Competition between magnetic catalysis effect and chiral rotation effect

In this work, we explore the competition between magnetic catalysis effect and chiral rotation effect in a general parallel electromagnetic field within the effective Nambu–Jona-Lasinio model. For a given electric field $E$ at zero temperature, the mass gap shows three different features with respect to an increasing magnetic field $B$: increasing monotonically, decreasing after increasing, and decreasing monotonically. By making use of strong magnetic field approximation, we illuminate that this is due to the competition between catalysis effect and chiral rotation effect induced both by the magnetic field, and a critical electric field $\sqrt{e{E}_c}=86.4\mathrm{MeV}$ is found beyond which the mass gap will eventually decrease at large $B$. As only large magnetic field is relevant for the derivation, the critical electric field does not depend on the temperature $T$ or chemical potential $\mu$.

Figure 1

The mass gap $m$ and pion condensate ${\pi }^0$ as functions of the parallel electric field $E$ and magnetic field $B$.

Figure 2

The mass gap $m$ as a function of magnetic field $B$ for different subcritical temperatures at vanishing chemical potential (upper panel) and for different chemical potentials at temperature $T=0.15\mathrm{GeV}$ (lower panel) for the subcritical electric field case.

Figure 3

The mass gap $m$ as a function of magnetic field $B$ for different subcritical temperatures at vanishing chemical potential (upper panel) and for different chemical potentials at temperature $T=0.15\mathrm{GeV}$ (lower panel) for the supercritical electric field case.

Figure 4

The pair production rate as a function of magnetic field $B$ at $T=0.15\mathrm{GeV}$ and $\mu =0.2\mathrm{GeV}$ for the subcritical electric field case.

Figure 5

The chiral mass $M$ and anomalous angle $\theta$ as functions of $I_2^{1/2}=E=B$ in the chiral rotation region for different subcritical temperatures. The upper limit of $\theta$ is $\frac{\pi }{2}$ in the lower panel.