338
21. CP Violation and Oscillation Phenomena
¯
B
0
B
0
¯
d
¯
b
b
u, c, t
d
¯
B
0
¯
d
b
u, c, t
¯
b
B
0
d
¯
u, ¯ c, ¯ t
¯
u, ¯ c, ¯ t
FIGURE 21.5
¯
Box diagram contributions to B
0 -B
0 mixing.
In the Standard Model, the B
0 -B ¯ 0 mixing amplitude occurs via the box
diagrams of figure 21.5. The box amplitude is approximately proportional
to the product of the masses of the internal quarks, and in this case the t
quark contribution dominates (the magnitudes of the CKM couplings are all
∗
comparable). The phase of M 12 is then that of (V V tb )
2 , which is the phase
td
of ((1 − ρ − iη)
∗ )
2 in the parametrization of (20.179), which in turn is equal
to the angle 2β. Hence
(q/p) = e
−2iβ ,
(21.42)
neglecting terms of order λ
4 . Equation (21.42) will be important in what
follows.
From (21.34) we can now read off that the probability that the state |B ¯ 0 ›
t
(which – we remind the reader – is the partner of the state tagged as a B
0 at
t = 0, and which is pure ¯ at t = 0) decays as a B ¯ 0 at t /
=
B
0
= 0, is |g + (t)|
2
2
exp(−Γt) cos
2 Δm t/2. Similarly, the probability that this state decays as a
B
B
0 at time t is exp(−Γt) sin
2 Δm B t/2, taking |(p/q)| = 1. Hence the difference
in these probabilities, normalized to their sum, is cos Δm B t. Measurements
of this flavour asymmetry yield the value of Δm B , currently (Nakamura et al.
2010)
Δm B = 3.3337 ± 0.033 × 10
−10 MeV.
(21.43)
More generally, we define decay amplitudes to final states |f › by
¯
B
0
A f = ‹f |H ˆ wk |B
0
› , A f = ‹f |H ˆ wk | ¯ ›
(21.44)
f ¯ | ˆ
¯
f ¯ | ˆ
A ¯ = ‹
› , A ¯ = ‹
›,
(21.45)
f
H wk |B
0
f
H wk |B
0
¯
where CP|f › = |f › and H ˆ wk is the weak interaction Hamiltonian responsible
for the transition. We can now calculate the rates for | ¯ › to go to |f ›, and
B
0
t
for |B
0
› to go to |f ›; up to a common normalization factor, which we omit,
t
these rates are (problem 21.2)
1
Γ(B ¯ 0
t → f ) = 2
¯
¯
e
−Γt
{|A f |
2 + |(p/q)A f |
2 + (|A f |
2
− |(p/q)A f |
2 ) cos Δm B t
q
¯ A
∗
+ 2Im( A f
f ) sin Δm B t},
(21.46)
p
21. CP Violation and Oscillation Phenomena
¯
B
0
B
0
¯
d
¯
b
b
u, c, t
d
¯
B
0
¯
d
b
u, c, t
¯
b
B
0
d
¯
u, ¯ c, ¯ t
¯
u, ¯ c, ¯ t
FIGURE 21.5
¯
Box diagram contributions to B
0 -B
0 mixing.
In the Standard Model, the B
0 -B ¯ 0 mixing amplitude occurs via the box
diagrams of figure 21.5. The box amplitude is approximately proportional
to the product of the masses of the internal quarks, and in this case the t
quark contribution dominates (the magnitudes of the CKM couplings are all
∗
comparable). The phase of M 12 is then that of (V V tb )
2 , which is the phase
td
of ((1 − ρ − iη)
∗ )
2 in the parametrization of (20.179), which in turn is equal
to the angle 2β. Hence
(q/p) = e
−2iβ ,
(21.42)
neglecting terms of order λ
4 . Equation (21.42) will be important in what
follows.
From (21.34) we can now read off that the probability that the state |B ¯ 0 ›
t
(which – we remind the reader – is the partner of the state tagged as a B
0 at
t = 0, and which is pure ¯ at t = 0) decays as a B ¯ 0 at t /
=
B
0
= 0, is |g + (t)|
2
2
exp(−Γt) cos
2 Δm t/2. Similarly, the probability that this state decays as a
B
B
0 at time t is exp(−Γt) sin
2 Δm B t/2, taking |(p/q)| = 1. Hence the difference
in these probabilities, normalized to their sum, is cos Δm B t. Measurements
of this flavour asymmetry yield the value of Δm B , currently (Nakamura et al.
2010)
Δm B = 3.3337 ± 0.033 × 10
−10 MeV.
(21.43)
More generally, we define decay amplitudes to final states |f › by
¯
B
0
A f = ‹f |H ˆ wk |B
0
› , A f = ‹f |H ˆ wk | ¯ ›
(21.44)
f ¯ | ˆ
¯
f ¯ | ˆ
A ¯ = ‹
› , A ¯ = ‹
›,
(21.45)
f
H wk |B
0
f
H wk |B
0
¯
where CP|f › = |f › and H ˆ wk is the weak interaction Hamiltonian responsible
for the transition. We can now calculate the rates for | ¯ › to go to |f ›, and
B
0
t
for |B
0
› to go to |f ›; up to a common normalization factor, which we omit,
t
these rates are (problem 21.2)
1
Γ(B ¯ 0
t → f ) = 2
¯
¯
e
−Γt
{|A f |
2 + |(p/q)A f |
2 + (|A f |
2
− |(p/q)A f |
2 ) cos Δm B t
q
¯ A
∗
+ 2Im( A f
f ) sin Δm B t},
(21.46)
p