Dual QCD insight into BSM hadronic matrix elements for K ⁰ –K ^{¯ 0} mixing from lattice QCD ^∗

We calculate BSM hadronic matrix elements for K ⁰ –K ^{¯ 0} mixing in the Dual QCD approach (DQCD). The ETM, SWME and RBC-UKQCD lattice collaborations find the matrix elements of the BSM density–density operators O _i with i = 2–5 to be rather different from their vacuum insertion values (VIA) with B ₂ ≈ 0.5, B ₃ ≈ B ₅ ≈ 0.7 and B ₄ ≈ 0.9 at µ = 3 GeV to be compared with B _i = 1 in the VIA. We demonstrate that this pattern can be reconstructed within the DQCD through the non-perturbative meson evolution from very low scales, where factorization of matrix elements is valid, to scales O(1 GeV) with subsequent perturbative quark–gluon evolution to µ = 3 GeV. This turns out to be possible in spite of a very different pattern displayed at low scales with B ₂ = 1.2, B ₃ = 3.0, B ₄ = 1.0 and B ₅ ≈ 0.2 in the large-N limit, N being the number of colours. Our results imply that the inclusion of meson evolution in the phenomenology of any non-leptonic transition like K ⁰ –K ^{¯ 0} mixing and K → ππ decays is mandatory. While meson evolution, as demonstrated in our paper, is hidden in lattice QCD results, to our knowledge, DQCD is the only analytic approach for non-leptonic transitions and decays which takes this important strong dynamics into account.