Cabibbo–Kobayashi–Maskawa matrix

The CKM matrix is strongly related to CP-symmetry. The Standard Model with three quark families can naturally generate CP violation in weak interaction.

Being the eigenstates of the strong Hamiltonian different from the eigenstates of the weak one, one can define a transformation matrix for these two bases. This matrix was introduced for six quarks by Kobayashi and Maskawa(cfr. km) and is called the Cabibbo-Kobayashi-Maskawa because one of the parameter was previously introduced by Cabibbo(cfr. Cabibbo).

The idea of Cabibbo originated from needs to explain two phenomena:

the transitions $u\leftrightarrow d,e\leftrightarrow \nu _{e}$ , $\mu \leftrightarrow \nu _{\mu }$ had similar amplitudes.
the transitions with variation of strangeness $\Delta$ S=1 had amplitudes equal to 1/4 of those with $\Delta$ S=0.

He proposed to consider the charged hadronic current as a sum of two currents, one with $\Delta$ S=1, the other with $\Delta$ S=0:

$J_{\mu }^{Hadronic}=J_{\mu }^{\Delta S=0}cos\theta _{c}+J_{\mu }^{\Delta S=1}sin\theta _{c}$

where $\theta _{c}$ is called the Cabibbo angle.

A fourth quark was not foreseen by Cabibbo. But taking in account the existence of the $charm$ , the Cabibbo angle can be seen as a mixing angle between two families of quark: 'up' and 'down', 'strange' and 'charm':

$J_{\mu }^{Quark}={\bar {u}}\gamma _{\mu }(1-\gamma _{5})d'+{\bar {c}}\gamma _{\mu }(1-\gamma _{5})s'$

where

$d'=d\ cos\theta _{c}+s\ sin\theta _{c}\quad {\mbox{and}}\quad s'=-d\ sin\theta _{c}+s\ cos\theta _{c}.$

In 1973, one year before the discovery of the 'charm', the existence of a third family of quarks was introduced by Kobayashi and Maskawa in order to explain the CP-violation phenomena observed in the neutral-kaon system. This was an 'ad hoc' hypothesis, because at that time no component of this third family was observed.

By introducing a mixing for the new family the parameters of the transformation matrix become four: three Eulero angles and one phase. It is this phase that can explain CP-violating phenomena in the neutral-kaon system. But other mechanisms has been proposed. Hence it is of great interest to study whether the pattern of CP-violating effects that can be observed in $B$~decays follows the predictions of the minimal standard model, or instead requires the introduction of beyond Standard Model.