Violation in Mixing

7
Analysis of the time-dependent
�È -violating asymmetry in
� � � � decays
This chapter describes the analysis of the time evolution in � � � � decays. In the Standard Model, the
time-dependent �È -violating asymmetry in � � � � is related to the angle « of the Unitarity Triangle
(Sec. 1.5.3.1). This decay mode has only recently been observed by CLEO [67] and confirmed by BaBar [6]
and Belle [68]. Due to the small decay rate (�Ê � � ¢ � ), large continuum ÕÕ background, and significant
cross-feed from � � à � decays, extraction of the �È asymmetry in � � is intimately related
to the branching fraction measurement. In addition, this measurement relies heavily on the infrastructure
(tagging [69] and vertexing [71] in particular) developed for the ×�Ò ¬ analysis [72].
7.1 �È analysis requirements
The formalism has been already described in the Sec. 1.5. Defining ¡Ø � Ø�È ØØ��, where Ø�È and ØØ��
are the proper decay times of the �È and tagged �’s, respectively, the decay rate distribution � � (in
Eq. 1.34) for ��È � � when �Ø�� is a � � can be rewritten as:
�¡Ø��� �
�¦ ¡Ø �
��
where � is the average � lifetime, ¡Ñ�� is the mixing frequency, and
� ¦ Ë� ×�Ò ¡Ñ�� ¡Ø § �� Ó× ¡Ñ�� ¡Ø ℄ � (7.1)
Ë� � ÁÑ�
��� �� �� � ���
���
� (7.2)
Interference effects are included in the physical quantity �, defined in Eq. 1.22 or in 1.41: it can be rewritten
as
� � Õ ��
� �� �
Ô ��
�� �� � (7.3)
��
where � �� is the weak mixing phase, �� �� is the amplitude for the decay � � � � � � , �� is the
�È eigenvalue of the final state, and the assumption of no �È violation in mixing (�Õ�Ô� � ) is implicit. In
this ¬ analysis, observable �È violation effects can arise from interference between different decay amplitudes
¬
¬ ( ¬� ��� � ¬ �� ) and interference between the mixing and decay weak phases (see Sec. 1.3.3).