On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

3.2. RS GIM Working 107
with ωZ LL = ωZ RR = 1 in the minimal model and if the electroweak gauge group is
extended in order to comprise a custodial symmetry, ωZ LL = 0 and ωZ RR = 3c2w s2 , as in
w
(3.15) and (3.16), compare also [167, App. B].
Corrections to b → sℓ + ℓ− arising from CRS ℓ3 and ˜ CRS ℓ3 are either O(mµmb/M 2 KK
O(v4 /M 4 KK ) and will consequentially also be ignored in the numerical analysis.
The branching ratios for the Bq → µ + µ − decays can be expressed as
B(Bq → µ + µ − ) = G2 F α2 m3 Bqf 2 BqτBq 64π3s4 w
�
� (qb) �
λ �
t
2
�
1 − 4m2 µ
m2 Bq
�
4m2 µ
×
m2 �
�CA − C
Bq
′ �
�
A
2 + m 2 �
Bq
1 − 4m2 µ
m 2 Bq
� ����
mb CS − mq C ′ S
mb + mq
�
�
�
�
2 �
) or
,
(3.36)
where mBq, fBq, and τBq are the mass, decay constant, and lifetime of the Bq meson
and λ (pr)
q
≡ V ∗
qpVqr. The electromagnetic coupling α entering the branching ratios
should be evaluated at mZ. The expressions for the coefficients CA,S and C ′ A,S read
CA = cA − s4wc 2 wm2 Z
α2λ (qb)
�
C
t
RS
ℓ1
�
− CRS ℓ2
, C ′ A = s4wc 2 wm2 Z
α2λ (qb)
�
˜C RS
ℓ1
t
− ˜ C RS
�
ℓ2 ,
CS = 2s4wc 2 wm2 Z
α2mbλ (qb)
C
t
RS
l3 , C′ S = 2s4wc 2 wm2 Z
α2mqλ (qb)
t
˜C RS
ℓ3 ,
(3.37)
where cA = 0.96 ± 0.02 denotes the SM contribution to the Wilson coefficient of the
axial-vector current [169, 170], and the coefficients CRS contain the 13
ℓ1−3 and ˜ CRS ℓ1−3
or 23 elements of the mixing matrices in the case of Bd → µ + µ − and Bs → µ + µ − ,
respectively. 7
The SM branching ratios of the Bq → µ + µ − decay channels evaluate to [171, 172]
B(Bd → µ + µ − )SM = (1.0 ± 0.1) · 10 −10 , (3.38)
B(Bs → µ + µ − )SM = (3.2 ± 0.2) · 10 −9 . (3.39)
These predictions are obtained by normalizing the decay rates to the well-measured
meson mass differences (∆mq)exp. This eliminates the dependence on CKM parameters
and the bulk of the hadronic uncertainties by trading the decay constants for
less uncertain hadronic parameters. The dominant source of error is nevertheless still
provided by the hadronic input.
The result is plotted in Figure 3.5 and shows again very good agreement with the
current bounds, which however start to cut into the parameter space. Experimental
bounds refer to the very precise limit from LHCb measurements, obtained including
the full 2011 dataset [166], and the two-sided bound from CDF [168] respectively. It is
7 While CA and C ′ A are scale independent, the coefficients CS and C ′ S have a non-trivial RG
evolution. We can ignore that because they are numerically insignificant.