349
21.3. CP violation in neutral K-meson decays
The merit of this manoeuvering is that the parameter ∊
′ involves only the CP
violation in the transition amplitude (‘direct CP violation’), while ∊ involves
both a transition phase and the mixing parameter ∊ ¯.
What can experiment tell us about ∊ and ∊
′ ? Consider first δ L . Assuming
that |A(K
0
→ e
+ ν e π
− )| = |A(K ¯ 0 → e
− ν ¯ e π
+ )| and that A(K
0
→ e
− ν ¯ e π
+ ) =
A(K ¯ 0 → e
+ ν e π
− ) = 0, we find
δ L = 2Re ¯
∊|
2 ) ≈ 2Re ∊,
∊/(1 + |¯
(21.93)
so that δ L is sensitive to the same parameter as appears in the K L → ππ
decays. An interesting observable is the ratio between the ratios of the decay
rates to π
+ π
− and π
0 π
0 of K S and K L . One finds
(
)
1
|η 00 |
2
1 −
≈ Re (∊
′ /∊),
(21.94)
6
|η +−| 2
which from equation (21.82) is another small number, approximately equal
to 1.64 × 10
−3 . In the years before the B factories opened, ∊
′ was the only
window into CP violation in the transition amplitude. But all the branching
ratios in (21.94) are of order 10
−3 , and establishing a non-zero value of ∊
′ was
very difficult. The first claim for non-zero ∊
′ was by the NA 31 experiment at
CERN (Barr et al. 1993), a 3.5 standard deviation effect. But a contemporary
experiment at Fermilab (Gibbons et al. 1993) found a result compatible with
zero. The next generation of experiments produced agreement:
Re (∊
′ /∊) = (2.07) ± 0.28) × 10
−3 Alavi-Harati et al. 2003 (KTeV)
(21.95)
Re (∊
′ /∊) = (1.47 ± 0.22) × 10
−3 Batley et al. 2002 (NA 48). (21.96)
The current world average is (1.65 ± 0.26) × 10
−3 . Fits to all the data also
yield (Nakamura et al. 2010)
|∊| = (2.228 ± 0.011) × 10
−3 .
(21.97)
The experimental value of δ L gives us Re ∊ ≃ 1.66 × 10
−3 , and we can deduce
that arg ∊ ≃ π/4. The phase of ∊
′ is π/2 + δ 2 − δ 0 which happens also to be
approximately π/4. It follows that ∊
′ /∊ is very nearly real.
Comparison of these small numbers with theoretical predictions is complicated by hadronic uncertainties, and it is beyond our scope to pursue that
issue.
In closing this discussion of mesonic mixing and CP violation, we briefly
¯
discuss the charm sector. First, we note that D
0 -D
0 mixing has been observed
(Aubert et al. 2007c, Staric et al. 2007, Aaltonen et al. 2008). CP-violating
effects in charm decays have been generally expected to be very small. A
rough estimate of the direct CP-violating asymmetries in D decays can be
made following the method of section 21.1. Consider, for example, the decays
21.3. CP violation in neutral K-meson decays
The merit of this manoeuvering is that the parameter ∊
′ involves only the CP
violation in the transition amplitude (‘direct CP violation’), while ∊ involves
both a transition phase and the mixing parameter ∊ ¯.
What can experiment tell us about ∊ and ∊
′ ? Consider first δ L . Assuming
that |A(K
0
→ e
+ ν e π
− )| = |A(K ¯ 0 → e
− ν ¯ e π
+ )| and that A(K
0
→ e
− ν ¯ e π
+ ) =
A(K ¯ 0 → e
+ ν e π
− ) = 0, we find
δ L = 2Re ¯
∊|
2 ) ≈ 2Re ∊,
∊/(1 + |¯
(21.93)
so that δ L is sensitive to the same parameter as appears in the K L → ππ
decays. An interesting observable is the ratio between the ratios of the decay
rates to π
+ π
− and π
0 π
0 of K S and K L . One finds
(
)
1
|η 00 |
2
1 −
≈ Re (∊
′ /∊),
(21.94)
6
|η +−| 2
which from equation (21.82) is another small number, approximately equal
to 1.64 × 10
−3 . In the years before the B factories opened, ∊
′ was the only
window into CP violation in the transition amplitude. But all the branching
ratios in (21.94) are of order 10
−3 , and establishing a non-zero value of ∊
′ was
very difficult. The first claim for non-zero ∊
′ was by the NA 31 experiment at
CERN (Barr et al. 1993), a 3.5 standard deviation effect. But a contemporary
experiment at Fermilab (Gibbons et al. 1993) found a result compatible with
zero. The next generation of experiments produced agreement:
Re (∊
′ /∊) = (2.07) ± 0.28) × 10
−3 Alavi-Harati et al. 2003 (KTeV)
(21.95)
Re (∊
′ /∊) = (1.47 ± 0.22) × 10
−3 Batley et al. 2002 (NA 48). (21.96)
The current world average is (1.65 ± 0.26) × 10
−3 . Fits to all the data also
yield (Nakamura et al. 2010)
|∊| = (2.228 ± 0.011) × 10
−3 .
(21.97)
The experimental value of δ L gives us Re ∊ ≃ 1.66 × 10
−3 , and we can deduce
that arg ∊ ≃ π/4. The phase of ∊
′ is π/2 + δ 2 − δ 0 which happens also to be
approximately π/4. It follows that ∊
′ /∊ is very nearly real.
Comparison of these small numbers with theoretical predictions is complicated by hadronic uncertainties, and it is beyond our scope to pursue that
issue.
In closing this discussion of mesonic mixing and CP violation, we briefly
¯
discuss the charm sector. First, we note that D
0 -D
0 mixing has been observed
(Aubert et al. 2007c, Staric et al. 2007, Aaltonen et al. 2008). CP-violating
effects in charm decays have been generally expected to be very small. A
rough estimate of the direct CP-violating asymmetries in D decays can be
made following the method of section 21.1. Consider, for example, the decays