Charge-dependent flow induced by magnetic and electric fields in heavy ion collisions

We investigate the charge-dependent flow induced by magnetic and electric fields in heavy-ion collisions. We simulate the evolution of the expanding cooling droplet of strongly coupled plasma hydrodynamically, using the iEBE-VISHNU framework, and add the magnetic and electric fields as well as the electric currents they generate in a perturbative fashion. We confirm the previously reported effect of the electromagnetically induced currents [Gursoy et al., Phys. Rev. C 89, 054905 (2014)], that is a charge-odd directed flow $\mathrm{\Delta }{v}_1$ that is odd in rapidity, noting that it is induced by magnetic fields (à la Faraday and Lorentz) and by electric fields (the Coulomb field from the charged spectators). In addition, we find a charge-odd $\mathrm{\Delta }{v}_3$ that is also odd in rapidity and that has a similar physical origin. We furthermore show that the electric field produced by the net charge density of the plasma drives rapidity-even charge-dependent contributions to the radial flow $\langle p_T\rangle$ and the elliptic flow $\mathrm{\Delta }{v}_2$. Although their magnitudes are comparable to the charge-odd $\mathrm{\Delta }{v}_1$ and $\mathrm{\Delta }{v}_3$, they have a different physical origin, namely the Coulomb forces within the plasma.

Figure 1

Schematic illustration of how the magnetic field $\vec{B}$ in a heavy-ion collision results in a directed flow of electric charge, $\mathrm{\Delta }{v}_1$. The collision occurs in the $z$ direction, meaning that the longitudinal expansion velocity $\vec{u}$ of the conducting QGP produced in the collision points in the $+z$ ($-z$) direction at positive (negative) $z$. We take the impact parameter vector to point in the $+x$ direction, choosing the nucleus moving toward positive (negative) $z$ to be located at negative (positive) $x$. The trajectories of the spectators that “miss” the collision because of the nonzero impact parameter are indicated by the red and blue arrows. This configuration generates a magnetic field $\vec{B}$ in the $+y$ direction, as shown. The directions of the electric fields (and hence currents) due to the Faraday, Lorentz, and Coulomb effects are shown. The two different Coulomb contributions are indicated, one due to the force exerted by the spectators and the other coming from Coulomb forces within the plasma. The dashed arrows indicate the direction of the directed flow of positive charge in the case where the Faraday + spectator Coulomb effects are on balance stronger than the Lorentz effect. Hence, the total directed flow in this example corresponds to $v_1<0$ ($v_1>0$) for positive charges at space-time rapidity ${\eta }_s>0$ (${\eta }_s<0$), and opposite for negative charges.

Figure 2

The electric (left) and magnetic (right) fields in the transverse plane at $z=0$ in the laboratory frame at a proper time $\tau =1$ fm/$c$ after a Pb + Pb collision with 20–30% centrality (corresponding to impact parameters in the range 6.24 fm $<b<$ 9.05 fm) and with a collision energy $\sqrt{s}=2.76$ ATeV. The fields are produced by the spectator ions moving in the $+z$ ($-z$) direction for $x<0$ ($x>0$) as well as by the ions that participate in the collision. In both panels, the contribution from the spectators is larger, however. The direction of the fields is shown by the black arrows. The strength of the field is indicated both by the length of the arrows and by the color. We see that the magnetic field is strongest at the center of the plasma, where it points in the $+y$ direction as anticipated in Fig. 1. The electric field points in a generally outward direction and is strongest on the periphery of the plasma. Its magnitude is not azimuthally symmetric: the field is on average stronger where it is pointing in the $\pm y$ directions than where it is pointing in the $\pm x$ directions.

Figure 3

Left: The $x$ component of the electric field in the local fluid rest frame at points on the freeze-out surface at space-time rapidity ${\eta }_s=0$, as a function of proper time. Each cross corresponds to a single fluid cell on the freeze-out surface, with the vertical line of crosses at any single $\tau$ corresponding to different points on the freeze-out surface at that $\tau$. Only the Coulomb electric field generated by the net charge in the plasma contributes at ${\eta }_s=0$, and by symmetry there for every point where $E_x^{\mathrm{lrf}}>0$ there is a point where $E_x^{\mathrm{lrf}}<0$. Right: Time dependence of the $y$ component of the magnetic field in the laboratory frame at ${\eta }_s=0$. Again, each cross corresponds to a single point on the freeze-out surface. We see that $B_y>0$ as diagramed in Fig. 1 and shown in Fig. 2, and here we can see how $B_y$ decreases with time.

Figure 4

Contributions to the electric field in the local rest frame of a unit cell in the fluid on the freeze-out surface at a specified, nonvanishing, space-time rapidity ${\eta }_s$: ${\eta }_s=1$ in the left panel and ${\eta }_s=3$ in the right panel. Each unit cell is represented in the figure by a black cross, a red cross, and a green cross. Black crosses denote the contribution to the electric field at a given fluid cell in its local rest frame coming from the Coulomb and Faraday effects. Red crosses denote the contribution from the Lorentz force. Green crosses represent the total electric field at the fluid cell, namely the sum of a black cross and a red cross. We observe that the Coulomb + Faraday and Lorentz contributions to the electric field point in opposite directions, as sketched in Fig. 1, and furthermore see that the two contributions almost cancel at large ${\eta }_s$, as we shall discuss in Sec. 4a. We shall see there that the Coulomb + Faraday contribution is slightly larger in magnitude than the Lorentz contribution.

Figure 5

To get a sense of how well the solution to relativistic viscous hydrodynamics upon which we build our calculation of electromagnetic fields and currents describes heavy-ion collisions, we compare our results for charged hadron multiplicities (left) and elliptic flow coefficients (right) to experimental measurements at the top RHIC and LHC energies from Refs. [30, 31, 32] and [33, 34, 35], respectively.

Figure 6

The solid black curves display the principal results of our calculations for 20–30% centrality Au + Au collisions at 200 GeV, as at RHIC. We show the contribution to the mean-$p_T$ of charged pions and the first three $v_n$ coefficients induced by the electromagnetic fields that we have calculated, isolating the electromagnetically induced effects by taking the difference between the calculated value of each observable for ${\pi }^{+}$ and ${\pi }^{-}$ mesons, namely the charge-odd or charge-dependent contributions that we denote $\mathrm{\Delta }\langle p_T\rangle$ and $\mathrm{\Delta }{v}_n$. We see rapidity-odd contributions $\mathrm{\Delta }{v}_1$ and $\mathrm{\Delta }{v}_3$ and rapidity-even contributions $\mathrm{\Delta }\langle p_T\rangle$ and $\mathrm{\Delta }{v}_2$. The red dashed curves show the results we obtain when we calculate the same observables in the presence of the electromagnetic fields produced by the spectators only. We see that the dominant contribution to the odd $v_n$'s is generated by these spectator-induced fields, whereas the even $v_n$'s also receive a significant contribution from the Coulomb force exerted on charges in the plasma by other charges in the plasma, originating from the participant nucleons.

Figure 7

The electromagnetically induced difference between the mean $p_T$ and $v_n$ coefficients of ${\pi }^{+}$ and ${\pi }^{-}$ mesons (solid lines) and between protons and antiprotons (dashed lines) as a function of particle rapidity for 20-30% Au+Au collisions at 200 GeV. Three different $p_T$ integration ranges are shown for each of the $\mathrm{\Delta }{v}_n$ as a function of particle rapidity.

Figure 8

The centrality dependence of the electromagnetically induced flow difference in ${\pi }^{+}$ vs ${\pi }^{-}$ as a function of particle rapidity in Au + Au collisions at 200 GeV.

Figure 9

The centrality dependence of the electromagnetically induced differences in the radial flow and anisotropic flow coefficients for positively and negatively charged hadrons, here at a fixed rapidity $y=-1$.

Figure 10

The collision energy dependence of the electromagnetically induced charge-odd contributions to flow observables. The difference of particle mean $p_T$ and $v_n$ between ${\pi }^{+}$ and ${\pi }^{-}$ are plotted as a function of particle rapidity for collisions at the top RHIC energy of 200 GeV and at two LHC collision energies.

Figure 11

The solid curves include the contributions to the electromagnetically induced charge-dependent flow observables of pions and protons produced after freeze-out by resonance decay, often referred to as resonance feed-down contributions. In the dashed curves, pions and protons produced from resonance feed-down are left out.

Figure 12

The dependence of the electromagnetically induced differences between the flow of protons and antiprotons on the choice of the drag coefficient $\mu \text{m}$ defined in Eq. (10). Elsewhere in this paper, we fix $\mu \text{m}$ by choosing the t'Hooft coupling in Eq. (10) to be $6\pi$. Here we explore the consequences of varying this parameter by factors of 2 and 1/2, thus varying $\mu \text{m}$ by factors of $\sqrt{2}$ and $1/\sqrt{2}$.

Figure 13

The dependence of the electromagnetically induced differences between the flow of protons and antiprotons on the choice of the electrical conductivity $\sigma$ in the Maxwell equations (7) and (8).