Download Casimir Effect in the Brane World

Joint Meeting of Pacific Region Particle Physics Communities
2006.10.29-11.03 Waikiki, Honolulu, Hawaii
Quantum Field Theory in AdS5
S. Ichinosea and A. Murayamab
aLab.Phys.,SFNS,Univ.Shizuoka
bDep.Education,Shizuoka Univ.
1. Introduction
String, M-theory, D-branes, ・・・
Higher Dimensional Models (Supergravity, Bulk-Boundary, ・・・)
’96 Horava-Witten 11D
’97 Mirabelli-Peskin 5D
’98 Randall-Sundrum 5D
cf. ’21 Kaluza 26’ Klein
’83 Appelquist-Chodos
Unsuccessful so far
• Unrenormalizable
• Few Physical Quantities
Vacuum Energy, Higgs mass, β-function
• Stability problem, Vacuum problem
• SUSY
Recent Progress
’00 Goldberger-Wise, 6D flat
‘communication’ between Bulk and Boundary
through Renormalization group
’01 Randall-Schwartz , 5D warped
Position/Momentum propagator
Position-dependent Cut-Off
2. Flat 5D Quantum Field Theory
5D Massless Scalar Theory
M4X S1/Z2
Kaluza-Klein Expansion
Field Equation
5D propagator
3. Pos./Mom. Propagator Approach
Finite Region
No periodicity at present
5D Propagator
Symmetries ABCC’
Position/Momentum Propagator
Symmetries of Gp(y,y’) w.r.t y and y’
８Regions
8 Regions divided by the symmetries
Fundamental Region: R1
0 = < y < y’ = < l
Step 1
Let us consider the region
R1 + R2 first
Dirichlet b.c. at fixed points
pp > 0 space-like
(i) y
≠
y’
Homogeneous eq.
From Sym(A), we may take
From Dirichlet b.c., we obtain
y＝｜ｙ‘
(ii) |y-y’|<<1
Jump Condition
Inhomogeneous term
+ Sym(A)
y=y’
Kp(y,y’)
In the combined region R1 + R2 , this is equivalent to
taking the absolute value of (y’-y)
 |y’-y|
_
in Kp above
arrange (y,y’)
Step 2. Extension to R1’ and R2’
We require Sym(B)
(y’+y)  |y’+y|
Step 3. Extension to All Other
Regions
same as Step 2
Above result is the complete
expression valid for all regions
sinh, cosh  sin, cos for pp < 0
Flat, p=0.03 << 1/L, space-like, boundary is strongly influenced
extra-axis size l= 3.14
Flat, p=3 >> 1/L, space-like, boundary is NOT influenced
Flat, p=1.7 > 1/L, time-like, boundary is slightly influenced
4. KK-expansion vs P/M propagator
Fourier expansion
5. 5D QFT on AdS5
AdS5 Space-Time Geometry
y coordinate
thickness or
bulk curvature
sign
func.
z coordinate
Conformal Flat
z vs y
w Planck
T Tev
2l period
y ( or z) finite region(s)
adsf4,6,7,8,9
5D Massive Scalar Theory
Kaluza-Klein Expansion
Bessel diff. eq.
Adsf10,11,12
Eigen Functions
5D Propagator
6. P/M Propagator Approach on AdS5
Field Equation
5D propagator
P/M propagator
adspro5,8 symmetries
For simplicity,
pp>0 space-like, pp<0 time-like
8regionsZ
Kp(z,z’)
5D Scalar, Dirichlet b.c., P= -1
pp>0 spacelike, Modified Bessel funcs.
Relation to Expansion Approach: Bessel Fourier expansion
Warped Scalar, p=0.005 <<T=0.04<< 1/L=0.3<w=1, spacelike, boundary and Wall effect are strongly influenced
Warped Scalar, T=0.04<< p=0.1<1/L=0.3<w=1, space-like,
boundary and Wall effect are influenced
Warped Scalar, T=0.04<< 1/L=0.3<w=p=1, space-like,
boundary Effect is small
Warped Scalar, T=0.04<< 1/L=0.3<w=1<<p=5.0 , space-like,
boundary and Wall effect are NOT influenced
7. Conclusion
Behaviour of P/M Propagators on 5D Flat and AdS5
are examined closely. It depends on
p 4D momentum
and the characteristic scales of the theory
l
size of the extra coordinate region
w
thickness (5D curvature)
determine the behaviour.
Especially
•flat versus warped
•space-like versus time-like
•Scalar(Dirichlet) versus Vector(Neumann)
A. Solution to Delta(0) problem
Delta(0) problem Horava ’96
Flat
Warped
x coth x
x= pL
Flat
pL
x= p/w
exp(kL)=1.5
p/w
pz <= L/k
flat cut-off
position
dep.cut-off
flat cut-off
New Regularization
New Uncertainty Principle ?
Warped Vector, p=0.005 <<T=0.04<< 1/L=0.3<w=1, spacelike, boundary and Wall effect are strongly influenced
Warped Vector, T=0.04<< 1/L=0.3<w=p=1, space-like,
boundary Effect is small
Warped Vector, T=0.04<< 1/L=0.3<w=1<<p=5.0 , space-like,
boundary and Wall effect are NOT influenced
Warped Vector, p=0.005 <<T=0.04<< 1/L=0.3<w=1, timelike, boundary and Wall effect are strongly influenced
Warped Vector, T=0.04<< p=0.1 <1/L=0.3<w=1, time-like,
boundary and Wall effect are influenced
Warped Vector, T=0.04<< 1/L=0.3<w=p=1, time-like,
boundary Effect is small
Warped Vector, T=0.04<< 1/L=0.3<w=1<<p=10 , time-like,
boundary and Wall effect are NOT influenced
`83 Appelquist and Chodos
5D bulk quantum effect
1-loop Effective Potential
gMN=gclMN+hMN
Veff =
L5
5b
+
8p
Cosmological term
Cut-off dependent
Quintically divergent
(2pR)5 fc5/3
Casimir Force
Cut-off independent
Gauge independent
(S.I. PL152B(’85)56)
finite
b=-
15
4p2
z (5)
= - 0.394 < 0
attractive
3. Mirabelli-Peskin Model `97
f+, f Compl.
Scalar
y Spinor
F Auxil.
BOUNDARY:
WZ model
P= +
- Z2 parity
Am, A5 Vector
F Scalar
1
l , l2 Spinor
X3, X1,X2 Auxil.
BULK : 5D Super YM
H
W
macro
parameters
X5 = 0
X5 = 2pR