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We investigate the soft mode at the QCD critical point (CP) on the basis of the functional renormalization group. We calculate the spectral functions in the meson channels in the two-flavor quark--meson model. Our result shows that the energy of the peak position of the particle--hole mode in the sigma channel becomes vanishingly small as the system approaches the QCD CP, which is a manifestation of the softening of the phonon mode. We also extract the dispersion curves of the mesonic and the phonon mode, a hydrodynamic mode which leads to a finding that the dispersion curve of the sigma-mesonic mode crosses the light-cone into the space-like momentum region, and then eventually merges into the phonon mode as the system approaches further close to the CP. This may suggest that the sigma-mesonic mode forms the soft mode together with the hydrodynamic mode at the CP.

Novel Picture of the Soft Modes at the QCD Critical Point Based on the FRG Method