616
K. Kleinknecht and U. Uwer
where the uncertainties are dominated by the lattice uncertainties of the hadronic
factors. Several uncertainties are reduced when calculating the ratio
ξ =
f B s
ˆ
B B s
/
f B d
ˆ
B B d
= 1.239 ± 0.046 .
The ratio |V td |/|V ts | is therefore stronger constraint [133],
|V td |/|V ts | = 0.210 ± 0.001 ± 0.008 .
Using the independent measurements of the CKM elements the unitarity of the
CKM matrix can be checked. One obtains for the first two rows [133],
|V ud |
2
+ |V us |
2
+ |V ub |
2
= 0.9994 ± 0.0005 ,
|V cd |
2
+ |V cs |
2
+ |V cb |
2
= 1.043 ± 0.034 ,
which agrees well with the unitarity assumption. In addition, the direct measurement
of |V tb | leaves little room for mixing of the top into unknown other states.
9.7.3 Global Analysis and Test of the Unitarity of the CKM
Matrix
The available information on the magnitude and the phases of the CKM elements
can be analysed by a global fit to all CKM parameters. The most precise information
is obtained imposing Standard Model constraints such as the unitarity of the CKM
matrix and the existence of exactly three quark generations. Input parameters of
the global analysis are, beside the experimental measurements, also theoretically
determined hadronic parameters with sometimes large errors.
Different approaches exist to combine the experimental data and to treat the
experimental and theoretical errors. The CKM-Fitter group [135, 147] is using
a frequentist’s framework based on a χ 2 analysis. In these fits the frequentist’s
treatment is also applied to the theoretical errors. The UTfit group [148] uses a
baysian approach for all errors. The two different statistical approaches lead to very
similar results and here only the results of [135] are presented.
Figure 9.39 shows the experimental constraints in the ( ¯
ρ, ¯
η) plane. Indicated
is the unitarity triangle with its angles α, β and γ . The different measurements
clearly limit the apex of the triangle to a small (dashed) region. For the Wolfenstein
parameters introduced in Eq. (9.217) the global fit [133, 135] gives
λ = 0.22453 ± 0.00044 ,
A= 0.836 ± 0.015 ,
(9.230)
¯
ρ = 0.122
+0.018
−0.017 ,
¯
η = 0.355
+0.012
−0.011 .
(9.231)
K. Kleinknecht and U. Uwer
where the uncertainties are dominated by the lattice uncertainties of the hadronic
factors. Several uncertainties are reduced when calculating the ratio
ξ =
f B s
ˆ
B B s
/
f B d
ˆ
B B d
= 1.239 ± 0.046 .
The ratio |V td |/|V ts | is therefore stronger constraint [133],
|V td |/|V ts | = 0.210 ± 0.001 ± 0.008 .
Using the independent measurements of the CKM elements the unitarity of the
CKM matrix can be checked. One obtains for the first two rows [133],
|V ud |
2
+ |V us |
2
+ |V ub |
2
= 0.9994 ± 0.0005 ,
|V cd |
2
+ |V cs |
2
+ |V cb |
2
= 1.043 ± 0.034 ,
which agrees well with the unitarity assumption. In addition, the direct measurement
of |V tb | leaves little room for mixing of the top into unknown other states.
9.7.3 Global Analysis and Test of the Unitarity of the CKM
Matrix
The available information on the magnitude and the phases of the CKM elements
can be analysed by a global fit to all CKM parameters. The most precise information
is obtained imposing Standard Model constraints such as the unitarity of the CKM
matrix and the existence of exactly three quark generations. Input parameters of
the global analysis are, beside the experimental measurements, also theoretically
determined hadronic parameters with sometimes large errors.
Different approaches exist to combine the experimental data and to treat the
experimental and theoretical errors. The CKM-Fitter group [135, 147] is using
a frequentist’s framework based on a χ 2 analysis. In these fits the frequentist’s
treatment is also applied to the theoretical errors. The UTfit group [148] uses a
baysian approach for all errors. The two different statistical approaches lead to very
similar results and here only the results of [135] are presented.
Figure 9.39 shows the experimental constraints in the ( ¯
ρ, ¯
η) plane. Indicated
is the unitarity triangle with its angles α, β and γ . The different measurements
clearly limit the apex of the triangle to a small (dashed) region. For the Wolfenstein
parameters introduced in Eq. (9.217) the global fit [133, 135] gives
λ = 0.22453 ± 0.00044 ,
A= 0.836 ± 0.015 ,
(9.230)
¯
ρ = 0.122
+0.018
−0.017 ,
¯
η = 0.355
+0.012
−0.011 .
(9.231)