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The decays $K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$, being thetheoretically cleanest rare decays of mesons, are very sensitive probes of NewPhysics. In view of the excellent prospects of reaching the Standard Modelsensitivity for $K^+\to\pi^+\nu\bar\nu$ by the NA62 experiment at CERN and for$K_L\to\pi^0\nu\bar\nu$ by the KOTO experiment at J-PARC, we study them in thesimplest extensions of the SM in which stringent correlations between these twodecays and other flavour observables are present. We first consider simplemodels with tree-level Z and Z' contributions in which either MFV or a $U(2)^3$symmetry is imposed on the quark flavour-violating couplings. We then comparethe resulting correlations with those present in generic models in which thelatter couplings are arbitrary, subject to constraints from $\Delta F=2$processes, electroweak and collider data. Of particular interest are thecorrelations with $\epsilon'/\epsilon$ and $K_L\to\mu^+\mu^-$ which limit thesize of NP contributions to $K^+\to\pi^+\nu\bar\nu$ and$K_L\to\pi^0\nu\bar\nu$, depending on the Dirac structure of couplings and therelevant operators. But in MFV also the constraint from $B_s\to\mu^+\mu^-$turns out to be important. We take into account the recent results from latticeQCD and large N approach that indicate $\epsilon'/\epsilon$ in the SM to besignificantly below the data. While in many models the enhancement of$\epsilon'/\epsilon$ implies a suppression of $K_L\to\pi^0\nu\bar\nu$, wepresent two models in which these observables can be simultaneously enhancedrelative to SM predictions. A correlation between $K^+\to\pi^+\nu\bar\nu$ and$B\to K(K^*)\mu^+\mu^-$, found by us in the simple models considered here,should be of interest for NA62 and LHCb experimentalists at CERN in the comingyears. The one with $B\to K(K^*)\nu\bar\nu$ will be tested at Belle II.

$K\to\pi\nu\bar\nu$ and $\epsilon'/\epsilon$ in Simplified New Physics Models