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

1. Balkan Workshop 2013, 25-29 April, 2013, Vrnjacka Banja, Serbia
Beyond the Standard Models
Vacuum polarization effects in
the cosmic string background and
primordial magnetic fields in
the Universe
Yu.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
1  

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

4. Topological defect of : Abrikosov-Nielsen-Olesen vortex
stress-energy tensor
and are nonvanishing inside the vortex core
1  

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

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

7. 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M.Fierz (unpublished)‫‏‬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

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

9. stress-energy tensor
scalar curvature
metric
tension
flux

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

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

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. Induced vacuum energy-momentum tensor in the cosmic string background
 
2
0 3 2
0 3 2 33 2
1
2 v
( ) = ( ) = v 2(1 4 ) (2 v) 2(1 4 ) v (2 v) (v; , ),
(2 ) v 1
m d
t r t r K mr mr K mr F
r

  


 
        
  

2
1 2
1 23 2
1
2 v
( ) = (v 4 ) (2 v) (v; , ),
(2 ) v 1
m d
t r K mr F
r

 


 
  
  

 
2
2 2
2 2 33 2
1
2 v
( ) = (v 4 ) (2 v) 2 v (2 v) (v; , ),
(2 ) v 1
m d
t 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 F
F
   

 
  


1
)4(1=
22
= 









 




G
gg
F

18. Conservation:
0=
  t
  0=)()()( 2
2
1
1
11
1 rtrtrrtr  
Trace:
  2
2 3 13 2
1
v 1
= (1 6 ) v (2 v) v (2 v) (2 v) (v; , )
2v v 1
m d m
g t K mr mr K mr m K mr F
r r



 




 
    
 

conformal invariance is achieved in the massless limit at 1
= :
6

0.=lim
1/6=0 





tg
m

19. For a global string
2
0 3 2 12
0 =0 3 =0 3 2
1
sin( ) v
( ) | = ( ) | = [cosh ( arccosh v) ( /2)]cos
(2 ) v 1
F F
m d
t r t r
r
 
 


 
  
  

 2
2 3[v 2(1 4 )] (2 v) 2(1 4 ) v (2 v) ,K mr mr K mr 
    
2 2
1
1 =0 23 2 22
1
sin( ) v v 4
( ) | = (2 v),
(2 ) cosh ( arccosh v) ( /2)cosv 1
F
m d
t r K mr
r
  
  
 
 
 
  

 
2
2
2 =0 2 33 2 22
1
sin( ) v v 4
( ) | = (2 v) 2 v (2 v)
(2 ) cosh ( arccosh v) ( /2)cosv 1
F
d
t r K mr mr K mr
  
  
 





20. For a vanishing string tension

2
0 3 2
0 =1 3 =1 23 2
1
2sin( ) v
( ) | = ( ) | = v 2(1 4 ) (2 v)
(2 ) v v 1
F m d
t r t r K mr
r
 




        
  

32(1 4 ) v (2 v) cosh[(2 1) arccosh v],mr K mr F  
2
1 2
1 =1 23 2
1
2sin( ) v
( ) | = (v 4 ) (2 v)cosh[(2 1)arccosh v],
(2 ) v v 1
F m d
t r K mr F
r





 
  
  

 
2
2 2
2 =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 d
t 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).

21. For a vanishing mass of scalar field:
 4 2 2
2 4
0
1 1
( ) = 1 1 30 (1 ) diag(1,1, 3,1)lim
12 60m
t r F F
r

 




       
 21
1 [1 6 (1 )] diag(2, 1,3,2)
6
F F 
 
       
  
Trace:
 )](16[11
6
1
2
1
=lim
2
42
0
FF
r
tg
m













V. P. Frolov and E. M. Serebriany, Phys. Rev. D 35, 3779 (1987).
J. S. Dowker, Phys. Rev. D 36, 3742 (1987).

22. -dimensional space-time1d
( 1)/2 (3 )/2
0
0 ( 3)/2 2
1
16 v
( ) = ( ) = v
(4 ) v 1
d d
j
j d
m
t 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 )/2
1 2
1 ( 1)/2( 3)/2 2
1
16 v
( ) = v (v 4 ) (2 v) (v; , ),
(4 ) v 1
d d
dd
m
t r d K mr F
r

 

  


 
  
  

( 1)/2 (3 )/2
2 2
2 ( 3)/2 2
1
16 v
( ) = v (v 4 )
(4 ) v 1
d d
d
m
t 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 F
F
   

 
  



23. 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

24. Conclusion
2
3 2 1 3 1
I 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 e
G F 
 
     
      
  
 
2 H
I
1
(1 ) ln ,
6 2
me
F F F
m


 
    
 
•Cosmic string induces current and magnetic field in the vacuum
The 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
e0
m e

25. Thank you