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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]

One half went to Yoichiro Nambu for discovering the mechanism of spontaneous symmetry breaking in subatomic physics. One quarter each went to Makoto Kobayashi and Toshihide Maskawa for explaining broken symmetry through a mechanism requiring at least three families of quarks.^[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]