Magnetic-field-induced squeezing effect at energies available at the BNL Relativistic Heavy Ion Collider and at the CERN Large Hadron Collider

In off-central heavy-ion collisions, quark-gluon plasma (QGP) is exposed to the strongest magnetic fields ever created in the universe. Because of the paramagnetic nature of the QGP at high temperatures, the spatially inhomogeneous magnetic field configuration exerts an anisotropic force density that competes with the pressure gradients resulting from purely geometric effects. In this paper, we simulate (3+1)-dimensional ideal hydrodynamics with external magnetic fields to estimate the effect of this force density on the anisotropic expansion of the QGP in collisions at the Relativistic Heavy Ion Collider and at the Large Hadron Collider (LHC). While negligible for quickly decaying magnetic fields, we find that long-lived fields generate a substantial force density that suppresses the momentum anisotropy of the plasma by up to $20\%$ at the LHC energy and also leaves its imprint on the elliptic flow $v_2$ of charged pions.

Figure 1

The squeezing force density $\left(F^x\right)$ along the $x$ axis at ${\tau }_0=0.2$ fm with the magnetic field given by settings (B) $t_d=0.1$ fm, (C) $t_d=0.5$ fm, (D) $t_d=1.0$ fm, and (E) $t_d=1.9$ fm, for Pb+Pb $\sqrt{s_{NN}}=2.76$ TeV collisions with local temperature from (3+1)D hydrodynamics.

Figure 2

The charged multiplicity for (a) Au+Au $\sqrt{s_{NN}}=200$ GeV collisions and (b) Pb+Pb $\sqrt{s_{NN}}=2.76$ TeV collisions with 3 different groups of configurations for initial thermalization time ${\tau }_0$ and maximum energy density ${\varepsilon }_0$ given in Table 3.

Figure 3

The pressure gradients for Pb+Pb $\sqrt{s_{NN}}=2.76$ TeV collisions with impact parameter $b=10$ fm.

Figure 4

The squeezing force density (a) along the $x$ direction and (b) along the $y$ direction, for Pb+Pb $\sqrt{s_{NN}}=2.76$ TeV collisions at ${\tau }_0=0.2$ fm, with the magnetic field given by setting E.

Figure 5

Time evolution of the momentum eccentricity ${\epsilon }_p$ for Pb+Pb $\sqrt{s_{NN}}=2.76$ TeV collisions with impact parameter $b=10$ fm, for (solid line) $e{B}_0=0$, setting E (long-dashed line, $e{B}_0=1.33{\mathrm{GeV}}^2,2{\sigma }_x={\sigma }_y=2.6\mathrm{fm})$, setting G (short-dashed line, $e{B}_0=1.0{\mathrm{GeV}}^2,2{\sigma }_x={\sigma }_y=4.8\mathrm{fm})$, and setting F (dash-dotted line, $e{B}_0=1.33{\mathrm{GeV}}^2,2{\sigma }_x={\sigma }_y=4.8\mathrm{fm})$.

Figure 6

$v_2$ for direct ${\pi }^{+}$ at Pb+Pb $\sqrt{s_{NN}}=2.76$ TeV collisions with impact parameter $b=10$ fm, for (solid line) $e{B}_0=0$, settings E (long-dashed line) $e{B}_0=1.33{\mathrm{GeV}}^2,2{\sigma }_x={\sigma }_y=2.6\mathrm{fm}$, G (short-dashed line) $e{B}_0=1.0{\mathrm{GeV}}^2,2{\sigma }_x={\sigma }_y=4.8\mathrm{fm}$, and F (dash-dotted line) $e{B}_0=1.33{\mathrm{GeV}}^2,2{\sigma }_x={\sigma }_y=4.8\mathrm{fm}$.

Figure 7

The effect of paramagnetic squeezing on the elliptic flow for ${\tau }_0=0.2,0.4,\mathrm{and}0.6$ fm.

Figure 8

The pressure gradients for Au+Au $\sqrt{s_{NN}}=200$ GeV collisions with impact parameter $b=10$ fm.

Figure 9

The squeezing force density for Au+Au $\sqrt{s_{NN}}=200$ GeV collisions at ${\tau }_0=0.2$ fm, with the magnetic field given by setting A.