Editing Cabibbo–Kobayashi–Maskawa matrix (section)

===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.