A Mathematica based Version of the CKMfitter Package

34 Chapter 5. Probing the Standard Model
5.1.12 Decay Constants and Bag Parameters
The observables ∆md, ∆ms and B(B + → l + νl) depend on a set of decay constants
and bag parameters:
fBd , fBs , Bd and Bs . (5.13)
They can be calculated using Lattice QCD, which leads to significant theoretical uncertainties
for the predictions of the observables discussed in Chapters 5.1.9 to 5.1.11.
Since some combinations of these parameters can be calculated more precisely, the
parameters:
fBd , Bd and ξ = fBs
√
Bs
√
fBd Bd
, (5.14)
have been used in the original CKMfitter package. Unfortunately, the LQCD calculation
of fBd has large theoretical uncertainties, originating from the chiral extrapolation
of the light lattice quark masses (mu,md) to the physical quark masses.
Hence, the theoretical uncertainties of fBd and ξ are anti-correlated. This is not the
case in the different but mathematically equivalent parametrization:
fBs ,
fBs
fBd
, Bs and Bs
Bd
, (5.15)
which is used in the Mathematica based version of the CKMfitter package. It is
free from the large correlation due to the uncertainty from the chiral extrapolation.
However, possible smaller correlations are neglected since their correlation matrix
has not been published.
5.1.13 ASL
Measurements of semileptonic B-meson decays, i.e. B → Xlν, allow the determination
of the magnitude of CP violation in neutral B-meson mixing.
The flavor specific CP asymmetry ASL as defined in Equation (2.42) is related to
the mixing matrix elements in the SM via:
� �
Γ12
ASL = Im . (5.16)
Since the absolute value of Γ12/M12 is suppressed to the percent level through
� � �
� Γ12 �
� �
m2 b
�M12
� = O
m2 �
, (5.17)
W
ASL is expected to be rather small. It is additionally suppressed by another order
of magnitude through the GIM3 factor z = m2 c/m2 b � 0.1. In the presence of New
3 In the SM, the Glashow-Iliopoulos-Maiani (GIM) mechanism leads to a cancellation of
FCNC processes at tree level.
M12