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

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

Z1

Casimir effect

vacuum polarization

Aharonov-Bohm effect

Casimir-Aharonov-Bohm effect

vacuum polarization

Casimir-Aharonov-Bohm effect

vacuum polarization

U(1) gauge Higgs model

upon quantization:

Topological defect of : Abrikosov-Nielsen-Olesen vortex

stress-energy tensor

and are nonvanishing inside the vortex core

1

Einstein-Hilbert equation

scalar curvature

Global 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 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• M.Fierz (unpublished)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

Generalization:d+1-dimensional space-time with a codimension-2 brane

ANO vortex

stress-energy tensor

scalar curvature

metric

tension

flux

Vacuum current induced by a codimension-2 brane

where

g/2π

Yu.A.S., N.D.Vlasii, Fortschr.Phys. 57, 705 (2009)

Induced vacuum magnetic fieldMaxwell equation

magnetic field strength

and its flux

• Identifying with the transverse size of the string core, then and

• Yu.A.S., N.D.Vlasii, Class. Quantum Grav. 26, 195009 (2009)

Energy-momentum tensor of charged massive scalar field

)(),(2

1)(),(

2

1=)( †† xxxxxT

)()(),(4

1)(),()(

4

1 2††2 xRmxgxxRmg

,)(),()(),()4

1( ††

xxRxxg

)()(

2

1=),( 22

xxxxxxxx RmgRmgxxD

.2

1

2

1

2

1

2

1

4

12 2

xxxxxxxxxx RRRRmgg

point-splitting regularization

,)()()()(),(=),( †† xxTxxTxxDxxT

where

Equation for Green's function

2 †0 | ( ) ( ) | 0 =x xm R i T x x

2 † 1= 0 | ( ) ( ) | 0 = ( )x xm R i T x x x x

g

decomposition of the VEV of the chronological operator product

† † †0 | ( ) ( ) | 0 = 0 | ( ) ( ) | 0 0 | ( ) ( ) | 0 .S RT x x T x x T x x

Renormalized VEV of the energy-momentum tensor

.0|)()()()(|0),(lim=)( ††R

xxxxTxxTxxDxt

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

2

0 3 20 3 2 33 2

1

2 v( ) = ( ) = v 2(1 4 ) (2 v) 2(1 4 ) v (2 v) (v; , ),

(2 ) v 1

m dt r t r K mr mr K mr F

r

2

1 21 23 2

1

2 v( ) = (v 4 ) (2 v) (v; , ),

(2 ) v 1

m dt r K mr F

r

2

2 22 2 33 2

1

2 v( ) = (v 4 ) (2 v) 2 v (2 v) (v; , ),

(2 ) v 1

m dt r K mr mr K mr F

r

where sin( ) 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

=

Ggg

F

Conservation: 0=

t

0=)()()( 22

11

111 rtrtrrtr

Trace:

22 3 13 2

1

v 1= (1 6 ) v (2 v) v (2 v) (2 v) (v; , )

2v v 1

m d mg t K mr mr K mr m K mr F

r r

conformal invariance is achieved in the massless limit at 1= :

6

0.=lim1/6=0

tg

m

For a global string

20 3 2 120 =0 3 =0 3 2

1

sin( ) v( ) | = ( ) | = [cosh ( arccosh v) ( /2)]cos

(2 ) v 1F F

m dt r t r

r

22 3[v 2(1 4 )] (2 v) 2(1 4 ) v (2 v) ,K mr mr K mr

2 211 =0 23 2 22

1

sin( ) v v 4( ) | = (2 v),

(2 ) cosh ( arccosh v) ( /2)cosv 1F

m dt r K mr

r

2

22 =0 2 33 2 22

1

sin( ) v v 4( ) | = (2 v) 2 v (2 v)

(2 ) cosh ( arccosh v) ( /2)cosv 1F

dt r K mr mr K mr

For a vanishing string tension

2

0 3 20 =1 3 =1 23 2

1

2sin( ) v( ) | = ( ) | = v 2(1 4 ) (2 v)

(2 ) v v 1

F m dt r t r K mr

r

32(1 4 ) v (2 v) cosh[(2 1) arccosh v],mr K mr F

21 21 =1 23 2

1

2sin( ) v( ) | = (v 4 ) (2 v) cosh[(2 1)arccosh v],

(2 ) v v 1

F m dt r K mr F

r

2

2 22 =1 2 33 2

1

2sin( ) v( ) | = (v 4 ) (2 v) 2 v (2 v) cosh[(2 1)arccosh v].

(2 ) v v 1

F m dt r K mr mr K mr F

r

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).

For a vanishing mass of scalar field:

4 2 22 4

0

1 1( ) = 1 1 30 (1 ) diag(1,1, 3,1)lim

12 60mt r F F

r

211 [1 6 (1 )] diag(2, 1,3,2)

6F F

Trace:

)](16[116

1

2

1=lim

242

0FF

rtg

m

V. P. Frolov and E. M. Serebriany, Phys. Rev. D 35, 3779 (1987).

J. S. Dowker, Phys. Rev. D 36, 3742 (1987).

-dimensional space-time 1d( 1)/2 (3 )/2

00 ( 3)/2 2

1

16 v( ) = ( ) = v

(4 ) v 1

d djj d

mt r t r d

r

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 2

1

16 v( ) = v (v 4 ) (2 v) (v; , ),

(4 ) v 1

d d

dd

mt r d K mr F

r

( 1)/2 (3 )/22 22 ( 3)/2 2

1

16 v( ) = v (v 4 )

(4 ) v 1

d d

d

mt r d

r

( 1)/2 ( 3)/2[ (2 v) 2 v (2 v)] (v; , ),d dK mr mr K mr F

where

sin( ) cosh[2(1 ) arccosh v] sin[(1 ) ]cosh(2 arccosh v)(v; , ) =

cosh(2 arccosh v) cos( )

F F F FF

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)

000 0 0 0vac | ( 1/ 4) ( ) | vact

2 1

1

2 2

d

en

Dimensionless energy density r3t00 as a function of mr (singular brane)

ξ = -1

ξ = 0

ξ = 1/8ξ = 1/4

ξ = 1/2ξ = 1

Dimensionless energy density r3t00 as a function of mr (the case of mr0=1)

ξ = -1

ξ = 0ξ = 1/8

ξ = 1/4

ξ = 1/2ξ = 1

Dimensionless energy density r3t00 as a function of mr (the case of mr0=10-1)

ξ = -1

ξ = 0

ξ = 1/4

ξ = 1/2

ξ = 1

Dimensionless energy density r3t00 as a function of mr (the case of mr0=10-2)

original

zoomedξ = -1

ξ = 0

ξ = 1/8

ξ = 1/2ξ = 1

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

0

002r

rdrtE

101 10944.6

0

mrE

000165.010/10mrE

0106.0100/10mrE

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)

Conclusion

23 2 1 3 1I 2 2

{ sin[(1 ) ] (1 ) sin( )}( ) e {1 [( ) ]}

2(4 ) sin ( / 2)mre F F F F

B r m r O mr

1 0 0(1 4 ) , ,2 2

e eG F

2 HI

1(1 ) ln ,

6 2

meF F F

m

• Cosmic string induces current and magnetic field in the vacuumThe magnetic field strength is directed along a cosmic string

where

where mH is the mass of the string-forming Higgs field.

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

e0

m e

Thank you