Violation in Mixing

40 �È Violation in the �� System
Table 1-2. Isospin decomposition for � � ��,� � �à in terms of the isospin amplitudes � ¡Á�Á�, where
¡Á and Á� are the isospin change of the transition and the final-state isospin, respectively.
Channel Decay Amplitudes
Ô
�� � � � � � �
� � �
Ô �� �� � � Ô � � �
Õ
Õ
� � � �
� � � � � � Ô � � � ��� � Õ
�à � � � � à � � � � � � � � � � Õ
�� �� à � � � � � � � � � � Õ
Ô �� �� à � � � � � � � � � � Õ
Ô �� �� à � � � � � � � � � �
1.5.3 Hadronic charmless two-body � decays
Hadronic decays resulting from � � Ù transitions are highly suppressed due to the factor �ÎÙ�� . Decays
such as � � �� and � � � can proceed either through a � � ٠spectator diagram or through a gluonic
penguin.
In a gluonic penguin, a � � � or � � × transition occurs through a virtual loop containing a Ï and either a
Ø, or Ù quark with the radiation of a gluon. The dominant contribution is expected to arise from the Ø-quark
intermediate state, but effects from the -quark are not necessarily negligible [32, 34].
A simple argument gives a rough idea of the possible relative contribution of the tree and penguin amplitudes
in � � �� and � � Ã�. Assuming that Ø-quark dominates the loop, the penguin contribution to � �
� � is suppressed relative to that for � � à � by �ÎØ��ÎØ×� � Ç � in the rate. Furthermore the
tree contribution to � � à � is suppressed relative to that � � � � by a factor �ÎÙ×�ÎÙ�� � in
the rate. By itself, all this tells us is that the tree diagram contributes more to �� than to � and the penguin
contributes more to � than to ��. Suppose that � � � � � �� � �à � , then � � à �
must be mainly penguin or else the � � �à � ratio would be larger. Even if all of the � � à � rate
were penguin, the assumed near equality of the branching fractions implies that the penguin contribution to
� � � � must be fairly small.
The tree process should contribute more to � � � � than to � � Ã � while the penguin process
(with intermediate Ø or quarks) should contribute more to � � Ã � than to � � � � . Both
the tree and penguin contributions to � � � � are Ç � . In a scenario where � � � � � �
� � � Ã � , we would conclude that � � Ã � was predominantly penguin and that � � � �
was predominantly a tree process. That is, the upper-left and lower-right diagrams must dominate. For, if
the � � Ã � were mainly a tree process (upper-right diagram, Ç � � ), then the � � � � tree
contribution (upper-left, Ç � ) would be even larger and we would observed � � � � � � � � �
MARCELLA BONA