitions inK→ππdecays

We consider the impact of isospin violation on the analysis of $\overrightarrow{K}\pi \pi$ decays. We scrutinize, in particular, the phenomenological role played by the additional weak amplitude, of $|\Delta I|=5/2\mathrm{}$ in character, incurred by the presence of isospin violation. We show that Watson’s theorem is appropriate in $\mathcal{O}{(m}_d-m_u),$ so that the inferred $\pi \pi$ phase shift at $\sqrt{s}{=m}_K$ determines the strong phase difference between the $I=0\mathrm{}$ and $I=2\mathrm{}$ amplitudes in $\overrightarrow{K}\pi \pi$ decay. We find the magnitude of the $|\Delta I|=5/2\mathrm{}$ amplitude thus implied by the empirical branching ratios to be larger than expected from estimates of isospin-violating strong and electromagnetic effects. We effect a new determination of the octet and 27-plet coupling constants with strong-interaction isospin violation and with electromagnetic effects, as computed by Cirigliano, Donoghue, and Golowich, and find that we are unable to resolve the difficulty. Exploring the role of $|\Delta I|=5/2\mathrm{}$ transitions in the $\mathrm{CP}$-violating observable ${\epsilon }^{\prime }/\epsilon ,$ we determine that the presence of a $|\Delta I|=5/2\mathrm{}$ amplitude impacts the empirical determination of ω, the ratio of the real parts of the $|\Delta I|=3/2\mathrm{}$ to $|\Delta I|=1/2\mathrm{}$ amplitudes, and that it generates a decrease in the estimation of ${\epsilon }^{\prime }/\epsilon .$