On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

2.5. Hierarchies in Quark Masses and Mixings and the RS GIM Mechanism 87
with the overlap integrals
(δq)ij = 2π
� 1
Lɛ ɛ
(δQ)ij = 2π
Lɛ
� 1
ɛ
(∆g q
h )ij = 2π
� 1
Lɛ ɛ
dt a
Q †
i
SQ
q †
dt ai Sq
i (t)SQ j (t) aQ
j
i (t)Sq j
(t) aq
j
q †
dt δ(t − 1) ai Sq
i (t)Y † q S Q
j (t) aQ
j . (2.179)
The last integral corresponds to the second Yukawa coupling in (2.177) and must be
evaluated with a properly regularized delta function, see [118, p.135f.]. It will represent
the leading term in Higgs mediated flavor violating processes, because it is not chirally
suppressed. However, for all processes considered it can still be neglected, because in
the case of ∆F = 1 currents the flavor-conserving vertex is chirally suppressed and for
∆F = 2 currents it is smaller then the tensor structure (2.176) by a factor v 2 /M 2 KK .
It is however important in Higgs physics [127].
In the ZMA, the above matrices simplify considerably,
∆Q → U q†
L diag
�
F 2 (cQi )
�
U
3 + 2cQi
q
L ,
∆q → U q†
R diag
�
F 2 (cqi )
�
U
3 + 2cqi
q
R ,
∆ ′ Q → U q†
L diag
�
5 + 2cQi
and
where
� �
�∆Q
mn⊗ � ∆q
� ′
�
∆ ′ q → U q†
R diag
m ′ n ′ → �
i,j
� q† �
UL mi
( � ∆Qq)ij = F 2 (cQi )
3 + 2cQi
2(3 + 2cQi )2 F 2 (cQi )
�
5 + 2cqi
2(3 + 2cqi )2 F 2 (cqi )
�
� q�
UL in ( � ∆Qq)ij
�
U q
L ,
U q
R , (2.180)
� q† �
UR m ′ j
� q �
UR jn ′ , (2.181)
3 + cQi + cqj F
2(2 + cQi + cqj )
2 (cqj )
, (2.182)
3 + 2cqj
and analogue for the remaining combinations of indices Q and q. Because of the
v/MKK suppression in (2.150), the ε structures vanish in the ZMA. Using the fact that
all ci parameters except cu3 are very close to −1/2, it is a reasonable approximation
to replace (3 + cQi + cqj )/(2 + cQi + cqj ) by 2, in which case we obtain the approximate
result
�∆A ⊗ � ∆B → ∆A ∆B , (2.183)