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We first review the method to calculate the spectral functions in thefunctional renormalization group (FRG) approach, which has been recentlydeveloped. We also provide the numerical stability conditions given by thepresent authors for a generic nonlinear evolution equation that are necessaryfor obtaining the accurate effective potential from the flow equation in theFRG. As an interesting example, we report the recent calculation of thespectral functions of the mesonic and particle-hole excitations using a chiraleffective model of Quantum Chromodynamics (QCD); we extract the dispersionrelations from them and try to reveal the nature of the soft modes at the QCDcritical point (CP) where the phase transition is second order. Our resultshows that a clear development and the softening of the phonon mode in thespace-like region as the system approaches the CP; furthermore it turns outthat the sigma mesonic mode once in the time-like region gets to merge with thephonon mode in the close vicinity of the CP, implying a novel possibility aboutthe nature of the soft mode of the QCD CP.

Spectral functions in functional renormalization group approach -- analysis of the collective soft modes at the QCD critical point --