Editing Cabibbo–Kobayashi–Maskawa matrix (section)

===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.]]