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Kaons are not eigenstates of $CP$: $$ CP|K\rangle =|\overline K\rangle\\ $$ Why do we need to mix them?

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One answer I read is "since they both decay into $2$ or $3$ pions". Couldn't they just decay in the same thing and that's it? There are many other particles that happen to decay the same way, but we don't mix them.

*Addendum. Citing from "Mixing and Oscillations of Particles" by Andrzej K. Wroblewski:

In 1955 Murray Gell-Mann and Abraham Pais analyzed the behavior of neutral particles under the operation $C$ of charge conjugation which changes every particle into its anti-particle. According to the proposed scheme of classification of $K$ mesons, the neutral kaon $K^0$ was assumed to possess an anti-particle $\overline{K}^0$ distinct from itself. Gell-Mann and Pais were able to show that in that case the neutral kaon must be considered to be a “particle mixture”, exhibiting two distinct lifetimes and different decay modes.

What were the reasons that brought Gell-Mann and Pais to this conclusion?

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    $\begingroup$ Why did you include those diagrams? You don't comment on them but it seems to me that they answer your question. $\endgroup$
    – octonion
    Commented Jul 26, 2022 at 14:48
  • $\begingroup$ Strangeness oscillations is an experimental phenomenon. It was observed, so it was inferred that flavor mixing occurred; copiously for this system, but much more weakly for other systems. What is your point, really? What are you asking? Your read what where? $\endgroup$
    – Cosmas Zachos
    Commented Jul 26, 2022 at 15:01
  • $\begingroup$ I'm asking if $K$ oscillations were known before considering the possibility that they could violate $CP$ parity OR if these oscillations were introduced to justify the $CP$ violation $\endgroup$
    – ric.san
    Commented Jul 26, 2022 at 15:13

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It is a terrible misconception to predicate flavor mixing (strangeness oscillations here) on CP violation. Flavor mixing was investigated (Gell-Mann―Pais, 1955, and the experiments probing it, 1957, 1960) for a decade (1955-1964) before the discovery of CP violation in 1964 (Fitch―Cronin).

CP violation is just a minuscule twist on the flavor oscillation of neutral mesons like the weak diagrams you are displaying, and is not a precondition for it: just a minute phenomenon riding on phases attached to small CKM matrix elements in your box diagrams.

Edit to address the altered question decoupling from CP violation. The point of the Gell-Mann―Pais paper, was that the masses of the odd and even CP eigenstates, KL, KS, were slightly different, with vastly different lifetimes, exactly what you'd get from a strong Hamiltonian with small doubly weak off-diagonal mixing terms, upon diagonalization of the mass matrix.

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