Vacuum Polarization by Topological Defects with Finite Core

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, 025017 (2006) E. R. Bezerra de Mello, A. A. Saharian, J. High Energy Phys. 10, 049 (2006) E. R. Bezerra de Mello, A. A. Saharian, Phys. Rev. D75, 065019, 2007 A. A. Saharian, A. L. Mkhitaryan, arXiv:0705.2245 [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. 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 on the surface of a unit sphere Line element for (D+1)-dim global monopole Solid angle deficit (1-σ2)SD

7. 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 fieldsquare and the energy-momentum tensor Wightman function determines the response of a particle detector of the Unruh-deWitt type

8. 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 Eigenfunctions

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

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. 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. 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: minimal minimal conformal conformal Flower-pot model

13. 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 Vacuum expectation values inside the core

14. Near the core boundary: At the centre of the corel=0mode contributes only to the VEV of the field square and the modes l=0,1contribute 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 minimal conformal conformal VEVs inside the core: Asymptotics

15. spin connection Field equation: Background geometry global monopole VEV of the energy-momentum tensor spinor spherical harmonics Eigenfunctions 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 Fermionic field

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

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

18. 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 Asymptotics

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

22. VEVs outside the string core part induced by the core VEV for the field square: VEV for a string with zero thickness 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. 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. Spacetime inside the core has constant curvature (ballpoint-pen model) Spacetime inside the core is flat (flower-pot model) Specific models for the string core

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

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

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

28. 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 Model with flat spacetime inside the brane

29. 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 Interior region

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

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

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

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