Violation in Mixing

22 �È Violation in the �� System
¯ �È violation in the interference between mixing and decay: it occurs in decays into final states that
are common to � and � . It often occurs in combination with the other two types but there are cases
when, to a very good approximation, it is the only effect.
1.3.1 �È Violation in Decay
In order to study this type of �È violation, for any final state �, the quantity � � �
� is defined since it is
��
independent of phase conventions and physically meaningful. There are two types of phases that may
appear in the amplitudes: complex parameters in any Lagrangian term that contributes to the amplitude
will appear in complex conjugate form in the �È conjugate amplitude. Therefore these phases appear in
�� and � with opposite signs. In the Standard Model these phases appear in the �ÃÅ matrix and are
�
called weak phases. The weak phase of any single term is dependent on the convention, but the difference
between the weak phases in two different terms in the amplitudes is convention independent. A second type
of phase can appear even when the Lagrangian is real: such phases come from the possible contribution
from intermediate on-shell states dominated by strong interactions and so they are called strong phases.
Since strong interactions conserve �È these phases appear in �� and �� with the same sign. Again only
the relative strong phases of different terms have physical meaning.
Contributions to the amplitudes can be factorized as:
- the magnitude ��;
- the weak phase term � ��� ;
- the strong phase term � �� .
If several amplitudes contribute to � � �, the amplitude �� (see Eq. 1.20) and the �È conjugate amplitude
� � are given by:
�� � �
��� � � �� � �� � � � �� �� �
��� � � �� � (1.35)
�
where �� and �� are defined in expressions like 1.17: �È �� � � � ��� �� � and �È � � � � � ��� � � �
(one should consider the complex conjugate of the latter expression � � � �È � � ��� � � �). If � is a �È
eigenstate then � ��� � ¦ being its �È eigenvalue. The convention-independent quantity is then
¬
�� ¬ � ¬ � � �� ��
��
¬
¬
È � ��� � Æ� ��
È � ��� � Æ� ��
¬
�
¬ � (1.36)
�È is conserved in decays when the magnitude of this ratio is , that means the rate of the decay must be
equal to the rate of the �È conjugate decay. This can happen only if all weak phases �� are the same phase
or if all the strong phases � are the same one. Therefore, from Eq. 1.36 one sees that
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