Violation in Mixing

1.5 Determination of « 39
Assuming negligible the penguins contributions, the previous method can be used to extract the angle «
through measurements of decays of � Ô�Ý× Ø and � Ô�Ý× Ø to final states like � � or � � . When
penguins are not negligible, this method measures a quantity, denoted «�«, which differs from the true
« by the unknown amount �� � . This quantity is mode-dependent because it depends on the ratio of treedominated
to penguin-only contributions.
1.5.2 Extraction of « in the Presence of Penguins
In most of the decays modes, more than one amplitude is present and in the expression of the total amplitude,
contributions from tree or penguin diagrams can be split: in the particular case of the channels of interest
here, the weak phase difference between these terms is «.
Going back to the case of a final state � being a �È eigenstate, we can factorize the decay amplitudes:
� � � ��È ����È �Ì���Ì � �ÆÌ È� ��È � �ÆÈ
� � � ��È ����È �Ì� ��Ì � �ÆÌ È� ��È � �ÆÈ (1.69)
where Ì , �Ì and ÆÌ (È , �È and ÆÈ ) are the magnitude, the weak phase and the strong phase of the treedominated
(penguin-only) amplitude.
Thus, assuming the presence of penguin contributions, if �Ì �� �È , ���È (Eq. 1.22) becomes a function of
tree and penguin diagram parameters and as a consequence it does not corresponds to a clean measure of
the CKM phase. The presence of non-negligible penguin contributions also leads to direct �È violation (see
Sec. 1.5.3), that is ����È � �� . In the presence of direct �È violation, the time-dependent �È asymmetry
contains a Ó× ¡Ñ�Ø term, the coefficient of which can also be measured. In case the strong phases are
equal, ÆÌ � ÆÈ , then ���È is a pure phase (i.e., ����È � � ) and so no direct �È violation is present.
However, like in the previous case, this phase depends on both tree and penguin parameters, so that there is
still a shift in « due to penguin contributions, even though there is no direct �È violation.
The method to separate the tree and penguin contributions is isospin analysis. Isospin amplitudes Á¡Á�Á�
can be labeled by the ¡Á value of the �-quark decay and by the Á� of the final state, which includes the
spectator quark. A gluon is pure Á � , so that the dominant gluonic � � � penguin diagrams are pure
¡Á � . On the other hand, the tree-level � � ÙÙ� decays have both ¡Á � and ¡Á � components. If
the ¡Á � part can be isolated, then the tree contribution, which contains the weak phase to be measured,
can be isolated. The inclusion of the spectator quark gives the final isospin value of or for the gluonic
penguin contributions, but , or for the tree contributions. The same arguments apply to � � × penguins
and � � ÙÙ× tree amplitudes.
Table 1-2 lists the isospin amplitudes for all relevant channels for these states. Note that, in all cases, there is
at least one isospin amplitude which can be reached only via tree diagrams: � � � for � � �� and � � �
for � � �Ã. Isolation of such isospin amplitudes allows the removal of penguin pollution.
�È VIOLATION IN THE �� SYSTEM