String Theory Demystified

274 String Theory Demystifi ed
i
µ
have induced metrics given by h = g ( x , y ), where i = 12 , µν µν i
for the visible and
hidden branes, respectively. The Randall-Sundrum action is
S dyd x g M 4
= ∫
3 ⎛ ⎞
5
− R− ⎝
⎜ Λ
⎠
⎟ +
2
2
4 () i
() i
∫d
x − h ( Λ + L i matter ) (16.21)
∑
i=
1 i
The index i on the second integral indicates that we integrate over each brane
separately. The additional terms included here are
• M 5 : The Planck mass in fi ve dimensions.
• Λ : The cosmological constant in the bulk.
• Λ and Λ : The cosmological constants on the visible and hidden branes.
1 2
• R: The scalar curvature in fi ve dimensions.
• L i ()
matter:
The lagrangian density for matter fi elds on the visible and hidden
branes. On the visible brane, it is the standard model fi elds but could be
different on the hidden brane.
The dimension y ranges over 0 ≤ y≤π r, where r is a constant and the two
c
c
branes are located at the boundaries. The visible brane is located at y = π r, while
1 c
the hidden brane is located at y = 0.
2
Imposing a requirement that Poincaré invariance is respected, the following
metric is chosen that is a slice of anti-de Sitter space:
The exponential term e ky −2
− ky µ ν
ds = e ηµνdx
dx + dy
2 2 2
(16.22)
is called the warp factor. We will see that the warp
factor connects mass scales in our 3 + 1 dimensional universe to fi ve-dimensional
mass parameters.
It can be shown that the cosmological constants in the bulk and on each of the
branes are given by
3 2
Λ =−6Mk
P
Λ =− Λ
3
=−6Mk
1 2
p
(16.23)
If k < M , this tells us that the space-time curvature of the bulk is small compared
p
to the Planck scale.
The exponential warp factor causes the large gap between the observed Planck
and electroweak scales. Moving to an effective four-dimensional theory, Randall
and Sundrum showed that the Planck mass in four dimensions could be derived
from the fi ve-dimensional Planck mass via
3
πr
2 3 c 2 M5
MP = M5 ∫ dye = 1−e
−πrc
k
( )
− ky −2k
rc
π (16.24)