Effective field theory for dark matter direct detection up to dimension seven

  • Joachim Brod
  • Aaron Gootjes-Dreesbach
  • Michele TammaroEmail author
  • Jure Zupan
Open Access
Regular Article - Theoretical Physics


We present the full basis of effective operators relevant for dark matter direct detection, up to and including operators of mass dimension seven. We treat the cases where dark matter is either a Dirac fermion, a Majorana fermion, a complex scalar, or a real scalar, allowing for dark matter to furnish a general representation of the electroweak gauge group. We describe the algorithmic procedure used to obtain the minimal set of effective operators and provide the tree-level matching conditions onto the effective theory valid below the electroweak scale.


Beyond Standard Model Effective Field Theories 


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]
    F. Bishara, J. Brod, B. Grinstein and J. Zupan, Chiral effective theory of dark matter direct detection, JCAP 02 (2017) 009 [arXiv:1611.00368] [INSPIRE].
  2. [2]
    J. Fan, M. Reece and L.-T. Wang, Non-relativistic effective theory of dark matter direct detection, JCAP 11 (2010) 042 [arXiv:1008.1591] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    A.L. Fitzpatrick et al., The effective field theory of dark matter direct detection, JCAP 02 (2013) 004 [arXiv:1203.3542] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    A.L. Fitzpatrick et al., Model independent direct detection analyses, arXiv:1211.2818 [INSPIRE].
  5. [5]
    N. Anand, A.L. Fitzpatrick and W.C. Haxton, Weakly interacting massive particle-nucleus elastic scattering response, Phys. Rev. C 89 (2014) 065501 [arXiv:1308.6288] [INSPIRE].
  6. [6]
    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].ADSCrossRefGoogle Scholar
  7. [7]
    G. Barello, S. Chang and C.A. Newby, A model independent approach to inelastic dark matter scattering, Phys. Rev. D 90 (2014) 094027 [arXiv:1409.0536] [INSPIRE].
  8. [8]
    R.J. Hill and M.P. Solon, Standard model anatomy of WIMP dark matter direct detection II: QCD analysis and hadronic matrix elements, Phys. Rev. D 91 (2015) 043505 [arXiv:1409.8290] [INSPIRE].
  9. [9]
    R. Catena and P. Gondolo, Global fits of the dark matter-nucleon effective interactions, JCAP 09 (2014) 045 [arXiv:1405.2637] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    J. Kopp, T. Schwetz and J. Zupan, Global interpretation of direct dark matter searches after CDMS-II results, JCAP 02 (2010) 014 [arXiv:0912.4264] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    R.J. Hill and M.P. Solon, WIMP-nucleon scattering with heavy WIMP effective theory, Phys. Rev. Lett. 112 (2014) 211602 [arXiv:1309.4092] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    R.J. Hill and M.P. Solon, Universal behavior in the scattering of heavy, weakly interacting dark matter on nuclear targets, Phys. Lett. B 707 (2012) 539 [arXiv:1111.0016] [INSPIRE].
  13. [13]
    A. Kurylov and M. Kamionkowski, Generalized analysis of weakly interacting massive particle searches, Phys. Rev. D 69 (2004) 063503 [hep-ph/0307185] [INSPIRE].
  14. [14]
    M. Pospelov and T. ter Veldhuis, Direct and indirect limits on the electromagnetic form-factors of WIMPs, Phys. Lett. B 480 (2000) 181 [hep-ph/0003010] [INSPIRE].
  15. [15]
    J. Bagnasco, M. Dine and S.D. Thomas, Detecting technibaryon dark matter, Phys. Lett. B 320 (1994) 99 [hep-ph/9310290] [INSPIRE].
  16. [16]
    V. Cirigliano, M.L. Graesser and G. Ovanesyan, WIMP-nucleus scattering in chiral effective theory, JHEP 10 (2012) 025 [arXiv:1205.2695] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Hoferichter, P. Klos and A. Schwenk, Chiral power counting of one- and two-body currents in direct detection of dark matter, Phys. Lett. B 746 (2015) 410 [arXiv:1503.04811] [INSPIRE].
  18. [18]
    M. Hoferichter, P. Klos, J. Menéndez and A. Schwenk, Analysis strategies for general spin-independent WIMP-nucleus scattering, Phys. Rev. D 94 (2016) 063505 [arXiv:1605.08043] [INSPIRE].
  19. [19]
    F. Bishara, J. Brod, B. Grinstein and J. Zupan, From quarks to nucleons in dark matter direct detection, JHEP 11 (2017) 059 [arXiv:1707.06998] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    J. Menendez, D. Gazit and A. Schwenk, Spin-dependent WIMP scattering off nuclei, Phys. Rev. D 86 (2012) 103511 [arXiv:1208.1094] [INSPIRE].
  21. [21]
    P. Klos, J. Menéndez, D. Gazit and A. Schwenk, Large-scale nuclear structure calculations for spin-dependent WIMP scattering with chiral effective field theory currents, Phys. Rev. D 88 (2013) 083516 [Erratum ibid. D 89 (2014) 029901] [arXiv:1304.7684] [INSPIRE].
  22. [22]
    L. Baudis et al., Signatures of dark matter scattering inelastically off nuclei, Phys. Rev. D 88 (2013) 115014 [arXiv:1309.0825] [INSPIRE].
  23. [23]
    L. Vietze et al., Nuclear structure aspects of spin-independent WIMP scattering off xenon, Phys. Rev. D 91 (2015) 043520 [arXiv:1412.6091] [INSPIRE].
  24. [24]
    J. Goodman et al., Gamma ray line constraints on effective theories of dark matter, Nucl. Phys. B 844 (2011) 55 [arXiv:1009.0008] [INSPIRE].
  25. [25]
    F. Bishara, J. Brod, B. Grinstein and J. Zupan, DirectDM: a tool for dark matter direct detection, arXiv:1708.02678 [INSPIRE].
  26. [26]
    L. Lehman and A. Martin, Hilbert series for constructing lagrangians: expanding the phenomenologist’s toolbox, Phys. Rev. D 91 (2015) 105014 [arXiv:1503.07537] [INSPIRE].
  27. [27]
    B. Henning, X. Lu, T. Melia and H. Murayama, Hilbert series and operator bases with derivatives in effective field theories, Commun. Math. Phys. 347 (2016) 363 [arXiv:1507.07240] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    B. Henning, X. Lu, T. Melia and H. Murayama, 2, 84, 30, 993, 560, 15456, 11962, 261485,. . .: higher dimension operators in the SM EFT, JHEP 08 (2017) 016 [arXiv:1512.03433] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    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].ADSCrossRefGoogle Scholar
  30. [30]
    G. Ovanesyan and L. Vecchi, Direct detection of dark matter polarizability, JHEP 07 (2015) 128 [arXiv:1410.0601] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    T. Appelquist et al., Detecting stealth dark matter directly through electromagnetic polarizability, Phys. Rev. Lett. 115 (2015) 171803 [arXiv:1503.04205] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    N. Weiner and I. Yavin, How dark are Majorana WIMPs? Signals from MiDM and Rayleigh dark matter, Phys. Rev. D 86 (2012) 075021 [arXiv:1206.2910] [INSPIRE].
  33. [33]
    M.T. Frandsen et al., Loop-induced dark matter direct detection signals from gamma-ray lines, JCAP 10 (2012) 033 [arXiv:1207.3971] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    W. Buchmüller and D. Wyler, Effective Lagrangian analysis of new interactions and flavor conservation, Nucl. Phys. B 268 (1986) 621 [INSPIRE].
  35. [35]
    B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-six terms in the standard model lagrangian, JHEP 10 (2010) 085 [arXiv:1008.4884] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  36. [36]
    F. Bishara, J. Brod, B. Grinstein and J. Zupan, Renormalization group effects in dark matter interactions, arXiv:1809.03506 [INSPIRE].
  37. [37]
    A. Denner, Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200, Fortsch. Phys. 41 (1993) 307 [arXiv:0709.1075] [INSPIRE].ADSGoogle Scholar
  38. [38]
    M.A. Fedderke, J.-Y. Chen, E.W. Kolb and L.-T. Wang, The fermionic dark matter Higgs portal: an effective field theory approach, JHEP 08 (2014) 122 [arXiv:1404.2283] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    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].
  40. [40]
    B. Gripaios and D. Sutherland, An operator basis for the standard model with an added scalar singlet, JHEP 08 (2016) 103 [arXiv:1604.07365] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    A. Gootjes-Dreesbach, Dimension-seven renormalization group effects in an effective field theory approach to fermionic dark matter, M.Sc. thesis, Technischen Universität Dortmund, Dortmund, Germany (2016).Google Scholar
  42. [42]
    C.C. Nishi, Simple derivation of general Fierz-like identities, Am. J. Phys. 73 (2005) 1160 [hep-ph/0412245] [INSPIRE].
  43. [43]
    V.I. Borodulin, R.N. Rogalyov and S.R. Slabospitskii, CORE 3.1 (COmpendium of RElations, Version 3.1), arXiv:1702.08246 [INSPIRE].
  44. [44]
    H.K. Dreiner, H.E. Haber and S.P. Martin, Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry, Phys. Rept. 494 (2010) 1 [arXiv:0812.1594] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    H. Simma, Equations of motion for effective Lagrangians and penguins in rare B decays, Z. Phys. C 61 (1994) 67 [hep-ph/9307274] [INSPIRE].
  46. [46]
    M. Neubert, Heavy quark symmetry, Phys. Rept. 245 (1994) 259 [hep-ph/9306320] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Joachim Brod
    • 1
    • 2
  • Aaron Gootjes-Dreesbach
    • 1
  • Michele Tammaro
    • 2
    Email author
  • Jure Zupan
    • 2
  1. 1.Fakultät für Physik, Technischen Universität DortmundDortmundGermany
  2. 2.Physics DepartmentUniversity of CincinnatiCincinnatiU.S.A.

Personalised recommendations