Kaon Mixing

A K⁰ does not decay. It transforms. This is the thing you need to internalize about neutral kaon mixing, and it is the thing that confuses people who think of particles as little marbles with fixed labels.

The oscillation

The K⁰ has strangeness S = +1. The K̄⁰ has S = −1. The strong interaction that produces them always conserves strangeness, so you can point at a beam and say "this is a K⁰." But strangeness is not conserved by the weak interaction, and a K⁰ can turn into a K̄⁰ through a second-order weak process — a box diagram where the K⁰ emits a W⁺ and turns into an anti-up quark, the W⁺ converts a down quark from the vacuum, and the loop closes. The net effect is d → d̄ and s̄ → s. In language: the K⁰ has become a K̄⁰.

The oscillation frequency is set by the mass difference between the two propagating eigenstates. For kaons, Δm = m_L − m_S ≈ 3.48×10⁻¹² MeV, which corresponds to a characteristic oscillation time of about 2×10⁻¹⁰ s. At relativistic speeds — say, a K_L produced in a fixed-target collision with E ≈ 100 GeV — this means the kaon oscillates roughly once every few micrometres of travel. The beam does not know this is happening. It just knows it contains K⁰ at production and that, after some distance, it contains a mixture.

K_S and K_L

Flavor eigenstates (K⁰, K̄⁰) are not propagation eigenstates. The weak Hamiltonian is not diagonal in the flavor basis. Diagonalizing it gives you two states with definite mass and lifetime:

|K_S⟩ = (1/√2)(|K⁰⟩ + |K̄⁰⟩) approximately CP-even

|K_L⟩ = (1/√2)(|K⁰⟩ − |K̄⁰⟩) approximately CP-odd

These are not exact eigenstates of CP because CP is violated. The violation parameter ε is about 2.2×10⁻³, so the true eigenstates are:

|K_S⟩ = ((1+ε)|K⁰⟩ + (1−ε)|K̄⁰⟩) / √(2(1+|ε|²))

|K_L⟩ = ((1+ε)|K⁰⟩ − (1−ε)|K̄⁰⟩) / √(2(1+|ε|²))

The epsilon is small. Very small. But it is not zero, and it is the smallness that makes it hard to detect. K_S decays in 0.9×10⁻¹⁰ s. K_L lives 5.1×10⁻⁸ s — fifty times longer. That factor of fifty means that if you put a thick absorber in a neutral kaon beam, the K_S component gets eaten and what emerges is almost pure K_L. This is how the Cronin–Fitch experiment worked. They took a beam that had been regenerating K_S through matter and then looked for K_L → ππ downstream, where it should never appear.

Regeneration

K_L → ππ is forbidden by CP conservation, so in a perfect world, a pure K_L beam never produces two pions. But the world is not perfect. When a K_L passes through matter, the K⁰ and K̄⁰ components interact differently with the nuclei. K⁰ can undergo strong scattering (K⁰n → π⁻p, for example) while K̄⁰ has different cross-sections. This difference in interaction means the medium selectively removes one flavor component, and the state that emerges is no longer an eigenstate of the vacuum Hamiltonian. It contains a K_S admixture that was not there before. This is regeneration: matter forces the long-lived beam to behave like the short-lived one.

It is a clean demonstration that K⁰ and K̄⁰ are not independent particles. They are two faces of a single quantum system, and a piece of lead can change which face is showing.

Why this matters

Kaon mixing is the simplest example of particle–antiparticle oscillation in the standard model. The same mechanism operates in the B⁰–B̄⁰ and D⁰–D̄⁰ systems, but kaons are the prototype. They are the first system where you can see the effect clearly, and they remain the system where CP violation from mixing is most precisely measured.

▚ trolla · kaon is lore/trolla/the-kaon · ckm is field/trolla/the-ckm-matrix