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We focus on 
decays of 
and 
. On general grounds,
there are 4 independent matrix elements, related to the (complex) form factors
of the transitions:
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:
:
Time-reversal relates each form factor to its complex conjugate, CP relates
K
to 
form factors. This suggests to parametrize the amplitudes
according to:
a and b (c and d) obey the same symmetry properties as the
non-leptonic amplitudes 
and 
(see Tab. 2), i.e.: b and d are CPT
violating, imaginary parts are all T-violating; c and d describe possible
violations of the 
rule. We consider
of order unity, and keep first order terms in all the
other quantities.
Of course, one should introduce analogous amplitudes for muonic decays, but we will leave this understood, in the following, to avoid a too heavy notation.
The following notations are also used [2]:
with:
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exact:
CPT exact:
T exact:
CP exact:
For convenience, we shall also define:
with 
defined in Eqs. (1.19) and (1.20) and:
The following relations are immediate:
There are in all four semileptonic rates, which can be expressed in terms
of the three combinations given above plus the average rate, which determines
. In addition, to study the correlated decays of
the 
-
pair produced at a 
factory, it is convenient to
introduce the complex quantities:
In the Standard Theory, CPT and CP are conserved in semileptonic processes
and the 
rule is obeyed to a very good precision [22],
with (
is the relative strength of the octet non-leptonic
amplitude):
In the current x current picture there is, in fact, little space for the violation of these symmetries, given our very good knowledge of the currents themselves.
Violation of CP or of the 
rule could arise from
contact interactions of quark and leptons (e.g. in composite models) and one
should keep an open mind on the possible presence of anomalies in the
semileptonic amplitudes. However, 
transitions require
hadronic operators transforming as 10 + 27 of flavour SU(3), see e.g. the
second paper of Ref. [22], that can be induced only by effective quark
and lepton operators of dimension higher than four. A typical example is:
with 
the compositeness scale, which leads to the (rather
generous) estimate:
The result (6.22) justifies the neglect of 
amplitudes,
still keeping open the possibility of CPT violation.
For the sake of brevity, the case in which semileptonic amplitudes are assumed
to conserve both CPT and the 
rule will be called
Scheme I, in the following. Scheme II will be the case in which CPT
is relaxed, still keeping exact the 
rule. We shall also
comment on Scheme III, where Eqs. (6.3) to (6.6) are considered in full
generality.