Physics Motivation

$\phi$-meson, as many times discussed (see. [1]) is a good source of correlated pairs of neutral kaons because it decays into correlated $K_S K_L$ pairs 32% of the time. By studying the equal decay modes of $K_L$ and $K_S$ quantum mechanics correlation can be seen in 0-8 $K_S$ lifetime region (c$\tau_S$=2.676cm). Final states can be the same due to CP-violating decay of $K_L$ into $\pi^0\pi^0$ or $\pi^+\pi^-$ with $2\times10^{-3}$ branching ratio but as for $K_S$ they are main decay modes. Semileptonic dacays also can be used for this interferometery, but B( $K_S\rightarrow\pi l \nu_l$) is expected at the level of $5\times 10^{-4}$.

The idea of experiments discussed for $\phi$-factories is in simultanious detection of $K_S K_L$ decays in $\pi^+\pi^-\pi^0\pi^0$ state and reconstruction of their decay points with accuracy better than one $K_S$ decay length. Due to fully asymmetric final state ( $J^{PC}=1^{- -}$) quantum mechanics prohibits decay of both kaons at the same distance from originated $\phi$ if they have the same decay mode.

In addition to this very exciting quantum mechanics test at macroscopic distances this interferometry can be used for a study of direct CP violation in $K_L$ decays ( $\epsilon '/\epsilon$) and for precision CPT tests. For these purposes one should measure the difference in number of events when $\pi^+\pi^-$ pair appeares before $\pi^0\pi^0$ ($N_{+-}$) with those when $\pi^0\pi^0$ pair is first ($N_{00}$).

$$A(t_{+-}-t_{00})=(N_{+-}-N_{00})/(N_{+-}+N_{00}),$$

where $t_{+-}, t_{00}$ are times (or decay lengths) from $\phi$ to decay point of kaon with neutral or charged pions.
This asymmetry integrated over all $K_L$ decay lengths (less acceptance gives less accuracy) is related as A= $3\cdot Re(\epsilon ' /\epsilon)$ with direct CP violation parameter.

The evaluation of this asymmetry in the interference region $(t_{+-}-t_{00}=1-5~ \tau_S$) gives an information about imaginary part of $\epsilon '/\epsilon$ and CPT conservation.

As it was shown in [1] with the $10^{11}$ produced $\phi$'s and reasonable detector acceptance (20% of $K_L$ decay length, c$\tau_L$=15.49 m and $\beta=0.22$ in case of $e^+e^-\rightarrow\phi$) the $\epsilon '/\epsilon$ value can be measured with about $10^{-4}$ accuracy. DAFNE claimes to have 50% acceptance for $K_L$ decays with a 4 m diameter of drift chamber.

If photon detector can reconstruct $\pi^0\pi^0$ vertex with accuracy better $\tau_S$ the interferometry test would give accuracy in CPT test parameter $\Delta\Phi = \Phi_{+-} - \Phi_{00}$ about 0.1 degree with a present limit about 1 degree.

By using $K_L$ decay or nuclear interaction in the forward system as a tag $K_S$ rare decays can be studied, including first observation of CP violated $K_S\rightarrow\pi^0\pi^0\pi^0$ and $K_S\rightarrow\pi^+\pi^-\pi^0$ modes, which are predicted at the level of $2\times10^{-9}$.

The electroproduction of $\phi$ at CEBAF also can be a source of correlated $K_S K_L$ pairs and below some preliminary analysis of posibility to perform similar experimets is presented.