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

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 $H_{\text{NJL}}=\bar{\psi }(-i{\gamma }^k{\partial }_k+m_0)\psi -G[(\bar{\psi }\psi {)}^2+(\bar{\psi }i{\gamma }_5\vec{\tau }\psi {)}^2]$ 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 $H_{\text{NJL}}$ 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.

10 pages, 6 figures

References in corpus (3)

• Symmetries and conservation laws in non-Hermitian field theories
• Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions
• Continuous quantum phase transition in the fermionic mass solutions of the Nambu-Jona-Lasinio model

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