On masses of unstable particles and their antiparticles in the CPT-invariant system

Abstract

We show that the diagonal matrix elements of the effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at $t > t_{0}$ ($t_{0}$ is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. To achieve this we use the transition amplitudes for transitions $|{\bf 1}> \to |{\bf 2}>$, $|{\bf 2}> \to |{\bf 1}>$ together with the identity expressing the effective Hamiltonian by these amplitudes and their derivatives with respect to time $t$. This identity must be fulfilled by any effective Hamiltonian (both approximate and exact) derived for the two state complex. 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 $t \gg t_{0}$ in the case of preserved CPT and violated CP symmetries.