Influence of the inverse magnetic catalysis and the vector interaction in the location of the critical end point

The effect of a strong magnetic field on the location of the critical end point (CEP) in the QCD phase diagram is discussed under different scenarios. In particular, we consider the contribution of the vector interaction and take into account the inverse magnetic catalysis obtained in lattice QCD calculations at zero chemical potential. The discussion is realized within the ($2+1$) Polyakov–Nambu–Jona-Lasinio model. It is shown that the vector interaction and the magnetic field have opposite competing effects, and that the winning effect depends strongly on the intensity of the magnetic field. The inverse magnetic catalysis at zero chemical potential has two distinct effects for magnetic fields above $\gtrsim 0.3{\mathrm{GeV}}^2$: it shifts the CEP to lower chemical potentials, hinders the increase of the CEP temperature and prevents a too large increase of the baryonic density at the CEP. For fields $eB<0.1{\mathrm{GeV}}^2$ the competing effects between the vector contribution and the magnetic field can move the CEP to regions of temperature and density in the phase diagram that could be more easily accessible to experiments.

Figure 1

Location of the CEP on a diagram $T$ vs the baryonic chemical potential (left), vs the baryonic density (middle), and vs magnetic field (right) for Case IA.

Figure 2

The critical chemical potential at $T=0\mathrm{MeV}$ vs the magnetic field (left panel) and the $u$, $d$-quark constituent masses at the CEP as a function of the magnetic field intensity.

Figure 3

Effect of the strength of the vector interaction on the location of the CEP on a diagram $T$ vs the baryonic chemical potential (left) and $T$ vs the baryonic density (right).

Figure 4

QCD phase diagram in a $T-{\mu }_B$ plot for Cases IA and IB, not including IMC effects, discussed in the text, left panel. The critical chemical potential as a function of the magnetic field intensity at $T=0$, right panel. The green thick lines correspond to no magnetic field, and dashed (full) lines were obtained with (without) the vector interaction.

Figure 5

The location of the CEP for Case IB, with the inclusion of the vector interaction and no IMC, the results for Case IB with $\alpha =0.25$ and 0.71 are compared with Case IA, with no vector interaction. The thin arrows indicate the direction of increasing magnetic field intensity.

Figure 6

Location of the CEP, $T^{\mathrm{CEP}}$ vs ${\mu }_B^{\mathrm{CEP}}$ (left panel) and $T^{\mathrm{CEP}}$ vs ${\rho }_B^{\mathrm{CEP}}$ (right panel), for different intensities of the magnetic field and excluding the vector interaction, without IMC effects $G_s=G_s^0$ (red curve) and with IMC effects $G_s=G_s(eB)$ (black curve).

Figure 7

Masses of the quarks as a function of ${\mu }_B$ for two intensities of the magnetic field: $eB=0.1{\mathrm{GeV}}^2$ (left) and $eB=0.5{\mathrm{GeV}}^2$ (right), at the respective $T^{\mathrm{CEP}}$.

Figure 8

The location of the CEP: (a) and (d) $T$ versus ${\mu }_B$, for different intensities of the magnetic field; (b) and (e) $T$ as a function of $eB$; (c) and (f) ${\mu }_B$ versus $eB$, for the vector interaction $\alpha =0.25$ [left, (a), (b) and (c) panels] and $\alpha =0.71$ [right, (d), (e) and (f) panels]. The red and blue curves were obtained with no IMC effects and correspond to Cases IA and IB. The IMC effects and vector interaction were included in the black curves ($G_V$ is fixed) and magenta ($G_V$ weakens with an increasing $eB$). For more details see the text at the end of Sec. 2.

Figure 9

The CEP location on the T-${\rho }_B$ plane. Cases IIA, IIB, and IIC with IMC effects are compared with Case IB without IMC effects. Case IIA is the only scenario excluding the vector interaction.