Balkan Workshop 2013, 25-29 April, 2013, Vrnjacka Banja, SerbiaBeyond the Standard ModelsVacuum polarization effects inthe...
Outline1. Spontaneous symmetry breaking and topological defects of2. Abrikosov-Nielsen-Olesen vortex and cosmic strings3. ...
U(1) gauge Higgs modelupon quantization:
Topological defect of : Abrikosov-Nielsen-Olesen vortexstress-energy tensorand are nonvanishing inside the vortex core1  
Einstein-Hilbert equationscalar curvatureGlobal characteristics of ANO vortex:linear energy density:flux of the gauge field:
The space-time metric outside the string core:where , a deficit angle is equal to
Cosmic strings were introduced inT.W.B.Kibble, J. Phys. A 9, 1387 (1976); Phys. Rep. 67, 183 (1980)‫‏‬A.Vilenkin, Phys. Re...
Generalization:d+1-dimensional space-time with a d-2-braneANO vortex
stress-energy tensorscalar curvaturemetrictensionflux
Vacuum current induced by a cosmic stringwhereg/2πYu.A.S., N.D.Vlasii, Fortschr.Phys. 57, 705 (2009)‫‏‬
Induced vacuum magnetic fieldMaxwell equationmagnetic field strengthand its flux
Identifying with the transverse size of the string core,then andYu.A.S., N.D.Vlasii, Class. Quantum Grav. 26, 195009 (2009...
Induced vacuum energy-momentum tensor in the cosmic string background 20 3 20 3 2 33 212 v( ) = ( ) = v 2(1 4 ) (2 v) 2(...
Conservation:0=  t  0=)()()( 2211111 rtrtrrtr  Trace:  22 3 13 21v 1= (1 6 ) v (2 v) v (2 v) (2 v) (v; , )2v ...
For a global string20 3 2 120 =0 3 =0 3 21sin( ) v( ) | = ( ) | = [cosh ( arccosh v) ( /2)]cos(2 ) v 1F Fm dt r t rr  ...
For a vanishing string tension20 3 20 =1 3 =1 23 212sin( ) v( ) | = ( ) | = v 2(1 4 ) (2 v)(2 ) v v 1F m dt r t r K mrr ...
For a vanishing mass of scalar field: 4 2 22 401 1( ) = 1 1 30 (1 ) diag(1,1, 3,1)lim12 60mt r F Fr      ...
-dimensional space-time1d( 1)/2 (3 )/200 ( 3)/2 2116 v( ) = ( ) = v(4 ) v 1d djj dmt r t r dr       2( 1...
Magnetic fields in galaxies and galactic clusters: 10-6÷10-5 GaussExtragalactic magnetic fields: 10-16÷10-10 Gaussgalactic...
Conclusion23 2 1 3 1I 2 2{ sin[(1 ) ] (1 )sin( )}( ) e {1 [( ) ]}2(4 ) sin ( / 2)mre F F F FB r m r O mr      ...
Thank you
Y. Sitenko: Vacuum Polarization Effects in the Cosmic String Background and Primordial Magnetic Fields in  the Universe
Y. Sitenko: Vacuum Polarization Effects in the Cosmic String Background and Primordial Magnetic Fields in  the Universe
Y. Sitenko: Vacuum Polarization Effects in the Cosmic String Background and Primordial Magnetic Fields in  the Universe
Y. Sitenko: Vacuum Polarization Effects in the Cosmic String Background and Primordial Magnetic Fields in  the Universe
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Y. Sitenko: Vacuum Polarization Effects in the Cosmic String Background and Primordial Magnetic Fields in the Universe

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Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia

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Y. Sitenko: Vacuum Polarization Effects in the Cosmic String Background and Primordial Magnetic Fields in the Universe

  1. 1. Balkan Workshop 2013, 25-29 April, 2013, Vrnjacka Banja, SerbiaBeyond the Standard ModelsVacuum polarization effects inthe cosmic string background andprimordial magnetic fields inthe UniverseYu.A.Sitenko(BITP, Kyiv, Ukraine)
  2. 2. Outline1. Spontaneous symmetry breaking and topological defects of2. Abrikosov-Nielsen-Olesen vortex and cosmic strings3. Induced vacuum current and magnetic field in the cosmicstring background4. Induced vacuum energy-momentum tensor in the cosmicstring background5. Primordial magnetic fields in the Universe1  
  3. 3. U(1) gauge Higgs modelupon quantization:
  4. 4. Topological defect of : Abrikosov-Nielsen-Olesen vortexstress-energy tensorand are nonvanishing inside the vortex core1  
  5. 5. Einstein-Hilbert equationscalar curvatureGlobal characteristics of ANO vortex:linear energy density:flux of the gauge field:
  6. 6. The space-time metric outside the string core:where , a deficit angle is equal to
  7. 7. Cosmic strings were introduced inT.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 inM.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 scaleof the grand unification of all interactions is characterized bythe values of the deficit angle
  8. 8. Generalization:d+1-dimensional space-time with a d-2-braneANO vortex
  9. 9. stress-energy tensorscalar curvaturemetrictensionflux
  10. 10. Vacuum current induced by a cosmic stringwhereg/2πYu.A.S., N.D.Vlasii, Fortschr.Phys. 57, 705 (2009)‫‏‬
  11. 11. Induced vacuum magnetic fieldMaxwell equationmagnetic field strengthand its flux
  12. 12. Identifying with the transverse size of the string core,then andYu.A.S., N.D.Vlasii, Class. Quantum Grav. 26, 195009 (2009)‫‏‬
  13. 13. Induced vacuum energy-momentum tensor in the cosmic string background 20 3 20 3 2 33 212 v( ) = ( ) = v 2(1 4 ) (2 v) 2(1 4 ) v (2 v) (v; , ),(2 ) v 1m dt r t r K mr mr K mr Fr             21 21 23 212 v( ) = (v 4 ) (2 v) (v; , ),(2 ) v 1m dt r K mr Fr       22 22 2 33 212 v( ) = (v 4 ) (2 v) 2 v (2 v) (v; , ),(2 ) v 1m dt r K mr mr K mr Fr       wheresin( )cosh[2(1 ) arccosh v] sin[(1 ) ]cosh(2 arccosh v)(v; , ) = ,cosh(2 arccosh v) cos( )F F F FF      1)4(1=22=  GggF
  14. 14. Conservation:0=  t  0=)()()( 2211111 rtrtrrtr  Trace:  22 3 13 21v 1= (1 6 ) v (2 v) v (2 v) (2 v) (v; , )2v v 1m d mg t K mr mr K mr m K mr Fr r       conformal invariance is achieved in the massless limit at 1= :60.=lim1/6=0 tgm
  15. 15. For a global string20 3 2 120 =0 3 =0 3 21sin( ) v( ) | = ( ) | = [cosh ( arccosh v) ( /2)]cos(2 ) v 1F Fm dt r t rr        22 3[v 2(1 4 )] (2 v) 2(1 4 ) v (2 v) ,K mr mr K mr     2 211 =0 23 2 221sin( ) v v 4( ) | = (2 v),(2 ) cosh ( arccosh v) ( /2)cosv 1Fm dt r K mrr          222 =0 2 33 2 221sin( ) v v 4( ) | = (2 v) 2 v (2 v)(2 ) cosh ( arccosh v) ( /2)cosv 1Fdt r K mr mr K mr     
  16. 16. For a vanishing string tension20 3 20 =1 3 =1 23 212sin( ) v( ) | = ( ) | = v 2(1 4 ) (2 v)(2 ) v v 1F m dt r t r K mrr           32(1 4 ) v (2 v) cosh[(2 1) arccosh v],mr K mr F  21 21 =1 23 212sin( ) v( ) | = (v 4 ) (2 v)cosh[(2 1)arccosh v],(2 ) v v 1F m dt r K mr Fr      22 22 =1 2 33 212sin( ) v( ) | = (v 4 ) (2 v) 2 v (2 v) cosh[(2 1)arccosh v].(2 ) v v 1F m dt r K mr mr K mr Fr      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).
  17. 17. For a vanishing mass of scalar field: 4 2 22 401 1( ) = 1 1 30 (1 ) diag(1,1, 3,1)lim12 60mt r F Fr         211 [1 6 (1 )] diag(2, 1,3,2)6F F           Trace: )](16[116121=lim2420FFrtgmV. P. Frolov and E. M. Serebriany, Phys. Rev. D 35, 3779 (1987).J. S. Dowker, Phys. Rev. D 36, 3742 (1987).
  18. 18. -dimensional space-time1d( 1)/2 (3 )/200 ( 3)/2 2116 v( ) = ( ) = v(4 ) v 1d djj dmt r t r dr       2( 1)/2 ( 3)/2[v 2(1 4 )] (2 v) 2(1 4 ) v (2 v) (v; , ),d dK mr mr K mr F        ( 1)/2 (3 )/21 21 ( 1)/2( 3)/2 2116 v( ) = v (v 4 ) (2 v) (v; , ),(4 ) v 1d dddmt r d K mr Fr        ( 1)/2 (3 )/22 22 ( 3)/2 2116 v( ) = v (v 4 )(4 ) v 1d ddmt r dr       ( 1)/2 ( 3)/2[ (2 v) 2 v (2 v)] (v; , ),d dK mr mr K mr F   wheresin( )cosh[2(1 ) arccosh v] sin[(1 ) ]cosh(2 arccosh v)(v; , ) =cosh(2 arccosh v) cos( )F F F FF      
  19. 19. Magnetic fields in galaxies and galactic clusters: 10-6÷10-5 GaussExtragalactic magnetic fields: 10-16÷10-10 Gaussgalactic dynamoSeed (primordial) magnetic field: ≥10-20 GaussSuperconducting cosmic string(Witten et al, 1986)Gauge cosmic string inducing vacuum polarization
  20. 20. Conclusion23 2 1 3 1I 2 2{ sin[(1 ) ] (1 )sin( )}( ) e {1 [( ) ]}2(4 ) sin ( / 2)mre F F F FB r m r O mr         1 0 0(1 4 ) , ,2 2e eG F                2 HI1(1 ) ln ,6 2meF F Fm      •Cosmic string induces current and magnetic field in the vacuumThe magnetic field strength is directed along a cosmic stringwherewhere mH is the mass of the string-forming Higgs field.is the gauge flux, is the coupling constant of the quantized field with thestring-forming gauge field, is the mass of the quantized field, is its electriccharge. Thus, owing to the vacuum polarization by a cosmic string, the latter isdressed in a shell of the force lines of the magnetic field. The flux of the magneticshell ise0m e
  21. 21. Thank you

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