Violation in Mixing

100 Strategy and Tools for Charmless Two-body � Decays Analysis
4.2.1 Topological Variables
Most of the background can be eliminated by simple kinematic cuts which make use of the differences
between �� and continuum events, which are the primary source of background. In the CM frame this
background typically exhibits a two-jet structure. In contrast, the low momentum and pseudo-scalar nature
of � mesons in the decay § �Ë � �� leads to a more spherically symmetric event. Some topological
variables can be used in order to distinguish between signal and background events. First of all, the event
sphericity Ë can be defined like [48]:
Ë � Ñ�Ò Ë Ò � Ñ�Ò
Ò Ò
where index � runs over Æ tracks of the event, the versor Ò spans over all the directions, Ô £ � is the component
of the momentum that is perpendicular to the versor Ò evaluated in the CM rest-frame of the § �Ë . The
versor Ò that satisfies the equation 4.1 is called sphericity axis. This function Ë Ò contains the information
on the way momenta are spatially distributed in the event.
Another useful discriminating variable is the event thrust, Ì , defined as:
where Ô £ �
Ì � Ñ�Ü Ì Ò � Ñ�Ü
Ò Ò
�
��
�
��
�
Ô £ ��
Ô £ �
� � Ò
is the component of the momentum that is parallel to the versor Ò and � Ò is the entire set of
tracks whose momenta have the component Ô � greater than zero. As before, momenta are evaluated in the
CM rest-frame of the § �Ë . The function Ì Ò represents the preferred direction of the momenta of the
tracks in the event: this variable contains information on the jet direction.
In the � � � ÕÕ production at high energies, the event in the CM rest-frame tends to assume a two-jet-like
structure: since all the hadrons in the final state come from hadronization of the high energy ÕÕ state, they
have to conserve the four-momentum and thus their flight directions are correlated to the initial ÕÕ line of
flight. This is the reason why variables like sphericity or thrust can be used to discriminate between signal
and background events.
These functions though are optimized in case the events really have a two-jet structure. On the other hand,
results from QCD show that more that of the events produced in a � � annihilation in the continuum
and at high energies should produce three or more jets in the CM rest-frame. Therefore topological variables
that do not depend on one specific event axis can be used to reject also non-two-jet continuum background:
an example of such variables are the Fox-Wolfram moments that can be written as [49]:
MARCELLA BONA
�
��
�Ô £ � �
�Ô £ � � �
(4.1)
(4.2)