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

The standard analytic solution of the renormalization group (rg) evolution for the δs = 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 ϵ ′ , the measure of direct cp violation in k → ππ decays. using our new rg evolution and the latest lattice results for the hadronic matrix elements, we calculate the ratio ϵ ′ /ϵ k (with ϵ k quantifying indirect cp violation) in the standard model (sm) at nlo to ϵ ′ /ϵ k  = (1.06 ± 5.07) × 10− 4, which is 2.8 σ 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 α em 2 /α 2 , which is numerically unimportant for the sm case but should be included in studies of high-scale new-physics.