Singularity-free next-to-leading order ΔS=1 renormalization group evolution and εK′/εK in the Standard Model and beyond

The standard analytic solution of the renormalization group (RG) evolution for the Delta S = 1 Wilson coefficients involves several singularities, which complicate analytic solutions. In this paper we derive a singularity-free solution of the next-to-leading order (NLO) RG equations, which greatly facilitates the calculation of epsilon(K)', the measure of direct CP violation in K -> pi pi decays. Using our new RG evolution and the latest lattice results for the hadronic matrix elements, we calculate the ratio epsilon(K)' (with epsilon(K) quantifying indirect CP violation) in the Standard Model (SM) at NLO to epsilon(K)'/epsilon(K)' = (1.06 +/- 5.07) x 10(-4), which is 2.8 sigma below the experimental value. We also present the evolution matrix in the high-energy regime for calculations of new physics contributions and derive easy-to-use approximate formulae. We find that the RG amplification of new-physics contributions to Wilson coefficients of the electroweak penguin operators is further enhanced by the NLO corrections: if the new contribution is generated at the scale of 1-10 TeV, the RG evolution between the new-physics scale and the electroweak scale enhances these coefficients by 50-100%. Our solution contains a term of order alpha(2)(EM)/alpha(2)(s), which is numerically unimportant for the SM case but should be included in studies of high-scale new-physics.