Violation in Mixing

1.4 �È Violation in the Standard Model 29
To a first approximation, the �ÃÅ matrix is simply the unit matrix, because the dominant transitions
are Ù � �, � × and Ø � �. In reality, none of the off-diagonal elements is exactly zero, leading to
generation-changing transitions between quarks and to the possibility of a �È -violating phase. The values
of both fermion masses and �ÃÅ matrix elements cannot be predicted since they are input parameters of
the Standard Model originating in the Higgs field.
In order to have a complete representation of the �ÃÅ matrix, only four real and independent parameters are
necessary: all nine �ÃÅ matrix elements can be expressed as functions of these four parameters. Generally
speaking, an Ò ¢ Ò unitary matrix has Ò real and independent parameters: a generic Ò ¢ Ò matrix would
have Ò and the unitary condition imposes Ò normalization constraints and Ò Ò conditions from the
orthogonality between each pair of columns: thus Ò Ò Ò Ò � Ò .
In the �ÃÅ matrix, not all of these parameters have a physical meaning since, given Ò quark generations,
Ò phases can be absorbed by the freedom to select the phases of the quark fields. A phase factor can
be applied to every quark operator so that the current  � could be written as:
 � � Ù� ��Ù � � �� �Ø� ��Ø ­ �
­�
ÎÙ� ÎÙ× ÎÙ�
Î � Î × Î �
ÎØ� ÎØ× ÎØ�
� �����
×� ��×
Each Ù, or Ø phase allows for multiplying a row of the �ÃÅ matrix by a phase, while each �, × or � phase
allows for multiplying a column by a phase: the Ù, and Ø phases can be chosen in order to make real one
element of each of the three rows (for example ÎÙ×, Î × and ÎØ×). Therefore all three elements of a column
(the second in the example) can be made real. In a similar way, the �, × and � phases can be chosen in
order to make real one element of each of the three columns (for example ÎÙ� and Î �). At the end of this
redefinition procedure, five of the �ÃÅ matrix phases have been re-absorbed with six quarks: in general,
with Ò quark families, Ò phases can be removed. So it is: Ò Ò � Ò . From the latter,
given quark families, � real and independent parameters are necessary.
If the �ÃÅ matrix were simply real and orthogonal, it would have Ò degrees of freedom from which one
should subtract the Ò normalization conditions and Ò Ò orthogonality conditions (the factor is due
to the fact that È � Î �Î � � is the same condition as È � Î �Î � � ): in this case, one would obtain a
number of degrees of freedom corresponding to Ò Ò real independent rotation parameters.
�� ���
Table 1-1. Degrees of freedom of �ÃÅ matrix as a function of the number Ò of quark families.
Ò(families) Total indep. params. Real rot. angles Complex phase factors
Ò Ò Ò ) Ò Ò
2 1 1 0
3 4 3 1
4 9 6 3
� �
�È VIOLATION IN THE �� SYSTEM