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We calculate for the first time all four-quark hadronic matrix elements oflocal operators possibly contributing to $K\to\pi\pi$ decays and in particularto the ratio $\epsilon'/\epsilon$ beyond the Standard Model (BSM). To this endwe use the Dual QCD (DQCD) approach. In addition to 7 new mirror operatorsobtained from the SM ones by flipping the chirality, we count 13 BSM four-quarkoperators of a given chirality linearly independent of each other and of theaforesaid 14 operators for which hadronic matrix elements are already known. Wepresent results in two bases for all these operators, one termed DQCD basisuseful for the calculation of the hadronic matrix elements in the DQCD approachand the other called SD basis suited to the short distance renormalizationgroup evolution above the 1~GeV scale. We demonstrate that the pattern of longdistance evolution (meson evolution) matches the one of short distanceevolution (quark-gluon evolution), a property which to our knowledge cannot bepresently achieved in any other analytical framework. The highlights of ourpaper are chirally enhanced matrix elements of tensor-tensor and scalar-scalarBSM operators. They could thereby explain the emerging $\epsilon'/\epsilon$anomaly which is strongly indicated within DQCD with some support from latticeQCD. On the other hand we do not expect the BSM operators to be relevant forthe $\Delta I=1/2$ rule.

BSM Hadronic Matrix Elements for $\epsilon'/\epsilon$ and $K\to\pi\pi$ Decays in the Dual QCD Approach