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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]

An $N\times N$ unitary matrix has $N^2$ real parameters. Its real rotation subgroup has $N(N-1)/2$ mixing angles, leaving $N(N+1)/2$ phases. The $N$ up-type and $N$ down-type quark fields allow $2N$ quantum field rephasings, but their common phase acts trivially on the CKM matrix. Removing $2N-1$ phases therefore leavesFor two quark generations, there is one mixing angle and no physical CP-violating phase. For three quark generations, there are three mixing angles and one physical CP-violating phase. The count describes generic, nondegenerate quark masses and generic mixing; a vanishing mixing angle can make the remaining phase unobservable, and mass degeneracies enlarge the allowed basis changes. Three quark generations permit CP violation but do not force it.

The Jarlskog invariant is a measure of CP violation that survives quantum field rephasings. Fix its sign by the quartetFor the exact three-generation CKM matrix, direct multiplication givesThe CKM unitarity triangle's illustrative parameters give $J=89991\sqrt{55}/3200000000\simeq 2.085597\times 10^{-4}$. Setting $s_{13}=0$ or $\delta =0$ makes $J=0$: the existence of three quark generations is necessary for this mechanism, but not sufficient for CP violation.

To see the invariance directly, the phases multiplying $V_{ud}{V}_{cs}{V}_{us}^{\ast }{V}_{cd}^{\ast }$ sum toUnder CP, the CKM matrix is complex conjugated and $J$ changes sign. In an exactly unitary three-generation CKM matrix, quartets formed from any distinct pair of rows and columns have imaginary parts equal to $J$ up to sign. This follows by multiplying a row orthogonality relation by a suitable pair of entries and taking its imaginary part: its diagonal product is real, forcing the two remaining imaginary parts to cancel. The same Jarlskog invariant therefore determines the areas of all six unscaled CKM unitarity triangles.

An experimentally observed example is $K_L\to {\pi }^{+}{\pi }^{-}$, a decay of the long-lived neutral kaon into a CP-even two-pion state. Without CP violation, the long-lived state would be CP-odd and this decay would be forbidden.

For the CKM matrix mechanism of CP violation, three quark generations are the minimum. Rephasings remove every complex phase for one or two generations, while three generations leave one physical CP-violating phase. This count concerns weak quark mixing. The Yang-Mills theta term in part (c) is a separate possible source of CP violation and does not require three generations.

The bottom quark has charge $-1/3$ and left-handed weak isospin $-1/2$, requiring a charge-$+2/3$ partner in a complete Standard Model weak doublet. Completing the third quark generation also maintains the quark/lepton cancellation of gauge anomalies. A three-generation CKM matrix permits an irreducible CP violation phase. Flavour constraints and loop corrections, including their strong top-mass dependence in electroweak precision observables, provided indirect evidence before direct production. These arguments identify the required partner and its interactions; they do not by themselves fix its mass.

The Tevatron collided protons and antiprotons at $\sqrt{s}=1.8\mathrm{TeV}$. Quantum chromodynamics produces $t\bar{t}$, mainly through $q\bar{q}$ annihilation there. Each top quark decays predominantly to $Wb$ before hadronization, giving $W^{+}{W}^{-}b\bar{b}$. The dilepton channel has two isolated oppositely charged leptons, missing transverse momentum from neutrinos, and bottom jets. The single-lepton channel has an isolated electron or muon, missing transverse momentum, and four leading jets, two consistent with bottom decay. The all-hadronic channel has more background. Secondary-vertex and soft-lepton tags identify bottom hadrons.

CDF and D0 reported independent excesses in 1995, with event kinematics and reconstructed masses consistent with a common very heavy quark. Fits assign jets to the two decay chains and compare with expected $W+$jets, multijet and instrumental backgrounds. Agreement of counts, bottom content and reconstruction across channels is stronger evidence than an excess in one event class alone. ^{[ref] [ref]}

For nondegenerate quark masses, changing the constant quantum phases of quark fields preserves their diagonal mass terms. Adopt the conventionSuch quantum field rephasings alter individual entries of the CKM matrix without altering physics. Physical CP symmetry in this mixing sector means that some allowed convention makes the CKM matrix real. Toshihide Maskawa emphasized the distinction between complex entries and an irremovable CP-violating phase.^[ref]

Here is a constructive two-generation argument. For a generic $2\times 2$ unitary matrix, use the three effective row/column phases to make $V_{11},V_{12},V_{22}$ real and positive. Column orthogonality then forces $V_{21}$ to be real and negative. Row normalization leaves just the real rotation specified by the Cabibbo angle. Zero entries are limiting cases and introduce no weak phase. Thus two-generation quark mixing cannot violate CP symmetry through this mechanism, even when its entries initially look complex.

For three quark generations, the Jarlskog invariant provides an obstruction to repeating that construction: all four phase factors cancel from its defining quartet. If its imaginary part is nonzero, no choice of quantum field rephasings can make every entry real. With nondegenerate quark masses, this proves physical CP violation in the mixing sector. This argument concerns the weak interaction; it does not dispose of the separate Strong CP problem.

For three quark generations, their corresponding leptons, and one complex Higgs doublet, use $Q=T_3+Y$ and $D_{\mu }\supset i{g}_{Y}Y{B}_{\mu }$. In $\mu d{g}_i/d\mu =b_i{g}_i^3/(16{\pi }^2)$ the coefficients are the displayed values. Per generation, the fermionic Dynkin index sums are two for both non-Abelian factors and the sum of squared hypercharges is $10/3$. The Higgs index for $SU(2)$ and its sum of squared hypercharges are each $1/2$. Gauge terms are $-22/3,-11$ for $SU(2),SU(3)$, respectively. For the grand-unified normalization $g_1=\sqrt{5/3}{g}_Y$, the Abelian coefficient instead is $41/10$.

The top quark is the up-type member of the third quark generation. Its mass exceeds the W boson mass, so a real W boson cannot decay into an on-shell top quark. A massless approximation for accessible decay products must not silently remove this channel threshold.