[edit] Origin in the Standard Model

In the standard model of particle physics the CKM matrix originates in the coupling of the quarks to the Higgs fields. The quark sector of the standard model Lagrangian contains the following Yukawa coupling terms

where v is the vacuum expectation value of the Higgs field, dⁱ(dⁱ) is the down(up)-type quark in the ith generation, and λ^ij_d and λ^ij_u are the coupling constants if the quarks to the Higgs field. If the λ^ij were diagonal the above terms would just be the standard mass terms, however naturalness requires that these coupling constants take generic values. For various reasons physists deem it convenient to redefine the quark fields in such a way that these couplings are diagonal. To do this it is used that a general matrix λ^ij can be written as U^ijm^jk(U^†)^kl, where m^ij is a diagonal matrix representing the masses and U^ij is a unitary matrix. Doing this for both the up and down-type quark-Higgs couplings, the terms above can be diagonalized by redefining the fields

and .

Most terms in the standard model lagrangian are invariant under such a transformation. The only exceptions are the weak interaction terms coupling up and down type quarks. In terms of the old fields these terms read:

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[edit] Weak interaction (draft)

Quarks can participate in the weak interaction in two different ways. Either by exchanging a Z-boson or a W-boson. Interactions induced by exchange of a Z-boson are in many ways similar to the electromagnetic interactions induced by exchange of a photon. There are however two important distinctions. First, the Z-boson (as well as the W-boson) is a massive particle, which causes strength of interactions caused by Z-boson exchange to fall of exponentially with distance. Moreover, interactions with the Z-boson affect left-handed quarks much more strongly than right-handed quarks. The weak interaction is the only interaction that violates parity (the symmetry that switches left and right) in this way.

A pictorial representation of the six quarks' decay modes, with mass increasing from left to right.

The exchange of a W-boson is again similar to the exchange of a Z-boson. The interactions violate parity and are exponentially suppressed because of the mass of the W-boson. However there is one property that makes the W-boson exchange distinct from all other interactions in the standard model – it is the only interaction that allows a quark to change flavor; by exchanging a W-boson an up-type quark can change into a down-type quark and vice versa. This is the interaction that allows the beta decay of some radioactive elements. For example one of the down quarks in a neutron (composition udd) can change into an up quark be emitting a W⁻-boson, transforming the neutron in a proton (composition uud). The W⁻ boson then decays into an electron (e⁻) and an electron antineutrino (ν_e).^[1]

n	→	p	+	e−	+	νe		(Beta decay, hadron notation)
udd	→	uud	+	e−	+	νe		(Beta decay, quark notation)

W-boson exchange can also allow quarks or hadrons to decay into completely different elementary particles through a process of annihiliation. For example, for the pi meson (composition ud), a decay into a corresponding quark–antiquark flavor pair such as uu or dd would result in an annihilation of the quark–antiquark pair. The release of energy therein could effect the creation of the new leptons, such as muons or neutrinos.^[2]

Any up-type quark can in principle change into any down-type quark. These transitions are however not all equally probable. The relative probabilities of are kept track of in the Cabibbo–Kobayashi–Maskawa matrix or (CKM-matrix),

where V_ud is related to the relative probability that an up quark changes into a down quark upon exchanging a W-boson. The entries of this matrix are constrained by various symmetries of the standard model. For example, weak universality require the matrix to be unitary (i.e. requires the inverse of the matrix to be equal to its hermitian conjugate). The similar matrix of the transitions of the antiquarks has a very simple relation to the CKM-matrix, namely V_du = V_ud^*. As a result any complex entry of the CKM-matrix will cause antiparticles to behave differently from normal particles, and thus breaks the symmetry (called CP symmetry) between the two. Kobayashi and Maskawa discovered that there must be at least three generations for the matrix to have complex entries, and postulated a third generation to explain the CP violation observed at the time. For this discovery they were awarded the 2008 Nobel prize in physics.

[edit] Weak interaction (old)

A pictorial representation of the six quarks' decay modes, with mass increasing from left to right.

A quark of one flavor can transform into a quark of a different flavor through the weak interaction, one of the four fundamental interactions through which particles interact with each other. A quark can decay into a lighter quark by emitting a virtual W boson, or can absorb a virtual W boson to turn into a heavier quark. This mechanism causes the radioactive process of beta decay, in which a neutron "splits" into a proton, an electron and an antineutrino. This occurs when one of the down quarks in the neutron (composition udd) decays into an up quark by emitting a virtual W⁻ boson, transforming the neutron into a proton (composition uud). The W⁻ boson then decays into an electron (e⁻) and an electron antineutrino (ν_e).^[3] A quark can also emit or absorb virtual Z bosons.

Weak interactions can also allow quarks or hadrons to decay into completely different elementary particles through a process of annihiliation. For example, for the pi meson (composition ud), a decay into a corresponding quark–antiquark flavor pair such as uu or dd would result in an annihilation of the quark–antiquark pair. The release of energy therein could effect the creation of the new leptons, such as muons or neutrinos.^[4]

As well as being the only interaction capable of causing flavor changes, the weak interaction is the only interaction violating parity symmetry. It affects only left-handed quarks and leptons, and right-handed antiquarks and antileptons, so its effects would be different if left and right were swapped.

[edit] Cabibbo angle and CKM matrix

In 1963, Nicola Cabibbo introduced the Cabibbo angle (θ_C) to keep track of how often certain weak interaction decays occurred in nature.^[5] In light of current knowledge (quarks were not yet theorized), the Cabibbo angle is related to the probability that down and strange quarks decay into up quarks. In particle physics parlance, the down and strange quarks were said to form a weak interaction eigenstate, a quantum superposition of down and strange quarks quantum states (|d〉 and |s〉 respectively).^[6] Mathematically this can be represented as:

where |d′〉 is the weak interaction eigenstate. The square of the magnitudes of V_ud and V_us (|V_ud|² and |V_us|²) represent the probability that down and strange quarks decay into up quarks. Using the currently accepted values for |V_ud| and |V_us| (see below), the Cabbibo angle can be calculated using

The modern equivalent of the Cabibbo angle is a mathematical table called the Cabibbo–Kobayashi–Maskawa matrix (or CKM matrix), developed by Makoto Kobayashi and Toshihide Maskawa in 1972. The CKM matrix keeps track of the weak decays of all six quarks.^[7]

The CKM matrix has additional features beyond the description of how often quarks of a flavor decay into quarks of other flavors. Kobayashi and Maskawa first build the CKM matrix to explain the violation of CP symmetry in weak interactions—weak interactions do not behave the same way if particles are replaced by their antiparticles (C symmetry) and if left is swapped with right (P symmetry).^[7] CP violation cannot be explained with one or two generations of quarks, but is possible with three or more generations (see CKM matrix for more details).^[7]

Currently, the best determination of the magnitude of the entries of the CKM matrix is approximately:^[8]