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

Abstract

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 ${\epsilon }_K^{\prime }$, the measure of direct $CP$ violation in $K\to \pi \pi$ decays. Using our new RG evolution and the latest 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 ${\epsilon }_K^{\prime }/{\epsilon }_K=(1.06\pm 5.07)\times 10^{-4}$, 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 ${\alpha }_{EM}^2/{\alpha }_s^2$, which is numerically unimportant for the SM case but should be included in studies of high-scale new-physics.

42 pages, 2 figures, 6 tables; formulae corrected, numerical results almost unchanged, to be published in JHEP

References in corpus (6)

• Big-Bang Nucleosynthesis
• Current status of the Standard Model CKM fit and constraints on $\Delta F=2$ New Physics
• Precise Measurements of Direct CP Violation, CPT Symmetry, and Other Parameters in the Neutral Kaon System
• $K\to \pi \pi$ $\Delta I=3/2$ decay amplitude in the continuum limit
• Z-Z' Mixing and Z-Mediated FCNCs in SU(3)_C x SU(3)_L x U(1)_X Models
• Renormalisation group evolution for the $\Delta S=1$ effective Hamiltonian with $N_f=2+1$

Cited by in corpus (4)

• Direct $CP$ Violation in $K\to {\mu }^{+}{\mu }^{-}$
• Chargino contributions in light of recent ${\epsilon }^{\prime }/\epsilon$
• Impact of Lattice QCD on CKM Phenomenology
• Direct CP violation in $K\to \pi \pi$ decays and supersymmetry