On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

68 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation
(2.90) holds and the BCs become
∂t χ a n(ɛ + ) = 0 , (2.102)
∂t χ a n(1 − ) = − L
2πrc
M 2 a
M 2 KK
χ a n(1 − ) = −L v2 4g2 4
4M 2 χ
KK
a n(1 − ) = −L m2a M 2 χ
KK
a n(1 − ) ,
where the masses of the zero modes ma, with a = W, Z have been introduced. By
comparison with (2.93), the propagator can be found from (2.82), and reads
rcD ξ µν(q, t; t ′ �
1 L � 2
) = ηµν + t< − t 2 − t ′2 + 1 ��
+ O(q 2 ) , (2.103)
2πm 2 W.Z
4π M 2 KK
and the propagator for the photon is given by (2.86).
The 4D effective theory is constructed by inserting the KK decomposition (2.99) into
(2.98), applying the EOM and the orthonormality relation, so that after integrating
out the fifth dimension one ends up with
SGauge,2 = �
�
n
d 4 �
x − 1 (n)
F µν F
4 µν(n) − 1
2ξ
− 1
4 Z(n) µν Z µν(n) − 1
2ξ
�
− 1 +(n)
W µν W
2 −µν(n) − 1
∂ µ Z (n)
µ
� 2
ξ ∂µ W +(n)
µ
�
∂ µ A (n)
µ
+ (mZ n ) 2
2
+ 1 (n)
∂µϕ A 2 ∂µ ϕ (n)
A − ξ(mAn ) 2
ϕ
2
(n)
A ϕ(n)
A
∂ ν W −(n)
ν
+ ∂µϕ +(n)
W ∂µ ϕ −(n)
W − ξ(mWn ) 2 ϕ +(n)
W ϕ−(n)
W
� 2
+ (mAn ) 2
2 A(n) µ A µ(n)
Z (n)
µ Z µ(n)
+ (m W n ) 2 W +(n)
µ W −µ(n)
1 (n)
+ ∂µϕ ϕ
2 2
(n)
Z ϕ(n)
Z
� �
+ d 4 �
1
x
2 ∂µh∂ µ h − λv 2 h 2
�
,
Z ∂µ ϕ (n)
Z − ξ(mZn ) 2
(2.104)
under the additional condition, that the Fourier coefficients in (2.99) and (2.100) are
a a n = − 1
ma , b
n
a n = Ma
√r
χ a n(1 − )
m a n
. (2.105)
Solving the EOM with the BCs (2.102) leads to
χ a �
L
n(t) = Nn
π t c+ n (t) , (2.106)
where only the UV BC is used in determining the unspecified coefficient in the homogeneous
solution, so that
c + n (t) = Y0(x a nɛ) J1(x a nt) − J0(x a nɛ) Y1(x a nt) , (2.107)
c − n (t) = 1
xa d � +
t c n (t)
nt dt
� = Y0(x a nɛ) J0(x a nt) − J0(x a nɛ) Y0(x a nt) . (2.108)