On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

52 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation
F G
H
UV brane IR brane
F
(ND)
G
(DD)
I
(NN)
H
(DN)
Figure 2.6: Illustration of the breaking by BCs at the UV and IR brane. In the dual
theory, the subgroup F is gauged by elementary fermions and the group H leaves
the composite sector invariant after confinement. The right hand side shows a Venn
diagram of the groups. The BCs of the fields living in the corresponding complements
are also given (For example, fields in G may have any combination of BCs, but fields
in G\(F ∪ H) must have Dirichlet BCs on both branes).
for all other combinations. We will consider the consequences of Goldstones theorem
in a situation in which breaking on both branes can appear, and its implications for
the dual theory.
In general, one has a bulk gauge group G, which is broken down to the subgroup
F on the UV brane by explicit breaking through weakly gauging this subgroup in
the elementary sector, see Figure 2.6. The dual theory is given by the Lagrangian
(2.41), where a denotes the indices of the generators of F . On the IR brane, G is
broken down to the subgroup H, which is the subgroup that still leaves the composite
sector invariant after confinement sets in. We will call the intersection of these groups
I = F ∩ H. One therefore expects dim G/H = dim G − dim H Goldstone bosons
from the spontaneous symmetry breaking. However, dim F/I of those will be eaten,
because there exist elementary gauge fields corresponding to the generators of this
spontaneously broken subgroup. In the end, one thus has dim I massless gauge bosons
in the elementary sector and NPNGB = dim G/H −dim F/I pseudo-Nambu Goldstone
bosons, since they correspond to explicitly broken generators. The general setup
is depicted in Figure 2.6, where the corresponding BCs for the vector part of the
bulk gauge fields have been given. Since the breaking takes place via BCs, there are
no scalar degrees of freedom on the brane. The longitudinal degrees of freedom of
the gauge bosons are in this setup provided by the A5 components of the residual
dim G/(F ∪ H), which must have Neumann BCs on both branes and thus develop
zero modes, compare Section 4 of [62]. Both approaches, symmetry breaking via a
brane field or via BCs, give the same number of degrees of freedoms. One can further
infer, that the A5 mode of a bulk gauge field is the holographic dual of a composite
pseudo Nambu Goldstone boson.
Higgsless TC is therefore described by an RS model without a brane scalar and the
following symmetries realized in the bulk and on the branes respectively (here, and
in the following scenarios, only the electroweak sector plays a role and the not noted