Neutral kaon mixing beyond the Standard Model with nf = 2 + 1 chiral fermions. Part 1: bare matrix elements and physical results

  • The RBC/UKQCD collaboration
  • Nicolas GarronEmail author
  • Renwick J. Hudspith
  • Andrew T. Lytle
Open Access
Regular Article - Theoretical Physics


We compute the hadronic matrix elements of the four-quark operators relevant for \( {K}^0-{\overline{K}}^0 \) mixing beyond the Standard Model. Our results are from lattice QCD simulations with n f = 2 + 1 flavours of domain-wall fermion, which exhibit continuum-like chiral-flavour symmetry. The simulations are performed at two different values of the lattice spacing (a ∼ 0.08 and a ∼ 0.11 fm) and with lightest unitary pion mass ∼ 300 MeV. For the first time, the full set of relevant four-quark operators is renormalised non-perturbatively through RI-SMOM schemes; a detailed description of the renormalisation procedure is presented in a companion paper. We argue that the intermediate renormalisation scheme is responsible for the discrepancies found by different collaborations. We also study different normalisations and determine the matrix elements of the relevant four-quark operators with a precision of ∼ 5% or better.


Beyond Standard Model CP violation Kaon Physics Lattice QCD 


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.


  1. [1]
    J.H. Christenson, J.W. Cronin, V.L. Fitch and R. Turlay, Evidence for the 2π Decay of the K 20 Meson, Phys. Rev. Lett. 13 (1964) 138 [INSPIRE].
  2. [2]
    S.L. Glashow, J. Iliopoulos and L. Maiani, Weak Interactions with Lepton-Hadron Symmetry, Phys. Rev. D 2 (1970) 1285 [INSPIRE].ADSGoogle Scholar
  3. [3]
    J.D. Bjorken and S.L. Glashow, Elementary Particles and SU(4), Phys. Lett. 11 (1964) 255 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    Y. Aoki et al., Continuum Limit of B K from 2 + 1 Flavor Domain Wall QCD, Phys. Rev. D 84 (2011) 014503 [arXiv:1012.4178] [INSPIRE].ADSGoogle Scholar
  5. [5]
    RBC/UKQCD collaboration, T. Blum et al., Domain wall QCD with physical quark masses, Phys. Rev. D 93 (2016) 074505 [arXiv:1411.7017] [INSPIRE].
  6. [6]
    S. Dürr et al., Precision computation of the kaon bag parameter, Phys. Lett. B 705 (2011) 477 [arXiv:1106.3230] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    ETM collaboration, M. Constantinou et al., B K -parameter from N f = 2 twisted mass lattice QCD, Phys. Rev. D 83 (2011) 014505 [arXiv:1009.5606] [INSPIRE].
  8. [8]
    Z. Ligeti and F. Sala, A new look at the theory uncertainty of ϵ K , JHEP 09 (2016) 083 [arXiv:1602.08494] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    S. Bertolini, A. Maiezza and F. Nesti, Present and Future K and B Meson Mixing Constraints on TeV Scale Left-Right Symmetry, Phys. Rev. D 89 (2014) 095028 [arXiv:1403.7112] [INSPIRE].ADSGoogle Scholar
  10. [10]
    A.J. Buras and J. Girrbach, Towards the Identification of New Physics through Quark Flavour Violating Processes, Rept. Prog. Phys. 77 (2014) 086201 [arXiv:1306.3775] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    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
  12. [12]
    F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, A Complete analysis of FCNC and CP constraints in general SUSY extensions of the standard model, Nucl. Phys. B 477 (1996) 321 [hep-ph/9604387] [INSPIRE].
  13. [13]
    C.R. Allton et al., B-parameters for ΔS = 2 supersymmetric operators, Phys. Lett. B 453 (1999) 30 [hep-lat/9806016] [INSPIRE].
  14. [14]
    M. Ciuchini et al., ΔM K and ϵ K in SUSY at the next-to-leading order, JHEP 10 (1998) 008 [hep-ph/9808328] [INSPIRE].
  15. [15]
    RBC/UKQCD collaboration, 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].
  16. [16]
    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].
  17. [17]
    SWME collaboration, T. Bae et al., Neutral kaon mixing from new physics: matrix elements in N f = 2 + 1 lattice QCD, Phys. Rev. D 88 (2013) 071503 [arXiv:1309.2040] [INSPIRE].
  18. [18]
    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].
  19. [19]
    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].
  20. [20]
    R.J. Hudspith, N. Garron and A.T. Lytle, Neutral Kaon mixing beyond the Standard Model, arXiv:1512.05398 [INSPIRE].
  21. [21]
    N. Garron, CP violation and Kaon weak matrix elements from Lattice QCD, PoS(CD15)034 [arXiv:1512.02440] [INSPIRE].
  22. [22]
    G. Martinelli, C. Pittori, C.T. Sachrajda, M. Testa and A. Vladikas, A General method for nonperturbative renormalization of lattice operators, Nucl. Phys. B 445 (1995) 81 [hep-lat/9411010] [INSPIRE].
  23. [23]
    RBC/UKQCD collaboration, N. Garron, R.J. Hudspith and A.T. Lytle, Neutral Kaon mixing beyond the Standard Model with n f = 2 + 1 Chiral Fermions. Part II: Non-Perturbative Renormalisation, in preparation.Google Scholar
  24. [24]
    A. Donini, V. Giménez, L. Giusti and G. Martinelli, Renormalization group invariant matrix elements of ΔS = 2 and ΔI = 3/2 four fermion operators without quark masses, Phys. Lett. B 470 (1999) 233 [hep-lat/9910017] [INSPIRE].
  25. [25]
    D. Becirevic and G. Villadoro, Remarks on the hadronic matrix elements relevant to the SUSY \( {K}^0-{\overline{K}}^0 \) mixing amplitude, Phys. Rev. D 70 (2004) 094036 [hep-lat/0408029] [INSPIRE].
  26. [26]
    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].
  27. [27]
    N. Garron, P.A. Boyle, R.J. Hudspith and A.T. Lytle, Weak Matrix Elements of Beyond the Standard Model Δs = 2 four-quark operators from n f = 2 + 1 Domain-Wall fermions, PoS(LATTICE 2012)108 [arXiv:1212.2871] [INSPIRE].
  28. [28]
    J.A. Bailey, H.-J. Kim, W. Lee and S.R. Sharpe, Kaon mixing matrix elements from beyond-the-Standard-Model operators in staggered chiral perturbation theory, Phys. Rev. D 85 (2012) 074507 [arXiv:1202.1570] [INSPIRE].ADSGoogle Scholar
  29. [29]
    Y. Iwasaki, Renormalization Group Analysis of Lattice Theories and Improved Lattice Action: Two-Dimensional Nonlinear O(N ) σ-model, Nucl. Phys. B 258 (1985) 141 [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    CP-PACS collaboration, M. Okamoto et al., Equation of state for pure SU(3) gauge theory with renormalization group improved action, Phys. Rev. D 60 (1999) 094510 [hep-lat/9905005] [INSPIRE].
  31. [31]
    Y. Shamir, Chiral fermions from lattice boundaries, Nucl. Phys. B 406 (1993) 90 [hep-lat/9303005] [INSPIRE].
  32. [32]
    RBC/UKQCD collaboration, R. Arthur et al., Domain Wall QCD with Near-Physical Pions, Phys. Rev. D 87 (2013) 094514 [arXiv:1208.4412] [INSPIRE].
  33. [33]
    RBC/UKQCD collaboration, R.J. Hudspith, Fourier Accelerated Conjugate Gradient Lattice Gauge Fixing, Comput. Phys. Commun. 187 (2015) 115 [arXiv:1405.5812] [INSPIRE].
  34. [34]
    RIKEN-BNL-Columbia-KEK collaboration, S. Sasaki, K. Orginos, S. Ohta and T. Blum, Nucleon axial charge from quenched lattice QCD with domain wall fermions, Phys. Rev. D 68 (2003) 054509 [hep-lat/0306007] [INSPIRE].
  35. [35]
    C. Sturm, Y. Aoki, N.H. Christ, T. Izubuchi, C.T. Sachrajda and A. Soni, Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point, Phys. Rev. D 80 (2009) 014501 [arXiv:0901.2599] [INSPIRE].ADSGoogle Scholar
  36. [36]
    M. Gockeler et al., Nonperturbative renormalization of composite operators in lattice QCD, Nucl. Phys. B 544 (1999) 699 [hep-lat/9807044] [INSPIRE].
  37. [37]
    P.F. Bedaque, Aharonov-Bohm effect and nucleon nucleon phase shifts on the lattice, Phys. Lett. B 593 (2004) 82 [nucl-th/0402051] [INSPIRE].
  38. [38]
    G.M. de Divitiis, R. Petronzio and N. Tantalo, On the discretization of physical momenta in lattice QCD, Phys. Lett. B 595 (2004) 408 [hep-lat/0405002] [INSPIRE].
  39. [39]
    C.T. Sachrajda and G. Villadoro, Twisted boundary conditions in lattice simulations, Phys. Lett. B 609 (2005) 73 [hep-lat/0411033] [INSPIRE].
  40. [40]
    RBC/UKQCD collaboration, R. Arthur, P.A. Boyle, N. Garron, C. Kelly and A.T. Lytle, Opening the Rome-Southampton window for operator mixing matrices, Phys. Rev. D 85 (2012) 014501 [arXiv:1109.1223] [INSPIRE].
  41. [41]
    RBC/UKQCD collaboration, A.T. Lytle, P.A. Boyle, N. Garron, R.J. Hudspith and C.T. Sachrajda, Kaon Mixing Beyond the Standard Model, PoS(LATTICE 2013)400 [arXiv:1311.0322] [INSPIRE].
  42. [42]
    A.J. Buras, M. Misiak and J. Urban, Two loop QCD anomalous dimensions of flavor changing four quark operators within and beyond the standard model, Nucl. Phys. B 586 (2000) 397 [hep-ph/0005183] [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • The RBC/UKQCD collaboration
  • Nicolas Garron
    • 1
    Email author
  • Renwick J. Hudspith
    • 2
  • Andrew T. Lytle
    • 3
  1. 1.Theoretical Physics Division, Department of Mathematical SciencesUniversity of LiverpoolLiverpoolU.K.
  2. 2.Department of Physics and AstronomyYork UniversityTorontoCanada
  3. 3.SUPA, School of Physics and AstronomyUniversity of GlasgowGlasgowU.K.

Personalised recommendations