Radiative bound-state formation in unbroken perturbative non-Abelian theories and implications for dark matter

  • Julia Harz
  • Kalliopi PetrakiEmail author
Open Access
Regular Article - Theoretical Physics


We compute the cross-sections for the radiative capture of non-relativistic particles into bound states, in unbroken perturbative non-Abelian theories. We find that the formation of bound states via emission of a gauge boson can be significant for a variety of dark matter models that feature non-Abelian long-range interactions, including multi-TeV scale WIMPs, dark matter co-annihilating with coloured partners and hidden-sector models. Our results disagree with previous computations, on the relative sign of the Abelian and non-Abelian contributions. In particular, in the case of capture of a particle-antiparticle pair into its tightest bound state, we find that these contributions add up, rather than partially canceling each other. We apply our results to dark matter co-annihilating with particles transforming in the (anti)fundamental of SU(3)c, as is the case in degenerate stop-neutralino scenarios in the MSSM. We show that the radiative formation and decay of particle-antiparticle bound states can deplete the dark matter density by (40 − 240)%, for dark matter heavier than 500 GeV. This implies a larger mass difference between the co-annihilating particles, and allows for the dark matter to be as heavy as 3.3 TeV.


Beyond Standard Model Cosmology of Theories beyond the SM Perturbative 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]
    M. Pospelov and A. Ritz, Astrophysical Signatures of Secluded Dark Matter, Phys. Lett. B 671 (2009) 391 [arXiv:0810.1502] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    J.D. March-Russell and S.M. West, WIMPonium and Boost Factors for Indirect Dark Matter Detection, Phys. Lett. B 676 (2009) 133 [arXiv:0812.0559] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    W. Shepherd, T.M.P. Tait and G. Zaharijas, Bound states of weakly interacting dark matter, Phys. Rev. D 79 (2009) 055022 [arXiv:0901.2125] [INSPIRE].ADSGoogle Scholar
  4. [4]
    B. von Harling and K. Petraki, Bound-state formation for thermal relic dark matter and unitarity, JCAP 12 (2014) 033 [arXiv:1407.7874] [INSPIRE].CrossRefGoogle Scholar
  5. [5]
    I. Baldes and K. Petraki, Asymmetric thermal-relic dark matter: Sommerfeld-enhanced freeze-out, annihilation signals and unitarity bounds, JCAP 09 (2017) 028 [arXiv:1703.00478] [INSPIRE].ADSGoogle Scholar
  6. [6]
    H. An, M.B. Wise and Y. Zhang, Effects of Bound States on Dark Matter Annihilation, Phys. Rev. D 93 (2016) 115020 [arXiv:1604.01776] [INSPIRE].ADSGoogle Scholar
  7. [7]
    H. An, M.B. Wise and Y. Zhang, Strong CMB Constraint On P-Wave Annihilating Dark Matter, Phys. Lett. B 773 (2017) 121 [arXiv:1606.02305] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    P. Asadi, M. Baumgart, P.J. Fitzpatrick, E. Krupczak and T.R. Slatyer, Capture and Decay of Electroweak WIMPonium, JCAP 02 (2017) 005 [arXiv:1610.07617] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    K. Petraki, M. Postma and J. de Vries, Radiative bound-state-formation cross-sections for dark matter interacting via a Yukawa potential, JHEP 04 (2017) 077 [arXiv:1611.01394] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    M. Cirelli, P. Panci, K. Petraki, F. Sala and M. Taoso, Dark Matter’s secret liaisons: phenomenology of a dark U(1) sector with bound states, JCAP 05 (2017) 036 [arXiv:1612.07295] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    C. Kouvaris, K. Langæble and N.G. Nielsen, The Spectrum of Darkonium in the Sun, JCAP 10 (2016) 012 [arXiv:1607.00374] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    I. Baldes, M. Cirelli, P. Panci, K. Petraki, F. Sala and M. Taoso, Asymmetric dark matter: residual annihilations and self-interactions, SciPost Phys. 4 (2018) 041 [arXiv:1712.07489] [INSPIRE].CrossRefGoogle Scholar
  13. [13]
    K. Petraki and R.R. Volkas, Review of asymmetric dark matter, Int. J. Mod. Phys. A 28 (2013) 1330028 [arXiv:1305.4939] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    R. Laha and E. Braaten, Direct detection of dark matter in universal bound states, Phys. Rev. D 89 (2014) 103510 [arXiv:1311.6386] [INSPIRE].ADSGoogle Scholar
  15. [15]
    A. Butcher, R. Kirk, J. Monroe and S.M. West, Can Tonne-Scale Direct Detection Experiments Discover Nuclear Dark Matter?, JCAP 10 (2017) 035 [arXiv:1610.01840] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    L. Pearce and A. Kusenko, Indirect Detection of Self-Interacting Asymmetric Dark Matter, Phys. Rev. D 87 (2013) 123531 [arXiv:1303.7294] [INSPIRE].ADSGoogle Scholar
  17. [17]
    J.M. Cline, Y. Farzan, Z. Liu, G.D. Moore and W. Xue, 3.5 keV x rays as the “21 cm line” of dark atoms and a link to light sterile neutrinos, Phys. Rev. D 89 (2014) 121302 [arXiv:1404.3729] [INSPIRE].
  18. [18]
    W. Detmold, M. McCullough and A. Pochinsky, Dark Nuclei I: Cosmology and Indirect Detection, Phys. Rev. D 90 (2014) 115013 [arXiv:1406.2276] [INSPIRE].ADSGoogle Scholar
  19. [19]
    L. Pearce, K. Petraki and A. Kusenko, Signals from dark atom formation in halos, Phys. Rev. D 91 (2015) 083532 [arXiv:1502.01755] [INSPIRE].ADSGoogle Scholar
  20. [20]
    K. Petraki, L. Pearce and A. Kusenko, Self-interacting asymmetric dark matter coupled to a light massive dark photon, JCAP 07 (2014) 039 [arXiv:1403.1077] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    S.J. Lonsdale and R.R. Volkas, Grand unified hidden-sector dark matter, Phys. Rev. D 90 (2014) 083501 [Erratum ibid. D 91 (2015) 129906] [arXiv:1407.4192] [INSPIRE].
  22. [22]
    S.J. Lonsdale, M. Schroor and R.R. Volkas, Asymmetric Dark Matter and the hadronic spectra of hidden QCD, Phys. Rev. D 96 (2017) 055027 [arXiv:1704.05213] [INSPIRE].ADSGoogle Scholar
  23. [23]
    S.J. Lonsdale and R.R. Volkas, Comprehensive asymmetric dark matter model, Phys. Rev. D 97 (2018) 103510 [arXiv:1801.05561] [INSPIRE].ADSGoogle Scholar
  24. [24]
    H. An, B. Echenard, M. Pospelov and Y. Zhang, Probing the Dark Sector with Dark Matter Bound States, Phys. Rev. Lett. 116 (2016) 151801 [arXiv:1510.05020] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    Z. Kang, Bound states via Higgs exchanging and heavy resonant di-Higgs, Phys. Lett. B 771 (2017) 313 [arXiv:1606.01531] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    G. Elor, H. Liu, T.R. Slatyer and Y. Soreq, Complementarity for Dark Sector Bound States, arXiv:1801.07723 [INSPIRE].
  27. [27]
    S.R. Coleman, Q Balls, Nucl. Phys. B 262 (1985) 263 [Erratum ibid. B 269 (1986) 744] [INSPIRE].
  28. [28]
    A. Kusenko and M.E. Shaposhnikov, Supersymmetric Q balls as dark matter, Phys. Lett. B 418 (1998) 46 [hep-ph/9709492] [INSPIRE].
  29. [29]
    A. Kusenko, Small Q balls, Phys. Lett. B 404 (1997) 285 [hep-th/9704073] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    A. Kusenko, Solitons in the supersymmetric extensions of the standard model, Phys. Lett. B 405 (1997) 108 [hep-ph/9704273] [INSPIRE].
  31. [31]
    A. Kusenko and P.J. Steinhardt, Q ball candidates for selfinteracting dark matter, Phys. Rev. Lett. 87 (2001) 141301 [astro-ph/0106008] [INSPIRE].
  32. [32]
    J. Harz, B. Herrmann, M. Klasen, K. Kovarik and Q.L. Boulc’h, Neutralino-stop coannihilation into electroweak gauge and Higgs bosons at one loop, Phys. Rev. D 87 (2013) 054031 [arXiv:1212.5241] [INSPIRE].ADSGoogle Scholar
  33. [33]
    J. Harz, B. Herrmann, M. Klasen, K. Kovařík and M. Meinecke, SUSY-QCD corrections to stop annihilation into electroweak final states including Coulomb enhancement effects, Phys. Rev. D 91 (2015) 034012 [arXiv:1410.8063] [INSPIRE].ADSGoogle Scholar
  34. [34]
    J. Harz, B. Herrmann, M. Klasen and K. Kovarik, One-loop corrections to neutralino-stop coannihilation revisited, Phys. Rev. D 91 (2015) 034028 [arXiv:1409.2898] [INSPIRE].ADSGoogle Scholar
  35. [35]
    M.J. Baker et al., The Coannihilation Codex, JHEP 12 (2015) 120 [arXiv:1510.03434] [INSPIRE].ADSGoogle Scholar
  36. [36]
    A. Ibarra, A. Pierce, N.R. Shah and S. Vogl, Anatomy of Coannihilation with a Scalar Top Partner, Phys. Rev. D 91 (2015) 095018 [arXiv:1501.03164] [INSPIRE].ADSGoogle Scholar
  37. [37]
    J. Harz, B. Herrmann, M. Klasen, K. Kovarik and P. Steppeler, Theoretical uncertainty of the supersymmetric dark matter relic density from scheme and scale variations, Phys. Rev. D 93 (2016) 114023 [arXiv:1602.08103] [INSPIRE].ADSGoogle Scholar
  38. [38]
    S.P. Liew and F. Luo, Effects of QCD bound states on dark matter relic abundance, JHEP 02 (2017) 091 [arXiv:1611.08133] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  39. [39]
    A. Pierce, N.R. Shah and S. Vogl, Stop Co-Annihilation in the Minimal Supersymmetric Standard Model Revisited, Phys. Rev. D 97 (2018) 023008 [arXiv:1706.01911] [INSPIRE].ADSGoogle Scholar
  40. [40]
    H.E. Haber, R. Hempfling and A.H. Hoang, Approximating the radiatively corrected Higgs mass in the minimal supersymmetric model, Z. Phys. C 75 (1997) 539 [hep-ph/9609331] [INSPIRE].
  41. [41]
    H.E. Haber and R. Hempfling, Can the mass of the lightest Higgs boson of the minimal supersymmetric model be larger than m Z ?, Phys. Rev. Lett. 66 (1991) 1815 [INSPIRE].
  42. [42]
    A. Hryczuk, I. Cholis, R. Iengo, M. Tavakoli and P. Ullio, Indirect Detection Analysis: Wino Dark Matter Case Study, JCAP 07 (2014) 031 [arXiv:1401.6212] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    M. Baumgart, I.Z. Rothstein and V. Vaidya, Constraints on Galactic Wino Densities from Gamma Ray Lines, JHEP 04 (2015) 106 [arXiv:1412.8698] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    M. Cirelli, T. Hambye, P. Panci, F. Sala and M. Taoso, Gamma ray tests of Minimal Dark Matter, JCAP 10 (2015) 026 [arXiv:1507.05519] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    M. Beneke, A. Bharucha, A. Hryczuk, S. Recksiegel and P. Ruiz-Femenia, The last refuge of mixed wino-Higgsino dark matter, JHEP 01 (2017) 002 [arXiv:1611.00804] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  46. [46]
    M. Baumgart et al., Resummed Photon Spectra for WIMP Annihilation, JHEP 03 (2018) 117 [arXiv:1712.07656] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    M. Beneke, Perturbative heavy quark-anti-quark systems, hep-ph/9911490 [INSPIRE].
  48. [48]
    S. Kim and M. Laine, Rapid thermal co-annihilation through bound states in QCD, JHEP 07 (2016) 143 [arXiv:1602.08105] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    S. Kim and M. Laine, On thermal corrections to near-threshold annihilation, JCAP 01 (2017) 013 [arXiv:1609.00474] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    S. Biondini and M. Laine, Re-derived overclosure bound for the inert doublet model, JHEP 08 (2017) 047 [arXiv:1706.01894] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    S. Biondini and M. Laine, Thermal dark matter co-annihilating with a strongly interacting scalar, JHEP 04 (2018) 072 [arXiv:1801.05821] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  52. [52]
    E. Braaten, E. Johnson and H. Zhang, Zero-range effective field theory for resonant wino dark matter. Part I. Framework, JHEP 11 (2017) 108 [arXiv:1706.02253] [INSPIRE].
  53. [53]
    E. Braaten, E. Johnson and H. Zhang, Zero-range effective field theory for resonant wino dark matter. Part II. Coulomb resummation, JHEP 02 (2018) 150 [arXiv:1708.07155] [INSPIRE].
  54. [54]
    E. Braaten, E. Johnson and H. Zhang, Zero-range effective field theory for resonant wino dark matter. Part III. Annihilation effects, JHEP 05 (2018) 062 [arXiv:1712.07142] [INSPIRE].
  55. [55]
    A. Mitridate, M. Redi, J. Smirnov and A. Strumia, Cosmological Implications of Dark Matter Bound States, JCAP 05 (2017) 006 [arXiv:1702.01141] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    W.-Y. Keung, I. Low and Y. Zhang, Reappraisal of dark matter co-annihilating with a top or bottom partner, Phys. Rev. D 96 (2017) 015008 [arXiv:1703.02977] [INSPIRE].ADSGoogle Scholar
  57. [57]
    K. Petraki, M. Postma and M. Wiechers, Dark-matter bound states from Feynman diagrams, JHEP 06 (2015) 128 [arXiv:1505.00109] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    J. Harz and K. Petraki, Higgs Enhancement for the Dark Matter Relic Density, Phys. Rev. D 97 (2018) 075041 [arXiv:1711.03552] [INSPIRE].ADSGoogle Scholar
  59. [59]
    J. Harz and K. Petraki, in preparation.Google Scholar
  60. [60]
    M. Geller, S. Iwamoto, G. Lee, Y. Shadmi and O. Telem, Dark quarkonium formation in the early universe, JHEP 06 (2018) 135 [arXiv:1802.07720] [INSPIRE].CrossRefGoogle Scholar
  61. [61]
    N. Brambilla, M.A. Escobedo, J. Ghiglieri and A. Vairo, Thermal width and gluo-dissociation of quarkonium in pNRQCD, JHEP 12 (2011) 116 [arXiv:1109.5826] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  62. [62]
    C. Itzykson and J. Zuber, Quantum field theory, McGraw-Hill (1980) [INSPIRE].
  63. [63]
    Y. Kats and M.D. Schwartz, Annihilation decays of bound states at the LHC, JHEP 04 (2010) 016 [arXiv:0912.0526] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  64. [64]
    A.D. Sakharov, Interaction of an Electron and Positron in Pair Production, Zh. Eksp. Teor. Fiz. 18 (1948) 631 [INSPIRE].Google Scholar
  65. [65]
    A. Sommerfeld, Über die Beugung und Bremsung der Elektronen, Annalen Phys. 403 (1931) 257.ADSCrossRefzbMATHGoogle Scholar
  66. [66]
    P.R. Manuel Drees, Rohini Godbole, Theory and Phenomenology of Sparticles, World Scientific (2005).Google Scholar
  67. [67]
    J. Edsjo and P. Gondolo, Neutralino relic density including coannihilations, Phys. Rev. D 56 (1997) 1879 [hep-ph/9704361] [INSPIRE].
  68. [68]
    S. El Hedri, A. Kaminska and M. de Vries, A Sommerfeld Toolbox for Colored Dark Sectors, Eur. Phys. J. C 77 (2017) 622 [arXiv:1612.02825] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    S. Cassel, Sommerfeld factor for arbitrary partial wave processes, J. Phys. G 37 (2010) 105009 [arXiv:0903.5307] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    S. Biondini, Bound-state effects for dark matter with Higgs-like mediators, JHEP 06 (2018) 104 [arXiv:1805.00353] [INSPIRE].CrossRefGoogle Scholar
  71. [71]
    A. Messiah, Quantum mechanics, North-Holland Pub. Co. (1962).Google Scholar
  72. [72]
    A.I. Akhiezer and N.P. Merenkov, The theory of lepton bound-state production, J. Phys. B 29 (1996) 2135.ADSGoogle Scholar
  73. [73]
    A.V. Manohar and I.W. Stewart, Running of the heavy quark production current and 1/v potential in QCD, Phys. Rev. D 63 (2001) 054004 [hep-ph/0003107] [INSPIRE].
  74. [74]
    A.V. Manohar and I.W. Stewart, Renormalization group analysis of the QCD quark potential to order v 2, Phys. Rev. D 62 (2000) 014033 [hep-ph/9912226] [INSPIRE].
  75. [75]
    J.J. Sakurai, Modern quantum mechanics, Addison-Wesley (1994).Google Scholar
  76. [76]
    A. Pineda and J. Soto, Effective field theory for ultrasoft momenta in NRQCD and NRQED, Nucl. Phys. Proc. Suppl. 64 (1998) 428 [hep-ph/9707481] [INSPIRE].
  77. [77]
    M.E. Luke, A.V. Manohar and I.Z. Rothstein, Renormalization group scaling in nonrelativistic QCD, Phys. Rev. D 61 (2000) 074025 [hep-ph/9910209] [INSPIRE].
  78. [78]
    B. Ioffe and M. Shifman, At the Frontier of Particle Physics: Handbook of QCD: Boris Ioffe Festschrift. vol. 4, World Scientific (2001).Google Scholar
  79. [79]
    A.H. Hoang and M. Stahlhofen, Ultrasoft NLL Running of the Nonrelativistic O(v) QCD Quark Potential, JHEP 06 (2011) 088 [arXiv:1102.0269] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589 CNRS & Sorbonne UniversitéParisFrance
  2. 2.NikhefAmsterdamThe Netherlands

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