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  • Open Access

Normal mode analysis within relativistic massive transport

Xin Lin1, Qiu-Ze Sun2, Xin-Hui Wu1, and Jin Hu1,*

  • *Contact author: hu-j23@fzu.edu.cn
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Phys. Rev. D 113, 076010 – Published 10 April, 2026

DOI: https://doi.org/10.1103/zj9g-2k12

Abstract

In this paper, we address the normal mode analysis of the linearized Boltzmann equation for massive particles in the relaxation time approximation. We find that the sound channel and the heat channel are coupled in the massive case, which contrasts with their decoupling in the massless relaxation time approximation. In the limit of zero mass, our results smoothly reduce to the known case where the two channels decouple. By utilizing the argument principle in complex analysis, we determine the existence condition for collective modes and find the onset transition behavior of collective modes previously observed in massless systems. We numerically determine the critical wave number κc for the existence of each mode under various values of the scaled mass. Within the range of scaled masses considered, the critical wave numbers for the heat and shear channels increase with increasing scaled mass, while that of the sound channel exhibits a nonmonotonic dependence on the scaled mass. In addition, we analytically derive the dispersion relations for these collective modes in the long-wavelength limit. Notably, kinetic theory also incorporates collisionless damping effects known as Landau damping. We find that the branch cut structure responsible for Landau damping differs from the massless case: Whereas the massless system features only two branch points, the massive system exhibits an infinite number of such points forming a continuous branch cut.

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Phys. Rev. D 113, 076010– Published 10 April, 2026
  • Received 2 December 2025
  • Accepted 24 March 2026
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