CKM matrix

In the particle physics of quarks, if the eigenstates of the physical Hamiltonian are different from the eigenstates of the weak one, a transformation matrix can be defined 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 parameters was previously introduced by Cabibbo .

The idea of Cabibbo originated from need 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 $Δ$ S=1 had amplitudes equal to 1/4 of those with $Δ$ S=0.

$J^{\mathrm{Hadronic}}_\mu=J^{\Delta S=0}_\mu \cos\theta_c + J^{\Delta S=1}_\mu \sin\theta_c$

where $θ c$ is called the Cabibbo angle.

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

$J^{\mathrm{Quark}}_\mu=\bar{u}\gamma_\mu (1-\gamma_5)d' + \bar{c}\gamma_\mu (1-\gamma_5)s'$

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 Euler angles and one phase. It is this phase that can explain CP-violating phenomena in the neutral-kaon system. But other mechanisms have 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 physics.

References

N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531.
M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652.

CKM matrix - Your Art History Reference Guide!

References

See also