Violation in Mixing

10 �È Violation in the �� System
Moreover one can demonstrate that, given the invariance for �È symmetry, the off-diagonal terms of Å and
should be real. As a matter of fact defining a Hermitian matrix À (which represents Å or ):
� � �À� � � � � � � �È �È À �È �È � � � � ���È � � � �À� � � � ���À��� £ � (1.4)
assuming that �È À �È � À and remembering that �È �È � (from which the eigenvalues
satisfy the ���È � � ) and that �È � � � � ��È � � �.
We can obtain one more constraint if we consider again the �ÈÌ invariance: the off-diagonal terms of Å
and have to be one the complex conjugate of the other. This can be shown using the generic Hermitian
matrix À:
� � �À� � � � � � � �ÈÌ �ÈÌ À �ÈÌ �ÈÌ ��� ����ÈÌ� ���À��� � ���À��� £
where we have been using �ÈÌ À �ÈÌ � À and �ÈÌ��� � ��ÈÌ���.
The À eigenvalues (complex in the most general case) can be written as Ñ� � � where one can demonstrate
that � � : the matrix is defined the decay matrix. As a matter of fact, if one defines the eigenstates ��
of the non-Hermitian matrix À, the evolution in time of such a state is given by (as Eq. 1.3 shows):
� Ø �� �ÀØ ���� �Ñ�Ø �Ø ��
from which one can extract the probability of the initial particle not to be decayed yet at a given time Ø:
�� Ø � � � �Ø � Ý � ���
This quantity depends only from � that can be considered the decay rate of the given À eigenstate ��. On
the other hand, the matrix Å is called the Hermitian part of the mass matrix.
1.2.2 The � system: general formalism
A generic neutral meson �È � together with its anti-particle �È � 2 can be considered as a set of eigenstates
of the imperturbed Hamiltonian À with eigenvalues Ñ and Ñ , respectively: assuming that À conserves
�ÈÌ, Ñ can be considered equal to Ñ and thus:
À �È � � Ñ �È �� À �È � � Ñ �È ��
These two states (particle and anti-particle) belong to Ñ that is the degenerate eigenvalue of À . Thus if
an arbitrary linear combination of them is considered:
2 Here È and È label each neutral meson anti-meson pair.
MARCELLA BONA