6 Measurements at the factory

According to the previous considerations, accurate measurements of the Dalitz plot for the individual channels would allow a stringent test of the current theoretical description of the nonleptonic weak interaction, in particular of the chiral Lagrangian realization to order . The determination of isospin amplitudes requires, in general, the combination of data on both and decays. DANE is a source of pure kaon beams, free from background contaminations, which in principle should allow high statistical accuracy (see Tab. 1), provided that detection efficiencies for decay products are also high. However, in some cases the present experimental accuracy is mostly limited by systematic errors that must be decreased accordingly, in order to take advantage of the quality of kaon beams. Another source of difficulty, to be taken into account in analyses, is the correlations among, e.g., linear and quadratic Dalitz plot slopes, which relate the accuracies obtainable for these parameters. The case of is a special one, since the quadratic slope is not contaminated by the linear amplitude which is zero for this mode (see Eq. (14)), and electromagnetic corrections are not required.

The direct observation of many hundreds of decay events, with practically no background (see Tab. 1), should represent a significant achievement obtainable at DANE and, as pointed out previously, will allow to carefully test the component of the weak Hamiltonian.

Another interesting analysis of is suggested by the use of DANE as an `interferometer', where the time dependence of - interference in vacuum can be accurately studied. Since the initial state from decay is the antisymmetric superposition

where is the direction of the kaons momenta in the c.m. system, the subsequent and decays are correlated, and their quantum interferences show up in relative time distributions and time asymmetries, which are of great interest in order to test CP (and CPT) violation [44]. In addition, as an alternative to the direct observation, also the CP conserving amplitude and the final state interaction imaginary parts could be measured via the time dependent interference of this decay with [45,12].

Specifically, a convenient observable is represented by the transition rate for the initial state to decay into the final states at time and at time , respectively (in the following, and are understood to be the proper times). Defining the `intensity' of time correlated events as:

where and t is the time difference , making use of the exponential time-dependence of the mass eigenstates K and K, and integrating over the phase space, one easily finds, with the notations :

and

In Eqs. (40) and (41), , , and for the amplitudes the expansions (14) or (15) must be introduced. >From these equations one can notice the possibility to study, in general, also negative `times', which is peculiar of the factory [46].

An important aspect of the interference in (40) and (41) is that, besides the real parts, the (expectedly) small final state interaction imaginary parts appear linearly. Instead, the width depends quadratically on them and thus has less sensitivity to such effects. Accordingly, by a fit to the full t-dependence of the interference, both the real and the imaginary parts could be determined (or at least, for the latter ones, a significant upper bound could be obtained, depending on the available statistics). This would be a desirable achievement, in view of the discussion in Sec. 3.

Using Eq. (14) or (15), the interference terms are easily seen to drop from the intensities (40) and (41) integrated over the full Dalitz plot, giving the total event rates. Considering that in the CP conserving case the Dalitz plot distributions are even in X for all channels, the interference can be extracted by integrating the intensities (40) and (41) over the phase space with odd-X cuts.

For example, with the phase space element, one can define the asymmetries

and

Using Eq. (14) to expand and up to second order in the kinematical variables, one finds to a good approximation

where

The amplitude (and ) can be measured from the rates. Therefore, Eq. (44) shows that the separate determination of the and dependences allows the measurement of the amplitude , as well as of the rescattering relative phase . In particular, the leading order predictions in Eq. (35) indicate .

The expression for is directly obtained from (44) by the changes , and but, as a function of time, the denominator would quickly become large and suppress the interference.

Analogously, is given by the more complicated expression:

where

and

If separately determined, the coefficients of the and terms could be useful to constrain the value of the quadratic amplitude and the combination of imaginary parts in (47). In this regard, we can notice that the expectedly small has a large coefficient proportional to (see Tab. 4). In fact, to leading order in , the numerator in Eq. (47) must be of order (the same counting applies to ), so that for theoretical consistency , which is of order , should not be included. In that case, Eq. (47) simplifies considerably, and using Eq. (35) we would predict [12].

In conclusion, studies of the time-dependent interference described above should provide alternative measurements of the CP conserving amplitude, and eventually could also give indications on rescattering phases and test the relevant predictions. A quantitative discussion for DANE, taking into account also the background from the CP even () state due to the radiative decay , is presented in Ref. [44].