Violation in Mixing

30 �È Violation in the �� System
So going back to the Ò real independent parameters of a generic unitary matrix, Ò Ò of these
parameters can be associated to real rotation angles: among the remaining Ò Ò Ò phases, Ò
can be removed with quark field re-definitions, as already said. So the number of independent phase factors
is:
Ò Ò Ò Ò � Ò Ò �
In table 1-1, the cases with , or � quark families are shown: note that at least three quark generations
are necessary in order to have, in the �ÃÅ matrix, a �È -violating phase factor. A two-generation theory
would not be able to accommodate �È violation without the addition of extra fields. It was this observation
that led Kobayashi and Maskawa to suggest a third quark generation long before there was any experimental
evidence for it [2, 3].
The unitarity of the �ÃÅ matrix can be made more explicit using a particular parameterization. There are
various useful ways to parameterize it, but the standard choice is based on the definition of four angles � ,
� , � e Æ [14, 20]:
Î �
× × � �Æ
× × × � �Æ × × × � �Æ ×
× × × � �Æ × × × � �Æ
� � (1.47)
where �� � Ó× ��� and ×�� � ×�Ò ���, ��� is the mixing angle between the � and � quark families and the
phase Æ is responsible for �È violation.
Considering that Î � when �È violation effect goes to zero, the representation 1.47 can be simplified:
for example one can use the fact that �ÎÙ�� � × � is very small and so is extremely close
to unity. As a consequence, one can neglect terms proportional to × relative to terms of order unity and
obtain:
Î �
× × � �Æ
× ×
× × × � �Æ ×
� � (1.48)
In this approximation, only ÎÙ� and ÎØ� carry phases. Another useful expansion of the �ÃÅ matrix has
been first given by Wolfenstein [21] in the small parameter � � ×�Ò �� � where �� is the Cabibbo
angle:
Î �
� � �� � ��
� � ��
�� � �� ��
� Ç � � � (1.49)
In this form, only four independent parameters remain: �, �, � and �. ToÇ � � � , the ¢ upper-left
portion of the �ÃÅ matrix is the matrix associated with Cabibbo rotations of the � and × quarks and it is
constructed to be nearly unitary:
MARCELLA BONA