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

We investigate the effects of non-Hermiticity on interacting fermionic systems. We do this by including non-Hermitian bilinear terms into the 3 + 1 dimensional Nambu-Jona-Lasinio (NJL) model. Two possible bilinear modifications give rise to PT-symmetric theories; this happens when the standard NJL model is extended either by a pseudovector background field ig (psi) over bar gamma B-5(mu)gamma(mu)psi or by an antisymmetric-tensor background field g (psi) over barF(mu nu)gamma(mu)gamma(nu)psi. The three remaining bilinears are anti-PT-symmetric in nature, ig (psi) over barB(mu)gamma(mu)psi, ig (psi) over bar gamma(5)psi and ig (psi) over bar1 psi, so that the Hamiltonian then has no overall symmetry. The pseudovector ig (psi) over bar gamma B-5(mu)gamma(mu)psi and the vector ig (psi) over bar gamma B-5(mu)gamma(mu)psi combinations, are, in addition, chirally symmetric. Thus, within this framework we are able to examine the effects that the various combinations of non-Hermiticity, PT symmetry, chiral symmetry, and the two-body interactions of the NJL model have on the existence and dynamical generation of a real effective fermion mass (a feature which is absent in the corresponding modified massless free Dirac models) as well as on the masses of the composite particles, the pseudoscalar and scalar mesonic modes (pi and sigma mesons). Our findings demonstrate that PT symmetry is not necessary for real fermion mass solutions to exist; rather, the two-body interactions of the NJL model supersede the non-Hermitian bilinear effects. The effects of chiral symmetry are evident most clearly in the meson modes, the pseudoscalar of which will always be Goldstone in nature if the system is chirally symmetric. Second solutions of the mesonic equations are also discussed.