Violation in Mixing

1.3 The Three Types of �È Violation in � Decays 21
In case the tag final state has � � , � ��Ø��, which identifies the second particle as a � at time ØØ��,
an expression similar to Eq. 1.32 applies, except that the signs of both the cosine and the sine terms are
reversed. The fact that �Õ�Ô� � means that the amplitudes for the two opposite tags are the same. Thus the
difference of these rates divided by their sum, which measures the time-dependent �È asymmetry, is given
by
���È � Ê � � �� ��Ø�� Ê � � �Ø��� � �
Ê � � �� ��Ø�� Ê � ��Ø��� � �
�
����È� ¡ Ó× ¡Ñ�Ø ÁÑ���È
����È� ×�Ò ¡Ñ�Ø
(1.33)
where Ø � Ø��È ØØ��. The above expressions has lost its dependence from the variable Ø Ø or Ø�È ØØ��:
this means that now one can fit the dependence on the variable Ø Ø without having to measure the § decay
time. This can be done also from equation 1.32 that can be integrated over the variable Ø Ø , substituting
the variables Ì � Ø Ø and Ø � Ø Ø and integrating over Ì which for Ø � and Ø � can take
values between �Ø� and infinity. This way, one gets an expression of Ê Ø which is only a function of the
time difference Ø�È ØØ�� and not of the § decay time:
¡ ¨
����È �
Ê Ø��È ØØ�� � �ØØ�� Ø��È �
¡ (1.34)
©
�
����È � ¡ Ó× �¡Ñ� Ø��È ØØ�� ℄ ×�Ò �¡Ñ� Ø��È ØØ�� ℄ ÁÑ ���È
The fact that the variable Ø Ø can be related to the distance between the decay vertices of the two �’s
is the main reason for building an energy-asymmetric collider for this kind of measurements (see Sec. 2.1).
By integrating also over this variable, all information on the coefficient of ×�Ò ¡Ñ� Ø Ø would be
lost and the experiment would be sensitive only to those �È -violating effects that give ��� �� . This is a
consequence of the coherent production of the two � states: in a hadronic environment, where the �’s are
produced incoherently, time-integrated rates are always integrals from Ø � to infinity so that they retain
information about the ×�Ò ¡Ñ�Ø term.
1.3 The Three Types of �È Violation in � Decays
�È violation can manifest itself in three different ways:
¯ �È violation in decay: also called direct �È violation, it occurs when a decay and its �È conjugate
process have different amplitudes. It can be studied in both charged and neutral decays.
¯ �È violation in mixing: also called indirect �È violation, it occurs when mixing provides interfering
amplitudes. In this case, the two neutral mass eigenstates cannot be �È eigenstates too.
�È VIOLATION IN THE �� SYSTEM