Violation in Mixing

16 �È Violation in the �� System
and the various quantities defined change as:
Å � � � �� Å � � � �� Õ�Ô � � � �� Õ�Ô � (1.19)
Also decay amplitudes are affected by the phase transformation in 1.18:
�� � � � �À�� �
�� � � � �À�� �
�� � � � �� ��
�� � � � �� ��
(1.20)
where ��� is the physical final state. From transformation in 1.18, and the transformation of Õ�Ô in 1.19, one
gets:
��Ä�� � � �� ��Ä� ��À�� � � �� ��À�� (1.21)
since one can always define the ratio Ô ��Ô as a pure phase. As a matter of fact only the ratio Õ�Ô and its
transformation are defined and one can write � Ô ��Ô℄� �� � � �� . Therefore equations 1.21 mean that both
the eigenstates are rotated by a common phase factor with no physical meaning.
Similar phase freedom exists in defining the �È transformation law for a possible final state � and its �È
conjugate � ��� �: the quantity �� depends on the flavour content of � and is related to the quark flavour
symmetries ( � � � �) of the strong interactions.
However, the freedom in defining the phase of the flavour eigenstates (which are defined through strong
interactions only) does not mean that the full Lagrangian, which involves also weak interactions, is invariant
under such phase re-definitions. Indeed, the differences of flavour redefinition phases appear as changes
in the phases of the quark mixing matrix elements and of the Yukawa couplings of quarks to Higgs fields
(or any other Lagrangian terms that cause couplings between different flavour eigenstates in more general
models).
While both Õ�Ô and �� acquire overall phase re-definitions when these phase rotations are made, the
quantity:
� � Õ
Ô
has a convention independent phase that has physical meaning.
1.2.5 Time Evolution of Neutral � � Mesons
��
��
(1.22)
The two neutral �� mesons are expected to have difference in lifetime at the level of Ç (see Sec. 1.2.2):
¡ �� has not been measured yet, but from general considerations it can be considered negligible with
respect to �� . As a matter of fact, the difference in width is produced by decay channels common to �
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