New Physics Patterns in $\varepsilon^\prime/\varepsilon$ and $\varepsilon_K$ with Implications for Rare Kaon Decays and $\Delta M_K$

TL;DR

This paper investigates new physics explanations for discrepancies in kaon decay parameters, analyzing how different models affect rare decay processes and meson mass differences, with implications for future experimental tests.

Contribution

It identifies how specific new physics operators influence kaon observables and explores model-dependent correlations, highlighting the importance of QCDP and EWP dominance scenarios.

Findings

QCDP operators can significantly enhance rare kaon decay rates.

Simultaneous enhancements require both LH and RH couplings in Z models.

Distinct signatures in $ ext{Δ}M_K$ can differentiate NP scenarios.

Abstract

The SM prediction for the ratio $\varepsilon^\prime/\varepsilon$ appears to be significantly below the experimental data. Also $\varepsilon_K$ in the SM tends to be below the data. Any NP removing these anomalies will first of all have impact on flavour observables in the $K$ meson system, in particular on rare $K$ decays and $\Delta M_K$. Restricting the operators contributing to $\varepsilon^\prime/\varepsilon$ to the SM ones and to the corresponding primed operators, NP contributions are quite generally dominated either by QCD penguin (QCDP) operators $Q_6(Q_6^\prime)$ or electroweak penguin (EWP) operators $Q_8(Q_8^\prime)$ with rather different implications for other flavour observables. We discuss general models with tree-level $Z$ and $Z^\prime$ flavour violating exchanges and few specific models. We find that simultaneous enhancements of $\varepsilon^\prime/\varepsilon$, $\varepsilon_K$, $\mathcal{B}(K_L\to\pi^0\nu\bar\nu)$ and $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$ in $Z$ scenarios are only possible in the presence of both LH and RH flavour-violating couplings. In $Z^\prime$ scenarios this is not required but the size of NP effects and the correlation between $\mathcal{B}(K_L\to\pi^0\nu\bar\nu)$ and $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$ depends strongly on whether QCDP or EWP dominate NP contributions to $\varepsilon^\prime/\varepsilon$. In the QCDP case possible enhancements of both branching ratios are much larger than for EWP scenario and take place only on the branch parallel to the Grossman-Nir bound, which is in the case of EWP dominance only possible in the absence of NP in $\varepsilon_K$. QCDP and EWP scenarios of NP in $\varepsilon^\prime/\varepsilon$ can also be uniquely distinguished by the size and the sign of NP contribution to $\Delta M_K$, elevating the importance of the precise calculation of $\Delta M_K$ in the SM.