2 Test Performed in the Past.

Tests of quantum mechanics using regeneration have been proposed long ago, [1]. They consist of transporting a neutral kaon beam over a distance long enough for all its --component to decay; letting it impinge on a piece of material called regenerator; and recording the number of decays into two charged pions as a function of distance to that regenerator.

There are two processes by which a kaon emerging from the regenerator in the very forward direction can decay into :
1) plain transmission of a and decay via the CP violating mode;
2) regeneration followed by a CP conserving decay.
The initial can be considered as a pure state in each momentum interval, if that interval is small enough. Because the kaon has spin zero, all configurations of its --decays can be considered as two pions in an S--wave state, which is a pure state. Thus, at each momentum, the two-pion state from the decay of a kaon in the very forward direction is a pure state. Whether the kaon is transmitted or coherently regenerated, the state of the regenerator is the same, after as well as before the kaon traversed it. Therefore the two processes interfere between one another. As a function of proper time t, the decay rate can be written as

where and are the total decay rates of the and of the mesons, respectively; is the -- mass difference; and , , , and are parameters that can be measured by the shape of the time distribution .

The third term in Eq. (4) is the interference term between the two processes. Quantum theory predicts . In accordance with Eq. (3), we define

which measures the amount of violation in the regeneration process.

A test of this kind was performed with neutral kaons of momentum ranging from 4 to 10 GeV/c and a carbon regenerator, [2]. The result was

The error was essentially statistical.

The same type of test could be performed at Dane . The accuracy would be improved. For instance, one may surround one of the two intersection points with a regenerator of a cylindric form, with its axis along the beam line, with a radius of 20 cm and a height of 20 cm too. The regenerator would intercept emitted within a solid angle of about 5 sr. The thickness of the cylinder should be adjusted to produce, just behind the regenerator, more --decays by regenerated than by transmitted . Then one makes sure that the -- interference pattern visible in the --mode is as long as possible. For instance, a beryllium cylinder 2.4 cm thick would be an adequate regenerator. It would regenerate one in 10 according to Ref. [3], i.e. an appropriate amount of regenerated .

At a luminosity , in one month calendar time with an efficiency of 30%, i.e. an integrated luminosity of 1000 pb, one may expect more than 10 \ through the regenerator according to Refs. [4] and [5]. Then the decoherence parameter would be measured with a statistical error of less 1%. If the angle of the kaon emerging from the regenerator can be determined to better than 20 mr, the effect of the incoherent regeneration and elastic scattering should introduce an uncertainty of less than 1%. Finally one should to be able to eliminate background in --decays down to less than 10 times the decay rate into \ at a machine like Dane \ designed to measure the CP violation parameter called . Therefore combining statistical and systematic uncertainties, the final error should of the order of 1%.

Since the value of of Ref. [2] is 1.5 standard deviation away from zero, it would be significant to repeat that test with an error of, let us say, 1%. Of course, this test is not one that can be considered as unique at a --factory. However it would establish a limit for spontaneous decoherence in regeneration of kaons comparable to the limit of 0.6% obtained for decoherence in neutron interferometry, [6].