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Time Evolution of K0 - anti K0, System in Spectral Formulation

Marek Nowakowski

Published Feb 4, 1996• 0 citations
Publisher

Abstract

We reanalyse the time evolution of the K0 -`K0 system in the language of certain spectral function whose Fourier transforms give the time dependent survival and transition amplitudes. Approximating the spectral function by an one-pole ansatz we give 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 restriction apply to the known formulae of Weisskopf-Wigner approximation as well. The present analysis can also be applied to the effect of vacuum regeneration of KL and KS, a possibility pointed out by Khalfin. As a result of this possible new contributions to the well known oscillatory terms will enter the time depend transition probanilities. These new terms are not associated with small/large time behaviour of the amplitudes and therefore their magnitude is apriori unknown. It will be shown that the order of magnitude of this new effect is very small and, in principle, its exact determination lies outside the scope of one-pole ansatz.

DOI
10.15161/oar.it/hkcr9-xsj94

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We reanalyse the time evolution of the K0 -`K0 system in the language of certain spectral function whose Fourier transforms give the time dependent survival and transition amplitudes. Approximating the spectral function by an one-pole ansatz we give 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 restriction apply to the known formulae of Weisskopf-Wigner approximation as well. The present analysis can also be applied to the effect of vacuum regeneration of KL and KS, a possibility pointed out by Khalfin. As a result of this possible new contributions to the well known oscillatory terms will enter the time depend transition probanilities. These new terms are not associated with small/large time behaviour of the amplitudes and therefore their magnitude is apriori unknown. It will be shown that the order of magnitude of this new effect is very small and, in principle, its exact determination lies outside the scope of one-pole ansatz.
TIME EVOLUTION OF K0−K0ˉK^0-\bar{K^0}K0−K0ˉ SYSTEM IN SPECTRAL FORMULATION
We reanalyse the time evolution of the K0−K0ˉK^0-\bar{K^0}K0−K0ˉ 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 KSK_SKS​ and KLK_LKL​. 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 4×10−44 \times 10^{-4}4×10−4. Whereas we agree with Khalfin on the general existence of vacuum regeneration of KSK_SKS​ and KLK_LKL​ 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.
BEYOND THE WEISSKOPF–WIGNER APPROXIMATION IN THE K0−Kˉ0K^0-\bar K^0K0−Kˉ0 SYSTEM
We reanalyze the time evolution of the [Formula: see text] system in the language of certain spectral function whose Fourier transforms give the time-dependent survival and transition amplitudes. Approximating the spectral function by a one-pole ansatz we obtain insight into the limitation of the validity of the one-pole approximation. It will be shown that the same validity restrictions apply to the known formulae of the Weisskopf–Wigner approximation as well. The present analysis can also be applied to the effect of so-called vacuum regeneration of KL and KS. As a result of this possibility new contributions to the well-known oscillatory terms will enter the time-dependent transition probabilities. It will be shown that the order of magnitude of this new effect is very small and, in principle, its exact determination lies outside the scope of the one-pole ansatz.
We establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation, the evolution of a total system subspace is not an exact semigroup for multichannel decay unless the projectors into eigenstates of the reduced evolution generator W(z)W(z)W(z) are orthogonal. With multichannel decay, the projectors must be evaluated at different pole locations ${z}_{\ensuremath{\alpha}}\ensuremath{\ne}{z}_{\ensuremath{\beta}}$, and since the orthogonality relation does not generally hold at different values of zzz, the semigroup evolution is a poor approximation for the multichannel decay, even for very weak coupling. Nevertheless, if the theory is generalized to take into account interactions with an environment, one can ensure orthogonality of the W(z)W(z)W(z) projectors regardless of the number of poles. Such a possibility occurs when W(z)W(z)W(z), and hence its eigenvectors, is independent of zzz, which corresponds to the Markovian limit of the coupling to the continuum spectrum.
The authors study the regeneration contribution to the decay law in Wigner-Weisskopf theory for times less than and up to the golden rule time. A power series expansion for the regeneration term and the part of the product of the amplitudes which has the semigroup property is carried out in second-order perturbation theory, the same order to which the Wigner-Weisskopf calculation is carried out in their estimate of the line widths in atomic decay. They show that the regeneration contribution as a smaller leading behaviour in t than the amplitudes at times of the order of the golden rule time, thus accounting for an approximate semigroup behaviour, on this scale, within the framework of the Wigner-Weisskopf theory. For very short times, the estimates of Misra and Sinha (1977) are obtained.
The Fock-Krylov formalism for the calculation of survival probabilities of unstable states is revisited paying particular attention to the mathematical constraints on the density of states, the Fourier transform of which gives the survival amplitude. We show that it is not possible to construct a density of states corresponding to a purely exponential survival amplitude. The survival probability P(t)P(t)P(t) and the autocorrelation function of the density of states are shown to form a pair of cosine Fourier transforms. This result is a particular case of the Wiener-Khinchin theorem and forces P(t)P(t)P(t) to be an even function of time which in turn forces the density of states to contain a form factor which vanishes at large energies. Subtle features of the transition regions from the nonexponential to the exponential at small times and the exponential to the power-law decay at large times are discussed by expressing P(t)P(t)P(t) as a function of the number of oscillations nnn performed by it. The transition at short times is shown to occur when the survival probability has completed one oscillation. The number of oscillations depends on the properties of the resonant state and a complete description of the evolution of the unstable state is provided by determining the limits on the number of oscillations in each region.
We consider a model of a unstable state defined by the truncated Breit-Wigner energy density distribution function. An analytical form of the survival amplitude a(t)a(t)a(t) of the state considered is found. Our attention is focused on the late time properties of a(t)a(t)a(t) and on effects generated by the non--exponential behavior of this amplitude in the late time region: In 1957 Khalfin proved that this amplitude tends to zero as ttt goes to the infinity more slowly than any exponential function of ttt. This effect can be described using a time-dependent decay rate γ(t)γ(t)γ(t) and then the Khalfin result means that this γ(t)γ(t)γ(t) is not a constant but at late times it tends to zero as ttt goes to the infinity. It appears that the energy E(t)E(t)E(t) of the unstable state behaves similarly: It tends to the minimal energy EminE_{min}Emin​ of the system as t→∞t \to \inftyt→∞. Within the model considered we find two first leading time dependent elements of late time asymptotic expansions of E(t)E(t)E(t) and γ(t)γ(t)γ(t). We discuss also possible implications of such a late time asymptotic properties of E(t)E(t)E(t) and γ(t)γ(t)γ(t) and cases where these properties may manifest themselves.