Normal ground state of dense relativistic matter in a magnetic field

The properties of the ground state of relativistic matter in a magnetic field are examined within the framework of a Nambu-Jona-Lasinio model. The main emphasis of this study is the normal ground state, which is realized at sufficiently high temperatures and/or sufficiently large chemical potentials. In contrast to the vacuum state, which is characterized by the magnetic catalysis of chiral symmetry breaking, the normal state is accompanied by the dynamical generation of the chiral shift parameter $\Delta$. In the chiral limit, the value of $\Delta$ determines a relative shift of the longitudinal momenta (along the direction of the magnetic field) in the dispersion relations of opposite chirality fermions. We argue that the chirality remains a good approximate quantum number even for massive fermions in the vicinity of the Fermi surface and, therefore, the chiral shift is expected to play an important role in many types of cold dense relativistic matter, relevant for applications in compact stars. The qualitative implications of the revealed structure of the normal ground state on the physics of protoneutron stars are discussed. A noticeable feature of the $\Delta$ parameter is that it is insensitive to temperature when $T\ll {\mu }_0$, where ${\mu }_0$ is the chemical potential, and increases with temperature for $T>{\mu }_0$. The latter implies that the chiral shift parameter is also generated in the regime relevant for heavy ion collisions.

Figure 1

Diagrammatic form of the gap equation in the Hartree-Fock (mean-field) approximation.

Figure 2

The dependence of the constituent mass $m$ on the bare mass $m_0$ at ${\mu }_0=0$ for several fixed values of temperature: $T=0$ (black), $T=3.75\times {10}^{-4}\Lambda =0.53m_{\mathrm{dyn}}$ (blue), $T=5\times {10}^{-4}\Lambda =0.70m_{\mathrm{dyn}}$ (red), $T={10}^{-3}\Lambda =1.41m_{\mathrm{dyn}}$ (green), $T=2\times {10}^{-3}\Lambda =2.81m_{\mathrm{dyn}}$ (dark brown), and $T=4\times {10}^{-3}\Lambda =5.62m_{\mathrm{dyn}}$ (light brown). The insert shows the details in the rectangular area around the origin.

Figure 3

The zero temperature results for the chiral shift parameter $\Delta$ (upper left panel), the mass $m$ (lower left panel), and the difference ${\mu }_0-\mu$ (upper right panel) on the chemical potential ${\mu }_0$ for several fixed values of the bare mass: $m_0=0$ (black solid line), $m_0={10}^{-4}\Lambda =0.14m_{\mathrm{dyn}}$ (blue long-dashed line), $m_0=3\times {10}^{-4}\Lambda =0.42m_{\mathrm{dyn}}$ (red short-dashed line), $m_0=5\times {10}^{-4}\Lambda =0.70m_{\mathrm{dyn}}$ (green dash-dotted line). The lower right panel shows the free energies for the solution with a nonzero dynamical mass (red dashed line) and the solution with a chiral shift (blue solid line) in the chiral limit, $m_0=0$, at zero temperature.

Figure 4

The nonzero temperature results for the chiral shift parameter $\Delta$ (left panel) and the mass $m$ (right panel) as function of the chemical potential ${\mu }_0$ for several fixed values of the temperature, $T=0$ (black), $T=1.41m_{\mathrm{dyn}}$ (blue lines), $T=2.81m_{\mathrm{dyn}}$ (red lines), and $T=5.62m_{\mathrm{dyn}}$ (green lines), and two values of the bare mass, $m_0=0.14m_{\mathrm{dyn}}$ (solid lines) and $m_0=0.70m_{\mathrm{dyn}}$ (short-dashed lines).

Figure 5

A schematic distribution of (negatively charged) particles in the ground state of dense relativistic matter in a magnetic field (pointing in the positive $z$ direction). The values of the quantum number $n$ (Landau levels) are shown along the horizontal axis, while the longitudinal momenta are shown along the vertical axis. The colored bars indicate the filled states of given chirality.

Figure 6

Diagrammatic form of the expression for the effective action in the Hartree-Fock mean-field approximation.