Isospin-Violating Contributions to ${ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ$

The known isospin-breaking contributions to the $K\to \pi \pi$ amplitudes are reanalyzed, taking into account our current understanding of the quark masses and the relevant non-perturbative inputs. We present a complete numerical reappraisal of the direct CP-violating ratio ${ϵ}^{\mathrm{\prime }}\mathrm{/}ϵ$, where these corrections play a quite significant role. We obtain the Standard Model prediction $$\text{Re}$\left(\epsilon’/\epsilon\right), =,\left(14,\pm,5\right)\cdot 10^{-4}$, which is in very good agreement with the measured ratio. The uncertainty, which has been estimated conservatively, is dominated by our current ignorance about $1\mathrm{/}{N}_C$-suppressed contributions to some relevant chiral-perturbation-theory low-energy constants.