Chirally Asymmetric Quark–Gluon Plasma

• Chirally asymmetric QGP is defined by unequal densities of right- and left-handed quarks, quantified by a nonzero axial chemical potential.
• It exhibits anomaly-induced transport effects such as CME, CESE, and CVE, which are measurable in heavy-ion collisions and extreme QCD states.
• Theoretical models combine hydrodynamics, lattice QCD, and field correlator methods to address experimental signatures and wave phenomena in QGP.

A chirally asymmetric quark–gluon plasma (QGP) is a deconfined phase of QCD where right- and left-handed quarks occur at unequal densities—a nonzero chiral chemical potential $\mu_5 = (\mu_R - \mu_L)/2$ quantifies this imbalance. In such a medium, fundamental quantum anomalies induce macroscopic transport effects that break parity locally and can be experimentally probed in relativistic heavy-ion collisions and other extreme environments. The following sections systematize the theoretical framework, experimental signatures, phase structure, radiative and collisional phenomena, and open questions in the study of chirally asymmetric QGP.

1. Quantum Anomaly, Axial Chemical Potential, and Hydrodynamic Realization

At the microscopic level, massless Dirac fermions subject to electromagnetic fields satisfy the anomaly relation

$$\partial_\mu J_5^\mu = -C\,\mathbf{E}\cdot\mathbf{B}$$

with anomaly coefficient $C = e^2/(2\pi^2)$ (single Dirac fermion) or $C = N_c\,\sum_f Q_f^2/(2\pi^2)$ in QCD (Shi et al., 2019). In thermal equilibrium, a chiral imbalance is encoded by a nonzero axial chemical potential, $\mu_5$.

A hydrodynamic description extends conserved currents to include anomaly-induced non-dissipative terms: $$J_5^\mu = n_5 u^\mu + \xi_5 B^\mu + \xi_\omega \omega^\mu + \ldots$$

$J^\mu = n u^\mu + \xi B^\mu + \ldots$

where $u^\mu$ is the fluid four-velocity, $$B^\mu = \tfrac12 \varepsilon^{\mu\nu\rho\sigma} u_\nu F_{\rho\sigma}$$ is the magnetic field in the fluid frame, and $$\omega^\mu = \varepsilon^{\mu\nu\rho\sigma} u_\nu \partial_\rho u_\sigma$$ is the vorticity. The transport coefficients are fixed by anomaly matching (Shi et al., 2019, Becattini, 2018).

The two-component interpretation ("chiral superfluid") arises naturally when bosonizing low-lying Dirac modes with a finite cut-off, yielding a collective axion-like field $\theta(x)$ whose gradient dynamics encode chiral transport. The Josephson relation,

$u^\mu\partial_\mu\theta + \mu_5 = 0,$

connects fluid kinematics and the chiral sector (Kalaydzhyan, 2014, Kalaydzhyan, 2012).

2. Chiral Magnetic, Electric, and Vortical Effects

Chirally asymmetric QGP manifests by several anomaly-driven transport phenomena:

• Chiral Magnetic Effect (CME):

An applied magnetic field induces a vector current:

$$\mathbf{J} = \sigma_5 \mathbf{B}, \quad \sigma_5 = C \mu_5 = \frac{e^2 \mu_5}{2\pi^2}$$

(Shi et al., 2019, Becattini, 2018)

• Chiral Electric Separation Effect (CESE):

An external electric field $e\mathbf{E}$ induces an axial current

$$\mathbf{J}_A = \sigma_{\chi e} (e\mathbf{E}), \quad \sigma_{\chi e} \propto \mu \mu_5$$

with $$\sigma_{\chi e} = 14.5163 \, T \, \mathrm{Tr}_f(Q_e Q_A)/(g^4 \ln(1/g)) \times \mu \mu_5/T^2$$ for $u, d$ flavors (Jiang et al., 2014).

• Chiral Vortical Effect (CVE):

Fluid vorticity induces a current with

$J^\mu = (\mu^2 + \mu_5^2) 4\pi^2 \omega^\mu$

(Becattini, 2018, Jiang et al., 2015).

• Chiral Dipole (Wave) Effect: Novel current terms arise in two-component models,

$$J_{\mathrm{dip}}^\mu = -C_{\mathrm{em}} u^\mu (\partial\theta \cdot B) = +C_{\mathrm{em}} \mu_5 (u\cdot B) u^\mu$$

corresponding to spatially modulated electric dipoles (Kalaydzhyan, 2014).

Table: Selected anomaly-induced conductivities in chirally asymmetric QGP

Effect	Current Structure	Conductivity formula
CME	$J \propto \mu_5 B$	$\sigma_5 = C \mu_5 = \frac{e^2 \mu_5}{2\pi^2}$
CESE	$J_A \propto \mu \mu_5 E$	$\sigma_{\chi e} = (\#) T \frac{\mu \mu_5}{T^2}$
CVE	$J \propto (\mu^2+\mu_5^2) \omega$	$4\pi^2 (\mu^2+\mu_5^2)$

3. Collective Modes, Instabilities, and Phase Structure

The interplay of anomaly transport and hydrodynamics produces collective excitations:

• Chiral Magnetic Wave (CMW): A coupled propagation of vector and axial charge densities in magnetic field, with dispersion

$$\omega(k) = \pm v_\chi k - i D_L k^2, \;\; v_\chi = \frac{N_c e B}{4\pi^2} \frac{\partial\mu_{R, L}}{\partial j^0_{R, L}}$$

(Burnier et al., 2011). The CMW induces an electric quadrupole moment in QGP, leading to splitting of charged pion elliptic flows $v_2(\pi^+) < v_2(\pi^-)$, with relative difference $\Delta v_2 \sim r A_\pm$ and $r \sim 0.04$ at low-energy RHIC.

• Chiral Vortical Wave (CVW): A gapless mode in rotating QGP with speed

$$V_{\Omega} = \frac{\mu_0 \omega}{2\pi^2 \chi_{\mu_0}}$$

induces flavor charge quadrupoles and leads to small but characteristic splitting in $\Lambda$ elliptic flow (Jiang et al., 2015).

• Chiral Plasma Instability: Berry-curvature kinetic theory predicts unstable plasma modes with exponential growth rate,

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

for $0 < k < \frac{\alpha \mu_5}{\pi}$ (Akamatsu et al., 2013). In QCD, color-damping yields time scales $$\tau_{\mathrm{QCD}} \sim 1/(\alpha_s^2 \mu_5 \ln(1/\alpha_s))$$.

Chiral-isospin chemical potentials ($\mu_{I5}$) induce charged pion condensation (PC) at finite baryon density and temperature, with a duality symmetry between chiral symmetry breaking and PC. The phase diagrams exhibit a persistent PC$_n$ domain for $\nu_5 > 0$ up to temperatures $T \sim 0.2-0.35\,m$ ($\sim 60$--$100$ MeV) (Khunjua et al., 2019).

4. Nonperturbative Suppression and Experimental Strategies

The nonperturbative QCD interactions strongly modulate anomaly transport. Field Correlator Method (FCM) analysis reveals that

• At high temperature $T \gg T_c$ (e.g., LHC/top RHIC), chromomagnetic confinement screens the CME conductivity, $S(T, \mu_B) \to 0$
• Only in a narrow strip $T \sim 1-1.2\,T_c$ ($160$--$200$ MeV) and rather large baryon chemical potential $\mu_B \gtrsim 600$ MeV does the CME remain unsuppressed, $S(T, \mu_B) \to 1$
• At low $T < T_c$, remnants of confinement also suppress anomaly transport (Abramchuk, 24 Mar 2025)

This suggests that QGP formed at elevated $\mu_B$ and moderate $T$—as in RHIC-BES, SPS, FAIR, NICA, J-PARC-HI—is optimal for CME studies, whereas collider energies producing high $T$ and low $\mu_B$ are not.

To isolate CME signal from vorticity- and flow-driven backgrounds, isobar-subtraction strategy (Ru+Ru vs. Zr+Zr) is employed: by matching charged multiplicity and elliptic flow ($N_{\mathrm{ch}}$, $v_2$) and exploiting a controlled difference in magnetic field ($B_{\mathrm{Ru}} \sim 10$–$25\%$ larger), the difference $\Delta \gamma_{Ru} - \Delta \gamma_{Zr}$ and the ratio $$\zeta^{EP}_{\mathrm{isobar}} = [\Delta\gamma_{Ru} - \Delta\gamma_{Zr}]/[\Delta\delta_{Ru} - \Delta\delta_{Zr}] \simeq -0.41 \pm 0.27$$ (EP) and $-0.90 \pm 0.45$ (RP) provide robust CME observables independent of $\mu_5$ uncertainty (Shi et al., 2019).

5. Radiation and Energy Loss: Chiral Cherenkov and Anomaly-Modified Bremsstrahlung

Chirally asymmetric QGP hosts emergent axion-like modes $\theta(x)$ from sphaleron-induced topological charge fluctuations. These couple anomalously to photons and gluons (axion electrodynamics/chromodynamics), changing their dispersion relations: $\omega^2 = k^2 - \lambda \sigma_\chi |k|$ where $\lambda$ is the circular polarization and $\sigma_\chi \sim C_A \mu_5$ (Hansen et al., 24 Sep 2024).

Key consequences:

• Chiral Cherenkov radiation: Free charged particles radiate even in vacuum ($\varepsilon=1$) if $\sigma_\chi \neq 0$. The quantum energy loss per unit length is

$\frac{dE}{dz} = \frac{1}{3} \alpha q^2 \sigma_\chi$

and color Cherenkov losses scale as $dE_{q \to qg}/dz \sim \alpha_s g^2 b_0 E$ (Hansen et al., 24 Sep 2024).

• Anomaly-modified bremsstrahlung: Scattering cross sections and energy loss become helicity dependent, with parametric corrections $\sim \sigma_\chi E / m^2$ or $\sigma_\chi E / \mu^2$ to standard Bethe–Heitler losses.
• Experimental relevance: The angular and polarization structure of emitted photons and gluons, and jet energy loss asymmetries, encode the presence of $\mu_5 \neq 0$, allowing direct access to QCD topological fluctuations.

6. Anisotropy, Mass Effects, and Lattice/QCD Model Evidence

Anisotropic QGP, as realized via holographic AdS backgrounds with nonzero spatial anisotropy parameter $a$, modifies the CME response for massive quarks. At fixed temperature, increasing $a$ enhances the magnitude of CME for quarks of finite mass (while remaining unchanged for massless quarks) (Ali-Akbari et al., 2014). The functional dependence is $$\langle J(a)/J(a=0) \rangle \simeq 1 + c_1 (a/T) + \cdots$$ for $m \neq 0$, with extension of the CME window to larger mass thresholds as anisotropy grows.

Lattice studies in the window $T_c < T < 2 T_c$ show spectral gaps between “near-zero” Dirac eigenmodes and the bulk, with low-lying coherent modes forming the chiral superfluid component and the gapped sector giving rise to standard thermalized QGP ("normal fluid") (Kalaydzhyan, 2014, Kalaydzhyan, 2012). The bosonization of IR modes yields the axion field $\theta(x)$, embedding all anomaly structures in the low-energy effective action.

7. Limitations, Open Problems, and Future Directions

Current theoretical constraints include:

• FCM and HTL approximations: Validity limited to certain regions of phase diagram; extrapolation into $T \sim (1-3) T_c$ and strong-coupling regimes requires nonperturbative tools (Abramchuk, 24 Mar 2025, Jiang et al., 2014).
• Back-reaction and time-dependence of $\mu_5$ and emergent axion domains: Their rapid evolution and spatial structure are not fully resolved in current hydrodynamic implementations.
• Quantitative modeling of experimental observables: Requires dynamic integration of anomaly transport coefficients, initial $\mu_5$ fluctuations, electromagnetic field evolution, and subleading backgrounds.
• Phenomenology of pion-condensed domains at high $T$ and $\mu_{I5} \neq 0$ calls for further studies in full (3+1)-dimensional QCD-based models, lattice simulations, and astrophysical settings (Khunjua et al., 2019).

A plausible implication is that, as experimental programs expand into lower collision energies and higher baryon densities, the prospect for direct observation of genuine chirally asymmetric QGP and robust confirmation of anomaly-induced transport increases. Future directions include refining initial condition models, implementing full spin hydrodynamics, and extending searches to differentiated species, differential rapidities, and other collision systems (Shi et al., 2019, Becattini, 2018).

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The synthesis above integrates the full anomaly-driven phenomenology, transport theory, collective-mode dynamics, phase structure, and experimental methodologies required for study of a chirally asymmetric quark–gluon plasma.