On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

3.5. Implications for Other Theories 125
would vanish, if the 5D doublet and singlet would decompose into purely left- and
right-handed 4D fields. The mixing induced by the couplings to a brane-localized
Higgs are the reason for the cancellation being not perfect, and those terms do also
appear for the example of a purely left-handed coupling above. This is however rooted
in the fermion implementation, which can be seen by the fact that one can not find
these terms based on the propagator (3.61) alone.
Moreover, the IR localized Higgs has another effect, not directly evident from (3.61).
Because it gives masses to the linear combination with axial vector couplings, the corresponding
composite mesons will have a different mass then the ones with vectorial
couplings which do not interact with the composite Higgs. The term in the last line
of (3.66) measures the amount by which this mass splitting sets off the cancellation,
which explains why it must be proportional to the Higgs vev. In the propagator we
encountered the responsible terms in the analysis of the (DD) BCs in Section 2.3.
There is good reason to believe that these conclusions carry over to a composite theory
which cannot be described by a holographic dual. Conformal Technicolor, which was
shortly discussed in section 1.1 is such a small N strongly coupled theory, which
features a composite Higgs but no composite fermions. It does therefore not explain
the hierarchies in the quark sector and has no built in RS-GIM, but it solves the
hierarchy problem by assuming that the scaling dimension of the Higgs mass operator
is ∆ H † H = 4, while ∆H � 1. If it were possible to describe CTC by a 5D theory, it
would correspond to a model in which the whole SM model field content is on the UV
brane and only the Higgs extends into the bulk, but stays close to the UV brane in
order to allow for it to have a small scaling dimension, so that the Yukawa interactions
are only slightly irrelevant [29]. We concluded, that it may be the only feasible way to
bring an explanation of the hierarchy problem based on a strongly coupled theory in
agreement with experimental constraints if it should turn out that quarks of all flavors
are elementary particles. It is however strongly constrained from numerical analyses
in combination with flavor constraints and already ruled out, if the new resonances
do not respect additional flavor symmetries [30].
Whether the results obtained in the last section will also hold for a small N theory
can not be computed. It is however encouraging that QCD, which isn’t exactly a
large N theory does not only exhibit rho-photon mixing in complete analogy with the
elementary-composite mixing in the dual descriptions of RS models, but also features
a vector meson octet, which corresponds to the global SU(3)V symmetry of (quantum)
QCD. In the holographic dual, this octet would be the first KK mode of a gauged
bulk flavor SU(3)V .
If the global symmetry of such a small N theory were SU(3)D × SU(3)S, it is not
too far fetched to assume that there would be mesons for these global charges as well.
The resulting loosening of the flavor bounds is shown by the available parameter space
for the extended model in the ∆ H † H − ∆H plane in Figure 3.8, shaded in green. It
is achieved by eliminating the coefficients of the dangerous mixed chirality operators
from the analysis. For comparison, the bound for minimal CTC as shown in Figure 1.4
is also plotted, the blue shaded region is generically excluded by numerical analyses.
Only a small corner of parameter space opens up, but the situation is actually better