Violation in Mixing

1.5 Determination of « 43
b
q
(a)
Z
q
q
q
q
(b)
b q
q
Z
C
P P
EW
EW
Figure 1-9. (a) Color-allowed �-penguin, (b) Color-suppressed �-penguin.
Using the above notations, the �È asymmetry in � � � � can be expressed as
ÁÑ� � � �ÁÑ
�
q
� �« � �
�
�
q
q
� (1.72)
The ratio � � �� is therefore the measure of the penguin pollution on the relationship between the angle
« and the measured asymmetry. The two-triangle construction gives the magnitude and phase, ���, of this
quantity, so that « can in principle be extracted cleanly, even in the presence of penguins.
One of the main issues in this analysis is the rate for � � � � . Some theoretical predictions of this
branching ratio are below or at the level of Ç � : a branching ratio at this order of magnitude would
make its measurement very challenging [33, 34, 31]. These estimates assume that color suppression is
significant in � decays to light mesons and small penguins, but if large rescattering effects are also present
in � � ��, the branching ratio of � � � � may be considerably larger than expectations and the isospin
analysis could then yield accurate results.
Theoretically, a problem that can arise in the isospin method is the presence of electroweak penguins
(EWP’s) [26]: the main EWP contributions to � � �� come from diagrams with virtual � exchange
(see Fig. 1-9). The couplings of the � contain both Á � and Á � terms and thus these diagrams
contribute also to ¡Á � so that their effects pollute the tree contributions. Though, the effects of these
electroweak penguins are expected to be small in this channel. Both the Ï and the � are color singlet
particles and thus two contributions, a color-allowed and a color-suppressed one, have to be considered for
each tree or electroweak-penguin diagram. If one includes both color-allowed and color-suppressed EWP’s,
the � � �� amplitudes become [27]:
� � � � � � Ô Ì � È�Ï È � �Ï
� � � � � � Ì È � È � �Ï
� � � � � � Ô � È � È�Ï È � �Ï
(1.73)
where Ì is the color-allowed tree contribution, � the color-suppressed tree contribution, È the gluonic
penguin, � the color-suppressed tree contribution, È�Ï the color-allowed EWP contribution and È � �Ï the
color-suppressed EWP contribution. The relative orders of magnitude of all these contributions are expected
to be:
�È VIOLATION IN THE �� SYSTEM