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
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