CPT symmetry and properties of the exact and approximate effective Hamiltonians for the neutral kaon complex

Abstract:
We start from a discussion of  the general form and general CP- and CPT-transformation properties of the Lee-Oehme-Yang (LOY) effective Hamiltonian for  the neutral kaon complex. Next we show that there exists an approximation which is more accurate than the LOY, and which leads to an effective Hamiltonian whose diagonal matrix elements posses CPT transformation properties, which differ from those of the LOY effective Hamiltonian. These properties of the mentioned effective Hamiltonians are compared with the properties of the exact effective Hamiltonian for the neutral kaon complex. 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 $t > t_{0}$ ($t_{0}$ 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 of type $<{\bf j}|\exp (-itH)|{\bf k}>$ (j=1,2) for transitions $|{\bf 1}> \rightarrow |{\bf 2}>$, $|{\bf 2}> \rightarrow |{\bf 1}>$ together with  an identity expressing the effective Hamiltonian by these amplitudes and their derivatives with respect to time $t$. (Here $|{\bf j}>$, $|{\bf k}>$,  $j,k =1,2,$ denote the states of type $|K_{0}>,$ $|{\overline{K}}_{0}>$, $H$ is the total Hamiltonian of the system considered). 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. It appears that the mentioned more accurate approximation  than the LOY, leads to an effective Hamiltonian whose diagonal matrix elements posses properties consistent with the conclusions for the exact effective Hamiltonian described above.