612
K. Kleinknecht and U. Uwer
precise determinations yield [136],
|V ud | = 0.97417 ± 0.00021 ,
where the error is dominated by the theoretical uncertainty stemming from the
nuclear structure and radiation correction.
The theoretical uncertainties in extracting a value of |V ud | from neutron decays
are significantly smaller than those for the superallowed decays, however the
value depends on the ratio between axial-vector and vector couplings (g A ≡
G A /G V ) and on the neutron lifetime. Using the most recent measurements [133]
the following value for |V ud | is obtained,
|V ud | = 0.9763 ± 0.0016 ,
with the error dominated by the g A uncertainty.
An alternative approach is the measurement of the very small (O(10 −8 ))
branching ratio of the pion beta-decay π + → π 0 e + ν. The value normalized
by a very precise theoretical prediction for π + → e + ν and yields [137]
|V ud | = 0.9749 ± 0.0026 .
The error here stems mainly from the measurement of the rare process.
2. |V us |. Earlier measurements of |V us | from kaon decays have used K L → πeν
to extract the product of |V us | and the form factor |V us |f + (0) at q 2 = 0. The
most recent data provide enough experimental constraints to also use decays to
muons as well as decays of K S and K ± . Averaging results from K L → πeν,
K L → πμν, K ± → π 0 e ± ν, K ± → π 0 μ ± ν and K S → πeν yields the value
|V us |f + (0) = 0.2165 ± 0.0004 [133]. Lattice QCD calculations of f + (0) have
been carried out for different numbers of quark flavors. The form-factor average,
f + (0) = 0.9704±0.0032, of the (2+1)-flavor lattice calculations [138] is in good
agreement with a classical calculation [139]. With this value one obtains,
|V us | = 0.2231 ± 0.0008 .
The lattice calculation of the ratio of kaon and pion decay constant, f K /f π =
1.1933 ± 0.0029 [138], allows the determination of the ratio |V us /V ud | from
K → μν and π → μν decays. Using the precise measurement of the K → μν
branching fraction by the KLOE collaboration [140] results in |V us | = 0.2253 ±
0.0007.
An alternative determination of |V us | uses hadronic τ decays to strange hadrons.
The average of the measured inclusive and exclusive branching fractions yields
|V us | = 0.2216 ± 0.0015 [134].
3. |V cd |. First determinations of |V cd | came from neutrino scattering experiments.
The difference of the ratio of double-muon to single-muon production for
neutrino and antineutrino scattering is depending on the charm production cross
K. Kleinknecht and U. Uwer
precise determinations yield [136],
|V ud | = 0.97417 ± 0.00021 ,
where the error is dominated by the theoretical uncertainty stemming from the
nuclear structure and radiation correction.
The theoretical uncertainties in extracting a value of |V ud | from neutron decays
are significantly smaller than those for the superallowed decays, however the
value depends on the ratio between axial-vector and vector couplings (g A ≡
G A /G V ) and on the neutron lifetime. Using the most recent measurements [133]
the following value for |V ud | is obtained,
|V ud | = 0.9763 ± 0.0016 ,
with the error dominated by the g A uncertainty.
An alternative approach is the measurement of the very small (O(10 −8 ))
branching ratio of the pion beta-decay π + → π 0 e + ν. The value normalized
by a very precise theoretical prediction for π + → e + ν and yields [137]
|V ud | = 0.9749 ± 0.0026 .
The error here stems mainly from the measurement of the rare process.
2. |V us |. Earlier measurements of |V us | from kaon decays have used K L → πeν
to extract the product of |V us | and the form factor |V us |f + (0) at q 2 = 0. The
most recent data provide enough experimental constraints to also use decays to
muons as well as decays of K S and K ± . Averaging results from K L → πeν,
K L → πμν, K ± → π 0 e ± ν, K ± → π 0 μ ± ν and K S → πeν yields the value
|V us |f + (0) = 0.2165 ± 0.0004 [133]. Lattice QCD calculations of f + (0) have
been carried out for different numbers of quark flavors. The form-factor average,
f + (0) = 0.9704±0.0032, of the (2+1)-flavor lattice calculations [138] is in good
agreement with a classical calculation [139]. With this value one obtains,
|V us | = 0.2231 ± 0.0008 .
The lattice calculation of the ratio of kaon and pion decay constant, f K /f π =
1.1933 ± 0.0029 [138], allows the determination of the ratio |V us /V ud | from
K → μν and π → μν decays. Using the precise measurement of the K → μν
branching fraction by the KLOE collaboration [140] results in |V us | = 0.2253 ±
0.0007.
An alternative determination of |V us | uses hadronic τ decays to strange hadrons.
The average of the measured inclusive and exclusive branching fractions yields
|V us | = 0.2216 ± 0.0015 [134].
3. |V cd |. First determinations of |V cd | came from neutrino scattering experiments.
The difference of the ratio of double-muon to single-muon production for
neutrino and antineutrino scattering is depending on the charm production cross