View source for Cabibbo–Kobayashi–Maskawa matrix

{{Short description|Unitary matrix containing information on the weak interaction}}
{{Flavour quantum numbers}}

In the [[Standard Model]] of [[particle physics]], the '''Cabibbo–Kobayashi–Maskawa matrix''', '''CKM matrix''', '''quark mixing matrix''', or '''KM matrix''' is a [[unitary matrix]] that contains information on the strength of the [[Flavour (particle physics)|flavour]]-changing [[weak interaction]]. Technically, it specifies the mismatch of [[quantum state]]s of [[quark]]s when they propagate freely and when they take part in the [[weak interaction]]s. It is important in the understanding of [[CP violation]]. This matrix was introduced for three generations of quarks by [[Makoto Kobayashi (physicist)|Makoto Kobayashi]] and [[Toshihide Maskawa]], adding one [[generation (particle physics)|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 – the Cabibbo matrix===
[[Image:Cabibbo angle.svg|thumb|270px|right|The Cabibbo angle represents the rotation of the mass eigenstate vector space formed by the mass eigenstates
<math> | d \rangle , \,| s \rangle </math>
into the weak eigenstate vector space formed by the weak eigenstates
<math> | d' \rangle \,, ~ | \, s' \rangle ~.</math> {{nowrap|{{mvar|θ}}{{sub|c}} {{=}} 13.02°  .}} ]]
In 1963, [[Nicola Cabibbo]] introduced the '''Cabibbo angle''' ({{mvar|θ}}{{sub|c}}) to preserve the universality of the [[weak interaction]].<ref name="Cabibbo">
{{cite journal
 |first=N. |last=Cabibbo
 |year=1963
 |title=Unitary Symmetry and Leptonic Decays
 |journal=[[Physical Review Letters]]
 |volume=10  |issue=12  |pages=531–533
 |doi=10.1103/PhysRevLett.10.531  |doi-access=free
 |bibcode=1963PhRvL..10..531C 
}}
</ref> 
Cabibbo was inspired by previous work by [[Murray Gell-Mann]] and  Maurice Lévy,<ref>
{{cite journal
 |first1=M. |last1=Gell-Mann |author1-link=Murray Gell-Mann
 |first2=M. |last2=Lévy
 |year=1960
 |title= The Axial Vector Current in Beta Decay
 |journal=[[Il Nuovo Cimento]]
 |volume=16 |issue=4 |pages=705–726
 |doi=10.1007/BF02859738
 |bibcode=1960NCim...16..705G
 |s2cid=122945049
}}
</ref>
on the effectively rotated nonstrange and strange vector and axial weak currents, which he references.<ref>
{{cite journal
 |first=L. |last=Maiani
 |year=2009
 |title=Sul premio Nobel per la fisica 2008
 |trans-title=On the Nobel prize in Physics for 2008
 |journal=Il Nuovo Saggiatore
 |volume=25  |issue=1–2  |pages=78
 |url=http://prometeo.sif.it:8080/papers/online/sag/025/01-02/pdf/78_opinioni.pdf
 |url-status=dead  |access-date=30 November 2010
 |archive-url=https://web.archive.org/web/20110722053046/http://prometeo.sif.it:8080/papers/online/sag/025/01-02/pdf/78_opinioni.pdf
 |archive-date=22 July 2011  |df=dmy-all
}}
</ref>

In light of current concepts (quarks had not yet been proposed), the Cabibbo angle is related to the relative probability that [[down quark|down]] and [[strange quark]]s decay into [[up quark]]s  (&nbsp;|{{mvar|V}}{{sub|ud}}|{{sup|2}} &nbsp; and &nbsp; |{{mvar|V}}{{sub|us}}|{{sup|2}}&nbsp;, respectively). In particle physics terminology, the object that couples to the up quark via charged-current weak interaction is a superposition of down-type quarks, here denoted by {{mvar|d′}}.<ref name="Hughes">
{{cite book
 |first=I.S. |last=Hughes
 |year=1991
 |chapter=Chapter 11.1 – Cabibbo Mixing
 |title=Elementary Particles
 |edition=3rd  |pages=242–243
 |publisher=[[Cambridge University Press]]
 |isbn=978-0-521-40402-0
 |chapter-url=https://books.google.com/books?id=JN6qlZlGUG4C&q=cabbibo+angle&pg=PA242
}}
</ref>
Mathematically this is:

:<math> d' = V_\mathrm{ud} \; d  ~~ + ~~ V_\mathrm{us} \; s ~,</math>

or using the Cabibbo angle:

:<math> d' = \cos \theta_\mathrm{c} \; d  ~~ + ~~ \sin \theta_\mathrm{c} \; s ~.</math>

Using the currently accepted values for &nbsp; |{{mvar|V}}{{sub|ud}}| &nbsp; and &nbsp; |{{mvar|V}}{{sub|us}}| &nbsp; (see below), the Cabibbo angle can be calculated using

:<math> \tan\theta_\mathrm{c} = \frac{\, |V_\mathrm{us}| \,}{|V_\mathrm{ud}|} = \frac{0.22534}{0.97427} \quad \Rightarrow \quad \theta_\mathrm{c}= ~13.02^\circ ~.</math>

When the [[charm quark]] was discovered in 1974, it was noticed that the down and strange quark could transition into either the up or charm quark, leading to two sets of equations:

:<math> d' = V_\mathrm{ud} \; d ~~ + ~~ V_\mathrm{us} \; s ~,</math>
:<math> s' = V_\mathrm{cd} \; d ~~ + ~~ V_\mathrm{cs} \; s ~;</math>

or using the Cabibbo angle:

:<math> d' = ~~~ \cos{\theta_\mathrm{c}} \; d  ~~+~~ \sin{\theta_\mathrm{c}} \; s ~,</math>
:<math> s' =  -  \sin{\theta_\mathrm{c}} \; d  ~~+~~ \cos{\theta_\mathrm{c}} \; s ~.</math>

This can also be written in [[matrix (mathematics)|matrix notation]] as:

:<math>
\begin{bmatrix}  d'  \\  s'  \end{bmatrix} =
\begin{bmatrix} V_\mathrm{ud} & V_\mathrm{us} \\ V_{cd} & V_{cs} \\ \end{bmatrix}
\begin{bmatrix}  d  \\  s  \end{bmatrix} ~,
</math>

or using the Cabibbo angle

:<math>
\begin{bmatrix} d'  \\  s'  \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}~,
</math>

where the various |{{mvar|V{{sub|ij}}}}|{{sup|2}} represent the probability that the quark of flavor {{mvar|j}} decays into a quark of flavor {{mvar|i}}. This 2×2&nbsp;[[rotation matrix]] is called the "Cabibbo matrix", and was subsequently expanded to the 3×3 CKM matrix.

[[Image:Quark weak interactions.svg|thumb|270px|right|A pictorial representation of the six quarks' decay modes, with mass increasing from left to right.]]

===CKM matrix===
[[Image:Weak_Decay_(flipped).svg|thumb|270px|right|A diagram depicting the decay routes due to the charged weak interaction and some indication of their likelihood. The intensity of the lines is given by the CKM parameters]]
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:<ref name="KM">
{{cite journal
 |first1=M. |last1=Kobayashi
 |first2=T. |last2=Maskawa
 |year=1973
 |title=CP-violation in the renormalizable theory of weak interaction
 |journal=[[Progress of Theoretical Physics]]
 |volume=49 |issue=2 |pages=652–657
 |doi=10.1143/PTP.49.652  |doi-access=free 
 |bibcode=1973PThPh..49..652K
|hdl=2433/66179|hdl-access=free}}
</ref>

:<math>\begin{bmatrix}  d'  \\  s'  \\  b'  \end{bmatrix} = \begin{bmatrix} V_\mathrm{ud} & V_\mathrm{us} & V_\mathrm{ub} \\ V_\mathrm{cd} & V_\mathrm{cs} & V_\mathrm{cb} \\ V_\mathrm{td} & V_\mathrm{ts} & V_\mathrm{tb} \end{bmatrix} \begin{bmatrix}  d  \\  s  \\  b  \end{bmatrix}~.</math>

On the left are the [[weak interaction]] doublet partners of down-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 flavour {{mvar|j}} quark to another flavour {{mvar|i}} quark. These transitions are proportional to |{{mvar|V{{sub|ij}}}}|{{sup|2}}.

As of 2023, the best determination of the individual [[absolute value|magnitude]]s of the CKM matrix elements was:<ref name="PDG2023">{{cite journal |last1=R.L. Workman et al. (Particle Data Group) |title=Review of Particle Physics (and 2023 update) |journal=Progress of Theoretical and Experimental Physics |date=August 2022 |volume=2022 |issue=8 |pages=083C01 |doi=10.1093/ptep/ptac097 |url=https://pdg.lbl.gov/ |access-date=12 September 2023 |ref=PDG2023|doi-access=free |hdl=20.500.11850/571164 |hdl-access=free }}</ref>
:<math>
\begin{bmatrix}
|V_{ud}| & |V_{us}| & |V_{ub}| \\
|V_{cd}| & |V_{cs}| & |V_{cb}| \\
|V_{td}| & |V_{ts}| & |V_{tb}|
\end{bmatrix} = \begin{bmatrix}
0.97435 \pm 0.00016 & 0.22500 \pm 0.00067 & 0.00369\pm 0.00011\\
0.22486 \pm 0.00067 & 0.97349 \pm 0.00016 & 0.04182^{+0.00085}_{-0.00074} \\
0.00857_{-0.00018}^{+0.00020} & 0.04110^{+0.00083}_{-0.00072}  & 0.999118^{+0.000031}_{-0.000036}
\end{bmatrix}.
</math>

Using those values, one can check the unitarity of the CKM matrix. In particular, we find that the first-row matrix elements give: <math> |V_\mathrm{ud}|^2 + |V_\mathrm{us}|^2 + |V_\mathrm{ub}|^2 = .999997 \pm .0007</math> 

making the experimental results in line with the theoretical value of 1.

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: The mass eigenstates {{math|u}}, {{math|c}}, and {{math|t}} of the up-type quarks can equivalently define the matrix in terms of ''their'' weak interaction partners {{math|u′}}, {{math|c′}}, and {{math|t′}}. Since the CKM matrix is unitary, its inverse is the same as its [[conjugate transpose]], which the alternate choices use; it appears as the same matrix, in a slightly altered form.

==General case construction==
To generalize the matrix, count the number of physically important parameters in this matrix {{mvar|V}} which appear in experiments. If there are {{sc|{{mvar|N}}}} generations of quarks (2{{sc|{{mvar|N}}}} [[flavour (particle physics)|flavour]]s) then

* An {{sc|{{mvar|N}}}}&nbsp;×&nbsp;{{sc|{{mvar|N}}}} unitary matrix (that is, a matrix {{mvar|V}} such that {{mvar|V{{sup|†}}V&nbsp;{{=}}&nbsp;I}}, where {{mvar|V{{sup|†}}}} is the [[conjugate transpose]] of {{mvar|V}} and {{mvar|I}} is the identity matrix) requires {{sc|{{mvar|N}}}}<sup>2</sup> real parameters to be specified.
* 2{{sc|{{mvar|N}}}}&nbsp;−&nbsp;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 the matrix is independent of a common phase. Hence, the total number of free variables independent of the choice of the phases of basis vectors is {{sc|{{mvar|N}}}}<sup>2</sup>&nbsp;−&nbsp;(2{{sc|{{mvar|N}}}}&nbsp;−&nbsp;1) = ({{sc|{{mvar|N}}}}&nbsp;−&nbsp;1)<sup>2</sup>.
** Of these, {{sfrac|1|2}}{{sc|{{mvar|N}}}}({{sc|{{mvar|N}}}}&nbsp;−&nbsp;1) are rotation angles called ''quark mixing angles''.
** The remaining {{sfrac|1|2}}({{sc|{{mvar|N}}}}&nbsp;−&nbsp;1)({{sc|{{mvar|N}}}}&nbsp;−&nbsp;2) are complex phases, which cause [[CP violation]].

=== {{sc|{{mvar|N}}}} = 2 ===
For the case {{sc|{{mvar|N}}}}&nbsp;=&nbsp;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]].

=== {{sc|{{mvar|N}}}} = 3 ===
For the [[Standard Model]] case ({{sc|{{mvar|N}}}}&nbsp;=&nbsp;3), there are three mixing angles and one CP-violating complex phase.<ref>
{{cite web
 |first=J.C. |last=Baez
 |date=4 April 2011
 |title=Neutrinos and the mysterious Pontecorvo-Maki-Nakagawa-Sakata matrix
 |url=http://math.ucr.edu/home/baez/neutrinos.html |access-date=2016-02-13
 |df=dmy-all
 |quote=In fact, the [[Pontecorvo–Maki–Nakagawa–Sakata matrix]] actually affects the behavior of all leptons, not just neutrinos. Furthermore, a similar trick works for quarks – but then the matrix ''U'' is called the Cabibbo–Kobayashi–Maskawa matrix.
}}
</ref>

==Observations and predictions==
Cabibbo's idea originated from a need to explain two observed phenomena:
#the transitions {{nowrap| {{math|u ↔ d}}, }} {{nowrap| {{math|e ↔ ν<sub>e</sub>}} ,}} and {{nowrap| {{math|μ ↔ ν<sub>μ</sub>}} }} had similar amplitudes.
#the transitions with change in strangeness {{nowrap| {{math|ΔS {{=}} 1}} }} had amplitudes equal to {{sfrac| 1 |4}} of those with {{nowrap| {{math|ΔS {{=}} 0}} .}}
Cabibbo's solution consisted of postulating ''weak universality'' (see below) to resolve the first issue, along with a mixing angle {{math|''θ''<sub>c</sub>}}, now called the ''Cabibbo angle'', between the {{math|d}} and {{math|s}} quarks to resolve the second.

For two generations of quarks, there can be no CP violating phases, as shown by the counting of the previous section. Since CP violations ''had'' already been seen in 1964, in neutral [[kaon]] decays, the [[standard model (basic details)|Standard Model]] that emerged soon after clearly indicated the existence of a third generation of quarks, as Kobayashi and Maskawa pointed out in 1973. The discovery of the [[bottom quark]] at [[Fermilab]] (by [[Leon Lederman]]'s group) in 1976 therefore immediately started off the search for the [[top quark]], the missing third-generation quark.

Note, however, that the specific values that the angles take on are ''not'' a prediction of the standard model: They are [[free parameter]]s. At present, there is no generally-accepted theory that explains why the angles should have the values that are measured in experiments.

==Weak universality==
The constraints of unitarity of the CKM-matrix on the diagonal terms can be written as

::<math>\sum_k |V_{jk}|^2 = \sum_k |V_{kj}|^2 = 1</math>

separately for each generation {{mvar|j}}. This implies that the sum of all couplings of any ''one'' 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 boson]]s 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

:<math>\sum_k V_{ik}V^*_{jk} = 0 ~.</math>

For any fixed and different {{mvar|i}} and {{mvar|j}}, this is a constraint on three complex numbers, one for each {{mvar|k}}, which says that these numbers form the sides of a triangle in the [[complex plane]]. There are six choices of {{mvar|i}} and {{mvar|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 violation|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''' (introduced by [[Cecilia Jarlskog]] in 1985), 
:<math> J = c_{12}c_{13}^2 c_{23}s_{12}s_{13}s_{23}\sin \delta \approx 3\cdot10^{-5} ~.</math> 
For Greek indices denoting up quarks and Latin ones down quarks, the 4-tensor <math>\;(\alpha,\beta;i,j)\equiv \operatorname{Im} (V_{\alpha i} V_{\beta j} V^*_{\alpha j} V_{\beta i}^{*}) \;</math> is doubly antisymmetric,
:<math>(\beta,\alpha;i,j) = -(\alpha,\beta;i,j)=(\alpha,\beta;j,i) ~.</math>
Up to antisymmetry, it only has {{nowrap| 9 {{=}} 3 × 3 }} non-vanishing components, which, remarkably, from the unitarity of {{mvar|V}}, can be shown to be ''all identical in magnitude'', that is,
:<math>
(\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} \;,
</math>
so that 
:<math>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) ~.</math>

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 experiment|BELLE]] and the American [[BaBar experiment|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 ({{thin space}}{{mvar|θ}}{{sub|1}}, {{mvar|θ}}{{sub|2}}, {{mvar|θ}}{{sub|3}}{{thin space}}) and a CP-violating phase angle ({{thin space}}{{mvar|δ}}{{thin space}}).<ref name="KM"/> {{mvar|θ}}{{sub|1}} is the Cabibbo angle. For brevity, the cosines and sines of the angles {{mvar|θ}}{{sub|k}} are denoted {{mvar|c}}{{sub|k}} and {{mvar|s}}{{sub|k}}, for {{nowrap|k {{=}} 1,{{thin space}}2,{{thin space}}3}} respectively. 
::<math>\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}. </math>

==="Standard" parameters===
A "standard" parameterization of the CKM matrix uses three [[Euler angles]] ({{thin space}}{{mvar|θ}}{{sub|12}}, {{mvar|θ}}{{sub|23}}, {{mvar|θ}}{{sub|13}}{{thin space}}) and one CP-violating phase ({{thin space}}{{mvar|δ}}{{sub|13}}{{thin space}}).<ref>{{cite journal |first1=L.L. |last1=Chau |first2=W.-Y. |last2=Keung |year=1984 |title=Comments on the Parametrization of the Kobayashi-Maskawa Matrix |journal=[[Physical Review Letters]] |volume=53 |pages=1802–1805 |doi=10.1103/PhysRevLett.53.1802 |bibcode=1984PhRvL..53.1802C |issue=19}}</ref> {{mvar|θ}}{{sub|12}} is the Cabibbo angle. This is the convention advocated by the [[Particle Data Group]]. Couplings between quark generations {{math|j}} and {{math|k}} vanish if {{nowrap|{{mvar|θ}}{{sub|jk}} {{=}} 0 }}. Cosines and sines of the angles are denoted {{mvar|c}}{{sub|jk}} and {{mvar|s}}{{sub|jk}}, respectively.
::<math> \begin{align} & \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{align} </math>

The 2008 values for the standard parameters were:<ref>Values obtained from values of Wolfenstein parameters in the 2008 ''[[Review of Particle Physics]]''.</ref>
:{{mvar|θ}}{{sub|12}} = {{val|13.04|0.05|u=°}}, {{mvar|θ}}{{sub|13}} = {{val|0.201|0.011|u=°}},  {{mvar|θ}}{{sub|23}} = {{val|2.38|0.06|u=°}}
and
:{{mvar|δ}}{{sub|13}} = {{val|1.20|0.08}}&nbsp;radians = {{val|68.8|4.5|u=°}}.

===Wolfenstein parameters===
A third parameterization of the CKM matrix was introduced by [[Lincoln Wolfenstein]] with the four real parameters {{mvar|λ}}, {{mvar|A}}, {{mvar|ρ}}, and {{mvar|η}}, which would all 'vanish' (would be zero) if there were no coupling.<ref>
{{cite journal
 |first=L. |last=Wolfenstein  |author-link=Lincoln Wolfenstein
 |year=1983
 |title=Parametrization of the Kobayashi-Maskawa Matrix
 |journal=[[Physical Review Letters]]
 |volume=51  |pages=1945–1947  |issue=21
 |doi=10.1103/PhysRevLett.51.1945  |bibcode=1983PhRvL..51.1945W
}}
</ref> The four Wolfenstein parameters have the property that all are of order 1 and are related to the 'standard' parameterization:
:{|
|-
| <math> \lambda = s_{12} ~, </math>
| <math> \lambda = s_{12} ~,</math>
|-
| <math> A \lambda^2 = s_{23} ~, </math>
| <math> A = \frac{s_{23} }{\; s_{12}^2 \;} ~,</math>
|-
| <math> A \lambda^3 ( \rho  - i \eta ) = s_{13} e^{-i\delta} ~, \quad </math>
| <math>  \rho = \operatorname\mathcal{R_e} \left\{ \frac{\; s_{13} \, e^{-i\delta} \;}{ s_{12} \, s_{23} } \right\} ~, \quad  \eta  = - \operatorname\mathcal{I_m} \left\{ \frac{\; s_{13} \, e^{-i\delta} \;}{ s_{12} \, s_{23} } \right\} ~. </math>
|}

Although the Wolfenstein parameterization of the CKM matrix can be as exact as desired when carried to high order, it is mainly used for generating convenient approximations to the standard parameterization. The approximation to order {{mvar|λ}}{{sup|3}}, good to better than 0.3% accuracy, is:
::<math>\begin{bmatrix} 1 - \tfrac{1}{2}\lambda^2 & \lambda & A\lambda^3(\rho-i\eta) \\
 -\lambda & 1-\tfrac{1}{2}\lambda^2 & A\lambda^2 \\
 A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1  \end{bmatrix} + O(\lambda^4) ~. </math>

Rates of [[CP violation]] correspond to the parameters {{mvar|ρ}} and {{mvar|η}}.

Using the values of the previous section for the CKM matrix, as of 2008 the best determination of the Wolfenstein parameter values is:<ref name="PDG2023">{{cite journal |last1=R.L. Workman et al. (Particle Data Group) |title=Review of Particle Physics (and 2023 update) |journal=Progress of Theoretical and Experimental Physics |date=August 2022 |volume=2022 |issue=8 |pages=083C01 |doi=10.1093/ptep/ptac097 |url=https://pdg.lbl.gov/ |access-date=12 September 2023 |ref=PDG2023|doi-access=free |hdl=20.500.11850/571164 |hdl-access=free }}</ref>
:{{mvar|λ}} =.22500 ± 0.0067, &nbsp; {{mvar|A}} = {{val|0.826|+0.018|-0.015}}, &nbsp; {{mvar|ρ}} = 0.159±0.010, &nbsp; and &nbsp; {{mvar|η}} = 0.348±0.010.

==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".<ref>
{{cite press release |publisher=[[The Nobel Foundation]] |date=7 October 2008 |title=The Nobel Prize in Physics 2008 |url=http://nobelprize.org/nobel_prizes/physics/laureates/2008/press.html |access-date=2009-11-24 |df=dmy-all}}</ref> Some physicists were reported to harbor bitter feelings about the fact that the Nobel Prize committee failed to reward the work of [[Nicola Cabibbo|Cabibbo]], whose prior work was closely related to that of Kobayashi and Maskawa.<ref>{{cite web |first=V. |last=Jamieson |date=7 October 2008 |title=Physics Nobel Snubs key Researcher |url=https://www.newscientist.com/article/dn14885-physics-nobel-snubs-key-researcher.html?DCMP=ILC-hmts&nsref=news8_head_dn14885 |work=[[New Scientist]] |access-date=2009-11-24 |df=dmy-all}}</ref> Asked for a reaction on the prize, Cabibbo preferred to give no comment.<ref>{{cite web |date=7 October 2008 |title=Nobel, l'amarezza dei fisici italiani |url=http://www.corriere.it/scienze_e_tecnologie/08_ottobre_07/nobel_fisica_italiani_traditi_d9993120-946d-11dd-a0d8-00144f02aabc.shtml |work=[[Corriere della Sera]] |language=it |access-date=2009-11-24 |df=dmy-all}}</ref>

==See also==
* [[Standard Model (mathematical formulation)|Formulation of the Standard Model]] and [[CP violation]]s
* [[Quantum chromodynamics]], [[flavour (particle physics)|flavour]] and [[CP-violation#Strong CP problem|strong CP problem]]
* [[Weinberg angle]], a similar angle for Z and photon mixing
* [[Pontecorvo–Maki–Nakagawa–Sakata matrix]], the equivalent mixing matrix for [[neutrino]]s
* [[Koide formula]]

==References==
{{Reflist|25em}}

==Further reading and external links==
* {{cite book
 |author=D.J. Griffiths
 |year=2008
 |title=Introduction to Elementary Particles
 |edition=2nd
 |publisher=[[John Wiley & Sons]]
 |isbn=978-3-527-40601-2}}

* {{cite book
 |author=B. Povh
 |year=1995
 |title=Particles and Nuclei: An Introduction to the Physical Concepts
 |publisher=[[Springer (publisher)|Springer]]
 |isbn=978-3-540-20168-7
 |display-authors=etal}}

* {{cite book
 |author=I.I. Bigi, A.I. Sanda
 |year=2000
 |title=CP violation
 |publisher=[[Cambridge University Press]]
 |isbn=978-0-521-44349-4}}

* {{cite web |url=http://pdg.lbl.gov/2013/reviews/rpp2012-rev-ckm-matrix.pdf |title=Particle Data Group: The CKM quark-mixing matrix}}
* {{cite web |url=http://pdg.lbl.gov/2013/reviews/rpp2012-rev-cp-violation.pdf |title=Particle Data Group: CP violation in meson decays}}
* {{cite web |url=http://www-public.slac.stanford.edu/babar/ |title=The Babar experiment}} at [[SLAC]], California, and {{cite web |url=http://belle.kek.jp |title=the BELLE experiment}} at [[KEK]], Japan.

{{Matrix classes}}
{{Standard model of physics}}
{{Use dmy dates|date=November 2018}}

{{DEFAULTSORT:Cabibbo-Kobayashi-Maskawa matrix}}
[[Category:Standard Model]]
[[Category:Electroweak theory]]
[[Category:Matrices]]