Violation in Mixing

32 �È Violation in the �� System
Figure 1-4. The Unitarity Triangle derived from Eq. 1.50: a) Î ��Î £
�� � which represents the orthogonality
condition between the first and the third column of the �ÃÅ matrix (the orientation depends on the phase
convention), b) re-scaled version where sides have been divided by �Î �Î £ �� and c) ÎÙ� ÎØ�
approximations have been applied. The form of the triangle is unchanged.
In the first two triangles, one side is much shorter than the other two, and so they almost collapse to a line.
This can give an intuitive explanation of why �È violation is small in the à system (the first triangle) and
in the �× system (the second triangle). The openness of the third triangle predicts large �È asymmetries in
� decays.
All triangles have the same area. One can define the quantity:
Â�È � �ÁÑ Î��Î £ £
�РΠ���Р�� � �� �� � �� �
where no sums over indices are implied. The term Þ � Î��Î £
�Рis one term in the sum of terms that gives
the inner product between column � and column Ð, while, in the same way, Þ £ � Î £
���Рis the complex
conjugate of another one of these terms. Since Þ and Þ are two of the sides of the unitary triangles, the
quantity ÁÑ Þ Þ £ is proportional to the sine of the angle between Þ and Þ (this can be demonstrated using
polar coordinates: Þ � � �� and Þ � � �� and thus ÁÑ Þ Þ £ ��� ×�Ò � � ) and so the area of
the unitary triangles ca be written like:
MARCELLA BONA