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The Cabibbo-Kobayashi-Maskawa matrix $V=U_{uL}^{†}{U}_{dL}$ 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 ${\bar{u}}_{Li}{\gamma }^{\mu }{d}_{Lj}{W}_{\mu }^{+}$ is $V_{ij}$. A single CKM matrix therefore connects quark masses, flavour-changing decay amplitudes, and CP violation.

Write $s_{ij}=\sin {\theta }_{ij}$ and $c_{ij}=\cos {\theta }_{ij}$. An exactly unitary matrix with three mixing angles and one CP-violating phase isThis 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]

A common weak phase makes CP violation across mesons testable through the BABAR time-dependent CP violation experiment. It does not explain the cosmic baryon asymmetry.^[ref]

Makoto Kobayashi co-developed a mechanism linking CP violation to quark mixing. With Toshihide Maskawa, he examined which phases of the weak-interaction mixing matrix can survive field redefinitions. Three fermion generations permit one independent CP-violating phase in the usual three-by-three Cabibbo-Kobayashi-Maskawa matrix. The proposal connected apparently separate questions about particle families and symmetry, and made comparison with flavor-changing decays a test of one common mixing structure.^[ref]

A physicist working on quark mixing.^[ref]

Toshihide Maskawa and Makoto Kobayashi investigated how CP violation could fit within a renormalizable theory of the weak interaction. A complex quark mixing matrix is not enough by itself: phases removable by redefining quark fields have no physical effect. With three fermion generations, a physical complex phase can remain in the Cabibbo-Kobayashi-Maskawa matrix. Their 1973 analysis connected an observed symmetry failure with the possible number and organization of quarks.^[ref]

A physicist working on quark mixing.^[ref]