Eur. Phys. J. C (2023) 83: 368
https://doi.org/10.1140/epjc/s10052-023-11474-2

Regular Article - Theoretical Physics

Kaon physics without new physics in ${\epsilon }_K$

¹ Physik-Institut, Universität Zürich, 8057, Zurich, Switzerland
² TUM Institute for Advanced Study, Lichtenbergstr. 2a, 85747, Garching, Germany
³ Theoretical Division, MS B283, Los Alamos National Laboratory, NM, 87545, Los Alamos, USA

^c jacky.kumar@lanl.gov

Received: 16 February 2023
Accepted: 3 April 2023
Published online: 5 May 2023

Abstract

Despite the observation of significant suppressions of $b\to s{\mu }^{+}{\mu }^{-}$ branching ratios no clear sign of New Physics (NP) has been identified in $\mathrm{\Delta }F=2$ observables $\mathrm{\Delta }{M}_{d,s}$, ${\epsilon }_K$ and the mixing induced CP asymmetries $S_{\psi K_S}$ and $S_{\psi \varphi }$. Assuming negligible NP contributions to these observables allows to determine CKM parameters without being involved in the tensions between inclusive and exclusive determinations of $|V_{\mathit{cb}}|$ and $|V_{\mathit{ub}}|$. Furthermore this method avoids the impact of NP on the determination of these parameters present likely in global fits. Simultaneously it provides SM predictions for numerous rare K and B branching ratios that are most accurate to date. Analyzing this scenario within $Z^{\prime }$ models we point out, following the 2009 observations of Monika Blanke and ours of 2020, that despite the absence of NP contributions to ${\epsilon }_K$, significant NP contributions to $K^{+}\to {\pi }^{+}\nu \overline{\nu }$, $K_L\to {\pi }^0\nu \overline{\nu }$, $K_S\to {\mu }^{+}{\mu }^{-}$, $K_L\to {\pi }^0{\ell }^{+}{\ell }^{-}$, ${\epsilon }^{\prime }/\epsilon$ and $\mathrm{\Delta }{M}_K$ can be present. In the simplest scenario, this is guaranteed, as far as flavour changes are concerned, by a single non-vanishing imaginary left-handed $Z^{\prime }$ coupling $g_{\mathit{sd}}^L$. This scenario implies very stringent correlations between the Kaon observables considered by us. In particular, the identification of NP in any of these observables implies automatically NP contributions to the remaining ones under the assumption of non-vanishing flavour conserving $Z^{\prime }$ couplings to $q\overline{q}$, $\nu \overline{\nu }$, and ${\mu }^{+}{\mu }^{-}$. A characteristic feature of this scenario is a strict correlation between $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ and $K_L\to {\pi }^0\nu \overline{\nu }$ branching ratios on a branch parallel to the Grossman-Nir bound. Moreover, $\mathrm{\Delta }{M}_K$ is automatically suppressed as seems to be required by the results of the RBC-UKQCD lattice QCD collaboration. Furthermore, there is no NP contribution to $K_L\to {\mu }^{+}{\mu }^{-}$ which otherwise would bound NP effects in $K^{+}\to {\pi }^{+}\nu \overline{\nu }$. Of particular interest are the correlations of $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ and $K_L\to {\pi }^0\nu \overline{\nu }$ branching ratios and of $\mathrm{\Delta }{M}_K$ with the ratio ${\epsilon }^{\prime }/\epsilon$. We investigate the impact of renormalization group effects in the context of the SMEFT on this simple scenario.