On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

action
�
S ∋ d 4 x r 2π
� 1 dt
L ɛ t
�
− 1
4 FµνF µν − 1
2ξ (∂µ Aµ) 2 + 1 �
∂µA5∂
2
µ A5 + M 2 KK ∂tAµ∂tA µ�
�
∂µZ5∂ µ Z5 + M 2 KK ∂tZµ∂tZ µ�
− 1
4 ZµνZ µν − 1
2ξ (∂µ Zµ) 2 + 1
2
− 1
2 W + µνW −µν − 1
+ k
δ(t − 1)
2
− ξ
�
1
MKK t∂t
2
− ξ
�
4
2.3. Profiles of Gauge Bosons 67
ξ ∂µ W + µ ∂ ν W − ν + � ∂µW + 5 ∂µ W − 5 + M 2 KK ∂tW + µ ∂tW −µ�
� 1
2 ∂µh∂ µ h − λv 2 h 2 + ∂µϕ + ∂ µ ϕ − + 1
2 ∂µϕ3∂ µ ϕ3
� 2
− ξ
�
8
+ M 2 Z
2 ZµZ µ + M 2 W W + µ W −µ
�2 t A5
1
δ(t − 1) kMZ ϕ3 + 2MKK t∂t
t Z5
δ(t − 1)kMW ϕ + ��
1
+ MKKt∂t δ(t − 1) kMW ϕ − 1
+ MKKt∂t
t W + 5
In the next step, analogue to (2.88), we decompose the fields in KK modes,
Aµ(x, t) = 1
√ rc
Zµ(x, t) = 1
√ rc
W ± µ (x, t) = 1
√ rc
�
n
�
n
�
n
A (n)
µ (x) χ A n (t) , A5(x, t) = MKK
√ rc
Z (n)
µ (x) χ Z n (t) , Z5(x, t) = MKK
√ rc
W ±(n)
µ (x) χ W n (t) , W ± MKK
5 (x, t) = √
rc
�
n
�
n
�
n
t W − 5
a A n ϕ (n)
A (x) ∂t χ A n (t) ,
a Z n ϕ (n)
Z (x) ∂t χ Z n (t) ,
(2.98)
�
��
.
a W n ϕ ±(n)
W (x) ∂t χ W n (t) ,
(2.99)
and expand the would-be Goldstone bosons in (2.96) in the same basis of mass eigenstates
as the scalars,
ϕ ± (x) = �
n
b W n ϕ ±(n)
W (x) , ϕ3(x) = �
We obtain the EOM (2.91) with the replacement
c 2 A → 1
2 δ(t − 1− )k M 2 a
M 2 KK
n
b Z n ϕ (n)
Z (x) . (2.100)
= δ(t − 1 − )L v2 4g2 4
4M 2 , (2.101)
KK
where the last equality holds for Ma = MW and also for Ma = MZ, if g2 4
by (g ′2
4 + g2 4 ). In the following, we denote both fields by Ma, with the corresponding
replacements for the couplings implied. Consequentially, the orthonormality relation
is replaced