Abstract
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 ′ K , 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 ′ K /ϵ K (with ϵ K quantifying indirect CP violation) in the Standard Model (SM) at NLO to ϵ ′ K /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 2EM /α 2 s , which is numerically unimportant for the SM case but should be included in studies of high-scale new-physics.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
F.J. Gilman and M.B. Wise, The ΔI = 1/2 Rule and Violation of CP in the Six Quark Model, Phys. Lett. B 83 (1979) 83 [INSPIRE].
B. Guberina and R.D. Peccei, Quantum Chromodynamic Effects and CP-violation in the Kobayashi-Maskawa Model, Nucl. Phys. B 163 (1980) 289 [INSPIRE].
J.S. Hagelin and F.J. Gilman, A lower bound on |ε ′ /ε|, Phys. Lett. B 126 (1983) 111 [INSPIRE].
A.J. Buras and J.M. Gérard, ϵ ′ /ϵ in the Standard Model, Phys. Lett. B 203 (1988) 272 [INSPIRE].
J.M. Flynn and L. Randall, The Electromagnetic Penguin Contribution to ϵ ′ /ϵ for Large Top Quark Mass, Phys. Lett. B 224 (1989) 221 [Erratum ibid. B 235 (1990) 412] [INSPIRE].
G. Buchalla, A.J. Buras and M.K. Harlander, The Anatomy of ϵ ′ /ϵ in the Standard Model, Nucl. Phys. B 337 (1990) 313 [INSPIRE].
E.A. Paschos and Y.L. Wu, Correlations between ϵ ′ /ϵ and heavy top, Mod. Phys. Lett. A 6 (1991) 93 [INSPIRE].
M. Lusignoli, L. Maiani, G. Martinelli and L. Reina, Mixing and CP-violation in K and B mesons: A Lattice QCD point of view, Nucl. Phys. B 369 (1992) 139 [INSPIRE].
A.J. Buras, M. Jamin, M.E. Lautenbacher and P.H. Weisz, Effective Hamiltonians for ΔS = 1 and ΔB = 1 nonleptonic decays beyond the leading logarithmic approximation, Nucl. Phys. B 370 (1992) 69 [Addendum ibid. B 375 (1992) 501] [INSPIRE].
A.J. Buras, M. Jamin, M.E. Lautenbacher and P.H. Weisz, Two loop anomalous dimension matrix for ΔS = 1 weak nonleptonic decays I: \( \mathcal{O}\left({\alpha}_s^2\right) \), Nucl. Phys. B 400 (1993) 37 [hep-ph/9211304] [INSPIRE].
M. Ciuchini, E. Franco, G. Martinelli and L. Reina, The ΔS = 1 effective Hamiltonian including next-to-leading order QCD and QED corrections, Nucl. Phys. B 415 (1994) 403 [hep-ph/9304257] [INSPIRE].
A.J. Buras, M. Jamin and M.E. Lautenbacher, Two loop anomalous dimension matrix for ΔS = 1 weak nonleptonic decays. (II) \( \mathcal{O}\left(\alpha {\alpha}_s\right) \), Nucl. Phys. B 400 (1993) 75 [hep-ph/9211321] [INSPIRE].
A.J. Buras, M. Jamin and M.E. Lautenbacher, The Anatomy of ϵ ′ /ϵ beyond leading logarithms with improved hadronic matrix elements, Nucl. Phys. B 408 (1993) 209 [hep-ph/9303284] [INSPIRE].
A.J. Buras, M. Gorbahn, S. Jäger and M. Jamin, Improved anatomy of ε ′ /ε in the Standard Model, JHEP 11 (2015) 202 [arXiv:1507.06345] [INSPIRE].
V. Cirigliano, A. Pich, G. Ecker and H. Neufeld, Isospin violation in ϵ ′, Phys. Rev. Lett. 91 (2003) 162001 [hep-ph/0307030] [INSPIRE].
V. Cirigliano, G. Ecker, H. Neufeld and A. Pich, Isospin breaking in K → ππ decays, Eur. Phys. J. C 33 (2004) 369 [hep-ph/0310351] [INSPIRE].
T. Blum et al., The K → (ππ) I=2 Decay Amplitude from Lattice QCD, Phys. Rev. Lett. 108 (2012) 141601 [arXiv:1111.1699] [INSPIRE].
T. Blum et al., Lattice determination of the K → (ππ) I=2 Decay Amplitude A 2 , Phys. Rev. D 86 (2012) 074513 [arXiv:1206.5142] [INSPIRE].
T. Blum et al., K → ππ ΔI = 3/2 decay amplitude in the continuum limit, Phys. Rev. D 91 (2015) 074502 [arXiv:1502.00263] [INSPIRE].
RBC/UKQCD collaboration, Z. Bai et al., Standard Model Prediction for Direct CP-violation in K → ππ Decay, Phys. Rev. Lett. 115 (2015) 212001 [arXiv:1505.07863] [INSPIRE].
M. Ciuchini, E. Franco, G. Martinelli and L. Reina, ϵ ′ /ϵ at the Next-to-leading order in QCD and QED, Phys. Lett. B 301 (1993) 263 [hep-ph/9212203] [INSPIRE].
T. Huber, E. Lunghi, M. Misiak and D. Wyler, Electromagnetic logarithms in \( \overline{B}\to {X}_s{l}^{+}{l}^{-} \), Nucl. Phys. B 740 (2006) 105 [hep-ph/0512066] [INSPIRE].
A.J. Buras, Asymptotic Freedom in Deep Inelastic Processes in the Leading Order and Beyond, Rev. Mod. Phys. 52 (1980) 199 [INSPIRE].
G. Buchalla, A.J. Buras and M.E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125 [hep-ph/9512380] [INSPIRE].
D.H. Adams and W. Lee, Renormalization group evolution for the ΔS = 1 effective Hamiltonian with N (f ) = 2 + 1, Phys. Rev. D 75 (2007) 074502 [hep-lat/0701014] [INSPIRE].
M. Gorbahn and U. Haisch, Effective Hamiltonian for non-leptonic |ΔF | = 1 decays at NNLO in QCD, Nucl. Phys. B 713 (2005) 291 [hep-ph/0411071] [INSPIRE].
A. Lenz, U. Nierste and G. Ostermaier, Penguin diagrams, charmless B decays and the missing charm puzzle, Phys. Rev. D 56 (1997) 7228 [hep-ph/9706501] [INSPIRE].
S. Herrlich and U. Nierste, Evanescent operators, scheme dependences and double insertions, Nucl. Phys. B 455 (1995) 39 [hep-ph/9412375] [INSPIRE].
Particle Data Group collaboration, K.A. Olive et al., Review of Particle Physics, Chin. Phys. C 38 (2014) 090001 [INSPIRE].
K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser, RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 133 (2000) 43 [hep-ph/0004189] [INSPIRE].
J. Charles et al., Current status of the Standard Model CKM fit and constraints on ΔF = 2 New Physics, Phys. Rev. D 91 (2015) 073007 [arXiv:1501.05013] [INSPIRE] and online updates on http://ckmfitter.in2p3.fr.
S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 74 (2014) 2890 [arXiv:1310.8555] [INSPIRE].
S. Gardner and G. Valencia, The Impact of |ΔI| = 5/2 transitions in K → ππ decays, Phys. Rev. D 62 (2000) 094024 [hep-ph/0006240] [INSPIRE].
V. Cirigliano, J.F. Donoghue and E. Golowich, K → ππ phenomenology in the presence of electromagnetism, Eur. Phys. J. C 18 (2000) 83 [hep-ph/0008290] [INSPIRE].
S. Alekhin, A. Djouadi and S. Moch, The top quark and Higgs boson masses and the stability of the electroweak vacuum, Phys. Lett. B 716 (2012) 214 [arXiv:1207.0980] [INSPIRE].
L.K. Gibbons et al., Measurement of the CP-violation parameter Re(ϵ ′ /ϵ), Phys. Rev. Lett. 70 (1993) 1203 [INSPIRE].
NA31 collaboration, G.D. Barr et al., A New measurement of direct CP-violation in the neutral kaon system, Phys. Lett. B 317 (1993) 233 [INSPIRE].
KTeV collaboration, A. Alavi-Harati et al., Observation of direct CP-violation in K S,L → ππ decays, Phys. Rev. Lett. 83 (1999) 22 [hep-ex/9905060] [INSPIRE].
NA48 collaboration, V. Fanti et al., A New measurement of direct CP-violation in two pion decays of the neutral kaon, Phys. Lett. B 465 (1999) 335 [hep-ex/9909022] [INSPIRE].
NA48 collaboration, J.R. Batley et al., A Precision measurement of direct CP-violation in the decay of neutral kaons into two pions, Phys. Lett. B 544 (2002) 97 [hep-ex/0208009] [INSPIRE].
KTeV collaboration, E. Abouzaid et al., Precise Measurements of Direct CP-violation, CPT Symmetry and Other Parameters in the Neutral Kaon System, Phys. Rev. D 83 (2011) 092001 [arXiv:1011.0127] [INSPIRE].
A.J. Buras and J.M. Gérard, Isospin Breaking Contributions to ϵ ′ /ϵ, Phys. Lett. B 192 (1987) 156 [INSPIRE].
W.A. Bardeen, A.J. Buras and J.M. Gérard, The K → ππ Decays in the Large-N Limit: Quark Evolution, Nucl. Phys. B 293 (1987) 787 [INSPIRE].
W.A. Bardeen, A.J. Buras and J.M. Gérard, A Consistent Analysis of the ΔI = 1/2 Rule for K Decays, Phys. Lett. B 192 (1987) 138 [INSPIRE].
A.J. Buras, J.-M. Gérard and W.A. Bardeen, Large-N Approach to Kaon Decays and Mixing 28 Years Later: ΔI = 1/2 Rule, \( {\widehat{B}}_K \) and ΔM K , Eur. Phys. J. C 74 (2014) 2871 [arXiv:1401.1385] [INSPIRE].
A.J. Buras and J.M. Gérard, Upper bounds on ε ′ /ε parameters B (1/2)6 and B (3/2)8 from large-N QCD and other news, JHEP 12 (2015) 008 [arXiv:1507.06326] [INSPIRE].
A.J. Buras and J.M. Gérard, Final State Interactions in K → ππ Decays: ΔI = 1/2 Rule vs. ε ′ /ε, arXiv:1603.05686 [INSPIRE].
E. Pallante and A. Pich, Strong enhancement of ϵ ′ /ϵ through final state interactions, Phys. Rev. Lett. 84 (2000) 2568 [hep-ph/9911233] [INSPIRE].
L. Lellouch and M. Lüscher, Weak transition matrix elements from finite volume correlation functions, Commun. Math. Phys. 219 (2001) 31 [hep-lat/0003023] [INSPIRE].
M. Constantinou et al., K → π matrix elements of the chromagnetic operator on the lattice, PoS(LATTICE2014)390 [arXiv:1412.1351] [INSPIRE].
S. Bertolini, M. Fabbrichesi and E. Gabrielli, The Relevance of the dipole Penguin operators in ϵ ′ /ϵ, Phys. Lett. B 327 (1994) 136 [hep-ph/9312266] [INSPIRE].
N.G. Deshpande, X.-G. He and S. Pakvasa, Gluon dipole penguin contributions to ϵ ′ /ϵ and CP-violation in hyperon decays in the Standard Model, Phys. Lett. B 326 (1994) 307 [hep-ph/9401330] [INSPIRE].
S. Bertolini, J.O. Eeg and M. Fabbrichesi, Studying ϵ ′ /ϵ in the chiral quark model: γ 5 -scheme independence and NLO hadronic matrix elements, Nucl. Phys. B 449 (1995) 197 [hep-ph/9409437] [INSPIRE].
R. Barbieri, R. Contino and A. Strumia, ϵ ′ from supersymmetry with nonuniversal A terms?, Nucl. Phys. B 578 (2000) 153 [hep-ph/9908255] [INSPIRE].
A.J. Buras, G. Colangelo, G. Isidori, A. Romanino and L. Silvestrini, Connections between ϵ′/ϵ and rare kaon decays in supersymmetry, Nucl. Phys. B 566 (2000) 3 [hep-ph/9908371] [INSPIRE].
A.J. Buras, F. De Fazio and J. Girrbach, ΔI = 1/2 rule, ε ′ /ε and \( K\to \pi \nu \overline{\nu} \) in Z ′(Z) and G ′ models with FCNC quark couplings, Eur. Phys. J. C 74 (2014) 2950 [arXiv:1404.3824] [INSPIRE].
A.J. Buras, D. Buttazzo and R. Knegjens, \( K\to \pi \nu \overline{\nu} \) and ε ′ /ε in simplified new physics models, JHEP 11 (2015) 166 [arXiv:1507.08672] [INSPIRE].
A.J. Buras, New physics patterns in ε ′ /ε and ε K with implications for rare kaon decays and ΔM K , JHEP 04 (2016) 071 [arXiv:1601.00005] [INSPIRE].
A.J. Buras, F. De Fazio and J. Girrbach-Noe, Z − Z ′ mixing and Z-mediated FCNCs in SU(3) C × SU(3) L × U(1) X models, JHEP 08 (2014) 039 [arXiv:1405.3850] [INSPIRE].
A.J. Buras and F. De Fazio, ε ′ /ε in 331 Models, JHEP 03 (2016) 010 [arXiv:1512.02869] [INSPIRE].
A.J. Buras and F. De Fazio, 331 Models Facing the Tensions in ΔF = 2 Processes with the Impact on ε ′ /ε, B s → μ + μ − and B → K ∗ μ + μ −, JHEP 08 (2016) 115 [arXiv:1604.02344] [INSPIRE].
M. Blanke, A.J. Buras and S. Recksiegel, Quark flavour observables in the Littlest Higgs model with T-parity after LHC Run 1, Eur. Phys. J. C 76 (2016) 182 [arXiv:1507.06316] [INSPIRE].
F. Goertz, J.F. Kamenik, A. Katz and M. Nardecchia, Indirect Constraints on the Scalar Di-Photon Resonance at the LHC, JHEP 05 (2016) 187 [arXiv:1512.08500] [INSPIRE].
M. Tanimoto and K. Yamamoto, Probing the SUSY with 10 TeV stop mass in rare decays and CP-violation of Kaon, arXiv:1603.07960 [INSPIRE].
T. Kitahara, U. Nierste and P. Tremper, Supersymmetric Explanation of CP-violation in K → ππ Decays, Phys. Rev. Lett. 117 (2016) 091802 [arXiv:1604.07400] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1607.06727
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kitahara, T., Nierste, U. & Tremper, P. Singularity-free next-to-leading order ΔS = 1 renormalization group evolution and ϵ ′K /ϵK in the Standard Model and beyond. J. High Energ. Phys. 2016, 78 (2016). https://doi.org/10.1007/JHEP12(2016)078
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2016)078