$ K\to \pi \nu \overline{\nu} $ and ε′/ε in simplified new physics models

The decays $$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$ and $$ {K}_L\to\ {\pi}^0\nu \overline{\nu} $$ , being the theoretically cleanest rare decays of mesons, are very sensitive probes of New Physics (NP). In view of the excellent prospects of reaching the Standard Model (SM) sensitivity for $$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$ by the NA62 experiment at CERN and for $$ {K}_L\to\ {\pi}^0\nu \overline{\nu} $$ 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 Minimal Flavour Violation (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 the constraints from ΔF = 2 processes, electroweak and collider data. Of particular interest are the correlations with e′/e and K L → μ + μ − which limit the size of NP contributions to $$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$ and $$ {K}_L\to\ {\pi}^0\nu \overline{\nu} $$ , depending on the Dirac structure of couplings and the relevant operators. But in MFV models also the constraint from B s → μ + μ − turns out to be important. We take into account the recent results from lattice QCD and large N approach that indicate e′/e in the SM to be significantly below the data. While in many models the enhancement of e′/e implies the suppression of $$ {K}_L\to\ {\pi}^0\nu \overline{\nu} $$ , we present two models in which e′/e and $$ {K}_L\to\ {\pi}^0\nu \overline{\nu} $$ can be simultaneously enhanced relative to SM predictions. A correlation between $$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$ and B → K(K *)μ + μ −, 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 $$ B\to K\left({K}^{\ast}\right)\nu \overline{\nu} $$ will be tested at Belle II.