Violation in Mixing

42 �È Violation in the �� System
6–98
8418A3
1
2
A(B° π + π – )
1
2
κππ
A(B° π + π – ~
)
A(B° π°π°)
A(B π – π°) = A(B + π + ~
π°)
Figure 1-8. Isospin analysis of � � �� decays.
~
A(B° π°π°)
Á � contribution. Therefore the three two-pion decay amplitudes depend only on two isospin amplitudes:
Ô �
� � � Ô �
� � �
(1.70)
where the amplitudes for the �È -conjugate processes � � � � , � � � � and � � � � are
obtained from the � amplitudes by simply changing the sign of the CKM phases; the strong phases remain
the same. Expressions in 1.70 can be seen as two triangles, as drawn in Fig. 1-8.
To have �� � the measurement of � � � � rate is needed, while �� � is taken from the � � � �
rate. �� � and �� � come from the measurement of the time-dependent decay rates for � Ø � � �
and � Ø � � � : this measurement also allows the extraction of the asymmetry and thus ÁÑ� � � .
Finally �� � and �� � can be obtain from the time-independent rate of � Ø � � � . The six magnitudes
determine the shapes of two isospin triangles.
Since the penguin diagram is purely ¡Á � � , the � amplitude takes contributions only from the tree
diagram and so �� � � �� �. This means that the two triangles have a base in common. However, due
to the fact that both the tree and penguin diagrams contribute to the Á � final state, �� ����� �and
�� ������. One can superimpose the triangles defined in Eqs. 1.70 by introducing
�� �� � � ��Ì � ��
i.e. � � � � � ��Ì �
(1.71)
where �Ì is the CKM phase of the tree diagram. This way one gets � � � � and can draw Fig. 1-8. The
angle ��� between � and � � can be determined up to a fourfold discrete ambiguity corresponding to
the choice of orientation of each of the triangles with respect to the other. Another two-fold ambiguity comes
from the fact that only the sine of the angle « ��� is measured: this leads to an eight-fold ambiguity in
the value of «. If a small error on each possible choice of ��� can be obtained, isospin analysis will be able
to significantly reduce the uncertainty in « extraction
MARCELLA BONA