On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

122 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories
which again considerably simplify if one keeps only the leading terms in the v 2 /M 2 KK
expansion,
C G+A
1
�C G+A
1
C G+A
4
= 2πL
M 2 �
αs
KK 2
= 2πL
M 2 KK
� αs
2
�
1 − 1
NC
�
1 − 1
NC
�� 1
�� 1
= − NCC RS
5 = 2πL
M 2 2αs
KK
c2 (
θ
� ∆D)12 ⊗ ( � ∆D)12 ,
s2 (
θ
� ∆d)12 ⊗ ( � ∆d)12 , (3.66)
�
− 1
c2 (
θ
� ∆D)12 ⊗ (�εd)12 − 1
s2 θ
− 1 vIR
2 2 + vIR
(∆D)12(∆d)12
( � ∆d)12 ⊗ (�εD)12
In (3.66), the first two terms in C4, C5 include the εQ,q structure and the last one is
proportional to the IR BC vIR, which both are of the order v2 /M 2 KK . Using (3.64)
and vIR < 1, this results in the relation
C G+A
4 ≈ αsπL
2NCc2 θs2 ξ
θ
v2 4
M 2 ⎧
⎨ 2
3
C4 ∼
⎩
KK
αsπ L ξ v2 4
M 4 C4 , θ = 45
KK
◦ ,
8
9αsπ L ξ v2 4
C4 , θ = 30◦ or θ = 60◦ (3.67)
,
where in the last step NC = 3 has been inserted and the value of the mixing angle
has been fixed for two examples. As noted above, in principle one is free to choose
the mixing angle because the cancellation of the leading order terms of C4 does not
depend on θ. From (3.67) one can see that also the term induced by the BCs is largely
independent, however in the parameter corner of very large or very small mixing angles,
the terms enhanced by c −2
θ and s−2
θ in the Wilson coefficients (3.66) become too
large. The smallest overall contributions are achieved for θ = 45◦ , which will be the
default setting for the following numerical analyses. Equation (3.67) has to be compared
with the equivalent relation for the scenario of aligned right-handed localization
parameters (3.49), for which we estimated, that the suppression is sufficient for a KK
scale of 1 − 2 TeV.
In the upper row and left panel of Figure 3.7, the scatter plot in the right panel
of Figure 3.6 has been redone with the gluon contributions in the Wilson coefficients
(3.20) replaced by the exact expressions in (3.65) and assuming the minimal RS model
without a custodial protection. For the numerical analysis the same parameter points
have been used and ξ = 1, θ = 45 ◦ as well as v4 = 246 GeV have been employed
for the parameters of the extended gauge sector. From the red curve, which is again
fitted to the median of eighteen 500 GeV bins of MKK one can see that the estimate
above was sharp. The approximate bound on MKK above which the parameter points
on average fulfill the ɛK constraint (3.45) is MKK > 2 TeV. In the left panel of the
lower row, the same plot has been made with the additional assumption of a custodial
protection induced by an extended electroweak gauge group. The bound is stronger
than from the minimal model for two reasons. Already without the extended color
gauge group, is the bound from ɛK stronger, because of the huge enhancement of the
˜C1, as discussed in Section 3.3. This is partially canceled by a cancellation of the gluon
M 4 KK
�
.