Kinetic Theory of Quasiparticles, Retarded Correlators and Hydrodynamics

Within the relaxation time approximation under a constant mass profile, we investigate the collective dynamics of a system of massive relativistic particles described by the Maxwell-Boltzmann equilibrium distribution. We analytically derive the two-point retarded correlation functions for both charge and energy-momentum tensor components at arbitrary momentum and frequency. We expand our results in the limits of very small and very large mass-to-temperature ratios ($m/T$). Similar to the massless case, we identify a critical threshold in ($k \tau$) below which the correlators permit physical solutions. This behavior arises from a logarithmic branch cut in the spectral function. At higher momenta, solutions emerge significantly below this cut, corresponding to non-hydrodynamic modes. Our analysis demonstrates that hydrodynamic poles dominate in the strong coupling regime, while the weak coupling regime features a logarithmic branch cut extending along $\omega = k$ and $\omega = -k$. Notably, in the sound channel, finite mass modifies the standard propagating sound mode, converting it into a purely imaginary mode. In contrast, the shear channel exhibits modes that asymptotically converge to their massless counterparts. Additionally, we compute the transport coefficients for shear and bulk viscosity, along with higher-order gradient corrections up to third order, expressed as perturbative expansions in both the small and large ($m/T$) regimes.