Shell-model Description in $^{99}$Rh and Systematics of Odd-$A$ Rh Isotopes

Abstract

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 nonperturbative 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.