– 1–
V
ud
,
V
us
, THE CABIBBO ANGLE,
AND CKM UNITARITY
Updated November 2009 by E. Blucher (Univ. of Chicago) and
W.J. Marciano (BNL)
The
CabibboKobayashiMaskawa
(CKM)
[1,2]
three
generation quark mixing matrix written in terms of the Wolfen
stein parameters (
λ, A, ρ, η
) [3] nicely illustrates the orthonor
mality constraint of unitarity and central role played by
λ
.
V
CKM
=
⎛
⎝
V
ud
V
us
V
ub
V
cd
V
cs
V
cb
V
td
V
ts
V
tb
⎞
⎠
=
⎛
⎝
1
−
λ
2
/
2
λ
Aλ
3
(
ρ
−
iη
)
−
λ
1
−
λ
2
/
2
Aλ
2
Aλ
3
(1
−
ρ
−
iη
)
−
Aλ
2
1
⎞
⎠
+
O
(
λ
4
)
.
(1)
That cornerstone is a carryover from the twogeneration Cabibbo
angle,
λ
= sin(
θ
Cabibbo
) =
V
us
. Its value is a critical ingredient
in determinations of the other parameters and in tests of CKM
unitarity.
Unfortunately, the precise value of
λ
has been somewhat
controversial in the past, with kaon decays suggesting [4]
λ
0
.
220, while hyperon decays [5] and indirect determinations via
nuclear
β
decays imply a somewhat larger
λ
0
.
225
−
0
.
230.
That discrepancy is often discussed in terms of a deviation from
the unitarity requirement

V
ud

2
+

V
us

2
+

V
ub

2
= 1
.
(2)
For many years, using a value of
V
us
derived from
K
→
πeν
(
K
e
3
) decays, that sum was consistently 2–2.5 sigma below
unity, a potential signal [6] for new physics effects. Below, we
discuss the current status of
V
ud
,
V
us
, and their associated
unitarity test in Eq. (2). (Since

V
ub

2
1
×
10

5
is negligibly
small, it is ignored in this discussion.)
V
ud
The value of
V
ud
has been obtained from superallowed
nuclear, neutron, and pion decays. Currently, the most precise
determination of
V
ud
comes from superallowed nuclear beta
decays [6]
(0
+
→
0
+
transitions). Measuring their halflives,
t
,
CITATION: K. Nakamura
et al.
(Particle Data Group), JPG
37
, 075021 (2010) (URL: http://pdg.lbl.gov)
July 30, 2010
14:34
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– 2–
and Q values which give the decay rate factor,
f
, leads to a
precise determination of
V
ud
via the master formula
[7–9]

V
ud

2
=
2984
.
48(5) sec
ft
(1 + RC)
(3)
where RC denotes the entire effect of electroweak radiative
corrections,
nuclear structure,
and isospin violating nuclear
effects. RC is nucleusdependent, ranging from about +3
.
0% to
+3
.
6% for the best measured superallowed decays. The most
recent analysis of Hardy and Towner [10, 11] gives a weighted
average (with errors combined in quadrature) of
V
ud
= 0
.
97425(22) (superallowed) ,
(4)
which, assuming unitarity, corresponds to
λ
= 0
.
2255(10). The
new average value of
V
ud
is shifted upward compared to our 2007
value of 0.97418(27) primarily because of improvements in the
experimental
ft
values and nuclear isospin breaking corrections
employed. We note, however, that the possibility of additional
nuclear coulombic corrections has been raised recently [12].
Combined measurements of the neutron lifetime,
τ
n
, and
the ratio of axialvector/vector couplings,
g
A
≡
G
A
/G
V
, via
neutron decay asymmetries can also be used to determine
V
ud
:

V
ud

2
=
4908
.
7(1
.
9) sec
τ
n
(1 + 3
g
2
A
)
,
(5)
where the error stems from uncertainties in the electroweak
radiative corrections [8] due to hadronic loop effects. Those
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 Spring '11
 Kutter
 Radioactive Decay, Quark, Rev. Lett., unitarity, Vud

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