On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

184 Appendix B. Neutral Meson Mixing
for the decay of a neutral meson into some final state f and its antistate ¯ f, and
analogous for the decay of the antimeson Af and A ¯ f . The variable
λf ≡ q
p
allows then to distinguish between the three types of CP violation,
1. CP Violation in Mixing is characterized by |q/p| �= 1,
2. CP Violation in Decay occurs for |A ¯ f /Af | �= 1, and
3. CP Violation in Interference of Mixing and Decays] is expected if Imλf �= 0.
Af
Af
(B.12)
A complementary way to categorize CP violation is by defining direct CP violation
for decays with |Af /A ¯ f | �= 1 and indirect CP violation if this ratio does not deviate
from one.
The observables in the Kaon and Bs − Bs system discussed in Chapter 3 correspond
to different measures of these types of CP violation. The ratio of the decay of longand
short-lived Kaons into pions in the strong isospin I = 0 eigenstates,
ɛK ≡ 〈(ππ)I=0|KL〉
〈(ππ)I=0|KS〉
(B.13)
measures the indirect CP violation in K − ¯ K mixing. By writing λ0 ≡ λ (ππ)I=0 and
inserting (B.2) in (B.13), one finds that
1 − λ0
ɛK = ≈
1 + λ0
1
2 (1 − λ0) ≈ 1
� � � �
�
1 + �
q �
�
2 �p
� + Imλ0 , (B.14)
where in the second to last step it was used that the experimental value of ɛK implies
that |λ0| ≈ 1, and in the last step the fact that direct CP violation is absent
was employed. Therefore, the real part of ɛK is sensitive to CP violation in mixing
and the imaginary part quantifies CP violation in the interference of mixing and decay.
Because of the characteristics discussed above, the decays of the Bs meson mass
eigenstates cannot clearly be identified with the decays of a short or a long lived
state. Consequentially, the relevant observables are different. Relevant for this thesis
is the time-dependent asymmetry, which measures the difference in of the Bs and Bs
into some final state. In the case of semileptonic final states, it will only measure
indirect CP violation,
A s SL = Γ(Bs(t) → ℓ + X) − Γ(Bs(t) → ℓ − X)
Γ(Bs(t) → ℓ + X) + Γ(Bs(t) → ℓ − X)
1 − |q/p|4
= , (B.15)
1 + |q/p| 4
because the only source of CP violation is the oscillation Bs → Bs → ℓ − X. It is
therefore sensitive to the CP violating phase in the CKM element Vts, which appears