Non-Hermitian extension of the Nambu–Jona-Lasinio model in 3+1 and 1+1 dimensions
| Autor: | Alireza Beygi, S. P. Klevansky, Alexander Felski |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Quantum chromodynamics
Physics 010308 nuclear & particles physics High Energy Physics::Phenomenology Fermion 01 natural sciences Hermitian matrix symbols.namesake Dirac equation Nambu–Jona-Lasinio model 0103 physical sciences symbols Symmetry breaking 010306 general physics Chiral symmetry breaking Hamiltonian (quantum mechanics) Mathematical physics |
| Zdroj: | Physical Review D. 101 |
| ISSN: | 2470-0029 2470-0010 |
| Popis: | This paper presents a non-Hermitian $\mathcal{PT}$-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 ${\mathcal{H}}_{\mathrm{NJL}}=\overline{\ensuremath{\psi}}(\ensuremath{-}i{\ensuremath{\gamma}}^{k}{\ensuremath{\partial}}_{k}+{m}_{0})\ensuremath{\psi}\ensuremath{-}G[(\overline{\ensuremath{\psi}}\ensuremath{\psi}{)}^{2}+(\overline{\ensuremath{\psi}}i{\ensuremath{\gamma}}_{5}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\tau}}\ensuremath{\psi}{)}^{2}]$ is extended by the non-Hermitian, $\mathcal{PT}$- and chiral-symmetric bilinear term $ig\overline{\ensuremath{\psi}}{\ensuremath{\gamma}}_{5}{B}_{\ensuremath{\mu}}{\ensuremath{\gamma}}^{\ensuremath{\mu}}\ensuremath{\psi}$; in $1+1$ dimensions, where ${\mathcal{H}}_{\mathrm{NJL}}$ is a form of the Gross-Neveu model, it is extended by the non-Hermitian $\mathcal{PT}$-symmetric but chiral symmetry breaking term $g\overline{\ensuremath{\psi}}{\ensuremath{\gamma}}_{5}\ensuremath{\psi}$. 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 $\mathcal{PT}$ 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 $\mathcal{PT}$ 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. Finally, we find that in both cases, a rich phase structure emerges from the gap equation as a function of the coupling strengths. |
| Databáze: | OpenAIRE |
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