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We investigate the effects of non-Hermiticity on interacting fermionicsystems. We do this by including non-Hermitian bilinear terms into the 3+1dimensional Nambu--Jona-Lasinio (NJL) model. Two possible bilinearmodifications give rise to $\mathcal{PT}$ symmetric theories; this happens whenthe standard NJL model is extended either by a pseudovector background field$ig \bar\psi\gamma_5 B_\mu \gamma^\mu \psi$ or by an antisymmetric-tensorbackground field $g \bar\psi F_{\mu\nu}\gamma^\mu \gamma^\nu \psi$. The threeremaining bilinears are {\it anti}-$\mathcal{PT}$-symmetric in nature, $ig\bar\psi B_\mu \gamma^\mu \psi, ig\bar\psi \gamma_5 \psi$ and $ig\bar\psi{1}\psi$, so that the Hamiltonian then has no overall symmetry. Thepseudovector $ig \bar\psi\gamma_5 B_\mu \gamma^\mu \psi$ and the vector $ig\bar\psi B_\mu \gamma^\mu \psi$ combinations, are, in addition, chirallysymmetric. Thus, within this framework we are able to examine the effects thatthe various combinations of non-Hermiticity, $\mathcal{PT}$ symmetry, chiralsymmetry and the two-body interactions of the NJL model have on the existenceand dynamical generation of a real effective fermion mass (a feature which isabsent in the corresponding modified massless free Dirac models) as well as onthe masses of the composite particles, the pseudoscalar and scalar mesonicmodes ($\pi$ and $\sigma$ mesons). Our findings demonstrate that $\mathcal{PT}$symmetry is not necessary for real fermion mass solutions to exist, rather thetwo-body interactions of the NJL model supersede the non-Hermitian bilineareffects. The effects of chiral symmetry are evident most clearly in the mesonmodes, the pseudoscalar of which will always be Goldstone in nature if thesystem is chirally symmetric. Second solutions of the mesonic equations arealso discussed.

Fermion and meson mass generation in non-Hermitian Nambu--Jona-Lasinio models