On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

58 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation
in the above equations by replacing
and
�
∂µA µ − ξ
c 2 A → L v2 4g2 4
4M 2 δ(t − 1
KK
− ) + L v2 4g2 4
4M 2 δ(t − ɛ
KK
+ ) . (2.75)
� 2
r2 ∂φe
c
−2σ Aφ
�
v4g5
�
∂µA µ − ξ
2rc
→
(δ(|φ| − π) + δ(|φ|)) ϕA + 1
r2 ∂φe
c
−2σ ��2 Aφ . (2.76)
This will only affect the propagator of the vector component, because in contrast to
the bulk Higgs, the kinetic term only needs couplings to the vector, in order to be
4D Lorentz invariant, as in (2.13). Note also, that here 〈H〉 = v5/ √ 2 and [ϕA] = 1,
because H is a brane localized scalar. As a consequence, there will be no Higgs KK
modes and only one Goldstone boson for the zero mode of the gauge field. Squaring
the delta function in (2.76) looks worrisome, but is unproblematic, because the resulting
terms will cancel in the KK decomposed theory, as discussed in Section 2.3.
Alternatively, a brane Higgs can also be implemented with brane gauge fixing terms,
as in [113, Sec. 2.3]. Note also, that the bulk solution has always ∆ = 3 in case of a
brane Higgs.
If the replacement (2.75) is made in (2.60), it basically corresponds to invoking BCs
for the gauge field. Similar to (2.70), they can be found by integrating over small
intervals around the branes (in φ-notation), or from a infinitesimally displaced point
in the bulk to the brane (in t−notation)
∂tD ξ µν(q, t; t ′ �
�
) �
t=ɛ + = L v2 4g2 4
∂tD ξ µν(q, t; t ′ �
�
) �
t=1− = −L v2 4g2 4
4M 2 D
KK
ξ µν(q, ɛ + ; t ′ ) , (2.77)
4M 2 D
KK
ξ µν(q, 1 − ; t ′ ) . (2.78)
The δ-function should be implemented in a way that does not clash with the boundary
conditions of the fields, which is induced by the orbifold symmetry – and guarantees
the vanishing of boundary terms from integration by parts. This is denoted by the superscripts
at 1− and ɛ + , which indicate a regularization of the δ function, for example
δ(t − 1− ) ≡ limη→0 δ(t − 1 + η), with
⎧
⎨ 1
η , for t ∈ [1 − η, 1] ,
δ(t − 1 + η) =
⎩0,
otherwise ,
(2.79)
so that the second equation in (2.77) makes sense, even though the orbifold symmetry
fixes the propagator or its derivative to vanish at the fixed points. See section 2.4 for
more details.
Symmetry breaking via BCs and by a brane Higgs field implies a different physical
spectrum, because the former will not generate a Higgs resonance in the spectrum.
However, for the following discussion, we will omit this distinction and choose a general