In this case one can study CP-odd charge asymmetries in the total rates and
in the linear slope of the Dalitz plot. Referring to [3] for more
details on the kinematics and the relevant definitions, we recall that,
due to the smallness of the outgoing pion energies, in general the
Dalitz plot distributions for 
can be expanded in powers of the
independent kinematical variables as follows (higher powers are not requested
by experimental data):
Here, for the process
, Y and X are the familiar
kinematical variables, 
and 
,
with 
, 
, and `3'
indicates the `odd charge' pion. The term linear in X in
Eq. (1) is CP violating and can appear only in the case of
, so that it is not relevant here.
For 
decays the charge asymmetries are then defined as
the relative differences
and
for both 
(i.e. 
) and 
(i.e. 
) decay modes.
We expand the relevant 
amplitudes up to linear terms. Assuming
, and including imaginary parts
originating from final state strong interaction effects, this expansion can
be written as
where 
, 
, 
can be chosen as real constants if CP is conserved,
and 
, 
and 
are also real.
With these definitions, 
is purely 
, so that the
rule would imply the condition:
Furthermore, lowest order in chiral perturbation theory and the
rule give:
In the presence of direct CP violation, the coefficients 
, 
and
are complex numbers, and nonvanishing charge asymmetries such as
(2) and (3) are generated. For example, specializing
to the centre of the Dalitz plot, for the 
-decay slope asymmetry
we obtain:
where the quantities in the denominators are taken in the CP-exact limit and
therefore are understood to be real. Notice that 
, which
appears in (4), does not contribute to the charge asymmetry
(7).
The r.h.s. of Eq. (7), and the analogous asymmetry for 
decay, explicitly vanish in the limit in which relations (5) and
(6) are obeyed, since in that case there is only one weak decay
amplitude whose phase can be transformed away [4,5].
The effective Hamiltonian at the constituent quark level consists, in fact,
of various components, transforming according to the 
,
and 
representations of chiral
.
If the t quark mass were small, the 
component,
which arises from the electroweak penguin diagrams, could be safely neglected.
In this case, in lowest order 
a non-vanishing asymmetry
would arise solely from the interference of the 
and 
amplitudes, and, in this limit, would be necessarily suppressed by the small
ratio
quite similar to 
. With 
very large, of the
order of 
, the effect of the electroweak operators becomes
non-negligible. This rise of the 
contribution with 
is in fact responsible for the decrease of the Standard Theory prediction of
(for more quantitative details, see
[1]).
As for the 
charge asymmetry, it turns out that the contribution
of the 
operator increases the leading order prediction of
for the Dalitz plot slope asymmetry, and decreases the
prediction for the asymmetry of the widths. Another source of corrections to
the lowest order 
prediction is the isospin breaking
u-d quark mass difference, which feeds contributions proportional
to the (large) coefficient of the 
component into the 
channel.
Making contact with Ref. [6], we describe now with some detail the
lowest order 
calculation, accounting for the electroweak
penguin diagram contribution and for isospin breaking corrections (see also
[7,8]). The first ingredient is the evaluation of strong
interaction rescattering phases. One finds in
[9,6] to one loop:
where 
.
As it has been introduced in [3], the 
effective Hamiltonian now consists of the familiar 
and
components, represented in leading order 
as
[10]
plus the 
component [9]:
where 
, 
, and
with 
the pseudoscalar
meson 
matrix.
In terms of the coefficients 
and 
one has:
where 
represents the effect of isospin breaking, and numerically
we shall use 
.
CP violation is related to the imaginary parts of the coefficients
, 
and 
. Introducing for convenience the dimensionless
quantities:
one can express the asymmetry as a superposition of the 
with
coefficients determined by the experimental (real) amplitudes. In turn,
the 
are derived from a theoretical determination of the matrix
elements of the various components of the effective Hamiltonian (
expansion [8], or Lattice QCD
[11,12,13,14]).
Eq. (7) then takes the form
where numerically:
The values of 
needed in (19) are obtained,
as in Ref. [6], by combining the short-distance, perturbative QCD
coefficients in the expansion of the nonleptonic 
Hamiltonian
in terms of four-quark operators, with the
nonperturbative B-factors computed in Lattice QCD [14].
As an indication, in Figs. 1 and 2 we represent the results which would be
obtained for the 
slope asymmetry (3), by applying
the same kind of numerical analysis performed for
in Ref. [1]. This analysis should
account for the most recent determinations of the Standard Model parameters
(
, CKM matrix elements, 
, etc.), including
present theoretical uncertainties on the relevant nonperturbative parameters
(such as 
), as well as the experimental statistical and systematical
uncertainties. Specifically, in the
histogram of Fig. 1, the solid line represents the predicted values of
without any `cut' on the possible values of 
(see Ref. [1] for details), while the dashed one is obtained by
imposing such a `cut' (namely, by constraining 
to the range of values
predicted by Lattice calculations). We recall that this `cut' would imply
as the most likely possibility, 
being
the CP violating CKM phase. As anticipated in Sec. 1, the predicted values of
are of the order of some units per million. Fig. 2 is a
combined plot of 
vs. 
, the
solid and dashed lines representing the allowed contours with 5% and
68% probability, respectively, limiting to the case of the 
`cut' which
privileges 
. One can remark from this figure that, for the
large values of 
[15],
there is little correlation between the values of 
and
, whereas a rather strong correlation would have
occurred for small values of 
[6]. Also, Fig. 2 indicates that, while 
can vanish, 
should be safely different from zero.
Finally, in Tab. 1 we summarize the numerical values for the various charge
asymmetries, as obtained from the procedure outlined above, in the case
.
Table 1: Experimental values and theoretical predictions for the CP
violating asymmetries in 
and 
modes at lowest order in
, including 
and 
effects.
This table shows that the estimated CP violating asymmetries
are much larger for the
Dalitz plot slopes than for the total decay widths. Qualitatively, similar
results and orders of magnitudes for the asymmetry have been obtained in
[8], using the 
expansion at leading order.
Eq. (7) is also useful for a simple discussion of possible higher
order 
effects, which have been advocated in [16] as
the source of an enhancement of the asymmetry at the level of two orders of
magnitude. Indeed, higher order chiral corrections could be important,
because they can introduce further octet effective operators, to give
independent phases to the amplitudes of the two I=1 states. CP violating
interference of these two amplitudes would avoid the suppression factor
in Eq. (8). Therefore, one could naively hope to gain an
enhancement factor 
.
Actually, essentially following the discussion of Ref. [17], let us
reconsider Eq. (7) limiting ourselves to the first term only
(the second one may be neglected as it involves the 
Hamiltonian in an essential way), and, after using Eq. (5),
rewrite it as
The first term in the square bracket is anyway suppressed by the combination of experimental amplitudes, which give
We can expand the second term, according to 
, into leading
and next-to-leading 
terms, as
Therefore, to a good approximation, the relative size of the correction is
represented by the factor 
:
Coefficients are such that:
and therefore one derives:
More precisely, it can be seen that in the decomposition of the 
amplitudes into loop and counterterm contributions,
only the weak counterterms are relevant to (31). Indeed, in principle
loops can generate large 
imaginary parts. However, these contributions
to the two 
amplitudes must have the same weak phase (that of
the unique 
operator in (11)), and
therefore cannot build up a large CP violating interference
[6,17,18].
The rigorous inclusion of chiral loops and local counterterms for CP violating
decays has not been accomplished yet, due to many unknown
constants. Nevertheless, chiral corrections are seen to be quite reasonable
for CP conserving amplitudes, introducing in that case effects of the order
of 40% or less (see [3]). Consequently, we can make the assumption
that the chiral expansion similarly works also for the CP violating
amplitudes (although not yet verified experimentally, this assumption seems
quite plausible). Thus, the order 
enhancement factor in (31)
should be expected to be of the order of 
or so
(the upper figure corresponding to
the extreme case 
),
which would enhance the predictions in Tab. 1 by an order of magnitude
at most. This is confirmed, e.g., by the findings of Ref. [19]
using the chiral 
-model, and of Ref. [8] in the framework of
the 
expansion.
