Influence of chiral chemical potential, parallel electric, and magnetic fields on the critical temperature of QCD

We study the influence of external parallel electric and magnetic fields and a chiral chemical potential ${\mu }_5$ on the chiral phase transition of quantum chromodynamics. Our theoretical framework is a Nambu–Jona-Lasinio model with a contact interaction. Within this model we compute the critical temperature of chiral symmetry restoration $T_c$ as a function of the chiral chemical potential and field strengths. We find that the fields inhibit and ${\mu }_5$ enhances chiral symmetry breaking, in agreement with previous studies.

Figure 1

Constituent quark mass versus temperature for $E=B=0$ (thin orange solid line), $E=0$ and $eB=8m_{\pi }^2$ (blue dashed line), $eE=8m_{\pi }^2$ and $B=0$ (green dot-dashed line), and $eE=eB=8m_{\pi }^2$ (thick maroon solid line).

Figure 2

Upper panel: Constituent quark mass versus temperature for $E=B=0$ (thin orange solid line), $eE=eB=8m_{\pi }^2$ and ${\mu }_5=0$ (thick maroon solid line), $eE=eB=8m_{\pi }^2$ and ${\mu }_5=150\mathrm{MeV}$ (maroon dashed line), and $eE=eB=8m_{\pi }^2$ and ${\mu }_5=300\mathrm{MeV}$ (maroon dot-dashed line). Lower panel: Condensation energy versus temperature (the line convention is the same as in the upper panel).

Figure 3

Constituent quark mass versus temperature for $eE=eB=8m_{\pi }^2$ and several values of ${\mu }_5$. The blue solid line corresponds to ${\mu }_5=0$, the maroon dashed lines correspond to ${\mu }_5=150\mathrm{MeV}$, and the orange dot-dashed lines correspond to ${\mu }_5=300\mathrm{MeV}$. Thick lines correspond to the solution of the full gap equation, while thin lines correspond to the perturbative solution of Ref. [24].

Figure 4

Constituent quark mass versus temperature for $E=B=0$ (thin orange solid line), $eE=eB=8m_{\pi }^2$ and ${\mu }_5=0$ (thick maroon solid line), $eE=eB=8m_{\pi }^2$ and ${\mu }_5=150\mathrm{MeV}$ (maroon dashed line), and $eE=eB=8m_{\pi }^2$ and ${\mu }_5=300\mathrm{MeV}$ (maroon dot-dashed line). IMC has been taken into account. The green thin dotted line corresponds to the quark mass for $eE=eB=8m_{\pi }^2$, ${\mu }_5=0$ and direct magnetic catalysis already shown in Fig. 2.

Figure 5

Constituent quark mass versus magnetic field with (blue lines) and without (red lines) the inverse magnetic catalysis for $E=0$ (dashed lines) and $E=B$ (dash-dotted lines). The upper panel corresponds to ${\mu }_5=150\mathrm{MeV}$, while the lower panel corresponds to ${\mu }_5=300\mathrm{MeV}$. Both panels correspond to $T=100\mathrm{MeV}$.

Figure 6

Chiral density versus chiral chemical potential for two temperatures: green lines correspond to $T=100\mathrm{MeV}$ and orange lines correspond to $T=200\mathrm{MeV}$. Solid lines denote $n_5$ obtained by using the full thermodynamic potential, while dashed lines denote the perturbative calculation of Ref. [24] in which only ${\mathrm{\Omega }}_0$ was used. Moreover, we have set $eE=eB=8m_{\pi }^2$.

Figure 7

Critical temperature versus ${\mu }_5$ for several values of $E$ and $B$. From top to bottom, we have shown data for $E=B=0$ (thin black solid line), $eE=eB=2.5m_{\pi }^2$ (green dotted line), $eE=eB=5m_{\pi }^2$ (orange dashed line), $eE=eB=7.5m_{\pi }^2$ (blue dot-dashed line), and $eE=eB=10m_{\pi }^2$ (maroon dot-dot-dashed line). Inverse magnetic catalysis has been taken into account in the calculations.