Journal of High Energy Physics

, 2016:83 | Cite as

A new look at the theory uncertainty of ϵ K

Open Access
Regular Article - Theoretical Physics


The observable ϵ K is sensitive to flavor violation at some of the highest scales. While its experimental uncertainty is at the half percent level, the theoretical one is in the ballpark of 15%. We explore the nontrivial dependence of the theory prediction and uncertainty on various conventions, like the phase of the kaon fields. In particular, we show how such a rephasing allows to make the short-distance contribution of the box diagram with two charm quarks, η cc , purely real. Our results allow to slightly reduce the total theoretical uncertainty of ϵ K , while increasing the relative impact of the imaginary part of the long distance contribution, underlining the need to compute it reliably. We also give updated bounds on the new physics operators that contribute to ϵK.


CP violation Kaon Physics 


Open Access

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  1. [1]
    T. Aushev et al., Physics at Super B Factory, arXiv:1002.5012 [INSPIRE].
  2. [2]
    LHCb collaboration, Implications of LHCb measurements and future prospects, Eur. Phys. J. C 73 (2013) 2373 [arXiv:1208.3355] [INSPIRE].
  3. [3]
    Particle Data Group collaboration, K.A. Olive et al., Review of Particle Physics, Chin. Phys. C 38 (2014) 090001 [INSPIRE].
  4. [4]
    J. Brod and M. Gorbahn, Next-to-Next-to-Leading-Order Charm-Quark Contribution to the CP-violation Parameter ϵ K and ΔM K, Phys. Rev. Lett. 108 (2012) 121801 [arXiv:1108.2036] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    A. Höcker, H. Lacker, S. Laplace and F. Le Diberder, A New approach to a global fit of the CKM matrix, Eur. Phys. J. C 21 (2001) 225 [hep-ph/0104062] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    CKMfitter Group collaboration, J. Charles et al., CP violation and the CKM matrix: Assessing the impact of the asymmetric B factories, Eur. Phys. J. C 41 (2005) 1 [hep-ph/0406184] [INSPIRE] and online update at
  7. [7]
    M. Ciuchini et al., 2000 CKM triangle analysis: A Critical review with updated experimental inputs and theoretical parameters, JHEP 07 (2001) 013 [hep-ph/0012308] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    UTfit collaboration, M. Bona et al., The 2004 UTfit collaboration report on the status of the unitarity triangle in the standard model, JHEP 07 (2005) 028 [hep-ph/0501199] [INSPIRE] and online update at
  9. [9]
    D. Derkach and L. Silvestrini, private communication.Google Scholar
  10. [10]
    A.J. Buras and J. Girrbach, Stringent tests of constrained Minimal Flavor Violation through ΔF = 2 transitions, Eur. Phys. J. C 73 (2013) 2560 [arXiv:1304.6835] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    J. Brod and M. Gorbahn, ϵ K at Next-to-Next-to-Leading Order: The Charm-Top-Quark Contribution, Phys. Rev. D 82 (2010) 094026 [arXiv:1007.0684] [INSPIRE].ADSGoogle Scholar
  12. [12]
    A.J. Buras, M. Jamin and P.H. Weisz, Leading and Next-to-leading QCD Corrections to ϵ Parameter and \( B0\hbox{--} {\overline{B}}^0 \) Mixing in the Presence of a Heavy Top Quark, Nucl. Phys. B 347 (1990) 491 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    G.C. Branco, L. Lavoura and J.P. Silva, CP Violation, Int. Ser. Monogr. Phys. 103 (1999) 1 [INSPIRE].Google Scholar
  14. [14]
    K. Anikeev et al., B physics at the Tevatron: Run II and beyond, hep-ph/0201071 [INSPIRE].
  15. [15]
    L. Lavoura, On the phase of ϵ and NA31 bound on CPT violation, Mod. Phys. Lett. A 7 (1992) 1367 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    Y. Grossman, B. Kayser and Y. Nir, The role of the vacuum insertion approximation in calculating CP asymmetries in B decays, Phys. Lett. B 415 (1997) 90 [hep-ph/9708398] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    A.J. Buras, Weak Hamiltonian, CP-violation and rare decays, hep-ph/9806471 [INSPIRE].
  18. [18]
    A.J. Buras, D. Guadagnoli and G. Isidori, On ϵ K Beyond Lowest Order in the Operator Product Expansion, Phys. Lett. B 688 (2010) 309 [arXiv:1002.3612] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A.J. Buras and D. Guadagnoli, Correlations among new CP-violating effects in ΔF = 2 observables, Phys. Rev. D 78 (2008) 033005 [arXiv:0805.3887] [INSPIRE].ADSGoogle Scholar
  20. [20]
    T. Blum et al., Kππ ΔI = 3/2 decay amplitude in the continuum limit, Phys. Rev. D 91 (2015) 074502 [arXiv:1502.00263] [INSPIRE].ADSGoogle Scholar
  21. [21]
    RBC and UKQCD collaborations, Z. Bai et al., Standard Model Prediction for Direct CP-violation in Kππ Decay, Phys. Rev. Lett. 115 (2015) 212001 [arXiv:1505.07863] [INSPIRE].
  22. [22]
    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].ADSCrossRefGoogle Scholar
  23. [23]
    Y. Nir, CP violation, Conf. Proc. C 9207131 (1992) 81 [INSPIRE] and online pdf version at
  24. [24]
    E.A. Andriyash, G.G. Ovanesyan and M.I. Vysotsky, Difference of \( \tilde{\epsilon} \) and ϵ in fitting the parameters of CKM matrix, Phys. Lett. B 599 (2004) 253 [hep-ph/0310314] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    E.A. Andriyash, G.G. Ovanesyan and M.I. Vysotsky, The Value of B K from the experimental data on CP-violation in K-mesons and up-to-date values of CKM matrix parameters, Phys. Atom. Nucl. 69 (2006) 286 [hep-ph/0502111] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    Z. Bai, K LK S mass difference computed with a 171 MeV pion mass, PoS(LATTICE2014)368 [arXiv:1411.3210] [INSPIRE].
  27. [27]
    Q. Liu, Kaon to two pions decays from lattice QCD: ΔI = 1/2 rule and CP violation, Ph.D. Thesis, Columbia University Academic Commons (2012) and online at
  28. [28]
    S. Alekhin, J. Blumlein and S. Moch, The ABM parton distributions tuned to LHC data, Phys. Rev. D 89 (2014) 054028 [arXiv:1310.3059] [INSPIRE].ADSGoogle Scholar
  29. [29]
    FLAG Working Group, S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 74 (2014) 2890 [arXiv:1310.8555] [INSPIRE] and online update at
  30. [30]
    S. Descotes-Genon, private communication.Google Scholar
  31. [31]
    Fermilab Lattice and MILC collaborations, A. Bazavov et al., B (s)0 -mixing matrix elements from lattice QCD for the Standard Model and beyond, Phys. Rev. D 93 (2016) 113016 [arXiv:1602.03560] [INSPIRE].
  32. [32]
    SWME collaboration, J.A. Bailey, Y.-C. Jang, W. Lee and S. Park, Standard Model evaluation of ε K using lattice QCD inputs for \( {\widehat{B}}_K \) and V cb, Phys. Rev. D 92 (2015) 034510 [arXiv:1503.05388] [INSPIRE].
  33. [33]
    RBC and UKQCD collaborations, N.H. Christ, T. Izubuchi, C.T. Sachrajda, A. Soni and J. Yu, Long distance contribution to the K LK S mass difference, Phys. Rev. D 88 (2013) 014508 [arXiv:1212.5931] [INSPIRE].
  34. [34]
    Z. Bai, N.H. Christ, T. Izubuchi, C.T. Sachrajda, A. Soni and J. Yu, K LK S Mass Difference from Lattice QCD, Phys. Rev. Lett. 113 (2014) 112003 [arXiv:1406.0916] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    J. Charles, S. Descotes-Genon, Z. Ligeti, S. Monteil, M. Papucci and K. Trabelsi, Future sensitivity to new physics in B d , B s and K mixings, Phys. Rev. D 89 (2014) 033016 [arXiv:1309.2293] [INSPIRE].ADSGoogle Scholar
  36. [36]
    RBC and UKQCD collaborations, T. Blum et al., Domain wall QCD with physical quark masses, Phys. Rev. D 93 (2016) 074505 [arXiv:1411.7017] [INSPIRE].
  37. [37]
    ETM collaboration, N. Carrasco et al., ΔS = 2 and ΔC = 2 bag parameters in the standard model and beyond from N f = 2 + 1 + 1 twisted-mass lattice QCD, Phys. Rev. D 92 (2015) 034516 [arXiv:1505.06639] [INSPIRE].
  38. [38]
    SWME collaboration, B.J. Choi et al., Kaon BSM B-parameters using improved staggered fermions from N f = 2 + 1 unquenched QCD, Phys. Rev. D 93 (2016) 014511 [arXiv:1509.00592] [INSPIRE].
  39. [39]
    RBC and UKQCD collaborations, P.A. Boyle, N. Garron and R.J. Hudspith, Neutral kaon mixing beyond the standard model with n f = 2 + 1 chiral fermions, Phys. Rev. D 86 (2012) 054028 [arXiv:1206.5737] [INSPIRE].
  40. [40]
    SWME collaboration, J. Leem et al., Calculation of BSM Kaon B-parameters using Staggered Quarks, PoS(LATTICE2014)370 [arXiv:1411.1501] [INSPIRE].
  41. [41]
    R.J. Hudspith, N. Garron and A.T. Lytle, Neutral Kaon mixing beyond the Standard Model, arXiv:1512.05398 [INSPIRE].
  42. [42]
    R. Babich, N. Garron, C. Hölbling, J. Howard, L. Lellouch and C. Rebbi, \( {K}^0-{\overline{K}}^0 \) mixing beyond the standard model and CP-violating electroweak penguins in quenched QCD with exact chiral symmetry, Phys. Rev. D 74 (2006) 073009 [hep-lat/0605016] [INSPIRE].ADSGoogle Scholar
  43. [43]
    CP-PACS collaboration, Y. Nakamura et al., Kaon B-parameters for Generic ΔS = 2 Four-Quark Operators in Quenched Domain Wall QCD, PoS(LAT2006)089 [hep-lat/0610075] [INSPIRE].
  44. [44]
    J.A. Bagger, K.T. Matchev and R.-J. Zhang, QCD corrections to flavor changing neutral currents in the supersymmetric standard model, Phys. Lett. B 412 (1997) 77 [hep-ph/9707225] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    M. Ciuchini, E. Franco, V. Lubicz, G. Martinelli, I. Scimemi and L. Silvestrini, Next-to-leading order QCD corrections to ΔF = 2 effective Hamiltonians, Nucl. Phys. B 523 (1998) 501 [hep-ph/9711402] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    ETM collaboration, V. Bertone et al., Kaon Mixing Beyond the SM from N f = 2 tmQCD and model independent constraints from the UTA, JHEP 03 (2013) 089 [Erratum ibid. 07 (2013) 143] [arXiv:1207.1287] [INSPIRE].
  47. [47]
    F. Mescia and J. Virto, Natural SUSY and Kaon Mixing in view of recent results from Lattice QCD, Phys. Rev. D 86 (2012) 095004 [arXiv:1208.0534] [INSPIRE].ADSGoogle Scholar
  48. [48]
    R. Contino, T. Kramer, M. Son and R. Sundrum, Warped/composite phenomenology simplified, JHEP 05 (2007) 074 [hep-ph/0612180] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    C. Csáki, A. Falkowski and A. Weiler, The Flavor of the Composite Pseudo-Goldstone Higgs, JHEP 09 (2008) 008 [arXiv:0804.1954] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    R. Barbieri, D. Buttazzo, F. Sala, D.M. Straub and A. Tesi, A 125 GeV composite Higgs boson versus flavour and electroweak precision tests, JHEP 05 (2013) 069 [arXiv:1211.5085] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    M. Redi and A. Weiler, Flavor and CP Invariant Composite Higgs Models, JHEP 11 (2011) 108 [arXiv:1106.6357] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  52. [52]
    R. Barbieri, D. Buttazzo, F. Sala and D.M. Straub, Flavour physics from an approximate U(2)3 symmetry, JHEP 07 (2012) 181 [arXiv:1203.4218] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    M. Redi, Composite MFV and Beyond, Eur. Phys. J. C 72 (2012) 2030 [arXiv:1203.4220] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    R. Barbieri, D. Buttazzo, F. Sala and D.M. Straub, Flavour physics and flavour symmetries after the first LHC phase, JHEP 05 (2014) 105 [arXiv:1402.6677] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    R.S. Chivukula and H. Georgi, Composite Technicolor Standard Model, Phys. Lett. B 188 (1987) 99 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    L.J. Hall and L. Randall, Weak scale effective supersymmetry, Phys. Rev. Lett. 65 (1990) 2939 [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    G. D’Ambrosio, G.F. Giudice, G. Isidori and A. Strumia, Minimal flavor violation: An Effective field theory approach, Nucl. Phys. B 645 (2002) 155 [hep-ph/0207036] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    R. Barbieri, G. Isidori, J. Jones-Perez, P. Lodone and D.M. Straub, U(2) and Minimal Flavour Violation in Supersymmetry, Eur. Phys. J. C 71 (2011) 1725 [arXiv:1105.2296] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    R. Barbieri, D. Buttazzo, F. Sala and D.M. Straub, Less Minimal Flavour Violation, JHEP 10 (2012) 040 [arXiv:1206.1327] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Ernest Orlando Lawrence Berkeley National LaboratoryUniversity of CaliforniaBerkeleyU.S.A.
  2. 2.LPTHE, CNRS, UMR 7589ParisFrance

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