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The standard analytic solution of the renormalization group (RG) evolutionfor the $\Delta S = 1$ Wilson coefficients involves several singularities,which complicate analytic solutions. In this paper we derive a singularity-freesolution of the next-to-leading order (NLO) RG equations, which greatlyfacilitates the calculation of $\epsilon_K^{\prime}$, the measure of direct$CP$ violation in $K\to \pi\pi$ decays. Using our new RG evolution and thelatest lattice results for the hadronic matrix elements, we calculate the ratio$\epsilon_{K}^{\prime}/\epsilon_{K}$ (with $\epsilon_{K}$ quantifying indirect$CP$ violation) in the Standard Model (SM) at NLO to$\epsilon_{K}^{\prime}/\epsilon_{K} = (1.06 \pm 5.07) \times 10^{-4} $, whichis $2.8\,\sigma$ below the experimental value. We also present the evolutionmatrix in the high-energy regime for calculations of new physics contributionsand derive easy-to-use approximate formulae. We find that the RG amplificationof new-physics contributions to Wilson coefficients of the electroweak penguinoperators is further enhanced by the NLO corrections: If the new contributionis generated at the scale of 1-10 TeV, the RG evolution between the new-physicsscale and the electroweak scale enhances these coefficients by 50-100 %. Oursolution contains a term of order $\alpha_{EM}^2/\alpha_s^2$, which isnumerically unimportant for the SM case but should be included in studies ofhigh-scale new-physics.

the journal of high energy physics/the journal of high energy physics

Singularity-free Next-to-leading Order $\Delta S= 1$ Renormalization  Group Evolution and $\epsilon_{K}^{\prime}/\epsilon_{K}$ in the Standard  Model and Beyond