On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

42 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation
ϕ(xµ),
Zbulk
� � � �
�
φ(x, z) � = ϕ(x) =
z=0
Dφ e iSAdS(φ) ≡ e iS(ϕ) , (2.20)
corresponds to the generating functional of the correlation functions of a boundary
CFT, where the boundary values of the bulk fields act as sources for the operators
O(x),
� � � � � �
�
d4x ϕ(x)O(x)
Zbulk φ(x, z) � = ϕ(x) = e
. (2.21)
z=0 CFT
In other words it is possible to derive all n-point functions of the strongly coupled
boundary CFT by varying (2.20) with respect to the boundary values ϕ of the bulk
fields φ, and setting them to zero
〈O1O2 . . . On〉 = 1 δ
n!
n �
ln
�
Zbulk �
� . (2.22)
δϕ1δϕ2 . . . δϕn
We will go through this exercise with the example of a free massive bulk scalar φ ≡
φ(x, z), with corresponding 5D action in euclidean signature,
SAdS = 1
�
2
d 4 x dz
� ϕ=0
� �5 R �GMN ∂Mφ ∂Nφ + M
z
2 φ 2� . (2.23)
Applying a Fourier transform along the flat space-time directions
�
φ(x, z) = d 4 p e ipx φ(p, � z) , (2.24)
leads to the bulk equations of motion (EOM),
�
−p 2 − c2
z2 + z3 1
∂z
z
3 ∂z
�
�φ(p, z) = 0 , (2.25)
where c ≡ MR denotes the dimensionless 5D mass parameter. With the substitution
�φ(p, z) = (zp) 2 f(zp), the equation of motion is recognizable as a Bessel PDE,
2 ∂2
∂
(pz) f(pz) + (pz)
∂(pz) 2 ∂(pz) f(pz) − � (pz) 2 + c 2 + 4 � f(pz) = 0 , (2.26)
with the solution
where
�φ(p, z) = p 2 z 2�
�
I∆−2(pz) C1 + K∆−2(pz) C2 , (2.27)
∆(∆ − 4) = c 2 , ∆± = 2 ± � 4 + c 2 . (2.28)
It should be noted, that one of the strange properties of Anti de Sitter space is that
bulk masses as low as the Breitenlohner Freedman bound c 2 > −4, are compatible
with unitarity [90]. This leaves us with the two solutions ∆+ ≥ 2 and ∆− < 2,