1. Vacuum Polarization by Topological Defects with Finite Core
Aram Saharian
Department of Theoretical Physics, Yerevan State University,
Yerevan, Armenia
International Centre for Theoretical Physics, Trieste, Italy
________________________________________________________
Based on: E. R. Bezerra de Mello, V. B. Bezerra, A. A. Saharian,
A. S. Tarloyan, Phys. Rev. D74, (2006)
E. R. Bezerra de Mello, A. A. Saharian,
J. High Energy Phys. 10, 049 (2006)
Phys. Rev. D75, , 2007
A. A. Saharian, A. L. Mkhitaryan, arXiv: [hep-th]

2. Topological defects
Investigation of topological defects (monopoles, strings, domain walls) is fast developing area, which includes various fields of physics, like low temperature condensed matter, liquid crystals, astrophysics and high energy physics
Defects are generically predicted to exist in most interesting models of particle physics trying to describe the early universe
Detection of such structures in the modern universe would provide precious information on events in the earliest instants after the Big Bang and
Their absence would force a major revision of current physical theories
Recently a variant of the cosmic string formation mechanism is proposed in the framework of brane inflation

3. Quantum effects induced by topological defects
In quantum field theory the non-trivial topology induced by defects leads to non-zero vacuum expectation values for physical observables (vacuum polarization)
Many of treatments of quantum fields around topological defects deal mainly with the case of idealized defects with the core of zero thickness
Realistic defects have characteristic core radius determined by the symmetry breaking scale at which they are formed

4. Aim:
Investigation of effects by non-trivial core on properties of quantum vacuum for a general static model of the core with finite thickness
Scalar field with general curvature coupling parameter
Global monopole, cosmic string, brane in Anti de Sitter (AdS) spacetime
Field:
Defect:

5. Plan
Positive frequency Wightman function
Vacuum expectation values (VEVs) for the field ...square and the energy-momentum tensor
Specific model for the core

6. Vacuum polarization by a global monopole with finite core
Line element on the surface of a unit sphere
Vacuum polarization by a global monopole with finite core
Global monopole is a spherical symmetric topological defect created by a phase transition of a system composed by a self coupling scalar field whose original global O(3) symmetry is spontaneously broken to U(1)
Background spacetime is curved (no summation over i)
Metric inside the core with radius
Line element for (D+1)-dim global monopole
Line element on the surface of a unit sphere
Solid angle deficit (1-σ2)SD

7. Complete set of solutions to the field equation
Scalar field
Field equation
(units ħ = c =1 are used)
Comprehensive insight into vacuum fluctuations is given by the Wightman function
Complete set of solutions to the field equation
Vacuum expectation values (VEVs) of the field square and the energy-momentum tensor
Wightman function determines the response of a particle detector of the Unruh-deWitt type

8. Trace of the surface energy-momentum tensor
Eigenfunctions
Eigenfunctions
hyperspherical harmonic
Radial functions
Notations:
Coefficients are determined by the conditions of continuity of the radial function and its derivative at the core boundary
In models with an additional infinitely thin spherical shell on the boundary of the core the junction condition for the derivative of radial function is obtained from the Israel matching conditions:
Trace of the surface energy-momentum tensor

9. WF for point like global monopole
Exterior Wightman function
Wightman function in the region outside the core
Notations:
ultraspherical polynomial
angle between directions
Part induced by the core
WF for point like global monopole
Rotate the integration contour by π/2 for s=1 and -π/2 for s=2
Notation:

10. Vacuum expectation values
VEV of the field square
For point-like global monopole and for massless field
μ - renormalization mass scale
B=0 for a spacetime of odd dimension
Part induced by the core
On the core boundary the VEV diverges:
At large distances (a/r<<1) the main contribution comes from l=0 mode and for massless field:
For long range effects of the core appear:

11. bilinear form in the MacDonald function and its derivative
VEV of the energy-momentum tensor
For point-like global monopole and for massless field
for D = even number
Part induced by the core
bilinear form in the MacDonald function and its derivative
On the core boundary the VEV diverges:
At large distances from the core and for massless field:
Strong gravitational field:
For ξ>0 the core induced VEVs are suppressed by the factor
(b) For ξ=0 the core induced VEVs behave as σ1-D
In the limit of strong gravitational fields the behavior of the VEVs is completely different for minimally and non-minimally coupled scalars

12. Flower-pot model
In the flower-pot model the spacetime inside the core is flat
Surface energy-momentum tensor
Interior radial function
In the formulae for the VEVs:
conformal
minimal

13. bilinear form in the modified Bessel function and its derivative
Vacuum expectation values inside the core
Subtracted WF:
Mikowskian WF
Notations:
VEV for the field square:
VEV for the energy-momentum tensor:
bilinear form in the modified Bessel function and its derivative

14. Core radius for an internal Minkowskian observer
VEVs inside the core: Asymptotics
Near the core boundary:
At the centre of the core l=0 mode contributes only to the VEV of the field square and the modes l=0,1 contribute only to the VEV of the energy-momentum tensor
In the limit the renormalized VEVs tend to finite limiting values
Core radius for an internal Minkowskian observer
minimal
conformal

15. spinor spherical harmonics
Fermionic field
Field equation:
spin connection
Background geometry
global monopole
VEV of the energy-momentum tensor
Eigenfunctions
spinor spherical harmonics
Eigenfunctions are specified by parity α=0,1, total angular momentum j=1/2,3/2,…, its projection M=-j,-j+1,…,j, and k2=ω2-m2
In the region outside the core

16. VEV of the EMT and fermionic condensate
induced by non-trivial core structure
part corresponding to point-like global monopole
Core-induced part
Decomposition of EMT
Notations:
radial part in the up-component eigenfunctions
Core induced part in the fermionic condensate
Bilinear form in the MacDonald function and its derivative

17. Flower-pot model: Exterior region
Interior line element
Vacuum energy density induced by the core
Notation:
Fermionic condensate

18. Asymptotics Near the core boundary
At large distances from the core for a massless field
In the limit of strong gravitational fields (σ << 1) main contribution comes from l = 1 mode and the core-induced VEVs are suppressed by the factor

19. Flower-pot model: Interior region
Renormalized vacuum energy density
Fermionic condensate
Near the core boundary
At the core centre term l = 0 contributes only:

20. Flower-pot model: Interior region

21. Vacuum polarization by a cosmic string with finite core
Background geometry:
points
and
are to be identified,
conical (δ-like) singularity
angle deficit
For D = 3 cosmic string
linear mass density

22. part induced by the core
VEVs outside the string core
VEV for the field square:
VEV for a string with zero thickness
part induced by the core
For a massless scalar in D = 3:
VEV of the field square induced by the core:
Notation:
- regular solution to the equation for the radial eigenfunctions inside the core
The corresponding exterior function is a linear combination of the Bessel functions

23. bilinear form in the MacDonald function and its derivative
VEV for the energy-momentum tensor
For a conformally coupled massless scalar in D = 3:
VEV of the energy-momentum tensor induced by the core:
bilinear form in the MacDonald function and its derivative
At large distances from the core:
Long-range effects of the core

24. Specific models for the string core
Spacetime inside the core has constant curvature (ballpoint-pen model)
Spacetime inside the core is flat (flower-pot model)

25. Vacuum densities for Z2 – symmetric thick brane in AdS spacetime
warp factor
brane
Background gemetry:
Line element:
We consider non-minimally coupled scalar field
Z2 – symmetry

26. Wightman function outside the brane
Radial part of the eigenfunctions
Notations:
Wightman function
WF for AdS without boundaries
part induced by the brane
Notation:

27. bilinear form in the MacDonald function and its derivative
VEVs outside the brane
VEV of the field square
Brane-induced part for Poincare-invariant brane ( u(y) = v(y) )
VEV of the energy-momentum tensor
Brane-induced part for Poincare-invariant brane
bilinear form in the MacDonald function and its derivative
Purely AdS part does not depend on spacetime point
At large distances from the brane

28. Model with flat spacetime inside the brane
Interior line element:
From the matching conditions we find the surface EMT
In the expressions for exterior VEVs
For points near the brane:
Non-conformally coupled scalar field
Conformally coupled scalar field
For D = 3 radial stress diverges logarithmically

29. Interior region Wightman function:
WF in Minkowski spacetime orbifolded along y - direction
part induced by AdS geometry in the exterior region
WF for a plate in Minkowski spacetime with Neumann boundary condition
Notations:
For a conformally coupled massless scalar field

30. VEV for the field square:
VEV in Minkowski spacetime orbifolded along y - direction
part induced by AdS geometry in the exterior region
Notation:
VEV for the EMT:
For a massless scalar:
Part induced by AdS geometry:

31. For points near the core boundary
Large values of AdS curvature:
For non-minimally coupled scalar field the VEVs coincide with the corresponding quantities induced by Dirichlet boundary in Minkowski spacetime orbifolded along y - direction
For minimally coupled scalar field the VEVs coincide with the corresponding quantities induced by Neumann boundary in Minkowski spacetime orbifolded along y - direction
Vacuum forces acting per unit surface of the brane are determined by
For minimally and conformally coupled scalars these forces tend to decrease the brane thickness

32. Brane-induced VEVs in the exterior region
Minimally coupled D = 4 massless scalar field
Energy density
Radial stress

33. Parts in the interior VEVs induced by AdS geometry
Minimally coupled D = 4 massless scalar field
Energy density
Radial stress

34. Conformally coupled D = 4 massless scalar field
Radial stress

35. For a general static model of the core with finite support we have presented the exterior Wightman function, the VEVs of the field square and the energy-momentum tensor as the sum zero radius defect part + core-induced part
The renormalization procedure for the VEVs of the field square and the energy-momentum tensor is the same as that for the geometry of zero radius defects
Core-induced parts are presented in terms of integrals strongly convergent for strictly exterior points
Core-induced VEVs diverge on the boundary of the core and to remove these surface divergences more realistic model with smooth transition between exterior and interior geometries has to be considered
For a cosmic string the relative contribution of the core-induced part at large distances decays logarithmically and long-range effects of the core appear
In the case of a global monopole long-range effects appear for special value of the curvature coupling parameter