On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

After integration by parts, one finds [112]
� |G|(LGauge+LGF + LMass) =
� �
rc
Aν ∂
2
2 η µν �
− 1 − 1
ξ
+Aφ
�
− e−2σ
r2 ∂
c
2 + ξ
r4 c
�
∂ µ ∂ ν − ∂φ
e −2σ ∂ 2 φe−2σ − e−4σ
r2 M
c
2 A
2.3. Profiles of Gauge Bosons 55
e−2σ r2 ∂φη
c
µν + M 2 Ae −2σ
�
Aµ
�
Aφ
�
. (2.53)
Boundary terms from partial integration vanish as long as the fields have either Dirichlet
or Neumann BCs and the fields are well behaved at 4D infinity.
We will once more change notation, because it is sensible for the numerics to have all
expressions in dimensionless variables. Therefore, let t ≡ MKK z = MKK e σ /k with
MKK = ɛk defined in (1.54), and ɛ ≡ ΛIR/ΛUV. The line element in this t-notation
reads
ds 2 =
�
ɛ
�2 �ηµνdx
t
µ dx ν − M −2
KKdt2� , (2.54)
and the positions of the branes are now intuitively expressed by ratios of scales, t = ɛ
for the UV and t = 1 for the IR brane. In addition, we define the volume of the extra
dimension as L = − ln ɛ = krcπ ≈ 36. Finally, we redefine the scalar component of
AM,
A5(xµ, t) = ɛ
Aφ(xµ, φ) , (2.55)
t rc
in order to adjust the mass dimension and simplify the following calculations. With
these conventions, equation (2.53) reads
�
S = d 4 �
x dφ � |G| (LGauge + LGF) = 1
�
d
2
4 � π
x rcdφ AMK
−π
MN
ξ AN
in which
⎛
K MN
ξ
= ⎝
�
∂2 − M 2
KKt∂t 1
t ∂t
� �
η µν −
= 1
2
�
1 − 1
ξ
d 4 x
2π rc
L
� 1
ɛ
dt
t
AMK MN
ξ AN , (2.56)
�
∂ µ ∂ ν + ɛ2
t 2 M 2 A ηµν 0
0 − ∂2 + ξM 2 1 ɛ2
KK∂tt∂t t − t2 M 2 A
(2.57)
is the inverse of the Feynman propagator in Rξ gauge. After a Fourier transformation
of the coordinates of the non-compact directions, pµ = i∂µ, the following two equations
⎞
⎠