Decays of neutral B mesons originating from the 
(4S) into a CP eigenstate f produce CP-violating asymmetries 
given by
where 
denotes the mass difference between the two 
mass eigenstates and 
, where 
and 
are the proper time for the and 
deacys, respectively.
The 
range is from 
to 
and the asymmetry vanishes in the time integrated rate. Therefore, in experiments at the (4S), the measurement of is essential to the observation of a mixing-enhanced CP asymmetry.
This is the motivation for the asymmetric beam energies in the B factory--the resulting motion of the CM enables the measurement of [20].
The angle 
is the phase difference between the 
mixing amplitude and the 
decay amplitude, which is directly related to the internal angles of the unitarity triangle.
Figure: (a) Tree and (b) penguin via t or c qurak,
and (c) penguin via u quark diagrams for 
Figure 
shows the quark diagrams responsible for the decays 
via the 
quark-level process.
The weak phases for the Cabibbo-allowed tree diagram in Fig. (a) and the penguin QCD-loop-induced diagram in Fig.(b) are equal.
The penguin diagram in Fig.(c) has a different weak phase, but is highly suppressed (by a factor of 
5%) [21], and only a small ambiguity is introduced into measurements of sin 2 
by this term.
The decay 
is the most promising mode for the measurement since the branching ratio for this decay has been measured and the signals are very clean with no appreciable back-ground.
However, the sensitivity can be increased by the inclusion of other decay modes.
Therefore, we have examined the feasibility of using the following additional decay modes: 
.
According to the CKM model, there are several other decay modes that provide measurements,
The verification of CP asymmetries in these modes, which include 
, is an important test of the CKM model.
Figure: (a) Tree and (b) penguin diagrams for 
One possible way to measure the 
angle is via the decay 
.
Figure. shows the quark diagrams responsible for this decay.
The weak phases for the Cabibbo-suppressed tree diagram in Fig. (a) and penguin diagram in Fig. are different, which introduces a theoretical uncertainty into the asymmetry that is estimated to be about 20% 
[21].
In principle, the effects of the penguin contribution can be extracted by means of an isospin analysis of the amplitudes for 
, and 
[22].
Recently, a potential method for CP-violation measurements using non-CP eigenstate decays of neutral B mesons has been proposed [23].
In this case, the final states are not CP eigenstates but CP self-conjugate at the quark level so both and can decay into the same final states.
Examples are 
and 
.
For these decays, CP asymmetries arise in the same way as for the CP eigenstate case, except for an additional dilution factor that appears because the final state is not a CP eigenstate.
Figure: Diagram contributing to (a) 
, and (b) 
In principle, 
can be determined from the CP asymmetry for 
in analogy to the measurement via 
.
There are, however, difficulties using 
decay modes at an asymmetric B factory.
Since mesons are not produced at the (4S), one has to use the (5S), where the production cross section for 
pairs is rather small.
Moreover, the Standard Model expectation for the mixing strngth 
is large ( 
8) and such mixing can mask the CP asymmetry produced by the modest boost of an asymmetric B Factory.
This mode also suffers ambiguities from penguins, and isospin analyses cannot be applied to decays.
One proposal is to determine from direct CP violation in 
decays [24].
Note that a B meson can decay into 
or 
through processes described by the Feynman diagrams shown in Figs. .
The decay amplitudes can be written as
Where 
and 
are hadronic phases.
Since the CP eigenstates of the 
are
the decay amplitudes can be expressed as
The amplitudes ( 
) and ( 
) form triangles in the complex plane(Fig. ).
An assumption that only a single diagram contributes in each process gives the constraints, 
, and 
.
Figure: Triangle (a) is formed by the amplitudes of decaying to 
. Triangle (b) is formed by the amplitudes of decaying to these three modes.
In the first triangle, the angle between 
and 
is 
while in the second, the angle between 
and 
is 
, where 
.
If the absolute values of four independent amplitudes ( 
) are measured, the two triangles are fixed and and 
can be obtained.
The absolute values of amplitudes can be obtained from the branching ratios.
An inequality between 
and 
is a signature of direct CP violation.