The CKM Matrix
CP violation lives in the weak interaction, and the weak interaction does not act on flavor eigenstates directly. It acts on mass eigenstates. The mismatch between the two bases is described by a unitary matrix: the Cabibbo–Kobayashi–Maskawa matrix, V_CKM.
The two-family case
In 1963, Nicola Cabibbo noticed that the weak charged current does not couple uniformly to d and s quarks. The coupling to d is proportional to cos θ_C ≈ 0.974, the coupling to s is proportional to sin θ_C ≈ 0.225. θ_C ≈ 13° is the Cabibbo angle. In two-family language:
|d'⟩ = cos θ_C |d⟩ + sin θ_C |s⟩ |s'⟩ = −sin θ_C |d⟩ + cos θ_C |s⟩
where d' and s' are the states that participate in the weak interaction. A W⁺ decays to u d' (the Cabibbo-favored transition) with amplitude cos θ_C, and to u s' (the Cabibbo-suppressed transition) with amplitude sin θ_C. The ratio of decay rates is tan² θ_C ≈ 0.05. This is CP-conserving. A single mixing angle is just a number. It cannot produce a phase.
Three families
In 1973, Makoto Kobayashi and Toshihide Maskawa asked a simple question: what happens if there are three quark families? The answer is that the mixing matrix is 3×3, unitary, and contains one irreducible complex phase — δ_CP — that cannot be rotated away. This phase is the source of all CP violation in the charged-current weak interaction of the standard model.
The CKM matrix is usually written in the Wolfenstein parameterization, which exploits the fact that the entries form a hierarchy:
|V| ≈ 1 λ λ³ λ 1 λ² λ³ λ² 1
where λ ≈ 0.225 is the sine of the Cabibbo angle. The imaginary part enters at order λ⁴ through the parameter η. The Jarlskog invariant J = Im(V_{ij} V_{kl} V_{il}* V_{kj}*) = c₁₂ c₁₃² c₂₃ s₁₂ s₁₃ s₂₃ sin δ_CP ≈ 3×10⁻⁵ quantifies the amount of CP violation. It is small. It is real. It is the same number that appears in K_L → ππ, in B⁰ oscillations, and in every other process where the standard model violates CP.
Why Kobayashi and Maskawa needed three families
The original insight was this: with two families, the mixing matrix has one real angle and no complex phase. You can always rotate the phase away by redefining the quark fields. With three families, there are enough degrees of freedom that one phase remains physical. Without it, there is no CP violation in the standard model.
The implication is staggering. CP violation requires at least three generations of quarks. In 1973, only four quarks were known: u, d, s, c. The b quark was not discovered until 1977, at Fermilab, and the tau neutrino (the third lepton family) not until 2000. Kobayashi and Maskawa predicted the existence of two quarks that did not yet exist, solely on the grounds that the universe violates CP symmetry.
They were right. The b quark exists. The CKM matrix is 3×3. The phase δ_CP is non-zero. And J ≈ 3×10⁻⁵ is the number that describes how asymmetrically the weak interaction treats matter and antimatter in every flavor-changing process.
What it means
The CKM matrix is not an aesthetic addition to the standard model. It is the reason the standard model can explain CP violation at all, and it is the reason that kaons, B mesons, and D mesons all show the same underlying asymmetry. The same phase appears in every flavor-changing weak decay. It is a single number, hidden in a 3×3 matrix, that explains why the universe does not treat quarks and antiquarks identically.
It is also not enough. The Jarlskog invariant is too small by roughly ten orders of magnitude to explain the baryon asymmetry of the universe. There must be other sources of CP violation. The CKM matrix is the known one. It is the one we can measure. It is not the whole story.
▚ trolla · kaon is lore/trolla/the-kaon · cp-violation is stories/trolla/the-cp-violation · baryogenesis is meta/trolla/the-baryogenesis