A kaon is produced at time as a pure

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Particle Physics & Quantum Fields · Quantum dynamics and its formulations

Symmetry and Flavour, Ch Neutral kaons: mixing, strangeness oscillation and CP violation

Problem

A kaon is produced at time

t=0

as a pure

K^0

. Work in the CP-conserving approximation

K_S\approx K_1

K_L\approx K_2

(the admixture parameter

\epsilon\sim2\times10^{-3}

is negligible for this problem), with

K_1=(K^0-\bar K^0)/\sqrt2

and

K_2=(K^0+\bar K^0)/\sqrt2

. ((a)) Starting from

|K^0(t)\rangle

written as a superposition of

K_1

and

K_2

, each evolving with its own mass and width, derive

P(K^0\to K^0;t)

and

P(K^0\to\bar K^0;t)

in terms of

\Gamma_S

\Gamma_L

\bar\Gamma=(\Gamma_S+\Gamma_L)/2

and

\Delta m=m_L-m_S

. ((b)) Using

\tau_S=0.8954\times10^{-10}\,\mathrm{s}

\tau_L=5.116\times10^{-8}\,\mathrm{s}

and

\Delta m\,\tau_S=0.474

, evaluate both probabilities at

t=20\,\tau_S

, and say what the result shows about the strangeness content of a kaon beam that is old compared with

\tau_S