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

Abstract

We present analytic formulae for the QCD renormalization group factors relating the Wilson coefficients C_i(mu_t) and C_i(mu), with mu_t = O(m_t) and mu < mu_t, of the Delta F=2 dimension six four-quark operators Q_i in the Standard Model and in all of its extensions. Analogous analytic formulae for the QCD factors relating the matrix elements <Q_i(2 GeV)> and <Q_i(mu_K)> with mu_K < 2GeV 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 - gamma_5)*(1 + gamma_5) and (1 - gamma_5)*(1 - gamma_5). We calculate the matrix elements <\bar K^0|Q_i|K^0> in the NDR scheme using the lattice results in the LRI scheme. We give expressions for the mass differences Delta M_K and Delta M_B and the CP-violating parameter epsilon_K in terms of the non-perturbative parameters B_i and the Wilson coefficients C_i(mu_t). The latter summarize the dependence on new physics contributions.