Transport coefficients of chiral fluid dynamics using low-energy effective models

We investigate the first-order transport coefficients of a fluid made of quasiparticles with a temperature-dependent mass extracted from chiral models. We describe this system using an effective kinetic theory, given by the relativistic Boltzmann equation coupled to a temperature-dependent background field determined from the thermal masses. We then simplify the collision term using the relaxation time approximation and implement a Chapman-Enskog expansion to calculate all first-order transport coefficients. In particular, we compute the bulk and shear viscosities using thermal masses extracted from the linear sigma model coupled with constituent quarks and the NJL model.

💡 Research Summary

The paper addresses the calculation of first‑order transport coefficients for a relativistic fluid composed of quasiparticles whose masses depend on temperature, a situation relevant for the quark–gluon plasma (QGP) near the chiral transition. The authors employ two widely used low‑energy chiral effective models—the linear sigma model coupled to constituent quarks (LSMq) and the Nambu–Jona‑Lasinio (NJL) model—to obtain temperature‑dependent effective quark masses M(T). These masses are interpreted as a background field in the relativistic Boltzmann equation, leading to a modified energy‑momentum tensor that includes an additional scalar term B(T). Energy‑momentum conservation then yields a differential relation ∂μB = –½∂μM²⟨1⟩₀, which can be integrated to give B(T) explicitly in terms of M(T) and Bessel functions.

Thermodynamic quantities (energy density ε₀, pressure P₀, entropy density s₀, trace anomaly I₀ = ε₀ – 3P₀) are computed from the modified tensor. Both models reproduce the expected high‑temperature limit of an ideal massless gas (P₀ = ε₀/3) after chiral symmetry restoration, but display markedly different behavior near the crossover. The LSMq shows a sharper drop of the condensate and consequently a more pronounced peak in B(T), a steeper rise of the trace anomaly, and a rapid decrease of the speed of sound c_s² just before the critical temperature. The NJL model yields smoother variations because its chiral restoration occurs over a broader temperature interval.

A central methodological contribution is the treatment of the collision term. The standard Anderson‑Witting relaxation‑time approximation (RTA) violates particle‑number and energy‑momentum conservation when the relaxation time τ_R depends on momentum. The authors adopt a recent improvement that introduces a momentum‑dependent τ_R(p) while adding a correction term that restores the conservation laws. In this framework the linearized Boltzmann operator becomes ˆL_R φ_p = –E_p τ_R(p) f⁽⁰⁾_p φ_p, where φ_p = (f – f⁽⁰⁾)/f⁽⁰⁾.

Using the Chapman‑Enskog expansion to first order, the non‑equilibrium correction to the distribution function is expressed in terms of gradients of temperature and flow velocity. Substituting this correction into the definitions of the stress tensor and the particle current yields analytic expressions for the shear viscosity η, bulk viscosity ζ, and the sound attenuation length. The shear viscosity scales essentially with the average relaxation time and typical momentum, while the bulk viscosity is proportional to the square of a combination of (1/3 – c_s²) and the temperature derivative of the mass, reflecting sensitivity to conformal symmetry breaking.

Numerical evaluation shows that η/T³ remains relatively smooth across the transition for both models, whereas ζ/T³ exhibits a pronounced peak near the chiral crossover, especially for the LSMq where the mass drops abruptly. The speed of sound squared c_s² follows the expected dip in the LSMq and a milder dip in the NJL case. These features are consistent with phenomenological extractions of transport coefficients from heavy‑ion collision data, which suggest a small shear viscosity and a bulk viscosity that may become sizable near the transition.

The authors conclude that incorporating temperature‑dependent quasiparticle masses together with a conservation‑law‑consistent RTA provides a robust kinetic‑theory framework for linking microscopic chiral dynamics to macroscopic transport properties. This approach can be extended to multi‑component systems, finite chemical potential, and higher‑order (second‑order) hydrodynamics, offering a promising path toward a more complete theoretical description of the QGP and related strongly interacting matter.