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

We first review the method to calculate the spectral functions in the functional renormalization group (FRG) approach, which has been recently developed. We also provide the numerical stability conditions given by the present authors for a generic nonlinear evolution equation that are necessary for obtaining the accurate effective potential from the flow equation in the FRG. As an interesting example, we report the recent calculation of the spectral functions of the mesonic and particle-hole excitations using a chiral effective model of Quantum Chromodynamics (QCD); we extract the dispersion relations from them and try to reveal the nature of the soft modes at the QCD critical point (CP) where the phase transition is second order. Our result shows that a clear development and the softening of the phonon mode in the space-like region as the system approaches the CP; furthermore it turns out that the sigma mesonic mode once in the time-like region gets to merge with the phonon mode in the close vicinity of the CP, implying a novel possibility about the nature of the soft mode of the QCD CP.