Dynamical Density Fluctuations around QCD Critical Point Based on Dissipative Relativistic Fluid Dynamics-possible fate of Mach cone at the critical point-

The purpose of this paper is twofold. Firstly, we study the dynamical density fluctuations around the critical point(CP) of Quantum Chromodynamics(QCD) using dissipative relativistic fluid dynamics in which the coupling of the density fluctuations to those of other conserved quantities is taken into account. We show that the sound mode which is directly coupled to the mechanical density fluctuation is attenuated and in turn the thermal mode becomes the genuine soft mode at the QCD CP. We give a speculation on the possible fate of a Mach cone in the vicinity of the QCD CP as a signal of the existence of the CP on the basis of the above findings. Secondly, we clarify that the so called first-order relativistic fluid dynamic equations have generically no problem to describe fluid dynamic phenomena with long wave lengths contrary to a naive suspect whereas even Israel-Stewart equation, a popular second-order equation, may not describe the hydrodynamic mode in general depending on the value of the relaxation time.Comment: 29pages, 4figures; accepted version for publication in Prog. Theor. Phys. Introduction and Sec.3 are somewhat modified to make clearer the purpose of this paper and the discussions on the critical behaviors, respectively. A few references are added. The conclusions are not changed at al

Regularized Renormalization Group Reduction of Symplectic Map

By means of the perturbative renormalization group method, we study a long-time behaviour of some symplectic discrete maps near elliptic and hyperbolic fixed points. It is shown that a naive renormalization group (RG) map breaks the symplectic symmetry and fails to describe a long-time behaviour. In order to preserve the symplectic symmetry, we present a regularization procedure, which gives a regularized symplectic RG map describing an approximate long-time behaviour succesfully

Precursor of Color Superconductivity in Hot Quark Matter

We investigate possible precursory phenomena of color superconductivity in quark matter at finite temperature T with use of a simple Nambu-Jona-Lasinio model. It is found that the fluctuating pair field exists with a prominent strength even well above the critical temperature T_c. We show that the collective pair field has a complex energy located in the second Riemann sheet, which approaches the origin as T is lowered to T_c. We discuss the possible relevance of the precursor to the observables to be detected in heavy ion collisions.Comment: 5 pages, 3 figures, version to appear in Phys. Rev. D. Discussions are enlarged on the physical origin of the large fluctuation of the pair field and its phenomenological consequences. References are adde

Chiral and Color-superconducting Phase Transitions with Vector Interaction in a Simple Model (Addenda)

In the preceding paper(Prog.Theor.Phys.108(2002)929 or hep-ph/0207255), we have shown that the critical line of the first order chiral transition of QCD can have two endpoints. In this addendum, we elucidate the mechanism to realize the two-endpoint structure in the QCD phase diagram and argue the robustness for the appearance of such an interesting phase structure

Dynamical Reduction of Discrete Systems Based on the Renormalization Group Method

The renormalization group (RG) method is extended for global asymptotic analysis of discrete systems. We show that the RG equation in the discretized form leads to difference equations corresponding to the Stuart-Landau or Ginzburg-Landau equations. We propose a discretization scheme which leads to a faithful discretization of the reduced dynamics of the original differential equations.Comment: LaTEX. 12pages. 1 figure include

Pseudogap of Color Superconductivity in Heated Quark Matter

We show that the pseudogap of the quark density of states is formed in hot quark matter as a precursory phenomenon of the color superconductivity on the basis of a low-energy effective theory. We clarify that the decaying process of quarks near Fermi surface to a hole and the diquark soft mode (qq)_{soft} is responsible for the formation of the pseudogap. Our result suggests that the pseudogap is a universal phenomenon in strong coupling superconductors.Comment: Introduction is largely rewritten and minor changes are made in other parts of the text. Some referenes with comments are added. Numerical errors in the figures are corrected. To appear in Phys. Rev.

Renormalization Group Method Applied to Kinetic Equations: roles of initial values and time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include Boltzmann equation in classical mechanics, Fokker-Planck equation, a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method is clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time $t_0$ to be on the averaged distribution function to be determined. The averaged distribution function may be thought as an integral constant of the solution of microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time $t_0$, thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in Fokker-Planck equation are also performed in a unified way in the present method.Comment: The detailed derivations are added to section 5 (fluiddynamical limit of Boltzmann equation) and to Appendix B (Adiabatic elimination of fast variables in Fokker-Planck equation) which is moved to the text as a section. Other minor corrections are made all over the text including typo

Precursor of Color Superconductivity

We investigate possible precursory phenomena of color superconductivity at finite temperature $T$ with an effective theory of QCD. It is found that the fluctuation of the diquark pair field exists with a prominent strength even well above the critical temperature $T_c$. We show that such a fluctuaiton forms a collective mode, the corresponding pole of which approaches the origin as $T$ is lowered to $T_c$ in the complex energy plane. We discuss the possible relevance of the precursor to the observables to be detected in heavy-ion collisions.Comment: 4 pages, 5 figures, Talk presented at the XVIth International Conference on Particles and Nuclei (PANIC02), Osaka, Japan, Sep.30 - Oct.4, 2002, Uses espcrc1.st