Vacuum Polarization by Topological Defects with Finite Core

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. D 74, 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. D 75, 065019, 2007 A. A. Saharian, A. L. Mkhitaryan, ar. Xiv: 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 Field: Scalar field with general curvature coupling parameter Defect: Global monopole, cosmic string, brane in Anti de Sitter (Ad. S) spacetime

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 • 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) Line element for (D+1)-dim global monopole Solid angle deficit (1 -σ2)SD • Background spacetime is curved (no summation over i) • Metric inside the core with radius Line element on the surface of a unit sphere

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-de. Witt type

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

Exterior Wightman function in the region outside the core ultraspherical polynomial angle between directions Notations: 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:

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 Mac. Donald function and its derivative On the core boundary the VEV diverges: At large distances from the core and for massless field: Strong gravitational field: (a) 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: minimal conformal

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

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 limiting values minimal the renormalized VEVs tend to finite Core radius for an internal Minkowskian observer minimal conformal

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 k 2=ω2 -m 2 In the region outside the core

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

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 are to be identified, angle deficit For D = 3 cosmic string and conical (δ-like) singularity linear mass density

VEVs outside the string core VEV for the field square: part induced by the core 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

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 Mac. Donald 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 Z 2 – symmetric thick brane in Ad. S spacetime Background gemetry: warp factor Z 2 – symmetry Ad. S space brane Line element: We consider non-minimally coupled scalar field

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

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

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 Ad. S 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 Notation: VEV for the EMT: For a massless scalar: Part induced by Ad. S geometry: part induced by Ad. S geometry in the exterior region

For points near the core boundary Large values of Ad. S 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 Ad. S 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