The Renormalization-Group Method Applied to Asymptotic Analysis of Vector Fields

The renormalization group method of Goldenfeld, Oono and their collaborators is applied to asymptotic analysis of vector fields. The method is formulated on the basis of the theory of envelopes, as was done for scalar fields. This formulation actually completes the discussion of the previous work for scalar equations. It is shown in a generic way that the method applied to equations with a bifurcation leads to the Landau-Stuart and the (time-dependent) Ginzburg-Landau equations. It is confirmed that this method is actually a powerful theory for the reduction of the dynamics as the reductive perturbation method is. Some examples for ordinary diferential equations, such as the forced Duffing, the Lotka-Volterra and the Lorenz equations, are worked out in this method: The time evolution of the solution of the Lotka-Volterra equation is explicitly given, while the center manifolds of the Lorenz equation are constructed in a simple way in the RG method.Comment: The revised version of RYUTHP 96/1. Submitted to Prog. Theor. Phys. (Kyoto) in Feb., 1996. 28 pages. LATEX. No figure

Overdamping Phenomena near the Critical Point in O(NN) Model

We consider the dynamic critical behavior of the propagating mode for the order parameter fluctuation of the O($N$) Ginzburg-Landau theory, involving the canonical momentum as a degree of freedom. We reexamine the renormalization group analysis of the Langevin equation for the propagating mode. We find the fixed point for the propagating mode as well as that for the diffusive one, the former of which is unstable to the latter. This indicates that the propagating mode becomes overdamped near the critical point. We thus can have a sufficient understanding of the phonon mode in the structural phase transition of solids. We also discuss the implication for the chiral phase transition.Comment: 5 pages, 1 figure;v3 modification for correcting a misleading description, conclusion unchange

Thermal unpairing transitions affected by neutrality constraints and chiral dynamics

We discuss the phase structure of homogeneous quark matter under the charge neutrality constraints, and present a unified picture of the thermal unpairing phase transitions for a wide range of the quark density. We supplement our discussions by developing the Ginzburg-Landau analysis.Comment: 3 pages, 3 plots, contributed to the Proceedings of PANIC'05 (Particles and Nuclei International Conference), Santa Fe, NM, 24-28 October 200

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

Structure of the sigma meson and the softening

We study the structure of the sigma meson, the lowest-lying resonance of the pi pi scattering in the scalar-isoscalar channel, through the softening phenomena associated with the partial restoration of chiral symmetry. We build dynamical chiral models to describe the pi pi scattering amplitude, in which the sigma meson is described either as the chiral partner of the pion or as the dynamically generated resonance through the pi pi attraction. It is shown that the internal structure is reflected in the softening phenomena; the softening pattern of the dynamically generated sigma meson is qualitatively different from the behavior of the chiral partner of the pion. On the other hand, in the symmetry restoration limit, the dynamically generated sigma meson behaves similarly to the chiral partner.Comment: 10 pages, 2 figures, 1 table, talk give at HNP09, November 16-19, 2009, RCNP, Osaka Universit