Cabibbo–Kobayashi–Maskawa matrix

The Cabibbo–Kobayashi–Maskawa (CKM) matrix is a fundamental concept in the field of particle physics, specifically in the study of the weak interaction and the quark sector of the Standard Model. It describes the mixing between the three generations of quarks and plays a crucial role in the phenomenon of flavor mixing as well as in the understanding of CP violation (charge-parity violation) in weak decays.

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Codex  0 Created 2026-09-24 Updated 2026-10-18

V = U_{uL}^{†} U_{d L}

compares the left-handed unitary matrices that diagonalize the up-type and down-type Yukawa matrices. Its rows label up quarks, charm quarks and top quarks; its columns label down quarks, strange quarks and bottom quarks. In the weak charged current, the coefficient of

\overset{u}{ˉ}_{L i} γ^{μ} d_{L j} W_{μ}^{+}

V_{ij}

. A single CKM matrix therefore connects quark masses, flavour-changing decay amplitudes, and CP violation.

Write

s_{ij} = sin θ_{ij}

and

c_{ij} = cos θ_{ij}

V = c_{12} c_{13} - s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i δ} s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i δ} s_{12} c_{13} c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i δ} - c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i δ} s_{13} e^{- i δ} s_{23} c_{13} c_{23} c_{13} .

This convention agrees with Makoto Kobayashi's account of three-generation quark mixing.^[ref]

The Christenson-Cronin-Fitch-Turlay kaon CP violation experiment established the phenomenon that required an explanation. Makoto Kobayashi and Toshihide Maskawa showed that two-generation quark mixing cannot supply it, whereas adding a third quark generation permits an irreducible weak phase. Their 1973 paper also considered additional fields as alternative mechanisms; the three-generation conclusion concerns CP violation from the minimal unitary quark mixing mechanism.^[ref]