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

We make an intensive investigation of the soft mode at the quantum chromodynamics (QCD) critical point on the basis of the functional renormalization group (FRG) method in the local potential approximation. We calculate the spectral functions $\rho_{\sigma, \pi}(\omega,\, p)$ in the scalar ($\sigma$) and pseudoscalar ($\pi$) channels beyond the random phase approximation in the quark--meson model. At finite baryon chemical potential $\mu$ with a finite quark mass, the baryon-number fluctuation is coupled to the scalar channel and the spectral function in the $\sigma$ channel has a support not only in the time-like ($\omega\,>\,p$) but also in the space-like ($\omega\,<\, p$) regions, which correspond to the mesonic and the particle--hole phonon excitations, respectively. We find that the energy of the peak position of the latter becomes vanishingly small with the height being enhanced as the system approaches the QCD critical point, which is a manifestation of the fact that the phonon mode is the {\em soft mode} associated with the second-order transition at the QCD critical point, as has been suggested by some authors. Moreover, our extensive calculation of the spectral function in the $(\omega, p)$ plane enables us to see that the mesonic and 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 eventually merges into the phonon mode as the system approaches the critical point more closely. This implies that the sigma-mesonic mode also becomes soft at the critical point. We also provide numerical stability conditions that are necessary for obtaining the accurate effective potential from the flow equation.