A Mathematica based Version of the CKMfitter Package

10 Chapter 2. Theory
b u,c,t
d
0
B
W W
u,c,t
0
B
d
b
b
d
0
B
W
u,c,t u,c,t
Figure 2.3: Box diagram contribution to B 0 - ¯ B 0 mixing
where B0 L is the lighter, and B0 H is the heavier eigenstate. Analogous to the
Kaon system, the oscillation parameters pB and qB are complex and normalized
by |qB| + |pB| = 1.
The mass and decay width differences of the mass eigenstates are defined by convention
as:
∆mB ≡ MH − ML � 2|M12| (2.33)
∆ΓB ≡ ΓH − ΓL � 2 Re (M12Γ ∗ 12 )
|M12|
W
0
B
d
b
(2.34)
where ∆ΓB ≪ ∆mB is assumed. The exact ratio qB/pB is given by:
�
2 M
qB
= −
pB
∗ i
12 −
2 Γ∗ �
12
∆mB − i
2 ∆ΓB
. (2.35)
The physical meaningful quantity, that is independent of the phase convention, is:
�
� � �
� qB �2
�M
� � �
�pB
� = �
�
�
∗ i
12 −
2 Γ∗12 M12 − i
�
�
�
�
�
�
�
. (2.36)
2 Γ12
The phase of qB/pB is convention dependent and, hence, not observable. Similar
definitions can be made in the Bs-meson system by replacing only the d-quark by a
s-quark.
2.5 CP Violation
The CP transformation is a combination of charge conjugation C and parity P.
Under C transformation, a particle transforms into its anti-particle, by conjugating
all internal quantum numbers, e. g. Q → −Q for the electromagnetic charge. P
transformation reflects the space coordinate �x into −�x. The combination of both
transforms a left-handed particle into its right-handed anti-particle, e. g. e −
L
→ e+
R .