A neutral kaon beam is produced with momentum in the laboratory frame

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Particle Physics & Quantum Fields · Symmetry, groups and the quark model

Particle Physics, Ch Parity, charge conjugation, time reversal and CPT

Problem

A neutral kaon beam is produced with momentum

p = 10\,\mathrm{GeV}/c

in the laboratory frame, using

m(K^0)c^2 = 497.6\,\mathrm{MeV}

. In the kaon's own rest frame,

c\tau(K_S) = 2.7\,\mathrm{cm}

and

c\tau(K_L) = 15.3\,\mathrm{m}

. ((a)) Find the Lorentz factor

\gamma

and speed

\beta

of the beam in the laboratory frame. ((b)) Find the mean decay length

\beta\gamma c\tau

in the laboratory frame for the

K_S

component and for the

K_L

component. ((c)) A detector for two-pion final states sits

20\,\mathrm{m}

downstream of production. Taking the surviving fraction after a distance

x

to be

e^{-x/L}

with

L

the mean decay length from part (b), estimate the fraction of the original

K_S

population still present at the detector, and explain why a

\pi^+\pi^-

signal seen there must come from

K_L

decays. ((d)) Given that this signal is observed in about

2\times10^{-3}

K_L

decays, and that

CP(\pi^+\pi^-) = +1

(a spin-0 parent forces

L=0

) while

K_L

is assigned

CP=-1

, state what this measurement establishes, and name one thing it leaves unexplained.