New Physics Patterns in ${\epsilon }^{\prime }/\epsilon$ and ${\epsilon }_K$ with Implications for Rare Kaon Decays and $\Delta M_K$

Abstract

The SM prediction for the ratio ${\epsilon }^{\prime }/\epsilon$ appears to be significantly below the experimental data. Also ${\epsilon }_K$ 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 ${\epsilon }^{\prime }/\epsilon$ 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 ${\epsilon }^{\prime }/\epsilon$, ${\epsilon }_K$, $\mathcal{B}(K_L\to {\pi }^0\nu \bar{\nu })$ 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 ${\epsilon }^{\prime }/\epsilon$. 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 ${\epsilon }_K$. QCDP and EWP scenarios of NP in ${\epsilon }^{\prime }/\epsilon$ 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.

67 pages, 6 Figures (12 Plots), references and clarifying comments on pages 3 and 9 and equation (24) added, conclusions unchanged

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• Two-Loop Electroweak Corrections for the K -> pi nu anti-nu Decays
• Current status of the Standard Model CKM fit and constraints on $\Delta F=2$ New Physics
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• Solving the flavour problem with hierarchical fermion wave functions
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• ${\epsilon }^{\prime }/\epsilon$ in 331 Models
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Cited by in corpus (28)

• Patterns of Flavour Violation in Models with Vector-Like Quarks
• Direct $CP$ Violation in $K\to {\mu }^{+}{\mu }^{-}$
• Yukawa enhancement of $Z$-mediated New Physics in $\Delta S=2$ and $\Delta B=2$ Processes
• Revisiting Kaon Physics in General $Z$ Scenario
• Direct CP violation in $K^0\to \pi \pi$: Standard Model Status
• Master formula for ${\epsilon }^{\prime }/\epsilon$ beyond the Standard Model
• Chargino contributions in light of recent ${\epsilon }^{\prime }/\epsilon$
• $Z^{\prime }$ models with less-minimal flavour violation
• Gluino-mediated electroweak penguin with flavor-violating trilinear couplings
• Singularity-free Next-to-leading Order $\Delta S=1$ Renormalization Group Evolution and ${\epsilon }_K^{\prime }/{\epsilon }_K$ in the Standard Model and Beyond
• Universal Unitarity Triangle 2016 and the Tension Between $\Delta M_{s,d}$ and ${\epsilon }_K$ in CMFV Models
• Probing the SUSY with $10$ TeV stop mass in rare decays and CP violation of Kaon
• Correlating ${\epsilon }^{\prime }/\epsilon$ to hadronic $B$ decays via $U(2{)}^3$ flavour symmetry
• The CKM Parameters
• Anatomy of ${\epsilon }^{\prime }/\epsilon$ beyond the Standard Model
• Isospin-Violating Contributions to ${\epsilon }^{\prime }/\epsilon$
• $K\to \pi \nu \overline{\nu }$ in the MSSM in Light of the ${\epsilon }_K^{\prime }/{\epsilon }_K$ Anomaly
• Left-handed color-sextet diquark in the Kaon system
• Lattice study of large $N_c$ QCD
• BSM Hadronic Matrix Elements for ${\epsilon }^{\prime }/\epsilon$ and $K\to \pi \pi$ Decays in the Dual QCD Approach
• ${\epsilon }_K^{\prime }/{\epsilon }_K$ and $K\to \pi \nu \bar{\nu }$ in a two-Higgs doublet model
• ${\epsilon }^{\prime }/\epsilon$ Anomaly and Neutron EDM in $SU(2{)}_L\times SU(2{)}_R\times U(1{)}_{B-L}$ model with Charge Symmetry
• 331 Models Facing the Tensions in $\Delta F=2$ Processes with the Impact on ${\epsilon }^{\prime }/\epsilon$, $B_s\to {\mu }^{+}{\mu }^{-}$ and $B\to K^{\ast }{\mu }^{+}{\mu }^{-}$
• PSI/UZH Workshop: Impact of $B\to {\mu }^{+}{\mu }^{-}$ on New Physics Searches
• ${\epsilon }_K$ and ${\epsilon }^{\prime }/\epsilon$ in a diquark model
• Another SMEFT Story: $Z^{\prime }$ Facing New Results on ${\epsilon }^{\prime }/\epsilon$, $\Delta M_K$ and $K\to \pi \nu \bar{\nu }$
• Kaon Flavour Physics Strikes Back
• The Revival of Kaon Flavour Physics