The dilepton events allow direct tests of T and CPT symmetries
[2,11].
A long time ago Kabir [26] showed that T violation implies
different probabilities for 
and
transitions, while CPT requires equal probabilities
for 
and 
transitions.
Then a T-violating asymmetry:
and a CPT-violating one:
can be defined.
Both these tests can be done at a 
factory, where the initial state
is an antisymmetric 
state, if the 
rule
holds.
If a neutral kaon decays into a positive lepton at a time t, the
other neutral kaon is at the same time a 
and
the sign of the lepton, emitted
in a subsequent semileptonic decay, signals if the 
has changed or
conserved its own flavour. Therefore,
if 
, the charge asymmetry
in equal-sign dilepton pairs measured at the 
factory will be
equal to 
. On the other hand, time asymmetry in opposite-sign
dilepton pairs signals CPT violation.
In the more general case, taking into account
also possible violations of the 
rule
one gets
:
and
where 
is the number of dilepton pairs
with the positive lepton emitted before (after) the negative one.
The number of equal-sign electron pairs (
)
and that of opposite-sign (
) expected
at DA
NE is about 
events/year,
therefore the T- and CPT-violating asymmetries can be measured with a statistical error of about
.
Violation of the 
rule does not affect eq. (30)
but the CPT violation in the decay amplitude contributes together with the
true T-violating term 
.
On the contrary in eq. (31)
the effects of CPT violation and 
transitions cannot be disentangled.
In the CPT limit the time asymmetry can be written as:
and inserting the experimental limits on 
[24] one has:
Thus a value of 
larger than 
indicates an actual
CPT violation either in the kaon mass matrix or in 
transition amplitudes.
More information can be obtained by
the study of the time dependence of opposite sign dilepton events.
Choosing for the final states of eq. (8)
and 
and integrating over the
phase space one gets:
where
The difference in the asymptotic limits (
) leads
to the determination of 
,
while the interference term singles out
.
The higher-order terms can be neglected (their upper bound is about
, smaller than the DA
NE sensitivity),
but the CPT-violating parameter 
and the 
contributions are still mixed.
An exact determination of the statistical error on 
and 
would require a simulation of the experimental apparatus, which is
beyond the purpose of this work. To give an idea of the DA
NE sensitivity
we report in Fig. 3 the asymmetry in opposite-sign dileptons as a function
of the time difference
for 
and 
.
As can be seen the asymptotic value is reached very soon and the three
curves are clearly distinct. Therefore we estimate 
. The value of
depends critically on the experimental resolution.
We estimate that, as happens for the real and the imaginary parts of 
,
will be about 20 times larger than 
.
Figure 3: The asymmetry 
as a function of the time difference
for 
. The full, dashed and dot-dashed lines correspond to
, 
and 
respectively.
As shown in Ref. [22], the inclusion in the analysis of
the 
decays allows us to
disentangle almost all the amplitudes. Indeed,
in the Wu-Yang phase convention, unitarity implies that
the phase of 
is equal to
and the phase of 
is
;
therefore, one has [11,22]:
The present experimental data on 
and 
[24]
give:
As can be seen, the CPT constraint is at present very well satisfied
and, assuming CPT conservation in decay amplitudes, the limit in 
mass difference is
close to the natural scale factor 
.
In addition, from the measured value of 
charge asymmetry one gets:
The future measurement of 
at DA
NE would lead also
to the determination of 
, while the
CPLEAR experiment will give direct measurement
of 
and of 
.
Therefore all the parameters that appear in the observables introduced above
can be disentangled, and some consistency relations must be satisfied.
The preliminary data of the CPLEAR collaboration [25]
have large errors and still do not give significant bounds.
We report in Table 2 the relations between the observables and the
theoretical parameters with the corresponding statistical errors
from present and future experiments, together with
the consistency equations and the corresponding sensitivity.
To simplify the notations of the table we define
Table 2: Table 2: Statistical errors on parameters and consistency
relations, using
present experimental data
[24] (for 
, 
and 
) together with
DA
NE (for 
, 
, 
and 
)
and CPLEAR (for 
and 
) future results.
The 
of the last equation in the table is only a guess.
