336
21. CP Violation and Oscillation Phenomena
state has been collapsed by the tagging at time t = 0 say; the partner meson
will be reconstructed by its decay products. Note that the partner meson
can decay earlier or later than the tagged one; its state vector has that time
dependence which ensures that it becomes the correlate of the tagged particle
at t = 0.
21.2.1 Time-dependent mixing formalism
We denote the neutral meson by B (which will usually be B
0 , but could also be
K
0 or D
0 ), and its CP-conjugate by ¯
B. According to the theory of Weisskopf
and Wigner (1930a, 1930b) (see also appendix A of Kabir 1968) a state that
is initially in some superposition of |B› and |B ¯ ›, say
|ψ(0)› = a(0)|B› + b(0)|B ¯ ›,
(21.27)
will evolve in time to a general superposition
|ψ(t)› = a(t)|B› + b(t)|B ¯ ›
(21.28)
governed by an effective Hamiltonian H with matrix elements, in the 2-state
subspace,
(
)
Γ
A p
2
H = M − i =
(21.29)
2
2
q
A
where M and Γ are Hermitian, and the equality M 11 −iΓ 11 /2 = M 22 −iΓ 22 /2 =
A follows from CPT invariance, which we shall assume. If CP is a good
symmetry, then
‹B ¯ |H|B› = ‹B ¯ |(CP)
−1 (CP)H(CP)
−1 CP|B›
= ‹B|H|B ¯ ›
(21.30)
so that p would equal q. Since M and Γ are both Hermitian, this would imply
that M 12 and Γ 12 are both real; in the CP non-invariant world, this is not
the case.
The eigenvalues of H are
ω L ≡ m L − iΓ L /2 = A + p q, ω H ≡ m H − iΓ H /2 = A − p q,
(21.31)
and the corresponding eigenstates are
|B L › = (p |B› + q|B ¯ ›)/(|p|
2 + |q|
2 )
1/2
|B H › = (p |B› − q|B ¯ ›)/(|p|
2 + |q|
2 )
1/2 .
(21.32)
The states |B L ›, |B H › have definite masses m H , m L and widths Γ L and Γ H . The
widths Γ L , Γ H are equal to a very good approximation for B and D mesons,
because the Q-values of both are large; in the case of K-mesons (see section
21.3), one state decays predominantly to 2π and the other to 3π, with different
Q-values, and the lifetimes are very different.
21. CP Violation and Oscillation Phenomena
state has been collapsed by the tagging at time t = 0 say; the partner meson
will be reconstructed by its decay products. Note that the partner meson
can decay earlier or later than the tagged one; its state vector has that time
dependence which ensures that it becomes the correlate of the tagged particle
at t = 0.
21.2.1 Time-dependent mixing formalism
We denote the neutral meson by B (which will usually be B
0 , but could also be
K
0 or D
0 ), and its CP-conjugate by ¯
B. According to the theory of Weisskopf
and Wigner (1930a, 1930b) (see also appendix A of Kabir 1968) a state that
is initially in some superposition of |B› and |B ¯ ›, say
|ψ(0)› = a(0)|B› + b(0)|B ¯ ›,
(21.27)
will evolve in time to a general superposition
|ψ(t)› = a(t)|B› + b(t)|B ¯ ›
(21.28)
governed by an effective Hamiltonian H with matrix elements, in the 2-state
subspace,
(
)
Γ
A p
2
H = M − i =
(21.29)
2
2
q
A
where M and Γ are Hermitian, and the equality M 11 −iΓ 11 /2 = M 22 −iΓ 22 /2 =
A follows from CPT invariance, which we shall assume. If CP is a good
symmetry, then
‹B ¯ |H|B› = ‹B ¯ |(CP)
−1 (CP)H(CP)
−1 CP|B›
= ‹B|H|B ¯ ›
(21.30)
so that p would equal q. Since M and Γ are both Hermitian, this would imply
that M 12 and Γ 12 are both real; in the CP non-invariant world, this is not
the case.
The eigenvalues of H are
ω L ≡ m L − iΓ L /2 = A + p q, ω H ≡ m H − iΓ H /2 = A − p q,
(21.31)
and the corresponding eigenstates are
|B L › = (p |B› + q|B ¯ ›)/(|p|
2 + |q|
2 )
1/2
|B H › = (p |B› − q|B ¯ ›)/(|p|
2 + |q|
2 )
1/2 .
(21.32)
The states |B L ›, |B H › have definite masses m H , m L and widths Γ L and Γ H . The
widths Γ L , Γ H are equal to a very good approximation for B and D mesons,
because the Q-values of both are large; in the case of K-mesons (see section
21.3), one state decays predominantly to 2π and the other to 3π, with different
Q-values, and the lifetimes are very different.