^{1}

^{2,3}

^{1}

^{4}

^{5}

^{3}.

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

Large magnetic fields

In previous work

The study in Ref.

The idea of Ref.

Schematic illustration of how the magnetic field

As illustrated in Fig.

(1)

(2)

(3)

As is clear from their physical origins, all three of these electric fields—and the consequent electric currents—have opposite directions at positive and negative rapidity. It is also clear from Fig.

In this paper we make three significant advances relative to the exploratory study of Ref.

Second, we find that the same mechanism that produces the charge-odd

Last but not least, we identify a new electromagnetic mechanism that generates another type of sideways current which generates a charge-odd,

(4)

The electric (left) and magnetic (right) fields in the transverse plane at

At the collision energies that we consider,

In the next section, we set up our model. In particular, we explain our calculation of the electromagnetic fields, the drift velocity, and the freeze-out procedure from which we read off the charge-dependent contributions to the radial

We simulate the dynamical evolution of the medium produced in heavy-ion collisions using the ^{1}

Starting hydrodynamics at a different thermalization time, between 0.2 and 0.6 fm/

The electromagnetic fields are generated by both the spectators and participant charged nucleons. The transverse distribution of the right-going (

The electromagnetic fields generated by the charges and currents evolve according to the Maxwell equations

With the evolution of the electromagnetic fields in hand, the next step is to compute the drift velocity

With the full, charge-dependent, fluid velocity

With the momentum distribution for hadrons with different charge in hand, the final step in the calculation is the evaluation of the anisotropic flow coefficients as function of rapidity:

In order to isolate the small contribution to the various flow observables that was induced by the electromagnetic fields, separating it from the much larger background hydrodynamic flow, we compute the difference between the value of a given flow observable for positively and negatively charged hadrons:

It is instructive to analyze the spatial distribution and the evolution of the electromagnetic fields in heavy-ion collisions. We shall do so in this section, before turning to a discussion of the results of our calculations in the next section.

Figure

The left panel in Fig.

Left: The

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

When solving the force-balance equation, Eq.

In this section we present our results for the charge-dependent contributions to the anisotropic flow induced by the electromagnetic effects introduced in Sec.

To provide a realistic dynamical background on top of which to compute the electromagnetic fields and consequent currents, we have calibrated the solutions to relativistic viscous hydrodynamics that we use by comparing them to experimental measurements of hadronic observables. To give a sense of the agreement that we have obtained, in Fig.

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.

To isolate the effect of electromagnetic fields on charged hadron flow observables, we study the difference between the

We should distinguish the charge-odd contributions to the odd flow moments, ^{2}

This electric field was called the Hall electric field in Ref.

On the other hand, the charge-dependent contributions to the even order anisotropic flow coefficients

In Fig.

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-

We compare the red dashed curves, arising from electromagnetic effects by spectators only, with the solid black curves that show the full calculation including the participants. Noting that the lines are significantly different, it follows that the Coulomb force exerted on charges in the plasma by charges in the plasma makes a large contribution to

The electromagnetically induced elliptic flow

Note that

The electromagnetically induced contributions to the odd flow harmonics

In Fig.

The electromagnetically induced difference between the mean

In Fig.

The centrality dependence of the electromagnetically induced flow difference in

Compared to any of the anisotropic flow coefficients

Figure

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

In Fig.

The collision energy dependence of the electromagnetically induced charge-odd contributions to flow observables. The difference of particle mean

Finally, in Fig.

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.

This concludes the presentation of our central results. In the remainder of this section, in two subsections we shall present a qualitative argument for why

As we have seen, the net effect on

One can find an expression for the total Faraday + Coulomb electric field ^{3}

To a very good approximation, one can in fact ignore the participant contribution

Throughout this paper, we have chosen fixed values for the two important material parameters that govern the magnitude of the electromagnetically induced contributions to flow observables, namely the drag coefficient

In Fig.

The dependence of the electromagnetically induced differences between the flow of protons and antiprotons on the choice of the drag coefficient

In Fig.

The dependence of the electromagnetically induced differences between the flow of protons and antiprotons on the choice of the electrical conductivity

The electromagnetically induced charge-odd contributions to the flow observables

We have described the effects of electric and magnetic fields on the flow of charged hadrons in noncentral heavy-ion collisions by using a realistic hydrodynamic evolution within the

In our calculations, we have treated the electrodynamics of the charged matter in the plasma in a perturbative fashion, added on top of the background flow, rather than attempting a full-fledged magnetohydrodynamical calculation. The smallness of the effects that we find supports this approach. However, we caution that we have made various important assumptions that simplify our calculations: (i) we treat the two key properties of the medium that enter our calculation, the electrical conductivity

Relaxing (i) necessitates solving the Maxwell equations on a medium with time- and space-dependent parameters, which would result in a more complicated profile for the electromagnetic fields. We expect that this would modify our results in a quantitative manner without altering main qualitative findings. We have tried to choose a value for

Relaxing (ii), which is to say adding event-by-event fluctuations in the initial conditions for the hydrodynamic evolution of the matter produced in the collision zone, as well as for the distribution of spectator charges, would have quite significant effects on the values of the charge-averaged

Relaxing assumption (iii) may bring new effects and, as we shall explain, could potentially flip the sign of the odd flow coefficients

The likely reduction in the magnitude of

Finally, let us consider relaxing our assumption (iv). This corresponds to considering a more general version of

This work was supported in part by the Netherlands Organisation for Scientific Research (NWO) under VIDI Grant No. 680-47-518, the Delta Institute for Theoretical Physics (D-ITP) funded by the Dutch Ministry of Education, Culture and Science (OCW), the Scientific and Technological Research Council of Turkey (TUBITAK), the Office of Nuclear Physics of the US Department of Energy under Contracts No. DE-SC0011090, No. DE-FG-88ER40388, and No. DE-AC02-98CH10886, and the Natural Sciences and Engineering Research Council of Canada. K.R. gratefully acknowledges the hospitality of the CERN Theory Group. C.S. gratefully acknowledges a Goldhaber Distinguished Fellowship from Brookhaven Science Associates. Computations were made in part on the supercomputer Guillimin from McGill University, managed by Calcul Québec and Compute Canada. The operation of this supercomputer is funded by the Canada Foundation for Innovation (CFI), NanoQuébec, RMGA and the Fonds de recherche du Québec–Nature et technologies (FRQ-NT). U.G. is grateful for the hospitality of the Boğaziçi University and the Mimar Sinan University in Istanbul. We gratefully acknowledge helpful discussions with G. Chen, U. Heinz, J. Margutti, R. Snellings, S. Voloshin, and F. Wang.