 u$ final state, while in charm decays it generates to a very good approximation the same amount of $d \bar d$ and $s \bar s$ states.

We write the amplitudes very generally and up to a normalization factor as $${\cal A} = 1 + r a e^{i(\phi+\delta)}\,, \label{eq:generic-ampl}$$ such that $r$ is real and depends on CKM matrix elements, $a$ is real and corresponds to the ratio of the respective hadronic matrix elements, $\phi$ is a weak phase and $\delta$ is a strong phase. For kaons $a$ is the ratio of matrix elements of the operators $Q^{\Delta I=1/2}$ over $Q^{\Delta I=3/2}$, while for charm it is the ratio of matrix elements of the operators $Q^{\Delta U=0}$ over $Q^{\Delta U=1}$.

We first consider the case where we neglect the third generation. In that limit for kaons we have the decomposition $${\cal A}_K = V_{us}V_{ud}^* (A_{1/2} + r_{CG} A_{3/2})\,,$$ where $r_{CG}$ is the CG coefficient that can be read from Eq. (\[eq:kaondata\]). For charm we have $${\cal A}_D = V_{cs}V_{us}^* A_1.$$ That means that in the two-generational limit for kaons we have $r=1$ and in charm $r=0$. If we switch on the third generation we get small corrections to these values in each case: $r\ll 1$ for charm and $|r-1|\ll 1$ for kaons. These effects come from the non-unitarity of the $2 \times 2$ CKM. For the kaon case there is an extra effect that stems from SD penguins that come with $V_{ts}V_{td}^*$. In both cases we have $\delta \sim
\mathcal{O}(1)$ from non-perturbative rescattering, as well as $\phi \sim \mathcal{O}(1)$.

The general formula for direct CP asymmetry is given as [@Tanabashi:2018oca] $$\begin{aligned}
A_{CP} &= -\frac{2 r a  \sin(\delta) \sin(\phi)
}{ 
1 + (ra)^2+ 2 ra \cos(\delta) \cos(\phi)
}
\approx
\begin{cases}
2 r a  \sin(\delta) \sin(\phi)&{\mbox{for $ra \ll 1$}},\\
2 (r a)^{-1}  \sin(\delta) \sin(\phi)&{\mbox{for $ra \gg 1$}}.\\
\end{cases} \label{eq:CPasym-general-formula} \end{aligned}$$ Non-perturbative effects enhance $a$ in both kaon and charm decays. This means the effect which is visible in the 