$K\to \pi \nu \bar{\nu }$ and ${ϵ}^{\prime }/ϵ$ in Simplified New Physics Models

Abstract

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 ${ϵ}^{\prime }/ϵ$ 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 ${ϵ}^{\prime }/ϵ$ in the SM to be significantly below the data. While in many models the enhancement of ${ϵ}^{\prime }/ϵ$ 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 $B\to K(K^{\ast }){\mu }^{+}{\mu }^{-}$, 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 $B\to K(K^{\ast })\nu \bar{\nu }$ will be tested at Belle II.

Comments: 32 pages, 6 figures, 1 table. v2: updated analysis in section 4, matches version published in JHEP