Violation in Mixing

1.5 Determination of « 45
� � ÙÙ� tree amplitude, the Cabibbo-suppressed � � ÙÙ× tree amplitude results in being smaller by a
factor of about 0.2 (the Cabibbo angle) than the � � × penguin amplitude. On the other hand, the � � ×
electroweak penguin is also suppressed by about 0.2 relative to the � � × gluonic penguin. Therefore, the
� � × electroweak penguin and the � � ÙÙ× tree amplitude could be comparable in magnitude. So the
electroweak-penguin contributions to � � �à could be non-negligible and an isospin analysis would not
be able to isolate the tree contribution [27].
In the case of � � à � , the penguin diagram produces the appropriate set of quarks, ×��Ù, while the
tree diagram produces ×ÙÙÙ: so in principle one could think that there can be absolutely no tree contribution,
in which case this signal would represent an unambiguous observation of a gluonic penguin process. In
reality, final-state interaction could convert the ÙÙ pair in the tree diagram into a ��: so under the assumption
that final-state interactions are likely to be small one can still consider � � Ã � channel a pure gluonic
penguin process. Moreover, the amplitude for the tree diagram is Ç � � , while that for the penguin is Ç � ,
so it seems even more unlikely that the tree contribution could be significant.
1.5.4 Direct �È Violation
As already said in Sec. 1.3.1, direct �È violation can occur in processes involving charged or neutral �’s.
In general, though, it is difficult to convert experimental observation of an asymmetry in a specific channel
into a quantitative determination of the basic parameters of the Standard Model.
We can observe �È -violating effects by comparing the amplitude È � � with È � � only if there
are both �È violating and non-�È violating phases: one could compare � � à � with � �
à � . These decays have both tree and penguin contributions which have different weak (and presumably
different strong) phases. Unfortunately it is not possible at present to calculate the strong phases and the
value of the weak phase would be ambiguous.
An example of what can be done with direct �È violation is the Fleischer-Mannel bound which is based on
an analysis of branching fractions for the various charged and neutral � � � modes. The decay � �
à � has both penguin and tree contributions, while, as previously discussed, to a good approximation,
� � Ã � is entirely a penguin process. Moreover, the penguin amplitudes for these processes should
be essentially identical, since the corresponding decays differ only in the isospin of the spectator quark. One
can therefore write the decay rates as
� � à � ��È �Ì � �­ � �Æ �
� � à � ��È �Ì � �­ � �Æ �
� � à � � � � à � ���
where �È and �Ì contain no CKM phases and Æ is the relative strong phase shift between the penguin and
tree processes. Computing the ratio:
� � à � � �à �
��È� Ö Ó× ­ Ó× Æ Ö
�È VIOLATION IN THE �� SYSTEM