In the Standard Model of particle physics,
the Cabibbo–Kobayashi–Maskawa matrix,
CKM matrix, quark mixing matrix, or KM matrix
is a unitary matrix which contains information
on the strength of the flavour-changing weak
interaction. Technically, it specifies the
mismatch of quantum states of quarks when
they propagate freely and when they take part
in the weak interactions. It is important
in the understanding of CP violation. This
matrix was introduced for three generations
of quarks by Makoto Kobayashi and Toshihide
Maskawa, adding one generation to the matrix
previously introduced by Nicola Cabibbo. This
matrix is also an extension of the GIM mechanism,
which only includes two of the three current
families of quarks.
== The matrix ==
=== Predecessor: Cabibbo matrix ===
In 1963, Nicola Cabibbo introduced the Cabibbo
angle (θc) to preserve the universality of
the weak interaction. Cabibbo was inspired
by previous work by Murray Gell-Mann and Maurice
Lévy, on the effectively rotated nonstrange
and strange vector and axial weak currents,
which he references.In light of current knowledge
(quarks were not yet theorized), the Cabibbo
angle is related to the relative probability
that down and strange quarks decay into up
quarks (|Vud|2 and |Vus|2 respectively). In
particle physics parlance, the object that
couples to the up quark via charged-current
weak interaction is a superposition of down-type
quarks, here denoted by d′. Mathematically
this is:
d
′
=
V
u
d
d
+
V
u
s
s
,
{\displaystyle d^{\prime }=V_{ud}d+V_{us}s,}
or using the Cabibbo angle:
d
′
=
cos
⁡
θ
c
d
+
sin
⁡
θ
c
s
.
{\displaystyle d^{\prime }=\cos \theta _{\mathrm
{c} }d+\sin \theta _{\mathrm {c} }s.}
Using the currently accepted values for |Vud|
and |Vus| (see below), the Cabibbo angle can
be calculated using
tan
⁡
θ
c
=
|
V
u
s
|
|
V
u
d
|
=
0.22534
0.97427
⇒
θ
c
=
13.02
∘
.
{\displaystyle \tan \theta _{\mathrm {c} }={\frac
{|V_{us}|}{|V_{ud}|}}={\frac {0.22534}{0.97427}}\Rightarrow
\theta _{\mathrm {c} }=~13.02^{\circ }.}
When the charm quark was discovered in 1974,
it was noticed that the down and strange quark
could decay into either the up or charm quark,
leading to two sets of equations:
d
′
=
V
u
d
d
+
V
u
s
s
;
{\displaystyle d^{\prime }=V_{ud}d+V_{us}s;}
s
′
=
V
c
d
d
+
V
c
s
s
,
{\displaystyle s^{\prime }=V_{cd}d+V_{cs}s,}
or using the Cabibbo angle:
d
′
=
cos
⁡
θ
c
d
+
sin
⁡
θ
c
s
;
{\displaystyle d^{\prime }=\cos {\theta _{\mathrm
{c} }}d+\sin {\theta _{\mathrm {c} }}s;}
s
′
=
−
sin
⁡
θ
c
d
+
cos
⁡
θ
c
s
.
{\displaystyle s^{\prime }=-\sin {\theta _{\mathrm
{c} }}d+\cos {\theta _{\mathrm {c} }}s.}
This can also be written in matrix notation
as:
[
d
′
s
′
]
=
[
V
u
d
V
u
s
V
c
d
V
c
s
]
[
d
s
]
,
{\displaystyle {\begin{bmatrix}d^{\prime }\\s^{\prime
}\end{bmatrix}}={\begin{bmatrix}V_{ud}&V_{us}\\V_{cd}&V_{cs}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}},}
or using the Cabibbo angle
[
d
′
s
′
]
=
[
cos
⁡
θ
c
sin
⁡
θ
c
−
sin
⁡
θ
c
cos
⁡
θ
c
]
[
d
s
]
,
{\displaystyle {\begin{bmatrix}d^{\prime }\\s^{\prime
}\end{bmatrix}}={\begin{bmatrix}\cos {\theta
_{\mathrm {c} }}&\sin {\theta _{\mathrm {c}
}}\\-\sin {\theta _{\mathrm {c} }}&\cos {\theta
_{\mathrm {c} }}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}},}
where the various |Vij|2 represent the probability
that the quark of j flavor decays into a quark
of i flavor. This 2 × 2 rotation matrix is
called the Cabibbo matrix.
=== CKM matrix ===
In 1973, Observing that CP-violation could
not be explained in a four-quark model, Kobayashi
and Maskawa generalized the Cabibbo matrix
into the Cabibbo–Kobayashi–Maskawa matrix
(or CKM matrix) to keep track of the weak
decays of three generations of quarks:
[
d
′
s
′
b
′
]
=
[
V
u
d
V
u
s
V
u
b
V
c
d
V
c
s
V
c
b
V
t
d
V
t
s
V
t
b
]
[
d
s
b
]
.
{\displaystyle {\begin{bmatrix}d^{\prime }\\s^{\prime
}\\b^{\prime }\end{bmatrix}}={\begin{bmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{bmatrix}}{\begin{bmatrix}d\\s\\b\end{bmatrix}}.}
On the left is the weak interaction doublet
partners of up-type quarks, and on the right
is the CKM matrix along with a vector of mass
eigenstates of down-type quarks. The CKM matrix
describes the probability of a transition
from one quark i 
to another quark j. These transitions are
proportional to |Vij|2.
As of 2010, the best determination of the
magnitudes of 
the CKM matrix elements was:
[
|
V
u
d
|
|
V
u
s
|
|
V
u
b
|
|
V
c
d
|
|
V
c
s
|
|
V
c
b
|
|
V
t
d
|
|
V
t
s
|
|
V
t
b
|
]
=
[
0.97427
±
0.00015
0.22534
±
0.00065
0.00351
−
0.00014
+
0.00015
0.22520
±
0.00065
0.97344
±
0.00016
0.0412
−
0.0005
+
0.0011
0.00867
−
0.00031
+
0.00029
0.0404
−
0.0005
+
0.0011
0.999146
−
0.000046
+
0.000021
]
.
{\displaystyle {\begin{bmatrix}|V_{ud}|&|V_{us}|&|V_{ub}|\\|V_{cd}|&|V_{cs}|&|V_{cb}|\\|V_{td}|&|V_{ts}|&|V_{tb}|\end{bmatrix}}={\begin{bmatrix}0.97427\pm
0.00015&0.22534\pm 0.00065&0.00351_{-0.00014}^{+0.00015}\\0.22520\pm
0.00065&0.97344\pm 0.00016&0.0412_{-0.0005}^{+0.0011}\\0.00867_{-0.00031}^{+0.00029}&0.0404_{-0.0005}^{+0.0011}&0.999146_{-0.000046}^{+0.000021}\end{bmatrix}}.}
The choice of usage of down-type quarks in
the definition is a convention, and does not
represent a physically preferred asymmetry
between up-type and down-type quarks. Other
conventions are equally valid, such as defining
the matrix in terms of weak interaction partners
of mass eigenstates of up-type quarks, u′,
c′ and t′, in terms of u, c, and t. Since
the CKM matrix is unitary, its inverse is
the same as its conjugate transpose.
== Counting ==
To proceed further, it is necessary to count
the number of parameters in this matrix, V
which appear in experiments, and therefore
are physically important. If there are N generations
of quarks (2N flavours) then
An N × N unitary matrix (that is, a matrix
V such that VV† = I, where V† is the conjugate
transpose of V and I is the identity matrix)
requires N2 real parameters to be specified.
2N − 1 of these parameters are not physically
significant, because one phase can be absorbed
into each quark field (both of the mass eigenstates,
and of the weak eigenstates), but an overall
common phase is unobservable. Hence, the total
number of free variables independent of the
choice of the phases of basis vectors is N2
− (2N − 1) = (N − 1)2.
Of these, N(N − 1)/2 are rotation angles
called quark mixing angles.
The remaining (N − 1)(N − 2)/2 are complex
phases, which cause CP violation.For the case
N = 2, there is only one parameter which is
a mixing angle between two generations of
quarks. Historically, this was the first version
of CKM matrix when only two generations were
known. It is called the Cabibbo angle after
its inventor Nicola Cabibbo.
For the Standard Model case (N = 3), there
are three mixing angles and one CP-violating
complex phase.
== Observations and predictions ==
Cabibbo's idea originated from a need to explain
two observed phenomena:
the transitions u ↔ d, e ↔ νe, and μ
↔ νμ had similar amplitudes.
the transitions with change in strangeness
ΔS = 1 had amplitudes equal to 1/4 of those
with ΔS = 0.Cabibbo's solution consisted
of postulating weak universality to resolve
the first issue, along with a mixing angle
θc, now called the Cabibbo angle, between
the d and s quarks to resolve the second.
For two generations of quarks, there are no
CP violating phases, as shown by the counting
of the previous section. Since CP violations
were seen in neutral kaon decays already in
1964, the emergence of the Standard Model
soon after was a clear signal of the existence
of a third generation of quarks, as pointed
out in 1973 by Kobayashi and Maskawa. The
discovery of the bottom quark at Fermilab
(by Leon Lederman's group) in 1976 therefore
immediately started off the search for the
missing third-generation quark, the top quark.
Note, however, that the specific values of
the angles are not a prediction of the standard
model: they are open, unfixed parameters.
At this time, there is no generally accepted
theory that explains why the measured values
are what they are.
== Weak universality ==
The constraints of unitarity of the CKM-matrix
on the diagonal terms can be written as
∑
k
|
V
i
k
|
2
=
∑
i
|
V
i
k
|
2
=
1
{\displaystyle \sum _{k}|V_{ik}|^{2}=\sum
_{i}|V_{ik}|^{2}=1}
for all generations i. This implies that the
sum of all couplings of any of the up-type
quarks to all the down-type quarks is the
same for all generations. This relation is
called weak universality and was first pointed
out by Nicola Cabibbo in 1967. Theoretically
it is a consequence of the fact that all SU(2)
doublets couple with the same strength to
the vector bosons of weak interactions. It
has been subjected to continuing experimental
tests.
== The unitarity triangles ==
The remaining constraints of unitarity of
the CKM-matrix can be written in the form
∑
k
V
i
k
V
j
k
∗
=
0.
{\displaystyle \sum _{k}V_{ik}V_{jk}^{*}=0.}
For any fixed and different i and j, this
is a constraint on three complex numbers,
one for each k, which says that these numbers
form the sides of a triangle in the complex
plane. There are six choices of i and j (three
independent), and hence six such triangles,
each of which is called a unitary triangle.
Their shapes can be very different, but they
all have the same area, which can be related
to the CP violating phase. The area vanishes
for the specific parameters in the Standard
Model for which there would be no CP violation.
The orientation of the triangles depend on
the phases of the quark fields.
A popular quantity amounting to twice the
area of the unitarity triangle is the Jarlskog
invariant,
J
=
c
12
c
13
2
c
23
s
12
s
13
s
23
sin
⁡
δ
≈
3
10
−
5
{\displaystyle J=c_{12}c_{13}^{2}c_{23}s_{12}s_{13}s_{23}\sin
\delta \approx 3~10^{-5}}
. For Greek indices denoting up quarks and
Latin ones down quarks, the 4-tensor
(
α
,
β
;
i
,
j
)
≡
Im
⁡
(
V
α
i
V
β
j
V
α
j
∗
V
β
i
∗
)
{\displaystyle (\alpha ,\beta ;i,j)\equiv
\operatorname {Im} (V_{\alpha i}V_{\beta j}V_{\alpha
j}^{*}V_{\beta i}^{*})}
is doubly antisymmetric,
(
β
,
α
;
i
,
j
)
=
−
(
α
,
β
;
i
,
j
)
=
(
α
,
β
;
j
,
i
)
.
{\displaystyle (\beta ,\alpha ;i,j)=-(\alpha
,\beta ;i,j)=(\alpha ,\beta ;j,i).}
Up to antisymmetry, it only has 9=3×3 non-vanishing
components, which, remarkably, from the unitarity
of V, can be shown to be all identical in
magnitude, that is,
(
α
,
β
;
i
,
j
)
=
J
[
0
1
−
1
−
1
0
1
1
−
1
0
]
α
β
⊗
[
0
1
−
1
−
1
0
1
1
−
1
0
]
i
j
,
{\displaystyle (\alpha ,\beta ;i,j)=J~{\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}}_{\alpha
\beta }\otimes {\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}}_{ij},}
so that
J
=
(
u
,
c
;
s
,
b
)
=
(
u
,
c
;
d
,
s
)
=
(
u
,
c
;
b
,
d
)
=
(
c
,
t
;
s
,
b
)
=
(
c
,
t
;
d
,
s
)
=
(
c
,
t
;
b
,
d
)
=
(
t
,
u
;
s
,
b
)
=
(
t
,
u
;
b
,
d
)
=
(
t
,
u
;
d
,
s
)
.
{\displaystyle J=(u,c;s,b)=(u,c;d,s)=(u,c;b,d)=(c,t;s,b)=(c,t;d,s)=(c,t;b,d)=(t,u;s,b)=(t,u;b,d)=(t,u;d,s).}
Since the three sides of the triangles are
open to direct experiment, as are the three
angles, a class of tests of the Standard Model
is to check that the triangle closes. This
is the purpose of a modern series of experiments
under way at the Japanese BELLE and the American
BaBar experiments, as well as at LHCb in CERN,
Switzerland.
== Parameterizations ==
Four independent parameters are required to
fully define the CKM matrix. Many parameterizations
have been proposed, and three of the most
common ones are shown below.
=== KM parameters ===
The original parameterization of Kobayashi
and Maskawa used three angles (θ1, θ2, θ3)
and a CP-violating phase (δ). Cosines and
sines of the angles are denoted ci and si,
respectively. θ1 is the Cabibbo angle.
[
c
1
−
s
1
c
3
−
s
1
s
3
s
1
c
2
c
1
c
2
c
3
−
s
2
s
3
e
i
δ
c
1
c
2
s
3
+
s
2
c
3
e
i
δ
s
1
s
2
c
1
s
2
c
3
+
c
2
s
3
e
i
δ
c
1
s
2
s
3
−
c
2
c
3
e
i
δ
]
.
{\displaystyle {\begin{bmatrix}c_{1}&-s_{1}c_{3}&-s_{1}s_{3}\\s_{1}c_{2}&c_{1}c_{2}c_{3}-s_{2}s_{3}e^{i\delta
}&c_{1}c_{2}s_{3}+s_{2}c_{3}e^{i\delta }\\s_{1}s_{2}&c_{1}s_{2}c_{3}+c_{2}s_{3}e^{i\delta
}&c_{1}s_{2}s_{3}-c_{2}c_{3}e^{i\delta }\end{bmatrix}}.}
=== "Standard" parameters ===
A "standard" parameterization of the CKM matrix
uses three Euler angles (θ12, θ23, θ13)
and one CP-violating phase (δ13). Couplings
between quark generation i and j vanish if
θij = 0. Cosines and sines of the angles
are denoted cij and sij, respectively. θ12
is the Cabibbo angle.
[
1
0
0
0
c
23
s
23
0
−
s
23
c
23
]
[
c
13
0
s
13
e
−
i
δ
13
0
1
0
−
s
13
e
i
δ
13
0
c
13
]
[
c
12
s
12
0
−
s
12
c
12
0
0
0
1
]
=
[
c
12
c
13
s
12
c
13
s
13
e
−
i
δ
13
−
s
12
c
23
−
c
12
s
23
s
13
e
i
δ
13
c
12
c
23
−
s
12
s
23
s
13
e
i
δ
13
s
23
c
13
s
12
s
23
−
c
12
c
23
s
13
e
i
δ
13
−
c
12
s
23
−
s
12
c
23
s
13
e
i
δ
13
c
23
c
13
]
.
{\displaystyle {\begin{aligned}&{\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta
_{13}}\\0&1&0\\-s_{13}e^{i\delta _{13}}&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta
_{13}}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta
_{13}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta
_{13}}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta
_{13}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta
_{13}}&c_{23}c_{13}\end{bmatrix}}.\end{aligned}}}
The currently best known values for the standard
parameters are:
θ12 = 13.04±0.05°, θ13 = 0.201±0.011°,
θ23 = 2.38±0.06°, and δ13 = 1.20±0.08
rad.
=== Wolfenstein parameters ===
A third parameterization of the CKM matrix
was introduced by Lincoln Wolfenstein with
the four parameters λ, A, ρ, and η. The
four Wolfenstein parameters have the property
that all are of order 1 and are related to
the "standard" parameterization:
λ = s12
Aλ2 = s23
Aλ3(ρ − iη) = s13e−iδThe Wolfenstein
parameterization of the CKM matrix, is an
approximation of the standard parameterization.
To order λ3, it is:
[
1
−
λ
2
/
2
λ
A
λ
3
(
ρ
−
i
η
)
−
λ
1
−
λ
2
/
2
A
λ
2
A
λ
3
(
1
−
ρ
−
i
η
)
−
A
λ
2
1
]
.
{\displaystyle {\begin{bmatrix}1-\lambda ^{2}/2&\lambda
&A\lambda ^{3}(\rho -i\eta )\\-\lambda &1-\lambda
^{2}/2&A\lambda ^{2}\\A\lambda ^{3}(1-\rho
-i\eta )&-A\lambda ^{2}&1\end{bmatrix}}.}
The CP violation can be determined by measuring
ρ − iη.
Using the values of the previous section for
the CKM matrix, the best determination of
the Wolfenstein parameters is:
λ = 0.2257+0.0009−0.0010, A = 0.814+0.021−0.022,
ρ = 0.135+0.031−0.016, and η = 0.349+0.015−0.017.
== Nobel Prize ==
In 2008, Kobayashi and Maskawa shared one
half of the Nobel Prize in Physics "for the
discovery of the origin of the broken symmetry
which predicts the existence of at least three
families of quarks in nature". Some physicists
were reported to harbor bitter feelings about
the fact that the Nobel Prize committee failed
to reward the work of Cabibbo, whose prior
work was closely related to that of Kobayashi
and Maskawa. Asked for a reaction on the prize,
Cabibbo preferred to give no comment.
== See also ==
Formulation of the Standard Model and CP violations.
Quantum chromodynamics, flavour and strong
CP problem.
Weinberg angle, a similar angle for Z and
photon mixing.
Pontecorvo–Maki–Nakagawa–Sakata matrix,
the equivalent mixing matrix for neutrinos.
Koide formula
