On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

120 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories
the Wilson coefficients C4 and C5 induced by the second term in (3.61) will become
of the same size as the terms which cancel out and thus reintroduce the flavor problem.
The value of vIR depends sensitively on the modeling of the extended scalar sector,
but with reference to (2.102) and Section 3.6, we can anticipate that it can be written
as
vIR = L
g2 s
2NCc2 θs2 v
θ
2 4
M 2 KK
ξ = 2παsL
NCc 2 θ s2 θ
v 2 4
M 2 KK
ξ, (3.64)
in which v4 denotes the vev of the scalar which couples to the strong sector (either
Hu and Hd in (3.60) or S in (3.59)), NC = 3 the number of colors, gs = gs5/ √ 2πrc
the strong coupling constant and ξ some for now unspecified order one factor, parameterizing
the effects from modeling the extended scalar sector. With the additional
assumption that v4 is proportional to the electroweak breaking scale (which will turn
out to be true for the scenario (3.60)), the contributions induced by the BCs are sup-
pressed by v 2 /M 2 KK
with respect to the gluon KK mode contributions to C4.
The combined contributions to the Wilson coefficients of the operators (3.18) from
the KK towers of the gluon (2.185) and the axigluon (3.63) read 9
C G+A
1
�C G+A
1
C G+A
4
= 2πL
M 2 � �
αs
1 −
KK 2
1
��
1
NC c2 �
θ
+ 1
s 2 θ
= 2πL
M 2 KK
+ 1
c 2 θ
( � ∆D)12 ⊗ ( � ∆D)12 − 2( � ∆D)12 ⊗ (�εD)12
(3.65)
(�εD)12 ⊗ (�εD)12 − 1
�
vIR
s
2 2 + vIR
2 θ (∆D) 2 12 − 2(∆D)12(εD)12 + 1
s2 (εD)
θ
2 �
12
�
,
� �
αs
1 −
2
1
��
1
NC s2 �
(
θ
� ∆d)12 ⊗ ( � ∆d)12 − 2( � ∆d)12 ⊗ (�εd)12
(�εd)12 ⊗ (�εd)12 − 1
�
vIR
c
2 2 + vIR
2 θ (∆d) 2 12 − 2(∆d)12(εd)12 + 1
c2 (εd)
θ
2 �
12
�
,
�
= − NCC RS
5 = 2πL
M 2 2αs
KK
+ 1
s 2 θ c2 θ
(�εD)12 ⊗ (�εd)12 − 1
�
vIR
2 2 + vIR
− 1
c2 (
θ
� ∆D)12 ⊗ (�εd)12 − 1
s2 θ
( � ∆d)12 ⊗ (�εD)12
(∆D)12(∆d)12 − 1
s2 (∆D)12(εd)12
θ
− 1
c2 (∆d)12(εD)12 +
θ
1
c2 θs2 (εD)12(εd)12
θ
9 The electroweak contributions are not repeated as they are unchanged from (3.20).
� �
,