The 
decay is a pure CP-violating transition,
while the 
decay receives both CP-conserving and
CP-violating contributions.
The CP-conserving 
decay amplitudes are well described by
Chiral Perturbation Theory (ChPT). They have been calculated, including
the next-to-leading-order corrections, in Ref. [27] and turn out
to be in good agreement with the experimental data.
The CP-conserving 
decay amplitude is odd under
momenta exchange and thus, neglecting final states
with high angular momenta, it is induced by
a 
transition. The ChPT calculation of Ref. [27]
leads to the prediction:
consistent with the recent data:
As in 
decays, for the CP-violating amplitudes
it is convenient to define the ratios:
The direct CP-violating parameters 
and 
have been evaluated at lowest order in ChPT [29] and turn out to
be of the same order as 
. As shown in [30], higher-order terms can substantially enhance
and 
, which are nevertheless
negligible
compared to 
. The
predicted branching ratios are:
much smaller then the present upper limits [24,25,31].
Due to the smallness of the branching ratios
it is very hard to detect 
decays, especially the
CP-violating ones. Tagging the 
as in the case of the semileptonic
decays (eqs. (25) and (26))
and inserting the numerical values,
one gets for 
final state:
The total number of events is very small (
6 per year)
and the ratio of right events (those with a 
decay)
to wrong ones (those with a 
decay)
is only 2.2.
In the case of the CP-conserving 
decay,
the expected number of events is about 440 with a negligible
background.
A more promising way to detect the CP-violating
decays is to study the
interference terms of 
, in eq. (8), choosing
and 
, as
suggested in Refs. [32,33].
For the 
it is useful to define the asymmetry:
which, integrating over the 
and 
Dalitz plots,
becomes:
For positive and large values of the time difference t, eq. (51)
reads: 
;
on the other hand, for negative value of t, one gets an interesting
interference effect between 
and 
, as shown in Fig. 4.
The asymmetry for t<0 is quite large, but the total number of events
is small, about 
per year.
Figure 4: The asymmetries 
(full line) and 
(
dashed line). We have fixed 
.
In the case of the 
final state, the CP-violating and
CP-conserving amplitudes have opposite symmetry under
momentum exchange. Therefore it is possible to select
the CP-violating and the CP-conserving part of the interference term in
eq.(8) with an even or an odd integration over the
Dalitz plot. Analogously to the 
case,
for the CP-violating part we define the asymmetry:
while the CP-conserving part can be singled out by the ratio:
where 
indicates the integration in
the region 
.
Figure 5: The ratio 
defined in eq.(53).
The full, dashed and dotted
lines correspond to 
, 
and
respectively.
The behaviour of 
is completely analogous to the one of
(Fig. 4). As discussed in refs. [32,33]
the study of 
will certainly lead to a determination of the
amplitude, performing an interesting test of
ChPT in the 
transitions, and perhaps could also lead
to a direct measurement of the 
rescattering functions.
The phase 
of eq. (53) can be written as:
where 
and 
are the first terms
in the expansion of the 
rescattering functions
of the I=2 and of the symmetric I=1 final states, respectively.
The ChPT prediction is 
[33]
and the first measurement [28] gives
.
With a different integration over the Dalitz plot, also the
rescattering function of the non-symmetric I=1 final state
could be selected [32,33].
Figure 5 shows the behaviour of 
for different values of
.
