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

TL;DR

This paper investigates the mass spectrum limitations in non-Hermitian PT-symmetric Dirac models with a gamma_5-dependent mass term, identifying regions of unbroken PT-symmetry and classifying particles as ordinary or exotic based on model parameters.

Contribution

It introduces a criterion for the mass spectrum limitation in PT-symmetric models with gamma_5 mass terms and classifies particle types based on parameter regions.

Findings

Mass spectrum is finite with a maximum mass m_max = m_1^2 / 2m_2.

Unbroken PT-symmetry region is defined by m ≤ m_max.

Identification of ordinary and exotic fermions based on parameter subregions.

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".