- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Black holes with electroweak hair: The detailed derivation

Romain Gervalle1,* and Mikhail S. Volkov1,2,†

  • 1Institut Denis Poisson, UMR-CNRS 7013, Université de Tours, Parc de Grandmont, 37200 Tours, France
  • 2CERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland

  • *Contact author: romain.gervalle@univ-tours.fr
  • †Contact author: mvolkov@univ-tours.fr
PDF

Phys. Rev. D 112, 064018 – Published 9 September, 2025

DOI: https://doi.org/10.1103/gzhg-62fz

Abstract

We present a very detailed derivation of solutions describing hairy black holes within the gravity-coupled Weinberg-Salam theory, which were previously reported in Gervalle and Volkov [Black holes with electroweak hair, Phys. Rev. Lett. 133, 171402 (2024)]. These black holes support a strong magnetic field that polarizes the electroweak vacuum and creates a condensate of massive fields carrying superconducting currents along the black hole horizon. The currents, in turn, generate a “corona” of magnetic vortex segments attached to the horizon at both ends. The condensate and corona together constitute the black hole hair. The extremal solutions approach, in the far field, the magnetic Reissner-Nordström configuration, with a total mass that is lower than the total charge, M<|Q|, due to the negative Zeeman energy of the condensate. This makes the removal of the hair energetically unfavorable. The maximally hairy black holes exhibit masses comparable to terrestrial values, with approximately 11% of their total mass stored in the hair. Given that these solutions arise within a well-tested theoretical framework, they are likely to have physical relevance.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (101)

  1. R. Ruffini and J. Wheeler, Introducing the black hole, Phys. Today 24, No. 1, 30 (1971).
  2. W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164, 1776 (1967).
  3. D. C. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Lett. 34, 905 (1975).
  4. P. O. Mazur, Proof of uniqueness of the Kerr-Newman black hole solution, J. Phys. A 15, 3173 (1982).
  5. M. Heusler, Black Hole Uniqueness Theorems (Cambridge University Press, Cambridge, England, 1996).
  6. J. E. Chase, Event horizons in static scalar-vacuum space-times, Commun. Math. Phys. 19, 276 (1970).
  7. J. D. Bekenstein, Nonexistence of baryon number for static black holes, Phys. Rev. D 5, 1239 (1972).
  8. J. D. Bekenstein, Transcendence of the law of baryon-number conservation in black hole physics, Phys. Rev. Lett. 28, 452 (1972).
  9. J. D. Bekenstein, Novel “no-scalar-hair” theorem for black holes, Phys. Rev. D 51, R6608 (1995).
  10. A. E. Mayo and J. D. Bekenstein, No hair for spherical black holes: Charged and nonminimally coupled scalar field with self-interaction, Phys. Rev. D 54, 5059 (1996).
  11. J. D. Bekenstein, Black hole hair: 25—years after, in Proceedings of the 2nd International Sakharov Conference on Physics (1996), pp. 216–219, arXiv:gr-qc/9605059.
  12. A. Saa, New no scalar hair theorem for black holes, J. Math. Phys. (N.Y.) 37, 2346 (1996).
  13. C. A. R. Herdeiro and E. Radu, Asymptotically flat black holes with scalar hair: A review, Int. J. Mod. Phys. D 24, 1542014 (2015).
  14. M. S. Volkov and D. V. Galtsov, NonAbelian Einstein Yang-Mills black holes, Pis’ma Zh. Eksp. Teor. Fiz. 50, 312 (1989) [JETP Lett. 50, 346 (1989), http://jetpletters.ru/ps/1130/article_17118.shtml].
  15. M. S. Volkov, Hairy black holes in the XX-th and XXI-st centuries, arXiv:1601.08230.
  16. H. P. Kuenzle and A. K. M. Masood-ul Alam, Spherically symmetric static SU(2) Einstein Yang-Mills fields, J. Math. Phys. (N.Y.) 31, 928 (1990).
  17. P. Bizon, Colored black holes, Phys. Rev. Lett. 64, 2844 (1990).
  18. S. Droz, M. Heusler, and N. Straumann, New black hole solutions with hair, Phys. Lett. B 268, 371 (1991).
  19. P. Bizon and T. Chmaj, Gravitating skyrmions, Phys. Lett. B 297, 55 (1992).
  20. P. Breitenlohner, P. Forgács, and D. Maison, Gravitating monopole solutions, Nucl. Phys. B383, 357 (1992).
  21. B. R. Greene, S. D. Mathur, and C. M. O’Neill, Eluding the no hair conjecture: Black holes in spontaneously broken gauge theories, Phys. Rev. D 47, 2242 (1993).
  22. G. Lavrelashvili and D. Maison, Regular and black hole solutions of Einstein-Yang-Mills dilaton theory, Nucl. Phys. B410, 407 (1993).
  23. S. Mignemi and N. R. Stewart, Charged black holes in effective string theory, Phys. Rev. D 47, 5259 (1993).
  24. P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, Dilatonic black holes in higher curvature string gravity, Phys. Rev. D 54, 5049 (1996).
  25. T. Torii, H. Yajima, and K.-i. Maeda, Dilatonic black holes with Gauss-Bonnet term, Phys. Rev. D 55, 739 (1997).
  26. B. Kleihaus and J. Kunz, Static black-hole solutions with axial symmetry, Phys. Rev. Lett. 79, 1595 (1997).
  27. B. Kleihaus and J. Kunz, Rotating hairy black holes, Phys. Rev. Lett. 86, 3704 (2001).
  28. M. S. Volkov and D. V. Gal’tsov, Gravitating nonAbelian solitons and black holes with Yang-Mills fields, Phys. Rep. 319, 1 (1999).
  29. U. Nucamendi and M. Salgado, Scalar hairy black holes and solitons in asymptotically flat space-times, Phys. Rev. D 68, 044026 (2003).
  30. K. G. Zloshchastiev, On co-existence of black holes and scalar field, Phys. Rev. Lett. 94, 121101 (2005).
  31. A. Anabalon, D. Astefanesei, and R. Mann, Exact asymptotically flat charged hairy black holes with a dilaton potential, J. High Energy Phys. 10 (2013) 184.
  32. M. Rinaldi, Black holes with non-minimal derivative coupling, Phys. Rev. D 86, 084048 (2012).
  33. T. P. Sotiriou and S.-Y. Zhou, Black hole hair in generalized scalar-tensor gravity: An explicit example, Phys. Rev. D 90, 124063 (2014).
  34. E. Babichev and C. Charmousis, Dressing a black hole with a time-dependent Galileon, J. High Energy Phys. 08 (2014) 106.
  35. T. Kobayashi and N. Tanahashi, Exact black hole solutions in shift symmetric scalar–tensor theories, Prog. Theor. Exp. Phys. 2014, 073E02 (2014).
  36. C. A. R. Herdeiro and E. Radu, Kerr black holes with scalar hair, Phys. Rev. Lett. 112, 221101 (2014).
  37. C. Herdeiro, E. Radu, and H. Rúnarsson, Kerr black holes with Proca hair, Classical Quantum Gravity 33, 154001 (2016).
  38. W. E. East and F. Pretorius, Superradiant instability and backreaction of massive vector fields around Kerr black holes, Phys. Rev. Lett. 119, 041101 (2017).
  39. D. D. Doneva, F. M. Ramazanoğlu, H. O. Silva, T. P. Sotiriou, and S. S. Yazadjiev, Spontaneous scalarization, Rev. Mod. Phys. 96, 015004 (2024).
  40. R. Brito, V. Cardoso, and P. Pani, Black holes with massive graviton hair, Phys. Rev. D 88, 064006 (2013).
  41. R. Gervalle and M. S. Volkov, Asymptotically flat hairy black holes in massive bigravity, Phys. Rev. D 102, 124040 (2020).
  42. E. Alonso-Monsalve and D. I. Kaiser, Primordial black holes with QCD color charge, Phys. Rev. Lett. 132, 231402 (2024).
  43. Y. Bai and M. Korwar, Hairy magnetic and dyonic black holes in the standard model, J. High Energy Phys. 04 (2021) 119.
  44. K.-M. Lee and E. J. Weinberg, Nontopological magnetic monopoles and new magnetically charged black holes, Phys. Rev. Lett. 73, 1203 (1994).
  45. S. A. Ridgway and E. J. Weinberg, Static black hole solutions without rotational symmetry, Phys. Rev. D 52, 3440 (1995).
  46. J. Maldacena, Comments on magnetic black holes, J. High Energy Phys. 04 (2021) 079.
  47. J. Ambjorn and P. Olesen, On electroweak magnetism, Nucl. Phys. B315, 606 (1989).
  48. J. Ambjorn and P. Olesen, Electroweak magnetism: Theory and application, Int. J. Mod. Phys. A 05, 4525 (1990).
  49. M. N. Chernodub, J. Van Doorsselaere, and H. Verschelde, Magnetic-field-induced superconductivity and superfluidity of W and Z bosons: In tandem transport and kaleidoscopic vortex states, Phys. Rev. D 88, 065006 (2013).
  50. M. N. Chernodub, V. A. Goy, and A. V. Molochkov, Phase structure of electroweak vacuum in a strong magnetic field: The lattice results, Phys. Rev. Lett. 130, 111802 (2023).
  51. Y. Bai, J. Berger, M. Korwar, and N. Orlofsky, Phenomenology of magnetic black holes with electroweak-symmetric coronas, J. High Energy Phys. 10 (2020) 210.
  52. Y. Bai, S. Lu, and N. Orlofsky, Searching for magnetic monopoles with the Earth’s magnetic field, Phys. Rev. Lett. 127, 101801 (2021).
  53. D. Ghosh, A. Thalapillil, and F. Ullah, Astrophysical hints for magnetic black holes, Phys. Rev. D 103, 023006 (2021).
  54. J. Estes, M. Kavic, S. L. Liebling, M. Lippert, and J. H. Simonetti, Stability and observability of magnetic primordial black hole-neutron star collisions, J. Cosmol. Astropart. Phys. 06 (2023) 017.
  55. M. D. Diamond and D. E. Kaplan, Constraints on relic magnetic black holes, J. High Energy Phys. 03 (2022) 157.
  56. Y. M. Cho, S.-W. Kim, and S. H. Oh, Electroweak primordial magnetic blackhole: Cosmic production and physical implication, arXiv:2408.05531.
  57. R. Gervalle and M. S. Volkov, Black holes with electroweak hair, Phys. Rev. Lett. 133, 171402 (2024).
  58. R. Gervalle and M. S. Volkov, Electroweak monopoles and their stability, Nucl. Phys. B984, 115937 (2022).
  59. R. Gervalle and M. S. Volkov, Electroweak multi-monopoles, Nucl. Phys. B987, 116112 (2023).
  60. E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, CODATA recommended values of the fundamental physical constants: 2018*, Rev. Mod. Phys. 93, 025010 (2021).
  61. Y. Nambu, String-like configurations in the Weinberg-Salam theory, Nucl. Phys. B130, 505 (1977).
  62. G. ’t Hooft, Magnetic monopoles in unified gauge theories, Nucl. Phys. B79, 276 (1974).
  63. P. A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. R. Soc. A 133, 60 (1931).
  64. B. Bertotti, Uniform electromagnetic field in the theory of general relativity, Phys. Rev. 116, 1331 (1959).
  65. I. Robinson, A solution of the Maxwell-Einstein equations, Bull. Acad. Pol. Sci., Ser. Sci., Math., Astron. Phys. 7, 351 (1959).
  66. S. Hod, Electroweak clouds supported by magnetically charged black holes: Analytic treatment along the existence line, Phys. Rev. D 110, 104045 (2024).
  67. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, New York, USA, 2007).
  68. J. Ambjorn and P. Olesen, A condensate solution of the electroweak theory which interpolates between the broken and the symmetric phase, Nucl. Phys. B330, 193 (1990).
  69. N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, The string landscape, black holes and gravity as the weakest force, J. High Energy Phys. 06 (2007) 060.
  70. Y. M. Cho and D. Maison, Monopoles in Weinberg-Salam model, Phys. Lett. B 391, 360 (1997).
  71. Relativity: The General theory, edited by J. L. Synge (North-Holland Publishing Company, Amsterdam, 1960).
  72. A. Komar, Covariant conservation laws in general relativity, Phys. Rev. 113, 934 (1959).
  73. C. Rebbi and P. Rossi, Multi—monopole solutions in the Prasad-Sommerfield limit, Phys. Rev. D 22, 2010 (1980).
  74. F. Hecht, New development in freefem++, J. Numer. Math. 20, 251 (2012).
  75. S. Deser, Absence of static solutions in source-free Yang-Mills theory, Phys. Lett. 64B, 463 (1976).
  76. G. H. Derrick, Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys. (N.Y.) 5, 1252 (1964).
  77. C. A. R. Herdeiro, J. a. M. S. Oliveira, A. M. Pombo, and E. Radu, Deconstructing scaling virial identities in general relativity: Spherical symmetry and beyond, Phys. Rev. D 106, 024054 (2022).
  78. R. P. Geroch, Multipole moments. II. Curved space, J. Math. Phys. (N.Y.) 11, 2580 (1970).
  79. R. Hansen, Multipole moments of stationary spacetimes, J. Math. Phys. (N.Y.) 15, 46 (1974).
  80. G. Fodor, C. Hoenselaers, and Z. Perjes, Multipole moments of axisymmetric systems in relativity, J. Math. Phys. (N.Y.) 30, 2252 (1989).
  81. G. Fodor, E. d. S. C. Filho, and B. Hartmann, Calculation of multipole moments of axistationary electrovacuum spacetimes, Phys. Rev. D 104, 064012 (2021).
  82. F. J. Ernst, New formulation of the axially symmetric gravitational potential, Phys. Rev. 167, 1175 (1968).
  83. F. J. Ernst, New formulation of the axially symmetric gravitational field problem. II, Phys. Rev. 168, 1415 (1968).
  84. P. Olesen, Vacuum structure of the electroweak theory in high magnetic fields, Phys. Lett. B 268, 389 (1991).
  85. J. Van Doorsselaere, Properties of symmetry restoration in the electroweak vacuum in very high magnetic fields, Phys. Rev. D 88, 025013 (2013).
  86. F. L. Bezrukov and M. Shaposhnikov, The standard model Higgs boson as the inflaton, Phys. Lett. B 659, 703 (2008).
  87. B. Kleihaus and J. Kunz, Monopole—anti-monopole solutions of Einstein-Yang-Mills-Higgs theory, Phys. Rev. Lett. 85, 2430 (2000).
  88. C. A. R. Herdeiro and E. Radu, Two Schwarzschild-like black holes balanced by their scalar hair, Phys. Rev. D 107, 064044 (2023).
  89. J. Urrestilla, A. Achucarro, J. Borrill, and A. R. Liddle, The evolution and persistence of dumbbells in electroweak theory, J. High Energy Phys. 08 (2002) 033.
  90. T. Patel and T. Vachaspati, Kibble mechanism for electroweak magnetic monopoles and magnetic fields, J. High Energy Phys. 01 (2022) 059.
  91. A. A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP 5, 1174 (1957), http://www.w2agz.com/Library/Classic%20Papers%20in%20Superconductivity/Abrikosov,%20Type%20II%20Superconductivity,%20Sov-Phys%20JETP%205,%201174%20(1957).pdf.
  92. V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).
  93. H. B. Nielsen and P. Olesen, Vortex line models for dual strings, Nucl. Phys. B61, 45 (1973).
  94. J. Garaud and M. S. Volkov, Superconducting non-Abelian vortices in Weinberg-Salam theory—electroweak thunderbolts, Nucl. Phys. B826, 174 (2010).
  95. T. Vachaspati, Vortex solutions in the Weinberg-Salam model, Phys. Rev. Lett. 68, 1977 (1992).
  96. T. Vachaspati and M. Barriola, A new class of defects, Phys. Rev. Lett. 69, 1867 (1992).
  97. W. B. Perkins, W condensation in electroweak strings, Phys. Rev. D 47, R5224 (1993).
  98. P. Olesen, A W dressed electroweak string, arXiv:hep-ph/9310275.
  99. A. Achucarro, R. Gregory, J. A. Harvey, and K. Kuijken, Cinderella strings, Phys. Rev. Lett. 72, 3646 (1994).
  100. T. Patel and T. Vachaspati, Structure of electroweak dumbbells, Phys. Rev. D 107, 093010 (2023).
  101. V. Domcke, Y. Ema, and T. Patel, Spontaneous magnetogenesis at the electroweak phase transition, arXiv:2505.10637.

Outline

Information

Phys. Rev. D 112, 064018– Published 9 September, 2025
  • Received 6 June 2025
  • Accepted 29 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation