vacuum polarization by topological defects with finite core aram saharian department of theoretical...
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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