Soft Modes at the Critical End Point in the Chiral Effective Models

At the critical end point in QCD phase diagram, the scalar, vector and entropy suscep-tibilities are known to diverge. The dynamic origin of this divergence is identified within the chiral effective models as softening of a hydrodynamic mode of the particle-hole–type motion, which is a consequence of the conservation law of the baryon number and the energy. Theoretical studies of QCD thermodynamics have been predicting rich phase structure in the plane of the temperature T and the chemical potential 3µ for baryon number, as discussed in the workshop, and there are heavy–ion programs now ongoing and planned in order to explore these new states of matter experimentally. Based on the lattice QCD results 1) as well as the effective model calculations 2) – 5) it is strongly argued that as one of the prominent features of the real diagram there exists a critical end point (CEP) *) of the first–order transition line, which is reviewed by M.A. Stephanov 6). Since it is a truly singular point, we expect clearer information of it from experiments. The basic property of CEP is the divergence of the suscep-tibilities of the baryon number and entropy densities as well as of the scalar density. Thus it is suggested that the fluctuations in the low–momentum pions and nucleons show anomalous behavior as functions of experimental parameters near CEP 7), 8). Theoretical estimates indicate the possibility that the CEP gives still significant effects on these observables although the finiteness of the experimental geometry in space and time will round off the critical growth of these fluctuations 8), 9). Near the critical point there will be at least one mode which becomes soft representing the criticality 10). In this report we address the issue of the soft mode associated with CEP, which causes these critical divergences, within the Nambu–Jona-Lasinio model (NJL) 11), 12) and the time–dependent Ginzburg–Landau (TDGL) approach 12). The correct identification of the soft mode associated with CEP is crucial to determine the time evolution of the critical fluctuations and to study the experimental signals. So far the static properties of CEP have been investigated, where the underlying (approximate) chiral symmetry is usually emphasized. The effective potential with the scalar density σ as the order parameter, becomes flat around a nonzero equilibrium value of σ at CEP, reflecting the divergence of the scalar susceptibility. *) In condensed matter physics the term critical end point is used …