Charged Kaon K → 3 pi CP Violating Asymmetries at NLO in CHPT

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We give the first full next-to-leading order analytical results in Chiral Perturbation Theory for the charged Kaon K → 3 pi slope g and decay rates CP-violating asymmetries. We have included the dominant Final State Interactions at NLO analytically and discussed the importance of the unknown counterterms. We find that the uncertainty due to them is reasonable just for ΔgC, i.e. the asymmetry in the K+ → pi+ pi+ pi- slope g; we get ΔgC = -(2.4 +- 1.2) 10⁻⁵. The rest of the asymmetries are very sensitive to the unknown counterterms. In particular, the decay rate asymmetries can change even sign. One can use this large sentivity to get valuable information on those counterterms and on Im(G₈) coupling --very important for the CP-violating parameter epsilon'ₖ-- from the eventual measurement of these asymmetries. We also provide the one-loop O(e² p²) electroweak octet contributions for the neutral and charged Kaon K → 3 pi decays.