12 φ$_{ϵ}$ < π/ 4 results in a suppression effect in ϵ$_{k}$ relative to the approximate formula with ζ = 0 and φ$_{ϵ}$ = π/ 4. In order to include these corrections we have used the parametrization where ¯ κ$_{ϵ}$ = 0 . 94 ± 0 . 02 and consequently κ$_{ϵ}$ = 0 . 92 ± 0 . 02, ¯ κ$_{ϵ}$ parameterizing the effect of ζ = 0 [61]. After some calculations it can be shown that [56] where λ$_{i}$ = λ ∗ $_{is}$λ$_{id}$ and x$_{q}$ = ( m 2 $_{q}$/M 2 $_{W}$) for all quarks q . Inserting (35) and (34) in (31) one finds where f$_{K}$ = 160 MeV. The value for B$_{K}$ has been taken from Ref. [62], in a recent analysis [63, 64] the error has been reduced to < ∼ 4%, however, in our analysis we use the more conservative value mentioned in Table. III from [62]. The effective Hamiltonian for K + → π $^{+}$ν ¯ ν can be written as First term is the contribution from the charm sector. The function X ( x ) is relevant for the top part, where x$_{q}$ = ( m 2 $_{q}$/M 2 $_{W}$) for all quarks q . Here X$_{0}$ ( x ) is the leading contribution given by X$_{0}$ ( x ) = x 8 [ − 2 + x 1 − x + 3 x − 6 (1 x ) 2 ln x ] , (39) X ( x ) = X$_{0}$ ( x ) + α$_{s}$ 4 π X$_{1}$ ( x ) , (38) H eff = G$_{F}$ √$_{2}$ α 2 π sin $^{2}$Θ$_{w}$ ∑ l = e,µ,τ [ V ∗ $_{cs}$V$_{cd}$ X l NL + V ∗ $_{ts}$V$_{td}$ X ( x$_{t}$ ) + V ∗ t ′ $_{s}$V$_{t}$ ′ $_{d}$X ( x$_{t}$ ′ ) ] (¯ sd )$_{V}$ − $_{A}$(¯ ν$_{l}$ν$_{l}$ )$_{V}$ − A . (37) G. K + → π $^{+}$ν ¯ ν decay ϵ$_{K}$ = G 2 F 12 π 2 √$_{2Δ}$$_{M}$$_{K}$ κ$_{ϵ}$f 2 $_{K}$B$_{K}$ m$_{K}$M 2 W Im [ λ ∗ c $^{2}$η$_{c}$ S$_{0}$ ( x$_{c}$ ) + λ ∗ t $^{2}$η$_{t}$ S$_{0}$ ( x$_{t}$ ) +2 λ ∗ $_{c}$λ ∗ $_{t}$η$_{ct}$ S$_{0}$ ( x$_{c}$, x$_{t}$ ) + λ ∗ t ′ $^{2}$η$_{t}$ ′ S$_{0}$ ( x$_{t}$ ′ ) + 2 λ ∗ $_{c}$λ ∗ t ′ η$_{ct}$ ′ S$_{0}$ ( x$_{c}$, x$_{t}$ ′ ) +2 λ ∗ $_{t}$λ ∗ t ′ η$_{tt}$ ′ S$_{0}$ ( x$_{t}$, x$_{t}$ ′ ) ] , (36) M$_{12}$ = G 2 F 12 π $_{2}$f 2 $_{K}$B$_{K}$ m$_{K}$M 2 W [ λ ∗ c $^{2}$η$_{c}$ S$_{0}$ ( x$_{c}$ ) + λ ∗ t $^{2}$η$_{t}$ S$_{0}$ ( x$_{t}$ ) + 2 λ ∗ $_{c}$λ ∗ $_{t}$η$_{ct}$ S$_{0}$ ( x$_{c}$, x$_{t}$ ) + λ ∗ t ′ $^{2}$η$_{t}$ ′ S$_{0}$ ( x$_{t}$ ′ ) + 2 λ ∗ $_{c}$λ ∗ t ′ η$_{ct}$ ′ S$_{0}$ ( x$_{c}$, x$_{t}$ ′ ) + 2 λ ∗ $_{t}$λ ∗ t ′ η$_{tt}$ ′ S$_{0}$ ( x$_{t}$, x$_{t}$ ′ ) ] , (35) κ$_{ϵ}$ = √$_{2 sin}$$_{φ}$$_{ϵ}$ ¯ κ$_{ϵ}$, (34)