In order to estimate 
, we have to constrain the CP-violating
phase 
in the CKM matrix, by using the available experimental
information. To this end, we consider the CP-violating term in the
--
mixing amplitude and the CP-conserving term for
--
mixing.
In the following, we present all the formulae used in our analysis,
namely the expressions of 
, 
and 
from the
, 
, 
effective Hamiltonian, respectively.
The effective Hamiltonian governing the 
amplitude is given by
where 
and
the functions 
and 
are the so-called
Inami--Lim functions [28], including QCD corrections
[2]; 
is known at the
next-to-leading order and has been included in our calculation.
From eqs. (18) and (59),
one can derive the CP-violation parameter
where
In eq. (60), 
, 
and 
, A, 
and 
are the
parameters of the CKM matrix in the Wolfenstein parametrization [19].
is the renormalization group invariant B-factor, the
definition of which at the leading order is
The 
effective Hamiltonian is given by
Here 
.
From eq. (63), one finds the 
--
mixing
parameter
where 
is the B-parameter relevant for 
mixing,
the definition of which is analogous to the 
one.
We can write 
as
With respect to eq. (14), we have here explicitly written
the isospin-breaking contribution 
, see for example ref.
[30],
To compute 
and 
, we need the
hadronic matrix elements of the operators 
between a kaon and two pions.
Usually they are given in terms of the so-called B-parameters:
where the subscripts VIA means that the matrix elements are computed in the vacuum insertion approximation. The relevant VIA matrix elements can be expressed in terms of three quantities
From 
, the expressions of 
and
in terms of Wilson coefficients and of the B-parameters
are obtained
Notice that the matrix elements of the electromagnetic left--right operators
, which belong to the 
representation of
, contain Y and do not vanish in the chiral limit.
The evaluation of the B-factors requires a non-perturbative technique.
The Wilson coefficients and the hadronic matrix elements both
depend on the regularization scheme.
In order to cancel this dependence (up to 
), it is necessary
to control the matching between the B-parameters and the coefficients at
the next-to-leading order.
Notice that many non-perturbative methods (e.g. 
expansion) do not fulfil
this requirement.
Two different approaches to the matrix element evaluation have been used
in recent next-to-leading 
analyses:
,
we have made educated guesses, which are discussed in detail in ref.
[9].
flavour symmetry and
deducing some constraints
relating hadronic matrix elements at the charm threshold.
Unfortunately, there is no way to determine the
most important
B-factors necessary to estimate 
, namely 
and 
, which
remain essentially unconstrained in this approach.
