[Paper Review] Magnetohydrodynamics, charged currents and directed flow in heavy ion collisions

Umut Gürsoy, Dmitri E. Kharzeev+1 more|Utrecht University Repository (Utrecht University)|Jan 16, 2014

High-Energy Particle Collisions Research17 citations

TL;DR

This paper proposes that strong, time-varying magnetic fields in non-central heavy ion collisions induce charged currents via Faraday and Hall effects in the quark-gluon plasma, leading to a charge-dependent directed flow ($v_1$) that is odd in rapidity and under charge conjugation. The key result is that these effects produce measurable correlations in the directed flow of oppositely charged hadrons at different rapidities, offering a clean signature for detecting early-time magnetic fields in heavy ion experiments.

ABSTRACT

The hot QCD matter produced in any heavy ion collision with a nonzero impact parameter is produced within a strong magnetic field. We study the imprint that these fields leave on the azimuthal distributions and correlations of the produced charged hadrons. The magnetic field is time-dependent and the medium is expanding, which leads to the induction of charged currents due to the combination of Faraday and Hall effects. We find that these currents result in a charge- dependent directed flow v1 that is odd in rapidity and odd under charge exchange. It can be detected by measuring correlations between the directed flow of charged hadrons at different rapidities, $\langle v_1^\pm(y_1)v_1^\pm(y_2) angle$.

Motivation & Objective

• To identify observable signatures of strong, transient magnetic fields in heavy ion collisions that arise from spectator charges.
• To analyze how time-dependent magnetic fields and expanding quark-gluon plasma generate charged currents via Faraday and Hall effects.
• To estimate the magnitude of charge-dependent directed flow ($v_1$) in final-state hadrons as a function of rapidity and transverse momentum.
• To propose a measurable observable—correlations between $v_1^\pm(y_1)$ and $v_1^\pm(y_2)$—to detect the imprint of initial magnetic fields.
• To motivate experimental searches for these effects at RHIC and the LHC, especially in low-energy collisions where effects may be enhanced.

Proposed method

• Model the magnetic field using the Biot-Savart law for non-central collisions, assuming spectators move along straight lines.
• Apply Faraday’s law to compute the electric field induced by the time-varying magnetic field, leading to a current density $\vec{J}_{\text{Faraday}} \propto \partial_t \vec{B}$.
• Compute the Hall current using the Lorentz force on moving charges in a magnetic field, resulting in $\vec{J}_{\text{Hall}} \propto \vec{u} \times \vec{B}$, where $\vec{u}$ is the plasma flow velocity.
• Combine both current contributions and integrate over spacetime to compute the net directed flow $v_1$ of charged hadrons.
• Use Gubser’s analytical solution to relativistic hydrodynamics to model the expanding plasma, assuming constant electrical conductivity $\sigma$ and drag parameter $\mu m$.
• Derive the correlation function $\langle v_1^\pm(y_1) v_1^\pm(y_2) \rangle$ as a probe of the magnetic field’s influence on charge-dependent flow.

Experimental results

Research questions

• RQ1 Can time-varying magnetic fields in non-central heavy ion collisions generate observable charged currents in the quark-gluon plasma?
• RQ2 How do Faraday and Hall effects combine to produce a net charge-dependent directed flow ($v_1$) in the final state?
• RQ3 What is the dependence of the resulting $v_1$ on rapidity $Y$ and transverse momentum $p_T$?
• RQ4 Can the correlation $\langle v_1^\pm(y_1) v_1^\pm(y_2) \rangle$ serve as a clean experimental signature for early-time magnetic fields?
• RQ5 How does the magnitude of the effect compare between RHIC and LHC energies, and is it enhanced at lower collision energies?

Key findings

• The net charged current in the quark-gluon plasma arises from a partial cancellation between Faraday and Hall effects, leading to a non-trivial rapidity and $p_T$ dependence of the directed flow.
• The resulting $v_1$ is odd in rapidity and odd under charge exchange, making it a distinctive signature of magnetic field effects.
• The magnitude of the charge-dependent $v_1$ is estimated to be on the order of $10^{-3}$ to $10^{-2}$, depending on rapidity and $p_T$, with larger effects expected at RHIC than at LHC energies.
• The correlation $\langle v_1^\pm(y_1) v_1^\pm(y_2) \rangle$ is proposed as a key observable, with a sign that depends on the dominance of Faraday vs. Hall effects.
• Preliminary data from RHIC Beam Energy Scan at $\sqrt{s} = 7.7$ and 11.5 AGeV show hints of the predicted $v_1$ behavior, supporting the need for higher-statistics measurements.
• The effects are expected to be larger at lower energies due to longer magnetic field persistence, but modeling becomes more complex due to phase transitions and non-ideal hydrodynamics.