Eur. Phys. J. C (2024) 84: 699
https://doi.org/10.1140/epjc/s10052-024-12942-z

Regular Article – Theoretical Physics

$\kappa$-Dirac stars

¹ School of Physical Science and Technology, Lanzhou University, 730000, Lanzhou, China
² Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, and Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, 730000, Lanzhou, China
³ Institute of Theoretical Physics and Research Center of Gravitation, Lanzhou University, 730000, Lanzhou, China
⁴ School of Physics and Technology, Nantong University, 226019, Nantong, China

^e yqwang@lzu.edu.cn

Received: 25 April 2024
Accepted: 22 May 2024
Published online: 15 July 2024

Abstract

In this paper, we construct a Dirac star model composed of $|\kappa |$ pairs of spinor fields. The azimuthal harmonic indices m of these spinor fields are half-integers, and they satisfy $-(|\kappa |-\frac{1}{2})\le m\le |\kappa |-\frac{1}{2}$. When $\kappa =1$, it corresponds to the conventional Dirac star model, formed by two spinor fields with $m=\frac{1}{2}$ and $m=-\frac{1}{2}$. When $|\kappa |>1$, among these $|\kappa |$ pairs of spinor fields, there exist spinor fields with azimuthal harmonic indices $m>\frac{1}{2}$, and all spinor fields still conform to the same Dirac field equation. Different families of solutions are distinguished by the value of $\kappa$, so we named these solutions $\kappa$-Dirac stars. We obtain solutions for $\kappa =\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6$ by using numerical methods. Additionally, we compute their ADM mass M, Noether charge Q, and binding energy E, and illustrate how these quantities change with the spinor field’s frequency $\omega$ for different $\kappa$. We observe significant differences between solutions for $|\kappa |>1$ and the $|\kappa |=1$ case. Furthermore, we provide the energy density distribution of the Dirac stars, wherein for $|\kappa |>1$ scenarios, the Dirac stars exhibit a spherical shell-like structure. Moreover, we employ three-dimensional diagrams to intuitively depict how $\kappa$ influences the combination of spinor fields to form a spherically symmetric configuration.