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### Description

The meaningful correlations between the zero-sound modes and the stiffness of the nuclear equation of state (EOS) are uncovered in symmetric nuclear matter with the relativistic mean-ﬁeld (RMF) theory. We found that the high-density zero-sound modes merely exist in models with the stiff EOS. While the soft RMF EOS are usually characteristic of the inclusion of the $\omega$-meson self-interactions (the $\omega^4$ term)$~^{[1]}$, the weakened coupling of the $\omega$-meson self-interactions can ensure the appearance of the zero sound at high density. These results suggest that the high-density zero-sound modes can be used to probe the stiffness of the EOS at supra-normal densities that is still subject to large uncertainty$~^{[2]}$. In asymmetric matter, the correlation also exist clearly between the zero-sound modes and the stiffness of the nuclear EOS. On the other hand, it is found that the symmetry energy has the impact on the behavior of zero sound modes. The symmetry energy in RMF models is modified by introducing the isoscalar-isovector ($\omega-\rho$) coupling term. A softer symmetry energy results in a narrower range for low-density zero sound, while the energy of the damped zero sound at high-density moves much closer to the undamped one as the symmetry energy softens. The implications and effects of zero sounds are also discussed in heavy ion collisions and neutron stars.

[1] R. Y. Yang, W. Z. Jiang, S. N. Wei, and D. R. Zhang, Sci. Rep.7, 1 (2017).

[2] J. Ye, J. Margueron, N. Li, and W. Z. Jiang, Phys. Rev. C 108, 044312 (2023).