Out-of-equilibrium chiral magnetic effect in QCD: first-principles determination of the real-time current response to chiral imbalance

The chiral magnetic effect (CME) — an electric current induced along a magnetic field by chiral imbalance — was measured on the lattice by Yamamoto (2011) as a finite but suppressed equilibrium current in two-flavor QCD with a chiral chemical potential, leaving open the continuum value of the coefficient in full dynamical QCD, the nonperturbative renormalization of the flavor-singlet local vector current, chiral-symmetry-preserving discretizations, interaction corrections to the analytic prefactor, and the real-time connection to heavy-ion charge-separation signals. Later work settled the equilibrium half of this question in an unexpected direction: the generalized Bloch theorem forbids equilibrium CME currents, and continuum-extrapolated full-QCD simulations (Brandt et al., 2024) show that the equilibrium coefficient is exactly zero, the 2011 nonzero values being artifacts of a non-conserved local vector current. The surviving open problem is the out-of-equilibrium CME response of QCD: determine from first principles, e.g. through the retarded vector-current–axial-charge correlator with controlled real-time reconstruction or equivalent real-time simulation, the magnitude and interaction corrections of the dynamical CME response in full QCD across the phase diagram, and its quantitative connection to the charge-separation signals searched for in heavy-ion collisions. Free-theory anomaly constraints, SU(2) gauge-theory results, and real-time classical simulations exist, but no full-QCD determination exists.

Background

The chiral magnetic effect (CME) is the generation of an electric current parallel to an external magnetic field $B$ in a medium with an imbalance between right- and left-handed fermions. In quantum chromodynamics (QCD) it is motivated by heavy-ion collisions: a noncentral collision produces an enormous magnetic field (of order $10^{18}$–$10^{19}$ Gauss at early times, created by the spectator protons), while topological fluctuations of the gluon field locally create chirality imbalance through the axial anomaly — the quantum breaking of axial-charge conservation, expressed by the relation $\partial_\mu J_5^\mu=(g^2N_f/16\pi^2)\mathrm{tr}\,F\tilde F+2im\bar\psi\gamma_5\psi$. The CME would then induce an event-by-event electric charge separation of the emitted hadrons, searched for via charge-dependent correlations at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC), including the dedicated Ru+Ru versus Zr+Zr isobar run designed to vary the magnetic field at fixed hadronic background.

For free massless fermions, solving the Dirac equation in a magnetic field — where transverse motion is quantized into Landau levels and, in the lowest of these levels, the fermion spin is locked to its momentum direction — gives an induced current $J=\frac{1}{2\pi^2}\mu_5 qB$ for a fermion of charge $q$ and chiral chemical potential $\mu_5$ (Fukushima, Kharzeev and Warringa, 2008), where $\mu_5$ is a parameter coupling to the axial charge density $N_R-N_L$; this free coefficient is independent of mass and temperature. Real-time studies by the same authors (2010) showed the effect is intrinsically dynamical: the current tracks a time-dependent chiral imbalance.

Lattice QCD — the discretized Euclidean path integral from which QCD observables are computed by Monte Carlo — cannot directly realize the dynamical interplay of magnetic field and topological transitions, but a static $\mu_5$ causes no sign problem (the fermion determinant stays positive, unlike for a quark chemical potential), so Yamamoto (2011) simulated two-flavor dynamical QCD with Wilson fermions (a discretization whose $O(a)$ term breaks chiral symmetry at finite lattice spacing $a$) and measured the local vector current induced along $B$. The current was finite and linear in $B$ and $\mu_5$, but suppressed: the best-fit coefficient was $C=0.013\pm0.001$ in full QCD and $C\simeq0.02$–$0.03$ after continuum extrapolation in the quenched approximation (in which fermion loops are dropped from the ensemble generation) versus the free value $1/(2\pi^2)\simeq0.05$. The paper left open: a controlled continuum extrapolation of the coefficient in full dynamical QCD; the nonperturbative renormalization factor of the flavor-singlet local vector current (whose disconnected diagrams are numerically noisy; the exactly conserved, point-split current is renormalization-group invariant instead); simulations with chiral-symmetry-preserving discretizations such as domain-wall or overlap fermions; possible QCD corrections to the analytic prefactor; and the relation between the measured equilibrium current and the real-time, out-of-equilibrium charge separation relevant to experiments.

Subsequent developments resolved the equilibrium part of this question in an unexpected direction. The Bloch theorem — the statement that persistent currents cannot flow in equilibrium — was generalized to relativistic quantum field theory by N. Yamamoto (2015): since the electromagnetic current is conserved, no CME current can flow in thermal equilibrium, and the CME is properly a nonequilibrium steady-state phenomenon, like the quantum Hall effect. A lattice study with overlap fermions (Buividovich, 2014) had found that the chiral magnetic conductivity vanishes in a gauge-invariant lattice regularization, tracing nonzero results to the order of the infrared limit versus the expansion in $\mu_5$. Definitively, Brandt, Endrődi, Garnacho-Velasco and Markó (2024) computed the CME coefficient $C_\mathrm{CME}=\partial\langle J_3\rangle/\partial\mu_5\,/(eB)$ using conserved currents: it vanishes for free fermions in a gauge-invariant Pauli–Villars regularization (an ultraviolet regularization that cancels divergences using fictitious heavy fields), in quenched QCD with both staggered and Wilson fermion discretizations, and in full $(2+1)$-flavor QCD at physical quark masses for temperatures from 100 to 400 MeV, after continuum extrapolation. Re-simulating the 2011 quenched setup, they reproduced Yamamoto's nonzero values only when using the non-conserved local vector current — establishing those values as an artifact of current nonconservation. The equilibrium items of the 2011 open question are thereby settled: the definitive continuum value of the equilibrium coefficient in full dynamical QCD is exactly zero, which also supersedes the renormalization and chiral-symmetry-preserving-fermion issues for the equilibrium problem. The contrasting chiral separation effect (CSE) — an axial current induced by a baryon chemical potential in a magnetic field — involves a non-conserved axial current, evades the theorem, and has been determined at the physical point (Brandt et al., 2024); for inhomogeneous magnetic fields a localized equilibrium CME current whose spatial average vanishes also exists (Brandt et al., 2025).

What survives is the experiment-facing part of the 2011 question: the out-of-equilibrium, real-time CME response of QCD. In linear response this is encoded in the retarded correlator of the vector current and the axial charge — the causal real-time response function; a Kubo formula expresses a transport coefficient as a limit of such a retarded correlator, and extracting the real-time response from Euclidean lattice correlators requires reconstructing a spectral function (the frequency-resolved density that encodes the real-time dynamics), an analytically ill-posed inversion whose systematics must be controlled, while the order of the frequency and momentum limits is known to matter. Real-time classical-statistical lattice simulations with dynamical fermions coupled to both non-Abelian and Abelian gauge fields demonstrate the dynamical CME and its interplay with the chiral separation effect (Müller, Schlichting and Sharma, 2016), but only in the classical-field regime. Buividovich (2024) proposed the Euclidean correlator of axial charge and electric current as a lattice-accessible observable of the out-of-equilibrium CME response: in the free theory only the lowest Landau level contributes, with a universal coefficient linear in $B$ and $T$, and the axial anomaly stringently limits interaction corrections; a first computation in $SU(2)$ gauge theory with two light flavors was given. A subsequent study across the finite-density $SU(2)$ phase diagram (Buividovich, von Smekal and Smith, 2026) finds this response close to the free-quark value in the quark-gluon-plasma regime. No full-QCD, continuum-limit determination of the out-of-equilibrium CME response exists, and recent reviews (Yamamoto, 2013; Garnacho Velasco et al., 2023; Endrődi, 2025) treat it as an open objective.

Problem Statement

Determine from first principles in QCD the out-of-equilibrium chiral magnetic effect: the real-time response of the electric current to chiral (axial) imbalance in the presence of a magnetic field — for example the retarded vector-current–axial-charge correlator with the order of its frequency and momentum limits pinned (the static and dynamic limits are known not to commute), or an equivalent explicitly specified real-time protocol. The required results, which jointly serve this single objective, are: (a) the continuum-limit value of the out-of-equilibrium CME response in full dynamical QCD as a function of temperature, magnetic-field strength, and quark masses, together with the interaction corrections to the free lowest-Landau-level response — identifying which corrections are excluded by the axial anomaly and which are allowed, and with what magnitude; and (b) the quantitative connection between this first-principles response and the out-of-equilibrium charge separation searched for in heavy-ion collisions, in particular whether the QCD response is compatible with the absence of an enhanced charge-separation signal in the RHIC isobar data. This is the surviving open core of the question posed by Yamamoto, Phys. Rev. D 84 (2011) 114504: the equilibrium items of that question — a definitive continuum value of the coefficient $C$ in full dynamical QCD, the nonperturbative renormalization of the flavor-singlet local vector current, and the role of chiral-symmetry-preserving discretizations — were resolved by Brandt et al., JHEP 09 (2024) 092, who showed that the equilibrium coefficient vanishes exactly, the 2011 nonzero value being an artifact of a non-conserved local current; the physically meaningful determination must therefore be out of equilibrium.

References

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