530
K. Kleinknecht and U. Uwer
The off-diagonal elements of the matrices are given by
Γ 21 = 2π
F N 0 |H W |F |H W |N
0
,
(9.20)
where the sum runs over all possible physical intermediate states F , which have a
phase space density F . Similarly,
M 21 = =N 0 |H W |N
0
+
n
N 0 |H W |n W |N 0
m
N 0 − m n
,
(9.21)
where the sum extends over all possible virtual intermediate states n.
The eigenvalue equations for X yield two eigenstates, which can be labeled by
their mass: h for the higher mass, l for the lower mass. These eigenstates are the
physical particles with a definite mass and an exponential lifetime distribution. The
eigenvalues M h and M l of the matrix X are
M h = m h −
i
2
Γ h ,
M l = m l −
i
2
Γ l
(9.22)
We denote the differences between the physical quantities by = Γ h − Γ l and
= m h − m l > 0 and denote the average values by
Γ =
Γ h + Γ l
2
,
(9.23)
m =
m h + m l
2
.
(9.24)
For the B 0 —B 0 system and the D 0 —D 0 system, the two decay widths Γ h and Γ l are
expected to be nearly equal (because the numbers of final states for the decay are
very similar). In these cases it is customary to introduce the dimensionless quantities
x =
m
Γ
(9.25)
and
y =
Γ
2Γ
.
(9.26)
Here x is positive by definition, and y varies between −1 and +1. For heavy systems
such as B 0 , |y| is expected to be much less than 1, while for the K 0 system, y is
found experimentally to be close to −1, since here the decay width of the lighter
state is 600 times larger then that of the heavier state. Therefore, for the K 0 system,
K. Kleinknecht and U. Uwer
The off-diagonal elements of the matrices are given by
Γ 21 = 2π
F N 0 |H W |F |H W |N
0
,
(9.20)
where the sum runs over all possible physical intermediate states F , which have a
phase space density F . Similarly,
M 21 = =N 0 |H W |N
0
+
n
N 0 |H W |n W |N 0
m
N 0 − m n
,
(9.21)
where the sum extends over all possible virtual intermediate states n.
The eigenvalue equations for X yield two eigenstates, which can be labeled by
their mass: h for the higher mass, l for the lower mass. These eigenstates are the
physical particles with a definite mass and an exponential lifetime distribution. The
eigenvalues M h and M l of the matrix X are
M h = m h −
i
2
Γ h ,
M l = m l −
i
2
Γ l
(9.22)
We denote the differences between the physical quantities by = Γ h − Γ l and
= m h − m l > 0 and denote the average values by
Γ =
Γ h + Γ l
2
,
(9.23)
m =
m h + m l
2
.
(9.24)
For the B 0 —B 0 system and the D 0 —D 0 system, the two decay widths Γ h and Γ l are
expected to be nearly equal (because the numbers of final states for the decay are
very similar). In these cases it is customary to introduce the dimensionless quantities
x =
m
Γ
(9.25)
and
y =
Γ
2Γ
.
(9.26)
Here x is positive by definition, and y varies between −1 and +1. For heavy systems
such as B 0 , |y| is expected to be much less than 1, while for the K 0 system, y is
found experimentally to be close to −1, since here the decay width of the lighter
state is 600 times larger then that of the heavier state. Therefore, for the K 0 system,