Vacuum densities for a thick brane in AdS spacetime

Aram SaharianDepartment of Theoretical Physics, Yerevan State University,

Yerevan, Armenia

_______________________________________________QFEXT07

University of Leipzig, September 16-21, 2007

Plan 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/0312092; Phys. Rev. D70, 064026 (2004), hep-th/0406211

Bulk and surface Casimir densities for higher dimensional branes with compact internal spaces

A.A.S., Phys. Rev. D73, 044012 (2006), hep-th/0508038; Phys. Rev. D73, 064019 (2006), hep-th/0508185; Phys. Rev. D74, 124009 (2006), hep-th/0608211

Casimir densities for a thick brane

A.A.S., A.L. Mkhitaryan, JHEP 08, 063 (2007), arXiv:0705.2245

• 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

DykD

kiikD

yk

kezx

dxdxzk

dydxdxeds

D

D

/

,)( 2

222

Bulk geometry: D+1-dim AdS spacetime

Branes: Minkowskian branes located at y=a and y=b)1,(DR

Scalar field

11DAdS

Branes

Braneworlds with warped internal spaces

Bulk geometry: (D+1)-dimensional spacetime 11DAdS

2222 dydxdxedxdxeds kiik

ykyk DD

Branes with a warped internal space (higher dimensional brane models))1,(DR

• Field: Scalar field with an arbitrary curvature coupling parameter

• Boundary conditions on the branes:

Field and boundary conditions

bayxBA yyy , ,0)()~~

(

0) ( 2 Rmii

Randall-Sundrum braneworld models

Orbifolded y-direction: S1/Z2

RS two-brane model

Orbifold fixed points (locations of the branes) y=0 y=L

Warp factor, ykDe

RS single brane model

y=0 y

brane

Warp factor||ykDe

Z2 identification)( yy

Boundary conditions in Randall-Sundrum braneworld

Boundary conditions in Randall-Sundrum braneworld

Vacuum fluctuations

Comprehensive insight into vacuum fluctuations is given by the Wightman function

)'()(0)'()(0)',( * xxxxxxWComplete set of solutions to the field equation

Vacuum expectation values (VEVs) of the field square and the energy-momentum tensor

)',(lim0)(0 '2 xxWx xx

0)(04

1

)',(lim0)(0

2

''

xRg

xxWxT

ikkil

lik

kixxik

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

),(),( 0 xxWxxW 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

),(ˆlim00 xxGDT MNxx

MN

Bulk energy-momentum tensor

MND̂

000 MNMN TT + Part induced by a

single brane at y=a+ Part induced by a

single brane at y=b

+ Interference part

)pp,(TMN ,...,...,diag00 ||

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, 1|| aykD

ayaykD , )( 2exp

ayaykD D , )( )2(exp

Strong gravitational fields suppress boundary-induced VEVs

Conformally coupled massless field

For a conformally coupled massless scalar 2/1ay ,

ay ,

• 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, 085005, 2004; hep-th/0308108)

2

1E

Surface energy-momentum tensor

00|| 0)(0

1)( vDv TgxdE

EE v )(

])2/12()[;( 2)( LL

MNMNs

MN nhKMxT

Induced metric

Unit normal

Extrinsic curvature

tensor

M

Ln

• 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

)()()()()( ,...),,...,(diag00 sj

sj

sj

sj

NsM ppT

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

2/121

1

)2(2

1

nD

D

mkkd

E

00 0)(0

)( vv TgdyE

baj

DjD

sj

s zkE,

)()( )/(

)()( sv EEE

00 )( DvDj TP

1

)()(

,

)()(

)(

DjD

sjDj

j

jbaj

jsj zk

EDkPn

z

EdSEPdVdE

• 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

DjG

Physics for an observer on the brane

1DG

)()2()()2(

1

1

)2( jbkDabkD

DDDj

D

De

e

GkDG

)()2()( 8~8 abkDDDjDj

sjDjDj

DeMGG

Cosmological constant on the

brane y=j

Effective Planck mass on the brane y=j

222 /)1(4/ DkmDDD

• 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

TeVMMTeVM PlDbD16

1 10~ ,~

40)( abkD

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 a thick brane AdS

Vacuum densities for Z2 – symmetric thick brane in AdS spacetime

Vacuum densities for Z2 – symmetric thick brane in AdS spacetime

AdS space

warp factor||2 ykDe

y

a2

brane

Background gemetry:

Line element:

aydyexdedte

aydydxdxeds

ywyvyu

ykD

|| ,

|| ,2)(22)(22)(2

2||22

We consider non-minimally coupled scalar field 0)( 2 Rmi

i

Z2 – symmetryyy

Wightman function outside the braneWightman function outside the brane

Radial part of the eigenfunctions

, )()(

),,,()( 2/

ayzYBzJAe

aykyRyf yDkD

Notations: ,/ ,/)1(4/ 222D

ykD kezkmDDD D

Wightman function

bS xxxxWxxW )()(2/),(),( WF for AdS without boundaries part induced by the brane

)cosh()()(

)(

)()(

)2()()( 22

22

2/1

ktk

zKzK

zK

zIdekdzz

kxx

a

a

k

xkiDD

DD

b

Notation: Dak

aaD

ayay

D

kezxFzixkaRk

zixkyR

kD

GDxFxxF D / ),(

)/,,(

|)/,,(

)1(

16

2)()(

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 braneVEVs outside the brane

VEV of the field squarebs

222 2/0000

Brane-induced part for Poincare-invariant brane ( u(y) = v(y) )

)()(

)(

)2/()4(2

0

12/

12 xzK

xzK

xzIxdx

D

zk

a

aDD

DDD

b

VEV of the energy-momentum tensor b

kis

ki

ki TTT 2/0000

Brane-induced part for Poincare-invariant brane

)]([)(

)(

)2/()4()(

0

12/

1

xzKFxzK

xzIxdx

D

zkT i

a

aDD

ki

DDD

b

ki

bilinear form in the MacDonald function

and its derivative

Purely AdS part does not depend on spacetime point

At large distances from the brane

aDb

kiDb

zzykTyk ), 2exp(~ ), 2exp(~2

)1()1()0( ... DFFF

Model with flat spacetime inside the braneModel with flat spacetime inside the brane

Interior line element: xeXdydxdxeds akak DD ,222

From the matching conditions we find the surface EMT

0 ,1,...,1,0 ,8

1 00

Dik

G

D kiD

ki

In the expressions for exterior VEVs

)()tanh()/(22/)()( xFamkmDDxFxxF D

For points near the brane:D

bay 22 )(~

D

b

DD

D

bayTayT 10

0 )(~ ,)(~Non-conformally coupled scalar field

Conformally coupled scalar field D

b

DD

D

bayTayT 320

0 )(~ ,)(~

For D = 3 radial stress diverges logarithmically

Interior regionInterior region

Wightman function: ),(),(),( 10 xxWxxWxxW WF in Minkowski spacetime orbifolded along y - direction

part induced by AdS geometry in the exterior region

2/),(),( )(0 xxWxxW N WF for a plate in Minkowski spacetime

with Neumann boundary condition

)cosh()(

))(cosh())(cosh(

)}(),)({cosh(

)}(,{

2),(

22

)(

1

tvkxx

yxyx

xzKaxC

xzKexCdxekd

kzxxW

k a

aax

xkiD

Da

Notations: )( /)()(22/)()()}(),({ vgakufuufDDvguvfvgufC D 222)( mexx akD

For a conformally coupled massless scalar field 0),(1 xxW

VEV for the field square:b

2

ren,0

2

ren

2

VEV in Minkowski spacetime orbifolded along y - direction

part induced by AdS geometry in the exterior region

m

DD

bxUxymxdx

D)()(cosh)(

)2/(

)4( 212/222/

2

Notation:)}/(),{cosh(

)}/(,{)(

22

22

D

Dax

kmxKaxC

kmxKeCxU

VEV for the EMT: b

ki

ki

ki TTT

ren,0ren

For a massless scalar: 0 ,2

1

)4(

)(ren,012/)1(ren,0

DDDD

Dii T

D

y

DT

m

iDD

b

ii xUyxFmxdx

DT )(),()(

)2/(

)4( )(12/222/

)(2

),( , 2/1)(cosh)14()(cosh

),(22

2)(2

22

22)(

xm

xyxFxy

mx

x

D

xyyxF Di

Part induced by AdS geometry:

For points near the core boundaryD

b

DD

D

b

ii

D

byaTyaTya )(~ ,)(~ ,)(~ 22

Large values of AdS curvature: Dkma /1,

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 byb

DDT

For minimally and conformally coupled scalars these forces tend to decrease the brane thickness

Brane-induced VEVs in the exterior regionBrane-induced VEVs in the exterior region

1

1.5

2

2.5

3

zza0

0.25

0.5

0.75

1

akD

0

0.1

0.2

T00 b

kDD1

1

1.5

2

2.5

3

zza1

1.5

2

2.5

3

zza0

0.25

0.5

0.75

1

akD

0

0.02

0.04

0.06

0.08

TDD b

kDD1

1

1.5

2

2.5

3

zza

100 / D

DbkT 1/ D

Db

DD kT

azz / azz /

DakDak

Energy density Radial stress

Minimally coupled D = 4 massless scalar field

Parts in the interior VEVs induced by AdS geometryParts in the interior VEVs induced by AdS geometry

1

2

3

akD

0

0.2

0.4

0.6

0.8

ya0

0.05

0.1

0.15

aD1T00bint

1

2

3

akD

Minimally coupled D = 4 massless scalar field

0

1

2

3

4

5

am2

4

6

8

10

akD

0

0.001

0.002

0.003

0.004

aD1TDDbint

0

1

2

3

4

5

am

b

D Ta 00

1

b

DD

D Ta 1

Energy density Radial stress

Dak Dakay /

am

Conformally coupled D = 4 massless scalar field

0

1

2

3

4

5

am2

4

6

8

10

akD

0

0.00005

0.0001

0.00015

aD1TDDbint

0

1

2

3

4

5

am

b

DD

D Ta 1

amDak

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

ConclusionConclusion

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

ConclusionConclusion

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

Solid angle deficit (1-σ2)SD

,,...,2 ,2 ,1

22

2

DiDnr

nR ii

222222Ddrdrdtds

Line element on the surface of a

unit sphere

ardedredteds Drwrvru ,2)(22)(22)(22

a

E. R. Bezerra de Mello, A.A.S., JHEP 10, 049 (2006), hep-th/0607036

E. R. Bezerra de Mello, A.A.S., Phys. Rev. D75, 065019 (2007), hep-th/0701143