Magnetoviscosity of relativistic plasma

Using first-principles quantum field-theoretical methods, we investigate the shear and bulk viscosities of strongly magnetized relativistic plasmas. The analysis is performed within the weak-coupling approximation and utilizes known results for the fermion damping rates in the Landau-level representation, $Γ_{n}(p_{z})$, which are dominated by one-to-two and two-to-one processes in the presence of a strong magnetic field. The transverse and longitudinal components of the viscosities are derived using Kubo’s linear response theory. Our results reveal a pronounced anisotropy in both shear and bulk viscosities induced by the magnetic field. In the case of an electron-positron plasma, where the weak-coupling approximation is well justified, the dimensionless longitudinal shear viscosity $η_{\parallel}/T^3$ increases rapidly with the magnetic field strength, while the transverse component $η_{\perp}/T^3$ decreases and can even drop below the KSS bound at sufficiently large fields. In contrast, both the dimensionless longitudinal and transverse bulk viscosities, $ζ_{\perp}/T^3$ and $ζ_{\parallel}/T^3$, initially rise from small values, reach a maximum, and then gradually decrease toward zero. We find that the bulk viscosity is highly sensitive to the longitudinal and transverse components of the sound velocity, particularly at high magnetic fields, indicating that its quantitative values should be interpreted with caution. We also calculate an additional cross viscosity, which is negative and whose magnitude increases with the magnetic field strength. Finally, we discuss the physical implications of these magnetoviscosity results in the contexts of magnetar physics and the strongly magnetized quark-gluon plasma produced in heavy-ion collisions.

💡 Research Summary

The paper presents a first‑principles quantum‑field‑theoretic calculation of shear and bulk viscosities in strongly magnetized relativistic plasmas, focusing on the weak‑coupling regime. Using the Kubo linear‑response formalism, the authors express the viscosity tensor components in terms of retarded correlators of the energy‑momentum tensor. Because a background magnetic field breaks spatial isotropy, the shear viscosity splits into a longitudinal component (η∥) parallel to the field and a transverse component (η⊥) perpendicular to it. Likewise, the bulk viscosity separates into longitudinal (ζ∥) and transverse (ζ⊥) parts, and a novel cross‑viscosity (ζ×) appears, coupling longitudinal stress to transverse pressure.

The microscopic input is the fermion propagator in the Landau‑level basis, whose spectral representation involves the damping rates Γₙ(p_z). These rates, derived in earlier work, are dominated by one‑to‑two and two‑to‑one processes that become kinematically allowed in a strong field. The authors carefully incorporate contributions from all Landau levels, showing that higher levels remain important even when eB≫T².

For an electron‑positron (QED) plasma, numerical evaluation over a wide range of magnetic‑field strengths (eB/T² from 10³ to 10⁶) reveals striking anisotropies. The longitudinal shear viscosity η∥/T³ grows rapidly with B, reaching values of order 10⁻¹ at the strongest fields considered, while the transverse shear viscosity η⊥/T³ declines and can fall below the Kovtun‑Son‑Starinets (KSS) bound 1/4π for eB/T² ≳10⁴. This reflects the fact that motion parallel to the field is less hindered, whereas transverse motion is strongly constrained by Landau quantization.

Both bulk viscosities exhibit a non‑monotonic dependence on B. Starting from very small values at weak fields, ζ∥/T³ and ζ⊥/T³ increase to a maximum at intermediate B, then gradually decrease toward zero as the magnetic field becomes dominant. The bulk viscosities are highly sensitive to the longitudinal and transverse sound speeds v_s,∥ and v_s,⊥, which themselves depend on the magnetization and the derivative of pressure with respect to energy density. Consequently, the quantitative bulk‑viscosity values must be interpreted with caution, especially at very high fields where the definition of transverse pressure involves the assumption of conserved magnetic flux.

The cross‑viscosity ζ× is found to be negative for all field strengths, and its magnitude |ζ×|/T³ grows roughly linearly with B. This term has no analogue in isotropic fluids; it represents a dissipative coupling between longitudinal stress and transverse pressure that becomes increasingly important as the magnetic field intensifies.

The authors also outline how the formalism extends to a strongly magnetized quark‑gluon plasma (QCD). While the color degrees of freedom and gluon contributions modify the damping rates, the overall structure of anisotropic viscosities persists. In the QCD case, the larger coupling leads to even stronger separation between η∥ and η⊥, suggesting that magnetohydrodynamic simulations of heavy‑ion collisions should incorporate these anisotropic transport coefficients.

Physical implications are discussed for magnetars, where the longitudinal shear viscosity could affect the damping of Alfvén‑type oscillations, and the reduced transverse shear viscosity may influence the evolution of magnetospheric currents. In heavy‑ion collisions, the anisotropic viscosities could modify the development of elliptic flow and charge‑dependent correlations, while the negative cross‑viscosity might generate novel signatures in the stress‑energy tensor evolution.

In conclusion, the paper demonstrates that, even within a weak‑coupling framework, strong magnetic fields induce pronounced anisotropies in both shear and bulk transport. The inclusion of higher Landau levels is essential for quantitative accuracy. The results provide a solid microscopic foundation for future magnetohydrodynamic modeling of astrophysical and laboratory relativistic plasmas, and they motivate further studies in the strong‑coupling regime and experimental validation through numerical simulations.