On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

48 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation
γ ρ γ
× ×
ϕ O ϕ
× ×
Figure 2.2: Illustration of γ − ρ mixing and the mixing of elementary and composite
states in the dual theory of the RS1 scenario.
1
1−|c−
L ∋ Lel + ω M 2 | � ψ¯ 0
LOR + h.c. � + LCFT , (2.43)
where M ∼ ΛUV is of order Planck scale and c is the localization parameter of the
bulk field which is dual to OR(x). In this notation the scaling dimension of the
composite fermion operator reads dim OR(x) = 4 − 3/2 − 1 + |c − 1/2| = 3/2 +
|c − 1/2|. After introducing an IR brane, and therefore spontaneous breaking of the
CFT, OR(x) represents a composite bound state in a strongly coupled theory with a
mass gap dictated by the IR scale ΛIR. In the following, we will therefore make use
of the symbolic notation introduced in Section 1.1 and write the Lagrangian at the
compositeness scale ΛIR as
L ∋ Lel + ω M
� � 1
|c−
ΛIR
2
M
| �
¯ψ 0
LOR + h.c. � + LCFT . (2.44)
This corresponds to attributing the varying scaling dimension of the composite operator
to the power law running in a strongly coupled theory close to the conformal
window, compare [37, p.4-5]. With the identification
γ = |c − 1/2| , (2.45)
the Lagrangian (2.44) can now directly be compared to the Lagrangian (1.24), which
shows how the localization in the bulk is connected to the mixing between elementary
and composite states. As illustrated in Figure 2.3, for a localization parameter
c < −1/2, the mixing operator becomes irrelevant and therefore there is next to no
mixing between the CFT and the elementary sector. As a consequence, in the mass
eigenbasis, the light eigenstate will correspond to a mostly elementary field. This is
the case for the light quarks and leptons in the RS model with an IR localized Higgs.
For a bulk mass −1/2 < c < 1/2, which corresponds to a IR localized fermion, the
mixing becomes relevant and maximal mixing is achieved for c = 1/2, which will be
dual to roughly a 50% composite, 50% elementary eigenstate. From (2.44), one would
expect, that even more IR localized fields the mixing decreases again, however for
localizations c > 1/2, the dual theory will look different and (2.44) does not hold.
We will not go into detail concerning the construction of the dual Lagrangian in this
case, but refer to the relevant literature [104]. We note, that in this case the spectrum
of the composite theory contains a light field, with negligible admixture from
the elementary sector, which in turn develops a mass term of the order of the cutoff.
One can therefore consider a fermion localized at c > 1/2 to be almost completely
composite [91].
This scenario is related to the two possible branches ∆± , which we found in the scalar