3 decays

The decay of the is in principle not forbidden by CP conservation. It is so, however, at the centre of the Dalitz plot, where symmetry in pion momenta forbids the appearance of odd values of the relative angular momenta. decay of is altogether CP violating.

Taking into account the mass-matrix mixing effect, one writes:

and similarly for the decay , with an analogous definition of and .

Theoretical analyses of the direct CP violation parameters, , proceed along much similar lines to those discussed previously for charged kaon decays (see, e.g., Refs. [4,7,8]). In lowest order , one finds [4]:

The more recent discussions [7,8,20] include the electroweak penguin contributions, as well as higher order chiral corrections. Analogously to the charged kaon case, it seems unlikely that the lowest order prediction (33) can be increased by more than one order of magnitude. In this case, the branching ratio is predicted to be:

to be compared with the present experimental limit [21]

For , the problem is the presence of the CP conserving amplitude which can obscure the CP violating one. Since CP conserving and CP violating amplitudes do not interfere in the total rate due to their different dependences on the kinematical variables, the effect of (mass-mixing) CP violation in the rate for this channel is of order , bringing it down to the level of .

A possible strategy to search for this effect at DANE should be the measurement of the time interference between and amplitudes, as discussed in [2]. Concerning the expected sensitivity on , it is difficult to assess it without a detailed simulation. As an indication, on statistical grounds the sensitivity should be of the order of , which is rather far from the expected value . On the other hand, the most recent experimental findings are [22]:

from an unconstrained fit to the data, and

if the theoretical constraint is used in such a fit. Thus, it might still be worthwhile to make an attempt to obtain an alternative experimental limit at DANE.

In the decay , the mass-mixing CP violating amplitude is suppressed by an angular momentum barrier and by the rule, analogously to the CP conserving amplitude of the decay into the same channel. This CP violation should manifest itself in the appearance of the term linear in X, i.e. , in the Dalitz plot (1). This term should arise from the interference between CP-even and CP-odd amplitudes. According to the estimate of [23], the order of magnitude of the CP-violating coefficient j should be , while the present determination is [21]. Thus, the statistics at DANE would be somewhat marginal to the predicted value for this parameter. Nonetheless, a significant improvement of the present experimental value might still be possible.