Non-Hermitian extension of the Nambu-Jona-Lasinio model in 3+1 and 1+1 dimensions - Felski, Alexander; Beygi, Alireza; Klevansky, Sandra Pamela (Physical review, 2020)
This paper presents a non-Hermitian PT-symmetric extension of the Nambu-Jona-Lasinio (NJL) model of quantum chromodynamics in 3+1 and 1+1 dimensions. In 3+1 dimensions, the SU(2)-symmetric NJL Hamiltonian HNJL=¯ψ(−iγk∂k+m0)ψ−G[(¯ψψ)2+(¯ψiγ5→τψ)2] is extended by the non-Hermitian, PT- and chiral-symmetric bilinear term ig¯ψγ5Bμγμψ; in 1+1 dimensions, where HNJL is a form of the Gross-Neveu model, it is extended by the non-Hermitian PT-symmetric but chiral symmetry breaking term g¯ψγ5ψ. In each case, the gap equation is derived, and the effects of the non-Hermitian terms on the generated mass are studied. We have several findings: in previous calculations for the free Dirac equation modified to include non-Hermitian bilinear terms, contrary to expectation, no real mass spectrum can be obtained in the chiral limit. In these cases, a nonzero bare fermion mass is essential for the realization of PT symmetry in the unbroken regime. Here, in the NJL model, in which four-point interactions are present, we do find real values for the mass spectrum also in the limit of vanishing bare masses in both 3+1 and 1+1 dimensions, at least for certain specific values of the non-Hermitian couplings g. Thus, the four-point interaction overrides the effects leading to PT symmetry breaking for these parameter values. Further, we find that in both cases, in 3+1 and in 1+1 dimensions, the inclusion of a non-Hermitian bilinear term can contribute to the generated mass. In both models, this contribution can be tuned to be small; we thus fix the fermion mass to its value when m0=0 in the absence of the non-Hermitian term, and then determine the value of the coupling required so as to generate a bare fermion mass. Finally, we find that in both cases, a rich phase structure emerges from the gap equation as a function of the coupling strengths.
Gesehen am 18.03.2021