Consistent with Eq. (10), also the 
transition
amplitudes are expanded in power series of the variables X and Y around
the centre of the Dalitz plot X=Y=0, assuming the absence of nearby poles,
up to quadratic terms. Limiting to 
and 
transitions, as also consistent with experimental data, there are three
possible three-pion final states with definite isospin:
; 
; and
. Due to the Bose symmetry of the three-pion final state,
the transition (of the 
) to the I=0 state, although possible in
principle, is strongly suppressed by a high angular momentum centrifugal
barrier, and we ignore it. Neglecting isospin breaking effects, the general
Bose-symmetric and CP conserving expansion can be written in terms of five
independent weak amplitudes as [8,9]:
In Eq. (11), 
are completely symmetric under permutations
of the indices 1, 2 and 3. Conversely, 
are symmetric only under
the exchange 
, and under permutations of indices obey
the relation
Finally, the amplitude for 
is antisymmetric under
the exchange 
. From the isospin point of view, the
amplitudes 
and 
contribute to 
and
transitions to the I=1 final state, while 
is the pure
transition to I=2. Also, one should notice that there are
two kinds of amplitudes to final I=1, which reflect the different pion
exchange symmetry properties of the corresponding three-pion states, i.e.,
the fully symmetric A's and the B's with mixed symmetry.
Taking the symmetry properties into account, A's and B's can be expanded up to quadratic terms in X and Y as follows:
In the absence of final state strong interactions, the CP conserving
amplitudes in (11) and (13) could be taken as real. As
required by unitarity, strong interaction rescattering of final state pions
produce imaginary parts, which can be taken into account by introducing
more phenomenological parameters (in addition to 
, 
, 
and
) in Eq. (13). Due to the smallness of the available
phase space, these effects are expected to be small and, to avoid
too many free parameters, in current fits to the data such strong phases
have been assumed to be negligible (within an uncertainty of 
).
This assumption is consistent with the available experimental
information, and also with theoretical expectations [5,10].
Nevertheless, as being sensitive to the properties of low-energy meson
dynamics, rescattering effects are theoretically quite interesting and
could eventually be in the reach of next-generation, high precision
experiments on 
. Consequently, their measurement would provide
alternative, and stringent, tests of the relevant theoretical description.
Specifically, of the pion-pion interaction if one neglects irreducible
rescattering diagrams that should be phase space-suppressed (barring
anomalously large 
vertices). Such tests would involve the
really low-energy range, where the framework of chiral perturbation theory
(
) can be most reliably applied.
Another, and very important, point of interest is the fact that these
imaginary parts crucially enter the determination of direct CP-violating
asymmetries in 
decays, and consequently are relevant to searches
for this, still unclear, phenomenon in a channel alternative to
[11].
Phenomenologically, momentum dependence of rescattering should be taken
into account for a consistent low-energy expansion of the amplitudes. This
is desirable also in view of the momentum expansions predicted
by chiral perturbation theory, which will be described in the sequel.
For the parametrization of such effects, there is the
complication that the two final I=1 three-pion states are coupled by (isospin
conserving) strong interactions, so that they can mix. This situation can be
dealt with, rather generally, by a coupled-channel formalism that introduces,
in the I=1 sector, a two-dimensional, momentum-dependent rescattering
matrix R common to both charged and neutral kaon decays, connecting
symmetric and nonsymmetric amplitudes [12]. Since R=I
corresponds to the limiting case of
no final state interaction, R can be expanded as 
,
where the elements of the (real) matrix 
are functions of X and
Y. Such functions can be expanded around the centre of the Dalitz plot,
similar to Eq. (13), and are expected to be small over the whole
allowed kinematical region. Moreover, in general the R-matrix elements are
not all independent, but are related by unitarity constraints following from
probability conservation.
Contrary to the I=1 case, for the decays to the I=2 three-pion state,
which is unique and cannot mix with the others via strong interactions,
there is just one amplitude with definite symmetry properties, and final state
interactions are simply taken into account by a phase function that is unique
for all decay modes, similar to the case of 
where I=0 and I=2
final states do not mix.
For simplicity, one can limit to include rescattering effects only in the
constant and linear slopes in Eq. (13), as it seems justified
by the smallness of experimental quadratic slopes. In
that case, using Eq. (13), the expansion of Eq. (11)
around X=Y=0 takes on the form:
The amplitudes for the decays of the 
are determined by taking
complex-conjugates of the coefficients 
with unchanged strong
interaction imaginary parts. CP conservation implies that these
coefficients should be real and consequently, in this case, the amplitudes
for 
and 
decays must be the same. In the presence of CP violation,
the coefficients 
are in general complex, so that amplitudes
for charge-conjugate processes can be different. Furthermore, in this
situation the 
amplitude could get additional,
imaginary constant and linear (in Y) terms, while an imaginary amplitude
linear in X could appear in 
[13].
A representation alternative to (14), which has been adopted in
fits to experimental data [5,10], is the expansion in terms of
amplitudes with definite isospin selection rules. With 
,
neglecting strong interaction imaginary parts, such an expansion can be
written as
In Eq. (15), the subscripts 1 and 3 refer to 
and
, respectively. The relation between the amplitudes in
Eq. (14) and those in Eq. (15) is easily found to
be:
In conclusion, the complex of 
modes is described by the set
of ten independent (real) isospin amplitudes in Eq. (15),
which are collected in Tab. 3 for convenience, or alternatively by the ten
amplitudes in Eq. (14). There are, in addition, the imaginary
parts from final state strong interactions, parametrized in leading order
by four real constants, as in Eq. (14).
Table 3: Independent isospin amplitudes for 
