Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions

• URL: http://arxiv.org/abs/2004.04011v1
• Date: Wed, 8 Apr 2020 14:29:36 GMT
• Title: Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions
• Authors: Alexander Felski, Alireza Beygi and S. P. Klevansky
• Abstract summary: We present a non-Hermitian PT-symmetric extension of the Nambu--Jona-Lasinio model of quantum chromodynamics in 3+1 and 1+1 dimensions. 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.
• Score: 68.8204255655161
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $H_{\textrm{NJL}} = \bar\psi (-i \gamma^k \partial_k + m_0) \psi - G [ (\bar\psi \psi)^2 + (\bar\psi i \gamma_5 \vec{\tau} \psi)^2 ]$ is extended by the non-Hermitian, PT- and chiral-symmetric bilinear term $ig\bar\psi \gamma_5 B_{\mu} \gamma^{\mu} \psi$; in 1+1 dimensions, where $H_{\textrm{NJL}}$ is a form of the Gross-Neveu model, it is extended by the non-Hermitian PT-symmetric but chiral symmetry breaking term $g \bar\psi \gamma_5 \psi$. 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 {\it 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 $m_0=0$ 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.

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