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.

arXiv comment42 pages, 2 figures, 6 tables; formulae corrected, numerical results almost unchanged, to be published in JHEP

References in corpus (44)

• Big-Bang Nucleosynthesis
• Weak Decays Beyond Leading Logarithms
• RunDec: a Mathematica package for running and decoupling of the strong coupling and quark masses
• The $\Delta S=1$ Effective Hamiltonian Including Next-to-Leading Order QCD and QED Corrections
• The top quark and Higgs boson masses and the stability of the electroweak vacuum
• Observation of Direct CP Violation in K_{S,L} to ππDecays
• Weak transition matrix elements from finite-volume correlation functions
• A new measurement of direct CP violation in two pion decays of the neutral kaon
• The Anatomy of ${\epsilon }^{\prime }/\epsilon$ Beyond Leading Logarithms with Improved Hadronic Matrix Elements
• The Two-Loop Anomalous Dimension Matrix for $\Delta S=1$ Weak Non-Leptonic Decays I: $\mathcal{O}({\alpha }_s^2)$
• Effective Hamiltonian for Non-Leptonic |Delta F| = 1 Decays at NNLO in QCD
• Current status of the Standard Model CKM fit and constraints on $\Delta F=2$ New Physics
• Two--Loop Anomalous Dimension Matrix for $\Delta S=1$ Weak Non-Leptonic Decays II: $\mathcal{O}({\alpha }_e{\alpha }_s)$
• A precision measurement of direct CP violation in the decay of neutral kaons into two pions
• Electromagnetic Logarithms in B -> X_s l+ l-
• Standard-model prediction for direct CP violation in $K\to \pi \pi$ decay
• Evanescent Operators, Scheme Dependences and Double Insertions
• ${\epsilon }^{\prime }/\epsilon$ at the next-to-leading order in QCD and QED
• 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
• Strong Enhancement of ${\epsilon }^{\prime }/\epsilon$ Through Final State Interactions
• Connections between epsilon'/epsilon and Rare Kaon Decays in Supersymmetry
• Quark flavour observables in the Littlest Higgs model with T-parity after LHC Run 1
• $K\to \pi \nu \bar{\nu }$ and ${\epsilon }^{\prime }/\epsilon$ in Simplified New Physics Models
• Improved anatomy of epsilon'/epsilon in the Standard Model
• Isospin violation in epsilon'
• The $K\to (\pi \pi {)}_{I=2}$ Decay Amplitude from Lattice QCD
• Z-Z' Mixing and Z-Mediated FCNCs in SU(3)_C x SU(3)_L x U(1)_X Models
• Lattice determination of the $K\to (\pi \pi {)}_{I=2}$ Decay Amplitude $A_2$
• Indirect Constraints on the Scalar Di-Photon Resonance at the LHC
• Isospin Breaking in K -> Pi Pi Decays
• Supersymmetric Explanation of CP Violation in $K\to \pi \pi$ Decays
• Studying ${\epsilon }^{\prime }/\epsilon$ in the Chiral Quark Model: ${\gamma }_5$-Scheme Independence and NLO Hadronic Matrix Elements
• Penguin Diagrams, Charmless B-Decays and the ``Missing Charm Puzzle''
• ${\epsilon }^{\prime }/\epsilon$ in 331 Models
• New Physics Patterns in ${\epsilon }^{\prime }/\epsilon$ and ${\epsilon }_K$ with Implications for Rare Kaon Decays and $\Delta M_K$
• Epsilon' from supersymmetry with non universal A terms?
• Gluon dipole penguin contributions to ${\epsilon }^{\prime }/\epsilon$ and CP violation in Hyperon decays in the Standard Model
• The Relevance of the Dipole Penguin Operators in ${\epsilon }^{\prime }/\epsilon$
• K -> pi pi Phenomenology in the Presence of Electromagnetism
• The Impact of |Delta I|=5/2 Transitions in K-> pi pi Decays
• K -> pi matrix elements of the chromagnetic operator on the lattice
• The chromomagnetic operator on the lattice
• Renormalisation group evolution for the $\Delta S=1$ effective Hamiltonian with $N_f=2+1$

Cited by in corpus (40)

• On the Importance of NNLO QCD and Isospin-breaking Corrections in ${\epsilon }^{\prime }/\epsilon$
• Right-handed charged currents in the era of the Large Hadron Collider
• Patterns of Flavour Violation in Models with Vector-Like Quarks
• Direct $CP$ Violation in $K\to {\mu }^{+}{\mu }^{-}$
• Yukawa enhancement of $Z$-mediated New Physics in $\Delta S=2$ and $\Delta B=2$ Processes
• The Emergence of the $\Delta U=0$ Rule in Charm Physics
• Revisiting Kaon Physics in General $Z$ Scenario
• Direct CP violation in $K^0\to \pi \pi$: Standard Model Status
• Leptoquarks meet ${\epsilon }^{\prime }/\epsilon$ and rare Kaon processes
• Chargino contributions in light of recent ${\epsilon }^{\prime }/\epsilon$
• Gluino-mediated electroweak penguin with flavor-violating trilinear couplings
• $Z^{\prime }$ models with less-minimal flavour violation
• Direct CP violation and the $\Delta I=1/2$ rule in $K\to \pi \pi$ decay from the Standard Model
• Correlating ${\epsilon }^{\prime }/\epsilon$ to hadronic $B$ decays via $U(2{)}^3$ flavour symmetry
• The CKM Parameters
• Final State Interactions in $K\to \pi \pi$ Decays: $\Delta I=1/2$ Rule vs. ${\epsilon }^{\prime }/\epsilon$
• Anatomy of ${\epsilon }^{\prime }/\epsilon$ beyond the Standard Model
• Isospin-Violating Contributions to ${\epsilon }^{\prime }/\epsilon$
• Implications for new physics from a novel puzzle in ${\bar{B}}_{(s)}^0\to D_{(s)}^{(\ast )+}\{{\pi }^{-},K^{-}\}$ decays
• $K\to \pi \nu \overline{\nu }$ in the MSSM in Light of the ${\epsilon }_K^{\prime }/{\epsilon }_K$ Anomaly
• ${\epsilon }^{\prime }/\epsilon$ in the Standard Model at the Dawn of the 2020s
• BSM Hadronic Matrix Elements for ${\epsilon }^{\prime }/\epsilon$ and $K\to \pi \pi$ Decays in the Dual QCD Approach
• Simultaneous interpretation of $K$ and $B$ anomalies in terms of chiral-flavorful vectors
• $K\to \pi \pi$ and $K-\pi$ Matrix Elements of the Chromomagnetic Operators from Dual QCD
• ${\epsilon }^{\prime }/\epsilon$ Anomaly and Neutron EDM in $SU(2{)}_L\times SU(2{)}_R\times U(1{)}_{B-L}$ model with Charge Symmetry
• The direct CP violation in a general two Higgs doublet model
• Effect of the Pauli exclusion principle in the electric dipole moment of ${}^9$Be with $|\Delta S|=1$ interactions
• Isospin-breaking in ${\epsilon }^{\prime }/\epsilon$: Impact of ${\eta }_0$ at the Dawn of the 2020s
• Theoretical status of ${\epsilon }^{\prime }/\epsilon$
• Probing SUSY effects in $K_S^0\to {\mu }^{+}{\mu }^{-}$
• ${\epsilon }_K$ and ${\epsilon }^{\prime }/\epsilon$ in a diquark model
• Updated Standard Model Prediction for ${\epsilon }^{\prime }/\epsilon$
• Exploring the flavour structure of the high-scale MSSM
• ${\epsilon }^{\prime }/\epsilon$ in and beyond the Standard Model
• Impact of Lattice QCD on CKM Phenomenology
• Anomalies from Non-Perturbative Standard Model Effects
• Lattice QCD: Flavor Physics and Spectroscopy
• Direct CP violation in $K\to \pi \pi$ decays and supersymmetry
• On the perturbative renormalisation of four-quark operators for new physics
• The Optimal Strategy for ${\epsilon }^{\prime }/\epsilon$ in the SM: 2019