Vacuum densities for a thick brane in AdS spacetime Aram Saharian Department of Theoretical Physics, Yerevan State University, Yerevan, Armenia _______________________________________________ QFEXT07 University of Leipzig, September 16-21, 2007

Plan Motivation Wightman function, bulk Casimir densities, and interaction forces for parallel plates in AdS spacetime Surface energy-momentum tensor and the energy balance Applications to Randall-Sundrum braneworlds: Radion stabilization and the generation of cosmological constant Exterior vacuum desnities for a thick brane in AdS Special model of the brane core and the interior vacuum densities

Bulk and surface Casimir densities for parallel branes in AdS spacetime A.A.S., Nucl. Phys. B712, 196 (2005), hep-th/ ; Phys. Rev. D70, (2004), hep-th/ Bulk and surface Casimir densities for higher dimensional branes with compact internal spaces A.A.S., Phys. Rev. D73, (2006), hep-th/ ; Phys. Rev. D73, (2006), hep-th/ ; Phys. Rev. D74, (2006), hep- th/ Casimir densities for a thick brane A.A.S., A.L. Mkhitaryan, JHEP 08, 063 (2007), arXiv:

Questions of principal nature related to the quantization of fields propagating on curved backgrounds AdS spacetime generically arises as a ground state in extended supergravity and in string theories AdS/Conformal Field Theory correspondence: Relates string theories or supergravity in the bulk of AdS with a conformal field theory living on its boundary Braneworld models: Provide a solution to the hierarchy problem between the gravitational and electroweak scales. Naturally appear in string/M-theory context and provide a novel setting for discussing phenomenological and cosmological issues related to extra dimensions Why AdS?

Quantum effects in braneworld models are of considerable phenomenological interest both in particle physics and in cosmology Braneworld corresponds to a manifold with boundaries and all fields which propagate in the bulk will give Casimir type contributions to the vacuum energy and stresses Vacuum forces acting on the branes can either stabilize or destabilize the braneworld Casimir energy gives a contribution to both the brane and bulk cosmological constants and has to be taken into account in the self- consistent formulation of the braneworld dynamics Quantum effects in braneworlds

Geometry of the problem AdS bulk Scalar field Parallel branes Bulk geometry: D+1-dim AdS spacetime Branes: Minkowskian branes located at y=a and y=b

Scalar field B ranes Braneworlds with warped internal spaces Bulk geometry: (D+1)-dimensional spacetime Branes with a warped internal space (higher dimensional brane models)

Field: Scalar field with an arbitrary curvature coupling parameter Boundary conditions on the branes: Field and boundary conditions

Randall-Sundrum braneworld models Orbifolded y-direction: S 1 /Z 2 RS two-brane model Orbifold fixed points (locations of the branes) y=0 y=L Warp factor, RS single brane model y=0 y brane Warp factor Z 2 identification

Boundary conditions in Randall-Sundrum braneworld

Vacuum fluctuations 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

Radial Kaluza-Klein modes are zeros of the combination of the Bessel and Neumann functions and the eigensum contains the summation over these zeros Generalized Abel-Plana formula allows to extract from the vacuum expectation values the boundary-free part and to present the brane induced part in terms of exponentially convergent integrals Wightman Function Part induced by a single brane at y=a Part induced by a single brane at y=b ++ + Interference part

Vacuum energy-momentum tensor in the bulk second order differential operator Decomposition of the EMT Vacuum EMT is diagonal Bulk energy-momentum tensor + Part induced by a single brane at y=a + Part induced by a single brane at y=b + Interference part Vacuum energy density Vacuum pressures in directions parallel to the branes Vacuum pressure perpendicular to the branes

Casimir densities induced by a single brane Notation: VEV in the region is obtained by the replacements proper distance from the brane Asymptotic at large distances from the brane, Strong gravitational fields suppress boundary-induced VEVs

Conformally coupled massless field For a conformally coupled massless scalar

Vacuum force acting per unit surface of the brane In dependence of the coefficients in the boundary conditions the vacuum interaction forces between the branes can be either attractive or repulsive Geometry of two branes = Self-action force + Force acting on the brane due to the presence of the second brane (Interaction force) In particular, the vacuum forces can be repulsive for small distances and attractive for large distances Stabilization of the interbrane distance (radion field) by vacuum forces

Total vacuum energy per unit coordinate surface on the brane Volume energy in the bulk In general Difference is due to the presence of the surface energy located on the boundary For a scalar field on manifolds with boundaries the energy- momentum tensor in addition to the bulk part contains a contribution located on the boundary (A.A.S., Phys. Rev. D69, , 2004; hep-th/ ) Surface energy-momentum tensor Induced metric Unit normal Extrinsic curvature tensor

Vacuum expectation value of the surface EMT on the brane at y=j This corresponds to the generation of the cosmological constant on the branes by quantum effects Induced cosmological constant is a function of the interbrane distance, AdS curvature radius, and of the coefficients in the boundary conditions: In dependence of these parameters the induced cosmological constant can be either positive or negative Surface energy-momentum tensor and induced cosmological constant

Energy Balance Total vacuum energy Volume energy Surface energy It can be explicitly checked that Perpendicular vacuum stress on the brane Energy balance equation

D-dimensional Newton’s constant measured by an observer on the brane at y=j is related to the fundamental (D+1)-dimensional Newton’s constant by the formula For large interbrane distances the gravitational interactions on the brane y=b are exponentially suppressed. This feature is used in the Randall- Sundrum model to address the hierarchy problem Same mechanism also allows to obtain a naturally small cosmological constant on the brane generated by vacuum fluctuations Physics for an observer on the brane Cosmological constant on the brane y=j Effective Planck mass on the brane y=j

For the Randall-Sundrum brane model D=4, the brane at y=a corresponds to the hidden brane and the brane at y=b corresponds to the visible brane Observed hierarchy between the gravitational and electrowek scales is obtained for For this value of the interbrane distance the cosmological constant induced on the visible brane is of the right order of magnitude with the value implied by the cosmological observations Hidden brane Visible brane AdS bulk Location of Standard Model fields Induced cosmological constant in the Randall-Sundrum model

Many of treatments of quantum fields in braneworlds deal mainly with the case of the idealized brane with zero thickness. This simplification suffers from the disatvantage that the curvature tensor is singular at the brane location From a more realistic point of view we expect that the branes have a finite thickness and the thickness can act as natural regulator for surface divergences In sting theory there exists the minimum length scale and we cannot neglect the thickness of the corresponding branes at the string scale. The branes modelled by field theoretical domain walls have a characteristic thickness determined by the energy scale where the symmetry of the system is spontaneously broken Vacuum densities for a thick brane AdS

Vacuum densities for Z 2 – symmetric thick brane in AdS spacetime AdS space warp factor brane Background gemetry: Line element: We consider non-minimally coupled scalar field Z 2 – 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: In the RS 1-brane model with the brane of zero thickness In the model under consideration the effective brane mass term is determined by the inner structure of the core

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 Energy densityRadial stress Minimally coupled D = 4 massless scalar field

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

Conformally coupled D = 4 massless scalar field Radial stress

The Casimir effect for parallel plates in the AdS bulk is exactly solvable Quantum fluctuations of a bulk scalar field induce surface densities of the cosmological constant type localized on the branes In the original Randall-Sundrum model for interbrane distances solving the hierarchy problem, the value of the cosmological constant on the visible brane by order of magnitude is in agreement with the value suggested by current cosmological observations without an additional fine tuning of the parameters There is a region in the space of Robin parameters in which the interaction forces are repulsive for small distances and are attractive for large distances, providing a possibility to stabilize interbrane distance by using vacuum forces Vacuum energy localized on the branes plays an important role in the consideration of the energy balance Conclusion

For a general static plane symmetric model of the brane core with finite support we have presented closed analytic formulae for the exterior Wightman function, the VEVs of the field square and the energy-momentum tensor Phenomenological parameters of the zero-thickness brane model are expressed in terms of the core structure 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 Strong gravitational fields suppress the boundary-induced VEVs Core-induced VEVs, in general, 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 Conclusion

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 sphereLine element on the surface of a unit sphere Line element for (D+1)-dim global monopole Solid angle deficit (1-σ 2 )S D Line element on the surface of a unit sphere E. R. Bezerra de Mello, A.A.S., JHEP 10, 049 (2006), hep-th/ E. R. Bezerra de Mello, A.A.S., Phys. Rev. D75, (2007), hep-th/