Unitarity of Neutral Kaon System

Abstract

In neutral kaon system, we always use non-hermitian Hamiltonian for convenience of treating decay process, unitarity seems to be lost. If we take decay channels (\pi\pi, \pi\pi\pi, \pi\ell\nu etc.) into account, however, Hamiltonian of the whole system must be hermitian. We attempt to derive an effective Hamiltonian with respect to only K^0, $\bar{K}^0$ states, starting from the hermitian Hamiltonian. For brevity, we take only a \pi\pi state into account as the decay channel in this paper. We can not avoid an oscillation between K^0, $\bar{K}^0$ and \pi\pi states if we start from a hermitian Hamiltonian whose states all have discrete energy levels. We therefore treat the \pi\pi state more appropriately to have a continuous energy spectrum to achieve the decay of K^0, $\bar{K}^0$ into \pi\pi. As the consequence, we find a different time evolution from what we expect in the conventional method immediately after the decay starts, though it recovers Fermi's golden rule for long enough time scale.