Tachyonic instability of the scalar mode prior to QCD critical point based on Functional renormalization-group method

We establish and elucidate the physical meaning of the appearance of an acausal mode in the sigma mesonic channel, found in the previous work by the present authors, when the system approaches the $\mathrm{Z}_{2}$ critical point. The functional renormalization group method is applied to the two--flavor quark--meson model with varying current quark mass $m_q$ even away from the physical value at which the pion mass is reproduced. We first determine the whole phase structure in the three-dimensional space $(T, \mu, m_q)$ consisting of temperature $T$, quark chemical potential $\mu$ and $m_q$, with the tricritical point, $\mathrm{O}(4)$ and $\mathrm{Z}_{2}$ critical lines being located; they altogether make a wing-like shape quite reminiscent of those known in the condensed matters with a tricritical point. We then calculate the spectral functions $\rho_{\sigma, \pi}(\omega, p)$ in the scalar and pseudoscalar channel around the critical points. We find that the sigma mesonic mode becomes tachyonic with a superluminal velocity at finite momenta before the system reaches the $\mathrm{Z}_{2}$ point from the lower density, even for $m_q$ smaller than the physical value. One of the possible implications of the appearance of such a tachyonic mode at finite momenta is that the assumed equilibrium state with a uniform chiral condensate is unstable toward a state with an inhomogeneous $\sigma$ condensate. No such an anomalous behavior is found in the pseudoscalar channel. We find that the $\sigma$-to-$2\sigma$ coupling due to finite $m_q$ play an essential role for the drastic modification of the spectral function.