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arXiv 2004.04011hep-phhep-thmath-phmath.MPquant-ph
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Nambu-Jona-Lasinio模型在3+1和1+1维中的非Hermite推广

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

Alexander Felski, Alireza Beygi, S. P. Klevansky

备注：
10 pages, 6 figures
发表信息：
Phys. Rev. D 101, 116001 (2020)
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摘要：

本文给出了3+1和1+1维量子色动力学Nambu-Jona-Lasinio（NJL）模型的非厄米PT对称推广。在3+1维中，SU（2）对称NJL哈密顿量$H_{\textrm{NJL}} = \barψ(-i γ^k \partial_k + m_0) ψ- G [ (\barψψ)^2 + (\barψi γ_5 \vecτ ψ)^2 ]$被非厄米的、PT对称和手征对称的双线性项$ig\barψγ_5 B_μ γ^μ ψ$扩展;在1+1维中，其中$H_{\textrm{NJL}}$是Gross-Neveu模型的一种形式，它被非厄米的PT对称但手征对称性缺项$g \barψγ_5 ψ$扩展。在每种情况下，间隙方程推导和非厄米项上产生的质量的影响进行了研究。我们有几个发现：在以前的计算中，自由狄拉克方程被修改为包括非厄米双线性项，与预期相反，在手征极限中不能获得真实的质谱;在这些情况下，非零的裸费米子质量对于在未破缺区域实现PT对称性是必不可少的。在这里，在NJL模型中，其中四点相互作用存在，我们{ \it do}发现，在3+1和1+1维中，至少对于非厄米特耦合$g$的某些特定值，在裸质量消失的极限下，质谱也是真实的。因此，四点相互作用覆盖了导致这些参数值的PT破坏的影响。此外，我们发现，在这两种情况下，在3+1和1+1维，包含一个非厄米双线性项可以产生的质量。在这两个模型中，这种贡献可以被调整到很小;因此，我们将费米子质量固定到它在没有非厄米项的情况下$m_0=0$时的值，然后确定所需的耦合值，以便产生裸费米子质量。

英文摘要：

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 $H_{\textrm{NJL}} = \barψ(-i γ^k \partial_k + m_0) ψ- G [ (\barψψ)^2 + (\barψi γ_5 \vecτ ψ)^2 ]$ is extended by the non-Hermitian, PT- and chiral-symmetric bilinear term $ig\barψγ_5 B_μ γ^μ ψ$; 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 $g \barψγ_5 ψ$. 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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