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 us, and can decay to π0e-ν. In this case the π0 can be considered to have the quark content uu, 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'⌉ = ⌈ cosθC sinθC⌉⌈d⌉
⌊s'⌋   ⌊-sinθC cosθC⌋⌊s⌋
```
where θC 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 θ12, θ13, and θ23. 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/ .

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