Subsystem of neutral mesons beyond the Lee–Oehme–Yang approximation

We begin with a discussion of the general properties of eigenvectors and eigenvalues for an effective Hamiltonian governing time evolution in a two state subspace of the state space of the total system under consideration. Next, the Lee, Oehme and Yang (LOY) theory of time evolution in such a subspace is considered. The CPT– and CP–symmetry properties of the LOY effective Hamiltonian are discussed. Next the CPT transformation properties of the exact effective Hamiltonian for two state subspace are discussed. Using the Khalfin Theorem we show that the diagonal matrix elements of the exact effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for t > t0 (t0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. (Suitable matrix elements of the LOY effective Hamiltonian are equal in such a case). The unusual consequence of this result is that, contrary to the properties of stable particles, the masses of the unstable particle ”1” and its antiparticle ”2” need not be equal for t ≫ t0 in the case of preserved CPT and violated CP symmetries. Also, basic assumptions necessary for the proof of the CPT Theorem are discussed. It is found that the CPT Theorem is not valid for a physical system with unstable particles decaying exponentially. From this property the conclusion is drawn that CPT–transformation cannot be a symmetry in a system which contains the LOY model as a subsystem, and, thus this model is shown to be incapable of describing possible CPT–violation effects correctly. Using an exact equation governing the time evolution in the subspace of the total state space we show that there exists an approximation which is more accurate than the LOY approximation, and which leads to an effective Hamiltonian whose diagonal matrix elements posses properties consistent with the conclusions obtained for the exact effective Hamiltonian. Using this more accurate approximation we show that the interpretation of the tests measuring the difference between the e–mail: [email protected]; [email protected]