Editing Cabibbo–Kobayashi–Maskawa matrix (section)

==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.