On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

3.1. Electroweak Precision Observables 95
parameter is related to ρ = m 2 W /(m2 Z c2 w) = 1 + αT , which is protected by the approximate
global SU(2)L × SU(2)R symmetry of the Higgs sector. This global symmetry,
is generally assumed to be realized in TC extensions of the SM as well, which leads
to small deviations.
In the minimal RS model, the leading contributions to the oblique parameters arise
from mixing of the KK excitations into the zero mode gauge bosons, [115, 140, 139],
S = 2πv2
M 2 �
1 −
KK
1
�
, T =
L
π2v 2c2 wM 2 �
L −
KK
1
�
. (3.6)
L
The result is shown in Figure 3.2, in which the cross is at the best fit point with
68%, 95% and 99% confidence level (CL) ellipses in shades of yellow, gray and blue
respectively, the star is the SM prediction and the orange (red) band rising in the
direction of a large T parameter shows (3.6) for a KK scale of 3 − 10 TeV (1 − 3)
TeV. In addition, the effect of varying the volume of the extra dimension L ∈ [5, 36]
is plotted with increasing L in the direction of the arrow. Two things are remarkable
here. First, the S parameter seems to be under control. This can be interpreted as
an unexpected nice result from the holographic theory, because it seems to contradict
the estimates relying on NDA or upscaled QCD (keep in mind that the RS model is
dual to a large Nc strongly coupled theory). However, it was shown for holographic
higgsless models, that if one localizes the fermions on the UV brane, a large positive
contribution to S in very good agreement with the naive estimates occurs, compare
[141, Sec.3]. On the other hand, models with IR brane localized fermions give large
(order L) negative contributions to S, as discovered in [142, Sec.3].
In [143] these results could be explained by showing that the contributions to the S
parameter are sensitive to the localization of the fermions and vanish for c ≈ −1/2.
This dependence is not the visible in (3.6), because we only consider universal corrections
there. It can be motivated in the dual theory described in Section 2.2, how a
different fermion localization may affect the S parameter. If the quarks are confined to
the UV brane, they are elementary fermions with direct couplings to the electroweak
gauge bosons and reproduce the SM quark contributions to S, while the composite
fields live in the IR and contribute in agreement with what we expect from a strongly
interacting theory. Confining the quarks to the IR brane makes them fully composite
and couplings to the electroweak gauge bosons can only occur through mixing
with composite vector bosons (there is no direct coupling due to the large anomalous
dimension of IR localized fields). Therefore, the contribution from the elementary
quarks to the reference value S = 0 is not present in this model, resulting in a negative
value of S, even though the technifermions still give positive corrections. The
possibility to achieve moderate S can therefore be viewed as another advantage of
partially composite fermions.
The second remark is, that the T parameter is enhanced by a volume factor and therefore
significantly larger than expected from NDA. An explanation can also be found
from the dual description. In Section 2.2 it was shown that a global symmetry in the
strongly coupled theory is described by a bulk gauge symmetry. As pointed out above,
the T parameter is small because it is protected by a global symmetry, the custodial
SU(2)L × SU(2)R. The minimal model as shown in Figure 2.1, with the SM gauge