Relaxation time approximation revisited and non-analytical structure in retarded correlators

In this paper, we give a rigorous mathematical justification for the relaxation time approximation (RTA) model. We find that only the RTA with an energy-independent relaxation time can be justified in the case of hard interactions. Accordingly, we propose an alternative approach to restore the collision invariance lacking in traditional RTA. Besides, we provide a general statement on the non-analytical structures in the retarded correlators within the kinetic description. For hard interactions, hydrodynamic poles are the long-lived modes. Whereas for soft interactions, commonly encountered in relativistic kinetic theory, the gapless eigenvalue spectrum of linearized collision operator leads to gapless branch-cuts. We note that particle mass and inhomogeneous perturbations would complicate the above-mentioned non-analytical structures.

💡 Research Summary

This paper provides a rigorous mathematical justification of the relaxation‑time approximation (RTA) within relativistic kinetic theory and clarifies the nature of non‑analytic structures that appear in retarded correlation functions. Starting from the Boltzmann equation, the authors linearize around a homogeneous equilibrium state and study the properties of the linearized collision operator (L_{0}). They show that (L_{0}) is self‑adjoint and positive‑semi‑definite in an appropriate Hilbert space, which guarantees a real spectrum that may contain both discrete eigenvalues and a continuous branch.

For hard (momentum‑independent) interactions the spectrum exhibits a clear gap: the smallest eigenvalue is isolated from the rest. In this situation the inverse of that eigenvalue can be identified with a single, energy‑independent relaxation time (\tau_{R}). Consequently, the traditional Anderson‑Witting (relativistic BGK) form of the RTA, (p!\cdot!\partial f = -(u!\cdot!p)/\tau_{R},(f-f_{\rm eq})), is mathematically exact. The authors emphasize that any energy dependence of (\tau_{R}) (parameterized as (\tau_{R}\propto (\beta,u!\cdot!p)^{\alpha})) would require a redefinition of the inner product and of the weight function; only for very small (|\alpha|) does the construction remain well‑behaved, and even then the resulting “RTA‑like” operator is not a proper approximation of the full collision term.

In contrast, for soft (e.g. gauge‑theory) interactions the eigenvalue spectrum of (L_{0}) becomes gapless: a continuum of low‑lying eigenvalues accumulates near zero. No isolated smallest eigenvalue exists, so a single constant (\tau_{R}) cannot capture the dynamics. The authors demonstrate that the retarded correlators in this regime do not possess isolated poles only; instead they develop branch‑cut singularities reflecting the infinite tower of non‑hydrodynamic modes. Hydrodynamic poles remain present as the longest‑lived excitations, but they are embedded in a dense spectrum that dominates the analytic structure.

The paper further discusses how particle mass and spatially inhomogeneous perturbations modify the collision operator’s symmetry, thereby complicating the eigenfunction basis and potentially altering the location and shape of branch cuts.

To address the shortcomings of the traditional RTA, the authors propose a novel truncation scheme: one retains the few slowest eigenmodes of (L_{0}) (including the hydrodynamic sector) explicitly and replaces the remaining fast sector by an effective relaxation time (\tau_{\rm eff}). This “spectral‑truncated RTA” respects collision invariants, works for both hard and soft interactions, and can accommodate modest energy dependence without violating the mathematical structure of the operator.

In summary, the work clarifies that (i) an energy‑independent relaxation time is the only mathematically justified form of RTA for hard interactions, (ii) soft interactions inevitably generate gapless spectra leading to branch‑cut structures in retarded correlators, and (iii) a spectral truncation of the full linearized collision operator offers a systematic improvement over the conventional BGK‑type models, opening a path toward more accurate kinetic‑theory descriptions of relativistic plasmas.