Journal of High Energy Physics

, 2018:59 | Cite as

Singlet Dirac fermion dark matter with mediators at loop

  • Junji Hisano
  • Ryo NagaiEmail author
  • Natsumi Nagata
Open Access
Regular Article - Theoretical Physics


We study the phenomenology of singlet Dirac fermion dark matter in the simplified models where the dark matter interacts with the Standard Model particles at loop-level with the help of either colored or non-colored mediators. We especially focus on the implications of non-zero CP phases in the dark sector, which induce the electric dipole moments of the Dirac fermion dark matter as well as those of electron and nucleon. It is then found that the dark matter direct detection searches and the measurements of the electric dipole moments are able to test the singlet Dirac fermion dark matter scenario in the forthcoming experiments.


Beyond Standard Model CP violation 


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]
    G. Arcadi et al., The waning of the WIMP? A review of models, searches and constraints, Eur. Phys. J. C 78 (2018) 203 [arXiv:1703.07364] [INSPIRE].
  2. [2]
    V. Silveira and A. Zee, Scalar phantoms, Phys. Lett. 161B (1985) 136 [INSPIRE].
  3. [3]
    J. McDonald, Gauge singlet scalars as cold dark matter, Phys. Rev. D 50 (1994) 3637 [hep-ph/0702143] [INSPIRE].
  4. [4]
    C.P. Burgess, M. Pospelov and T. ter Veldhuis, The minimal model of nonbaryonic dark matter: a singlet scalar, Nucl. Phys. B 619 (2001) 709 [hep-ph/0011335] [INSPIRE].
  5. [5]
    H. Davoudiasl, R. Kitano, T. Li and H. Murayama, The new minimal standard model, Phys. Lett. B 609 (2005) 117 [hep-ph/0405097] [INSPIRE].
  6. [6]
    P. Hut and K.A. Olive, A cosmological upper limit on the mass of heavy neutrinos, Phys. Lett. 87B (1979) 144 [INSPIRE].CrossRefGoogle Scholar
  7. [7]
    S. Nussinov, Technocosmology: could a technibaryon excess provide a ‘natural’ missing mass candidate?, Phys. Lett. 165B (1985) 55 [INSPIRE].CrossRefGoogle Scholar
  8. [8]
    S.M. Barr, R.S. Chivukula and E. Farhi, Electroweak fermion number violation and the production of stable particles in the early universe, Phys. Lett. B 241 (1990) 387 [INSPIRE].
  9. [9]
    S.M. Barr, Baryogenesis, sphalerons and the cogeneration of dark matter, Phys. Rev. D 44 (1991) 3062 [INSPIRE].
  10. [10]
    D.B. Kaplan, A single explanation for both the baryon and dark matter densities, Phys. Rev. Lett. 68 (1992) 741 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Dodelson, B.R. Greene and L.M. Widrow, Baryogenesis, dark matter and the width of the Z, Nucl. Phys. B 372 (1992) 467 [INSPIRE].
  12. [12]
    V.A. Kuzmin, A simultaneous solution to baryogenesis and dark matter problems, Phys. Part. Nucl. 29 (1998) 257 [hep-ph/9701269] [INSPIRE].
  13. [13]
    D. Hooper, J. March-Russell and S.M. West, Asymmetric sneutrino dark matter and the Ω(b)/Ω(DM ) puzzle, Phys. Lett. B 605 (2005) 228 [hep-ph/0410114] [INSPIRE].
  14. [14]
    R. Kitano and I. Low, Dark matter from baryon asymmetry, Phys. Rev. D 71 (2005) 023510 [hep-ph/0411133] [INSPIRE].
  15. [15]
    G.R. Farrar and G. Zaharijas, Dark matter and the baryon asymmetry, Phys. Rev. Lett. 96 (2006) 041302 [hep-ph/0510079] [INSPIRE].
  16. [16]
    D.E. Kaplan, M.A. Luty and K.M. Zurek, Asymmetric dark matter, Phys. Rev. D 79 (2009) 115016 [arXiv:0901.4117] [INSPIRE].
  17. [17]
    N. Nagata, K.A. Olive and J. Zheng, Asymmetric dark matter models in SO(10), JCAP 02 (2017) 016 [arXiv:1611.04693] [INSPIRE].
  18. [18]
    Y.G. Kim, K.Y. Lee and S. Shin, Singlet fermionic dark matter, JHEP 05 (2008) 100 [arXiv:0803.2932] [INSPIRE].Google Scholar
  19. [19]
    S. Kanemura, O. Seto and T. Shimomura, Masses of dark matter and neutrino from TeV scale spontaneous U(1)BL breaking, Phys. Rev. D 84 (2011) 016004 [arXiv:1101.5713] [INSPIRE].
  20. [20]
    M. Lindner, D. Schmidt and T. Schwetz, Dark matter and neutrino masses from global U(1)BL symmetry breaking, Phys. Lett. B 705 (2011) 324 [arXiv:1105.4626] [INSPIRE].
  21. [21]
    S. Kanemura, T. Nabeshima and H. Sugiyama, TeV-scale seesaw with loop-induced Dirac mass term and dark matter from U(1)BL gauge symmetry breaking, Phys. Rev. D 85 (2012) 033004 [arXiv:1111.0599] [INSPIRE].
  22. [22]
    S. Baek, P. Ko and W.-I. Park, Search for the Higgs portal to a singlet fermionic dark matter at the LHC, JHEP 02 (2012) 047 [arXiv:1112.1847] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    P. Ko, N. Nagata and Y. Tang, Hidden charged dark matter and chiral dark radiation, Phys. Lett. B 773 (2017) 513 [arXiv:1706.05605] [INSPIRE].
  24. [24]
    N. Okada and O. Seto, Higgs portal dark matter in the minimal gauged U(1)BL model, Phys. Rev. D 82 (2010) 023507 [arXiv:1002.2525] [INSPIRE].
  25. [25]
    K. Cheung and T.-C. Yuan, Hidden fermion as milli-charged dark matter in Stueckelberg Z model, JHEP 03 (2007) 120 [hep-ph/0701107] [INSPIRE].
  26. [26]
    Y. Mambrini, The ZZ kinetic mixing in the light of the recent direct and indirect dark matter searches, JCAP 07 (2011) 009 [arXiv:1104.4799] [INSPIRE].
  27. [27]
    H. An, X. Ji and L.-T. Wang, Light dark matter and Z dark force at colliders, JHEP 07 (2012) 182 [arXiv:1202.2894] [INSPIRE].
  28. [28]
    V. Barger, D. Marfatia and A. Peterson, LHC and dark matter signals of Z bosons, Phys. Rev. D 87 (2013) 015026 [arXiv:1206.6649] [INSPIRE].
  29. [29]
    X. Chu, Y. Mambrini, J. Quevillon and B. Zaldivar, Thermal and non-thermal production of dark matter via Z -portal(s), JCAP 01 (2014) 034 [arXiv:1306.4677] [INSPIRE].
  30. [30]
    G. Arcadi, Y. Mambrini, M.H.G. Tytgat and B. Zaldivar, Invisible Z and dark matter: LHC vs LUX constraints, JHEP 03 (2014) 134 [arXiv:1401.0221] [INSPIRE].
  31. [31]
    O. Lebedev and Y. Mambrini, Axial dark matter: the case for an invisible Z , Phys. Lett. B 734 (2014) 350 [arXiv:1403.4837] [INSPIRE].
  32. [32]
    Y. Mambrini, S. Profumo and F.S. Queiroz, Dark matter and global symmetries, Phys. Lett. B 760 (2016) 807 [arXiv:1508.06635] [INSPIRE].
  33. [33]
    P. Agrawal, S. Blanchet, Z. Chacko and C. Kilic, Flavored dark matter and its implications for direct detection and colliders, Phys. Rev. D 86 (2012) 055002 [arXiv:1109.3516] [INSPIRE].
  34. [34]
    N. Weiner and I. Yavin, UV completions of magnetic inelastic and Rayleigh dark matter for the Fermi Line(s), Phys. Rev. D 87 (2013) 023523 [arXiv:1209.1093] [INSPIRE].
  35. [35]
    Y. Bai and J. Berger, Fermion portal dark matter, JHEP 11 (2013) 171 [arXiv:1308.0612] [INSPIRE].CrossRefGoogle Scholar
  36. [36]
    A. DiFranzo, K.I. Nagao, A. Rajaraman and T.M.P. Tait, Simplified models for dark matter interacting with quarks, JHEP 11 (2013) 014 [Erratum ibid. 1401 (2014) 162] [arXiv:1308.2679] [INSPIRE].
  37. [37]
    J. Kopp, L. Michaels and J. Smirnov, Loopy constraints on leptophilic dark matter and internal bremsstrahlung, JCAP 04 (2014) 022 [arXiv:1401.6457] [INSPIRE].CrossRefGoogle Scholar
  38. [38]
    Y. Bai and J. Berger, Lepton portal dark matter, JHEP 08 (2014) 153 [arXiv:1402.6696] [INSPIRE].CrossRefGoogle Scholar
  39. [39]
    S. Chang, R. Edezhath, J. Hutchinson and M. Luty, Leptophilic effective WIMPs, Phys. Rev. D 90 (2014) 015011 [arXiv:1402.7358] [INSPIRE].
  40. [40]
    P. Agrawal, Z. Chacko and C.B. Verhaaren, Leptophilic dark matter and the anomalous magnetic moment of the muon, JHEP 08 (2014) 147 [arXiv:1402.7369] [INSPIRE].CrossRefGoogle Scholar
  41. [41]
    P. Agrawal, B. Batell, D. Hooper and T. Lin, Flavored dark matter and the galactic center gamma-ray excess, Phys. Rev. D 90 (2014) 063512 [arXiv:1404.1373] [INSPIRE].
  42. [42]
    A. Hamze et al., Lepton-flavored asymmetric dark matter and interference in direct detection, Phys. Rev. D 91 (2015) 035009 [arXiv:1410.3030] [INSPIRE].
  43. [43]
    Z.-H. Yu, X.-J. Bi, Q.-S. Yan and P.-F. Yin, Tau portal dark matter models at the LHC, Phys. Rev. D 91 (2015) 035008 [arXiv:1410.3347] [INSPIRE].
  44. [44]
    C. Kilic, M.D. Klimek and J.-H. Yu, Signatures of top flavored dark matter, Phys. Rev. D 91 (2015) 054036 [arXiv:1501.02202] [INSPIRE].
  45. [45]
    R. Primulando, E. Salvioni and Y. Tsai, The dark penguin shines light at colliders, JHEP 07 (2015) 031 [arXiv:1503.04204] [INSPIRE].CrossRefGoogle Scholar
  46. [46]
    P. Ko, A. Natale, M. Park and H. Yokoya, Simplified DM models with the full SM gauge symmetry: the case of t-channel colored scalar mediators, JHEP 01 (2017) 086 [arXiv:1605.07058] [INSPIRE].CrossRefzbMATHGoogle Scholar
  47. [47]
    W. Chao, H.-K. Guo and H.-L. Li, Tau flavored dark matter and its impact on tau Yukawa coupling, JCAP 02 (2017) 002 [arXiv:1606.07174] [INSPIRE].CrossRefGoogle Scholar
  48. [48]
    A. Ibarra and S. Wild, Dirac dark matter with a charged mediator: a comprehensive one-loop analysis of the direct detection phenomenology, JCAP 05 (2015) 047 [arXiv:1503.03382] [INSPIRE].CrossRefGoogle Scholar
  49. [49]
    M.J. Baker and A. Thamm, Leptonic WIMP coannihilation and the current dark matter search strategy, JHEP 10 (2018) 187 [arXiv:1806.07896] [INSPIRE].CrossRefGoogle Scholar
  50. [50]
    J. Herrero-Garcia, E. Molinaro and M.A. Schmidt, Dark matter direct detection of a fermionic singlet at one loop, Eur. Phys. J. C 78 (2018) 471 [arXiv:1803.05660] [INSPIRE].
  51. [51]
    T. Banks, J.-F. Fortin and S. Thomas, Direct detection of dark matter electromagnetic dipole moments, arXiv:1007.5515 [INSPIRE].
  52. [52]
    S. Weinberg, Larger Higgs exchange terms in the neutron electric dipole moment, Phys. Rev. Lett. 63 (1989) 2333 [INSPIRE].CrossRefGoogle Scholar
  53. [53]
    F.J. Botella, G.C. Branco and M. Nebot, The hunt for new physics in the flavour sector with up vector-like quarks, JHEP 12 (2012) 040 [arXiv:1207.4440] [INSPIRE].CrossRefGoogle Scholar
  54. [54]
    A.K. Alok, S. Banerjee, D. Kumar and S. Uma Sankar, Flavor signatures of isosinglet vector-like down quark model, Nucl. Phys. B 906 (2016) 321 [arXiv:1402.1023] [INSPIRE].
  55. [55]
    C. Bobeth, A.J. Buras, A. Celis and M. Jung, Patterns of flavour violation in models with vector-like quarks, JHEP 04 (2017) 079 [arXiv:1609.04783] [INSPIRE].CrossRefGoogle Scholar
  56. [56]
    T. Morozumi, Y. Shimizu, S. Takahashi and H. Umeeda, Effective theory analysis for vector-like quark model, PTEP 2018 (2018) 043B10 [arXiv:1801.05268] [INSPIRE].
  57. [57]
    Planck collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 [INSPIRE].
  58. [58]
    K. Griest and D. Seckel, Three exceptions in the calculation of relic abundances, Phys. Rev. D 43 (1991) 3191 [INSPIRE].
  59. [59]
    J. Hisano, S. Matsumoto and M.M. Nojiri, Explosive dark matter annihilation, Phys. Rev. Lett. 92 (2004) 031303 [hep-ph/0307216] [INSPIRE].
  60. [60]
    J. Hisano, S. Matsumoto, M.M. Nojiri and O. Saito, Non-perturbative effect on dark matter annihilation and gamma ray signature from galactic center, Phys. Rev. D 71 (2005) 063528 [hep-ph/0412403] [INSPIRE].
  61. [61]
    J. Hisano, S. Matsumoto, M. Nagai, O. Saito and M. Senami, Non-perturbative effect on thermal relic abundance of dark matter, Phys. Lett. B 646 (2007) 34 [hep-ph/0610249] [INSPIRE].
  62. [62]
    J. Ellis, F. Luo and K.A. Olive, Gluino coannihilation revisited, JHEP 09 (2015) 127 [arXiv:1503.07142] [INSPIRE].CrossRefGoogle Scholar
  63. [63]
    S.P. Liew and F. Luo, Effects of QCD bound states on dark matter relic abundance, JHEP 02 (2017) 091 [arXiv:1611.08133] [INSPIRE].CrossRefzbMATHGoogle Scholar
  64. [64]
    ATLAS collaboration, Combination of the searches for pair-produced vector-like partners of the third generation quarks at \( \sqrt{s}=13 \) TeV with the ATLAS detector, ATLAS-CONF-2018-032 (2018).
  65. [65]
    CMS collaboration, Search for vector-like T and B quark pairs in final states with leptons at \( \sqrt{s}=13 \) TeV, JHEP 08 (2018) 177 [arXiv:1805.04758] [INSPIRE].
  66. [66]
    ATLAS collaboration, Search for pair production of a new heavy quark that decays into a W boson and a light quark in pp collisions at \( \sqrt{s}=8 \) TeV with the ATLAS detector, Phys. Rev. D 92 (2015) 112007 [arXiv:1509.04261] [INSPIRE].
  67. [67]
    CMS collaboration, Search for vectorlike light-flavor quark partners in proton-proton collisions at \( \sqrt{s}=8 \) TeV, Phys. Rev. D 97 (2018) 072008 [arXiv:1708.02510] [INSPIRE].
  68. [68]
    ATLAS collaboration, Search for squarks and gluinos in final states with jets and missing transverse momentum using 36 fb −1 of \( \sqrt{s}=13 \) TeV pp collision data with the ATLAS detector, Phys. Rev. D 97 (2018) 112001 [arXiv:1712.02332] [INSPIRE].
  69. [69]
    CMS collaboration, Search for supersymmetry in multijet events with missing transverse momentum in proton-proton collisions at 13 TeV, Phys. Rev. D 96 (2017) 032003 [arXiv:1704.07781] [INSPIRE].
  70. [70]
    J. Hisano, R. Nagai and N. Nagata, Effective theories for dark matter nucleon scattering, JHEP 05 (2015) 037 [arXiv:1502.02244] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    J. Hisano, K. Ishiwata and N. Nagata, QCD effects on direct detection of WINO dark matter, JHEP 06 (2015) 097 [arXiv:1504.00915] [INSPIRE].CrossRefGoogle Scholar
  72. [72]
    A. Crivellin, F. D’Eramo and M. Procura, New constraints on dark matter effective theories from standard model loops, Phys. Rev. Lett. 112 (2014) 191304 [arXiv:1402.1173] [INSPIRE].CrossRefGoogle Scholar
  73. [73]
    F. D’Eramo and M. Procura, Connecting dark matter UV complete models to direct detection rates via effective field theory, JHEP 04 (2015) 054 [arXiv:1411.3342] [INSPIRE].CrossRefGoogle Scholar
  74. [74]
    J. Brod, B. Grinstein, E. Stamou and J. Zupan, Weak mixing below the weak scale in dark-matter direct detection, JHEP 02 (2018) 174 [arXiv:1801.04240] [INSPIRE].CrossRefGoogle Scholar
  75. [75]
    F. Bishara, J. Brod, B. Grinstein and J. Zupan, Renormalization group effects in dark matter interactions, arXiv:1809.03506 [INSPIRE].
  76. [76]
    J. Ellis, N. Nagata and K.A. Olive, Uncertainties in WIMP dark matter scattering revisited, Eur. Phys. J. C 78 (2018) 569 [arXiv:1805.09795] [INSPIRE].
  77. [77]
    M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Remarks on Higgs boson interactions with nucleons, Phys. Lett. B 78 (1978) 443.Google Scholar
  78. [78]
    J.D. Lewin and P.F. Smith, Review of mathematics, numerical factors and corrections for dark matter experiments based on elastic nuclear recoil, Astropart. Phys. 6 (1996) 87 [INSPIRE].CrossRefGoogle Scholar
  79. [79]
    R.H. Helm, Inelastic and elastic scattering of 187 MeV electrons from selected even-even nuclei, Phys. Rev. 104 (1956) 1466 [INSPIRE].
  80. [80]
    XENON collaboration, E. Aprile et al., Dark matter search results from a one ton-year exposure of XENON1T, Phys. Rev. Lett. 121 (2018) 111302 [arXiv:1805.12562] [INSPIRE].
  81. [81]
    XENON collaboration, E. Aprile et al., Physics reach of the XENON1T dark matter experiment, JCAP 04 (2016) 027 [arXiv:1512.07501] [INSPIRE].
  82. [82]
    LUX-ZEPLIN collaboration, D.S. Akerib et al., Projected WIMP sensitivity of the LUX-ZEPLIN (LZ) dark matter experiment, arXiv:1802.06039 [INSPIRE].
  83. [83]
    T. Abe, J. Hisano and R. Nagai, Model independent evaluation of the Wilson coefficient of the Weinberg operator in QCD, JHEP 03 (2018) 175 [arXiv:1712.09503] [INSPIRE].CrossRefzbMATHGoogle Scholar
  84. [84]
    D.A. Demir, M. Pospelov and A. Ritz, Hadronic EDMs, the Weinberg operator and light gluinos, Phys. Rev. D 67 (2003) 015007 [hep-ph/0208257] [INSPIRE].
  85. [85]
    G. Degrassi, E. Franco, S. Marchetti and L. Silvestrini, QCD corrections to the electric dipole moment of the neutron in the MSSM, JHEP 11 (2005) 044 [hep-ph/0510137] [INSPIRE].
  86. [86]
    C.A. Baker et al., An Improved experimental limit on the electric dipole moment of the neutron, Phys. Rev. Lett. 97 (2006) 131801 [hep-ex/0602020] [INSPIRE].
  87. [87]
    A. Lehrach et al., Precursor experiments to search for permanent Electric Dipole Moments (EDMs) of protons and deuterons at COSY, arXiv:1201.5773 [INSPIRE].
  88. [88]
    ACME collaboration, J. Baron et al., Order of magnitude smaller limit on the electric dipole moment of the electron, Science 343 (2014) 269 [arXiv:1310.7534] [INSPIRE].
  89. [89]
    D.M. Kara et al., Measurement of the electron’s electric dipole moment using YbF molecules: methods and data analysis, New J. Phys. 14 (2012) 103051 [arXiv:1208.4507] [INSPIRE].CrossRefGoogle Scholar
  90. [90]
    D. Kawall, Searching for the electron EDM in a storage ring, J. Phys. Conf. Ser. 295 (2011) 012031 [INSPIRE].
  91. [91]
    S.M. Barr and A. Zee, Electric dipole moment of the electron and of the neutron, Phys. Rev. Lett. 65 (1990) 21 [Erratum ibid. 65 (1990) 2920] [INSPIRE].
  92. [92]
    V. Fock, Proper time in classical and quantum mechanics, Phys. Z. Sowjetunion 12 (1937) 404 [INSPIRE].zbMATHGoogle Scholar
  93. [93]
    J.S. Schwinger, Particles, sources and fields. Volume II, Frontiers in Physics, Westview Press, U.S.A. (1973).Google Scholar
  94. [94]
    C. Cronstrom, A simple and complete Lorentz covariant gauge condition, Phys. Lett. B 90 (1980) 267.Google Scholar
  95. [95]
    V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Calculations in external fields in quantum chromodynamics. Technical review, Fortsch. Phys. 32 (1984) 585 [INSPIRE].
  96. [96]
    J. Hisano, K. Ishiwata and N. Nagata, Gluon contribution to the dark matter direct detection, Phys. Rev. D 82 (2010) 115007 [arXiv:1007.2601] [INSPIRE].
  97. [97]
    L. Vecchi, WIMPs and un-naturalness, arXiv:1312.5695 [INSPIRE].
  98. [98]
    C. Savage, K. Freese and P. Gondolo, Annual modulation of dark matter in the presence of streams, Phys. Rev. D 74 (2006) 043531 [astro-ph/0607121] [INSPIRE].
  99. [99]
    C. McCabe, The astrophysical uncertainties of dark matter direct detection experiments, Phys. Rev. D 82 (2010) 023530 [arXiv:1005.0579] [INSPIRE].
  100. [100]
    V. Barger, W.-Y. Keung and D. Marfatia, Electromagnetic properties of dark matter: dipole moments and charge form factor, Phys. Lett. B 696 (2011) 74 [arXiv:1007.4345] [INSPIRE].
  101. [101]
    XENON100 collaboration, E. Aprile et al., Likelihood approach to the first dark matter results from XENON100, Phys. Rev. D 84 (2011) 052003 [arXiv:1103.0303] [INSPIRE].
  102. [102]
    XENON collaboration, E. Aprile et al., Lowering the radioactivity of the photomultiplier tubes for the XENON1T dark matter experiment, Eur. Phys. J. C 75 (2015) 546 [arXiv:1503.07698] [INSPIRE].
  103. [103]
    P. Barrow et al., Qualification tests of the R11410-21 photomultiplier tubes for the XENON1T detector, 2017 JINST 12 P01024 [arXiv:1609.01654] [INSPIRE].
  104. [104]
    XENON100 collaboration, E. Aprile et al., Dark matter results from 100 live days of XENON100 data, Phys. Rev. Lett. 107 (2011) 131302 [arXiv:1104.2549] [INSPIRE].
  105. [105]
    M. Cirelli, E. Del Nobile and P. Panci, Tools for model-independent bounds in direct dark matter searches, JCAP 10 (2013) 019 [arXiv:1307.5955] [INSPIRE].CrossRefGoogle Scholar
  106. [106]
    S. Yellin, Finding an upper limit in the presence of unknown background, Phys. Rev. D 66 (2002) 032005 [physics/0203002] [INSPIRE].
  107. [107]
    S.J. Witte and G.B. Gelmini, Updated constraints on the dark matter interpretation of CDMS-II-Si data, JCAP 05 (2017) 026 [arXiv:1703.06892] [INSPIRE].CrossRefGoogle Scholar
  108. [108]
    S.P. Martin, Two loop effective potential for a general renormalizable theory and softly broken supersymmetry, Phys. Rev. D 65 (2002) 116003 [hep-ph/0111209] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Kobayashi-Maskawa Institute for the Origin of Particles and the UniverseNagoya UniversityNagoyaJapan
  2. 2.Department of PhysicsNagoya UniversityNagoyaJapan
  3. 3.Kavli IPMU (WPI), UTIAS, University of TokyoKashiwaJapan
  4. 4.Department of PhysicsTohoku UniversitySendaiJapan
  5. 5.Department of PhysicsUniversity of TokyoTokyoJapan

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