We apply the functional renormalization-group (FRG) equation to analyze the nature of the QCD critical point beyond the mean-field approximation by taking into consideration the fact that the soft mode associated with the QCD critical point is a line
ar combination of fluctuations of the chiral condensate and the quark-number density, rather than the pure chiral fluctuations. We first construct an extended quark-meson model in which a new field corresponding to quark-number density is introduced to the conventional one composed of the chiral fields sigma, pi and the quarks. The fluctuations of the quark-number density as well as the chiral condensate are taken into account by solving the FRG equation which contains sigma and the new field as coupled dynamical variables. It is found that the mixing of the two dynamical variables causes a kind of level repulsion between the curvature masses, which in turn leads to an expansion of the critical region of the QCD critical point, depending on the coupling constants in the model yet to be determined from microscopic theories or hopefully by experiments.
The partial restoration of chiral symmetry in nuclear medium is investigated in a model independent way by exploiting operator relations in QCD. An exact sum rule is derived for the quark condensate valid for all density. This sum rule is simplified
at low density to a new relation with the in-medium quark condensate <bar{q}q>*, in-medium pion decay constant F_{pi}^t and in-medium pion wave-function renormalization Z_{pi}*. Calculating Z_{pi}*at low density from the iso-scalar pion-nucleon scattering data and relating F_{pi}^t to the isovector pion-nucleus scattering length b_1^*, it is concluded that the enhanced repulsion of the s-wave isovector pion-nucleus interaction observed in the deeply bound pionic atoms directly implies the reduction of the in-medium quark condensate. The knowledge of the in-medium pion mass m_{pi}* is not necessary to reach this conclusion.
Exploiting operator relations in QCD, we derive a novel and model-independent formula relating the in-medium quark condensate <bar-q q>* to the decay constant F*_t and the wave function renormalization constant Z* of the pion in the nuclear medium. E
valuating Z* at low density from the iso-scalar pion-nucleon scattering data, it is concluded that the enhanced repulsion of the s-wave isovector pion-nucleusinteraction observed in the deeply bound pionic atoms implies directly the reduction of the in-medium quark condensate. The knowlege of the in-medium pion mass is not necessary to reach this conclusion.
