9 Symmetry Violations and Quark Flavour Physics
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where x = g/f is the ratio of the Q = −S to the Q = S amplitude
(Sects. 9.5.3.2 and 9.5.5.6).
9.5.3 Interference Between Decay Amplitudes of K L and K S
An arbitrary coherent mixture of K L and K S states will show interference phenomena when decaying into 2π and in other common decay channels. According to
Sect. 9.3.2.1 the eigentime development of K L is
|K L → |K L e
−iM L τ ,
(9.109)
where M L = m L − (i/2)Γ L , and correspondingly for K S . An arbitrary mixture
|ψ(0) = a S |K S + a L |K L
(9.110)
will develop into
|ψ(τ ) = a S e
−iM S τ
|K S + a L e
−iM L τ
|K L .
(9.111)
We call the ratio a S /a L = V .
9.5.3.1 2π Decay
The 2π decay amplitude is therefore
2π|T |ψ(τ ) = a S e
−iM S τ
2π|T |K S + a L e
−iM L τ
2π|T |K L
= =2π|T |K S a S e
−iM S τ
+ a L η e
−iM L τ ,
(9.112)
where η = η +− for π + π − decay and η = η 00 for π 0 π 0 decay. The observed decay
rate is proportional to
R(τ ) = |a S |
2 e
−Γ S τ
+ |a L η|
2 e
−Γ L τ
+ 2|a S ||a L ||η| e
−(Γ L +Γ S )(τ/2) cos ( τ + Φ) .
(9.113)
where Φ = arg(a S ) − arg(ηa L ). We obtain for various initial conditions of the
mixture:
1. For an initially pure K 0 state (a S = 1 = a L ),
R 1 (τ ) = e
−Γ S τ
+ |η|
2 e
−Γ L τ
+ 2|η| e
−(Γ L +Γ S )(τ/2) cos ( τ − arg η) .
(9.114)