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

The decays $K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$, being the theoretically cleanest rare decays of mesons, are very sensitive probes of New Physics. In view of the excellent prospects of reaching the Standard Model sensitivity 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 the simplest extensions of the SM in which stringent correlations between these two decays and other flavour observables are present. We first consider simple models 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 compare the resulting correlations with those present in generic models in which the latter couplings are arbitrary, subject to constraints from $\Delta F=2$ processes, electroweak and collider data. Of particular interest are the correlations with $\epsilon'/\epsilon$ and $K_L\to\mu^+\mu^-$ which limit the size 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 the relevant 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 lattice QCD and large N approach that indicate $\epsilon'/\epsilon$ in the SM to be significantly below the data. While in many models the enhancement of $\epsilon'/\epsilon$ implies a suppression of $K_L\to\pi^0\nu\bar\nu$, we present two models in which these observables can be simultaneously enhanced relative 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 coming years. The one with $B\to K(K^*)\nu\bar\nu$ will be tested at Belle II.