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) Phys. Rev. D75, 065019, 2007 A. A. Saharian, A. L. Mkhitaryan, arXiv:0705.2245 [hep-th]

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

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

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:

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

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

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

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

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:

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:

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

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

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

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

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

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

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

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

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

Flower-pot model: Interior region

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

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

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

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)

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

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

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

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

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

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:

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

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

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

Conformally coupled D = 4 massless scalar field Radial stress

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