P002 · Mixing, decay and CP violation

Neutral Kaon Interferometer

How can a neutral kaon change flavour while its probability of survival falls? A neutral kaon evolves as a combination of short-lived and long-lived components. Their relative phase changes its flavour as they decay. A small CP-violating amplitude lets the long-lived component decay to two pions.

Move the observation gate along a neutral-kaon beam. Follow flavour mixing as particles decay, then prepare a long-lived beam and reveal its rare two-pion channel.

Kaon flavour probability $0$

Antiparticle flavour probability $0$

Survival probability $0$

Two-pion rate per million per short lifetime $0$

Flavour probabilities and decay loss

Two-pion rate on a logarithmic scale

Calculated probabilities refer to the original beam. The two-pion rate is per short lifetime; its logarithmic axis reveals the small long-lived contribution.

Model and scientific sources

A coherent, monoenergetic vacuum beam with CPT-symmetric mixing. The reference values are from the 2024 PDG: short lifetime 89.54 picoseconds, long lifetime 51.16 nanoseconds, and indirect CP amplitude 0.002228 at 43.51 degrees. The two-pion amplitude ratio is approximated by this indirect CP amplitude; direct CP violation, regeneration in matter, wave-packet separation and detector backgrounds are omitted. The displayed two-pion rate includes the short-lived charged-pion branching fraction. The adjustable CP scale is a teaching comparison. Beam tracks show schematic decay paths, not a simulated detector sample.

g_{\pm}(t)=\frac{e^{-i m_S t-\Gamma_S t/2}\pm e^{-i m_L t-\Gamma_L t/2}}{2}

Start with the essentials

Focus question: How do flavour mixing, particle loss and the rare two-pion channel fit together?
One-sentence intuition: An unstable quantum superposition can change flavour while its total surviving probability decreases.

Core mathematical model

Vacuum propagation of the two components

g_\pm(t)=\frac{e^{-i m_S t-\Gamma_S t/2}\pm e^{-i m_L t-\Gamma_L t/2}}{2}

The widths set the disappearance rates. The mass difference sets the relative phase; natural units are used in this expression.

Two-pion decay from an initially tagged kaon

\frac{dP_{\pi\pi}}{d(t/\tau_S)}=\frac{B_{S\to\pi^+\pi^-}}{4|p|^2}\left|e_S(t)+\eta_{+-}e_L(t)\right|^2

The short-lived and long-lived decay amplitudes can interfere. In this model the small two-pion amplitude ratio is approximated by the indirect CP-violation parameter.

Common difficulties

Probability relative to the initial beam

Typical misconceptionThe two flavour probabilities must always add to one.

Better mental modelTheir sum is the surviving fraction. The missing probability belongs to particles that have decayed.

Mixing and CP violation

Typical misconceptionAny flavour oscillation means CP symmetry is broken.

Better mental modelMixing also occurs in the CP-conserving comparison. The long-lived two-pion channel is sensitive to the small CP-violating amplitude.

Run the experiment

01
Follow the beam
Drag the observation gate through the beam.
What to observe: The flavour balance changes, and the total surviving fraction falls.
02
Prepare the long-lived state
Choose Rare long-lived two-pion decay.
What to observe: A small nonzero two-pion rate remains after the short-lived contribution has disappeared.
03
Restore the symmetry
Choose Restore CP symmetry.
What to observe: The two-pion rate of the long-lived preparation becomes zero in this approximation.