The long- and short-lived kaons are and

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Flavour, neutrinos and CP violation · Flavour, neutrinos and CP violation

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

Problem

The long- and short-lived kaons are

K_L=(K_2+\epsilon K_1)/\sqrt{1+|\epsilon|^2}

and

K_S=(K_1+\epsilon K_2)/\sqrt{1+|\epsilon|^2}

. Let

A(K_1\to f)

be the CP-conserving decay amplitude of

K_1

to a two-pion final state

f

(either

\pi^+\pi^-

\pi^0\pi^0

), and let

A(K_2\to f)

be the small amplitude for the CP-odd state

K_2

to reach the same final state, non-zero only if CP is violated in the decay amplitude itself. ((a)) Write

A(K_L\to f)

and

A(K_S\to f)

in terms of

A(K_1\to f)

A(K_2\to f)

and

\epsilon

, form

\eta_f\equiv A(K_L\to f)/A(K_S\to f)

, and keep only the leading term when both

\epsilon

and

A(K_2\to f)/A(K_1\to f)

are treated as small. Define

\epsilon'_f\equiv A(K_2\to f)/A(K_1\to f)

. ((b)) Explain, from the result of part (a), why

\eta_{+-}=\eta_{00}

for every final state would follow if CP violation were confined to the mixing, and why the measured

\eta_{+-}\ne\eta_{00}

proves that CP is also violated in the decay amplitude.

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