of 62
Experimental Status of the CKM Matrix
Frank C. Porter
Lauritsen Laboratory of Physics
California Institute of Technology
Pasadena, California, USA
Abstract
The CKM matrix,
V
, relates the quark mass and flavor bases. In the standard model,
V
is unitary 3
×
3,
and specified by four arbitrary parameters, including a phase allowing for
CP
violation. We review the
experimental determination of
V
, including the four parameters in the standard model context. This is
an active field; the precision of experimental measurements and theoretical inputs continues to improve.
The consistency of the determination with the standard model unitarity is investigated. While there
remain some issues the overall agreement with standard model unitarity is good.
Contents
1 Introduction
2
2 Parameters
3
3 Magnitudes of CKM elements
6
3.1
|
V
ud
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.2
|
V
us
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.3
|
V
cs
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.4
|
V
cd
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.5
|
V
cb
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.6
|
V
ub
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.7
|
V
tb
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.8
|
V
ts
|
and
|
V
td
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4 Phases
21
4.1
CP
violation in kaon mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.2 Mixing-induced
CP
violation in
B
decays . . . . . . . . . . . . . . . . . . . . . . . . .
23
4.2.1
The angle
β
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2.2
The angle
α
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.2.3
The angle
γ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.2.4
The angle
β
s
(and
φ
s
) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
5 Global Fits
35
6 Summary and Discussion
37
Preprint submitted to Progress in Particle and Nuclear Physics
June 24, 2016
arXiv:1604.04940v2 [hep-ex] 22 Jun 2016
1. Introduction
According to the standard model, quarks come in three families, with “up”-type and “down”-type
flavors represented in the mass basis by:
U
=
u
c
t
, D
=
d
s
b
(1)
The standard model weak charged current for quarks may be written
J
μ
+
qW
=
1
2
̄
U
L
γ
μ
V D
L
.
(2)
The hermitian conjugate gives
J
μ
qW
. The
L
subscript indicates the left-handed projection:
Q
L
=
1
2
(1
γ
5
)
Q
. The quantity of interest to us here is the matrix
V
, called the Cabibbo-Kobayashi-Maskawa
(CKM) matrix [1, 2], describing how the mass states are mixed in the weak interaction.
The standard model does not predict
V
; it must be evaluated experimentally. Otherwise, some
physics beyond the standard model is necessary to predict
V
. However, it is not an arbitrary 3
×
3
complex matrix. In the standard model with 3 generations, it relates two bases, and must be unitrary.
Nine real parameters are required to describe an arbitrary 3
×
3 unitary matrix. Not all of these
parameters are of physical significance, five of them can be absorbed as arbitrary phases in the definition
of the quark fields. This leaves four real physical parameters required to completely specify the CKM
matrix in the standard model. As only three parameters are sufficient to specify a real orthogonal matrix
(i.e., a rotation matrix), the matrix is in general complex. This yields a standard model mechanism for
CP
violation.
The unitarity constraint permits testing the self-consistency of the standard model through precise
measurements of the elements of
V
. If any deviation from unitarity is detected, that is evidence for new
physics, such as the presence of additional generations. Alternatively, if two measurements, via different
processes, of the same element in the standard model yield different results, that is also evidence for new
physics. An important class of such searches for new physics is the possibility of new physics at a high
energy scale contributing to some processes via new virtual particle exchange, perhaps in semileptonic
decays or in “penguin diagrams”. For example, this could show up in
CP
asymmetry measurements.
Thus, the measurement of
V
has two thrusts: First, the four parameters are fundamental in the standard
model, and must be determined by measurement. Second, measurements of the CKM matrix elements
provide a means to search for physics beyond the standard model, or to constrain such theories.
In this article we review the present experimental status of our knowledge of
V
. For reliability,
we include in our averages only published, or accepted for publication, results, with a nominal cutoff
of December 31, 2015, although we often mention preliminary work. This remains an active area,
both experimentally and theoretically. Theoretical issues are included only to the extent that they are
relevant to the measurements of
V
. There are many excellent theoretical reviews; we reference several
at appropriate places.
The Review of Particle Properties [3] is used for constants such as
G
F
and particle properties such as
masses, lifetimes, etc., except as otherwise noted. In addition, there are several other averaging groups
providing extremely useful services, including FLAG (Flavor Lattice Averaging Group) [4], FlaviaNet
Kaon Working Group [5], and HFAG (Heavy Flavor Averaging Group) [6, 7, 8]. In some cases, we
have used unpublished averages from these groups, as long as the primary results are published. A
comprehensive review of the physics of the
B
-factories (BaBar and Belle) is available in [9].
In the next section, we’ll discuss the parameterization of the CKM matrix. It will be convenient to
separate the discussion of the measurements into the magnitudes of the elements (Section 3) and the
phases via
CP
violation (Section 4). Then in Section 5 we will briefly discuss three extensive efforts at
providing global fits to available information concerning the CKM matrix and possible extensions. We
conclude with some fits of our own with discussion in Section 6.
2
2. Parameters
There are several useful parameterizations of the CKM matrix. We review here the ones that we
shall be concerned with.
The conventional labeling for the general 3
×
3 flavor mixing matrix is
V
=
V
ud
V
us
V
ub
V
cd
V
cs
V
cb
V
td
V
ts
V
tb
(3)
With this labelling, the charged current vertex gets a factor
V
ji
as in Fig. 1. In terms of the flavor
labels,
V
ji
=
V
ij
, for example,
V
du
=
V
ud
.

W
i
j
V
ji

W
b
q
t
q
V
tb

W
b
t
q
q
V
tb

t
b
g
W
t
V
tb

b
b
g
W
t
V
tb
1
Figure 1: Appearance of CKM matrix element
V
ji
in the Feynman rule for a charged current vertex, where
i
and
j
are
quark flavor labels.
Standard model unitarity implies nine independent equations relating the elements of
V
:
The sum of the absolute squares of the elements in each row (or column) is one:
|
V
ud
|
2
+
|
V
us
|
2
+
|
V
ub
|
2
= 1
(4a)
|
V
cd
|
2
+
|
V
cs
|
2
+
|
V
cb
|
2
= 1
(4b)
|
V
td
|
2
+
|
V
ts
|
2
+
|
V
tb
|
2
= 1
(4c)
The dot product of a column with the complex conjugate of a different column is zero. This yields
the remaining six equations (considering real and imaginary terms separately):
V
ud
V
us
+
V
cd
V
cs
+
V
td
V
ts
= 0
(5a)
V
ud
V
ub
+
V
cd
V
cb
+
V
td
V
tb
= 0
(5b)
V
us
V
ub
+
V
cs
V
cb
+
V
ts
V
tb
= 0
(5c)
Alternatively, we could have taken the dot products by rows, but this does not yield new inde-
pendent equations (since
V
V
=
I
⇐⇒
V V
=
I
).
3
These conditions reduce the number of parameters to nine real parameters required to define an
arbitrary unitary 3
×
3 matrix. However, there are five arbitrary phases defining the relative quark
fields. When these are chosen, there remain four physical real parameters needed to specify the CKM
matrix.
Since a rotation (orthogonal) matrix in three dimensions is specified by three angles (e.g., the
Euler angles), it is intuitive to think of the CKM matrix as described by three angles
θ
ij
;
i
= 1
,
2;
i <
j
3 corresponding to a rotation, plus an additional phase angle,
δ
, giving complex elements to the
matrix. Conventionally, the
θ
ij
are chosen to be in the first quadrant. The complex phase provides a
mechanism for
CP
violation in the standard model. Perhaps the most common convention for such a
parameterization of the matrix is [10, 3]:
V
=
c
12
c
13
s
12
c
13
s
13
e
s
12
c
23
c
12
s
23
s
13
e
c
12
c
23
s
12
s
23
s
13
e
s
23
c
13
s
12
s
23
c
12
c
23
s
13
e
s
23
c
12
s
12
c
23
s
13
e
c
23
c
13
,
(6)
where
s
ij
sin
θ
ij
and
c
ij
cos
θ
ij
.
Empirically,
V
is approximately diagonal, and this suggests a useful parameterization as an ex-
pansion in powers of one of the parameters [11]. Following [12, 13], we define a new set of four real
parameters (
λ
,
A
,
ρ
,
η
, which we shall refer to as Wolfenstein parameters) according to:
λ
s
12
2
s
23
3
(
ρ
)
s
13
e
.
(7)
The parameter
λ
0
.
22 then functions as an expansion parameter for describing
V
, and we have (e.g.,
[14]):
V
=
1
λ
2
2
λ
4
8
λ
3
(
ρ
)
λ
[
1 +
A
2
λ
4
(
ρ
+
1
2
)
]
1
1
2
λ
2
1
8
λ
4
(1 + 4
A
2
)
2
3
[
1
(
ρ
+
)(1
1
2
λ
2
)
]
2
[
1 +
λ
2
(
ρ
+
1
2
)
]
1
1
2
A
2
λ
2
+
O
(
λ
6
)
,
(8)
Measurements of rates are typically sensitive to magnitudes of particular CKM matrix elements,
|
V
ij
|
.
Thus, we shall discuss these magnitudes extensively. In the standard model, they are related according
to the unitarity constraints above, but experimentally they provide nine independent quantities, hence
allowing for tests of the standard model. If the experimental results do not fit the constraints, that is
evidence for new physics.
On the other hand, measurements of
CP
-violating processes are sensitive to the phase
δ
, or alterna-
tively the parameter
η
. The connection with experiment is often done in the context of the “unitarity
triangles”. Eqs. 5 describe triangles in the complex plane. In the limit of no
CP
-violation, the relative
phases of all the elements would be 0 or 180
, and the triangles would have zero area. In fact, all of
the triangles have the same area, related to a phase-convention invariant quantity called the Jarlskog
invariant [15]:
J
= (
1)
a
+
b
=
(
V
ij
V
kl
V
kj
V
il
)
(9)
where one row (index
a
) and one column (index
b
) of
V
is crossed out to obtain the 2
×
2 matrix
V
(
ab
)
=
(
V
ij
V
il
V
kj
V
kl
)
,
(10)
defining indices
i,j,k,l
. The magnitude of
J
, or equivalently the area of the triangles, is a measure of
how much
CP
violation there is in the standard model. It has more recently been pointed out that the
Jarlskog invariant can also be expressed as [16, 17]:
J
=
=
(
V
31
V
22
V
13
)
,
(11)
4
assuming
V
has been expressed in a form with determinant one by multiplying by an overall phase as
needed. In the Wolfenstein parameterization, at the order of Eq. 8,
J
=
A
2
λ
6
η.
(12)
One triangle in particular, Eq. 5b, has the feature that all of the sides have length of order
3
.
This is usually what we mean when we say the (standard) “unitarity triangle”. Dividing through by
V
cd
V
cb
gives:
1 =
V
ud
V
ub
V
cd
V
cb
V
td
V
tb
V
cd
V
cb
(13)
It is conventional to define the complex apex point as
̄
ρ
+
i
̄
η
≡−
V
ud
V
ub
V
cd
V
cb
=
1
λ
2
(
ρ
+
)
1
A
2
λ
4
+
1
λ
2
A
2
λ
4
(
ρ
+
)
(
ρ
+
)(1
λ
2
/
2)
.
(14)
We may graph this triangle, as in Fig. 2. Three angles are thus defined:
α
arg
(
1
̄
ρ
i
̄
η
̄
ρ
+
i
̄
η
)
= arg
(
V
td
V
tb
V
ud
V
ub
)
(15)
β
arg
(
1
1
̄
ρ
i
̄
η
)
= arg
(
V
cd
V
cb
V
td
V
tb
)
(16)
≈−
arg
V
td
(17)
γ
arg ( ̄
ρ
+
i
̄
η
) = arg
(
V
ud
V
ub
V
cd
V
cb
)
.
(18)
The angles
β
,
α
, and
γ
are also commonly called
φ
1
,
φ
2
, and
φ
3
, respectively.
(0
,
0)
( ̄
ρ,
̄
η
)
(1
,
0)
γ
β
α
̄
ρ
+
i
̄
η
1
̄
ρ
i
̄
η
1
Figure 2: The vectors forming the standard unitarity triangle.
A second unitarity triangle has more recently become experimentally accessible. This is the relation
in Eq. 5c, which, dividing by
V
cs
V
cb
, is:
1 +
V
ts
V
tb
V
cs
V
cb
+
V
us
V
ub
V
cs
V
cb
= 0
.
(19)
Considering the Wolfenstein parameterization in powers of
λ
, the first two terms are of order one, while
the third term is
O
(
λ
2
). Hence, this triangle has at least one very small angle, called
β
s
:
β
s
arg
(
V
ts
V
tb
V
cs
V
cb
)
λ
2
̄
η
(20)
arg(
V
ts
)
.
(21)
5
3. Magnitudes of CKM elements
As already noted, rates for weak transitions provide for relatively direct measurements of the mag-
nitudes of the CKM matrix elements. In the following, we review what is known about each of the nine
magnitudes in turn, without assuming unitarity of the matrix. This is a convenient organization also
because the results are largely uncorrelated. We consider the upper left 2
×
2 submatrix elements first,
as the best known quantities, followed by
|
V
ub
|
and
|
V
cb
|
, then finish with the top-quark elements.
3.1.
|
V
ud
|
The magnitude of
V
ud
is measured in weak
u
d
transitions. Thus, the most promising candidates to
study are nuclei, neutrons, and pions. The principal pion decay,
π
+
μ
+
ν
μ
(
γ
), has a rate proportional
to
|
V
ud
|
2
, given by:
Γ
(
π
+
μ
+
ν
μ
(
γ
)
)
=
G
2
F
8
π
f
2
π
m
2
μ
m
π
(
1
m
2
μ
m
2
π
)
2
|
V
ud
|
2
(
1 +
α
π
C
π
)
,
(22)
where
G
F
is the Fermi constant,
f
π
is the pion decay constant,
m
π
is the charged pion mass,
m
μ
is the
muon mass, and
C
π
allows for radiative corrections, including both virtual and real photons. Radiative
corrections are not negligible considering available experimental precision, and there is a large literature
on this subject. A convenient discussion, though by no means the last word, is provided in [18]. Our
main concern here is the dependence on
f
π
, which is not perturbatively calculable. The present situation
on the lattice calculation of
f
π
is reviewed in [19], where a precision of 1.3% is quoted. Thus, while
the pion lifetime,
τ
π
= (2
.
6033
±
0
.
0005)10
8
s [3], is precisely known, it measures the combination
f
2
π
|
V
ud
|
2
, and the extraction of
|
V
ud
|
is hampered by the inability to precisely calculate
f
π
. Instead, the
pion decay is typically used to measure
f
π
given other more precise determinations of
|
V
ud
|
.
A method that is theoretically clean is to look at the rare pion beta decay process
π
+
π
0
e
+
ν
e
[20].
The world average branching fraction is dominated by the most recent measurement from the PIBETA
experiment at the Paul Scherrer Institute, yielding
|
V
ud
|
= 0
.
9728(30)[21]. Using the theoretical value
for the normalizing branching fraction for
π
+
e
+
ν
e
(
γ
) instead of the less precise world average,
Towner and Hardy [22] recommend the value
|
V
ud
|
= 0
.
9742
±
0
.
0026. In spite of the robustness of this
approach, the small branching fraction has so far kept it from being as precise as the measurement via
nuclear transitions. It provides, however, a valuable consistency check, being independent of nuclear
structure.
The lifetime of the neutron is measured to be
τ
n
= 880
.
3
±
1
.
1 s [3]. There has been substantial
movement in the neutron lifetime in recent years, a synopsis of the situation appears in [3]. Due to this
fluidity, it seems prudent to wait for further developments before reaching firm conclusions based on
this quantity.
Presently, the most precise value for
|
V
ud
|
comes from superallowed 0
+
0
+
nuclear beta transitions.
This is largely because these are vector transitions and the conserved vector current hypothesis provides
reduced hadronic uncertainties. Over the last several decades there has been considerable experimental
effort to improve the rate measurements, and concomitant improvements in the theoretical evaluation
of the few percent corrections for symmetry breaking and radiative diagrams. The corrected
ft
values
(comparative half-lifes) are independent of nucleus, and inversely proportional to
|
V
ud
|
2
. The most
recent update [23] uses an average of the 14 most precise
ft
values measured on different nuclei, and
quotes
|
V
ud
|
= 0
.
97417(21)
(23)
|
V
ud
|
2
= 0
.
94900(42)
,
(24)
where we quote also the square as being more directly related to the measurements. This represents
a slight shift and slight improvement in precision from the five-year earlier evaluation [22] of
|
V
ud
|
=
6
0
.
97425
±
0
.
00022. The dominant uncertainty is theoretical, from the nucleus-independent portion of
the radiative corrections [24].
The measured value of
|
V
ud
|
2
is many standard deviations different from one, the value for a world
with a single generation. Hence, this measurement alone implies at least two generations are required.
3.2.
|
V
us
|
The measurement of the magnitude of
V
us
requires
s
d
transitions. The most promising sources
are kaon and hyperon decays, as well as
τ
decays with strangeness in the final state. So far, kaon decays
provide the most precise measurements. They may be classified as the purely leptonic decays,
K
μν
(including radiative decays) and semileptonic decays,
K
0
L
π`ν
,
K
0
S
π`ν
, and
K
±
π
0
`
±
ν
. The
leptonic
K
decay rate is suppressed relative to
K
μν
by (
m
e
/m
μ
)
2
, as seen by considering the
kaon decay equation analogous to Eq. 22.
Extraction of
|
V
us
|
using the leptonic decay suffers from the same uncertainty, now for the kaon
decay constant
f
K
, as for the pion in our discussion of
V
ud
. However, the ratio
f
K
/f
π
may be computed
rather precisely using lattice QCD (LQCD), and the precise value of
|
V
ud
|
may then be used to evaluate
|
V
us
|
:
|
V
us
|
2
=
|
V
ud
|
2
Γ (
K
μν
(
γ
))
Γ (
π
μν
(
γ
))
(
f
π
f
K
)
2
m
π
m
K
1
(
m
μ
m
π
)
2
1
(
m
μ
m
K
)
2
1 +
α
π
C
π
1 +
α
π
C
K
.
(25)
The ratio of radiative correction factors is taken from [5], using the chiral perturbation theory results
in [25]. This corresponds to the same central value, but half the uncertainty of the ratio in [26]. Thus,
we use
1 +
α
π
C
π
1 +
α
π
C
K
= 1
.
0070(18)
,
(26)
the dominant corrections being the same for the pion and kaon. Two recent four flavor (
N
f
= 2 + 1 + 1)
lattice caculations of
f
K
/f
π
quote a precision of around 0
.
2% (see also [27]):
f
K
+
/f
π
+
=
{
1
.
1916(21)
HPQCD [28]
1
.
1956
+(27)
(24)
Fermilab Lattice and MILC [29]
.
(27)
Symmetrizing the second interval and averaging, we use
f
K
+
/f
π
+
= 1
.
1935(21)
,
(28)
where we have not reduced the uncertainty below the smaller error because of potential systematic
correlations.
The measured input is summarized in Table 1. Using Eq. 25 we find:
V
us
V
ud
2
= 0
.
05347(14)
Γ
(10)
C
(19)
f
,
|
V
us
|
2
= 0
.
05074(13)
Γ
(2)
V
ud
(9)
C
(18)
f
,
(
K
μν
)
|
V
us
|
= 0
.
22526(29)
Γ
(5)
V
ud
(20)
C
(40)
f
,
(29)
or, quadratically combining the uncertainties,
|
V
us
|
2
= 0
.
05074(24) and
|
V
us
|
= 0
.
2253(5). This result
is close to other determinations, such as in [3], but the quoted uncertainty is smaller due to the use
of [28, 29] for the ratio of decay constants. Nevertheless, this ratio remains the dominant source of
uncertainty in
|
V
us
|
. This method introduces a correlation with
|
V
ud
|
. However, the contribution to
the uncertainty from
|
V
ud
|
is small, and the linear correlation coefficient is only
ρ
(
|
V
us
|
2
,
|
V
ud
|
2
) =
cov(
|
V
us
|
2
,
|
V
ud
|
2
)
σ
|
V
ud
|
2
σ
|
V
us
|
2
=
|
V
us
V
ud
|
2
σ
|
V
ud
|
2
σ
|
V
us
|
2
0
.
09.
7
Table 1: Experimental rates in the evaluation of
|
V
us
|
from leptonic decays. “Fit” refers to the constrained fit performed
by the Particle Data Group.
Quantity
Value
Reference
τ
(
π
+
)
2
.
6033(5)
×
10
8
s
[3]
B
(
π
+
μ
+
ν
(
γ
))
0
.
9998770(4)
[3] (fit)
Γ (
π
+
μ
+
ν
(
γ
))
3
.
8408(7)
×
10
7
s
1
Γ (
K
+
μ
+
ν
(
γ
)) 5
.
133(13)
×
10
7
s
1
[3] (fit)
The value of
|
V
us
|
is also measured in
K
π`ν
,
`
=
e,μ
decays, avoiding the dependence on the
decay constant, but introducing the form factor parameter
f
+
(0). The total decay rate for this process
may be expressed in terms of the product
|
V
us
|
f
+
(0) according to [30, 5, 31]:
Γ
K
`
3
=
G
2
F
m
5
K
192
π
3
C
2
K
|
V
us
|
2
f
+
(0)
2
I
K`
S
EW
(1 +
δ
EM
+
δ
SU
(2)
)
,
(30)
where
C
2
K
is 1 or 1/2 for
K
0
or
K
+
, respectively, and
I
K`
is a form factor dependent phase space integral
(see [5] for discussion). The remaining
S
EW
(1 +
δ
EM
+
δ
SU
(2)
)
O
(1) factor includes corrections for
short distance electroweak, long distance electromagnetic, and isospin-breaking effects. The correction
depends on whether it is a neutral or charged kaon, and whether the lepton is
e
or
μ
. The form factor
f
+
(0) is by convention that of the neutral kaon decay.
The FlaviaNet Working Group on Kaon Decays reviewed the
K
`
3
measurement in 2010 ([5] and
references therein). We use the most recent update from [32], which includes more recent results from
KLOE [33], KTeV [34], and NA48/2 [35]. It should be remarked that the NA48/2 results are actually
unpublished, in violation of our selection criteria. However, the main difference with the 2010 average
is actually in the correction for strong isospin breaking rather than in the additional data, and the
influence of the new NA48/2 analysis on the result is very small. The result after radiative corrections
is:
f
+
(0)
|
V
us
|
= 0
.
2165(4)
.
(31)
This is an average over neutral and charged kaon decays. The largest uncertainty is from measurement,
in the lifetime for
K
0
L
, and in the branching fractions for
K
0
S
and
K
±
.
We use the recent lattice result [36]
f
+
(0) = 0
.
9704(32) to obtain
|
V
us
|
2
= 0
.
04978(38)
,
|
V
us
|
= 0
.
2231(8)
.
(
K
`
3
)
(32)
The uncertainty is dominated by the uncertainty in
f
+
(0). This result is about 2.5 standard deviations
below the
K
`
2
result.
We may average the
K
`
2
and
K
`
3
results to obtain:
|
V
us
|
2
= 0
.
05046(20)
,
|
V
us
|
= 0
.
2247(5)
,
(33)
In this average, the correlation with
|
V
ud
|
reduces to
ρ
(
|
V
us
|
2
,
|
V
ud
|
2
)
0
.
08. Including the Particle Data
Group scaling factor procedure [3] the uncertainties enlarge with a scale factor of
S
= 2
.
2 to (44) and
(10), respectively.
The
|
V
us
|
matrix element is also measured in hyperon decays and in tau decays to strangeness (e.g.,
see Blucher and Marciano in [3] and references therein). The precision and theoretical understanding of
8
these measurements is not competitive with the
K
`
2
and
K
`
3
measurements at this time. In particular,
there has been a long-standing discrepancy between
|
V
us
|
determined from inclusive
τ
X
s
ν
τ
decays
compared with the kaon results. An evaluation in [37], based on sum rule and flavor breaking theoretical
work described in [38], yields
|
V
us
|
(
τ
X
s
ν
τ
) = 0
.
2176(21), which is 3.3 standard deviations smaller
than the
K
`
2
and
K
`
3
average. This value would also imply a 3.6
σ
deviation from three generation
unitarity when combined with the values for
|
V
ud
|
2
and
|
V
ub
|
2
in this review. Improvements in the
theoretical framework have recently been suggested [39], and a new evaulation of the
τ
data yields
|
V
us
|
(
τ
X
s
ν
τ
) = 0
.
2228(23)
exp
(5)
thy
. This value (which uses some unpublished preliminary data
from BaBar on
τ
K
π
0
ν
[40] that also helps to improve the agreement) is consistent with the kaon
determination. The experimental uncertainty dominates, and there is room for the
τ
determination of
|
V
us
|
to improve, e.g., with Belle-II.
With
|
V
ud
|
and
|
V
us
|
we may ask whether a third generation is required. We find (without using the
scaled error for
|
V
us
|
):
1
−|
V
ud
|
2
−|
V
us
|
2
= (0
.
00054
±
0
.
00047)
.
(34)
This is consistent with zero, hence there is no evidence that a third generation exists based on these
values. However, a constraint is obtained on how large the mixing with the third generation could be
or on new physics scenarios such as additional generations.
3.3.
|
V
cs
|
The
c
s
transition is the “Cabibbo-favored” decay channel for charm. The value of
|
V
cs
|
is best
measured in
D
and
D
s
decays analogous to the kaon decays discussed for
|
V
us
|
. The analog of the
K
μ
2
decay is
D
s
with
`
=
μ
or
τ
and the analog of the
K
`
3
decay is
D
K`ν
. There are other
processes dependent on
|
V
cs
|
, such as charmed baryon decays to strangeness and
W
c
̄
s
, but the
available precision as measurements of
|
V
cs
|
is not competitive.
The Heavy Flavor Averaging Group (HFAG) [7] has averaged the published results from BaBar [41],
Belle [42], and CLEO-c [43, 44, 45] for
D
s
, corrected for
τ
branching fractions in [3] where relevant,
obtaining:
B
(
D
s
μν
) = 5
.
57(24)
×
10
3
B
(
D
s
τν
) = 5
.
55(24)
×
10
2
(35)
Using the appropriately relabled Eq. 22, neglecting the radiative corrections, but including experimental
correlations, these results are combined by HFAG [7], obtaining:
f
D
s
|
V
cs
|
= 250
.
6
±
4
.
5 MeV
.
(36)
The value of the
f
D
s
decay constant is evaluated in lattice QCD. We use the HPQCD collaboration
result from [46] and the Fermilab Lattice (FNAL)/MILC collaboration result from [29]. We do not
include the also recent but less accurate HPQCD result in [47]. The average is performed assuming both
no correlation and completely correlated systematics. The central value we quote is the no correlation
value, and the uncertainty is increased linearly by the difference between the correlated and uncorrelated
averages, with the result
f
D
s
= 248
.
6
±
1
.
6 MeV
.
(37)
Combining Eqs. 36 and 37 gives
|
V
cs
|
= 1
.
008
±
0
.
019
.
(38)
The lattice calculation has improved to the point where the dominant uncertainty comes from the
experimental measurement.
We may also determine the value of
|
V
cs
|
using the semileptonic
D
K`ν
process and the analog
of Eq. 30. Several methods have been employed and compared to evaluate the form factor dependent
9
Table 2: Experimental measurements of
|
V
cs
|
f
DK
+
(0). The first error is statistical, the second systematic. The notation
“10 GeV” refers to data taken in the
B
̄
B
threshold region, mostly on the Υ(4
S
).
Quantity
Data
|
V
cd
|
f
DK
+
(0)
Reference
D
0
K
e
+
ν
10 GeV, 75 fb
1
0
.
720(7)(7)
BaBar [53, 7]
D
0
K
(
e
+
+
)
ν
10 GeV, 282 fb
1
0
.
692(7)(22)
Belle [54, 7]
D
+
K
L
e
+
ν
ψ
(3770), 2.92 fb
1
0
.
728(6)(11)
BESIII [55]
D
0
K
e
+
ν
ψ
(3770), 2.92 fb
1
0
.
7172(25)(35) BESIII [52]
D
0
K
e
+
ν
ψ
(3770), 281 pb
1
0
.
747(9)(9)
CLEO-c untagged [56]
D
+
̄
K
0
e
+
ν
ψ
(3770), 281 pb
1
0
.
733(14)(11)
CLEO-c untagged [56]
D
Ke
+
ν
ψ
(3770), 818 pb
1
0
.
719(6)(5)
CLEO-c tagged [57]
D
0
K
e
+
ν ψ
(3770), 818 pb
1
0
.
726(8)(4)
CLEO-c tagged [57]
D
+
̄
K
0
e
+
ν ψ
(3770), 818 pb
1
0
.
707(10)(9)
CLEO-c tagged [57]
D
K`
+
ν
0
.
7208(33)
Our average
integral
I
D`
, including dispersion relations [48], pole parameterizations [49], “
z
-expansion” [50] (and
references therein), and ISGW2 [51]. Recent analyses typically settle on the
z
-expansion for quoting
results. The basic idea of the
z
-expansion is to map
t
=
q
2
to a variable (
z
) such that a Taylor series
with good convergence properties can be used. The chosen mapping is
z
(
t,t
0
)
t
+
t
t
+
t
0
t
+
t
+
t
+
t
0
,
(39)
where
t
+
= (
m
D
+
m
K
)
2
is the threshold for
DK
production and
t
0
(
−∞
,t
+
) is the value of
t
corresponding to
z
= 0, and may be chosen for desirable properties. The form factor is then expanded
in a Taylor series in
z
. Three terms in the expansion is usually found to be sufficient.
The most precise published results on the semileptonic decay
D
K`
+
ν
(
`
=
e,μ
) come from
BaBar, Belle, BESIII, and CLEO-c, see Table 2. On a 2.9 fb
1
dataset taken at the
ψ
(3770), BESIII
reports a very precise result [52],
f
DK
+
(0)
|
V
cd
|
= 0
.
7172(25)(35) from
D
0
K
e
+
ν
decays.
In Table 2 the 818 pb
1
dataset from CLEO-c includes the 281 pb
1
dataset, and there is some
correlation between the tagged and untagged analyses in both statistical and systematic errors. The
correlations, including as well those between the charged and neutral
D
channels, have been derived by
CLEO-c on the 281 pb
1
dataset [58]. We use the correlation information (including the assumption
that all of the 818 pb
1
dataset has the same correlation in the systematic errors with the untagged
analysis) to obtain an average of the data in Table 2:
|
V
cd
|
f
DK
+
(0) = 0
.
7208(33). The
p
value for the
χ
2
statistic is 0.004. The correlation between the two BESIII results is small, and is neglected. This result
is close to the HFAG average [6, 7] (0.728(5)), except that we now use the published BESIII result for
D
0
K
e
+
ν
and we also include the BESIII
D
+
K
L
e
+
ν
measurement.
It is noted in [5] that the HFAG averaging does not remove the final-state Coulomb correction in
the
D
0
channel prior to averaging. This and other corrections are potentially important now that the
measurements have become precise. However, this determination of
|
V
cs
|
is limited by the uncertainty
in
f
DK
+
(0), hence neglect of these corrections is presently safe enough.
The smallness of the
p
value is of some concern. It is possible that we are misestimating the
correlations in our treatment of the CLEO data. However, it may also be an indication that the neglect
of the different electromagnetic corrections between
D
+
and
D
0
is no longer justified. The incorporation
of such corrections should be performed as part of the analysis. However, they naively could be as large
as
O
(%) [59]. To investigate the possible effect, “correcting” the
D
0
numbers by a factor of 0.99 improves
the
p
value to 2%, while changing the average to 0.7194(33); a factor of 0.98 improves it to 5%, while
10
changing the average to 0.7181(33). As already noted the uncertainty for
|
V
cs
|
remains dominated by
the uncertainty in
f
DK
+
(0), so this remains a secondary issue for our discussion. Nevertheless, we suggest
that future evaluations of
f
DK
+
(0)
|
V
cs
|
include such corrections in the analysis over the Dalitz plot.
To extract
|
V
cs
|
, the form factor at zero recoil,
f
DK
+
(0), is required, and lattice calculations are
available. The FLAG evaluation [4] is
f
DK
+
(0) = 0
.
747
±
0
.
019
.
(40)
Due to the FLAG quality and publication requirements, this is just the HPQCD evaluation from
Ref. [60]. This yields
|
V
cs
|
= 0
.
965(25).
The leptonic and semileptonic results are consistent (
p
(
χ
2
) = 0
.
24), averaging them yields:
|
V
cs
|
2
= 0
.
983(30)
,
|
V
cs
|
= 0
.
992(15)
.
(41)
The use of the detailed
q
2
dependence in
D
K`ν
decays to improve the measurement of
|
V
cs
|
is dis-
cussed in a preprint [61] from the HPQCD Collaboration, in which a value of
|
V
cs
|
= 0
.
973(5)
exp
(14)
lattice
is obtained. It should be noted that the lattice QCD calculations are also improving for semileptonic de-
cays to vector mesons, for example
D
s
φ`ν
. A recent evaluation [62] extracts
|
V
cs
|
= 1
.
017(63) using
the branching fraction measured by BaBar [63]. The error is dominated by the theoretical uncertainties,
but not by much.
3.4.
|
V
cd
|
Completing the upper left 2
×
2 submatrix is the “Cabibbo-suppressed”
c
d
transition. Early
measurements of
|
V
cd
|
were performed in neutrino production of charm, and this method remains in
principle competitive. Lattice calculation of form factors has advanced such that the semileptonic
D
π`ν
decay is also useful, analogous to
D
K`ν
for
|
V
cs
|
. Also similarly with other elements, the
leptonic decay
D
+
`
+
ν
provides a measurement of
|
V
cd
|
if lattice calculations are used for the
f
D
decay constant.
The neutrino measurements consist in measuring di-muon production, where one muon is the result
of a charged current interaction (providing a
d
c
transition, hence dependence on
|
V
cd
|
), and the
second muon tags the decay of a charmed hadron. The measurement is reported as the product
B
μ
|
V
cd
|
2
,
where
B
μ
is the semileptonic branching fraction of charmed hadrons, as appropriate to the experimental
conditions.
There has not been much recent development in this area in the neutrino experiments. The 2004
Review of Particle Properties (RPP) [64] quotes an average for the CDHS [65], CCFR [66, 67], and
CHARM II [68] measurements of
B
μ
|
V
cd
|
2
= 0
.
00463(34). An evaluation [69] of
B
μ
for the nominal
kinematic regime (visible energy
>
30 GeV) is combined in the 2014 RPP [3] with a measurement
from CHORUS (nuclear emulsion) [70] obtaining
B
μ
= 0
.
087(5), and thence
|
V
cd
|
= 0
.
230(11). There is
a further result from CHORUS based on events produced in the lead-scintillating fiber calorimeter [71],
which has not been included in this average.
However, the review in [69] notes the inconsistency in combining leading order (LO) and next-to-
leading order (NLO) determinations, and quotes separate results. The LO results from CDHS, CCFR,
and CHARM II are averaged obtaining
|
V
cd
|
LO
= 0
.
232(10), while the NLO result from CCFR yields
|
V
cd
|
NLO
= 0
.
246(16). We do not include the neutrino in our average for
|
V
cd
|
, opting instead for the in
any event presently more precise results from
D
meson decays.
As with earlier elements,
|
V
cd
|
may be measured in leptonic and semileptonic decays, in particular,
D
and
D
π`ν
. For the leptonic channel, the
f
D
+
decay constant is required, and this
11