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We establish and elucidate the physical meaning of the appearance of anacausal mode in the sigma mesonic channel, found in the previous work by thepresent authors, when the system approaches the $\mathrm{Z}_{2}$ criticalpoint. The functional renormalization group method is applied to thetwo--flavor quark--meson model with varying current quark mass $m_q$ even awayfrom the physical value at which the pion mass is reproduced. We firstdetermine 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 linesbeing located; they altogether make a wing-like shape quite reminiscent ofthose known in the condensed matters with a tricritical point. We thencalculate the spectral functions $\rho_{\sigma, \pi}(\omega, p)$ in the scalarand pseudoscalar channel around the critical points. We find that the sigmamesonic mode becomes tachyonic with a superluminal velocity at finite momentabefore the system reaches the $\mathrm{Z}_{2}$ point from the lower density,even for $m_q$ smaller than the physical value. One of the possibleimplications of the appearance of such a tachyonic mode at finite momenta isthat the assumed equilibrium state with a uniform chiral condensate is unstabletoward a state with an inhomogeneous $\sigma$ condensate. No such an anomalousbehavior is found in the pseudoscalar channel. We find that the$\sigma$-to-$2\sigma$ coupling due to finite $m_q$ play an essential role forthe drastic modification of the spectral function.

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