= Direct CP violation in two-pion kaon decay
{title2=$\eta_{+-}\simeq\epsilon+\epsilon',\quad\eta_{00}\simeq\epsilon-2\epsilon'$}
Direct <CP violation> concerns the decay transition itself. At leading order, the charged and neutral two-<pion> <decay amplitude> ratios separate a common $\epsilon$ from channel-dependent $\epsilon'$.https://www.nevis.columbia.edu/~haas/documents/physics_nobel_lectures/physics71-80-3.pdf[Cronin, pp.573–574, distinction between epsilon and epsilon-prime]{ref}
For the corresponding double ratio of partial <decay widths>,
$$
R=\frac{\Gamma_L^{00}/\Gamma_S^{00}}{\Gamma_L^{+-}/\Gamma_S^{+-}}
\simeq\left|\frac{1-2x}{1+x}\right|^2
=1-6\operatorname{Re}x+O(|x|^2),\qquad x=\frac{\epsilon'}{\epsilon}.
$$
The expansion follows from $(1-2x)/(1+x)=1-3x+O(x^2)$. A single nonzero long-lived two-<pion> yield does not isolate $x$: it can already occur with $\epsilon'=0$. Thus the 1964 evidence for <CP violation> must be distinguished from the later experimental establishment of direct <CP violation> through $\epsilon'$.
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