CKM Matrix

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CKM MATRIX

The CKM matrix contains information on all quark flavor transitions and is the source of the violation of CP symmetry in the Standard Model.

Before electroweak symmetry breaking occurs, the Standard Model fermions are massless, and they obey the symmetries of the Standard Model. The quarks and leptons appear in pairs known as doublets under the left-handed weak interactions. A charge +⅔ and a charge -⅓ quark compose a left-handed weak doublet, and three such doublets, known as families or generations, are known to exist. When the electroweak symmetry is spontaneously broken by the Higgs mechanism, masses are generated for the fermions via their interactions with the Higgs field. These massive states are the physical particles that are observed in the laboratory.

The quark-Higgs interactions are not diagonal under the weak interactions. This means that quark states that carry the weak charge are mixtures of the physical quark states. If the quarks were massless (or had equal masses), they would not mix. By convention, this mixing is mathematically ascribed to the charged -⅓ quark states: down, strange, and bottom. The wavefunctions of the weak flavor quark states, denoted here by primes, are then expressed in terms of the physical quark wavefunctions (unprimed) by

The nine quantities V_ij are numbers, which are in principle complex. They describe the weighting of each physical quark wavefunction in the mixture and are labeled by the quark states that they link. These V_ij are the elements of the 3 × 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix.

This matrix is named after Nicola Cabibbo, who first developed these ideas in 1963 when only three quark flavors (up, down, and strange) were known, and Makoto Kobayashi and Toshihide Maskawa, who extended it to the six quark flavors that are now known to exist. Although only three quark flavors were known in 1973 when Kobayashi and Maskawa hypothesized this extension, they realized that CP violation could be naturally accommodated in the Standard Model if one incorporated six quarks within this framework.

The CKM matrix is unitary, meaning that the product of the matrix with its complex conjugate must be the unit matrix. This implies that the values of the elements are interconnected. For example, the sum of the squares of the elements in a row or column equals unity for three generations. In addition, the multiplication of two different rows or columns must equal zero: where the asterisk denotes the complex conjugate of the element. This last relationship can be represented as a triangle in the complex plane (see Figure 1), where the length of the sides of the unitary triangle is given by the magnitudes of the elements, and the angles of the triangle are related to the phases in the matrix. A 2 × 2 mixing matrix consists of only a single parameter: one mixing angle that is known as the Cabibbo angle and that mixes the down and strange quarks. The 3 × 3 CKM matrix can be

FIGURE 1

This mixing of quark flavors, as well as the generation of quark masses through the Higgs field, has no fundamental explanation in the Standard Model. The values of the quark masses and mixing angles are arbitrary and not predicted. Numerous theoretical attempts have been made to derive these quantities from a basic theory of flavor, but as of 2002 no compelling model exists. The numerical values of the quark masses and mixings are determined from laboratory measurements.

The values of the individual CKM matrix elements determine the rates of transitions, or weak decays, of the quarks. For example, a heavy quark of charge -⅓, such as the strange quark, can decay to a lighter quark of charge +⅔, such as an up quark, by emitting a W^- particle. The strength of this sample transition is specified by the value of the quantity V_us. There are nine such transitions, corresponding to the nine elements of the CKM matrix. These transitions are known as charged current decays of the quarks, as the charged weak gauge bosons W^± are emitted. Since the W^± particles are heavier than all the quarks except for the top quark, they subsequently decay to a pair of lighter quarks or leptons after being emitted. Neutral current couplings, that is, quark couplings to the neutral weak gauge boson Z⁰, are flavor diagonal in the Standard Model. Flavor changing neutral current decays, such as b → sγ , only occur when quantum corrections to the Standard Model are included.

The most accurately measured CKM matrix element is V_ud, which is determined from comparing superallowed nuclear β decay to muon decay. Its value is close to unity. The elements V_{us,ub,cd,cs,cb} are all measured in three-body semileptonic decays (where the emitted W^± boson decays into a lepton pair) of the heavier of the linked quarks. For example, the transition b → cl^- v̄₁ determines V_cb . Vcd,cs can also be determined in neutrino nucleon collisions, ν_μ N → ν + c + X . All these measurements are subject to theoretical uncertainties (in addition to experimental errors) introduced by the fact that the quarks are not free particles but are bound inside of hadrons. The quantity V_tb is measured directly in the two-body decay t → bW⁺ since the top quark is massive enough to decay into a W⁺ boson. Its value is also close to unity. The elements Vtd,ts have yet to be measured directly, but a range for their values can be inferred from the unitarity properties of the CKM matrix.

Present knowledge of the CKM matrix elements is given at the web site of the Particle Data Group, which summarizes the measurements of particle properties. In 2002 the absolute values of the matrix elements lie in the following ranges:

This includes the constraints imposed from unitarity. Note that the matrix is nearly diagonal and that first-second generation mixing is strong, whereas the first-third generation mixing is very weak. The explanation of CP violation in the Standard Model requires a nonvanishing value of V_ub, which is satisfied by the data.

An active research topic within high-energy physics is improving the determination of the CKM matrix elements. This is being accomplished via better experimental precision with larger data samples and refined experimental resolution with modern detectors, and with more precise theoretical calculations. An accurate determination of the fundamental parameters of the Standard Model will hopefully provide insight to the underlying theory of flavor.