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We make an intensive investigation of the soft mode at the quantumchromodynamics (QCD) critical point on the basis of the functionalrenormalization group (FRG) method in the local potential approximation. Wecalculate the spectral functions $\rho_{\sigma, \pi}(\omega,\, p)$ in thescalar ($\sigma$) and pseudoscalar ($\pi$) channels beyond the random phaseapproximation in the quark--meson model. At finite baryon chemical potential$\mu$ with a finite quark mass, the baryon-number fluctuation is coupled to thescalar channel and the spectral function in the $\sigma$ channel has a supportnot only in the time-like ($\omega\,>\,p$) but also in the space-like($\omega\,<\, p$) regions, which correspond to the mesonic and theparticle--hole phonon excitations, respectively. We find that the energy of thepeak position of the latter becomes vanishingly small with the height beingenhanced as the system approaches the QCD critical point, which is amanifestation of the fact that the phonon mode is the {\em soft mode}associated with the second-order transition at the QCD critical point, as hasbeen suggested by some authors. Moreover, our extensive calculation of thespectral function in the $(\omega, p)$ plane enables us to see that the mesonicand phonon modes have the respective definite dispersion relations$\omega_{\sigma.{\rm ph}}(p)$, and it turns out that $\omega_{\sigma}(p)$crosses the light-cone line into the space-like region, and then eventuallymerges into the phonon mode as the system approaches the critical point moreclosely. This implies that the sigma-mesonic mode also becomes soft at thecritical point. We also provide numerical stability conditions that arenecessary for obtaining the accurate effective potential from the flowequation.

Functional renormalization group analysis of the soft mode at the QCD critical point