CKM

The CKM matrix, named after Cabibbo, Kobayashi, and Maskawa, is a unitary 3×3 matrix that encodes the mixing between the three generations of quarks in charged weak interactions. It relates the weak-interaction eigenstates of up-type quarks (u, c, t) to down-type quarks (d, s, b). The matrix elements Vij describe the strength of transitions between i-th up-type and j-th down-type quarks in processes mediated by W bosons, and thus determine the amplitudes for quark decays and related phenomena.

The matrix is unitary, leading to relations among its elements that can be visualized as unitarity triangles. A complex phase within the matrix is the source of CP violation in the quark sector of the Standard Model, and this CP-violating effect requires at least three generations of quarks.

Parametrization of the CKM matrix can be given in terms of three mixing angles, θ12, θ23, θ13, and a CP-violating phase, δ. A widely used approximate form is the Wolfenstein parameterization, characterized by λ ≈ 0.226, A ≈ 0.8, and ρ and η that encode CP violation.

Experimentally, the CKM matrix is determined from a wide range of processes in K, D, and B meson systems, including decay rates and meson mixing. The diagonal elements are close to unity, while the off-diagonal elements are smaller, with magnitudes of order a few tenths or smaller (e.g., |V_ud| ≈ 0.974, |V_us| ≈ 0.225, |V_cb| ≈ 0.041, |V_ub| ≈ 0.004).

The CKM mechanism is central to flavor physics, enabling quantitative tests of the Standard Model and providing a framework to search for new physics through potential deviations from its unitarity and phase structure.