Violation in Mixing

18 �È Violation in the �� System
From Eqs. 1.12 and 1.9, one can then deduce that
is a good approximation.
Õ
Ô
Å £ � �
� ÁÑ
�Š�
Å
¬
¬
¬ Õ
Ô¬
�
1.2.6 Time Formalism for Coherent �� States
�� � (1.26)
In an � � collider operating at the § �Ë resonance (called a � factory, see the following chapter 2), the
� and � mesons produced from the decay of the § are in a coherent Ä � state. The § �Ë is a resonant
� � state with quantum numbers  � � and it can decay into � � or � � pairs: � mesons are
scalars (Â È � ) and so, because of the total angular momentum conservation, the � � pair has to be
produced in a Ä � state. Since the § �Ë decays strongly, � mesons are produced in the two flavour
eigenstates � and � .
After the production, one can imagine that each of the two particles evolve in time as described above for
a single �. However they evolve in phase, so that at any time there is always exactly one � and one �
present, at least until one particle decays. As a matter of fact, if at a given time Ø one � could oscillate
independently from the other, they could become a state made up of two identical mesons: but this cannot
happen since the Ä � state is anti-symmetric, while a system of two identical mesons (that are bosons)
must be completely symmetric for the two particle exchange, However once one of the particles decays the
other continues to evolve, and thus there are possible events with two � or two � decays, whose probability
is governed by the time between the two decays.
Identifying the two particles from the § �Ë decay by the angle � that they form with the � beam direction
in the § �Ë rest frame (and thus calling them a backward and a forward �), the two-� state can be written
as [12]:
Ë Ø� �Ø� � Ô �
�Š� �
Ò � Ø� ���� � Ø��� �� � �
� � ���� � ��� �� � �
Ó ×�Ò � (1.27)
where � is the proper time of the ��, the � particle in the forward half-space at angle �� ��� ��� and
� is the proper time for the backward-moving ��, at � �� ��� �).
In Eq. 1.27, it has been taken into account that the § �Ë is a spatially asymmetric state (it is a parity
eigenstate with eigenvalue ) and it is a � eigenstate with eigenvalue . Therefore also the system
to which it decays, must be spatially anti-symmetric as well as charge-conjugation anti-symmetric at a
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