9 Symmetry Violations and Quark Flavour Physics
533
or in the form of the corresponding relations
|N
0
=
1
2p
(|N h + |N l ) ,
|N 0 =
−1
2q
(|N h − |N l ) .
(9.35)
Unitarity requires |p| 2 + |q 2 | = 1, and CP invariance would mean p = q = 1/
√
2.
In the case of CP noninvariance, an asymmetry parameter can be defined by
ε =
p − q
p + q
or
p
q
=
1 + ε
1 − ε
(9.36)
and we obtain
ε =
1
2 m Γ 12 + i m M 12
−
i
2
,
(9.37)
where m M 12 m Γ 12 for the K meson system, and therefore
arg ε arctan
2 m
Γ S
(9.38)
In this case the two physical states are not orthogonal, and we obtain
l |N h =
2 e ε
1 + |ε| 2 .
(9.39)
For the eigenstates of the time-dependent Schrödinger equation, the time evolution
obeys an exponential decay law, as can be shown in the Wigner–Weisskopf
approximation. Here, the time t is measured in the rest frame given by the common
mass defined by the strong and electromagnetic interactions. The time evolution is
given by
|N h (t) = e
−im h t −
1
2 Γ h t
|N h (0) ,
|N l (t) = e
−im l t −
1
2 Γ l t
|N l (0) .
(9.40)
In this way, N h decays as exp(−Γ h t) and N l as exp(−Γ l t), while the phases of the
two states evolve with different frequency, and this difference will show up in any
interference effect between the two decaying mesons.