New physics patterns in ε ′/ ε and ε K with implications for rare kaon decays and ∆ M K

The standard model (sm) prediction for the ratio ε′/ε appears to be significantly below the experimental data. also ε k in the sm tends to be below the data. any new physics (np) removing these anomalies will first of all have impact on flavour observables in the k meson system, in particular on rare decays $$ {k}^{+}to {pi}^{+}nu overline{nu} $$ , $$ {k}_lto {pi}^0nu overline{nu} $$ , k l → μ + μ − and k l → π 0 ℓ + ℓ − and δm k . restricting the operators contributing to ε′/ε to the sm ones and to the corresponding primed operators, np contributions to ε′/ε are quite generally dominated either by qcd penguin (qcdp) operators q 6(q 6 ′ ) or electroweak penguin (ewp) operators q 8(q 8 ′ ) with rather different implications for other flavour observables. our presentation includes general models with tree-level z and z′ flavour violating exchanges for which we summarize known results and add several new ones. we also briefly discuss few specific models. the correlations of ε′/ε with other flavour observables listed above allow to differentiate between models in which ε′/ε can be enhanced. various dna-tables are helpful in this respect. we find that simultaneous enhancements of ε′/ε, ε k , $$ mathrm{mathcal{b}}left({k}_lto {pi}^0nu overline{nu}right) $$ and $$ mathrm{mathcal{b}}left({k}^{+}to {pi}^{+}nu overline{nu}right) $$ in z scenarios are only possible in the presence of both left-handed and right-handed flavour-violating couplings. in z′ scenarios this is not required but the size of np effects and the correlation between $$ mathrm{mathcal{b}}left({k}_lto {pi}^0nu overline{nu}right) $$ and $$ mathrm{mathcal{b}}left({k}^{+}to {pi}^{+}nu overline{nu}right) $$ depends strongly on whether qcdp or ewp dominate np contributions to ε′/ε. 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 ε k .we point out that qcdp and ewp scenarios of np in ε′/ε can also be uniquely distinguished by the size and the sign of np contribution to δm k , elevating the importance of the precise calculation of δm k in the sm. we emphasize the importance of the theoretical improvements not only on ε′/ε, ε k and δm k but also on k l → μ + μ −, k l → π 0 ℓ + ℓ −, and the k → ππ isospin amplitudes rea 0 and rea 2 which would in the future enrich our analysis.