CKM mixing is a process that occurs in fundamental particle
physics. Specifically, it occurs in the interaction of quarks
through the weak nuclear force. It is an important part of the
Standard Model and is still the subject of large-scale
experiments.

The six quarks are organised into three 'doublets', each containing
an up-type quark and a down-type quark. These are (u,
d), (c, s), and (t, b). The weak nuclear force changes one member
of the doublet into the other member, carrying away a unit
charge. This is a wonderfully simple model and explains a large
fraction of hadron decays.

The problem occurs when you consider a decay such as the decay of a
kaon into a pion. A K^{-} has the quark content
~~u~~s, and can decay to
π^{0}e^{-}~~ν~~.
In this case the π^{0} can be considered to have the quark
content u~~u~~, so in the decay the strange quark transforms into
an up quark. Other than this crossing of doublet boundaries, the decay
works exactly like a weak decay as described above.

Since this decay and others like it act just like weak decays
connecting the members of different doublets, it should be possible to
modify the weak force theory to explain them. The mechanism that was
added was CKM mixing. In this scheme, the down-type states that
interact via the weak force are mixtures of
the actual down-type states, called 'flavour eigenstates'. Thus, the
flavour eigenstates can be expressed as combinations of the weak eigenstates.

The original theory was proposed by Nicola Cabibbo, at a time when only
two quark doublets were known: (u,d) and (c,s). For the purposes of
the weak interaction, these were replaced by (u,d') and (c,s'), where
d' and s' are orthogonal, normalised linear combinations of d and s. In matrix form, this
can be expressed as:

⌈d'⌉ = ⌈ cosθ_{C} sinθ_{C}⌉⌈d⌉
⌊s'⌋ ⌊-sinθ_{C} cosθ_{C}⌋⌊s⌋

where θ

_{C} is an angle called the Cabibbo angle and is
approximately 13 degrees, making the off-diagonal elements much
smaller than the diagonal elements These d' and s' states are both present in s and d
quarks, so although the strange quark usually interacts as an s', it can
sometimes interact as a d' and change into an up quark.

There are, however, *three* quark doublets. M. Kobayashi and T. Maskawa adapted the Cabibbo theory to this situation, defining a unitary matrix, `V`, which is usually called the CKM mixing matrix after Cabibbo, Kobayashi, and Maskawa, to relate d', s', and b' to d, s, and b. If the CKM matrix is taken to be real, it has three degrees of freedom, usually expressed as the angles θ_{12}, θ_{13}, and θ_{23}. However, CP violation requires that the CKM matrix have a complex component, which at present can be expressed with a single phase angle δ. Various experiments are in progress to find the elements of the CKM matrix. The established ranges for the CKM matrix elements are, as of 2002:

⌈ 0.9741 to 0.9756 0.219 to 0.226 0.0025 to 0.0048 ⌉
| 0.219 to 0.226 0.9732 to 0.9748 0.038 to 0.044 |
⌊ 0.004 to 0.014 0.037 to 0.044 0.9990 to 0.9993 ⌋

The CKM mechanism is an important effect in modern particle
physics, explaining many of the decays encountered in high-energy
collision experiments. The CKM matrix's counterpart for leptons is
the MNS matrix and is responsible for neutrino oscillation.

Sources include my senior undergraduate particle physics course
and the Particle Data Group website at http://pdg.lbl.gov/ ,
particularly the report at http://pdg.lbl.gov/2002/kmmixrpp.pdf .

**(CC)**
This writeup is copyright 2004 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .