Chiral Plasma Instability (CPI)

• Chiral Plasma Instability (CPI) is a plasma instability triggered by a chiral imbalance (nonzero μ₅) that converts microscopic chirality into macroscopic, helical magnetic fields.
• Theoretical and numerical analyses reveal that CPI exhibits exponential growth, saturation via chiral charge depletion, and an inverse cascade that shifts energy to larger scales.
• CPI plays a key role in generating seed magnetic fields in astrophysical contexts like magnetars and the early Universe, though its dynamics are moderated by chirality-flipping processes and plasma conductivity.

The chiral plasma instability (CPI) is a collective electromagnetic or non-Abelian gauge-field instability in relativistic plasmas carrying a net chiral imbalance. Specifically, CPI arises when a plasma contains unequal populations of left- and right-handed fermions, quantified through a nonzero chiral chemical potential μ₅. This imbalance triggers a parity-odd, helicity-selective current—often referred to as the chiral magnetic effect (CME). The feedback between this current and gauge-field dynamics causes certain helical gauge-field modes to grow exponentially at the expense of chiral imbalance, converting microscopic chirality into macroscopic, helical magnetic or chromomagnetic fields. The CPI has seen rigorous theoretical development via kinetic theory with Berry curvature, anomalous magnetohydrodynamics (MHD), and large-scale numerical simulations, revealing it as a key mechanism in the dynamics of astrophysical and cosmological plasmas, as well as in laboratory analogs.

1. Theoretical Foundations: Chiral Anomaly, CME, and MHD

In quantum field theory, the chiral anomaly leads to nonconservation of axial current:

$$\partial_\mu j_5^\mu = -\frac{e^2}{2\pi^2} \mathbf{E} \cdot \mathbf{B}$$

In a plasma, the difference in right-handed and left-handed fermion populations is characterized by the chiral chemical potential μ₅ = (μ_R − μ_L)/2. The CME dictates that a chiral imbalance induces an electric current along the magnetic field,

$$\mathbf{j}_{\text{CME}} = \frac{e^2}{2\pi^2} \mu_5 \mathbf{B}$$

Maxwell's equations are modified accordingly. In the resistive MHD regime (σ ≫ ω), the induction equation becomes

$$\partial_t \mathbf{B} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta \nabla^2 \mathbf{B} + v_5 \nabla \times \mathbf{B},$$

where $\eta = 1/\sigma$ is the magnetic diffusivity, $\mathbf{u}$ is the bulk fluid velocity, $v_5 = \frac{e^2}{2\pi^2 \sigma} \mu_5$ is the chiral velocity. In the absence of fluid motion, evolution is governed by the competition between CME-induced growth and resistive diffusion (Kamada et al., 2022, Mace et al., 2019, Brandenburg et al., 2023).

The anomaly equation links the dissipation of chiral charge to electromagnetic field configurations:

$$\partial_t n_5 = -\frac{e^2}{2\pi^2}\int \mathbf{E} \cdot \mathbf{B}\, d^3x - \Gamma_f n_5,$$

where $\Gamma_f$ is the rate for chirality-flipping processes (e.g., due to finite fermion mass interactions).

2. Linear Analysis: Dispersion Relation and Instability Condition

The CPI manifests in the linear regime as an exponentially growing band of helical magnetic modes. Decomposing $\mathbf{B}$ into helicity eigenmodes and working in Fourier space, the linearized induction equation for a single mode gives (Kamada et al., 2022, Anand et al., 2018):

$\gamma_\pm(k) = v_5 k \mp \eta k^2$

(for right- and left-handed helicities, with the $+$ sign corresponding to the growing mode when $\mu_5 > 0$).

The growth rate is maximal at $k_* = v_5/2\eta$, with maximum

$\gamma_{\rm max} = \frac{v_5^2}{4\eta}$

This band of instability exists for $0<k<k_c = v_5/\eta$.

Including nonlinear electromagnetic response and plasma screening, kinetic-theory formulations provide a more refined dispersion relation:

$$\gamma(k) = \frac{4\alpha \mu_5}{\pi^2 m_D^2} k^2 \left[1-\frac{\pi k}{\alpha \mu_5}\right]$$

where $\alpha$ is the fine-structure constant and $m_D$ is the Debye mass, showing the growth is most rapid at intermediate $k$ and sharply drops off outside the window $0<k<k_c$ (Kumar et al., 2016, Akamatsu et al., 2013, Kumar et al., 2016, Kumar et al., 2014).

CPI in non-Abelian plasmas (e.g., QCD) is governed by analogous expressions but involves a chiral magnetic conductivity $\sigma_{\rm anom} = (N_f g^2 \mu_5)/(4\pi^2)$ and color conductivity $\sigma_c \sim T/\ln(1/g)$, with the instability band at $k < \sigma_{\rm anom}$ and maximal growth at $k_* = \sigma_{\rm anom}/2$ (Akamatsu et al., 2014, Schlichting et al., 2022).

3. Nonlinear Evolution: Helicity Transfer and Inverse Cascade

As CPI-excited modes grow, they deplete the axial charge through the anomaly (E•B term), with conservation of the sum of chiral charge and magnetic helicity:

$\frac{d}{dt}(Q_5 + H_{\rm mag}) = 0$

Numerical simulations show CPI leads to the formation of maximally helical magnetic fields, saturating when the chiral chemical potential is sufficiently depleted (Mace et al., 2019, Mace et al., 2020, Buividovich et al., 2015, Brandenburg et al., 2023). After saturation, a turbulent inverse cascade ensues in which magnetic helicity and energy are transferred from the instability band (small scales) to larger scales, characterized by self-similar scaling:

$$H_M(k,t) \sim t^\alpha F(k t^\beta),\quad \text{often}\,\, \beta \sim 0.37,\,\, \alpha \sim 1.14$$

For fully helical turbulence, the correlation length grows as $\xi_M \propto t^{2/3}$, while in the chiral MHD regime with ongoing anomaly transfer, the scaling is typically shallower, $\xi_M \propto t^{4/9}$, until spin-flip processes become important and restore pure helicity conservation (Brandenburg et al., 2023).

CPI-generated turbulence enhances transport—such as shear viscosity—via “anomaly-driven” eddy viscosity, scaling in the cold regime as $\eta_A \propto \mu_5^3$ (Kumar et al., 2016).

4. Astrophysical and Cosmological Realizations

Core-Collapse Supernovae and Magnetars: Weak-interaction processes in proto–neutron stars and magnetar progenitors produce order-unity electron/neutral-lepton chirality imbalances, which drive CPI via effective chiral currents even in the absence of electron μ₅ (Matsumoto et al., 2022, Ohnishi et al., 2014). Simulations and analytical estimates yield maximal field strengths $B_{\rm CPI} \sim \mu_5^2 \sim 10^{18}\,\text{G}$ on small scales, with inverse cascade capable of building $B\sim 10^{15}$–$10^{18}$ G fields on macroscopic length scales needed for magnetar stability.

Primordial Magnetogenesis and Gravitational Waves: In the early Universe, CPI can operate with hypercharge fields, producing maximally helical hypermagnetic fields if the right-handed electron Yukawa rate is out of equilibrium (for $T\gtrsim 80$ TeV), and potentially sourcing observable relic gravitational wave backgrounds with unique circular polarization (Brandenburg et al., 2023, Gurgenidze et al., 9 Dec 2025, Anand et al., 2018). The chiral anomaly constrains the transfer between chiral number and magnetic helicity—relevant both for baryogenesis and the present-day strength and helicity of intergalactic magnetic fields.

Recent work has established that with an out-of-equilibrium source of chiral charge, CPI remains operative even when Standard-Model chirality flipping would otherwise erase μ₅, yielding seed fields for galactic dynamos of order $10^{-13}$ G on Mpc scales (Gurgenidze et al., 9 Dec 2025).

Other Astrophysical Contexts: In magnetar magnetospheres CPI can be continuously triggered in plasma gaps, yielding circularly polarized emission across a broad frequency range (radio to near-IR) (Gorbar et al., 2021).

5. Competing Instabilities and Anisotropy Effects

CPI operates in isotropic plasmas with μ₅ ≠ 0, but in anisotropic distributions, competition arises with the Weibel instability. The latter dominates for large momentum-space anisotropy $\xi$ unless the chiral asymmetry is sufficiently large or special propagation angles are selected. Analytical treatment shows CPI's maximal growth rate dominates only for $\xi < \xi_c \sim (μ_5/T)^2$ (Kumar et al., 2016, Kumar et al., 2014). Anisotropy can enhance CPI along the anisotropy direction and suppress it transversely; jet-plasma systems introduce new parity-odd unstable branches.

6. Theory, Simulations, and Scaling Laws

Analytical and first-principles lattice simulations (classical statistical, Wilson–Dirac fermions, and noncompact lattice gauge theory) confirm the rich temporal evolution of CPI:

1. Exponential growth in the linear regime,
2. Saturation via anomaly-induced depletion of chiral charge,
3. Nonlinear turbulent inverse cascade and potential self-similar scaling of the helicity and energy spectra (Mace et al., 2019, Mace et al., 2020, Buividovich et al., 2015).

In Abelian plasmas, this leads to persistent, large-scale helical fields; in non-Abelian settings, nonperturbative sphaleron transitions tend to wash out axial charge before large-scale fields can fully develop (Schlichting et al., 2022, Akamatsu et al., 2014). The saturation scale and dynamic evolution depend sensitively on plasma parameters, the persistence of chiral sources, and the competition with chirality-flipping processes.

A concise tabulation of archetypal CPI features across contexts:

Context	Instability Driver	Saturation Field	Inverse Cascade?	Scaling Law(s)
QED plasma	μ₅, axial anomaly	μ₅²	Yes	ξ_M ~ t^{2/3}, t^{4/9}
QCD plasma	μ₅, axial anomaly, sphalerons	μ₅², then washed	No (sphalerons)	n_B ~ 1/g², τ_sph ∼ 1/T
Supernovae/magnetars	μ_{5,eff} (ν-driven CME)	μ_{5,eff}²∼10^{18}G	Yes	τCPI ~ (μ{5,eff}² η)^{-1}
Early universe	μ₅^Y/hypercharge, out-of-eq.	μ₅^2	Yes	ξ_M ~ t^{2/3}

7. Limitations, Open Questions, and Outlook

Several physical effects limit or modulate CPI:

• Chirality-flipping: Finite fermion mass induces rapid μ₅ decay, constraining CPI's duration, especially in electron plasmas where timescales can be ~10^{-12} s (Kaplan et al., 2016, Ohnishi et al., 2014).
• Conductivity and Damping: Finite conductivity slows growth and alters spectra (Anand et al., 2018, Kamada et al., 2022).
• Non-Abelian Effects: In SU(2) or QCD plasmas, non-Abelian self-interactions and sphaleron transitions dominate the fate of axial charge, generally precluding persistent helical field formation (Schlichting et al., 2022, Akamatsu et al., 2014).

The precise interplay between CPI-generated turbulence, cosmological or astrophysical observables, and mechanisms such as baryogenesis and pulsar kicks remains an area of active analytical and numerical investigation. Further quantitative progress requires high-fidelity, large-scale simulations that incorporate or systematically vary key microphysical rates, multi-fluid transport coefficients, and potential chirality sources or sinks (Gurgenidze et al., 9 Dec 2025, Matsumoto et al., 2022, Brandenburg et al., 2023, Brandenburg et al., 2023).