On limitation of mass spectrum in non-Hermitian $\cal PT$-symmetric models with the $\gamma_5$-dependent mass term

Abstract

The modified Dirac equations for the massive particles with the replacement of the physical mass $m$ with the help of the relation $m\rightarrow m_1 + \gamma_5 m_2$ are investigated. It is shown that for a free fermion theory with a $\gamma_5$ mass term, the finiteness of the mass spectrum at the value $ m_{max}= {m_1}^2/2m_2$ takes place. In this case the region of the unbroken $\cal PT$-symmetry may be expressed by means of the simple restriction of the physical mass $m\leq m_{max}$. Furthermore, we have that the areas of unbroken $\cal PT$-symmetry $m_1\geq m_2\geq 0$, which guarantees the reality values of the physical mass $m$, consists of three different parametric subregions: i) $0\leq m_2 < m_1/\sqrt{2}$, \,\,ii) $m_2=m_1/\sqrt{2}=m_{max},$ \,\,(iii)$m_1/\sqrt{2}< m_2 \leq m_1$. It is vary important, that only the first subregion (i) defined mass values $m_1,m_2,$ which correspond to the description of traditional particles in the modified models, because this area contain the possibility transform the modified model to the ordinary Dirac theory. The second condition (ii) is defined the "maximon" - the particle with maximal mass $m=m_{max}$. In the case (iii) we have to do with the unusual or "exotic" particles for description of which Hamiltonians and equations of motion have no a Hermitian limit. The formulated criterions may be used as a major test in the process of the division of considered models into ordinary and "exotic fermion theories".