Abstract
The neutral Kaon system has both CP violation in the mass matrix and a non-vanishing lifetime difference in the width matrix. This leads to an effective Hamiltonian which is not a normal operator, with incompatible (non-commuting) masses and widths. In the Weisskopf–Wigner Approach (WWA), by diagonalizing the entire Hamiltonian, the unphysical non-orthogonal “stationary” states KL,S are obtained. These states have complex eigenvalues whose real (imaginary) part does not coincide with the eigenvalues of the mass (width) matrix. In this work we describe the system as an open Lindblad-type quantum mechanical system due to Kaon decays. This approach, in terms of density matrices for initial and final states, provides a consistent probabilistic description, avoiding the standard problems because the width matrix becomes a composite operator not included in the Hamiltonian. We consider the dominant decay channel to two pions, so that one of the Kaon states with definite lifetime becomes stable. This new approach provides results for the time dependent decay rates in agreement with those of the WWA.
Highlights
The neutral Kaon system has both CP violation in the mass matrix and a non-vanishing lifetime difference in the width matrix
The simultaneous presence of CP violation in the mass matrix M and a difference of lifetimes in the antihermitian matrix iΓ/2 leads to a quantum incompatibility between M and Γ, M, Γ = 0, i.e. one cannot define states of definite mass and lifetime simultaneously, because H is not a normal operator
Ignoring CPT Violation and CP violation in the decay, the choice of a real B leads to the result that the K1,2 states are the ones with definite lifetimes, so that the width operator in the |K1,2 basis is given by the following 2 × 2 diagonal matrix with eigenvalues 0 and γ: ΓWWA =
Summary
The neutral Kaon system has both CP violation in the mass matrix and a non-vanishing lifetime difference in the width matrix.
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