Vacuum Polarization by Topological Defects with Finite Core

Aram SaharianDepartment of Theoretical Physics, Yerevan State University,

Yerevan, ArmeniaInternational 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]

Topological defectsTopological 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 defectsQuantum 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 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)

• 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

Scalar field

Field equation 0)( 2 Rmii

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

(units ħ = c =1 are used)

Eigenfunctions

2 ),,...,,( ,...,2,1,0 ,),,()()( 21 DnlemYrfx nti

kl

hyperspherical harmonic

Radial functions

arrYBrJAr

arrRrf

ll lln

l

l , )()(

),,()( 2/

Notations:2/1

2

2

22

1

2

4

1)1()1(

2

1 ,

D

Dnn

nlm l

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:

)(

1

16)()( af

D

Gafaf lll

Trace of the surface energy-momentum tensor

Eigenfunctions

Exterior Wightman functionWightman function in the region outside the core

22)'(

220

22

2/

02/

1

)()(

)',(),()(cos

)'(

2

2)',( mttin

ll

nD

D

eaYaJ

ragrag

m

dC

rr

nl

nSxxW

ll

ll

Notations: )()()()(),( aJrYaYrJraglllll

)()/,(

)/,(

1

16

2)()( zF

azaR

azaRaa

D

GnzFzzF

l

l

ultraspherical polynomial angle between directions ),( and ),(

Part induced by the core ),(),()()( xxWxxWxx mc

WF for point like global monopole

)()()(

)(

2

1)()(

)()(

),(),( )()(

2,1)(22

rHrHaH

aJrJrJ

aYaJ

ragrag ss

ss

])'cosh[()()(

)(~

)(~

)(cos)'(

2)'()( 22

22

2

02/

1

mzttmz

rzKzrK

zaK

zaIdzzC

rr

nl

nSxx

m

n

ll

nD

D

c

ll

l

l

Rotate the integration contour by π/2 for s=1 and -π/2 for s=2

Notation: )()/,(

)/,(

1

16

2)()(

~2/

2/

zFazeaR

azeaRaa

D

GnzFzzF

il

il

Vacuum expectation values

VEV of the field squarec

2

renm,

2

ren

2

For point-like global monopole and for massless field

,/)ln( 1

renm,

2 DrrBA μ - renormalization mass scale

B=0 for a spacetime of odd dimension

Part induced by the core

!)1(

)2()22( ,

)(

)(~

)(~

22

2

0

12

lD

DlDlD

mz

zrK

zaK

zaIdzzD

Sr ll m

l

Dn

D

c

l

l

l

On the core boundary the VEV diverges: 1 ,)(~2 Darc

At large distances (a/r<<1) the main contribution comes from l=0 mode and for massless field: 0 ,)/(~ 0

122 0 D

cra

For long range effects of the core appear:

0 ,)/ln(

)/(~ 0

12

ra

ra D

c00

VEV of the energy-momentum tensor cikikik TTT renm,ren

For point-like global monopole and for massless field

,/)ln( 1)2()1(

renm,

Dikikik rrqqT 0)2( ikq for D = even number

Part induced by the core

armz

zrKF

zaK

zaIdzzD

SrT ll

l

l

i

l ml

Dn

ki

D

c

ki

,)]([

)(~

)(~

2 22

)(

0

31

bilinear form in the MacDonald function

and its derivative

On the core boundary the VEV diverges: 1 ,)(~ 22 DarT

c

ki

At large distances from the core and for massless field:

0 ,)/ln(

)/(~ and 0 ,)/(~ 0

1

012 0

ra

raTraT

D

c

ki

D

c

ki

Strong gravitational field: 1(a) For ξ>0 the core induced VEVs

are suppressed by the factor )/ln()1()/2(exp arnn (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 flat22222 ])1([ Ddardrdtds

Surface energy-momentum tensor

DiD

D

Ga

D ki

ki ,...,3,2 ,

1

2 ,

8

11

1 00

00

Interior radial function arrrrJCrR nnlll )1(~ ,~/)~(),( 2/

2/

In the formulae for the VEVs:

)()(

)())1(4(

11

2

1)()(

~

2/

2/ zFzI

zIznnzFzzF

nl

nl

1.5 2 2.5 3 3.5 4

-0.0002

0

0.0002

0.0004

0.0006

0.0008

1.2 1.4 1.6 1.8 2

-0.002

0

0.002

0.004

0 ,3 ,21 mDac

D

conformal

minimal

0 ,3 ,00

1 mDTac

D

minimal

conformal

ar /

Vacuum expectation values inside the core

Subtracted WF: )',()',()'()( MsubxxWxxWxx Mikowskian WF

)'(cosh)'~()~(

),()(cos

)'~~(

21)'()(

222/2/

220

2/

2/sub

ttmrzIrzI

mz

zazUdzC

rr

nl

nSxx

nlnl

m

l

l

nln

D

Notations:)}(),({)}(),({

)}(),({)}(),({/1),(

2/2/

2/2/

zKzICzIzIC

zIzKCzKzICzU

ll

ll

nlnl

nlnl

l

)()()()()()}(),({ zgzfzzfzgzzfzgzfC

2/)1(4/11 nn

VEV for the field square: )~(),(

~1 2

2/220

ren

2 rzImz

zazUdzD

rS nlm

l

lln

D

VEV for the energy-momentum tensor:

)]~([),(

~2 2/)(

2/22

3

0ren

rzIFmz

zaUzdzD

rST nl

inl

m

l

lln

D

kik

i

bilinear form in the modified Bessel function and its derivative

VEVs inside the core: Asymptotics

Near the core boundary:1

ren

11ren

2

ren

2 )/(1~ ,,...,2,0 ,)/(1~ ,)/(1~ DDii

D raTDiraTra

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

fixed ,0 aCore radius for an internal Minkowskian observer

0.2 0.4 0.6 0.8 1

0.04

0.06

0.08

0.1

0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

ar /~

5.0 ,0

,3 ,ren

21

m

DaD

minimal

conformal

5.0 ,0

,3 ,ren

00

1

m

DTaD

minimal

conformal

Fermionic field

Field equation: ,0 mi

spin connection

Background geometry

arddrdrdt

arddedredteds

rwrvru

,sin

,sin2222222

222)(22)(22)(22

global monopole

VEV of the energy-momentum tensor },{00 TT

Eigenfunctions

),()(

),()(

Mlj

Mjlti

rgn

rfe

spinor spherical harmonics

),,,( Mjk

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

2/ ,2/ ,)1(

njlnjln In the region outside the core

)(/)()()(

/)()()(

21

21

mrkrYckrJcknrg

rkrYckrJcrf

nn

VEV of the EMT and fermionic condensate

cmTTT 00

induced by non-trivial core structure

part corresponding to point-like global monopole

Core-induced part

Decomposition of EMT

)](,[)(

~)(

~

2)(

2,1)(

)(

122

3

22xrKxF

xaK

xaI

mx

xdxl

rT

lls

l

s

l

sl mc

Notations: 2/1/ ,)1( 1 lls

s

)(2/12)/,(

)/,()()(

~2/

2/)( xfGa

axeaR

axeaRaxfxxf

i

i

s

ss

),()( krRrf radial part in the up-component eigenfunctions

Core induced part in the fermionic condensate

)()(

)(~

)(~

2

1

2122

2

22

2,1)(

)(

122

xrKimx

mxrKi

mx

m

xaK

xaIdxxl

r

ll

s

l

s

l

ss

sl mc

Bilinear form in the MacDonald function

and its derivative

Flower-pot model: Exterior region

Interior line element

arrddrdrdtds )1(~ ),sin(~ 2222222

Vacuum energy density induced by the core

)()()}(),({

)}(),({22

12/1

2/122

12

20

0 xrKxrKxaKxaIC

xaIxaICmxxdxl

rT

ll

l

l

l

l

l mc

Notation:

)()(2/)()1/1()()()}(),({ xgxfxxfxgxxfxgxfC

Fermionic condensate

)()()}(),({

)}(),({2

12

2/1

2/1

221

2

2

xrKxrKxaKxaIC

xaIxaIC

mx

xdxl

r

mll

l

l

l

l

l mc

Asymptotics

Near the core boundary

)(12

)1/1( ,

)(120

/1122

2321

12

20

0ara

m

araT

ar

aTT

cccc

At large distances from the core for a massless field3/25/2 )/(~ ,)/(~

raraTcc

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 )]/ln()/2(exp[ ar

1.2 1.4 1.6 1.8 2

-0.005

0

0.005

0.01

2

0.5

2

0.5

0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.002

-0.001

0

0.001

0.002

ar /

cTa 0

04

cTa 1

14

5.1/ ar

Flower-pot model: Interior region

Renormalized vacuum energy density

)~()~()}(),({

)}(),({~

22/1

22/1

2/1

2/122

12

10

0 rxIrxIxaKxaIC

xaKxaKCmxxdxl

rT ll

l

l

l m l

l

Fermionic condensate

)()()}(),({

)}(),({~

22/1

22/1

2/1

2/1

221

2

1

xrIxrIxaKxaIC

xaKxaKC

mx

xdxl

r

mll

l

l

l m l

l

Near the core boundary

aa

raa

m

raaT

ar

aTT

~ ,

)~~(~12

)1/1( ,

)~~(~120

/11~~

~22

2321

12

20

0

At the core centre term l = 0 contributes only:

3,2,1 ,1)( ),1/(3)(

)}(),({

)}(),({)(

3

2

)(222)0(

2/1/12/1

2/1/12/1

222

)(3

430~

xCxamxC

xKxIC

xKxKC

amx

xCxdx

aT

mar

Flower-pot model: Interior region

0.2 0.4 0.6 0.8 1

-0.04

-0.02

0

0.02

0.04

2

0.5 2

0.5

0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.015

-0.01

-0.005

0

0.005

0.01

00

4~ Ta

11

4~ Ta

ar ~/~

5.0~/~ ar

Vacuum polarization by a cosmic string with finite core

Background geometry:

N

i i

N

i i

ardzdrdrarPdt

ardzdrdrdtds

1222222

122222

2

,)/(

,

00 /2 ,1for 1)( ,/1)(lim qxxPqxPx

,0 0 ),...,,,( 1 Nzzr points ),...,,,( 10 Nzzr and

are to be identified,

conical (δ-like) singularityangle deficit

02

For D = 3 cosmic string G82 0 linear mass density

VEVs outside the string core

VEV for the field square:cs

222 0000

VEV for a string with zero thickness

part induced by the core

For a massless scalar in D = 3: 22

222(s)

ren

2

48

10000

r

qMs

VEV of the field square induced by the core:

)2/)1((

2 ),(

)(

)()( '

2/)1(322/)3(22

00

2

DAxrK

xaK

xaImxxdx

A DD

Dqnqn

qnD

mn

D

c

Notation: )(),1(

),1()()( zf

izaR

izaRzfzzf

n

n

),/( aarRn - regular solution to the equation for the radial eigenfunctions inside the core

The corresponding exterior function is a linear combination of the Bessel functions )( , )( rYrJ qnqn

VEV for the energy-momentum tensorc

kis

ki

ki TTT 0000

For a conformally coupled massless scalar in D = 3:

)1,3,1,1(diag1440

142

4(s)

ren

r

qT ki

VEV of the energy-momentum tensor induced by the core:

)],([)(

)()( ' )(2/)3(223

00

xrKFxaK

xaImxxdx

AT qn

iqn

qn

qnD

mn

Dkic

ki

c

DDccTTT ...3

30

0bilinear form in the MacDonald

function and its derivative

At large distances from the core:)/ln(

1~ ,

)/ln(

1~

112

arrT

arr Dc

kiDc

)/ln(/1~.../...(s)

renar

c Long-range effects of the core

Specific models for the string coreSpecific models for the string core

Spacetime inside the core has constant curvature (ballpoint-pen model)

12)1( ,

)/1(

)/1('1

),1(

),1(2

222

q

az

qP

qPq

izaR

izaRn

n

n

n

Spacetime inside the core is flat (flower-pot model)

)1(2)/(

)/(

),1(

),1(

q

qzaI

qzaIza

izaR

izaR

n

n

n

n

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)()(

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

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