Journal of High Energy Physics

, 2016:78

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

Open AccessRegular Article - Theoretical Physics

DOI: 10.1007/JHEP12(2016)078

Cite this article as:
Kitahara, T., Nierste, U. & Tremper, P. J. High Energ. Phys. (2016) 2016: 78. doi:10.1007/JHEP12(2016)078


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 ϵK, the measure of direct CP violation in Kππ decays. Using our new RG evolution and the latest lattice results for the hadronic matrix elements, we calculate the ratio ϵKK (with ϵK quantifying indirect CP violation) in the Standard Model (SM) at NLO to ϵKK = (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 αEM2/αs2, which is numerically unimportant for the SM case but should be included in studies of high-scale new-physics.


Renormalization Group CP violation Kaon Physics 
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© The Author(s) 2016

Authors and Affiliations

  1. 1.Institute for Theoretical Particle Physics (TTP), Karlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Institute for Nuclear Physics (IKP), Karlsruhe Institute of TechnologyEggenstein-LeopoldshafenGermany