9 Conclusions

From the previous analysis it is clear that a factory is very suitable for an accurate study of the origin of CP violation in and decays and to test the Standard Model predictions.

The real part of the ratio which is a clear signal of direct CP violation, can be measured with high precision, about . A non-vanishing value of the imaginary part of which would imply CPT violation, can be detected up to some units in . Even if the fixed target experiments will reach a similar sensitivity, the KLOE apparatus has a completely different systematics and such experimental result will be very important.

The presence of a pure beam will allow, for the first time, the direct detection of CP violation in decays. Moreover many interesting tests of T and CPT symmetries (in addition to ) can be performed.

The combined analysis of DANE , CPLEAR, E731 and NA31 experiments will allow to disentangle the CPT-violating contributions in decay amplitudes from those in mass matrix and also the transitions can be singled out. All the real parts of the parameters could be bounded up to while a lower sensitivity (--) is expected for the imaginary parts.

More doubtful is the situation for decays. We think that the CP-conserving decays will certainly be measured and some information on the rescattering phases could be obtained. For the CP-violating ones we observe, without doing a complete statistical analysis, that the shape of the interference effect of Fig. 4 is very characteristic and could easily be detected over a flat background. The analysis of the possibility to measure leads to similar conclusions.

Also the recent suggestions on possible quantum mechanics violations might be tested at DANE .

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Appendix

The KLOE detector measures the positions and of the two decay vertices. To compare the theoretical expressions of time dependent asymmetries with the experimental distributions, the decay times and introduced in the text must be transformed to the corresponding decay distances. Usually this was simply done by using the classical equations of motion, i.e. through the identity

where is the mean velocity of the two kaons and . This leads to the replacement , as in [20].

Recently, the possibility that the different classical velocities of and could modify Eq. (A.1) has been suggested in Ref. [42]. However, such conclusion seems to follow from an incomplete account of the particle quantum properties, which ultimately overlooks the distinction between the possible values of a quantum observable and the corresponding classical value. Indeed, we present here the proof of Eq. (A.1) based on the description of the state, in terms of localized wave-packets, of Ref. [43].

The plane wave state of eq. (1), is replaced with a wave packet peaked at and similarly for the other state, :

We take, for simplicity, gaussian wave packets, but any other, well-localized, function would give similar results. At time t, one has:

where ellipses represent further terms in the expansion, which can be neglected for a sufficiently narrow packet.

We have to introduce a complex to describe an unstable particle. The relativistically invariant propagator for a weakly unstable particle reads:

from which, at the pole, we derive:

The gaussian integral is easily performed by completing the square, with the result:

where

g is a gaussian wave-packet with a time-dependent size:

and m is the particle mass. With respect to the stable particle case, there is a contribution to the real part of A from the particle width which, in our case, turns out to be very small.

We consider now the two particle state at time t=0, replacing the plane wave state given in eq. (1) of the text with:

where the subscripts indicate which particle ( or ) is in which wave packet. At later times, the two packets evolve independently, according to the rules given above.

The amplitude for a pair to appear at time and location , and a pair at time and location , is given by:

(we are assuming and to be on opposite sides of the interaction region, so that the amplitude that, say, the particle in gives rise to an event at is negligible). According to the previous discussion:

and similarly for the other cases. Numerically, in the decay , with approximately at rest:

The total number of events (for one initial pair) with the pair in and the pair in is found by integrating the probability over and :

In the direct terms, we may replace each factor by a function, thereby obtaining, after integration, the result of eq. (A.1). In the interference term, we can do the same, provided that the difference in the peak position of g and , due to the different velocities of and , is negligible with respect to the width of the wave-packet:

The l.h.s of the inequality is less than a Fermi for , which is where the interference term starts being suppressed by decay. Neglecting the velocity difference in the interference is thus justified in all the region of interest, for any reasonable value of the size of the wave-packet. Therefore the total number of events with the pair at distance from the interaction vertex is given by:

with .

Considerations leading to analogous results, for the case of single Kaon and B oscillations, have been recently presented in Ref. [44]. A few comments. i) In the case of stable particles, e.g. solar neutrino oscillations, we may reach sufficiently large distances, where the overlap factor:

vanishes. In the far-distant region, coherent oscillations would disappear to gives rise to an incoherent superposition of a and beam.

ii) the velocity dependent terms in the exponents have reconstructed the correct Lorentz factors. We have obtained exponents of the form:

where is the particle proper time, as one could have expected on general ground.

iii) One may have preferred to assume exactly vanishing total momentum, rather then independently distributed momenta for the two particles, as done in eq. (A.3). This, however, is incorrect. A zero- momentum state is translation invariant. If this were the case, the intensity would depend only upon the difference, , which is obviously wrong. Electrons and positrons annihilate in the interaction region and not elsewhere, and the interaction region is small with respect to the general dimensions of the experiment. As a consequence, must depend from the individual distances of the two points from the interaction region, as in eq. (A.5).

Clearly, the initial wave packet may be not in the factorized form of eq. (A.3), generally will be a superposition of all the possible factorized forms (one possibility is to factorize the c.m. and the relative motion). However an explicit calculation shows that the result of eq. (A.6) holds for any form of the initial state, provided it satisfies the physical constraints: and it has reasonable widths. Should the result depend on more particular assumptions, the outcome experiment would be entirely unpredictable.