333
21.1. Direct CP violation in B decays
it contains the factor V ub V
∗
∼ Aλ
3 (ρ − iη); (iii) it will have a different strong
cs
interaction phase. With these factors in mind, we write
i(δB−γ)
˜
A B = r B A B e
(21.15)
where δ B is the difference in strong phases between A ˜ B and A B , and r B is
the magnitude ratio of the amplitudes. Since |ρ − iη| ∼ 0.38, r B is of order
0.1–0.2, allowing for the colour suppression.
Once again, the asymmetry is proportional to
|1 + r B exp[i(δ B − γ)] |
2
− |1 + r B exp[i(δ B + γ)] |
2
≈ 4r B sin δ B sin γ. (21.16)
This involves γ, but the relative smallness of r B tends to reduce the sensitivity
to γ. An alternative determination of γ can be made (Attwood et al. 2001,
Giri et al. 2003) by making use of three-body decays (to a common channel)
¯
of D
0 and D ¯ 0 , such as D
0 , D
0
→ K S π
+ π
− . If we denote the amplitude for
D
0
→ K S π
+ π
− by A(s − , s + ) (see figure 21.3), where s − = (p K + p π − )
2 and
s + = (p K + p π + )
2 are the indicated invariant masses, then the amplitude for
the B
− to decay to K
− K S π
+ π
− via the D
0 path is
1
A − = A B D[A(s − , s + ) + r B e
i(δB−γ) A(s + , s − )],
(21.17)
and the amplitude for the charge conjugate reaction B
+
→ K S π
− π
+ is
A + = A B D[A(s + , s − ) + r B e
i(δB+γ) A(s + − s + )],
(21.18)
where D is the D meson propagator. The event rate for the B
− decay is then
Γ − (s − , s + ) where (Aubert et al. 2008)
2
2
2
Γ − (s − , s + ) ∝ |A(s − , s + )| + r B |A(s + , s − )| +
2 [x − Re{A(s − , s + )A
∗ (s + , s − )} + y − Im{A(s − , s + )A
∗ (s + , s − )}] (21.19)
and the rate for B
+ decay is Γ + (s − , s + ) where
2
2
2
Γ + (s − , s + ) ∝ |A(s + , s − )| + r B |A(s − , s + )| +
2 [x + Re{A(s + , s − )A
∗ (s − , s + )} + y + Im{A(s + , s − )A
∗ (s − , s + )}] . (21.20)
Here
x − = r B cos(δ B − γ), y − = r B sin(δ B − γ)
(21.21)
x + = r B cos(δ B + γ), y + = r B sin(δ B + γ).
(21.22)
The geometry of the CP-violating parameters is shown in figure 21.4. Note
that the separation of the B
− and B
+ positions in the (x, y) plane is equal to
1 We are neglecting D 0 -D ¯ 0 mixing and CP asymmetries in D decays, which are at the
1% or less level (Grossman et al. 2005).
21.1. Direct CP violation in B decays
it contains the factor V ub V
∗
∼ Aλ
3 (ρ − iη); (iii) it will have a different strong
cs
interaction phase. With these factors in mind, we write
i(δB−γ)
˜
A B = r B A B e
(21.15)
where δ B is the difference in strong phases between A ˜ B and A B , and r B is
the magnitude ratio of the amplitudes. Since |ρ − iη| ∼ 0.38, r B is of order
0.1–0.2, allowing for the colour suppression.
Once again, the asymmetry is proportional to
|1 + r B exp[i(δ B − γ)] |
2
− |1 + r B exp[i(δ B + γ)] |
2
≈ 4r B sin δ B sin γ. (21.16)
This involves γ, but the relative smallness of r B tends to reduce the sensitivity
to γ. An alternative determination of γ can be made (Attwood et al. 2001,
Giri et al. 2003) by making use of three-body decays (to a common channel)
¯
of D
0 and D ¯ 0 , such as D
0 , D
0
→ K S π
+ π
− . If we denote the amplitude for
D
0
→ K S π
+ π
− by A(s − , s + ) (see figure 21.3), where s − = (p K + p π − )
2 and
s + = (p K + p π + )
2 are the indicated invariant masses, then the amplitude for
the B
− to decay to K
− K S π
+ π
− via the D
0 path is
1
A − = A B D[A(s − , s + ) + r B e
i(δB−γ) A(s + , s − )],
(21.17)
and the amplitude for the charge conjugate reaction B
+
→ K S π
− π
+ is
A + = A B D[A(s + , s − ) + r B e
i(δB+γ) A(s + − s + )],
(21.18)
where D is the D meson propagator. The event rate for the B
− decay is then
Γ − (s − , s + ) where (Aubert et al. 2008)
2
2
2
Γ − (s − , s + ) ∝ |A(s − , s + )| + r B |A(s + , s − )| +
2 [x − Re{A(s − , s + )A
∗ (s + , s − )} + y − Im{A(s − , s + )A
∗ (s + , s − )}] (21.19)
and the rate for B
+ decay is Γ + (s − , s + ) where
2
2
2
Γ + (s − , s + ) ∝ |A(s + , s − )| + r B |A(s − , s + )| +
2 [x + Re{A(s + , s − )A
∗ (s − , s + )} + y + Im{A(s + , s − )A
∗ (s − , s + )}] . (21.20)
Here
x − = r B cos(δ B − γ), y − = r B sin(δ B − γ)
(21.21)
x + = r B cos(δ B + γ), y + = r B sin(δ B + γ).
(21.22)
The geometry of the CP-violating parameters is shown in figure 21.4. Note
that the separation of the B
− and B
+ positions in the (x, y) plane is equal to
1 We are neglecting D 0 -D ¯ 0 mixing and CP asymmetries in D decays, which are at the
1% or less level (Grossman et al. 2005).