We present for the first time a model-independent anatomy of the ratio $varepsilon/varepsilon$ in the context of the $Delta S = 1$ effective theory with operators invariant under QCD and QED and in the context of the Standard Model Effective Field Theory (SMEFT) with the operators invariant under the full SM gauge group. Our goal is to identify the new physics scenarios that are probed by this ratio and which could help to explain a possible deviation from the SM that is hinted by the data. To this end we derive a master formula for $varepsilon/varepsilon$, which can be applied to any theory beyond the Standard Model (BSM) in which the Wilson coefficients of all contributing operators have been calculated at the electroweak scale. The relevant hadronic matrix elements of BSM operators are from the Dual QCD approach and the SM ones from lattice QCD. Within SMEFT, the constraints from $K^0$ and $D^0$ mixing as well as electric dipole moments limit significantly potential new physics contributions to $varepsilon/varepsilon$. Correlations of $varepsilon/varepsilon$ with $Ktopi ubar u$ decays are briefly discussed. Building on our EFT analysis and the model-independent constraints, we discuss implications of a possible deviation from the SM in $varepsilon/varepsilon$ for model building, highlighting the role of the new scalar and tensor matrix elements in models with scalar mediators.
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