Probing CPT violation in meson mixing by non-cyclic phase

The presence of non-cyclic phases is revealed in the time evolution of mixed meson systems. Such phases are related to the parameter $z$ describing the $CPT$ violation; moreover, a non zero phase difference between particle and antiparticle arises only in presence of $CPT$ symmetry breaking. Thus, a completely new test for the $CPT$ invariance can be provided by the study of such phases in mixed mesons. Systems which are particularly interesting for such an analysis are the $B_{s}^{0}-ar{B}_{s}^{0}$ and the $K^{0}-ar{K}^{0}$ ones. In order to introduce non-cyclic phases, some aspects of the formalism describing the mixed neutral mesons are analyzed. Since the effective Hamiltonian of systems like $K^{0}-ar{K}^{0}$, $B^{0}-ar{B}^{0}$, $B_{s}^{0}-ar{B}_{s}^{0}$, $D^{0}-ar{D}^{0}$ is non-Hermitian and non-normal, it is necessary to diagonalize it by utilizing the rules of non-Hermitian quantum mechanics.