TIME EVOLUTION OF $K^0-\bar{K^0}$ SYSTEM IN SPECTRAL FORMULATION

Abstract

We reanalyse the time evolution of the $K^0-\bar{K^0}$ system in the language of certain spectral function whose Fourier transforms give the time dependent survival and transition amplitudes. The reanalysis turned out to be necessary in view of the astonishing theorem by Khalfin on the possibility of vacuum regeneration of $K_S$ and $K_L$. The main reason for this unexpected behaviour is the non-orthogonality of $\ks$ and $\kl$. As a result of this theorem new contributions to the well known oscillatory terms will enter the time dependent transition probabilities. These new terms are not associated with small/large time behaviour of the amplitudes and therefore their magnitude is apriori unknown. Approximating the spectral functions by an one-pole ansatz Khalfin estimated the new effect in transition probabilities to be $4 \times 10^{-4}$. Whereas we agree with Khalfin on the general existence of vacuum regeneration of $K_S$ and $K_L$ we disagree on the size of the effect. A careful analysis of the one-pole approximation reveals that the effect is eleven orders of magnitude smaller than Khalfin's estimate and, in principle, its exact determination lies outside the scope of the one-pole ansatz. The present paper gives also insight into the limitation of the validity of one-pole approximation, not only for small/large time scales, but also for intermediate times where new effects, albeit small, are possible. It will be shown that the same validity restrictions apply to the known formulae of Weisskopf-Wigner approximation as well.