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Outline 1. Spontaneous symmetry breaking and topological defects of

XII International School-seminar “The Actual Problems of Microworld Physics” July 22 – August 2, 2013, Gomel, Belarus Vacuum polarization effects in the cosmic string background and primordial magnetic fields in the Universe Yu. A. Sitenko ( BITP, Kyiv, Ukraine ) . Outline

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Outline 1. Spontaneous symmetry breaking and topological defects of

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  1. XII International School-seminar “The Actual Problems of Microworld Physics”July 22 – August 2, 2013, Gomel, BelarusVacuum polarization effects in the cosmic string background and primordial magnetic fields in the UniverseYu. A. Sitenko(BITP, Kyiv, Ukraine)

  2. Outline 1. Spontaneous symmetry breaking and topological defects of 2. Abrikosov-Nielsen-Olesen vortex and cosmic strings 3. Induced vacuum current and magnetic field in the cosmic string background 4. Induced vacuum energy-momentum tensor in the cosmic string background 5. Primordial magnetic fields in the Universe

  3. vacuum polarization Casimir effect

  4. Aharonov-Bohm effect

  5. vacuum polarization Casimir-Aharonov-Bohm effect

  6. vacuum polarization Casimir-Aharonov-Bohm effect

  7. upon quantization: U(1) gauge Higgs model

  8. stress-energy tensor and are nonvanishing inside the vortex core Topological defect of : Abrikosov-Nielsen-Olesen vortex

  9. scalar curvature Global characteristics of ANO vortex: linear energy density: flux of the gauge field: Einstein-Hilbert equation

  10. The space-time metric outside the string core: where , a deficit angle is equal to

  11. Cosmic strings were introduced in T.W.B.Kibble, J. Phys. A 9, 1387 (1976); Phys. Rep. 67, 183 (1980)‏ A.Vilenkin, Phys. Rev. D 23, 852 (1981); D 24, 2082 (1981)‏ Earlier studies in M.Fierz (unpublished)‏ J.Weber, J.A.Wheeler, Rev.Mod.Phys. 29, 509 (1957)‏ L.Marder, Proc. Roy. Soc. London A 252, 45 (1959)‏ A cosmic string resulting from a phase transition at the scale of the grand unification of all interactions is characterized by the values of the deficit angle

  12. Generalization: d+1-dimensional space-time with a codimension-2 brane ANO vortex

  13. stress-energy tensor scalar curvature metric tension flux

  14. where g/2π Yu.A.S., N.D.Vlasii, Fortschr.Phys. 57, 705 (2009)‏ Vacuum current induced by a codimension-2 brane

  15. Maxwell equation magnetic field strength and its flux Induced vacuum magnetic field

  16. Identifying with the transverse size of the string core, then and Yu.A.S., N.D.Vlasii, Class. Quantum Grav. 26, 195009 (2009)‏

  17. point-splitting regularization Energy-momentum tensor of charged massive scalar field where

  18. Equation for Green's function decomposition of the VEV of the chronological operator product Renormalized VEV of the energy-momentum tensor

  19. Induced vacuum energy-momentum tensor in the cosmic string background Yu.A.S., Vlasii. Class. Quantum Grav. 29, 095002 (2012) where

  20. Conservation: Trace: conformal invariance is achieved in the massless limit at

  21. For a global string

  22. For a vanishing string tension Yu. A. S. and V. M. Gorkavenko, Phys. Rev. D 67, 085015 (2003). Yu. A. S. and A. Yu. Babansky, Mod. Phys. Lett. A 13, 379 (1998). Yu. A. S. and A. Yu. Babansky, Phys. At. Nucl. 61, 1594 (1998).

  23. For a vanishing mass of scalar field: Trace: V. P. Frolov and E. M. Serebriany, Phys. Rev. D 35, 3779 (1987). J. S. Dowker, Phys. Rev. D 36, 3742 (1987).

  24. -dimensional space-time where

  25. Transverse size of a codimension-2 brane temporal component of the vacuum energy-momentum tensor (vacuum energy density) V.M.Gorkavenko,Yu.A.S., O.B.Stepanov. Intern.J.Mod.Phys.A 26, 3889 (2011)

  26. Dimensionless energy density r3t00 as a function of mr (singular brane) ξ= 1 ξ= 1/2 ξ= 1/4 ξ= 1/8 ξ= 0 ξ= -1

  27. Dimensionless energy density r3t00 as a function of mr (the case of mr0=1) ξ= 1 ξ= 1/2 ξ= 1/4 ξ= 1/8 ξ= 0 ξ= -1

  28. Dimensionless energy density r3t00 as a function of mr (the case of mr0=10-1) ξ= 1 ξ= 1/2 ξ= 1/4 ξ= 0 ξ= -1

  29. Dimensionless energy density r3t00 as a function of mr (the case of mr0=10-2) original ξ= 1 ξ= 1/2 ξ= 1/8 zoomed ξ= 0 ξ= -1

  30. Total energyV.M.Gorkavenko,Yu.A.S., O.B.Stepanov. Intern.J.Mod.Phys.A 26, 3889 (2011)

  31. Magnetic fields in galaxies and galactic clusters: 10-6÷10-5 Gauss Extragalactic magnetic fields: 10-16÷10-10 Gauss galactic dynamo Seed (primordial) magnetic field: ≥10-20 Gauss Superconducting cosmic string (Witten et al, 1986) Gauge cosmic string inducing vacuum polarization L.V.Zadorozhna, B.I.Hnatyk, Yu.A.S. Ukrain.J.Phys.58,398(2013)

  32. Conclusion • Cosmic string induces current and magnetic field in the vacuum The magnetic field strength is directed along a cosmic string where is the gauge flux, is the coupling constant of the quantized field with the string-forming gauge field, is the mass of the quantized field, is its electric charge. Thus, owing to the vacuum polarization by a cosmic string, the latter is dressed in a shell of the force lines of the magnetic field. The flux of the magnetic shell is where mH is the mass of the string-forming Higgs field.

  33. Thank you

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