Master formulae for ΔF=2 NLO-QCD factors in the Standard Model and beyond Original Research Article

We present analytic formulae for the QCD renormalization group factors relating the Wilson coefficients Ci(μt) and Ci(μ), with μt=O(mt) and μ<μt, of the ΔF=2 dimension six four-quark operators Qi in the Standard Model and in all of its extensions. Analogous analytic formulae for the QCD factors relating the matrix elements 〈Qi(2 GeV)〉 and 〈Qi(μK)〉 with μK<2 GeV are also presented. The formulae are given in the NDR scheme. The strongest renormalization-group effects are found for the operators with the Dirac structures (1−γ5)⊗(1+γ5) and (1−γ5)⊗(1−γ5). We calculate the matrix elements 〈 K̄ 0|Qi|K0〉 in the NDR scheme using the lattice results in the LRI scheme. We give expressions for the mass differences ΔMK and ΔMB and the CP-violating parameter ϵK in terms of the non-perturbative parameters Bi and the Wilson coefficients Ci(μt). The latter summarize the dependence on new physics contributions.