On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

96 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories
group only, does not correspond to a theory with such a protection and therefore the
T parameter turns out to be large. One can therefore consider a custodial symmetry
indispensable for any extension of the SM, and in the custodially protected RS model
with an corresponding extended bulk group one finds [114],
S = 2πv2
M 2 �
1 −
KK
1
�
, T = −
L
π2v 2c2 wM 2 1
, (3.7)
KK L
as shown in Figure 3.2 by the orange band extending from the SM prediction towards
larger S.
One can think of other solutions, for example a version of the RS model in which
the volume factor L is smaller than 36. The effect of this modification is shown in
Figure 3.2 as well, in which the volume factor grows from L = 5 to L = 36 along the
black arrows. The physical meaning of such a truncation is that the gauge hierarchy
problem will only be cured up to some intermediate scale ΛUV = e L TeV. Based on
little Higgs models, these models are called little RS (LRS) models and just like the
little Higgs model they need some UV completion already at ΛUV = e L TeV. Since
lowering the volume factor will affect many observables also in the flavor sector, we
will come back to this option when we discuss FCNCs.
Corrections to Z → b ¯ b
The only observables with at least moderate deviations from the SM in 3.1 are sensitive
to corrections to the Zb¯b vertex. Since the left-handed b quark is the weak isospin
partner of the top, they share the same 5D mass or localization parameter cQ3 in the
RS model. In order to generate the large top mass, cQ3 is shifted towards the IR
compared to the light quark localizations. Consequentially, one should expect sizable
corrections from the RS model.
From (2.113) and the structures defined in (2.175) it follows that the Z−vertex correction
for general external quark flavors can be written in the form
in which
L4D ∋
g
cos θW
× � ��gq �
L
g q
L = � T q
3 − sin2 �
θW Qq
�
1 − m2 Z
g q
R = − sin2 θW Qq
ij
�
1 − m2 Z
2M 2 KK
�
1 + m2 Z
2M 2 KK
4M 2 KK
�
1 − 1
��
Z
L
0 µ
ij ¯qL,iγ µ qL,j + � g q
R
�
ij ¯qR,iγ µ qR,j
�
L ∆Q − ∆ ′ �
Q
�
− T q
�
3 δQ − m2 Z
� L ∆q − ∆ ′ q
� �
+ T q
�
3 δq − m2 Z
2M 2 KK
�
, (3.8)
2M 2 KK
� L εq − ε ′ q
�
L εQ − ε ′ �
Q
�
,
� �
, (3.9)
with i, j = 1, 2, 3 denoting flavor indices and T d 1
3 = − 3 the corresponding
weak isospin and charge for down type quarks. Relevant for the Zb¯b vertex are the 33
2 and Qd = − 1