On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

36 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation
and see what choices we have, given a few sensible conditions. 2 Because φ is a dimensionless
variable, dimensional analysis tells us, that the mass dimensions of [a(φ)] = 0
and [b(φ)] = −2. Another condition we can surely impose is that in the decoupling
limit rc → 0, we must recover flat Minkowski spacetime, so limrc→0 b(φ) = 0 and
limrc→0 a(φ) = 1. Further b(φ) can be chosen constant, since every function of φ
can be absorbed in the differential element with a suitable coordinate transformation.
With only the dimensionful quantities rc and k at hand, this leaves only the choice
b ∼ r 2 c . For the other coefficient function, we can conclude that up to constant coefficients
ai, one can write a(φ) = 1 + a1 krc|φ| + . . ., where the ellipsis stands for
higher powers of krc |φ| with the corresponding coefficients, and the absolute value is
dictated by the orbifold symmetry. These arguments are not a rigorous derivation,
but after all, (1.49) does not seem all that strange.
One result of [55] was, that (1.49) is a solution to the 5D Einstein equations with
a large, O(MPl), negative cosmological constant, which corresponds to an extremely
curved Anti de Sitter space. This sounds very puzzling, because we assume a stable
compact extra dimension with flat four dimensional boundaries. Wouldn’t the two
branes not immediately be smashed together? After all we know, our universe is
slightly de Sitter and expands with an observable rate. And what tells us that the
brane metric should be flat?
Answers for these questions can be found by considering Einsteins equations. Making
use of such a coordinate transformation as mentioned above, |z| = e σ (φ)/k, the metric
can be put in the form
ds 2 =
� �2 R �ηµνdx
|z|
µ dx ν − dz 2� . (2.2)
The UV brane is localized at z = 1/k ≡ R and the IR brane at z = ΛUV/(kΛIR) ≡ R ′
in these coordinates. They are particular well suited for the calculation of the Einstein
tensor, because the metric is conformally flat, that is it can be written as GMN =
Ω 2 ηMN with ηMN the flat five dimensional metric and Ω a smooth function, and are
therefore often called conformal coordinates. In order to differentiate between the
notations, we will refer to the notation introduced in (1.49) as φ-notation. Applying
[78] and A(z) = log(|z|/R), the Einstein tensor G becomes a one-liner,
GMN = RMN − 1
2 GMN R
= 3 � �
∂MA∂NA + ∂M∂NA − ηMN ∂K ∂ K A − ∂KA∂ K A �� . (2.3)
The only non-vanishing components are
G55 = 6(A ′ ) 2 , (2.4)
� ′′ ′ 2
Gµν = −3ηµν A − (A ) � . (2.5)
Even if we assume the SM fields to propagate in the bulk, in the presence of an order
Planck scale cosmological constant one can neglect the matter contributions to the
2 If not stated otherwise, the convention for the 5D metric signature in this thesis is (+, −, −, −, −).