Isospin-Violating Contributions to $\epsilon'/\epsilon$

The known isospin-breaking contributions to the $K\rightarrow \pi\pi$ 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 $\epsilon'/\epsilon$, 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 $1/N_C$-suppressed contributions to some relevant chiral-perturbation-theory low-energy constants.