Restriction of the fermion mass spectrum in $PT$-symmetric systems and its implications for studying dark matter

We formulate principal positions of a non-Hermitian model with a $\gamma_5$-extension of the fermion mass, which are often neglected in investigating this subject. A consistent approach to this problem requires the constraint $m\le M$, where $M$ bounds the entire fermion mass spectrum. An analogous approach was proposed in the geometric model, which can be regarded as the first $PT$-symmetric non-Hermitian fermion model with a $\gamma_5$-extension of mass. Exotic particles appear in both these theories. A detailed consideration of the properties of these particles allows conjecturing that they are possible candidates in the structure of dark matter. We also discuss a simple estimate for determining the maximum admissible value of the fermion mass $M$.